broadband wireless communications

Multi- Carrier Techniques for Broadband Wireless

Communications A Signal Processing Perspective

Communications and Signal Processing

Editors: Prof. A. Manikas & Prof. A. G. Constantinides(Imperial College London, UK)

Vol. 1: Joint Source-Channel Coding of Discrete-Time Signals withContinuous Amplitudesby Norbert Goertz

Vol. 2: Quasi-Orthogonal Space-Time Block Codeby Chau Yuen, Yong Liang Guan and Tjeng Thiang Tjhung

Vol. 3: Multi-Carrier Techniques for Broadband Wireless Communications:A Signal Processing Perspectiveby C-C Jay Kuo, Michele Morelli and Man-On Pun

KwangWei - Multi-Carrier Techniques.pmd 2/18/2008, 3:35 PM2

Multi- Carrier Techniques for Broadband Wireless

Communications A Signal Processing Perspective

Man-On Pun Princeton University, USA

Michele Morelli

C-C Jay Kuo University of Pisa, Italy

University of Southern California, USA

Imperial College Press

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

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Copyright © 2007 by Imperial College Press

Communications and Signal Processing — Vol. 3MULTI-CARRIER TECHNIQUES FOR BROADBAND WIRELESSCOMMUNICATIONSA Signal Processing Perspective

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June 15, 2007 10:2 World Scientific Book - 9in x 6in book

To my wife Ying and my mother.(Man-On Pun)

To my wife Monica and my son Tommaso.(Michele Morelli)

To my parents, my wife Terri and my daughter Allison.(C.-C. Jay Kuo)

v


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Preface

The demand for multimedia wireless communications is growing today at anexplosive pace. One common feature of many current wireless standards forhigh-rate multimedia transmission is the adoption of a multicarrier air in-terface based on either orthogonal frequency-division multiplexing (OFDM)or orthogonal frequency-division multiple-access (OFDMA). The latest ex-amples of this trend are represented by the IEEE 802.11 and IEEE 802.16families of standards for wireless local area networks (WLANs) and wire-less metropolitan area networks (WMANs). Although the basic principleof OFDM/OFDMA is well established among researchers and communica-tion engineers, its practical implementation is far from being trivial as itrequires rather sophisticated signal processing techniques in order to fullyachieve the attainable system performance.

This book is intended to provide an accessible introduction to OFDM-based systems from a signal processing perspective. The first part providesa concise treatment of some fundamental concepts related to wireless com-munications and multicarrier systems, whereas the second part offers acomprehensive survey of recent developments on a variety of critical designissues including synchronization techniques, channel estimation methods,adaptive resource allocation and practical schemes for reducing the peak-to-average power ratio of the transmitted waveform. The selection andtreatment of topics makes this book quite different from other texts indigital communication engineering. In most books devoted to multicarriertransmissions the issue of resource assignment is not discussed at all whilesynchronization and channel estimation are only superficially addressed.This may give the reader the erroneous impression that these tasks arerather trivial and the system can always operate close to the limiting caseof ideal synchronization and channel estimation. However, as discussed

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viii Multi-Carrier Techniques for Broadband Wireless Communications

in this book, special design attentions are required for successfully accom-plishing these tasks. In many cases, the trade-off between performanceand system complexity has to be carefully taken into consideration in thepractical implementation of multicarrier systems.

Most of the presented material originates from several projects and re-search activities conducted by the authors in the field of multicarrier trans-missions. In order to keep the book concise, we do not cover advancedtopics in multiple-input multiple-output (MIMO) OFDM systems as wellas latest results in the field of resource assignment based on game theory.Also, we do not include a description of current wireless standards employ-ing OFDM or OFDMA which are available in many other texts and journalpapers.

The book is written for graduate students, design engineers in telecom-munications industry as well as researchers in academia. Readers are as-sumed to be familiar with the basic concepts of digital communication the-ory and to have a working knowledge of Fourier transforms, stochasticprocesses and estimation theory. Whenever possible, we have attempted tokeep the presentation as simple as possible without sacrificing accuracy. Wehope that the book will contribute to a better understanding of most criti-cal issues encountered in the design of a multicarrier communication systemand may motivate further investigation in this exciting research area.

The authors acknowledge contributions of several people to the writingof this book. Many thanks go to Prof. Umberto Mengali who reviewed sev-eral portions of the manuscript and suggested valuable improvements to itsoriginal version. Without his advice and encouragement, this book wouldnever have seen the light of day. We would also like to express appreciationto our co-workers and friends Antonio D’Amico, Marco Moretti and LucaSanguinetti who reviewed the manuscript in detail and offered correctionsand insightful comments. To all of them we owe a debt of gratitude. Specialthanks go to Ivan Cosovic from NTT-DoCoMo, who critically read a firstdraft of the manuscript and provided invaluable suggestions.

M. Pun would like to thank his former colleagues at the SONY corpo-ration, particularly Takahiro Okada, Yasunari Ikeda, Naohiko Iwakiri andTamotsu Ikeda for first teaching him about the principle of OFDM. M. Punwould also like to acknowledge the Sir Edward Youde Foundation and theCroucher Foundation for supporting him in his research activity. M. Morelliwould like to thank his wife Monica and son Tommaso for their supportand understanding during the time he devoted to writing this book, and tohis parents for their endless sacrifices. C.-C. J. Kuo would like to thank his


Preface ix

parents, his wife Terri and daughter Allison for their encouragement andsupport for years.

Man-On PunMichele MorelliC.-C. Jay Kuo


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Contents

Preface vii

1. Introduction 1

1.1 Aim of this book . . . . . . . . . . . . . . . . . . . . . . . 11.2 Evolution of wireless communications . . . . . . . . . . . 3

1.2.1 Pioneering era of wireless communications . . . . 41.2.2 First generation (1G) cellular systems . . . . . . . 51.2.3 Second generation (2G) cellular systems . . . . . . 61.2.4 Third generation (3G) cellular systems . . . . . . 71.2.5 Wireless local and personal area networks . . . . . 81.2.6 Wireless metropolitan area networks . . . . . . . . 111.2.7 Next generation wireless broadband systems . . . 13

1.3 Historical notes on multicarrier transmissions . . . . . . . 141.4 Outline of this book . . . . . . . . . . . . . . . . . . . . . 15

2. Fundamentals of OFDM/OFDMA Systems 17

2.1 Mobile channel modeling . . . . . . . . . . . . . . . . . . . 172.1.1 Parameters of wireless channels . . . . . . . . . . 182.1.2 Categorization of fading channels . . . . . . . . . 27

2.2 Conventional methods for channel fading mitigation . . . 332.2.1 Time-selective fading . . . . . . . . . . . . . . . . 342.2.2 Frequency-selective fading . . . . . . . . . . . . . 34

2.3 OFDM systems . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 System architecture . . . . . . . . . . . . . . . . . 372.3.2 Discrete-time model of an OFDM system . . . . . 40

2.4 Spectral efficiency . . . . . . . . . . . . . . . . . . . . . . 44

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xii Multi-Carrier Techniques for Broadband Wireless Communications

2.5 Strengths and drawbacks of OFDM . . . . . . . . . . . . . 452.6 OFDM-based multiple-access schemes . . . . . . . . . . . 462.7 Channel coding and interleaving . . . . . . . . . . . . . . 48

3. Time and Frequency Synchronization 51

3.1 Sensitivity to timing and frequency errors . . . . . . . . . 523.1.1 Effect of timing offset . . . . . . . . . . . . . . . . 543.1.2 Effect of frequency offset . . . . . . . . . . . . . . 58

3.2 Synchronization for downlink transmissions . . . . . . . . 613.2.1 Timing acquisition . . . . . . . . . . . . . . . . . . 623.2.2 Fine timing tracking . . . . . . . . . . . . . . . . . 673.2.3 Frequency acquisition . . . . . . . . . . . . . . . . 693.2.4 Frequency tracking . . . . . . . . . . . . . . . . . 72

3.3 Synchronization for uplink transmissions . . . . . . . . . . 763.3.1 Uplink signal model with synchronization errors . 783.3.2 Timing and frequency estimation for systems with

subband CAS . . . . . . . . . . . . . . . . . . . . 813.3.3 Timing and frequency estimation for systems with

interleaved CAS . . . . . . . . . . . . . . . . . . . 843.3.4 Frequency estimation for systems with generalized

CAS . . . . . . . . . . . . . . . . . . . . . . . . . 883.4 Timing and frequency offset compensation in uplink trans-

missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.4.1 Timing and frequency compensation with subband

CAS . . . . . . . . . . . . . . . . . . . . . . . . . 963.4.2 Frequency compensation through interference can-

cellation . . . . . . . . . . . . . . . . . . . . . . . 1003.4.3 Frequency compensation through linear multiuser

detection . . . . . . . . . . . . . . . . . . . . . . . 1013.4.4 Performance of frequency correction schemes . . . 104

4. Channel Estimation and Equalization 107

4.1 Channel equalization . . . . . . . . . . . . . . . . . . . . . 1084.2 Pilot-aided channel estimation . . . . . . . . . . . . . . . 111

4.2.1 Scattered pilot patterns . . . . . . . . . . . . . . . 1124.2.2 Pilot distances in time and frequency directions . 1134.2.3 Pilot-aided channel estimation . . . . . . . . . . . 1144.2.4 2D Wiener interpolation . . . . . . . . . . . . . . 115

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Contents xiii

4.2.5 Cascaded 1D interpolation filters . . . . . . . . . . 117

4.3 Advanced techniques for blind and semi-blind channel es-

timation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.3.1 Subspace-based methods . . . . . . . . . . . . . . 126

4.3.2 EM-based channel estimation . . . . . . . . . . . 129

4.4 Performance comparison . . . . . . . . . . . . . . . . . . . 133

5. Joint Synchronization, Channel Estimation and Data

Symbol Detection in OFDMA Uplink 135

5.1 Uncoded OFDMA uplink . . . . . . . . . . . . . . . . . . 136

5.1.1 Signal model . . . . . . . . . . . . . . . . . . . . . 136

5.1.2 Iterative detection and frequency synchronization 137

5.1.3 Practical adjustments . . . . . . . . . . . . . . . . 144

5.1.4 Performance assessment . . . . . . . . . . . . . . . 146

5.2 Trellis-coded OFDMA uplink . . . . . . . . . . . . . . . . 150

5.2.1 Signal model for coded transmissions . . . . . . . 150

5.2.2 Iterative detection and frequency synchronization

with coded transmissions . . . . . . . . . . . . . . 152

5.2.3 Performance assessment . . . . . . . . . . . . . . . 157

6. Dynamic Resource Allocation 159

6.1 Resource allocation in single-user OFDM systems . . . . . 160

6.1.1 Classic water-filling principle . . . . . . . . . . . . 161

6.1.2 Rate maximization and margin maximization . . 166

6.1.3 Rate-power function . . . . . . . . . . . . . . . . . 167

6.1.4 Optimal power allocation and bit loading under

BER constraint . . . . . . . . . . . . . . . . . . . 168

6.1.5 Greedy algorithm for power allocation and bit

loading . . . . . . . . . . . . . . . . . . . . . . . . 171

6.1.6 Bit loading with uniform power allocation . . . . 173

6.1.7 Performance comparison . . . . . . . . . . . . . . 176

6.1.8 Subband adaptation . . . . . . . . . . . . . . . . . 178

6.1.9 Open-loop and closed-loop adaptation . . . . . . . 179

6.1.10 Signaling for modulation parameters . . . . . . . 180

6.2 Resource allocation in multiuser OFDM systems . . . . . 182

6.2.1 Multiaccess water-filling principle . . . . . . . . . 184

6.2.2 Multiuser rate maximization . . . . . . . . . . . . 188

6.2.3 Max-min multiuser rate maximization . . . . . . . 190

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xiv Multi-Carrier Techniques for Broadband Wireless Communications

6.2.4 Multiuser margin maximization . . . . . . . . . . 192

6.2.5 Subcarrier assignment through average channel

signal-to-noise ratio . . . . . . . . . . . . . . . . . 194

6.3 Dynamic resource allocation for MIMO-OFDMA . . . . . 197

6.4 Cross-layer design . . . . . . . . . . . . . . . . . . . . . . 199

7. Peak-to-Average Power Ratio (PAPR) Reduction 201

7.1 PAPR definitions . . . . . . . . . . . . . . . . . . . . . . . 202

7.2 Continuous-time and discrete-time PAPR . . . . . . . . . 203

7.3 Statistical properties of PAPR . . . . . . . . . . . . . . . 206

7.4 Amplitude clipping . . . . . . . . . . . . . . . . . . . . . . 208

7.4.1 Clipping and filtering of oversampled signals . . . 209

7.4.2 Signal-to-clipping noise ratio . . . . . . . . . . . . 214

7.4.3 Clipping noise mitigation . . . . . . . . . . . . . . 217

7.5 Selected mapping (SLM) technique . . . . . . . . . . . . . 219

7.6 Partial transmit sequence (PTS) technique . . . . . . . . 223

7.7 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.8 Tone reservation and injection techniques . . . . . . . . . 232

7.8.1 Tone reservation (TR) . . . . . . . . . . . . . . . 232

7.8.2 Tone injection (TI) . . . . . . . . . . . . . . . . . 234

7.9 PAPR reduction for OFDMA . . . . . . . . . . . . . . . . 237

7.9.1 SLM for OFDMA . . . . . . . . . . . . . . . . . . 238

7.9.2 PTS for OFDMA . . . . . . . . . . . . . . . . . . 238

7.9.3 TR for OFDMA . . . . . . . . . . . . . . . . . . . 238

7.10 Design of AGC unit . . . . . . . . . . . . . . . . . . . . . 239

Bibliography 243

Index 255


Chapter 1

Introduction

1.1 Aim of this book

The pervasive use of wireless communications is more and more condition-ing lifestyle and working habits in many developed countries. Examplesof this trend are the ever increasing number of users that demand Inter-net connection when they are traveling, the use of cellular phones to checkbank accounts and make remote payments, or the possibility of sharing mo-ments in our lives with distant friends by sending them images and videoclips. In the last few years, the proliferation of laptop computers has ledto the development of wireless local area networks (WLANs), which arerapidly supplanting wired systems in many residential homes and businessoffices. More recently, wireless metropolitan area networks (WMANs) havebeen standardized to provide rural locations with broadband Internet ac-cess without the costly infrastructure required for deploying cables. A newgeneration of wireless systems wherein multimedia services like speech, au-dio, video and data will converge into a common and integrated platformis currently under study and is expected to become a reality in the nearfuture.

The promise of portability is clearly one of the main advantages of thewireless technology over cabled networks. Nevertheless, the design of awireless communication system that may reliably support emerging mul-timedia applications must deal with several technological challenges thathave motivated an intense research in the field. One of this challenge isthe harsh nature of the communication channel. In wireless applications,the radiated electromagnetic wave arrives at the receiving antenna after be-ing scattered, reflected and diffracted by surrounding objects. As a result,the receiver observes the superposition of several differently attenuated and

1


2 Multi-Carrier Techniques for Broadband Wireless Communications

delayed copies of the transmitted signal. The constructive or destructivecombination of these copies induces large fluctuations in the received signalstrength with a corresponding degradation of the link quality. In addition,the characteristics of the channel may randomly change in time due to un-predictable variations of the propagation environment or as a consequenceof the relative motion between the transmitter and receiver. A second chal-lenge is represented by the limited amount of available radio spectrum,which is a very scarce and expensive resource. It suffices to recall thatEuropean telecommunication companies spent over 100 billion dollars toget licenses for third-generation cellular services. To obtain a reasonablereturn from this investment, the purchased spectrum must be used as ef-ficiently as possible. A further impairment of wireless transmissions is therelatively high level of interference arising from channel reuse. Althoughadvanced signal processing techniques based on multiuser detection haverecently been devised for interference mitigation, it is a fact that mobilewireless communications will never be able to approach the high degreeof stability, security and reliability afforded by cabled systems. Neverthe-less, it seems that customers are ready to pay the price of a lower datathroughput and worse link quality in order to get rid of wires.

The interest of the communication industry in wireless technology is wit-nessed by the multitude of heterogeneous standards and applications thathave emerged in the last decade. In the meantime, the research communityhas worked (and is still working) toward the development of new broad-band wireless systems that are expected to deliver much higher data ratesand much richer multimedia contents than up-to-date commercial products.The ability to provide users with a broad range of applications with dif-ferent constraints in terms of admissible delay (latency), quality of serviceand data throughput, demands future systems to exhibit high robustnessagainst interference and channel impairments, as well as large flexibility inradio resource management. The selection of a proper air-interface revealscrucial for achieving all these features. The multicarrier technology in theform of orthogonal frequency-division multiplexing (OFDM) is widely rec-ognized as one of the most promising access scheme for next generationwireless networks. This technique is already being adopted in many appli-cations, including the terrestrial digital video broadcasting (DVB-T) andsome commercial wireless LANs. The main idea behind OFDM is to split ahigh-rate data stream into a number of substreams with lower rate. Thesesubstreams are then transmitted in parallel over orthogonal subchannelscharacterized by partially overlapping spectra. Compared to single-carrier


Introduction 3

transmissions, this approach provides the system with increased resistanceagainst narrowband interference and channel distortions. Furthermore, itensures a high level of flexibility since modulation parameters like constella-tion size and coding rate can independently be selected over each subchan-nel. OFDM can also be combined with conventional multiple-access tech-niques for operation in a multiuser scenario. The most prominent schemein this area is represented by orthogonal frequency-division multiple-access(OFDMA), which has become part of the emerging standards for wirelessMANs.

Even though the concept of multicarrier transmission is simple in itsbasic principle, the design of practical OFDM and OFDMA systems is farfrom being a trivial task. Synchronization, channel estimation and radioresource management are only a few examples of the numerous challengesrelated to multicarrier technology. As a result of continuous efforts of manyresearchers, most of these challenging issues have been studied and severalsolutions are currently available in the open literature. Nevertheless, theyare scattered around in form of various conference and journal publications,often concentrating on specific performance and implementation issues. Asa consequence, they are hardly useful to give a unified view of an otherwiseseemingly heterogeneous field. The task of this book is to provide thereader with a harmonized and comprehensive overview of new results in therapidly growing field of multicarrier broadband wireless communications.Our main goal is to discuss in some detail several problems related to thephysical layer design of OFDM and OFDMA systems. In doing so we shallpay close attention to different trade-offs that can be achieved in terms ofperformance and complexity.

1.2 Evolution of wireless communications

Before proceeding to a systematic study of OFDM and OFDMA, we thinkit useful to review some basic applications of such schemes and highlightthe historical reasons that led to their development. The current section isdevoted to this purpose, and illustrates the evolution of wireless communi-cation systems starting from the theoretical works of Maxwell in the nine-teenth century till the most recent studies on broadband wireless networks.Some historical notes on multicarrier transmissions are next provided inthe last section of this introductory chapter.



1.2.1 Pioneering era of wireless communications

The modern era of wireless communications began with the mathemati-cal theory of electromagnetic waves formulated by James Clerk Maxwellin 1873. The existence of these waves was later demonstrated by HeinrichHertz in 1887, when for the first time a radio transmitter generated a sparkin a receiver placed several meters away. Although Nikola Tesla was thefirst researcher who showed the ability of electromagnetic waves to conveyinformation, Guglielmo Marconi is widely recognized as the inventor of wire-less transmissions. His first publicized radio experiment took place in 1898from a boat in the English Channel to the Isle of Wight, while in 1901 hisradio telegraph system sent the first radio signal across the Atlantic Oceanfrom Cornwall to Newfoundland. Since then, the wireless communicationidea was constantly investigated for practical implementation, but until the1920s mobile radio systems only made use of the Morse code. In 1918 Ed-win Armstrong invented the superheterodyne receiver, thereby opening theway to the first broadcast radio transmission that took place at Pittsburghin 1920. In the subsequent years the radio became widespread all overthe world, but in the meantime the research community was studying thepossibility of transmitting real-time moving images through the air. Theseefforts culminated in 1929 with the first experiment of TV transmissionmade by Vladimir Zworykin. Seven years later the British BroadcastingCorporation (BBC) started its TV services.

Although radio and TV broadcasting were the first widespread wire-less services, an intense research activity was devoted to develop practicalschemes for bi-directional mobile communications, which were clearly ap-pealing for military applications and for police and fire departments. Thefirst mobile radio telephones were employed in 1921 by the Detroit PoliceDepartment’s radio bureau, that began experimentation for vehicular mo-bile services. In subsequent years, these early experiments were followed bymany others. In the 1940s, radio equipments called “carphones” occupiedmost of the police cars. These systems were powered by car batteries andallowed communications among closed group of users due to lack of inter-connection with the public switched telephone network (PSTN). In 1946,mobile telephone networks interconnected with the PSTN made their firstappearance in several cities across the United States. The main shortcom-ing of these systems was the use of a single access point to serve an entiremetropolitan area, which limited the number of active users to the numberof allocated frequency channels. This drawback motivated investigations as


Introduction 5

how to enlarge the number of users for a given allocated frequency band.A solution was found in 1947 by the AT&T’s Bell Labs with the adventof the cellular concept [131], which represented a fundamental contributionin the development of wireless communications. In cellular communicationsystems, the served area is divided into smaller regions called cells. Due toits reduced dimension, each cell requires a relatively low power to be cov-ered. Since the power of the transmitted signal falls off with distance, usersbelonging to adequately distant cells can operate over the same frequencyband with minimal interference. This means that the same frequency bandcan be reused in other (most often non adjacent) cells, thereby leading toa more efficient use of the radio spectrum.

In 1957, the Union Soviet launched its first satellite Sputnik I and theUnited States soon followed in 1958 with Explorer I. The era of spaceexploration and satellite communications had begun. Besides being usedfor TV services, modern satellite networks provide radio coverage to widesparsely populated areas where a landline infrastructure is absent. Typicalapplications are communications from ships, offshore oil drilling platformsand war or disaster areas.

1.2.2 First generation (1G) cellular systems

Despite its theoretical relevance, the cellular concept was not widelyadopted during the 1960s and 1970s. To make an example, in 1976 theBell Mobile Phone had only 543 paying customers in the New York Cityarea, and mobile communications were mainly supported by heavy ter-minals mounted on cars. Although the first patent describing a portablemobile telephone was granted to Motorola in 1975 [25], mobile cellular sys-tems were not introduced for commercial use until the early 1980s, when theso-called first generation (1G) of cellular networks were deployed in mostdeveloped countries. The common feature of 1G systems was the adoptionof an analog transmission technology. Frequency modulation (FM) wasused for speech transmission over the 800-900 MHz band and frequency-division multiple-access (FDMA) was adopted to separate users’ signalsin the frequency domain. In practice, a fraction of the available spectrum(subchannel) was exclusively allocated to a given user during the call set-upand retained for the entire call.

In the early 1980s, 1G cellular networks experienced a rapid growth inEurope, particularly in Scandinavia where the Nordic Mobile Telephony(NMT) appeared in 1981, and in United Kingdom where the Total Access



Communication System (TACS) started service in 1985. The AdvancedMobile Phone Service (AMPS) was deployed in Japan in 1979, while in theUnited States it appeared later in 1983. These analog systems created acritical mass of customers. Their main limitations were the large dimensionsof cellphones and the reduced traffic capacity due to a highly inefficient useof the radio spectrum.

At the end of the 1980s, progress in semiconductor technology and de-vice miniaturization allowed the production of small and light-weight hand-held phones with good speech quality and acceptable battery lifetime. Thismarked the beginning of the wireless cellular revolution that took almosteveryone by surprise since in the meantime many important companies hadstopped business activities in cellular communications, convinced that mo-bile telephony would have been limited to rich people and would have neverattracted a significant number of subscribers.

1.2.3 Second generation (2G) cellular systems

The limitations of analog radio technology in terms of traffic capacity be-came evident in the late 1980s, when 1G systems saturated in many bigcities due to the rapid growth of the cellular market. Network operatorsrealized that time was ripe for a second generation (2G) of cellular systemsthat would have marked the transition from analog to digital radio technol-ogy. This transition was not only motivated by the need for higher networkcapacity, but also by the lower cost and improved performance of digitalhardware as compared to analog circuitry.

Driven by the success of NMT, in 1982 the Conference of EuropeanPosts and Telecommunications (CEPT) formed the Group Special Mobile(GSM) in order to develop a pan-European standard for mobile cellularradio services with good speech quality, high spectral efficiency and theability for secure communications. The specifications of the new standardwere approved in 1989 while its commercial use began in 1993. Unlike1G systems, the GSM was developed as a digital standard where users’analog signals are converted into sequences of bits and transmitted on aframe-by-frame basis. Within each frame, users transmit their bits onlyduring specified time intervals (slots) that are exclusively assigned at thecall setup according to a time-division multiple-access (TDMA) approach.Actually, the GSM is based on a hybrid combination of FDMA and TDMA,where FDMA is employed to divide the available spectrum into 200 kHz-wide subchannels while TDMA is used to separate up to a maximum of


Introduction 7

eight users allocated over the same subchannel. In Europe the operatingfrequency band is 900 MHz, even though in many big cities the 1800 MHzband is also being adopted to accommodate a larger number of users. Manymodern European GSM phones operate in a “dual-band” mode by selectingeither of the two recommended frequencies. In the United States, the 1900MHz frequency band is reserved to the GSM service.

In addition to circuit-switched applications like voice, the adoption ofa digital technology enabled 2G cellular systems to offer low-rate data ser-vices including e.mail and short messaging up to 14.4 kbps. The successof GSM was such that by June 2001 there were more than 500 millionsGSM subscribers all over the world while in 2004 the market penetrationexceeded 80% in Western Europe. The reasons for this success can be foundin the larger capacity and many more services that the new digital stan-dard offered as compared to previous 1G analog systems. Unfortunately,the explosive market of digital cellphones led to a proliferation of incom-patible 2G standards that sometimes prevent the possibility of roamingamong different countries. Examples of this proliferation are the DigitalAdvanced Mobile Phone Services (D-AMPS) which was introduced in theUnited States in 1991 and the Japanese Pacific Digital Cellular (PDS) [67].The Interim Standard 95 (IS-95) became operative in the United Statesstarting from 1995 and was the first commercial system to employ thecode-division multiple-access (CDMA) technology as an air interface.

1.2.4 Third generation (3G) cellular systems

At the end of the 1990s it became clear that GSM was not sufficient to indef-initely support the explosive number of users and the ever-increasing datarates requested by emerging multimedia services. There was the need for anew generation of cellular systems capable of supporting higher transmis-sion rates with improved quality of service as compared to GSM. After longdeliberations, two prominent standards emerged: the Japanese-EuropeanUniversal Mobile Telecommunication System (UMTS) [160] and the Amer-ican CDMA-2000 [161]. Both systems operate around the 2 GHz frequencyband and adopt a hybrid FDMA/CDMA approach. In practice, groupsof users are allocated over disjoint frequency subbands, with users shar-ing a common subband being distinguished by quasi-orthogonal spreadingcodes. The CDMA technology has several advantages over TDMA andFDMA, including higher spectral efficiency and increased flexibility in radioresource management. In practical applications, however, channel distor-



tions may destroy orthogonality among users’ codes, thereby resulting inmultiple-access interference (MAI). In the early 1990s, problems related toMAI mitigation spurred an intense research activity on CDMA and otherspread-spectrum techniques. This led to the development of a large numberof multiuser detection (MUD) techniques [164], where the inherent struc-ture of interfering signals is exploited to assist the data detection process.

The introduction of 3G systems offered a wide range of new multimediaapplications with the possibility of speech, audio, images and video trans-missions at data rates of 144-384 kbps for fast moving users up to 2 Mbpsfor stationary or slowly moving terminals. In addition to the increased datarate, other advantages over 2G systems are the improved spectral efficiency,the ability to multiplex several applications with different quality of servicerequirements, the use of variable bit rates to offer bandwidth on demandand the possibility of supporting asymmetric services in the uplink anddownlink directions, which is particularly useful for web browsing and high-speed downloading operations. Unfortunately, the impressive costs paid bytelecom providers to get 3G cellular licenses slackened the deployment ofthe 3G infrastructure all over the world and led to a spectacular crash ofthe telecom stock market during the years 2000/2001. As a result, manystartup companies went bankrupt while others decreased or stopped at alltheir investments in the wireless communication area. This also produceda significant reduction of public funding for academic research.

1.2.5 Wireless local and personal area networks

In the first years of the new millennium, the development of personal areanetworks (PANs) and wireless local area networks (WLANs) has suscitateda renewed interest in the wireless technology. These products provide wire-less connectivity among portable devices like laptop computers, cordlessphones, personal digital assistants (PDAs) and computer peripherals. Com-pared to wired networks they promise portability, allow simple and fastinstallation and save the costs for deploying cables. Because of their rel-atively limited coverage range, both technologies are mainly intended forindoor applications.

Several standards for PAN products have been developed by the IEEE802.15 working group [62]. Among them, Bluetooth is perhaps the mostpopular scheme. The first release of Bluetooth appeared in 1999 while thefirst headset was produced by Ericsson in the year 2000. This technologyenables low-powered transmissions with short operating ranges up to 10


Introduction 9

meters. It provides wireless connection among closely spaced portable de-vices with limited battery power and must primarily be considered as asubstitute for data transfer cables. Typical applications are the intercon-nection between a hands-free headset and a cellular phone, a DVD playerand a television set, a desktop computer and some peripheral devices like aprinter, keyboard or mouse. Bluetooth operates over the unlicensed Indus-trial, Scientific and Medical (ISM) frequency band, which is centered around2.4 GHz. The allocated spectrum is divided into 79 adjacent subchannelswhich are accessed by means of a frequency-hopping spread-spectrum (FH-SS) technique. Each subchannel has a bandwidth of 1 MHz for a data rateapproaching 1 Mbps [44].

WLANs have a wider coverage area as compared to PANs and aremainly used to distribute the Internet access to a bunch of portable devices(typically laptop computers) dislocated in private homes or office buildings.A typical application is represented by a user who needs to be able to carryout a laptop into a conference room without losing network connection.WLANs are also being used in hotels, airports or coffee shops to create“hotspots” for public access to the Internet. The number of users that cansimultaneously be served is usually limited to about 10, even though inprinciple more users could be supported by lowering the individual datarates. The typical network topology of commercial WLANs is based ona cellular architecture with cell radii up to 100 meters. In this case, sev-eral user terminals (UTs) establish a wireless link with a fixed access point(AP) which is connected to the backbone network as illustrated in Fig. 1.1.An alternative configuration is represented in Fig. 1.2, where an ad-hocnetwork is set up for peer-to-peer communications without involving anyAP.

Internet

User terminal

User terminal

Access point

Fig. 1.1 Illustration of a WLAN with fixed access point.



Peer

Peer

Peer

Fig. 1.2 Illustration of a WLAN for peer-to-peer communications.

The most successful class of WLAN products is based on the IEEE802.11 family of standards. The first 802.11 release appeared in 1997 [58]and was intended to provide data rates of 1 and 2 Mbps. Three differentphysical layer architectures were recommended. The first two operate overthe 2.4 GHz band and employ either a direct-sequence spread-spectrumor frequency-hopping technology. The third operational mode is based oninfrared light and has rarely been used in commercial products. A firstamendment called 802.11b was ratified in 1999 to improve the data rate upto 11 Mbps [60]. This product was adopted by an industry group calledWiFi (Wireless Fidelity) and became soon very popular. In the same year anew amendment called 802.11a recommended the use of OFDM to furtherincrease the data rate up to 54 Mbps [59]. This standard operates over the5 GHz band, which is unlicensed in the US but not in most other countries.A TDMA approach is used to distinguish users within a cell while FDMAis employed for cell separation. A further evolution of the 802.11 familywas approved in 2003 and is called 802.11g [61]. This standard is similar to802.11a, except that it operates over the ISM band, which is license-exemptin Europe, United States and Japan.

Other examples of WLAN standards include the Japanese multimediamobile access communication (MMAC) and the European high performanceLAN (HiperLAN2) [41]. The physical layers of these systems are based onOFDM and only present minor modifications with respect to IEEE 802.11a.The major differences lie in the MAC layer protocols. Actually, HiperLAN2employs a reservation based access scheme where each UT sends a requestto the AP before transmitting a data packet, while 802.11 adopts Carrier-Sense Multiple-Access with Collision Avoidance (CSMA-CA), where each


Introduction 11

UT determines whether the channel is currently available and only in thatcase it starts transmitting data. As for MMAC, it supports both of theaforementioned protocols.

The current generation of WLANs offers data rates of tens of Mbps andis characterized by low mobility and relatively limited coverage areas. Thechallenge for future WLANs is to extend the radio coverage and supportnew services like real-time video applications that are highly demanding interms of data rate and latency.

1.2.6 Wireless metropolitan area networks

SS

Internet

Base station

Fig. 1.3 Illustration of a WMAN providing wireless Internet access to a remote SS.

Wireless metropolitan area networks (WMANs) represent the naturalevolution of WLANs. The purpose of these systems is to provide networkaccess to residential or enterprise buildings through roof-top antennas com-municating with a central radio base station, thereby replacing the wired“last mile” connection by a wireless link. This offers an appealing alterna-tive to cabled access networks or digital subscriber line (DSL) links, andpromises ubiquitous broadband access to rural or developing areas wherebroadband is currently unavailable for lack of a cabled infrastructure. Fig-ure 1.3 depicts a typical scenario where the WMAN provides wireless Inter-net access to a Subscriber Station (SS) placed within a building. A WLANor a backbone local network is used inside the building to connect the SS tothe user terminals. In a more challenging application, the SS is mounted ona moving vehicle like a car or a train to provide passengers with continuous



Internet connectivity.Several options for the WMAN air interface and MAC protocols are

specified by the IEEE 802.16 Working Group, who started its activity in1998. The goal was to deliver high data rates up to 50 Mbps over metropoli-tan areas with cell radii up to 50 kilometers. At the beginning, the interestof the Group focused on the 10-66 GHz band where a large amount of un-licensed spectrum is available worldwide. The first 802.16 release appearedin 2002 [63] and was specifically intended for line-of-sight (LOS) appli-cations due to the severe attenuations experienced by short wavelengthswhen passing through walls or other obstructions. This standard adoptssingle-carrier (SC) modulation in conjunction with a TDMA access scheme.Transmission parameters like modulation and coding rates are adaptivelyadjusted on a frame-by-frame basis depending on the actual interferencelevel and channel quality. The LOS requirement was the main limitationof this first release since rooftop antennas mounted on residential buildingsare typically too low for a clear sight line to the base station antenna. Forthis reason, in the same year 2002 a first amendment called 802.16a wasapproved to support non line-of-sight (NLOS) operations over the 2-11 GHzband [112]. This novel standard defines three different air interfaces and acommon MAC protocol with a reservation based access. The first air in-terface relies on SC transmission, the second employs OFDM-TDMA whilethe third operates according to the OFDMA principle in which users’ sep-aration is achieved at subcarrier level. Among the three recommended airinterfaces, those based on OFDM and OFDMA seem to be favored by thevendor community due to their superior performance in NLOS applications.The last evolution of the 802.16 family is represented by the 802.16e speci-fications, whose standardization process began in the year 2004 [113]. Thisemerging standard adopts a scalable OFDMA physical layer and promisesmobility at speeds up to 120 km/h by using adaptive antenna arrays andimproved inter-cell handover. Its main objective is to provide continuousInternet connection to mobile users moving at vehicular speed.

In order to ensure interoperability among all 802.16-based devices andrapidly converge to a worldwide WMAN standard, an industry consortiumcalled WiMax (Worldwide Interoperability for Microwave Access) Forumhas been created. However, due to the large variety of data rates, cover-age ranges and potential options specified in the standards, it is currentlydifficult to predict what type of performance WiMax-certified devices willreasonably provide in the near future.


Introduction 13

1.2.7 Next generation wireless broadband systems

The demand for novel high-rate wireless communication services is growingtoday at an extremely rapid pace and is expected to further increase in thenext years. This trend has motivated a significant number of research anddevelopment projects all over the world to define a fourth generation (4G) ofwireless broadband systems that may offer increased data rates and betterquality of service than current 3G products. The new wireless technologywill support multimedia applications with extremely different requirementsin terms of reliability, bit rates and latency. The integration of the existingmultitude of standards into a common platform represents one of the majorgoals of 4G systems, which can only be achieved through the adoption of aflexible air interface with high scalability and interoperability [57,138].

Software Defined Radio (SDR) represents a viable solution to pro-vide 4G systems with the necessary level of flexibility and reconfigurabil-ity [4,159,170]. The main concept behind SDR is that different transceiverfunctions are executed as software programs running on suitable processors.Once the software corresponding to existing standards has been pre-loadedon the system, the SDR platform guarantees full compatibility among dif-ferent wireless technologies. In addition, SDR can easily incorporate newstandards and protocols by simply loading the specific application software.

A second challenge for next generation systems is the conflict betweenthe increasing demand for higher data rates and the scarcity of the radiospectrum. This calls for an air interface characterized by an extremelyhigh spectral efficiency. Recent advances in information theory has shownthat large gains in terms of capacity and coverage range are promised bymultiple-input multiple-output (MIMO) systems, where multiple antennasare deployed at both ends of the wireless link [46]. Based on these re-sults, it is likely that the MIMO technology will be widely adopted in 4Gnetworks. An alternative way for improving the spectral efficiency is theuse of flexible modulation and coding schemes, where system resources areadaptively assigned to users according to their requested data rates andchannel quality. As mentioned previously, the multicarrier technique isrecognized as a potential candidate for next generation broadband wire-less systems thanks to its attractive features in terms of robustness againstchannel distortions and narrowband interference, high spectral efficiency,high flexibility in resource management and ability to support adaptivemodulation schemes. Furthermore, multicarrier transmissions can easilybe combined with MIMO technology as witnessed by recent advances on



MIMO-OFDM [149] and MIMO-OFDMA.

1.3 Historical notes on multicarrier transmissions

The first examples of multicarrier (MC) modems operating in the High-Frequency (HF) band date back to the 1950s. In these early experiments,the signal bandwidth was divided into several non-overlapping frequencysubchannels, each modulated by a distinct stream of data coming from acommon source. On one hand, the absence of any spectral overlap betweenadjacent subchannels helped to eliminate interference among different datastreams (interchannel interference). On the other, it resulted into a veryinefficient use of the available spectrum. The idea of orthogonal MC trans-mission with partially overlapping spectra was introduced by Chang in 1966with his pioneering paper on parallel data transmission over dispersive chan-nels [15]. In the late 1960s, the MC concept was adopted in some militaryapplications such as KATHRYN [184] and ANDEFT [120]. These systemsinvolved a large hardware complexity since parallel data transmission wasessentially implemented through a bank of oscillators, each tuned on a spe-cific subcarrier. As a consequence, in that period much of the researcheffort was devoted to find efficient modulation and demodulation schemesfor MC digital communications [121, 139]. A breakthrough in this sensecame in 1971, when Weinstein and Ebert eliminated the need for a bankof oscillators and proposed the use of the Fast Fourier Transform (FFT)for baseband processing. They also introduced the guard band concept toeliminate interference among adjacent blocks of data. The new FFT-basedtechnique was called orthogonal frequency-division multiplexing (OFDM).Despite its reduced complexity with respect to previously developed MCschemes, practical implementation of OFDM was still difficult at that timebecause of the limited signal processing capabilities of the electronic hard-ware. For this reason, OFDM did not attract much attention until 1985,when was suggested by Cimini for high-speed wireless applications [21].

Advances in digital and hardware technology in the early 1990s enabledthe practical implementation of FFTs of large size, thereby making OFDMa realistic option for both wired and wireless transmissions. The ability tosupport adaptive modulation and to mitigate channel distortions withoutthe need for adaptive time-domain equalizers made OFDM the selected ac-cess scheme for asymmetric digital subscriber loop (ADSL) applications inthe USA [19]. In Europe, Digital Audio Broadcasting (DAB) standardized


Introduction 15

by ETSI was the first commercial wireless system to use OFDM as an airinterface in 1995 [39]. This success continued in 1997 with the adoptionof OFDM for terrestrial Digital Video Broadcasting (DVB-T) [40] and in1999 with the release of the WLAN standards HiperLAN2 [41] and IEEE802.11a [59], both based on OFDM-TDMA. More recently, OFDM has beenused in the interactive terrestrial return channel (DVB-RCT) [129] and inthe IEEE 802.11g WLAN products [61]. In 1998 a combination of OFDMand FDMA called orthogonal frequency-division multiple-access (OFDMA)was proposed by Sari and Karam for cable TV (CATV) networks [140].The main advantages of this scheme over OFDM-TDMA are the increasedflexibility in resource management and the ability for dynamic channel as-signment. Compared to ordinary FDMA, OFDMA offers higher spectralefficiency by avoiding the need for large guard bands between users’ signals.A hybrid combination of OFDMA and TDMA has been adopted in the up-link of the DVB-RCT system while both OFDM-TDMA and OFDMA arerecommended by the IEEE 802.16a standard for WMANs [112]. An intenseresearch activity is currently devoted to study MIMO-OFDM and MIMO-OFDMA as promising candidates for 4G wireless broadband systems.

1.4 Outline of this book

The remaining chapters of this book are organized in the following way.Chapter 2 lays the groundwork material for further developments and

is divided into three parts. The first is concerned with the statistical char-acterization of the wireless channel. Here, some relevant parameters areintroduced ranging from the channel coherence bandwidth and Dopplerspread to the concept of frequency-selective and time-selective fading. Thesecond part illustrates the basic idea of OFDM and how this kind of modu-lation can be implemented by means of FFT-based signal processing. TheOFDMA principle is described in the third part of the chapter, along withsome other popular multiple-access schemes based on OFDM.

Chapter 3 provides a comprehensive overview of synchronization meth-ods for OFDMA applications. A distinction is made between downlink anduplink transmissions, with a special attention to the uplink situation whichis particularly challenging due to the presence of many unknown synchro-nization parameters. Several timing and frequency recovery schemes arepresented, and comparisons are made in terms of system complexity andestimation accuracy. Some methods for compensating the synchronization



errors in an uplink scenario are illustrated in the last part of this chapter.Chapter 4 deals with channel estimation and equalization in OFDM

systems. After illustrating how channel distortions can be compensated forthrough a bank of one-tap complex-valued multipliers, we present a largevariety of methods for estimating the channel frequency response over eachsubcarrier. A number of these schemes are based on suitable interpolationof pilot symbols which are inserted in the transmitted frame following somespecified grid patterns. Other methods exploit the inherent redundancyintroduced in the OFDM waveform by the use of the cyclic prefix and/orvirtual carriers. The chapter concludes by illustrating recent advances inthe context of joint channel estimation and data detection based on theexpectation-maximization (EM) algorithm.

Chapter 5 extends the discussions of the previous two chapters andpresents a sophisticated receiver structure for uplink OFDMA transmissionswhere the tasks of synchronization, channel estimation and data detectionare jointly performed by means of advanced iterative signal processing tech-niques. At each iteration, tentative data decisions are exploited to improvethe synchronization and channel estimation accuracy which, in turn, pro-duces more reliable data decisions in the next iteration. Numerical resultsdemonstrate the effectiveness of this iterative architecture.

Chapter 6 covers the topic of dynamic resource allocation in multicar-rier systems, where power levels and/or data rates are adaptively adjustedover each subcarrier according to the corresponding channel quality. Webegin by reviewing the rate-maximization and margin-maximization con-cepts and discuss several bit and power loading techniques for single-userOFDM. The second part of the chapter presents a survey of state-of-the-artallocation techniques for OFDMA applications. In this case, the dynamicassignment of subcarriers to the active users provides the system with someform of multiuser diversity which can be exploited to improve the overalldata throughput.

Finally, Chapter 7 provides a thorough discussion of the peak-to-averagepower ratio (PAPR) problem, which is considered as one of the main ob-stacles to the practical implementation of OFDM/OFDMA. After provid-ing a detailed statistical characterization of the PAPR, a large numberof PAPR reduction schemes are presented, starting from the conventionalclipping technique till some sophisticated encoding approaches based onReed-Muller codes and Golay complementary sequences.


Chapter 2

Fundamentals of OFDM/OFDMASystems

This chapter lays the groundwork for the material in the book and ad-dresses several basic issues. Section 2.1 describes the main features of thewireless communication channel and introduces the concept of frequency-selective and time-selective fading. In Sec. 2.2 we review conventional ap-proaches to mitigate the distortions induced by the wireless channel onthe information-bearing signal. Section 2.3 introduces the principle of Or-thogonal Frequency-Division Multiplexing (OFDM) as an effective meansfor high-speed digital transmission over frequency-selective fading channels.We conclude this chapter by illustrating how OFDM can be combined withconventional multiple-access techniques to provide high-rate services to sev-eral simultaneously active users. In particular, we introduce the conceptof Orthogonal Frequency-Division Multiple-Access (OFDMA), where eachuser transmits its own data by modulating an exclusive set of orthogonalsubcarriers. The advantages of OFDMA are highlighted through compar-isons with other popular multiplexing techniques.

2.1 Mobile channel modeling

In a mobile radio communication system, information is conveyed by adigitally modulated band-pass signal which is transmitted through the air.The band-pass signal occupies an assigned portion of the radio frequency(RF) spectrum and is mathematically expressed as

sRF (t) = <es(t)ej2πfct

, (2.1)

where <e · denotes the real part of the enclosed quantity, s(t) is the com-plex envelope of sRF (t) and fc is the carrier frequency. Since only theamplitude and phase of s(t) are modulated by the information symbols, in

17



the ensuing discussion we can restrict our attention to s(t) without any lossof generality. Furthermore, in order to highlight the performance degrada-tion caused by channel impairments, we temporarily neglect the effect ofthermal noise and other disturbance sources. This enables a better under-standing of the OFDM ability to cope with severe channel distortions.

2.1.1 Parameters of wireless channels

Reflectors

Base station

Mobile station

Fig. 2.1 The basic principle of multipath propagation.

Figure 2.1 depicts a typical wireless communication environment whereradio waves are scattered, reflected and diffracted from surrounding objectslike buildings, trees or hills. In such a scenario, the transmitted waveformarrives at the receiving antenna after traveling through several distinctpaths, each characterized by a specific attenuation, phase and propagationdelay. The received signal is thus the superposition of a possibly largenumber of attenuated, phase-shifted and delayed versions of the transmittedwaveform known as multipath components. This results into a linear (andpossibly time-varying) distortion of the information-bearing signal whileit propagates through the transmission medium. A schematic situation isdepicted in Fig. 2.2, where a narrow pulse is spread over a relatively large


Fundamentals of OFDM/OFDMA Systems 19

time interval as a consequence of multipath propagation.

Time Time

Wireless channel

Transmitted pulse Received distorted signal

Fig. 2.2 Distortion introduced by multipath propagation.

At the receiving antenna, the multipath components may overlap in aconstructive or destructive fashion depending on their relative phase shifts.Therefore, the received signal strength is subject to unpredictable fluctu-ations due to random variations of the propagation scenario or in conse-quence of the relative motion between the transmitter and receiver. Sinceeach multipath component undergoes a phase shift of 2π over a travel dis-tance as short as one wavelength, power fluctuations induced by multipathpropagation occur over a very small time-scale and, for this reason, they arenormally referred to as small-scale fading . In addition, the mean receivedpower (averaged over small-scale fading) may still randomly fluctuate be-cause of several obstructions (walls, foliage or other obstacles) encounteredby radio waves along their way. These fluctuations occur over distances upto a few hundreds of wavelengths (tens of meters), and result in large-scalefading .

From the ongoing discussion it should be clear that wireless propaga-tion is mostly governed by a large number of unpredictable factors whichcan hardly be described in a rigorous fashion. For this reason, it is oftenpreferable to characterize the wireless channel from a statistical viewpointusing some fundamental parameters that are now introduced.

2.1.1.1 Path loss

The path loss is a statistical measure of the attenuation incurred by thetransmitted signal while it propagates through the channel. Assume thatthe transmitter and the receiver are separated by a distance d and let PT

and PR be the average transmitted and received powers, respectively. Then,



in the absence of any shadowing effect, it has been empirically found that

PR = βd−nPT (2.2)

where n is the path-loss exponent and β is a parameter that depends onthe employed carrier frequency, antenna gains and other environmentalfactors. For free-space propagation the path-loss exponent is 2, while inurban environment it takes values between 4 and 6.

The path loss Lpath(d) at a specified distance d is defined as the ratioPR/PT expressed in decibel (dB). From Eq. (2.2) it follows that

Lpath(d) = Lpath(d0) + 10n log10

(d

d0

), (2.3)

where d0 is an arbitrarily chosen reference distance. It is worth noting thatpower fluctuations induced by large-scale fading are not contemplated inEq. (2.3). The common approach to take these fluctuations into account isto assume a Gaussian distribution of the received power around the valuein Eq. (2.3). This amounts to setting

Lpath(d) = Lpath(d0) + 10n log10

(d

d0

)+ Z, (2.4)

where Z is a Gaussian random variable with zero-mean and standard devi-ation σZ (measured in dB). Since the path loss expressed in logarithmic dBscale follows a normal distribution, the model Eq. (2.4) is usually referredto as log-normal shadowing. Typical values of σZ lie between 5 and 12 dB.

2.1.1.2 Excess delay

The wireless channel is fully described by its channel impulse response(CIR) h(τ, t). This represents the response of the channel at time t to aDirac delta function applied at time t− τ , i.e., τ seconds before. DenotingNp the number of resolvable multipath components, we may write

h(τ, t) =Np∑

`=1

α`(t)ejθ`(t)δ (τ − τ`(t)) , (2.5)

where α`(t), θ`(t) and τ`(t) are the time-varying attenuation, phase shiftand propagation delay of the `th path, respectively. Without loss of gener-ality, we assume that the path delays are arranged in an increasing order ofmagnitude and define the `th excess delay ∆τ`(t) as the difference betweenτ`(t) and the delay τ1(t) of the first arriving multipath component, i.e.,∆τ`(t) = τ`(t)− τ1(t). At the receiver side, it is a common practice to use



a time scale such that τ1(t) = 0. In this case, the excess delays reduce to∆τ`(t) = τ`(t) for ` > 1.

If a signal sRF (t) is transmitted over a wireless channel characterized bythe CIR given in Eq. (2.5), the complex envelope of the received waveformtakes the form

r(t) =Np∑

`=1

α`(t)ejθ`(t)s (t− τ`(t)) . (2.6)

2.1.1.3 Power delay profile

The power delay profile (PDP) is a statistical parameter indicating howthe power of a Dirac delta function is dispersed in the time-domain asa consequence of multipath propagation. The PDP is usually given as atable where the average power associated with each multipath componentis provided along with the corresponding delay. In particular, the averagepower p(τ`) of the `th path is defined as

p(τ`) = E|α`(t)|2, (2.7)where | · | is the magnitude of the enclosed complex-valued quantity whileE· denotes statistical expectation. Clearly, summing all quantities p(τ`)provides the total average received power PR. In practice, however, thePDP is normalized so that the sum of p(τ`) is unity, i.e.,

Np∑

`=1

p(τ`) = 1. (2.8)

In this case, the CIR h(τ, t) in Eq. (2.5) must be multiplied by a factor√A, where A is a log-normal random variable which takes into account the

combined effect of path loss and large-scale fading.

Table 2.1 The PDP of a typical urban (TU) channel

Path number `Typical Urban Channel

Delay τ` (µs) Average power p(τ`)

0 0.0 0.18971 0.2 0.37852 0.5 0.23883 1.6 0.09514 2.3 0.06005 5.0 0.0379

Table 2.1 provides the PDP of a typical urban (TU) wireless channel[89]. A pictorial illustration of the same PDP is given in Fig. 2.3.



0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time delay, τ (µs)

Ave

rage

pow

er,

p(τ

)

Fig. 2.3 PDP of the TU channel in Table 2.1.

2.1.1.4 Root-mean-squared (RMS) delay spread

The root-mean-squared (RMS) delay spread provides a measure of the timedispersiveness of a multipath channel. This parameter is defined as

τrms =√

τ2 − (τ)2, (2.9)

where τ and τ2 are obtained from the PDP of the channel in the form

τ =Np∑

`=1

τ`p(τ`) (2.10)

and

τ2 =Np∑

`=1

τ2` p(τ`). (2.11)

From the normalization condition Eq. (2.8), it appears evident that thequantities p(τ`) for ` = 1, 2, . . . , Np can be interpreted as a probabilitymass function. In this respect, τrms represents the standard deviation ofthe path delays τ`.



Typical values of τrms are in the order of nanoseconds for indoor appli-cations and of microseconds for outdoor environments. For example, usingthe PDP in Table 2.1 it is found that τrms = 1.0620 µs for the TU chan-nel. This statistical parameter is an important indicator for evaluating theimpact of multipath distortion on the received signal. Actually, the distor-tion is negligible if the symbol duration Ts is adequately larger than τrms,say Ts > 10τrms. Otherwise, appropriate techniques must be employed tocompensate for the disabling effects of multipath distortion on the systemperformance. For example, in the IEEE 802.11a/g standards for wirelesslocal area networks (WLANs) the symbol duration is Ts,WLAN = 50 ns.Since in a typical urban channel we have τrms = 1.0620 µs, it follows thatTs,WLAN ¿ τrms. As a result, some compensation procedures are requiredat the receiver to avoid severe performance degradations.

2.1.1.5 Coherence bandwidth

The channel frequency response at time t is defined as the Fourier transformof h(τ, t) with respect to τ , i.e.,

H(f, t) =∫ ∞

−∞h(τ, t)e−j2πfτ dτ. (2.12)

To characterize the variations of H(f, t) with f at a given time instantt, we introduce the concept of coherence bandwidth Bc as a measure of the“flatness” of the channel frequency response. More precisely, two samplesof H(f, t) that are separated in frequency by less than Bc can be assumedas highly correlated. It is well-known that Bc is inversely proportional toτrms. In particular, for a 0.5-correlation factor it is found that

Bc ≈ 15τrms

. (2.13)

If the bandwidth Bs of the transmitted signal is smaller than Bc, thechannel frequency response can be considered as approximately flat overthe whole signal spectrum. In this case the spectral characteristics of thetransmitted signal are preserved at the receiver. Vice versa, if Bs is muchlarger than Bc, the signal spectrum will be severely distorted and the chan-nel is said to be frequency-selective. From the above discussion it turns outthat it is not meaningful to say that a given channel is flat or frequency-selective without having any information about the transmitted signal. Re-calling that the signal bandwidth is strictly related to the speed at whichinformation is transmitted, a given channel may appear as flat or frequency-selective depending on the actual transmission rate.



Example 2.1 The RMS delay spread of the TU channel in Table 2.1 hasbeen found to be 1.0620 µs. Hence, the 0.5-correlation coherence bandwidthis given by

Bc ≈ 15× 1.0620 µs

= 0.2 MHz. (2.14)

This means that the frequency response of the TU channel can be con-sidered as nearly flat over frequency intervals not larger than 0.2 MHz.This fact can also be inferred by inspecting Fig. 2.4, which illustrates theamplitude |H(f)| of the frequency response as a function of f .

0 1 2 3 4 5 6 7 8 9 10−14

−12

−10

−8

−6

−4

−2

0

Frequency f (MHz)

| H(

f )| (

dB)

Coherence bandwidth ≈ 0.2 MHz

Fig. 2.4 Frequency response of the TU channel in Table 2.1.

2.1.1.6 Doppler spread

In a mobile communication environment, the physical motion of the trans-mitter, receiver and surrounding objects induces a Doppler shift in eachmultipath component. To fix the ideas, assume that a pure sinusoid of fre-quency fc is transmitted over the channel and received by a mobile antennatraveling at a speed of v m/s. Defining ψ` the angle between the direction



of the receiver motion and the direction of arrival of the `th multipathcomponent, the corresponding Doppler shift is given by

fD,` =fcv

ccos(ψ`), (2.15)

where c = 3×108 m/s is the speed of light in the free space. In the presenceof several multipath components, the received signal is a superposition ofmany sinusoidal waveforms, each affected by an unpredictable frequencyshift due to the random nature of the angles ψ`. This phenomenon re-sults into a spectral broadening of the received spectrum known as Dopplerspread. The maximum Doppler shift is obtained from Eq. (2.15) by settingthe cosine function to unity and reads

fD,max =fcv

c. (2.16)

In practice, fD,max provides information about the frequency interval overwhich a pure sinusoid is received after propagating through the channel.Specifically, if fc is the transmitted frequency, the received Doppler spec-trum will be confined in the range [fc − fD,max, fc + fD,max].

Example 2.2 Assume that a laptop computer is moving at a speed of 20km/h in a IEEE 802.11g local area network operating around the 2.2 GHzfrequency band. From Eq. (2.16) it follows that the maximum Doppler shiftis given by

fD,max =2.2× 109 · (20× 103/3600)

3× 108≈ 40.7 Hz. (2.17)

Figure 2.5 illustrates the power of the received signal r(t) as a functionof t when fD,max = 40.7 Hz. We see that the power occasionally dropsfar below its expected value. This is a manifestation of the small-scalefading, which is caused by non-coherent superposition of the multipathcomponents at the receiving antenna. Inspection of Fig. 2.5 indicates thatin the presence of destructive superposition the received power may dropsdramatically. When this happens, we say that the channel is experiencinga deep fade.

The rate of occurrence of fade events is measured by the so-called levelcrossing rate (LCR). This parameter is defined as the expected rate atwhich the received power goes beyond a preassigned threshold level κ. Thefrequency of threshold crossings is a function of κ and is expressed by [64]

Nκ = fD,maxκ√σ2

r/πe− κ2

2σ2r , (2.18)



0 20 40 60 80 100−20

−15

−10

−5

0

5

10

Time (msec)

Rec

eive

d po

wer

(dB

)

Fig. 2.5 Fluctuations of the received signal power with fD,max = 40.7 Hz.

where σ2r =E|r(t)|2. The maximum of Nκ is found by computing the

derivative of Eq. (2.18) with respect to κ and setting it to zero. Thisyields Nκ,max = fD,maxe

−1/2√

π ' 1.07 ·fD,max, meaning that the expectednumber of fade events is approximately equal to the maximum Dopplershift fD,max. This result is validated by computer simulations shown inFig. 2.5, where four deep fades are observed over a time interval of 0.1 swhen fD,max = 40.7 Hz.

2.1.1.7 Coherence time

The coherence time Tc is a measure of how fast the channel characteristicsvary in time. From a theoretical viewpoint, this parameter is defined as themaximum time lag between two highly correlated channel snapshots. In amore practical sense, Tc can be regarded as the time interval over whichthe CIR is time-invariant.

The coherence time is proportional to the inverse of the maximum



Doppler shift. For a correlation threshold of 0.5, it is well approximated by

Tc =9

16πfD,max. (2.19)

If the signaling period Ts is smaller than Tc, each data symbol is subjectto stationary propagation conditions. In such a case we say that the channelis slowly fading. Vice versa, if Ts > Tc the propagation environment maysignificantly vary over a symbol period and the channel is thus affected byfast fading. We conclude that the same channel can appear as slowly orfast fading depending on the actual signaling rate.

Example 2.3 Assuming a maximum Doppler shift of 40.7 Hz as in Exam-ple 2.2, from Eq. (2.19) we find

Tc =9

16π · 40.7s ≈ 4.4 ms. (2.20)

Since the duration of each data block in the IEEE 802.11a/g standardsis about 4.0 µs, the TU channel can be considered as time invariant overone block.

2.1.2 Categorization of fading channels

As discussed earlier, the impact of multipath propagation on the reliabilityof a wireless link is strictly related to the characteristics of the transmittedsignal. In general, we can distinguish four distinct types of channels. Thelatter are summarized in Fig. 2.6 and are now discussed in some detail.

Frequency-nonselective slowly-fading

Frequency-selective fading

Time-selective fading

Frequency and time- selective fading

B c

B s > B c

B s <

>

T c

T s

<

T c

T s

Fig. 2.6 Categorization of fading channels.



2.1.2.1 Frequency-nonselective and slowly-fading channels

In many practical applications such as fixed communications within localareas, the coherence time Tc is much greater than the symbol duration Ts.In this case, the channel is affected by slowly-fading and the multipathparameters in Eq. (2.5) may be regarded as approximately invariant overmany signaling intervals. As a result, the CIR becomes independent of t

and can be rewritten as

h(τ) =Np∑

`=1

αèjθ`δ (τ − τ`) , (2.21)

while the corresponding channel frequency response is given by

H(f) =Np∑

`=1

αèjθè−j2πfτ` . (2.22)

If the path delays are much smaller than the symbol duration, then we mayreasonably set τ` ≈ 0 into Eqs. (2.21) and (2.22). This yields

h(τ) ≈ ρejϕδ(τ) (2.23)

and

H(f) ≈ ρejϕ, (2.24)

where we have defined

ρejϕ =Np∑

`=1

αèjθ` . (2.25)

Inspection of Eq. (2.24) reveals that H(f) is practically constant over thewhole signal bandwidth, and the channel is therefore frequency-nonselectiveor flat. In this case the complex envelope of the received signal takes theform

r(t) = ρejϕs(t) (2.26)

and is simply an attenuated and phase-rotated version of s(t).As indicated in Eq. (2.25), the multiplicative factor ρejϕ is the sum of

Np statistically independent contributions, each associated with a distinctmultipath component. Thus, invoking the central limit theorem [2], thereal and imaginary parts of ρejϕ can reasonably be approximated as twostatistically independent Gaussian random variables with the same varianceσ2 and expected values ηR and ηI , respectively. In the absence of any line-of-sight (LOS) path between the transmitter and receiver, no dominant



multipath component is present and we have ηR = ηI = 0. In such a casethe phase term ϕ is found to be uniformly distributed over [−π, π), whilethe amplitude ρ follows a Rayleigh distribution with probability densityfunction (pdf)

p(ρ) =ρ

σ2exp

(− ρ2

2σ2

), ρ ≥ 0. (2.27)

In some applications including satellite or microcellular mobile radiosystems, a LOS is normally present in addition to a scattered component.In this case ρ has a Rician distribution and its pdf is given by

p(ρ) =2ρ(K + 1)

Pρexp

−

[K +

(K + 1)ρ2

Pρ

]I0

(2ρ

√K(K + 1)

Pρ

),

(2.28)where ρ ≥ 0 and Pρ =Eρ2 = 2σ2 + η2

R + η2I while K = (η2

R + η2I )/(2σ2)

is the Rician factor, which is defined as the ratio between the power of theLOS path and the average power of the scattered component. Moreover,I0(x) is the zeroth-order modified Bessel function of the first kind, whichreads

I0(x) =12π

∫ 2π

0

ex cos αdα. (2.29)

Note that in the absence of any LOS component (K = 0) the Riciandistribution in Eq. (2.28) boils down to the Rayleigh pdf in Eq. (2.27)because of the identities Pρ = 2σ2 and I0(0) = 1.

2.1.2.2 Frequency-selective fading channels

Assume for simplicity that the channel is slowly-fading and consider itsfrequency response as given in Eq. (2.22). If the transmitted signal has abandwidth Bs larger than the channel coherence bandwidth, its spectralcomponents will undergo different attenuations while propagating from thetransmitter to the receiver. In this case the channel is frequency-selectiveand the received waveform is a linearly distorted version of the transmittedsignal. The frequency selectivity of a channel can also be checked in thetime-domain. Bearing in mind that Bs and Bc are inversely proportionalto Ts and τrms, respectively, the channel appears as frequency-selective ifTs < τrms and frequency-nonselective (or flat) otherwise. The most promi-nent impairment caused by frequency-selective fading is the insurgence ofintersymbol interference (ISI) in the received signal. A schematic illustra-tion of the ISI phenomenon is shown in Fig. 2.7, where a train of pulses



separated by Ts seconds is transmitted over a frequency-selective channel.If Ts is shorter than the channel delay spread, each received pulse overlapswith neighboring pulses, thereby producing ISI.

Time Time

Frequency -selective channel

Transmitted pulses Received distorted signal

Intersymbol interference (ISI)

T s

Fig. 2.7 Illustration of the intersymbol interference (ISI) phenomenon.

Figure 2.8 depicts a frequency-selective and slowly-fading channel wherethe channel frequency response keeps approximately constant over eachsymbol interval, but slowly varies from one interval to another.

Frequency

Time

H ( f , t )

Fig. 2.8 Illustration of a frequency-selective and slowly-fading channel.



2.1.2.3 Time-selective fading channels

The concept of time-selective fading is typically introduced by consideringa frequency-flat channel in which the delay spread is much smaller than thesymbol duration. As discussed previously, in this case we may reasonablysubstitute τ` = 0 into Eq. (2.5) to obtain

h(τ, t) = ρ(t)ejϕ(t)δ(τ) (2.30)

with

ρ(t)ejϕ(t) =Np∑

`=1

α`(t)ejθ`(t). (2.31)

The corresponding channel frequency response is given by

H(f, t) = ρ(t)ejϕ(t) (2.32)

and its amplitude is schematically depicted in Fig. 2.9 at some differenttime instants t.

Frequency

| H ( f,t )

|

t 0

t = t + t 1

t = t + 2 t 0 2

0

Fig. 2.9 Illustration of a time-selective fading channel.

Substituting τ` = 0 into Eq. (2.6) and using Eq. (2.31), yields

r(t) = ρ(t)ejϕ(t)s(t), (2.33)

from which it follows that the received signal is a replica of the transmittedwaveform s(t) except for a time-varying multiplicative distortion.



If the symbol period is greater than the channel coherence time, themultiplicative factor ρ(t)ejϕ(t) may significantly vary over a signaling inter-val. In such a case the channel is said to be time-selective and produces aDoppler spread of the received signal spectrum. A classical model to statis-tically characterize the multiplicative distortion induced by time-selectivefading is due to Jakes [64]. This model applies to a scenario similar to thatillustrated in Fig. 2.10, where an omni-directional antenna receives a largenumber of multipath contributions in the horizontal plane from uniformlydistributed scatterers.

o 69 . 27

V

Fig. 2.10 A typical scenario for application of the Jakes model.

In the above hypothesis, the quadrature components of ρ(t)ejϕ(t) arestatistically independent zero-mean Gaussian processes with power σ2 andautocorrelation function

R(τ) = σ2J0(2πfD,maxτ), (2.34)

where J0(x) is the zeroth-order Bessel function of the first kind while fD,max

denotes the maximum Doppler shift. In this case ρ(t) follows a Rayleighdistribution and the corresponding Doppler power spectrum (which is de-fined as the Fourier transform of 2R(τ)) is given by

P (f) =

2σ2

π√

f2D,max−f2

|f | ≤ fD,max

0 otherwise.(2.35)



Function P (f) exhibits the classical “bowl-shaped” form depicted inFig. 2.11. However, in many practical situations the Doppler power spec-trum can considerably deviate from the Jakes model.

- f D,max f

P ( f )

D,max

Fig. 2.11 The “bowl-shaped” Doppler power spectrum of the Jakes model.

The main impairment of a time-selective Rayleigh fading channel is thatρ(t) may occasionally drop to very low values (deep fades). When this hap-pens, the signal-to-noise ratio (SNR) becomes poor and the communicationsystem is thus vulnerable to the additive noise.

2.1.2.4 Frequency- and time-selective fading channels

In some applications it may happen that the symbol period and transmis-sion bandwidth of the information-bearing signal are larger than the chan-nel coherence time and coherence bandwidth, respectively. In this case thetransmitted signal undergoes frequency-selective as well as time-selectivefading (often referred to as doubly-selective fading), and the received wave-form is the superposition of several time-varying multipath components,each characterized by a non-negligible path delay as indicated in Eq. (2.6).In general, compensating the distortions induced by doubly-selective fadingis a rather difficult task.

2.2 Conventional methods for channel fading mitigation

Channel fading represents a major drawback in digital wireless commu-nications. Numerous research efforts have been devoted to combating its



detrimental effects and different solutions have been devised depending onwhether the channel can be categorized as time- or frequency-selective.

2.2.1 Time-selective fading

As mentioned previously, signals experiencing time-selective fading are oc-casionally plagued by deep fades which lead to severe attenuation of thereceived signal power. In this case data symbols are highly vulnerable tothe additive noise and “bursts” of errors are likely to occur. Channel cod-ing can be used to cope with the drop of SNR associated with deep fades.The main idea is to introduce some redundancy in the transmitted datastream so as to protect the information symbols against additive noise [26].Since channel coding is more effective in the presence of sparse errors, timeinterleaving is typically employed to break up error bursts. In addition tointerleaving and channel coding, diversity techniques have been proposedto combat time-selective fading.

2.2.2 Frequency-selective fading

The main impairment induced by frequency-selective fading is the occur-rence of ISI in the received signal. A classical approach to compensatefor ISI is to pass the received signal into a properly designed linear filtercalled channel equalizer. Several approaches have been proposed for thefilter design. Figure 2.12 illustrates the zero-forcing (ZF) solution, wherethe frequency response of the equalizer is taken as the inverse of the chan-nel frequency response H(f). In this case ISI is completely removed atthe expense of some noise enhancement. Better results are obtained withthe classical minimum mean-square error (MMSE) solution, which aims atminimizing the mean-square error (MSE) between the received samples andthe transmitted data symbols. In this way the equalizer can reduce the ISIwith much lower noise enhancement as compared to the ZF equalizer.

Example 2.4 We consider a wireless channel with three multipath com-ponents and the following frequency response

H(f) = 0.815− 0.495e−j2πfTs − 0.3e−j4πfTs . (2.36)

If we neglect the contribution of thermal noise, the nth received sample isgiven by

r(n) = 0.815 · c(n)− 0.495 · c(n− 1)− 0.3 · c(n− 2), (2.37)



From transmitter

Noise

H ( f ) Data detection

Equalizer

1 H ( f )

RECEIVER

Fig. 2.12 Structure of a conventional zero-forcing (ZF) equalizer.

where c(n) is the nth transmitted symbol. A ZF equalizer is used to com-pensate for the linear distortion produced by H(f). As shown in Fig. 2.13,the equalizer is implemented as a finite impulse response (FIR) filter oflength M and with weighting coefficients

pm = 1.143 · (0.981)m− 0.631 · (−0.542)m, m = 0, 1, . . . ,M − 1. (2.38)

z 1_

p0

p1

pM-1

y(n)

z 1_z 1_

Σ

r(n)

Fig. 2.13 FIR implementation of the ZF equalizer in Example 2.4.

The performance of the equalizer is usually given in terms of the outputMSE. This parameter is defined as

MSE = E|y(n)− c(n)|2, (2.39)

where y(n) is the equalizer output and represents a soft estimate of c(n)



Figure 2.14 illustrates the impact of the equalizer length M on the out-put MSE as obtained through Monte-Carlo simulations. These results in-dicate that efficient ISI compensation requires an equalizer with at least 70weighting coefficients. A longer filter is necessary if the propagation chan-nel comprises more multipath components with larger path delays, therebyincreasing the complexity of the receiving terminal. This is clearly unde-sirable since mobile receivers have usually limited computational resourcesand strict power constraints. A straightforward solution to reduce the ISIis to make the symbol duration adequately longer than the maximum chan-nel delay spread. However, since τrms is only determined by the physicalcharacteristics of the propagation channel, this approach translates into asuitable enlargement of the symbol period with a corresponding reduction ofthe achievable throughput. All these facts indicate that frequency-selectivefading is in general a serious obstacle for broadband wireless communica-tions.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Equalizer length, M

Mea

n−sq

uare

err

or (

MS

E)

Fig. 2.14 Output MSE as a function of the equalizer length M .



2.3 OFDM systems

2.3.1 System architecture

Orthogonal frequency-division multiplexing (OFDM) is a signaling tech-nique that is widely adopted in many recently standardized broadbandcommunication systems due to its ability to cope with frequency-selectivefading. Figure 2.15 shows the block diagram of a typical OFDM system.

a ) c i

Insert CP

Encoded symbols

S/P

VC

IDFT P/S

s i

s i ( cp )

b )

Discard CP

r i

r i

( cp ) Equalizer

VC

DFT S/P

R i

P/S

c i To the decoding

unit

Fig. 2.15 Block diagram of a typical OFDM system: a) transmitter; b) receiver.

The main idea behind OFDM is to divide a high-rate encoded datastream into Nu parallel substreams that are modulated onto Nu orthogonalcarriers (referred to as subcarriers). This operation is easily implementedin the discrete-time domain through an N -point inverse discrete Fouriertransform (IDFT) unit with N > Nu. The N − Nu unused inputs of theIDFT are set to zero and, in consequence, they are called virtual carri-ers (VCs). In practice, VCs are employed as guard bands to prevent thetransmitted power from leaking into neighboring channels. By modulatingthe original data onto N subcarriers, OFDM increases the symbol dura-



tion by a factor of N , thereby making the transmitted signal more robustagainst frequency-selective fading. The essence of this process is illustratedin Fig. 2.16 through a simple example where the symbol duration is doubledby dividing the original data stream into two parallel substreams. A com-parison with Fig. 2.7 reveals that lengthening the symbol duration providesan effective means to cope with ISI.

Time

Time

Fading channel

Transmitted impulse train

Received distorted signal

T s

2T s

2T s

Time

Carrier 1

Carrier 2

Fig. 2.16 Time-domain illustration of the benefits arising from lengthening the symbolduration.

The same conclusion can be drawn by examining the signal spectrumat the IDFT output. As shown in Fig. 2.17, the whole bandwidth is di-vided into two subchannels. If the latter are narrow enough compared tothe channel coherence bandwidth, the channel frequency response turnsout to be approximately flat over each subchannel. Hence, we may saythat OFDM converts a frequency-selective channel into several adjacentflat fading subchannels.

Bandwidth Channel

frequency response

Carrier #1 Carrier #2 Frequency

Bandwidth Channel

frequency response

Single Carrier

Frequency

Fig. 2.17 Frequency-domain illustration of the benefits arising from lengthening thesymbol duration.



From the ongoing discussion it appears that data transmission in OFDMsystems is accomplished in a block-wise fashion, where each block conveysa number Nu of (possibly coded) data symbols. As a consequence of thetime dispersion associated with the frequency-selective channel, contigu-ous blocks may partially overlap in the time-domain. This phenomenonresults into interblock interference (IBI), with ensuing limitations of thesystem performance. The common approach to mitigate IBI is to intro-duce a guard interval of appropriate length among adjacent blocks. Inpractice, the guard interval is obtained by duplicating the last Ng samplesof each IDFT output and, for this reason, is commonly referred to as cyclicprefix (CP). As illustrated in Fig. 2.18, the CP is appended in front ofthe corresponding IDFT output. This results into an extended block ofNT = N + Ng samples which can totally remove the IBI as long as Ng isproperly designed according to the channel delay spread.

Cyclic prefix IDFT output

Ng_ NgN N g

Fig. 2.18 Structure of an OFDM block with CP insertion.

Returning to Fig. 2.15 b), we see that the received samples are dividedinto adjacent segments of length NT , each corresponding to a different blockof transmitted data. Without loss of generality, in the ensuing discussionwe concentrate on the ith segment. The first operation is the CP removal,which is simply accomplished by discarding the first Ng samples of theconsidered segment. The remaining N samples are fed to a discrete Fouriertransform (DFT) unit and the corresponding output is subsequently passedto the channel equalizer. Assuming that synchronization has already beenestablished and the CP is sufficiently long to eliminate the IBI, only a one-tap complex-valued multiplier is required to compensate for the channeldistortion over each subcarrier. To better understand this fundamentalproperty of OFDM, however, we need to introduce the mathematical modelof the communication scheme depicted in Fig. 2.15.



2.3.2 Discrete-time model of an OFDM system

We denote ci = [ci(0), ci(1), . . . , ci(N − 1)]T the ith block of data at thetransmitter input, with (·)T representing the transpose operator. Symbolsci(n) are taken from either a phase-shift keying (PSK) or quadrature am-plitude modulation (QAM) constellation, while those corresponding to VCsare set to zero. After serial-to-parallel (S/P) conversion, vector ci is fed tothe IDFT unit. The corresponding output is given by

si = F Hci, (2.40)

where F is the N -point DFT matrix with entries

[F ]n,k =1√N

exp(−j2πnk

N

), for 0 ≤ n, k ≤ N − 1 (2.41)

while the superscript (·)H represents the Hermitian transposition.Vector si is next parallel-to-serial (P/S) converted and its last Ng ele-

ments are copied in front of it as shown in Fig. 2.18. The resulting vectors(cp)i is modeled as

s(cp)i = T (cp)si, (2.42)

where

T (cp) =[

PNg×N

IN

]. (2.43)

In the above equation, IN represents the N × N identity matrix whilePNg×N is an Ng×N matrix collecting the last Ng rows of IN . The entries ofs(cp)i are then fed to the D/A converter, which consists of an interpolation

filter with signaling interval Ts. The latter produces a continuous-timewaveform which is up-converted to a carrier frequency fc and launchedover the channel.

For presentational convenience, we consider a time-invariant frequency-selective channel with discrete-time impulse responseh = [h(0), h(1), . . . , h(L− 1)]T , with L denoting the channel length ex-pressed in signaling intervals. In practice, h represents the composite CIRencompassing the transmission medium as well as the transmit and receivefilters. After down-conversion and low-pass filtering, the received waveformis sampled at rate fs = 1/Ts. The resulting samples are mathematicallyexpressed as the convolution between the transmitted blocks s(cp)

i andh. Assuming that the block duration is longer than the maximum delay



spread and neglecting for simplicity the contribution of thermal noise, wecan write the ith block of received samples as

r(cp)i = B(l)s

(cp)i + B(u)s

(cp)i−1 , (2.44)

where B(l) and B(u) are NT ×NT Toeplitz matrices given by

B(l) =

h(0) 0 0 · · · 0h(1) h(0) 0 · · · 0h(2) h(1) h(0) · · · 0

......

......

...h(L− 1) h(L− 2) h(L− 3) · · · 0

0 h(L− 1) h(L− 2) · · · 0...

......

......

0 0 · · · 0 h(0)

(2.45)

and

B(u) =

0 · · · 0 h(1) h(2) · · · h(L− 1)0 · · · 0 0 h(1) · · · h(L− 2)...

......

. . . . . . . . ....

0 · · · · · · · · · · · · 0 h(1)0 · · · · · · · · · · · · · · · 0...

......

. . . . . . . . ....

0 · · · · · · · · · · · · · · · 0

. (2.46)

The second term in the right-hand-side of Eq. (2.44) is the IBI contribu-tion, which is eliminated after discarding the CP. Defining the CP removalmatrix as R(cp) = [0N×Ng IN ] and using the identity R(cp)B(u) = 0N×NT ,we have

ri = R(cp)r(cp)i = BcF

Hci (2.47)

where Bc = R(cp)B(l)T (cp) is an N×N circulant matrix whose first columnis

[hT 0T

N−L

]T .Vector ri is serial-to-parallel converted and fed to the receive DFT unit.

This produces

Ri =c FBcFHci. (2.48)

Recalling the well-known diagonalization property of circulant matrices[92], we have

FBcFH = DH , (2.49)



where DH is a diagonal matrix with H =√

NFh on its main diagonal.Hence, we may rewrite the DFT output as

Ri = DHci, (2.50)

or, in scalar form,

Ri(n) = H(n)ci(n), 0 ≤ n ≤ N − 1 (2.51)

where Ri(n) and ci(n) are the nth entries of Ri and ci, respectively, whileH(n) is the channel frequency response over the nth subcarrier, which reads

H(n) =L−1∑

`=0

h(`)e−j2πn`/N . (2.52)

Inspection of Eq. (2.51) indicates that OFDM can be viewed as a set of N

non-interfering (orthogonal) parallel transmissions with different complex-valued attenuation factors H(n). The transmitted symbols are recoveredafter pre-multiplying Ri by the inverse of DH , i.e.,

ci = D−1H Ri. (2.53)

Recalling that DH is a diagonal matrix, the above equation can be rewrit-ten in scalar form as

ci(n) =Ri(n)H(n)

, 0 ≤ n ≤ N − 1 (2.54)

from which it is seen that channel equalization in OFDM is simply accom-plished through a bank of one-tap complex-valued multipliers 1/H(n). Inpractice, due to the unavoidable presence of thermal noise and/or inter-ference, the equalizer only provides soft estimates of the transmitted datasymbols. The latter are eventually retrieved by passing the equalizer outputto a data detection/decoding unit.

In the OFDM literature, the sequences at the IDFT input and DFToutput are usually referred to as frequency-domain samples while those atthe IDFT output and DFT input are called time-domain samples.

Example 2.5 For illustration purposes, we consider an OFDM system withonly N = 4 subcarriers. The CP has length Ng = 2 and no VC is present.Transmission takes place over a multipath channel of length L = 3 andimpulse response as in Example 2.4. The following two blocks of binarydata symbols are fed to the IDFT unit

c0 =

1−1−1

1

, c1 =

−1−1−1

1

. (2.55)



Then, the CP is appended in front of each IDFT output, thereby producingthe vectors

s(cp)0 =

01 + j

01− j

01 + j

, s(cp)1 =

−1j

−1−j

−1j

. (2.56)

The received signal is distorted by frequency-selective fading. The time-domain samples corresponding to the second received OFDM block areexpressed by

r(cp)1 =

0.815 0 0 0 0 0−0.495 0.815 0 0 0 0−0.3 −0.495 0.815 0 0 0

0 −0.3 −0.495 0.815 0 00 0 −0.3 −0.495 0.815 00 0 0 −0.3 −0.495 0.815

−1j

−1−j

−1j

+

0 0 0 0 −0.495 −0.30 0 0 0 0 −0.4950 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

01 + j

01− j

01 + j

=

−1.31− 0.495j

0.195 + 0.515j

−0.515− 0.495j

0.495− 1.115j

−0.515 + 0.495j

0.495 + 1.115j

. (2.57)

After CP removal, the received samples are fed to the DFT unit. FromEq. (2.52) we know that

H =√

NFh =

0.021.115 + 0.495j

1.011.115− 0.495j

, (2.58)

and the data block c1 is thus retrieved as indicated in Eq. (2.53), i.e.,

c1 =

0.02 0 0 00 1.115 + 0.495j 0 00 0 1.01 00 0 0 1.115− 0.495j

−1

−0.02−1.115− 0.495j

−1.011.115− 0.495j

=

−1−1−1

1

. (2.59)



The above equation reveals that the transmitted symbols can ideally berecovered from the DFT output as long as the receiver has perfect knowl-edge of the channel response and the noise is vanishingly small. Also, weobserve that channel distortion is easily compensated through a bank offour complex-valued multipliers while a time-domain equalizer with tens oftaps is required in a conventional single-carrier system as that consideredin Example 2.4.

2.4 Spectral efficiencyY ( f ) FDM Y ( f ) OFDM

(a) Frequency-division multiplexing (b) OFDM

f f W W 0 W W 0 W

2 W 2

W 2

W 2

Fig. 2.19 Comparison between the spectral efficiencies of FDM and OFDM systems.

In addition to being robust against frequency-selective fading, anotheradvantage of OFDM is the relatively high spectral efficiency as comparedto conventional frequency-division multiplexing (FDM) systems. In theseapplications, the whole available bandwidth is divided into several subchan-nels and one data stream is transmitted over each subchannel. Figure 2.19(a) depicts the spectrum of a typical FDM system employing four parallelsubchannels. Here, the rectangular box spanning the frequency interval[−W,W ] represents the ideal signal spectrum that fully exploits the as-signed bandwidth. It appears that FDM scheme suffers from some spectralinefficiency, as indicated by the large shaped area within the rectangularbox.

As shown in Fig. 2.19 (b), in OFDM systems adjacent subchannelspartially overlap in the frequency domain. As a result, OFDM has muchhigher spectral efficiency than conventional FDM schemes. To cope with the



interference caused by spectra overlapping, carriers of different subchannelsare mutually orthogonal. As we have seen, this goal is efficiently achievedby means of FFT/IFFT operations. It is evident from Fig. 2.19 (b) thatthe spectral efficiency improves as the number of subcarriers increases. Onthe other hand, employing more subcarriers on a fixed bandwidth resultsinto narrower subchannels and longer OFDM blocks. This may greatlycomplicate the synchronization and channel equalization tasks since blocksof long duration are exposed to time-selective fading.

2.5 Strengths and drawbacks of OFDM

The main advantages of OFDM can be summarized as follows:

(1) Increased robustness against multipath fading, which is obtained bydividing the overall signal spectrum into narrowband flat-fading sub-channels. As a result, channel equalization is accomplished through asimple bank of complex-valued multipliers, thereby avoiding the needfor computationally demanding time-domain equalizers.

(2) High spectral efficiency due to partially overlapping subchannels in thefrequency-domain.

(3) Interference suppression capability through the use of the cyclic prefix.(4) Simple digital implementation by means of DFT/IDFT operations.(5) Increased protection against narrowband interference which, if present,

is expected to affect only a small percentage of the overall subcarriers.(6) Opportunity of selecting the most appropriate coding and modulation

scheme on each individual subcarrier according to the measured channelquality (adaptive modulation). In practice, higher order constellationsare normally used on less attenuated subcarriers in order to increasethe data throughput, while robust low-order modulations are employedover subcarriers characterized by low SNR values.

On the other hand, OFDM suffers from the following drawbacks ascompared to conventional single-carrier (SC) transmissions:

(1) It is very sensitive to phase noise and frequency synchronization errors,which translates into more stringent specifications for local oscillators.

(2) It needs power amplifiers that behave linearly over a large dynamicrange because of the relatively high peak-to-average power ratio(PAPR) characterizing the transmitted waveform.



(3) There is an inherent loss in spectral efficiency related to the use of thecyclic prefix.

2.6 OFDM-based multiple-access schemes

Conventional multiple-access techniques can be combined with OFDMto provide high-speed services to a number of simultaneously activeusers. Three prominent OFDM-based multiple-access schemes are avail-able in the technical literature. They include OFDM with time-divisionmultiple-access (OFDM-TDMA) [133], OFDM with code-division multiple-access (MC-CDMA) [53] and orthogonal frequency-division multiple-access(OFDMA) [141]. The main ideas behind these techniques are illustratedin Fig. 2.20 and are now briefly reviewed in order to highlight their mainfeatures.

OFDM-TDMA

In OFDM-TDMA, data transmission occurs into several consecutive time-slots, each comprehending one or more OFDM blocks. Since each slot isexclusively assigned to a specific user, no multiple-access interference (MAI)is present in the received data stream as long as a sufficiently long CP is ap-pended in front of the transmitted blocks. A possible drawback of OFDM-TDMA is the need for very high power amplifiers at the transmit side dueto the following reasons. First, because of its inherent TDMA structure,an OFDM-TDMA transmitter demands much higher instantaneous powerthan a frequency-division multiple-access (FDMA) system. Second, thetransmit amplifier must exhibit a linear characteristic over a wide dynamicrange due to the relatively high PAPR of the OFDM waveform [8]. Clearly,the need for highly linear power amplifiers increases the implementationcost of OFDM-TDMA transmitters.

MC-CDMA

MC-CDMA exploits the additional diversity gain provided by spread-spectrum techniques while inheriting the advantages of OFDM. In MC-CDMA systems, users spread their data symbol over M chips, which arethen mapped onto a set of M distinct subcarriers out of a total of N . Eachset of subcarriers is typically shared by a group of users which are sepa-rated by means of their specific spreading codes [42]. In order to achieve

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15,2007

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FD

M/O

FD

MA

System

s47

Frequency

Time

Signal power

Frequency

Time

Signal power

Frequency

Time

Signal power

User 1

User 2

(a) OFDM-TDMA (b) MC-CDMA (c) OFDMA

Fig. 2.20 Illustration of OFDM-based multiple-access schemes.



some form of frequency diversity, the M subcarriers can be interleaved overthe whole signal spectrum so as to maximize their separation distance inthe frequency domain. Similarly to CDMA, MC-CDMA signals are nor-mally plagued by MAI when transmitted over a frequency-selective fadingchannel. Since subcarriers are subject to different channel attenuations,orthogonality among users will be destroyed even though an orthogonalcode set is employed at the transmit side for spreading purposes. To alle-viate the MAI problem, sophisticated channel estimation and interferencecancellation techniques are needed in MC-CDMA systems [35].

OFDMA

The OFDMA concept is based on the inherent orthogonality of the OFDMsubcarriers. The latter are divided into several disjoint clusters which arenormally referred to as subchannels, and each user is exclusively assignedone or more subchannels depending on its requested data rate. Since allcarriers are perfectly orthogonal, in case of ideal synchronization no MAIis present at the output of the receiver DFT unit. This property greatlysimplifies the design of an OFDMA receiver by avoiding the need for com-putationally demanding detection techniques based on multiuser interfer-ence cancellation. In addition, the adoption of a dynamic subchannel as-signment strategy offers to OFDMA systems an effective means to exploitthe user-dependent frequency diversity. Actually, a specific carrier whichappears in a deep fade to one user may exhibit a relatively small atten-uation for another user. As a result, OFDMA can exploit channel stateinformation to provide users with the “best” subcarriers that are currentlyavailable, thereby leading to remarkable gains in terms of achievable datathroughput [172]. Thanks to its favorable features, OFDMA is widely rec-ognized as a promising technique for fourth generation broadband wirelessnetworks [149].

2.7 Channel coding and interleaving

Channel coding and interleaving are fundamental parts of any OFDM sys-tem as they allow to exploit the frequency diversity offered by the wirelesschannel.



Outer Encoder

Outer Interleaver

Mapper Binary source data

Inner Encoder

Inner Interleaver

Encoded symbols

Fig. 2.21 Channel coding and interleaving in an OFDM transmitter.

Encoding

Figure 2.21 illustrates the generation process of the encoded symbols at theinput of an OFDM system. The sequence of binary source data is dividedinto segments of k bits and fed to the outer encoder, where n − k redun-dant bits are added to each segment to protect information against channelimpairments and thermal noise. The encoder output is then passed to theouter interleaver, which is followed by the inner encoder. The output ofthe inner encoder is further interleaved before the encoded bits are mappedonto modulation symbols taken from a designated constellation. The mostcommonly used inner and outer coding architectures employ Reed–Solomon(RS) codes and convolutional codes, respectively [123]. The concatenatedcoding scheme of Fig. 2.21 is attractive due to its improved error correctioncapability and low decoding complexity.

Decoding

At the receiver, channel decoding and de-interleaving are accomplished asdepicted in Fig. 2.22.

Outer Decoder

Outer De-interleaver

De- Mapper

Detected symbols

Inner Decoder

Inner De-interleaver

Estimated source data

Fig. 2.22 Channel decoding and de-interleaving in an OFDM receiver.

The de-mapper converts the detected symbols into a sequence of bits.Since convolutional codes are very sensitive to burst errors, it is importantthat the inner de-interleaver can scatter the erroneous bits over the wholeinterleaving range before applying inner decoding. The convolutional innerdecoder is efficiently implemented by means of the Viterbi algorithm [123].After inner decoding, most bit errors in the received stream will be cor-rected. The output of the inner decoder is then de-interleaved before beingpassed to the outer decoder.



We recall that an RS code can correct up to dn−k2 e erroneous bits in one

encoded block of size n, where dxe denotes the highest integer not largerthan x. Therefore, if the outer de-interleaver scatters the remaining biterrors over multiple blocks and no more than dn−k

2 e bit errors are left ineach block, all source data are correctly retrieved.

The above discussion indicates that bit interleaving and de-interleavingare essential in OFDM systems to fully exploit the correction capabilityof the employed code structures. However, these operations may resultinto large storage requirements, which are clearly undesirable in terms ofimplementation cost.


Chapter 3

Time and Frequency Synchronization

Synchronization plays a major role in the design of a digital communica-tion system. Essentially, this function aims at retrieving some referenceparameters from the received signal that are necessary for reliable data de-tection. In a multicarrier network, the following synchronization tasks canbe identified.

(1) sampling clock synchronization: in practical systems the sampling clockfrequency at the receiver is slightly different from the correspondingfrequency at the transmitter. This produces interchannel interference(ICI) at the output of the receive DFT with a corresponding degrada-tion of the system performance. The purpose of symbol clock synchro-nization is to limit this impairment to a tolerable level.

(2) timing synchronization: the goal of this operation is to identify the be-ginning of each received OFDMA block so as to find the correct positionof the DFT window. In burst-mode transmissions timing synchroniza-tion is also used to locate the start of the frame (frame synchronization).

(3) frequency synchronization: a frequency error between the received car-rier and the local oscillator used for signal demodulation results ina loss of orthogonality among subcarriers with ensuing limitations ofthe system performance. Frequency synchronization aims at restor-ing orthogonality by compensating for any frequency offset caused byoscillator inaccuracies or Doppler shifts.

We limit our discussion to timing and frequency synchronization with-out addressing the problem of sampling clock recovery in this chapter. Thereason is that nowadays the accuracy of modern oscillators is in the or-der of some parts per million (ppm) and sample clock variations below 50ppm have only marginal effects on the performance of practical multicarrier

51



systems [118].In the ensuing discussion the synchronization task is separately ad-

dressed for the downlink and uplink case. As we shall see, while synchro-nization in the downlink can be achieved with the same methods employedin conventional OFDM transmissions, the situation is much more compli-cated in the uplink due to the possibly large number of parameters that thebase station (BS) has to estimate and the inherent difficulty in correctingthe time and frequency errors of each active user.

This chapter is organized as follows. The sensitivity of a multicarriersystem to timing and frequency errors is discussed in Sec. 3.1. In Sec. 3.2 weillustrate several synchronization algorithms explicitly designed for down-link transmissions. The uplink case is treated in Sec. 3.3 and Sec. 3.4.In particular, timing and frequency estimation is studied in Sec. 3.3 whilesome schemes for compensating the synchronization errors at the BS areillustrated in Sec. 3.4.

3.1 Sensitivity to timing and frequency errors

Timing and frequency errors in multicarrier systems destroy orthogonalityamong subcarriers and may result in large performance degradations. Tosimplify the analysis, in the following we concentrate on a downlink trans-mission but we point out that the final results essentially apply also to theuplink case.

The time-domain samples of the ith OFDM block are given by

s(cp)i (k) =

1√N

∑

n∈Ici(n) ej2πnk/N , −Ng ≤ k ≤ N − 1 (3.1)

where N is the size of the transmit IDFT unit, I denotes the set of mod-ulated subcarriers, Ng is the length of the cyclic prefix (CP) in samplingperiods and ci(n) is the symbol transmitted over the nth subcarrier. Fornotational simplicity, the superscript (·)(cp) is neglected throughout thischapter.

The baseband-equivalent discrete-time signal transmitted by the BS isthus represented by

sT (k) =∑

i

si(k − iNT ), (3.2)

where i counts the OFDM blocks and NT = N + Ng is the block length(included the CP).


Time and Frequency Synchronization 53

r(k )

Timingestimation

Frequencyestimation

Analogfront-end A/D

LO

DFT

j2π εk/Ne_

to channelequalization anddata detection

θ

rR F

(t)

Fig. 3.1 Block diagram of an OFDM receiver.

The block diagram of the receiver is depicted in Fig. 3.1. In the analogfront-end, the incoming waveform rRF (t) is filtered and down-convertedto baseband using two quadrature sinusoids generated by a local oscillator(LO). The baseband signal is then passed to the A/D converter, where itis sampled with frequency fs = 1/Ts.

Due to Doppler shifts and/or oscillator instabilities, the frequency fLO

of the LO is not exactly equal to the received carrier frequency fc. Thedifference fd = fc − fLO is referred to as carrier frequency offset (CFO).In addition, since the time scales at the transmit and receive sides are notperfectly aligned, at the start-up the receiver does not know where theOFDM blocks start and, accordingly, the DFT window will be placed ina wrong position. As shown later, since small (fractional) timing errorsdo not produce any degradation of the system performance, it suffices toestimate the beginning of each received OFDM block within one samplingperiod.

In the following we denote θ the number of samples by which the receivetime scale is shifted from its ideal setting. The samples from the A/D unitare thus expressed by

r(k) = ej2πεk/N∑

i

L−1∑

`=0

h(`)si(k − θ − `− iNT ) + w(k), (3.3)

where ε = NfdTs is the frequency offset normalized to the subcarrier spac-ing fcs = 1/(NTs), h = [h(0), h(1), . . . , h(L− 1)]T is the discrete-timechannel impulse response (CIR) encompassing the physical channel as wellas the transmit/receive filters and, finally, w(k) is complex-valued AWGNwith variance σ2

w. Since a carrier phase shift can be encapsulated into



the CIR, it is normally compensated for during the channel equalizationprocess.

The frequency and timing synchronization units shown in Fig. 3.1 em-ploy the received samples r(k) to compute estimates of ε and θ, say ε andθ. The former is used to adjust the frequency of the LO in a closed loopfashion or, alternatively, to counter-rotate r(k) at an angular speed 2πε/N

(frequency correction), while the timing estimate is exploited to achieve thecorrect positioning of the receive DFT window (timing correction). Specif-ically, the samples r(k) with indices iNT + θ ≤ k ≤ iNT + θ + N − 1 arefed to the DFT device and the corresponding output is used to detect thedata symbols conveyed by the ith OFDM block. The DFT output can alsobe exploited to track and compensate for small short-term variations of thefrequency error (fine-frequency estimation).

In the rest of this Section we assess the impact of uncompensated timingand frequency errors on the system performance.

3.1.1 Effect of timing offset

dataCP

dataCP

IBI-free partof the CP

Transmittedblocks

Receivedblocks

tail of the

L 1_

_(i 1)th block ith block

Ng

(i 1)th block_

Fig. 3.2 Partial overlapping of received blocks due to multipath dispersion.



We assume perfect frequency synchronization (i.e., ε = 0) and consideronly the effect of a timing error ∆θ = θ−θ. As shown in Fig. 3.2, the tail ofeach received block extends over the first L− 1 samples of the subsequentblock as a consequence of multipath dispersion. Since in a well designedsystem we must ensure that Ng ≥ L, at the receiver a certain range of theguard interval is not affected by the previous block. As long as the DFTwindow starts anywhere in this range, no interblock interference (IBI) willbe present at the DFT output.

To better explain this point, we see from Eqs. (3.1) and (3.3) that themth received block (apart from thermal noise) is expressed by

sm,R(k) =L−1∑

`=0

h(`)sm(k − θ − `−mNT ), (3.4)

and is non-zero for k′m ≤ k ≤ k′′m, where k′m = θ + mNT − Ng and k′′m =θ + (m + 1)NT −Ng + L− 2.

This means that the last sample of the (i−1)th received block has indexk′′i−1 = θ + iNT −Ng + L − 2 while the first sample of the (i + 1)th blockoccurs at k′i+1 = θ + iNT + N . Accordingly, samples r(k) with index k inthe set [θ + iNT −Ng + L− 1; θ + iNT + N − 1] are only contributed bythe ith OFDM block and, in consequence, do not suffer from IBI. Recallingthat the DFT window for the detection of the ith block spans the intervaliNT + θ ≤ k ≤ iNT + θ + N − 1, it follows that IBI is not present as longas −Ng + L − 1 ≤ ∆θ ≤ 0. In this case the DFT output over the nthsubcarrier can be represented as

Ri(n) = ej2πn∆θ/NH(n)ci(n) + Wi(n), (3.5)

where Wi(n) is the noise contribution with power σ2w and

H(n) =L−1∑

`=0

h(`) e−j2π`n/N (3.6)

is the channel frequency response over the considered subcarrier.Inspection of Eq. (3.5) reveals that the timing offset appears as a lin-

ear phase across the DFT outputs and is compensated for by the channelequalizer, which cannot distinguish between phase shifts introduced by thechannel and those deriving from the timing offset. This means that nosingle correct timing synchronization point exists in OFDM systems, sincethere are Ng − L + 2 values of θ for which interference is not present.

On the other hand, if the timing error is outside the interval −Ng +L − 1 ≤ ∆θ ≤ 0, the DFT output will be contributed not only by the ith



OFDM block, but also by the (i − 1)th or (i + 1)th block, depending onwhether ∆θ < −Ng + L − 1 or ∆θ > 0. In addition to IBI, this resultsinto a loss of orthogonality among subcarriers which, in turn, generatesICI. In this case the nth DFT output is affected by interference causedby data symbols transmitted over adjacent subcarriers and/or belonging toneighboring blocks, and reads

Ri(n) = ej2πn∆θ/Nα(∆θ)H(n)ci(n) + Ii(n,∆θ) + Wi(n), (3.7)

where Ii(n,∆θ) accounts for IBI and ICI while α(∆θ) is an attenuationfactor which is well approximated by [148]

α(∆θ) =L−1∑

`=0

|h(`|2 N −∆θ`

N, (3.8)

with

∆θ` =

∆θ − `,

`−Ng −∆θ,

0,

if ∆θ > `

if ∆θ < `−Ng

otherwise.(3.9)

The term Ii(n, ∆θ) can reasonably be modeled as a zero-mean randomvariable whose power σ2

I (∆θ) depends on the channel delay profile andtiming error according to the following relation

σ2I (∆θ) = C2

L−1∑

`=0

|h(`|2[2∆θ`

N+

(∆θ`

N

)2]

, (3.10)

where C2 =E|ci(n)|2 is the average power of the transmitted data sym-bols.

A useful indicator to evaluate the effect of timing errors on the systemperformance is the loss in signal-to-noise ratio (SNR). This quantity isdefined as

γ(∆θ) =SNR(ideal)

SNR(real), (3.11)

where SNR(ideal) is the SNR across subcarriers in a perfectly synchronizedsystem, while SNR(real) is the SNR in the presence of a timing offset. Inthe ideal case, the DFT output is given by

R(ideal)i (n) = H(n)ci(n) + Wi(n), (3.12)

so that, for a channel with unit average power (i.e., E|H(n)|2 = 1), wehave

SNR(ideal) = C2/σ2w. (3.13)



On the other hand, recalling that the three terms in the right-hand-sideof Eq. (3.7) are statistically uncorrelated, it follows that

SNR(real) = C2α2(∆θ)/

[σ2

w + σ2I (∆θ)

]. (3.14)

Substituting the above results into Eq. (3.11) yields

γ(∆θ) =1

α2(∆θ)

[1 +

σ2I (∆θ)σ2

w

]. (3.15)

It is useful to express the SNR loss in terms of Es/N0, where Es is theaverage received energy over each subcarrier while N0/2 is the two-sidedpower spectral density of the ambient noise. For this purpose we collectEqs. (3.10) and (3.15) and observe that C2/σ2

w = Es/N0. This produces

γ(∆θ) =1

α2(∆θ)

1 +

Es

N0

L−1∑

`=0

|h(`|2[2∆θ`

N+

(∆θ`

N

)2]

. (3.16)

0 2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

Timing error, ∆θ

γ(∆θ

), d

B

E

s/N

0 = 5 dB

Es/N

0 = 10 dB

Es/N

0 = 15 dB

Fig. 3.3 SNR loss due to timing errors.

Figure 3.3 illustrates γ(∆θ) (in dB) versus the timing error ∆θ for N =256 and some values of Es/N0. The CIR has length L = 8 and the channel



taps are modeled as circularly symmetric independent Gaussian randomvariables with zero-mean (Rayleigh fading) and power E|h(`)|2 = βe−`/8,where β is a suitable factor that normalizes the average energy of the CIRto unity. At each simulation run, a new channel snapshot is generated andthe results are obtained by numerically averaging the right-hand-side ofEq. (3.16) with respect to the channel statistics.

For a given timing error, we see that γ(∆θ) increases with Es/N0. Thiscan be explained by observing that at low SNRs the system performanceis mainly limited by thermal noise so that the impact of synchronizationerrors becomes less and less evident. The results in Fig. 3.3 indicate that inorder to keep the SNR degradation to a tolerable level of less than 1.0 dB,the error ∆θ after timing correction should be smaller than a few percentsof the block length. As discussed earlier, the presence of the CP providesintrinsic protection against timing errors since no performance degradationoccurs as long as −Ng + L − 1 ≤ ∆θ ≤ 0. The requirement of the timingsynchronizer is thus determined by the number of samples by which the CPexceeds the CIR duration. This provides the designer with a trade-off tool.Using a longer CP results into a relaxation of the timing synchronizationrequirements, but inevitably increases the system overhead.

3.1.2 Effect of frequency offset

We now assess the impact of a frequency error on the system performance.For simplicity, we assume ideal timing synchronization and let θ = θ = 0.At the receiver, the DFT output for the ith OFDM block is computed as

Ri(n) =1√N

N−1∑

k=0

r(k + iNT ) e−j2πnk/N , 0 ≤ n ≤ N − 1 (3.17)

and is not affected by IBI as long as Ng ≥ L − 1. Substituting Eq. (3.3)into Eq. (3.17) and performing standard manipulations yields

Ri(n) = ejϕi

∑

m∈IH(m)ci(m) ejπ(N−1)(ε+m−n)/NfN (ε + m− n) + Wi(n),

(3.18)where Wi(n) is thermal noise, ϕi = 2πiεNT /N and

fN (x) =sin(πx)

N sin(πx/N). (3.19)

We begin by considering the situation in which the frequency offset is amultiple of the subcarrier spacing fcs. In this case ε is integer-valued and



Eq. (3.18) reduces to

Ri(n) = ejϕiH (|n− ε|N ) ci (|n− ε|N ) + Wi(n), (3.20)

where |n− ε|N is the value of n− ε reduced to the interval [0, N − 1]. Thisequation indicates that an integer frequency offset does not destroy orthog-onality among subcarriers and only results into a shift of the subcarrierindices by a quantity ε. In this case the nth DFT output is an attenuatedand phase-rotated version of ci (|n− ε|N ) rather than of ci(n). Vice versa,when ε is not integer-valued the subcarriers are no longer orthogonal andICI does occur. In this case it is convenient to isolate the contribution ofci(n) in the right-hand-side of Eq. (3.18) to obtain

Ri(n) = ej[ϕi+πε(N−1)/N ]H(n)ci(n) fN (ε) + Ii(n, ε) + Wi(n), (3.21)

where Ii(n, ε) accounts for ICI and reads

Ii(n, ε) = ejϕi

∑

m6=n

H(m)ci(m) ejπ(N−1)(ε+m−n)/NfN (ε + m− n). (3.22)

Letting E|H(n)|2 = 1 and assuming independent and identically dis-tributed data symbols with zero-mean and power C2, from Eq. (3.22) wesee that Ii(n, ε) has zero-mean and power

σ2I (ε) = C2

∑

m6=n

f2N (ε + m− n). (3.23)

A more concise expression of σ2I (ε) is found when all N available sub-

carriers are used for data transmission, i.e., I = 0, 1, . . . , N − 1. In thiscase the above equation becomes

σ2I (ε) = C2

[1− f2

N (ε)], (3.24)

where we have used the identityN−1∑m=0

f2N (ε + m− n) = 1, (3.25)

which holds true independently of ε.The impact of the frequency error on the system performance is still

assessed in terms of the SNR loss, which is defined as

γ(ε) =SNR(ideal)

SNR(real), (3.26)

where SNR(ideal) is the SNR of a perfectly synchronized system as givenin Eq. (3.13), while

SNR(real) = C2f2N (ε)/

[σ2

w + σ2I (ε)

](3.27)



is the SNR in the presence of a frequency offset ε. Substituting Eqs. (3.13)and (3.27) into Eq. (3.26) and recalling that C2/σ2

w = Es/N0, we have

γ(ε) =1

f2N (ε)

[1 +

Es

N0(1− f2

N (ε))]

, (3.28)

where we have also borne in mind Eq. (3.24). For small values of ε, theabove equation can be simplified using the Taylor series expansion of f2

N (ε)around ε = 0. This produces

γ(ε) ≈ 1 +13

Es

N0(πε)2, (3.29)

from which it follows that the loss in SNR is approximately related to thesquare of the normalized frequency offset.

10−2

10−1

0

1

2

3

4

5

6

7

Normalized frequency error, ε

γ(ε)

, dB

E

s/N

0 = 5 dB

Es/N

0 = 10 dB

Es/N

0 = 15 dB

Fig. 3.4 SNR loss due to frequency errors.

Equation (3.28) is plotted in Fig. 3.4 as a function of ε for some val-ues of Es/N0 and N = 256. This diagram indicates that the frequencyoffset should be kept as low as 4-5% of the subcarrier distance to avoid asevere degradation of the system performance. For example, in the IEEE



802.16 standard for wireless MANs, the subcarrier spacing is 11.16 kHzand the maximum tolerable frequency error is thus in the order of 500 Hz.Assuming a carrier frequency of 5 GHz, this corresponds to an oscillatorinstability of 0.1 ppm. Since the accuracy of low-cost oscillators for mo-bile terminals usually does not meet the above requirement, an estimateε of the frequency offset must be computed at each terminal and used tocounter-rotate the samples at the input of the DFT device so as to reducethe residual frequency error ∆ε = ε− ε within a tolerable range.

3.2 Synchronization for downlink transmissions

Synchronization for OFDMA downlink transmissions is a relatively sim-ple task that can be accomplished with the same methods employed inconventional single-user OFDM systems. Here, each terminal exploits thebroadcast signal transmitted by the BS to get timing and frequency esti-mates, which are then exploited to control the position of the DFT windowand to adjust the frequency of the local oscillator.

The synchronization process is typically split into an acquisition stepfollowed by a tracking phase. During acquisition, pilot blocks with a par-ticular repetitive structure are normally exploited to get initial estimatesof the synchronization parameters [76,95,96,99,142,146,178]. Since in thisphase the time- and frequency-scales of the receiving terminal are still to bealigned to the incoming signal, synchronization algorithms must be foundthat can cope with large synchronization errors. The tracking phase is de-voted to the refinement of the initial timing and frequency estimates as wellas to counteract short-term variations that may occur due to oscillator driftsand/or time-varying Doppler shifts. For this purpose, several techniques ex-ploiting either the redundancy of the CP or pilot tones multiplexed in thefrequency-domain are available in the literature [24,29,163]. Alternatively,blind methods operating over the DFT output can be used [30,98].

In this Section we investigate timing and frequency estimation in adownlink scenario. Both the acquisition and tracking phases are consideredand separately discussed. As standardized in many commercial systemsincluding DAB [39], DVB-T [40] and HIPERLAN/II [41], the transmissionis organized in frames, each containing some known reference blocks toassist the synchronization process.

A possible example of frame structure is depicted in Fig. 3.5. Here,a null block where nothing is transmitted (no signal power) is placed at



reference blocks

data blocks null block

FRAME

Fig. 3.5 Example of frame structure in the downlink.

the beginning of the frame, followed by a given number of reference anddata blocks. In addition, some pilot tones carrying known symbols arenormally placed within data blocks at some specified subcarriers in orderto track possible variations of the synchronization parameters. The nullblock is exploited for interference and noise power estimation. Furthermore,it provides a simple means to achieve coarse frame synchronization. Inthis case, the drop of power corresponding to the null block is revealedby a power detector and used as a rough estimate of the start of a newframe [107]. Fine frame synchronization is next achieved using informationprovided by the timing synchronization unit.

3.2.1 Timing acquisition

In most multicarrier applications, timing acquisition represents the firststep of the downlink synchronization process. This operation has two mainobjectives. First, it detects the presence of a new frame in the receiveddata stream. Second, once the frame has been detected, it provides acoarse estimate of the timing error so as to find the correct position of thereceive DFT window. Since the CFO is usually unknown in this phase, it isdesirable that the timing recovery scheme be robust against possibly largefrequency offsets.

One of the first timing acquisition algorithms for OFDM transmissionswas proposed by Nogami and Nagashima [107], and was based on the ideaof searching for a null reference block in the received frame. Unfortunately,this method provides highly inaccurate timing estimates. Also, it is notsuited for burst-mode applications since the null block cannot be distin-guished by the idle period between neighboring bursts. A popular approachto overcome these difficulties makes use of some reference blocks exhibitinga repetitive structure in the time domain. In this case, a robust timing es-timator can be designed by searching for the peak of the correlation among



the repetitive parts. This approach was originally proposed by Schmidl andCox (S&C) in [142], where a reference block with two identical halves oflength N/2 is transmitted at the beginning of each frame and exploited fortiming and frequency acquisition.

subcarriers

0 0 c ( 0 ) 0

0 1 2 3 N 2 _

first half second half

time-domain samples

( a )

( b )

c (2 ) c ( ) _ N 2

Fig. 3.6 S&C reference block in the frequency-domain (a) and in the time-domain (b).

As shown in Fig. 3.6, the reference block can easily be generated inthe frequency domain by modulating the subcarriers with even indices bya pseudonoise (PN) sequence c = [c(0), c(2), . . . , c(N − 2)]T while settingto zero the remaining subcarriers with odd indices. As long as the CP isnot shorter than the CIR duration, the two halves of the reference blockwill remain identical after passing through the transmission channel exceptfor a phase difference caused by the CFO. Hence, if the received samplescorresponding to the first half are given by

r(k) = sR(k)ej2πεk/N + w(k), θ ≤ k ≤ θ + N/2− 1 (3.30)

with sR(k) being the useful signal and w(k) denoting the thermal noise,then the samples in the second half take the form

r(k+N/2) = sR(k)ej2πεk/Nejπε+w(k+N/2), θ ≤ k ≤ θ+N/2−1. (3.31)

In this case, the magnitude of a sliding window correlation of lag N/2provides useful information about the timing error since a peak is expectedwhen the sliding window is perfectly aligned with the reference block. Thisapproach leads to the timing estimate [142]

θ = arg maxθ

∣∣∣Γ(θ)∣∣∣

, (3.32)



where Γ(θ) is the following normalized N/2-lag autocorrelation of the re-ceived samples

Γ(θ) =

N/2−1∑q=0

r(q + N/2 + θ)r∗(q + θ)

N/2−1∑q=0

∣∣∣r(q + N/2 + θ)∣∣∣2

. (3.33)

−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tim

ing

met

ric

δθ

Fig. 3.7 Example of timing metric for the S&C algorithm.

Figure 3.7 shows an example of timing metric∣∣∣Γ(θ)

∣∣∣ as a function of the

difference δθ = θ− θ. The results are obtained numerically over a Rayleighmultipath channel with L = 8 taps. The number of subcarriers is N = 256and the CP has length Ng = 16. The signal-to-noise ratio over the receivedsamples is defined as SNR = σ2

s/σ2w with σ2

s =E|sR(k)|2, and is set to20 dB.

As mentioned before, the first step of the timing acquisition process isrepresented by the detection of a new frame in the received data stream.



For this purpose,∣∣∣Γ(θ)

∣∣∣ is continuously monitored and the start of a frameis declared whenever it overcomes a given threshold λ. The latter mustproperly be designed by taking into account the statistics of the timingmetric so as to achieve a reasonably trade-off between false alarm and mis-detection probabilities. Once the presence of a new frame has been detected,a timing estimate θ is computed by searching for the maximum of

∣∣∣Γ(θ)∣∣∣

as indicated in Eq. (3.32).Unfortunately, we see from Fig. 3.7 that the timing metric of the S&C

algorithm exhibits a large “plateau” that may greatly reduce the esti-mation accuracy. Solutions to this problem are proposed in some recentworks, where reference blocks with suitably designed patterns are exploitedto obtain sharper timing metric trajectories [95, 146]. For instance, Shiand Serpedin (S&S) use a training block composed of four repetitive parts[+B + B −B + B] with a sign inversion in the third segment [146]. Asdepicted in Fig. 3.8, a sliding window of length N spans the received time-domain samples with indices θ ≤ k ≤ θ+N −1, and collects them into fourvectors rj(θ) = r(k + jN/4 + θ) ; 0 ≤ k ≤ N/4− 1 with j = 0, 1, 2, 3.

Sliding window (N samples)

Time-domainsamples

r0 θ( ) r1 θ( ) r2 θ( ) r3 θ( )

Fig. 3.8 Sliding window used in the S&S timing acquisition scheme.

The timing metric is then computed as

ΓSS(θ) =

∣∣∣Λ1(θ)∣∣∣ +

∣∣∣Λ2(θ)∣∣∣ +

∣∣∣Λ3(θ)∣∣∣

32

3∑

j=0

∥∥∥rj(θ)∥∥∥

2, (3.34)

where

Λ1(θ) = rH0 (θ)r1(θ)− rH

1 (θ)r2(θ)− rH2 (θ)r3(θ), (3.35)

Λ2(θ) = rH1 (θ)r3(θ)− rH

0 (θ)r2(θ), (3.36)

Λ3(θ) = rH0 (θ)r3(θ). (3.37)

Figure 3.9 illustrates ΓSS(θ) as obtained in the same operating condi-tions of Fig. 3.7. Since the plateau region associated with the S&C metric



−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δθ

Tim

ing

met

ric

Fig. 3.9 Example of timing metric for the S&S algorithm.

is significantly reduced, more accurate timing estimates are expected. Asindicated in [95], reference blocks with more than four repetitive segmentscan be designed to further increase the sharpness of the timing trajectory.

Simulation results obtained with both S&C and S&S algorithms indi-cate that the residual timing error ∆θ takes on positive values with non-negligible probability. In this case the system performance may severelybe degraded by IBI since the DFT window includes samples of the currentOFDM block as well as of the next block. Appending a short cyclic postfixat the end of each transmitted block is a viable solution to mitigate theeffect of small positive timing errors. Alternatively, we can pre-advancethe estimate θ by some samples θc to obtain a final timing estimate in theform [95]

θ(f) = θ − θc, (3.38)

where θ is still given in Eq. (3.32) while θc is designed so as to maximize theprobability that ∆θ(f) = θ(f)−θ lies in the interval Ng +L−1 ≤ ∆θ(f) ≤ 0in order to mitigate IBI.



3.2.2 Fine timing tracking

If the transmit and receive clock oscillators are adequately stable, the timingestimate computed at the beginning of the downlink frame on the basis ofthe reference block can be used for data detection over the entire frame.In certain applications, however, the presence of non-negligible errors inthe sampling clock frequency results in a short-term variation of the timingerror ∆θ which must be tracked in some way.

One straightforward solution is found by considering ∆θ as intro-duced by the physical channel rather than by the oscillator drift. Thisamounts to absorbing ∆θ into the CIR vector or, equivalently, to replac-ing h = [h(0), h(1), . . . , h(L− 1)]T by its time-shifted version h′(∆θ) =[h(∆θ), h(1 + ∆θ), . . . , h(L− 1 + ∆θ)]T . Therefore, in the presence ofsmall sampling frequency offsets, channel estimates computed over differentOFDMA blocks are differently delayed as a consequence of the long-termfluctuations of ∆θ. A possible method to track these fluctuations is to lookfor the delay of the first significant tap in the estimated CIR vector. Thisapproach is adopted in [178], where the integer part of the timing esti-mate is used by the DFT controller to adjust the DFT window position,while the fractional part appears as a linear phase across subcarriers andis compensated for by the channel equalization unit.

Alternative schemes to track residual timing errors make use of suitablecorrelations computed either in the time- or frequency-domain. For in-stance, the method proposed in [168] exploits known pilot tones multiplexedinto the transmitted data stream, which are correlated at the output of thereceive DFT with the transmitted pilot pattern. A time-domain approachis discussed in [163] and [76], where the autocorrelation properties inducedby the use of the CP on the received time-domain samples is exploitedfor fine timing tracking. In this case, the following N -lag autocorrelationfunction is used as a timing metric

γ(k) =Ng−1∑q=0

r(k − q)r∗(k − q −N), (3.39)

where k is the time index of the currently received sample.Since the CP is just a duplication of the last Ng samples of the OFDM

block, we expect that γ(k) may periodically exhibit peaks whenever thesamples r(k− q−N) with 0 ≤ q ≤ Ng − 1 belong to the CP. This intuitionis confirmed by the experimental results of Fig. 3.10, where γ(k) is shownversus the time index k for a Rayleigh multipath channel with CIR duration



L = 8 and SNR = 20 dB. The number of subcarriers is N = 256 whileNg = 16.

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20

Time index, k

γ (k

)

Fig. 3.10 Timing metric based on the CP correlation properties.

Figure 3.10 indicates the presence of peaks at a regular distance of NT

samples, which can be used to continuously track the residual timing offset.It should be observed that accurate timing estimation may be difficult inthe presence of strong interference and/or noise due to the relatively shortintegration window employed in Eq. (3.39). A possible remedy to thisdrawback is suggested in [163], where the timing metric is smoothed bymeans of a first-order infinite impulse response (IIR) filter. This yields thefollowing modified metric

γ(k) = αγ(k −NT ) + (1− α)γ(k), (3.40)

in which γ(k) is still given in Eq. (3.39) and 0 < α < 1 is a forgettingfactor which is designed so as to achieve a reasonable trade-off betweenestimation accuracy and tracking capabilities. The location of the peaks inγ(k) indicate the start of the received blocks and are used to control theposition of the DFT window.



3.2.3 Frequency acquisition

After frame detection and timing acquisition, each terminal must computea coarse frequency estimate to align its local oscillator to the received car-rier frequency. This operation is referred to as frequency acquisition andis normally accomplished at each new received frame by exploiting thesame reference blocks used for timing acquisition, in addition to possiblyother dedicated blocks. As mentioned previously, the reference blocks arenormally composed by some repetitive parts which remain identical afterpassing through the channel except for a phase shift caused by the fre-quency error. The latter is thus estimated by measuring the induced phaseshift. This approach has been employed by Moose in [96], where the phaseshift between two successive identical blocks is measured in the frequency-domain at the DFT output. More precisely, assume that timing acquisitionhas already been achieved and let R1(n) and R2(n) be the nth DFT outputcorresponding to the two reference blocks. Then, we may write

R1(n) = SR(n) + W1(n), (3.41)

and

R2(n) = SR(n)ej2πεNT /N + W2(n), (3.42)

where SR(n) is the signal component (the same over the two blocks as longas the channel is static) while W1(n) and W2(n) are noise terms. The aboveequations indicate that an estimate of ε can be derived as

ε =1

2π(NT /N)arg

N−1∑n=0

R2(n)R∗1(n)

. (3.43)

One major drawback of this scheme is the relatively short acquisition range.Actually, since the arg · function returns values in the range [−π, π), wesee from Eq. (3.43) that |ε| ≤ N/(2NT ), which is less than one half of thesubcarrier spacing.

A viable method to enlarge the frequency acquisition range is proposedby Schmidl and Cox (S&C) in [142]. Similarly to Moose, they performfrequency acquisition by exploiting two reference blocks which are suitablydesigned so as to guarantee an acquisition range of several subcarrier spac-ings.

As depicted in Fig. 3.11, the first block is the same used for timing ac-quisition and is composed of two identical halves in the time-domain (eachof length N/2). The second block contains a differentially encoded pseudo-noise sequence PN1 on the even subcarriers and another pseudo-noise se-quence PN2 on the odd subcarriers. In describing the S&C method, we



CP

first half second half

CP first reference block second reference block

PN1 and PN2 sequences

Fig. 3.11 Reference blocks employed by the S&C frequency acquisition scheme.

assume for simplicity that the timing acquisition phase has been success-fully completed and the receiver has perfect knowledge of the timing offsetθ. Also, we decompose the frequency error into a fractional part, less than1/T in magnitude, plus an integer part which is multiple of 2/T , whereT = NTs is the length of the OFDM block (excluded the CP). Hence, wemay write the normalized frequency error as

ε = ν + 2η, (3.44)

where ν ∈ (−1, 1] and η is an integer.The S&C algorithm exploits the first reference block to get an estimate

of ν. For this purpose, the following N/2-lag autocorrelation is computed

Ψ =θ+N/2−1∑

k=θ

r(k + N/2)r∗(k), (3.45)

where r(k) and r(k + N/2) are time-domain samples in the two halves ofthe first reference block as expressed in Eqs. (3.30) and (3.31), respectively.Apart from thermal noise, r(k) and r(k + N/2) are identical except for aphase shift of πν. Hence, an estimate of ν is obtained as

ν =1π

arg

θ+N/2−1∑

k=θ

r(k + N/2)r∗(k)

. (3.46)

This equation indicates that timing information is necessary to computeν. In practice, the quantity θ in Eq. (3.46) is replaced by its correspondingestimate θ as given in Eq. (3.32).

In order to compensate for the fractional part of the CFO, the time-domain samples are counter-rotated at an angular speed 2πν/N and fed tothe DFT unit. We denote R1(n) and R2(n) (n = 0, 1, . . . , N − 1) the DFToutputs corresponding to the first and second reference blocks, respectively.Although no ICI will be present on R1(n) and R2(n) as long as ν ≈ ν, theDFT outputs will be shifted from their correct position if η 6= 0 due to the



uncompensated integer frequency error. Bearing in mind Eq. (3.20), wemay write

R1(n) = ejϕ1H (|n− 2η|N ) c1 (|n− 2η|N ) + W1(n), (3.47)

and

R2(n) = ej(ϕ1+4πηNT /N)H (|n− 2η|N ) c2 (|n− 2η|N ) + W2(n), (3.48)

where |n− 2η|N is the value n − 2η reduced to the interval [0, N − 1],H(n) is the channel response and ci(n) the symbol transmitted over thenth subcarrier and belonging to the ith block. Neglecting for simplicity thenoise terms and calling d(n) = c2(n)/c1(n) the differentially-modulated PNsequence on the even subcarriers of the second block, from Eqs. (3.47) and(3.48) we see that R2(n) ≈ ej4πηNT /Nd (|n− 2η|N )R1(n) if n is even. Anestimate of η is thus calculated by looking for the integer η that maximizesthe following metric

B(η) =

∣∣∑n∈J R2(n)R∗1(n)d∗ (|n− 2η|N )

∣∣∑

n∈J |R2(n)|2 , (3.49)

where J is the set of indices for the even subcarriers and η varies overthe range of possible frequency offsets. Bearing in mind Eq. (3.44), theestimated CFO is finally given by

ε = ν + 2η, (3.50)

and its mean-square error (MSE) can reasonably be approximated as [142]

MSE ε =2(SNR)−1

π2N, (3.51)

where SNR = σ2s/σ2

w is the signal-to-noise ratio over the received time-domain samples.

Appealing features of the S&C method are its simplicity and robustness,which make it well suited for burst-mode transmissions where accurateestimates of the synchronization parameters must be obtained as fast aspossible. An extension of the S&C algorithm has been proposed by Morelliand Mengali (M&M) in [99] by considering a reference block composed byQ ≥ 2 repetitive parts, each comprising P = N/Q time-domain samples.In the M&M algorithm the estimated CFO is computed as

ε =Q

2π

Q/2∑q=1

χ(q) arg Ψ(q)Ψ∗(q − 1) , (3.52)



where χ(q) are suitable weighting coefficients given by

χ(q) =12(Q− q)(Q− q + 1)−Q2

2Q(Q2 − 1), (3.53)

while Ψ(q) is the following qP -lag autocorrelation

Ψ(q) =θ+N−qP−1∑

k=θ

r(k + qP )r∗(k) q = 1, 2, . . . , Q/2. (3.54)

The M&M scheme gives unbiased estimates as long as |ε| ≤ Q/2 andthe SNR is adequately high. Hence, if Q is designed such that the possiblefrequency offsets lie in the interval [−Q/2, Q/2], the CFO can be estimatedwithout the need for a second reference block as required by the S&Cmethod, thereby allowing a substantial reduction of the system overhead.

The MSE of the estimate Eq. (3.52) is given by [99]

MSE ε =3(SNR)−1

2π2N(1− 1/Q2), (3.55)

and for Q > 2 is lower than the corresponding result Eq. (3.51) obtainedwith the S&C method.

Figure 3.12 compares the S&C and M&M algorithms in terms of MSEversus SNR. The number of available subcarriers is N = 256 and thechannel has L = 8 taps. The latter are Gaussian distributed with zero-mean and an exponentially decaying power delay profile. Parameter Q

with the M&M scheme has been fixed to 8. The dashed lines representtheoretical analysis as given by Eqs. (3.51) and (3.55) while marks indicatesimulation results. We see that the theoretical MSEs are validated only atlarge SNR values. The best results are obtained with the M&M algorithm,which achieves a gain of approximately 0.8 dB over the S&C.

3.2.4 Frequency tracking

The CFO estimate obtained during the acquisition phase is used to adjustthe frequency of the LO or, alternatively, to counter-rotate the basebandreceived samples r(k) at an angular speed 2πε/N so as to produce the newsequence r′(k) = r(k)e−j2πkε/N . Due to estimation inaccuracies and/ortime-varying Doppler shifts, r′(k) may still be affected by a residual fre-quency error ∆ε = ε−ε. The latter induces a phase shift that varies linearlyin time with a slope proportional to ∆ε. As long as ∆ε is adequately small,the phase shift can be absorbed into the channel frequency response andcompensated for during the channel equalization process. However, if ∆ε



0 2 4 6 8 10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

SNR (dB)

MS

E (ε

)

S&C SimulationS&C theoryM&M SimulationM&M theory

Fig. 3.12 Accuracy of the frequency estimates vs. SNR with S&C and M&M algorithms.

exceeds a few percent of the subcarrier spacing, the DFT output will beaffected by non-negligible ICI. In such a case frequency tracking becomesmandatory to avoid severe degradation of the system performance. Thisoperation is typically accomplished on a block-by-block basis using a closed-loop architecture as that depicted in Fig. 3.13.

Here, the sequence r′i(m) (−Ng ≤ m ≤ N−1) collects the samples r′(k)belonging to the ith received OFDM block (included the CP) while ei isan error signal which is proportional to the residual frequency offset. Thissignal is computed at each new received block and fed to the loop filter,which updates the frequency estimate according to the following recursiveequation

∆εi+1 = ∆εi + αei, (3.56)

where ∆εi is the estimate of ∆ε over the ith block and α is a designparameter (step-size) that controls the convergence speed of the track-ing loop. Increasing α improves the tracking capabilities but inevitablydegrades the estimation accuracy in the steady-state. Thus, convergence



i discardprefix

N

DFT

loopfilter

errorgenerator

r'(m)ix (m)

ei∆ε i

R (n)i

jψ (m)e_

i

CO

Fig. 3.13 Closed-loop architecture for tracking a residual CFO.

speed and tracking accuracy are contrasting goals which must be traded-offby a proper selection of the step-size.

Returning to Fig. 3.13, we see that ∆εi is fed to a numerically controlledoscillator (NCO) that generates the exponential term e−jψi(m). The phaseψi(m) is recursively computed as

ψi(m) = ψi(m− 1) + 2π∆εi/N, −Ng ≤ m ≤ N − 1 (3.57)

where ψi(−Ng − 1) is set equal to ψi−1(N − 1) in order to avoid any phasejump between the last sample of the (i− 1)th block and the first sample ofthe ith block. Inspection of Eq. (3.57) indicates that ψi(m) varies linearly intime with a slope proportional to the current frequency estimate ∆εi. Theexponential term is then used to obtain the frequency-corrected samplesxi(m) = r′i(m)e−jψi(m) for −Ng ≤ m ≤ N − 1. After discarding the CP,the latter are finally fed to the DFT device which generates the frequency-domain samples Ri(n) (0 ≤ n ≤ N − 1).

Several frequency tracking schemes available in the literature rely onthe closed-loop structure of Fig. 3.13 and only differ in the adopted errorsignal ei [29, 30, 98, 163]. In particular, we distinguish between frequency-domain and time-domain algorithms, depending on whether ei is computedusing the DFT output Ri(n) or the samples xi(m) at the input of the DFTdevice. For example, the schemes proposed in [29] and [163] operate in thetime-domain and exploit the redundancy offered by the CP to obtain an



error signal of the type

ei =1

Ng=m

−1∑

m=−Ng

xi(m + N)x∗i (m)

, (3.58)

where xi(m) (−Ng ≤ m ≤ −1) are samples taken from the CP of the ithreceived block.

To explain the rationale behind Eq. (3.58), we temporarily neglect theeffect of thermal noise as well as any interference on xi(m) caused by chan-nel echoes. Then, in the presence of a residual frequency offset ∆ε −∆εi,the samples xi(m) and xi(m + N) only differ for a phase shift and we canreasonably write xi(m + N) ≈ xi(m)ej2π(∆ε−∆εi) for −Ng ≤ m ≤ −1.Substituting this relation into Eq. (3.58) indicates that ei is proportionalto sin [2π(∆ε−∆εi)] and can be used in Eq. (3.56) to improve the accuracyof the frequency estimate as it is now explained. To fix the ideas, assumethat ∆εi is (slightly) smaller than the true offset ∆ε. Since in this case ei

is a positive quantity, from Eq. (3.56) it follows that ∆εi+1 > ∆εi, whichresults into a reduction of the estimation error. The situation ∆εi > ∆ε

can be dealt with similar arguments and leads to the same final conclu-sion. The equilibrium point is achieved in a perfectly synchronized systemwhere ∆εi = ∆ε. Indeed, in this case ei = 0 and from Eq. (3.56) we have∆εi+1 = ∆εi, meaning that the frequency estimate is kept fixed at its cur-rent value. In practice, the estimate will fluctuate around the equilibriumpoint due to the unavoidable presence of thermal noise and interference.

As mentioned previously, the error signal can also be computed in thefrequency-domain by exploiting the quantities Ri(n) at the output of theDFT unit (see Fig. 3.13). An example in this sense is given in [30], whereei is derived using a maximum likelihood (ML) approach and reads

ei = <e

∑

n∈IR∗i (n) [Ri(n + 1)−Ri(n− 1)]

. (3.59)

A similar method with improved performance is proposed in [98] andemploys the following error signal

ei = <e

∑

n∈I

R∗i (n) [Ri(n + 1)−Ri(n− 1)]1 + β |Ri(n)|2

, (3.60)

where β is a suitable parameter that depends on the operating SNR.It is worth noting that all the considered schemes for computing ei are

blind in that they do not exploit any pilot symbols embedded into thetransmitted data stream.



3.3 Synchronization for uplink transmissions

In a typical multiuser system, each terminal computes timing and frequencyestimates by exploiting the downlink signal broadcasted by the BS. Thisoperation reduces the synchronization errors to a tolerable level and, incase of multicarrier transmissions, can easily be accomplished using thetechniques described in the previous section. The estimated parameters arethen employed by each user not only to detect the downlink data stream,but also as synchronization references for the uplink transmission. Due toDoppler shifts and propagation delays, however, the uplink signals arrivingat the BS may still be affected by residual frequency and timing errors. Tosee how this comes about, we denote TB = NT Ts the length of each OFDMblock (including the CP) and assume that the BS starts to transmit the `thdownlink block at t = `TB (` = 0, 1, 2, . . .) on the carrier frequency fc. Theblock is received by the mth user at t = `TB + τm on the frequency fc +∆fm, where τm and ∆fm are the line-of-sight (LOS) propagation delay andDoppler shift of the considered user, respectively. The latter are expressedby

τm =dm

c, (3.61)

and

∆fm =fcvm

c, (3.62)

where c is the speed of light, vm represents the speed of the mth mobileterminal and dm is the separation distance between the considered terminaland the BS.

During the uplink phase, each user transmits according to the timingand frequency references established on the basis of the downlink broad-cast channel. Assuming that the synchronization parameters have beenperfectly estimated, the OFDM uplink blocks are transmitted by the mthuser at instants t = iTB +τm (i = 0, 1, 2, . . .) on the frequency fc+∆fm+F ,where F is the nominal separation between the uplink and downlink carrierfrequencies (clearly, F = 0 in time-division-duplex systems). Because of thepropagation delay and Doppler shift, the BS receives the blocks from themth user at instants iTB + 2τm on the frequency fc + 2∆fm + F , whichresults into timing and frequency errors of 2τm and 2∆fm, respectively.The foregoing discussion indicates that synchronization performed at eachterminal during the downlink phase may be sufficient to avoid any furthersynchronization in the uplink as long as the Doppler shift is adequately



smaller than the subcarrier spacing and the duration of the CP is so largeto accommodate both the CIR duration and the two-way propagation de-lay 2τm. If the above conditions are not simultaneously met, however,the uplink signals loose their orthogonality and multiple-access interference(MAI) arises in addition to ICI and IBI. In such a case synchronizationat the BS becomes mandatory to avoid severe degradations of the systemperformance.

Intuitively speaking, synchronization in a multiuser uplink scenario ismuch more challenging than in the downlink. The reason is that whilein the downlink each terminal must estimate and compensate only for itsown synchronization parameters, the uplink waveform arriving at the BSis a mixture of signals transmitted by different users, each characterizedby different timing and frequency offsets. The latter cannot be estimatedwith the same methods employed in the downlink because each user must beseparated from the others before the synchronization process can be started.The separation method is closely related to the particular carrier assignmentscheme (CAS) adopted in the system, i.e., the strategy according to whichsubcarriers are distributed among the active users.

151 4131 21 11 09876543210

(a)

user 1 user 2 user 3 user 4

(b)

(c)

Fig. 3.14 Examples of subcarrier allocation schemes: subband CAS (a), interleavedCAS (b) and generalized CAS (c).

Commonly adopted carrier assignment schemes are the subband andinterleaved CAS as depicted in Fig. 3.14 (a) and (b), where a total of



N = 16 subcarriers is assumed for illustration purposes. As is seen, inthe subband CAS users are provided with groups of adjacent subcarrierswhile in the interleaved CAS the subcarriers of each user are interleavedover the signal bandwidth in order to fully exploit the frequency diversityof the multipath channel. However, the current trend in OFDMA favors amore flexible allocation scheme called generalized CAS (see Fig. 3.14 (c)),in which users can select the best subcarriers (namely, those exhibiting thehighest channel gains) that are currently available.

In the rest of this section, the problem of timing and frequency estima-tion in the OFDMA uplink is addressed separately for systems employingsubband, interleaved or generalized CAS. How to use the estimated syn-chronization parameters for MAI mitigation is the subject of Sec. 3.4.

3.3.1 Uplink signal model with synchronization errors

Without loss of generality, we adopt a baseband-equivalent discrete-timesignal model with sampling period Ts. The time-domain samples of themth user during the ith OFDM block are expressed by

sm,i(k) =1√N

∑

n∈Im

cm,i(n) ej2πnk/N , −Ng ≤ k ≤ N − 1 (3.63)

where Im is the set of subcarriers assigned to the considered user whilecm,i(n) is the symbol transmitted over the nth subcarrier. To avoid thata given subcarrier can be shared by different users, we must ensure thatIm ∩ Ij = ∅ if m 6= j. Clearly, the signal transmitted by the mth terminalconsists of several adjacent blocks and is given by

sm(k) =∑

i

sm,i(k − iNT ). (3.64)

We assume that M users are simultaneously active in the system andtransmit their data streams to the BS receiver. Each stream sm(k) (m =1, 2, . . . , M) propagates through a multipath channel with impulse responsehm = [hm(0), hm(1), . . . , hm(Lm − 1)]T and arrives at the BS with a timingoffset θm and a frequency error εm (normalized to the subcarrier spacing).After baseband conversion and sampling, the received samples are modeledas

r(k) =M∑

m=1

rm(k) + w(k), (3.65)



where w(k) represents complex-valued AWGN with variance σ2w while rm(k)

is the signal from the mth user and reads

rm(k) = ej2πεmk/NLm−1∑

`=0

hm(`)sm(k − θm − `). (3.66)

As mentioned previously, timing and frequency errors cause the loss oforthogonality among subcarriers of different users and give rise to multiple-access interference. Since the latter significantly degrades the system per-formance, the BS must compute estimates of θm and εm for each active user.The estimates are then used to restore orthogonality among the uplink sig-nals. As is intuitively clear, this multiple-parameter estimation problemcan be solved only after the users’ signals are properly separated at the BS.

A simple way to counteract the effects of users’ timing errors is to selectthe length of the CP so as to accommodate both the channel delay spreadand timing offsets. This results into a quasi-synchronous scenario [6] wherethe two-way propagation delays are viewed as part of the channel impulseresponses and the received samples can thus be rewritten as

rm(k) = ej2πεmk/NL−1∑

`=0

h′m(`)sm(k − `), (3.67)

where h′m = [h′m(0), h′m(1), . . . , h′m(L− 1)]T is the mth extended channelvector, with entries

h′m(`) = hm(`− θm), 0 ≤ ` ≤ L− 1 (3.68)

and length L = maxm

Lm + θm. In practice, a quasi-synchronous system isequivalent to a perfectly time-synchronized network in which the durationof the mth CIR (expressed in sampling periods) is artificially extended fromLm to L.

The situation is depicted in Fig. 3.15, where OFDMA blocks of differentusers arrive at the receiver with different delays depending on the distancesbetween the user terminals and the BS. As is seen, each CP is decomposedinto two segments. The first one (colored in black) has length Lm−1 and isaffected by interference from the previous block due to channel dispersion.The second segment (colored in gray) accommodates the last Ng − Lm +1 samples of the CP and is free from IBI. The vertical line on the leftrepresents the starting point of the ith OFDMA block in the BS time-scale, while the ith receive DFT window starts at t = iNT . If the lengthNg of the CP is not shorter than L− 1, the samples rm,i(k) = rm(k + iNT )



user 1

TiNTNg

_iN +

DFT window(N samples)

CP( samples)Ng

samples affected by IBI

IBI-free part of the CP

user 2

user 3

user 4

T1N

_iN

Fig. 3.15 Uplink received signals and DFT window in a quasi-synchronous scenario.

(0 ≤ k ≤ N − 1) falling within the ith DFT window are immune to IBIand, accordingly, are expressed by

rm,i(k) = ej2πεm(k+iNT )/NL−1∑

`=0

h′m(`)sm,i(k − `), 0 ≤ k ≤ N − 1 (3.69)

with sm,i(k) as given in Eq. (3.63). Substituting Eq. (3.63)into Eq. (3.69)yields

rm,i(k) =1√N

ej2πεmk/N∑

n∈Im

Hm,i(n)cm,i(n) ej2πnk/N , (3.70)

for 0 ≤ k ≤ N − 1, where Hm,i(n) = H ′m(n) ej2πεmiNT /N and

H ′m(n) =

L−1∑

`=0

h′m(`) e−j2πn`/N , 0 ≤ n ≤ N − 1 (3.71)

is the N -point DFT of h′m(`). Finally, from Eq. (3.65) we see that thesamples ri(k) = r(k + iNT ) (0 ≤ k ≤ N − 1) of the superimposed uplinksignals within the ith receive DFT window are given by

ri(k) =M∑

m=1

rm,i(k) + wi(k), (3.72)



with wi(k) = w(k + iNT ).The fact that propagation delays are absorbed by the extended channel

vectors makes quasi-synchronous systems extremely appealing since timingerrors simply appear as phase shifts at the DFT output and are compen-sated for by the channel equalization process. Timing estimation is thusunnecessary and the BS has only to estimate the frequency offsets εm,thereby reducing the number of synchronization parameters by a factor oftwo. The price for this simplification is a certain loss of efficiency due to theextended CP. To keep the loss to a tolerable level, the length of the CP mustbe maintained within a small fraction of the block duration. This poses anupper limit to the maximum admissible value of θm, say θmax, which mustbe adequately smaller than N. Since each θm is proportional to the two-waypropagation delay, the distances between the users’ terminals and the BSreceiver cannot exceed a certain value dmax. In particular, recalling thatθm ≈ 2τm/Ts and bearing in mind Eq. (3.61), we obtain dmax = cTsθmax/2.

3.3.2 Timing and frequency estimation for systems with

subband CAS

In OFDMA systems with subband CAS, the available spectrum is dividedinto several groups of adjacent subcarriers (subbands) and each user is ex-clusively assigned to one ore more groups. In the presence of frequencyerrors, subbands of different users are shifted in frequency from their nom-inal positions so that subcarriers located at the edges of a given groupmay experience significant ICI. To mitigate this problem, it is expedient toseparate subbands pertaining to different users by means of suitable guardintervals comprising a specified number of unmodulated subcarriers.

Assigning groups of adjacent subcarriers to each user facilitates the taskof separating the uplink signals at the BS. As shown in Fig. 3.16, it sufficesto pass the received samples through a bank of digital band-pass filters,each selecting one group of subcarriers. If the users’ frequency offsets areadequately smaller than the guard intervals among adjacent subbands, thefiltering operation roughly separates the uplink signals and allows the BSto perform timing and frequency estimation independently for each user.Clearly, perfect users’ separation is not possible since this would requireideal brickwall filters and/or very large guard intervals. Hence, the outputfrom the filter tuned on the mth subband takes the form

xm(k) = rm(k) + Im(k) + wm(k), (3.73)



estimatorx (k)1

x (k)M

estimator

r(k ) filter

bank

ε1θ1( , )

εMθM( , )

Fig. 3.16 Timing and frequency estimation for an OFDMA uplink receiver with sub-band CAS.

where rm(k) is the mth uplink signal as given in Eq. (3.66), wm(k) is thecontribution of thermal noise and, finally, Im(k) is an interference term thataccounts for imperfect users’ separation. As is intuitively clear, estimates ofθm and εm can be obtained from xm(k) applying any timing and frequencyestimation schemes suitable for single-user OFDM systems. One possibilityis to adopt the method discussed in [163], which exploits the correlationinduced on xm(k) by the use of the CP. In this case timing and frequencyestimates are obtained in the form

θm = arg maxθγm(θ), (3.74)

and

εm =12π

argγm(θm), (3.75)

where

γm(θ) =θ−1∑

k=θ−Ng

xm(k + N)x∗m(k) (3.76)

is the N -lag autocorrelation of the sequence xm(k).A slightly modified version of this algorithm is used in [162], where it

is shown that the estimator’s performance is heavily affected by the num-ber of subcarriers in one subband and deteriorates as this number becomessmaller and smaller due to the increased correlation among the received



time-domain samples. A second factor that may limit the estimation accu-racy is the amount of residual MAI and ICI arising from imperfect separa-tion of the users’ signals. A simple way to improve the system performanceconsists of averaging γm(θ) over Q successive OFDM blocks. This yields anew metric

γm(θ) =Q−1∑q=0

γm(θ + qNT ), (3.77)

which can be used in Eqs. (3.74) and (3.75) in place of γm(θ). In spite of itseffectiveness, this solution may provide the receiver with outdated estimatesof the synchronization parameters due to the enlarged estimation window.In practice, it can be adopted on condition that timing and frequency offsetsdo not change significantly over a time interval comprising Q OFDM blocks.

An alternative scheme to obtain estimates of θm and εm from the se-quence xm(k) is discussed in [6]. This method exploits unmodulated(virtual) subcarriers inserted in each user subband and updates the timingand frequency estimates until the average energy of the DFT outputs cor-responding to the virtual carriers achieves a minimum. Mathematically, wehave (

θm, εm

)= arg min

θm,εm

J(θm, εm)

, (3.78)

where θm and εm represent trial values of θm and εm, respectively, while thecost function J(θm, εm) is proportional to the average energy of the time-and frequency-corrected samples xm(k + θm)ej2πεmk/N falling across thevirtual carriers. As is seen, computing θm and εm directly from Eq. (3.78)requires a complicated bidimensional (2D) grid search over the set spannedby θm and εm. A certain reduction of complexity is possible if the minimumof J(θm, εm) is approached through a 2D steepest-descent algorithm.

As mentioned previously, the main advantage of the subband CAS is thepossibility of separating signals from different users through a simple filterbank even in a completely asynchronous scenario with arbitrarily large tim-ing errors. On the other hand, grouping the subcarriers together preventsthe possibility of optimally exploiting the channel diversity since a deepfade might hit a substantial number of subcarriers of a given user if theyare close together. Interleaving the subcarriers over the available spectrumis a viable method to provide the users with some form of frequency diver-sity. As it is now shown, however, this approach greatly complicates thesynchronization task.



3.3.3 Timing and frequency estimation for systems with in-

terleaved CAS

In OFDMA systems with interleaved CAS, the N available subcarriersare divided into R subchannels, where R is the maximum number ofusers that the system can simultaneously support. Each subchannel hasP = N/R subcarriers that are uniformly spaced in the frequency do-main at a distance R from each other. In particular, the subchannel as-signed to the mth user is composed of subcarriers with indices in the setIm = im + pR ; 0 ≤ p ≤ P − 1, where im may be any integer in the in-terval [0, R− 1].

Compared to the subband CAS, the interleaved CAS is clearly morerobust against frequency-selective fading by exploiting the frequency di-versity. However, separating the uplink signals in an interleaved OFDMAsystem is much more difficult than in subband transmissions. The rea-son is that in the presence of frequency errors the users’ signals overlapin the frequency-domain and cannot simply be separated through a filterbank. As it is now shown, however, the interleaved CAS provides the up-link signals with an inherent periodic structure that can be exploited forsynchronization purposes.

For simplicity, in the following the timing and frequency estimationtasks are separately addressed. The reason is that in an interleavedOFDMA system the joint estimation of all synchronization parameters ap-pears as a formidable problem for which no feasible solution is available inthe open literature. Accordingly, for the time being we consider a quasi-synchronous scenario and limit our attention to the frequency estimationproblem. A method for estimating the timing offsets of the active users isillustrated later.

We concentrate on the ith received OFDMA block and consider thesamples rm,i(k) (0 ≤ k ≤ N −1) of the mth uplink signal falling within theith receive DFT window. Since cm,i(n) is non-zero only for n = im + pR

(0 ≤ p ≤ P − 1), we may rewrite Eq. (3.70) in the equivalent form

rm,i(k) =1√N

ej2πξmk/PP−1∑p=0

Sm,i(p) ej2πpk/P , (3.79)

where Sm,i(p) = Hm,i(im + pR)cm,i(im + pR), while ξm is defined as

ξm =im + εm

R. (3.80)

Inspection of Eq. (3.79) reveals thatrm,i(k) = ej2π`ξmrm,i(k + `P ), (3.81)



from which it follows that each OFDMA block has a periodic structurethat repeats every P samples. This inner structure can be exploited forfrequency estimation. A solution in this sense is proposed in [11] by re-sorting to subspace-based methods. The resulting procedure is called theCao-Tureli-Yao Estimator (CTYE) and operates in the following way:

The Cao-Tureli-Yao Estimator (CTYE)

(1) arrange the received samples ri(k) (k = 0, 1, . . . , N − 1) into thefollowing R× P matrix

Mi =

ri(0) · · · ri(P − 1)ri(P ) · · · ri(2P − 1)

.... . .

...ri(N − P ) · · · ri(N − 1)

; (3.82)

(2) Compute the R×R sample-correlation matrix

Zi =1P

MiMHi ; (3.83)

(3) Determine the noise subspace by finding the R−M smallest eigen-values of Zi and arrange the corresponding eigenvectors into anR× (R−M) matrix Ui;

(4) Compute estimates

ξm

M

m=1of the quantities ξm by locating the

M largest peaks of the following metric

Γ(ξ) =1∥∥∥UH

i a(ξ)∥∥∥

2 , (3.84)

where a(ξ) =[1, ej2πξ, ej4πξ, . . . , ej2π(R−1)ξ

]T

;

(5) Use Eq. (3.80) and the quantities

ξm

M

m=1to compute frequency

estimates in the form

εm = Rξm − im, 0 ≤ m ≤ M − 1. (3.85)

This structure-based algorithm is reminiscent of the multiple signal clas-sification (MUSIC) technique [143], and provides estimates of the users’CFOs without requiring neither training blocks nor channel knowledge.The only requirement is that the CFOs cannot exceed one half of the sub-carrier spacing since otherwise the uncertainty intervals of the quantities



ξm are partially overlapping and in such a case there is no way of matchingeach ξm with the corresponding user. Luckily, the above requirement doesnot represent a serious problem since the uplink CFOs are mainly due toDoppler shifts and in a well-designed system they are typically confinedwithin 20 or 30% of the subcarrier spacing.

The main drawback of the CTYE is that in its original form it cannotbe applied to a fully-loaded system in which the number M of active usersis equal to the number R of subchannels. The reason is that the rank of theR× (R−M) matrix Ui must be at least one, which means that M ≤ R−1.This limitation may be overcome by extending the length of the CP fromNg to Ng + hP , where h is a suitable integer. The first Ng samples areused as a guard interval among blocks to avoid IBI. The last hP samplesare free from IBI and are exploited by CTYE together with the remainingN samples to estimate the frequency offsets. This results into a matrix Ui

of dimensions (R+h)× (R+h−M) and the algorithm can thus work evenwith M = R.

It is shown in [11] that the performance of CTYE degrades as the num-ber of active users becomes large. A simple way to improve the estimationaccuracy is to enlarge the observation window so as to comprehend a speci-fied number I of adjacent OFDMA blocks. In this case the CTYE proceedsas indicated earlier, except that the sample correlation matrix Zi is nowcomputed as

Zi =1

PI

i+I−1∑

k=i

MkMHk . (3.86)

A major assumption for the application of the CTYE is that theOFDMA uplink signals are quasi-synchronous. As discussed previously,this poses an upper limit to the maximum distance between the BS andthe mobile terminals, which may prevent the use of CTYE in a number ofapplications, including cellular networks with relatively large cell radii (onthe order of some kilometers). A possible solution to this problem relies onthe transmission of some training blocks at the beginning of each uplinkframe. These blocks are exploited for synchronization purposes and can beequipped with long CPs comprising both the channel delay spread and thepropagation delay. In this way the uplink signals are quasi-synchronousduring the training period, thereby allowing the use of CTYE for frequencyestimation. To reduce unnecessary overhead, however, it is desirable thatdata blocks have a shorter prefix (on the order of the channel responseduration). Thus, accurate estimation of the timing offsets is necessary to



align all users in time and avoid IBI over the data section of the frame.A simple method for obtaining timing estimates is based on knowledge ofthe users’ channel responses and is now explained by reconsidering the mthextended channel vector h′m defined in Eq. (3.68).

We begin by observing that

h′m =[0T

θmhT

m 0TL−θm−Lm

]T, (3.87)

where hm = [hm(0), hm(1), . . . , hm(Lm − 1)]T while 0K is a K -dimensionalcolumn vector with all zero entries. Next, we assume that an estimate ofh′m is available at the BS receiver in the form

h′m = h′m + ηm, (3.88)

where ηm accounts for the estimation error. In practice, h′m can be com-puted by exploiting the training blocks transmitted at the beginning ofthe uplink frame using one of the methods described in the next chapter.Combining Eqs. (3.87) and ( 3.88) produces

h′m = Am(θm)hm + ηm, (3.89)

where Am(θm) is an L× Lm matrix with entries

[Am(θm)]`,k =

1 if `− k = θm

0 otherwise .(3.90)

Vector h′m is now exploited to compute estimates of θm and hm bylooking for the minimum of the following least-squares (LS) cost function

Λ(θ, h) =∥∥∥h′m −Am(θ)h

∥∥∥2

. (3.91)

Minimizing with respect to h and observing that ATm(θ)Am(θ) is the iden-

tity matrix yields hm(θ) = ATm(θ)h′m. Inserting this result back into

Eq. (3.91) and minimizing with respect to θ gives an estimate of θm inthe form

θm = arg maxθ

∥∥∥ATm(θ)h′m

∥∥∥2

, (3.92)

or equivalently,

θm = arg maxθ

Lm+θ−1∑

`=θ

∣∣∣h′m(`)∣∣∣2

. (3.93)

The above equation indicates that the timing estimator looks for the max-imum of the energy of h′m over a sliding window of length Lm equal to theduration of the mth CIR hm.



3.3.4 Frequency estimation for systems with generalized

CAS

The generalized CAS is a dynamic resource allocation scheme in whichsubchannels are assigned to users according to their actual channel qualityand requested data rates. The fact that each user can select the bestsubcarriers that are currently available makes this allocation strategy moreflexible than subband or interleaved schemes. In particular, the generalizedCAS provides the system with some form of multiuser diversity [87] since asubcarrier that appears in a deep fade to one user may exhibit a relativelylarge gain for another user. On the other hand, the absence of any rigidstructure in the allocation policy makes the synchronization task even morechallenging than with interleaved CAS.

A method for estimating the timing and frequency errors of a new userentering an OFDMA network with generalized CAS has been proposedin [97]. This scheme has potentially good performance but relies on the factthat all other active users have already been synchronized, an assumptionthat may be too stringent in practical applications. Alternative solutionsdescribed in [125] and [126] are based on the ML principle and provideestimates of the synchronization parameters by exploiting a training blocktransmitted by each user at the beginning of the uplink frame. Thesemethods are now revisited assuming a quasi-synchronous scenario whereinthe CP of the training block is made sufficiently long to comprise both thechannel delay spread and propagation delays incurred by users’ signals. Inthe ensuing discussion we limit our attention to the joint ML estimationof the channel responses and frequency errors. If needed, timing estimatescan be obtained from the channel responses as indicated in the previoussection.

Without loss of generality, we assume that the training block has indexi = 0 and denote pm(n) (n ∈ Im) the pilot symbols transmitted by the mthuser over its assigned subcarriers. The corresponding time-domain samplescan thus be written as

bm(k) =1√N

∑

n∈Im

pm(n) ej2πnk/N , −Ng ≤ k ≤ N − 1. (3.94)

At the BS receiver, the CP is removed and the remaining samples areexpressed by

r(k) =M∑

m=1

ej2πεmk/NL−1∑

`=0

h′m(`)bm(k − `) + w(k), 0 ≤ k ≤ N − 1 (3.95)



where w(k) represents thermal noise, h′m(`) is defined in Eq. (3.68) and Mis the number of simultaneously active users.

Collecting the received samples into an N -dimensional vector r =[r(0), r(1), . . . , r(N − 1)]T , we may rewrite Eq. (3.95) into the equivalentform

r =M∑

m=1

rm + w, (3.96)

where w = [w(0), w(1), . . . , w(N − 1)]T is a Gaussian vector with zero-mean and covariance matrix σ2

wIN , while

rm = Γ(εm)Bmh′m, (3.97)

where

h′m = [h′m(0), h′m(1), . . . , h′m(L− 1)]T (3.98)

is the mth extended channel vector given in Eq. (3.87) and Γ(εm) is adiagonal matrix

Γ(εm) = diag

1, ej2πεm/N , . . . , ej2π(N−1)εm/N

, (3.99)

and Bm is an N × L matrix with known entries [Bm]k,` = bm(k − `) for0 ≤ k ≤ N − 1 and 0 ≤ ` ≤ L− 1.

The received vector r is now exploited to jointly estimate the frequencyoffsets ε = [ε1, ε2, . . . , εM ]T and channel responses h′ = [h′

T

1 , h′T

2 , . . . , h′T

M ]T

of all active users. In doing so we adopt an ML approach and rewriteEqs. (3.96) and (3.97) in a more concise form as

r = Q(ε)h′ + w, (3.100)

with

Q(ε) = [Γ(ε1)B1 Γ(ε2)B2 · · · Γ(εM )BM ] . (3.101)

Then, the log-likelihood function for the unknown set of parameters isgiven by

Λ(ε,h′) = −N ln(πσ2w)− 1

σ2w

∥∥∥r −Q(ε)h′∥∥∥

2

, (3.102)

where ε and h′ are trial values of ε and h′, respectively.The joint ML estimates of ε and h′ are obtained by searching for the

global maximum of Λ(ε,h′). This yields

ε = arg maxε

‖P (ε)r‖2

, (3.103)



and

h′ =[QH(ε)Q(ε)

]−1QH(ε)r, (3.104)

with P (ε) being defined as

P (ε) = Q(ε)[QH(ε)Q(ε)

]−1QH(ε). (3.105)

From the above equations it appears that the estimates of ε and h′ aredecoupled, meaning that the former is computed first and is then exploitedto get the latter. Unfortunately, the maximization in Eq. (3.103) requires agrid-search over the multidimensional domain spanned by ε, which wouldbe too intense even in the presence of few active users. A viable solutionto this problem is proposed in [125] and [126] by resorting to the space-alternating projection expectation-maximization algorithm (SAGE) [45] .

Similarly to the well known EM algorithm [34] , this technique operatesin an iterative fashion where the original measurements are replaced withsome complete data set from which the original measurements can be ob-tained through a many-to-one mapping. The SAGE algorithm alternatesbetween an E-step, calculating the log-likelihood function of the completedata, and an M-step, maximizing that expectation with respect to the un-known parameters. At any iteration the parameter estimates are updatedand the process continues until no significant changes in the updates areobserved. Compared to the classical EM algorithm, the SAGE has theadvantage of a faster convergence rate. The reason is that the maximiza-tions in the EM are simultaneously performed with respect to all unknownparameters, which results into a slow process that requires searches overspaces with many dimensions. Vice versa, the maximizations required inthe SAGE are performed varying small groups of parameters at a time. Inthe following the SAGE algorithm is applied to our problem without furtherexplanations. The interested reader is referred to [45] for details.

Returning to the estimation of ε and h′ , we apply the SAGE so as toreduce the M -dimensional maximization problem in Eq. (3.103) to a seriesof simpler maximizations. The resulting procedure consists of iterationsand cycles. An iteration is made of M cycles and each cycle updates theparameters of a single user while keeping those of the others at their mostupdated values. Specifically, we call ε

(j)m and h

′(j)m the estimates of εm and

h′m after the j th iteration, respectively. Given initial estimates ε(0)m and

h′(0)m , the BS computes the following M vectors, one for each user

r(0)m = Γ(ε(0)

m )Bmh′(0)m , 1 ≤ m ≤ M. (3.106)



Then, during the mth cycle of the jth iteration the SAGE algorithmproceeds as follows:

SAGE-based frequency estimator

• E-step Compute

y(j)m = r−

m−1∑

k=1

r(j)k −

M∑

k=m+1

r(j−1)k , (3.107)

where a notation of the type∑u

` is zero whenever u < `.• M-step Compute estimates of εm and h′m by locating the minimum

of the following cost function

Λ(j)(εm,h′m) =∥∥∥y(j)

m − Γ(εm)Bmh′m∥∥∥

2

(3.108)

with respect to εm and h′m. This yields

ε(j)m = arg max

εm

∥∥∥PmΓH(εm)y(j)m

∥∥∥2

, (3.109)

and

h′(j)m =(BH

mBm

)−1BH

mΓH(ε(j)m )y(j)

m , (3.110)

where Pm = Bm

(BH

mBm

)−1BH

m is a matrix that can be pre-computed and stored in the receiver as it only depends on the pilotsymbols transmitted by the mth user . The estimated parametersare used to obtain the following vector

r(j)m = Γ(ε(j)

m )Bmh′(j)m , (3.111)

which is then exploited in the E-step of the next cycle or iteration.

In the ensuing discussion, the estimator based on Eq. (3.109) is referredto as the Alternating-Projection Frequency Estimator (APFE). A physicalinterpretation of this algorithm is of interest. From Eqs. (Eq. (3.96)) and(3.97) we see that the signal component in r results from the contributionsrk of several users (1 ≤ k ≤ M), each depending on a set of parameters(εk, h′k). If all the sets were known except for (εm, h′m), the contributionsof the users with indices k 6= m could be subtracted from r, yielding aMAI-free vector

ym = r −∑

k 6=m

rk (3.112)



or, bearing in mind Eqs. (3.96) and (3.97),

ym = Γ(εm)Bmh′m + w. (3.113)

Then, the issue would arise of estimating (εm, h′m) based on the ob-servation of ym. Unfortunately, ym is not available at the BS since fromEq. (3.112) we see that its computation would entail perfect knowledge ofthe interfering signals rk. However, a comparison between Eqs. (3.107) and(3.112) reveals that y

(j)m can be considered as a reasonable approximation

of ym. In this respect, we may write

y(j)m = Γ(εm)Bmh′m + dm + w, (3.114)

where dm is a disturbance term that accounts for imperfect cancellation ofthe interfering signals. Vector y

(j)m is thus used in place of the true ym to

compute LS estimates of (εm, h′m) as indicated in Eqs. (3.109) and (3.110).In light of the above arguments, the algorithm based on Eqs. (3.109) and(3.110) is recognized as a recursive approximation to the ML estimator inwhich previous estimates of the synchronization parameters are exploitedto cancel out the MAI. Compared to the true ML estimator, the APFE ismuch simpler to implement as it splits the multidimensional maximizationproblem Eq. (3.103) into a series of mono-dimensional grid searches.

A possible shortcoming of EM-type algorithms comes from the fact thatthe log-likelihood function Λ(ε,h′) is not guaranteed to have a unique ab-solute maximum. Indeed, it might exhibit several local peaks that canattract the APFE toward spurious locks. False locks occur since the algo-rithm tends to settle on the local peak immediately uphill from the initialestimates ε(0) = [ε(0)

1 , ε(0)2 , . . . , ε

(0)M ]T . This indicates that the APFE has

a higher chance to converge to the global maximum of Λ(ε,h′) if an ac-curate estimate ε(0) is used for the initialization task. Two methods canbe used to obtain ε(0). One possibility is to simply initialize the frequencyestimates to zero. Alternatively, one can compute the N -point DFT of r

and select the DFT outputs corresponding to the set Im of subcarriers as-signed to the mth user while putting to zero all the others. After returningin the time-domain through an IDFT operation, the resulting samples areexploited to get an estimate ε

(0)m by resorting to the frequency estimator

proposed in [100] and suitable for single-user transmissions. As is intu-itively clear, computing the DFT of r and forcing to zero the subcarriersallocated to interfering users is a viable method to partially mitigate theMAI. Albeit more computationally demanding, this approach is expected toprovide better initialization values and faster convergence rate than simplyputting ε

(0)m = 0.



0 1 2 3 4

10-5

10-4

10-3

10-2

10-1

Number of iterations, Ni

Freq

uenc

y M

SE

APFE (M = 2)APFE (M = 3)APFE (M = 4)

Fig. 3.17 Convergence rate of APFE.

The performance of APFE has been assessed for an OFDMA systemwith N = 128 subcarriers operating in the 5 GHz frequency band. Thechannel response of each user has length Lm = 8, and the channel co-efficients are modeled as independent and complex-valued Gaussian ran-dom variables with zero-mean (Rayleigh fading) and an exponential powerdelay profile. The normalized CFOs are uniformly distributed over theinterval [−0.3, 0.3] and vary at each new simulation run. We assume aquasi-synchronous system where the CP of the training block is sufficientlylong to accommodate both the channel response and the maximum propa-gation delay. Each user transmits data over 32 distinct subcarriers, whichare randomly assigned in order to demonstrate the applicability of APFEin conjunction with a generalized CAS. Without loss of generality, onlyresults for the first user are illustrated.

Figure 3.17 shows the MSE of the frequency estimates E[ε1 − ε1]2 as

a function of the number Ni of iterations in case of M =2, 3 or 4 activeusers. The latter have equal power with Es/N0 = 20 dB and the frequency



estimates are initialized to zero to reduce the system complexity. We seethat APFE achieves convergence in only two iterations and no further gainsare observed with Ni > 2.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

ε1

Ave

rage

freq

uenc

y es

timat

es

IdealAPFE

Fig. 3.18 Average frequency estimates of APFE vs. ε1.

The average frequency estimates are shown in Fig. 3.18 as a function ofε1 assuming that three users are active in the system. Here, ε1 is kept fixedat each new simulation run while the frequency offsets of the other usersvary independently over the range [−0.3, 0.3]. The ideal line Eε1 = ε1

is also drawn for comparison. These results indicate that APFE providesunbiased estimate over the interval |ε1| < 0.5.

Figure 3.19 illustrates the frequency MSE as a function of Es/N0 in caseof two active users. The tick solid line represents the Cramer–Rao lowerbound (CRLB) for frequency estimation in quasi-synchronous OFDMA up-link transmissions [125] and is shown as a benchmark. The simulation set upis the same as in Fig. 3.17, except that now an interleaved CAS is adoptedin order to make comparisons with the CTYE discussed in the previoussubsection. We see that APFE achieves the CRLB for Es/N0 > 15 dB.



0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

Es/N

0 (dB)

Fre

quen

cy M

SE

CRLBAPFECTYE

Fig. 3.19 Accuracy of APFE and CTYE vs. Es/N0.

The CTYE exhibits good performance at high SNR values, but a certaindegradation is observed with respect to APFE for Es/N0 < 15 dB.

3.4 Timing and frequency offset compensation in uplinktransmissions

Once the uplink timing and frequency offsets have been estimated, theymust be employed by the BS receiver to restore orthogonality among sub-carriers. This operation is known as timing and frequency correction andrepresents the final stage of the synchronization process. In downlink trans-missions, frequency correction is typically achieved by counter-rotating thetime-domain samples at an angular speed 2πεm/N , while timing adjust-ment is accomplished by shifting the DFT window by a number θm ofsampling intervals. Unfortunately, these methods cannot be used in anuplink scenario. The reason is that the uplink signals arriving at the BSare affected by different synchronization errors, so that the correction of



one user’s time and frequency offset would misalign other initially alignedusers. A solution to this problem is presented in [162] and [97], whereestimates of the users’ offsets are returned to the active terminals via adownlink control channel and exploited by each user to properly adjust itstransmitted signal. In a time-varying scenario, however, users should beperiodically provided with updated estimates of their synchronization pa-rameters, which may result into an excessive extra load for the downlinktransmission and outdated adjustment due to the intrinsic feedback delay.An interesting alternative is to use advanced signal processing techniquesto compensate for synchronization errors directly at the BS, i.e., withoutthe need of returning timing and frequency estimates back to the activeterminals. Solutions derived along this line of reasoning are largely inher-ited from the multiuser detection area and are subject to the particularsubcarrier allocation scheme adopted in the system.

In the rest of this section we first concentrate on the problem of timingand frequency correction for an OFDMA system with subband CAS. Amore flexible generalized CAS is next considered to illustrate how linearmultiuser detection and interference cancellation schemes can be employedto compensate for the users’ CFOs.

3.4.1 Timing and frequency compensation with subband

CAS

In OFDMA systems with subband CAS the uplink signals arriving at the BScan be separated by a bank of band-pass filters if suitable guard intervalsare inserted between adjacent subbands. The receiver can thus estimateand correct the synchronization errors independently for each active user.A solution in this sense is depicted in Fig. 3.20. After users’ separation,each uplink signal xm(k) (1 ≤ m ≤ M) is exploited to get estimates θm andεm of the timing and frequency offsets using one of the methods describedin Sec. 3.3.2. The estimated parameters are then employed to compensatefor the synchronization errors of each signal by resorting to conventionalsingle-user techniques. In particular, the samples xm(k) are multiplied bythe exponential term e−j2πkεm/N to cancel out any phase rotation inducedby the CFO whereas the timing estimate θm is used to select the N samplesthat are next processed by the DFT unit. After channel equalization (notshown in the figure), the DFT outputs corresponding to the mth subchannelare finally passed to the data detection unit.

The receiver architecture shown in Fig. 3.20 relies on the fact that the



r(k) filter

bank

k/Nj2π εMe_

x (k )1

estimatorθ 1

k/Ne_j2π ε1

DFT

x (k)M

θΜestimator

DFT

to channelequalization anddata detection

Fig. 3.20 Timing and frequency synchronization for an OFDMA uplink receiver withsubband CAS.

uplink signals are perfectly separated at the output of the filter bank. Inpractice, however, perfect separation is not possible even in the presenceof ideal brick-wall filters due to the frequency leakage among adjacent sub-channels caused by synchronization errors. This means that some residualMAI will be present at the DFT output, with ensuing limitations of theerror-rate performance. In addition, compensating for the frequency er-rors in the time-domain as depicted in Fig. 3.20 requires an N -point DFToperation for each active user. Since the complexity involved with theDFT represents a major concern for system implementation, the receiverstructure of Fig. 3.20 may be too computationally demanding in practicalapplications, especially when the number M of simultaneously active usersand/or the number N of available subcarriers are relatively large.

An alternative scheme for uplink frequency correction in subbandOFDMA systems is sketched in Fig. 3.21. This solution has been pro-posed in [18] and is referred to as the Choi–Lee–Jung–Lee (CLJL) methodin the ensuing discussion. Its main advantage is that it avoids the need formultiple DFT operations, but can only be applied to a quasi-synchronoussystem where the uplink signals are time aligned within the length of theCP and timing correction is thus unnecessary.

To explain the rationale behind CLJL, we reconsider the N samplesri(k) (0 ≤ k ≤ N−1) falling within the ith receive DFT window. Collecting



DFT

to channelequalization

anddata detection

iRir (k )

M,iY

1,iX 1,iY

circularconvolutionMP M,iX

circularconvolution1P

1)_

C( M)_ε

C( ε

Fig. 3.21 Frequency correction by means of circular convolutions applied at the DFToutput.

Eqs. (3.70) and (3.72) we may write

ri(k) =M∑

m=1

zm,i(k) ej2πεmk/N + wi(k), 0 ≤ k ≤ N − 1 (3.115)

with

zm,i(k) =1√N

∑

n∈Im

Hm,i(n)cm,i(n) ej2πnk/N . (3.116)

For convenience, the N -point DFT of the sequences ri(k), zm,i(k) andwi(k) are arranged into three N -dimensional vectors Ri, Zm,i and Wi, re-spectively. Then, recalling that a multiplication in the time-domain corre-sponds to a circular convolution in the frequency-domain, from Eq. (3.115)we have

Ri =M∑

m=1

Zm,i ⊗C(εm) + Wi, (3.117)

where ⊗ denotes the N -point circular convolution, Zm,i has entries

Zm,i(n) =

Hm,i(n)cm,i(n)

0if n ∈ Im

otherwise(3.118)

and, finally, C(εm) is the N -point DFT ofej2πεmk/N ; 0 ≤ k ≤ N − 1

with entries

C(εm, n) =sin [π (n− εm)]

sin [π (n− εm) /N ]e−jπ(N−1)(n−εm)/N , 0 ≤ n ≤ N − 1.

(3.119)



Returning to Fig. 3.21, we see that for each active user an N -dimensionalvector Xm,i (1 ≤ m ≤ M) is obtained from the DFT output by puttingto zero all entries of Ri that do not correspond to the subcarriers of theconsidered user. This amounts to setting Xm,i = PmRi, where Pm is adiagonal matrix with entries

[Pm]n,n =

10

if n ∈ Im

otherwise.(3.120)

In practice, Pm acts as a band-pass filter that aims at isolating thecontribution of the mth uplink signal at the DFT output. Bearing in mindEq. (3.117) and assuming perfect signal separation, we may write

Xm,i ≈ Zm,i ⊗C(εm) + Wm,i, (3.121)

where Wm,i = PmWi is the noise contribution. The above equation indi-cates that Xm,i can reasonably be assumed free from MAI. However, it isstill affected by residual ICI due to the uncompensated frequency error εm.

Instead of performing frequency correction in the time-domain as illus-trated in Fig. 3.20, we can equivalently compensate for εm in the frequency-domain using a suitable circular convolution followed by band-pass filter-ing [18]. This produces

Ym,i = Pm [Xm,i ⊗C(−εm)] , (3.122)

where C(−εm) is a vector that collects the N -point DFT of the se-quence

e−j2πεmk/N ; 0 ≤ k ≤ N − 1

and whose entries are obtained from

Eq. (3.119) after replacing εm by −εm.Substituting Eq. (3.121) into Eq. (3.122) and assuming ideal frequency

estimation (i.e., εm = εm), yields

Ym,i = Zm,i + Pm [Wm,i ⊗C(−εm)] , (3.123)

where we have used the identity Zm,i ⊗ C(εm) ⊗ C(−εm) = Zm,i. Theabove equation, together with Eq. (3.118), indicates that Ym,i is free frominterference except for channel distortion and thermal noise. In practice,however, non-ideal frequency compensation and imperfect users’ separationwill generate residual ICI and MAI on Ym,i, thereby resulting in someperformance degradation with respect to the ideal setting described byEq. (3.123).

As mentioned previously, a favorable feature of CLJL is that it onlyneeds a single DFT operation. This result is achieved by operating overthe frequency-domain samples Ri and leads to a significant reduction ofcomplexity as compared to the receive architecture of Fig. 3.20, where aseparate DFT operation is required for each user.



3.4.2 Frequency compensation through interference cancel-

lation

The CLJL scheme discussed in the previous subsection is only suited forOFDMA systems with subband CAS. The reason is that the bank of ma-trices Pm (1 ≤ m ≤ M) in Fig. 3.21 provides accurate users’ separation aslong as the subcarriers of a given user are grouped together and sufficientlylarge guard intervals are inserted among adjacent subchannels. When usedin conjunction with an interleaved or a generalized CAS, however, the CLJLcannot significantly reduce the MAI induced by frequency errors. In thiscase, alternative approaches must be resorted to. One possibility is offeredby the concept of multiuser detection [164]. The latter includes all ad-vanced signal processing techniques for the joint demodulation of mutuallyinterfering data streams.

Multiuser detection schemes are largely categorized into linear or in-terference cancellation (IC) architectures. In this subsection we limit ourattention to the latter class. In particular, we show how the IC concept canbe applied to CLJL in order to reduce the residual interference present onYm,i. The resulting scheme has been derived by Huang and Letaief (HL)in [55] and operates in an iterative fashion.

Calling Y(j)

m,i the mth restored signal after the j th iteration, the HLproceeds as follows:

The HL algorithm

• InitializationCompute the CLJL vectors defined in Eq. (3.122), i.e.,

Ym,i = Pm [(PmRi)⊗C(−εm)] , 1 ≤ m ≤ M (3.124)

and set Y(0)

m,i = Ym,i for m = 1, 2, . . . , M .• j th iteration (j=1,2,...)

For each active user (m = 1, 2, . . . ,M) perform interference cancel-lation in the form

Y(j)

m,i = Ri −M∑

k=1,k 6=m

Y(j−1)

k,i ⊗C(εk), 1 ≤ m ≤ M (3.125)

and remove the effect of εm following a CLJL approach

Y(j)

m,i = Pm

[(PmY

(j)m,i

)⊗C(−εm)

], 1 ≤ m ≤ M. (3.126)



As indicated in Eq. (3.125), at each iteration circular convolutions areemployed to regenerate interference, which is then subtracted from theoriginal DFT output Ri. The expurgated vectors Y

(j)m,i are next used to

obtain the restored signals Y(j)

m,i according to Eq. (3.126). In this respect,the HL can be regarded as a parallel interference cancellation (PIC) scheme.In contrast to the conventional PIC, however, HL does not suffer from errorpropagation since orthogonality among the received signals is tentativelyrestored without employing any data decision.

Simulation results reported in [55] indicate that HL performs much bet-ter than CLJL after just a few iterations. In particular, its increased robust-ness against ICI and MAI makes it suited for any CAS, whereas CLJL canonly be used in conjunction with a subband CAS. It is worth noting thatthe windowing function Pm employed in Eqs. (3.124) and (3.126) aims atremoving all the energy present on subcarriers allocated to other users. Al-beit useful to reduce interference, this operation entails some performanceloss in the presence of relatively large CFOs since in this case the undes-ignated subcarriers might contain a significant portion of the user’s energywhich is definitely discarded by HL.

3.4.3 Frequency compensation through linear multiuser de-

tection

Linear multiuser detection can be used as an alternative to IC-based solu-tions for mitigating interference caused by uplink CFOs. An example in thissense is provided by the Cao-Tureli-Yao-Honan (CTYH) scheme discussedin [12]. This method is suited for any CAS, but can only operate in a quasi-synchronous scenario where no IBI is present. The CTYH is now derivedfollowing a two-step procedure. We begin by establishing a new convenientsignal model for the DFT output Ri. Orthogonality among subcarriers issubsequently restored by means of linear transformations applied to Ri.

In deriving the new signal model we make the following assumptionswithout loss of generality:

(1) each user transmits its data over P = N/R subcarriers, where R is themaximum number of simultaneously active users in the system underconsideration;

(2) the indices of subcarriers assigned to the mth user belong to the setIm = qm(p); 0 ≤ p ≤ P − 1 .

Bearing in mind Eq. (3.70), we may rewrite the samples rm,i(k) of the



mth received uplink signal as

rm,i(k) =1√N

ej2πεmk/NP−1∑p=0

Sm,i(p) ej2πqm(p)k/N , 0 ≤ k ≤ N − 1

(3.127)where

Sm,i(p) = Hm,i(qm(p))cm,i(qm(p)) (3.128)

is an attenuated and phase-rotated version of the symbol transmittedover the qm(p)th subcarrier. For convenience, we define a vector Rm,i =[Rm,i(0), Rm,i(1), . . . , Rm,i(N − 1)]T whose entries are the DFT of rm,i(k),i.e.,

Rm,i(n) =1√N

N−1∑

k=0

rm,i(k) e−j2πnk/N , 0 ≤ n ≤ N − 1. (3.129)

Then, substituting Eq. (3.127) into Eq. (3.129) and letting Sm,i =[Sm,i(0), Sm,i(1), . . . , Sm,i(P − 1)]T , yields

Rm,i = Πm(εm)Sm,i, (3.130)

where Πm(εm) is an N × P matrix with elements

[Πm(εm)]n,p = fN [qm(p) + εm − n] ejπ(N−1)(qm(p)+εm−n)/N , (3.131)

for 0 ≤ n ≤ N − 1 and 0 ≤ p ≤ P − 1, with fN (x) defined as in Eq. (3.19).As shown in Eq. (3.72), the samples ri(k) of the ith received OFDMA

block are the superposition of all uplink signals plus thermal noise. Theoutput of the receive DFT unit is thus given by

Ri =M∑

m=1

Rm,i + Wi, (3.132)

where Wi is a complex-valued Gaussian vector with zero-mean and covari-ance matrix σ2

wIN . Finally, substituting Eq. (3.130) into Eq. (3.132) andletting Si =

[ST

1,i ST2,i · · · ST

M,i

]T , we obtain the desired signal model forRi in the form

Ri = Π(ε)Si + Wi, (3.133)

where Π(ε) = [Π1(ε1) Π2(ε2) · · · ΠM (εM )]T is an N×MP matrix whoseelements are related to the users’ frequency offsets ε = [ε1, ε2, . . . , εM ]T .

Inspection of Eq. (3.128) reveals that the entries of Si are the trans-mitted data symbols multiplied by the corresponding channel frequencyresponse. Accordingly, Si is the vector that would be present at the DFT



output in the absence of any interference and thermal noise. The purposeof CTYH is to obtain an estimate of Si starting from Ri. As illustratedin Fig. 3.22 , this goal is achieved by means of a linear transformation ap-plied to Ri. The estimated vector Si is then fed to the channel equalizerand data detection unit, which provides decisions on the transmitted datasymbols.

to channel equalization and

data detection DFT

Linear transformation

R i S i r i ( k )

Fig. 3.22 Frequency correction by means of a linear transformation at the DFT output.

Two possible methods for computing Si are illustrated in [12]. The firstone is based on the LS approach and is equivalent to the well known lineardecorrelating detector (LDD) [164]

Si,LDD = Π†(ε)Ri, (3.134)

where Π†(ε) =[ΠH( ε)Π(ε)

]−1ΠH(ε) denotes the Moore-Penrose gener-

alized inverse of Π(ε).Substituting Eq. (3.133) into Eq. (3.134) yields

Si,LDD = Si + Π†(ε)Wi, (3.135)

meaning that the decorrelating detector can totally suppress any interfer-ence caused by frequency errors. As it is known, the price for this result isa certain enhancement of the output noise level.

The second solution is based on the MMSE approach and aims at min-imizing the overall effect of interference plus ambient noise. The resultingscheme is known as the linear MMSE detector [164] and reads

Si,MMSE = Q(ε,σ2w)Ri, (3.136)

with Q(ε,σ2w) =

[ΠH(ε)Π(ε) + σ2

wIMP

]−1ΠH(ε). Although the output

of the MMSE detector is still affected by some residual MAI, the noiseenhancement phenomenon is greatly reduced as compared to the LDD.

The main drawback of CTYH is the relatively huge complexity requiredto evaluate Π†(ε) or Q(ε,σ2

w). Note that these matrices cannot be pre-computed and stored in the receiver as they do depend on the actual CFOs



and noise power. Since the quantities ε and σ2w are not perfectly known at

the BS, in practice they are replaced by suitable estimates ε and σ2w. It is

observed in [12] that Π†(ε) and Q(ε,σ2w) are banded matrices with non-zero

elements only in the vicinity of their main diagonal. This property can beexploited to reduce the complexity involved with their computation.

3.4.4 Performance of frequency correction schemes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

ρ

BE

R

HL (Ni = 5)

CTYHIdeal

Fig. 3.23 BER performance of HL and CTYH vs. ρ for an uncoded QPSK transmissionwith Es/N0 = 20 dB.

It is interesting to compare the performance of HL and CTYH in termsof bit-error-rate (BER) in a quasi-synchronous uplink scenario. For thispurpose, we consider an OFDMA system with N = 128 subcarriers and ageneralized carrier assignment policy. Each subchannel is composed by 32subcarriers, so that the maximum number of simultaneously active usersis limited to R = 4. We assume a fully-loaded system in which M =R = 4 and let ε = ρ [1,−1, 1,−1]T , where ρ is a deterministic parameterbelonging to the interval [0, 0.5] and known as frequency attenuation factor



[55]. A new channel snapshot is generated at each simulation run andkept fixed over an entire frame. Ideal frequency and channel estimatesare assumed throughout simulations. Five iterations are performed by HLwhile CTYH employs the decorrelating matrix Π†(ε) as in Eq. (3.134).

Figure 3.23 illustrates the BER performance as a function of ρ for anuncoded QPSK transmission. Users have equal power with Es/N0 = 20 dB.The curve labeled “ideal” is obtained by assuming that all CFOs have beenperfectly corrected at the mobile terminals, i.e., εm = 0 for m = 1, 2, 3, 4.This provides a benchmark for the BER performance since in this casethe users’ signals are perfectly orthogonal and no interference is presentat the DFT output. We see that the BER degrades with ρ due to theincreased amount of ICI and MAI. As mentioned previously, the latteris mitigated by CTYH at the price of non-negligible noise enhancement,while the windowing functions used by HL leads to a significant loss ofsignal energy in the presence of relatively large CFOs.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−4

10−3

10−2

10−1

100

Es/N

0 (dB)

BE

R

HL (N

i = 5)

CTYHIdeal

Fig. 3.24 BER performance of HL and CTYH vs. Es/N0 for an uncoded QPSK trans-mission with ρ = 0.3.



Figure 3.24 shows the BER of the considered schemes vs. Es/N0 for anuncoded QPSK transmission. Users have equal power and ρ is set to 0.3.Again, we see that CTYH provides the best performance. In particular, atan error rate of 10−2 the loss of CTYH with respect to the ideal system isapproximately 4 dB. As for HL, it performs poorly and exhibits an errorfloor at high SNR values.


Chapter 4

Channel Estimation and Equalization

In OFDM transmissions, the effect of channel distortion on each subcar-rier is represented by a single complex-valued coefficient that affects theamplitude and phase of the relevant information symbol. Coherent detec-tion of the transmitted data can be performed only after this multiplicativedistortion has been properly compensated for. This operation is knownas channel equalization, and can easily be accomplished in the frequency-domain if an estimate of the channel response is available at the receiver.An alternative to coherent detection is offered by the use of differential en-coding techniques. In this case information data are transmitted as phasevariations between adjacent subcarriers and are recovered at the receiverthrough differential demodulation, thereby eliminating the need for chan-nel knowledge. The price for this simplification is a certain loss of powerefficiency as compared to coherent detection.

In this Chapter we present some popular schemes to recover channelstate information (CSI) in OFDM systems. One common approach is basedon the periodic insertion of pilot symbols within the transmitted signal.This idea has been adopted in many OFDM standards and has led to thedevelopment of so-called pilot-aided schemes. Although the use of pilotsymbols may facilitate the channel estimation task to a great extent, itinevitably leads to some reduction of the data throughput because of therequired extra overhead. This problem has motivated intense research ac-tivity on blind channel identification and equalization techniques, wherethe inherent redundancy present in the transmitted signal is exploited atthe receiver to get CSI with the aid of only a few pilots or using no pilotsat all.

The Chapter has the following outline. Section 4.1 illustrates the con-cept of frequency-domain channel equalization. Combining schemes are also

107



presented for receivers equipped with multiple antenna elements. The ideaof pilot-aided channel estimation is discussed in Sec. 4.2. After illustratingsome popular pilot insertion patterns adopted in commercial systems, weshow how the minimum allowable distance between pilots is related to thestatistical parameters of the wireless channel. Several techniques for pilots’interpolation are also discussed. Section 4.3 illustrates recent advances inthe area of blind and semi-blind channel estimation and equalization. Here,two different approaches are considered. The first one relies on the conceptof subspace decomposition, while in the other the expectation-maximization(EM) algorithm is applied to couple the channel estimation/equalizationtask with the decision making process.

4.1 Channel equalization

Channel equalization is the process through which a coherent receiver triesto compensate for any distortion induced by frequency-selective fading.For the sake of simplicity, ideal timing and frequency synchronization isconsidered throughout this chapter. The channel is assumed static over eachOFDM block, but can vary from block to block. Under these assumptions,the output of the receive DFT unit during the ith block is given by

Ri(n) = Hi(n)ci(n) + Wi(n), 0 ≤ n ≤ N − 1 (4.1)

where Hi(n) is the channel frequency response over the nth subcarrier, ci(n)is the relevant data symbol and, finally, Wi(n) represents the frequency-domain noise contribution with zero-mean and variance σ2

w.One appealing feature of OFDM is that channel equalization can in-

dependently be performed over each subcarrier by means of a bank ofone-tap multipliers. In practice, the nth DFT output Ri(n) is weightedby a complex-valued quantity pi(n) in an attempt of compensating for thechannel-induced attenuation and phase rotation. As shown in Fig. 4.1, theequalized sample Yi(n) = pi(n)Ri(n) is subsequently passed to the detec-tion unit, which delivers final decisions ci(n) on the transmitted data.

A popular approach for the design of the equalizer coefficients relies onthe minimum mean-square error (MMSE) criterion . In this case pi(n) ischosen so as to minimize the following quantity

Ji(n) = E|pi(n)Ri(n)− ci(n)|2

, (4.2)

which represents the mean-square error (MSE) between the equalizer out-put Yi(n) and the transmitted symbol ci(n).


Channel Estimation and Equalization 109

decision device

i R ( n ) i Y ( n )

i p ( n )

channel equalizer

c i ( n )

Fig. 4.1 Equalization and data detection over the nth subcarrier.

From the orthogonality principle [72], we know that the optimal weightspi(n) are such that the error Yi(n) − ci(n) is orthogonal to the relevantDFT output, i.e.,

E [pi(n)Ri(n)− ci(n)]R∗i (n) = 0. (4.3)

Substituting Eq. (4.1) into Eq. (4.3) and computing the expectationwith respect to thermal noise and data symbols (the latter are assumed tobe statistically independent with zero-mean and power C2), yields

pi(n) =H∗

i (n)|Hi(n)|2 + ρ

, (4.4)

where ρ = σ2w/C2 is the inverse of the operating signal-to-noise ratio (SNR).

As indicated by Eq. (4.4), computing the MMSE equalization coeffi-cients requires knowledge of Hi(n) and σ2

w. A suboptimum solution isobtained by designing parameter ρ for a fixed nominal noise power σ2

w,thereby allowing the equalizer to operate in a mismatched mode wheneverσ2

w 6= σ2w. The resulting scheme dispenses from knowledge of σ2

w and onlyneeds channel state information. This simplified approach also includes thewell-known Zero-Forcing (ZF) equalization criterion, which corresponds tosetting σ2

w = 0. In this case the equalizer performs a pure channel inversionand its coefficients are given by

pi(n) =1

Hi(n), (4.5)

while the DFT output takes the form

Yi(n) = ci(n) +Wi(n)Hi(n)

, 0 ≤ n ≤ N − 1. (4.6)

This equation indicates that ZF equalization is capable of totally com-pensating for any distortion induced by the wireless channel. However, the



noise power at the equalizer output is given by σ2w/ |Hi(n)|2 and may be

excessively large over deeply faded subcarriers characterized by low channelgains.

It is worth noting that the equalization coefficients in Eqs. (4.4) and(4.5) only differ for a positive multiplicative factor 1 + ρ/ |Hi(n)|2, so thatthe phase of the equalized sample Yi(n) is the same in both cases. Aninteresting consequence of this fact is that ZF and MMSE equalizers areperfectly equivalent in the presence of a pure phase modulation (as occurswith PSK data symbols) since in this case the decision on ci(n) is solelybased on the argument of Yi(n).

All the above results can easily be extended to OFDM receiversequipped with Q > 1 antenna elements for diversity reception. In sucha situation, the contributions from all receive antennas may properly becombined to improve the reliability of data decisions. As is intuitivelyclear, the best performance is obtained when the combining strategy is in-tegrated with the channel equalization process in a single functional unit.To see how this comes about, denote H

(q)i (n) the frequency response of the

channel viewed by the qth receiving antenna and let

R(q)i (n) = H

(q)i (n)ci(n) + W

(q)i (n), 0 ≤ n ≤ N − 1 (4.7)

be the DFT output over the corresponding diversity branch.

decision device

i Y ( n )

Equalization & combination unit

( Q )

i R ( n ) (1)

i p ( n ) ( Q )

i p ( n ) ( 1 )

c i ( n )

i R ( n )

Fig. 4.2 Equalization and data detection over the nth subcarrier in the presence ofmultiple receiving antennas.



As illustrated in Fig. 4.2, the decision statistic for ci(n) is obtained bylinearly combining the DFT outputs from the Q available antennas, i.e.,

Yi(n) =Q∑

q=1

p(q)i (n)R(q)

i (n). (4.8)

The weighting coefficients p(q)i (n) can be selected according to various

optimality criteria. Among them, the MMSE strategy aims at minimizingthe following MSE

Ji(n) = E

∣∣∣∣∣Q∑

q=1

p(q)i (n)R(q)

i (n)− ci(n)

∣∣∣∣∣

2 . (4.9)

Assuming for simplicity that the noise power σ2w is the same at each

branch, the optimum weights are found to be

p(q)i (n) =

[H(q)i (n)]∗

ρ +∑Q

`=1

∣∣∣H(`)i (n)

∣∣∣2 , (4.10)

where ρ = σ2w/C2. Interestingly, setting ρ = 0 in the above equation results

into the well-known maximum-ratio-combining (MRC) strategy, which hasthe appealing property of maximizing the SNR at the output of the com-bining/equalization unit.

4.2 Pilot-aided channel estimation

In multicarrier systems the transmission is normally organized in frames,each containing a specified number of OFDM blocks. As mentioned inChapter 3, some reference blocks carrying known data are usually appendedin front of the frame to assist the synchronization process as well as to pro-vide initial estimates of the channel frequency response. If the channelremains static over the frame duration, the estimates obtained from thereference blocks can be used to coherently detect the entire payload. Thissituation is typical of WLAN systems, where the user terminals are charac-terized by low mobility and, in consequence, the channel coherence time isexpected to be much greater than the packet length. On the other hand, inapplications characterized by relatively high mobility as those envisionedby the IEEE 802.16e standard for WMANs, the channel response undergoessignificant variations over one frame and must continuously be tracked tomaintain reliable data detection. In this case, in addition to initial reference



blocks, known symbols called pilots are normally inserted into the payloadsection of the frame at some convenient positions. These pilots are scat-tered in both the time and frequency directions (i.e., they are positionedover different blocks and different subcarriers), and are used as referencevalues for channel estimation and tracking. In practice, the channel transferfunction is first estimated at the positions where pilots are placed. Inter-polation techniques are next employed to obtain the channel response overinformation-bearing subcarriers. This approach is usually referred to aspilot-aided channel estimation and is the subject of this Section.

4.2.1 Scattered pilot patterns

0 12

Block Index

-21 7 -7 21 -32 31

(b) DAB & DVB

(a) IEEE 802.11a

Carrier Index

Data Pilot

(Time)

(Freq.)

Carrier Index (Freq.) Block

Index (Time)

23

Fig. 4.3 Pilot arrangements in commercial systems: IEEE 802.11a WLAN standard(a); DAB and DVB systems (b).

Figure 4.3 illustrates two major examples of pilot arrangements in thetime- and frequency-domains adopted in commercial applications. In par-ticular, Fig. 4.3 (a) refers to the IEEE 802.11a standard for WLANs [41,59],while the pattern of Fig. 4.3 (b) is employed in digital audio broadcasting(DAB) [39] and digital video broadcasting (DVB) systems [40]. The ver-tical axis represents the time direction and spans over the OFDM blocks,while the horizontal axis indicates the frequency direction and counts theindices of subcarriers in a given block.

As is seen, in the WLAN some specified subcarriers (called pilot tones)



are exclusively reserved for pilot insertion. In these systems, initial channelacquisition is performed at the beginning of each frame by exploiting tworeference blocks (not shown in the figure) carrying known symbols overall subcarriers. During the payload section, pilot tones can be exploitedfor channel tracking, even though in the IEEE 802.11a standard they arespecifically employed to track any residual frequency error that may remainafter initial frequency acquisition.

Generally speaking, the arrangement of Fig. 4.3 (a) is advantageous interms of system complexity because of the fixed positions occupied by pilottones in the frequency-domain. On the other hand, it is not robust againstpossible deep fades that might hit some of these pilot tones for the entireframe duration. As shown in Fig. 4.3 (b), in DAB and DVB systems thisproblem is mitigated by shifting the pilot positions in the frequency-domainat each new OFDM block. Compared to the pilot insertion strategy adoptedin the WLAN, this approach offers increased robustness against deep fadesand provides the system with improved channel tracking capabilities.

4.2.2 Pilot distances in time and frequency directions

A fundamental issue in the design of the pilot grid is the determinationof the time and frequency distances between adjacent pilots. These pa-rameters are strictly related to the rapidity of channel fluctuations in boththe time- and frequency-domains, and their selection is driven by the two-dimensional sampling theorem.

Let fD,max be the maximum expected Doppler frequency and assumethat, at any given frequency f , the channel response H(f, t) can be modeledin the time direction as a narrow-band stochastic process whose powerspectral density is confined within the interval [−fD,max, fD,max]. Then,from the sampling theorem we know that the distance ∆p,t (measured inOFDM blocks) between neighboring pilots in the time-domain must satisfythe inequality

∆p,t ≤ d 12fD,maxTB

e, (4.11)

where TB = NT Ts is the length of the OFDM block (including the cyclicprefix) and dxe is the largest integer not exceeding x.

On the other hand, at any given instant t, the rate of variation of H(f, t)with respect to f is related to the channel delay spread or, equivalently, tothe length of the channel impulse response (CIR) h(τ, t) over the τ -axis.Thus, assuming that h(τ, t) has support [0, τmax], the frequency spacing



between pilots is subject to the following constraint

∆p,f ≤ d 1τmaxfcs

e, (4.12)

where ∆p,f is normalized to the subcarrier spacing fcs = 1/(NTs). A prac-tical criterion for the design of ∆p,t and ∆p,f is to fix them to approximatelyone-half of their maximum allowable values given in Eqs. (4.11) and (4.12).This approach corresponds to two-times oversampling of H(f, t) and helpsto relax the requirements of the interpolation filters used for channel esti-mation.

The optimal arrangement of pilot symbols in both the time and fre-quency directions has extensively been studied in the literature [36,93,106].One major result is that in many cases a uniform pilot distribution repre-sents a good choice as it maximizes the channel estimation accuracy for agiven number of pilots.

Example 4.1 In this example we evaluate the maximum time and fre-quency distances among pilots in the DAB system. We consider a typicalurban (TU) channel with τmax = 5 µs and fD,max = 180 Hz, which corre-sponds to a mobile speed of approximately 100 km/h if the carrier frequencyis fixed to 2 GHz. The subcarrier spacing is fcs = 992 Hz while the du-ration of the OFDM block is TB = 1.3 ms. Substituting these parametersinto Eqs. (4.11) and (4.12) produces

∆p,t ≤ d 12× 180× 1.3× 10−3

e = 2, (4.13)

and

∆p,f = d 15× 10−6 × 992

e = 201. (4.14)

Actually, the pilot arrangement specified in the DAB system is charac-terized by ∆p,t = 1 and ∆p,f = 12, as shown in Fig. 4.3 (b). This meansthat, in principle, the DAB system can correctly operate in multipath envi-ronments with delay spreads much larger than 5 µs and with user terminalsmoving at speeds greater than 100 km/h.

4.2.3 Pilot-aided channel estimation

Channel estimation by means of scattered pilots is normally accomplished intwo successive steps. Let i′ and n′ be the coordinates of the pilot positionsin the time/frequency grid of Fig. 4.3 (a) or (b), and denote P the setof all ordered pairs (i′, n′). Then, in the first step an estimate Hi′(n′)



of the channel transfer function is computed for each pair (i′, n′) ∈ P byexploiting the corresponding DFT output Ri′(n′). During the second step,the quantities Hi′(n′) are interpolated in some way to obtain channel stateinformation over data-bearing subcarriers.

One simple method to compute Hi′(n′) results from application of theleast-squares (LS) approach to the signal model Eq. (4.1). This produces

Hi′(n′) =Ri′(n′)ci′(n′)

, for (i′, n′) ∈ P (4.15)

where ci′(n′) is the corresponding pilot symbol. Substituting Eq. (4.1) intoEq. (4.15) yields

Hi′(n′) = Hi′(n′) +Wi′(n′)ci′(n′)

, (4.16)

from which it follows that Hi′(n′) is unbiased with variance σ2w/σ2

p, whereσ2

p = |ci′(n′)|2 is the pilot power. If information about the channel covari-ance matrix and noise power is available, channel estimation at the pilotpositions can be performed according to the MMSE optimality criterion.Compared to the LS solution in Eq. (4.15), the MMSE approach is expectedto achieve better performance at the price of higher complexity. The latteris somewhat reduced by resorting to low-rank techniques available in theliterature [37].

As mentioned previously, channel estimates over information-bearingsubcarriers are obtained by suitable interpolation of the quantities Hi′(n′).Two alternative approaches can be adopted for this purpose. The first oneis based on two-dimensional (2D) filtering in both the time and frequencydirections. This technique provides optimum performance at the expenseof heavy computational load [54]. A better trade-off between complexityand estimation accuracy is achieved by the second approach, where the2D interpolator is replaced by the cascade of two one-dimensional (1D)filters working sequentially and performing independent interpolations inthe time- and frequency-domains. The design of 2D and 1D interpolatingfilters is discussed hereafter under some specified optimality criterions.

4.2.4 2D Wiener interpolation

With 2D Wiener filtering, the estimated channel frequency response overthe nth subcarrier of the ith OFDM block is given by

Hi(n) =∑

(i′,n′)∈Pq(i, n; i′, n′)Hi′(n′), (4.17)



where Hi′(n′) is the channel estimate at the pilot position (i′, n′) ∈ P asgiven in Eq. (4.15), while q(i, n; i′, n′) are suitable coefficients minimizingthe mean-square channel estimation error

Ji(n) = E∣∣∣Hi(n)−Hi(n)

∣∣∣2

. (4.18)

Equation (4.17) can be rewritten in matrix form as

Hi(n) = qT (i, n)H, (4.19)

where q(i, n) and H are column vectors of dimension Np equal to the cardi-nality of P and collect the quantities q(i, n; i′, n′) and Hi′(n′), respectively.From the orthogonality principle [123], we know that Ji(n) achieves itsglobal minimum when the error Hi(n)−Hi(n) is orthogonal to the obser-vations Hi′(n′) for each pair (i′, n′) ∈ P, i.e.,

E[

Hi(n)−Hi(n)]HH

= 0T . (4.20)

Substituting Eq. (4.19) into Eq. (4.20) leads to the following set ofWiener–Hopf equations

qT (i, n)RH = θT (i, n), (4.21)

where RH = EHHH is the autocorrelation matrix of H

while θT (i, n) = EHi(n)HH. The entries of RH are givenby RH(i′′, n′′; i′, n′) = EHi′′(n′′)H∗

i′(n′) with both (i′′, n′′) and

(i′, n′) belonging to P, while θT (i, n) is a row-vector with elementsθ(i, n; i′, n′) =EHi(n)H∗

i′(n′).

Bearing in mind Eq. (4.16) and assuming that the channel response andthermal noise are statistically independent, we may write

RH(i′′, n′′; i′, n′) = RH(i′′, n′′; i′, n′) +σ2

w

σ2p

· δ(i′′ − i′)δ(n′′ − n′), (4.22)

and

θ(i, n; i′, n′) = RH(i, n; i′, n′), (4.23)

where δ(`) is the Kronecker delta function and RH(i, n; j, m) =EHi(n)H∗

j (m) the two-dimensional channel autocorrelation function. In[90] it is shown that for a typical mobile wireless channel RH(i, n; j, m) canbe separated into the multiplication of a time-domain correlation Rt(·) bya frequency-domain correlation Rf (·), i.e.,

RH(i, n; j, m) = Rt(i− j) ·Rf (n−m). (4.24)



Clearly, Rf (·) depends on the multipath delay spread and power delayprofile, while Rt(·) is related to the vehicle speed or, equivalently, to theDoppler frequency.

The optimum interpolating coefficients for the estimation of Hi(n) arecomputed from Eq. (4.21) and read

qT (i, n) = θT (i, n)R−1

H. (4.25)

A critical issue in 2D Wiener filtering is the inversion of the Np -dimensional matrix RH , which may be prohibitively complex for largeNp values. Also, computing RH and θ(i, n) requires information aboutthe channel statistics and noise power, which are typically unknown at thereceiver. One possible strategy is to derive suitable estimates of these pa-rameters, which are then used in Eqs. (4.22) and (4.23) in place of theirtrue values. In general, this approach provides good results but requiresthe on-time inversion of RH .

An alternative method relies on some a-priori assumptions about thechannel statistics and optimizes the filter coefficients for specified values ofthe noise power and channel correlation functions. In practice, the Wienercoefficients are often designed for a uniform Doppler spectrum and powerdelay profile [90]. This amounts to assuming a wireless channel with thefollowing time- and frequency-correlation functions

Rt(i) = sinc(2fDiTB

), (4.26)

andRf (n) = sinc (nfcsτ) e−jπnfcsτ , (4.27)

in which fD and τ are conservatively chosen a bit larger than the maximumexpected Doppler frequency and multipath delay spread, respectively.

This approach leads to a significant reduction of complexity because thefilter coefficients are now pre-computed and stored in the receiver. Clearly,the price for this simplification is a certain degradation of the system perfor-mance due to a possible mismatch between the assumed operating param-eters and their actual values. However, theoretical analysis and numericalresults indicate that the mismatching effect is tolerable if the interpolatingcoefficients are designed on the basis of the autocorrelation functions givenin Eqs. (4.26) and (4.27).

4.2.5 Cascaded 1D interpolation filters

A simple method to avoid the complexity of 2D Wiener filtering is based onthe use of two cascaded 1D filters which perform independent interpolation



decision device

i p ( n )

i ' ( n' R )

i ' ( n' H ) i ( n' H )

i ( n ) H

Frequency domain

interpolation

Time domain

interpolation

Compute equalizer

coefficients

Pilot extraction

i Y ( n ) DFT outputs over the

entire frame

1/ c i ' ( n' )

i R ( n )

c i ( n )

Fig. 4.4 A typical equalizer structure with two-cascaded 1D interpolation filters.

in the time and frequency directions. This idea is illustrated in Fig. 4.4,where interpolation in the time-domain precedes that in the frequency-domain, even though the opposite ordering could be used as well due tothe linearity of the filters. Regardless of the actual filtering order, theessence of the first interpolation is to compute channel estimates over somespecific data subcarriers that are subsequently used as additional pilots forthe second interpolation stage.

Consider a specific subcarrier n′ (represented by a column in the time-frequency grids of Fig. 4.3) and assume that the latter conveys pilot sym-bols over a number Np,t of OFDM blocks specified by the indices i′ ∈Pt(n′). For example, the WLAN pilot arrangement of Fig. 4.3 (a) resultsinto Pt(n′) = 1, 2, 3, . . . for n′ = ±7 or ±21 and Pt(n′) = ∅ for theremaining subcarriers. In the DAB/DVB system of Fig. 4.3 (b) we havePt(n′) = ∅ if n′ is not multiple of three while Pt(3m′) = |m′|4 + 4`,where m′ and ` are non-negative integers and |m′|4 denotes the remainderof the ratio m′/4.

As indicated in Fig. 4.4, pilot tones are extracted from the DFT out-put and used to compute the quantities Hi′(n′) specified in Eq. (4.15).The latter are then interpolated by the time-domain filter to obtain thefollowing channel estimates over the n′th subcarrier of each OFDM block(i = 1, 2, . . .)

Hi(n′) =∑

i′∈Pt(n′)

qt(i; i′, n′)Hi′(n′), n′ ∈ Pf (4.28)



where qt(i; i′, n′) are suitable coefficients designed according to some opti-mality criterion while the set Pf collects the indices of pilot-bearing sub-carriers and has cardinality Np,f . Clearly, Pf = ±7,±21 in Fig. 4.3 (a)while Pf = 0, 3, 6, . . . in Fig. 4.3 (b).

0 12 432

Data

Pilots

Subcarrier index ( Freq. )

Block index ( Time )

Additional pilots after time-domain interpolation

Fig. 4.5 Increase of effective pilots after time-domain interpolation.

Figure 4.5 illustrates the position of the time-interpolated channel esti-mates Hi(n′) in the DAB frame. As mentioned previously, these quantitiesare viewed by the second interpolation filter as additional pilots, and usedto obtain the channel transfer function over the entire time-frequency grid.In particular, the estimate of Hi(n) is computed as

Hi(n) =∑

n′∈Pf

qf (n;n′)Hi(n′), (4.29)

where the weights qf (n; n′) are independent of the time index i and, accord-ingly, are the same over all OFDM blocks. Popular approaches for designingthe filtering coefficients qt(i; i′, n′) and qf (n; n′) are discussed hereafter.

4.2.5.1 Cascaded 1D Wiener interpolators

Wiener interpolators are based on the MMSE optimality criterion. Specif-ically, for a given n′ the coefficients qt(i, n′) = qt(i; i′, n′); i′ ∈ Pt(n′) ofthe time-domain Wiener filter are designed so as to minimize the followingMSE:

Ji(n′) = E∣∣Hi(n′)−Hi(n′)

∣∣2

, (4.30)



with Hi(n′) as given in Eq. (4.28). After invoking the orthogonality prin-ciple, we find that

qTt (i, n′) = θT

t (i, n′)R−1t , (4.31)

where θt(i, n′) is a column vector of length Np,t whose entries are relatedto the time-domain channel correlation function Rt(·) by

[θt(i, n′)]i′ = Rt(i− i′), i′ ∈ Pt(n′) (4.32)while Rt is a matrix of order Np,t with elements

[Rt]i′′,i′ = Rt(i′′ − i′) +σ2

w

σ2p

· δ(i′′ − i′), i′′, i′ ∈ Pt(n′). (4.33)

It is worth noting that Rt is independent of n′ and i, whereas θt(i, n′)may depend on n′ through i′ ∈ Pt(n′). However, if the pilot arrangementis such that the same set Pt(n′) is used for each n′ ∈ Pf as in Fig. 4.3(a), vector θt(i, n′) becomes independent of n′ and the same occurs to thefilter coefficients in Eq. (4.31). This property is clearly appealing because insuch a case the same set of time-interpolation coefficients are used over allsubcarriers n′ ∈ Pf , thereby reducing the computational effort and storagerequirement of the channel estimation unit.

The orthogonality principle is also used to obtain the interpolation coef-ficients qf (n) = qf (n; n′); n′ ∈ Pf of the frequency-domain Wiener filter.This yields

qTf (n) = θT

f (n)R−1f , (4.34)

where θf (n) is a vector of length Np,f and Rf a matrix of the same or-der. Their entries are related to the frequency-domain channel correlationfunction Rf (·) by [

θTf (n)

]n′

= Rf (n− n′), n′ ∈ Pf (4.35)and

[Rf ]n′′,n′ = Rf (n′′ − n′) +σ2

w

σ2p

· δ(n′′ − n′), n′′, n′ ∈ Pf . (4.36)

Although much simpler than 2D Wiener filtering, the use of two-cascaded 1D Wiener interpolators may still be impractical for a couple ofreasons. The first one is the dependence of the filtering coefficients on thechannel statistics and noise power. As discussed previously, a robust filterdesign based on the sinc-shaped autocorrelation functions in Eqs. (4.26)and (4.27) can mitigate this problem to some extent. The second difficultyis that time-domain Wiener interpolation cannot be started until all blockscarrying pilot symbols have been received. This results into a significantfiltering delay, which may be intolerable in many practical applications. Apossible solution to this problem is offered by piecewise polynomial inter-polation, as it is now discussed.



4.2.5.2 Cascaded 1D polynomial-based interpolators

The concept of piecewise polynomial interpolation is extensively coveredin the digital signal processing literature [28, 136]. One of the main con-clusions is that excellent interpolators can be implemented with a smallnumber of taps, say either two or three. The limited amount of complexityassociated with polynomial-based filters makes them particularly attractivein a number of applications. In the ensuing discussion, they are applied toOFDM systems in order to find practical schemes for interpolating channelestimates in both the time- and frequency-domains [132].

For illustration purposes, we concentrate on the DAB pilot arrangementof Fig. 4.3 (b) and observe that, for any given pilot-bearing subcarrier withindex n′ ∈ Pf = 0, 3, 6, . . ., two neighboring pilots are separated in thetime direction by three OFDM blocks. In other words, if a pilot is presenton the n′th subcarrier of the i′th block, the next pilot on the same subcarrierwill not be available until reception of the (i′ + 4)th block.

0 12

Subcarrier index ( Freq. )

Block index ( Time )

i' i'+ 4 i'+ 8

(a) Zero-order hold filter

(b) First-order linear filter

H (12)

i'

i'+ 4

i' + 8

Block index

i' i'+ 4 i'+ 8 Block index i'

H (12) i'

H (12) i'

Fig. 4.6 Time-domain interpolation by means of (a) zero-order and (b) first-order poly-nomial filters.

The simplest form of piecewise polynomial interpolation is representedby the zero-order hold filter. When applied in the time direction over then′th subcarrier, this filter receives a channel estimate Hi′(n′) and keeps it



fixed until the arrival of the next pilot. Mathematically, we have

Hi(n′) = Hi′(n′), for i′ ≤ i ≤ i′ + pt − 1 and n′ ∈ Pf (4.37)

where pt = 4 is the time-distance between adjacent pilots. The concept oftime-domain zero-order interpolation is illustrated in Fig. 4.6 (a) for n′ =12. This technique does not introduce any filtering delay but can only beused in those applications where the channel transfer function Hi(n) keepsalmost unchanged between adjacent pilots. Channel variations occurringin high-mobility systems are better handled by first-order interpolation. Inthis case Hi(n′) varies in a piecewise-linear fashion as depicted in Fig. 4.6(b), and is computed as

Hi(n′) =1pt

[(pt + i′ − i) Hi′(n′) + (i′ − i) Hi′+pt(n

′)], (4.38)

for i′ ≤ i ≤ i′ + pt − 1 and n′ ∈ Pf .Intuitively speaking, first-order interpolation is expected to provide

more accurate estimates than zero-order filtering. However, it results intoan inherent filtering delay since the estimate Hi(n′) in Eq. (4.38) cannotbe computed before reception of the (i′ + pt)th OFDM block. Polynomialfilters based on second or higher order interpolation provide even betterperformance at the price of increased delays. For this reason, they arerarely used in practice.

The idea of piecewise polynomial filtering can also be applied in thefrequency direction to obtain final channel estimates Hi(n). Contrarilyto time-domain interpolation, however, in this case the filtering delay isnot a critical issue. The reason is that the frequency-domain interpolatoroperates on a block-by-block basis, so that in principle the quantities Hi(n′)are filtered as soon as the ith OFDM block has been received. It followsthat low-order filters with a small number of taps are not strictly necessaryfor frequency-domain interpolation. More sophisticated schemes based onLS reasoning can be resorted to as it is now illustrated.

4.2.5.3 LS-based interpolation in frequency domain

The quantity Hi(n′) produced by the time-domain interpolation filter aremodeled as

Hi(n′) = Hi(n′) + W i(n′), n′ ∈ Pf (4.39)

where W i(n′) is a disturbance term that accounts for thermal noise andpossible interpolation errors. We denote hi = [hi(0), hi(1), . . . , hi(L− 1)]T



the Ts-spaced samples of the CIR during the ith OFDM block, and recallthat the channel transfer function is obtained by taking the DFT of hi, i.e.,

Hi(n) =L−1∑

`=0

hi(`) e−j2πn`/N . (4.40)

Substituting Eq. (4.40) into Eq. (4.39) produces

Hi = Fhi + W i, (4.41)

where Hi and W i are Np,f -dimensional vectors with elements Hi(n′) andW i(n′), respectively, while F∈ CNp,f×L is a matrix with entries e−j2πn′`/N

for 0 ≤ ` ≤ L − 1 and n′ ∈ Pf . The quantities Hi in Eq. (4.41) are nowexploited to derive an estimate of hi. For this purpose, we adopt a LSapproach and obtain

hi = (FH

F )−1FH

Hi. (4.42)

Note that a necessary condition for the invertibility of FH

F inEq. (4.42) is that Np,f ≥ L. This amounts to saying that the numberof pilots in the frequency direction cannot be less than the number of chan-nel taps, otherwise the observations Hi(n′) are not sufficient to estimateall unknown parameters hi(`).

From Eq. (4.40), an estimate of the channel transfer function is obtainedas

Hi(n) =L−1∑

`=0

hi(`) e−j2πn`/N , 0 ≤ n ≤ N − 1. (4.43)

After substituting Eq. (4.42) into Eq. (4.43), we get the final channelestimate in the form

Hi(n) =∑

n′∈Pf

qLSf (n; n′)Hi(n′), (4.44)

where the LS coefficients qLSf (n; n′) are given by

qLSf (n; n′) =

L−1∑

`1=0

L−1∑

`2=0

[(F HF )−1

]`1,`2

ej2π(n′`2−n`1)/N . (4.45)

In [101] it is shown that the accuracy of the estimator Eq. (4.44) isoptimized when the pilot symbols are uniformly spaced in the frequency-domain with a separation interval ∆p,f = N/Np,f . In this case F

HF =

Np,f · IL and the filtering coefficients in Eq. (4.45) take the form

qLSf (n;n′) =

1Np,f

ejπ(L−1)(n′−n)/N sin [πL (n′ − n) N ]sin [π (n′ − n) /N ]

. (4.46)



It is worth noting that in many commercial systems a specified numberof subcarriers at both edges of the signal spectrum are left unmodulated(virtual or null subcarriers) so as to reduce out-of-band emission. If thisnumber is greater than N/Np,f , a uniform distribution of pilots in thefrequency-domain is not possible. In this case, the optimum pilots’ po-sitions can only be determined through a numerical search. Simulationresults reported in [101] indicate that in the presence of virtual subcarriers(VCs) it is convenient to adopt a non-uniform pilot arrangement with asmaller separation distance in the neighborhood of the spectrum edges. Analternative method is depicted in Fig. 4.7. Here, the transmitter insertsuniformly spaced pilots only within the signal spectrum while leaving thesuppressed bandwidth empty. At the receiving terminal, the pilot sym-bols closest to the spectrum boundaries are artificially duplicated over thesuppressed bandwidth and used by the interpolation filters as if they wereregular pilots. Clearly, this approach is more practical then using non-uniformly spaced pilots, even though channel estimates in the vicinity ofthe suppressed bandwidth are expected to be less accurate than those inthe middle of the signal spectrum.

#0 #6 #1 #2 #3 #4 #5 #7 #8 #9 #0 #6 #1 #2 #3 #4 #5 #7 #8 #9

Estimated CIR True CIR Duplicated pilot

Subcarrier index

Subcarrier index

Frequency-domain interpolation

Pilot

Fig. 4.7 Channel estimation in the vicinity of suppressed carriers.

In a sparse multipath environment where only a few multipath compo-nents are present with relatively large differential delays, most of the CIRcoefficients hi(`) are expected to be vanishingly small. In such a scenario,the accuracy of the LS estimator can be improved by adopting a parametricchannel model characterized by a reduced number of unknown parameters.This approach is suggested in [179], where the minimum description length(MDL) criterion [169] is employed to detect the number of paths in thechannel. After recovering the path delays through rotational invariant tech-niques (ESPRIT) [135], estimates of the path gains are eventually obtainedusing LS or MMSE methods.



4.3 Advanced techniques for blind and semi-blind channelestimation

The insertion of pilot symbols into the transmitted data stream simplifiesthe channel estimation task to a large extent, but inevitably reduces thespectral efficiency of the communication system. This problem has inspiredconsiderable interest in blind or semi-blind channel estimation techniqueswhere only a few pilots are required. These schemes are largely categorizedinto subspace-based or decision-directed (DD) methods. In the former case,the intrinsic redundancy provided by the cyclic prefix (CP) or by VCsis exploited as a source of channel state information. A good sample ofthe results obtained in this area are found in [86, 103, 167] and referencestherein. Although attractive because of the considerable saving in trainingoverhead, the subspace approach is effective as long as a large amount ofdata is available for channel estimation. This is clearly a disadvantagein high-mobility applications, since in this case the time-varying channelmight preclude accumulation of a large data record.

In DD methods, tentative data decisions are exploited in addition to afew pilots to improve the channel estimation accuracy. An example of thisidea is presented in [91], where trellis decoding is employed for joint equal-ization and data detection of differentially-encoded PSK signals. Differen-tial encoding is performed in the frequency direction while trellis decodingis efficiently implemented through a standard Viterbi processor. The latteroperates in a per-survivor fashion [128] wherein a separate channel estimateis computed for each surviving path.

The idea of exploiting data decisions to improve the channel estimationaccuracy is also the rationale behind EM-based methods [102, 176]. Theseschemes operate in an iterative mode with channel estimates at a given stepbeing derived from symbol decisions obtained at the previous step. In thisway, data detection and channel estimation are no longer viewed as separatetasks but, rather, are coupled together and accomplished in a joint fashion.Other blind approaches for channel estimation in OFDM systems exploiteither the cyclostationarity property induced by the CP on the receivedtime-domain samples [70] or the fact that the information-bearing symbolsbelong to a finite alphabet set [183].

It is fair to say that strictly blind channel estimation techniques exploit-ing no pilots at all are hardly usable in practice as they are plagued by aninherent scalar ambiguity. This amounts to saying that, even in the absenceof noise and/or interference, the channel response can only be estimated



up to a complex-valued factor. The only way to solve the ambiguity is toinsert a few pilot symbols into the transmitted blocks in order to providea phase reference for the receiving terminal. The use of pilots in combina-tion with blind algorithms results into semi-blind schemes with improvedestimation accuracy. Compared to the pilot-aided methods discussed pre-viously, the semi-blind approach suffers from some drawbacks in terms ofcomputational complexity and prolonged acquisition time.

4.3.1 Subspace-based methods

time

1 st block 2 nd block N B

th block

observation window

SUPERBLOCK

CP CP CP

Fig. 4.8 Observation of a superblock for subspace-based channel estimation.

Subspace-based methods derive channel information from the inherentredundancy introduced in the transmitted signal by the use of the CPand/or VCs. To explain the basic idea behind this class of blind estimationtechniques, we define a superblock as the concatenation of NB successiveOFDM blocks, where NB is a suitably chosen design parameter. As de-picted in Fig. 4.8, at the receiver side the observation window spans anentire superblock, except for the CP of the first OFDM block which isintentionally discarded to avoid IBI from the previously transmitted su-perblock. The total number of time-domain samples falling within the kthobservation window is thus MT = NBNT−Ng. These samples are arrangedinto a vector

r(k) = sR(k) + w(k), (4.47)

where sR(k) is the signal component while w(k) accounts for thermal noise.We assume that some VCs are present in the signal spectrum, so that only P

subcarriers out of a total of N are actually employed for data transmission.This means that each superblock conveys NBP data symbols, which are



collected into a vector c(k). Hence, we can rewrite sR(k) in the form

sR(k) = G(h)c(k), (4.48)

where h = [h(0), h(1), . . . , h(L− 1)]T is the CIR vector (assumed static forsimplicity) while G(h) ∈ CMT×NBP is a tall matrix whose entries dependon the indices of the modulated subcarriers and are also linearly related toh. It is worth noting that the mapping c(k) −→ sR(k) in Eq. (4.48) can beinterpreted as a sort of coding scheme wherein G(h) is the code generatormatrix and the introduced redundancy is proportional to the differencebetween the dimensions of sR(k) and c(k) , say Nr = MT − NBP . Thisredundancy originates from the use of VCs and CPs, and can be exploitedfor the purpose of channel estimation as it is now explained.

Returning to Eq. (4.48), we observe that sR(k) is a linear combinationof the columns of G(h), each weighted by a given transmitted symbol. Asa result, sR(k) belongs to the subspace of CMT spanned by the columns ofG(h), which is referred to as the signal subspace. If G(h) is full-rank (anevent which occurs with unit probability), the signal subspace has dimen-sion NBP . Its orthogonal complement in CMT is called the noise subspaceand has dimension Nr. To proceed further, we consider the correlationmatrix Rrr of the received vector r(k). After substituting Eq. (4.48) intoEq. (4.47) we obtain

Rrr = V (h) + σ2wIMT

, (4.49)

where σ2w is the noise power and V (h) = G(h)RccG

H(h), withRcc =Ec(k)cH(k) denoting the correlation matrix of the data vector. Atthis stage we observe that rank V (h) = min MT , NBP = NBP . Thismeans that V (h) has only NBP non-zero eigenvalues µj (1 ≤ j ≤ NBP )out of a total of MT . Thus, from Eq. (4.49) it follows that the eigenvaluesof Rrr (arranged in a decreasing order of magnitude) are given by

λj =

µj + σ2w, 1 ≤ j ≤ NBP,

σ2w, NBP + 1 ≤ j ≤ MT .

(4.50)

A fundamental property of Rrr is that the set U = u1,u2, . . . , uNrof Nr eigenvectors associated to the smallest eigenvalues σ2

w constitute abasis for the noise subspace, while the remaining NBP eigenvectors lie inthe signal subspace. Since the latter is spanned by the columns of G(h)and is also orthogonal to the noise subspace (hence, to each vector uj inthe basis U), we may write

uHj G(h) = 0T

NBP , 1 ≤ j ≤ Nr (4.51)



where 0NBP is a column vector of NBP zeros.Recalling that the entries of G(h) are related to the unknown channel

vector h in a linear fashion, we may interpret the constraints Eq. (4.51)as a set of NrNBP linear homogeneous equations in the variables h(`).Hence, they can equivalently be rewritten as

hHB(U) = 0TNrNBP , (4.52)

where B(U) is a suitable matrix of dimensions L×NBPNr whose entriesdepend on the basis U of the noise subspace. Solving the set of equations inEq. (4.52) and discarding the trivial solution h = 0L provides an estimateof the CIR vector up to a complex scaling factor.

From the above discussion it turns out that subspace-based methods relyon the decomposition of the observation space CMT into a signal subspaceplus a noise subspace, and determine the channel estimate by exploitingthe reciprocal orthogonality among them. This decomposition is performedover the correlation matrix Rrr which, however, is typically unknown. Inpractice, Rrr is replaced by the so-called sample-correlation matrix, whichis obtained by averaging the received time-domain samples over a specifiednumber KB of superblocks, i.e.,

Rrr =1

KB

KB∑

k=1

r(k)rH(k). (4.53)

The eigenvectors of Rrr associated with the Nr smallest eigenvaluesare taken as an estimate U of the noise subspace, which is then used inEq. (4.52) in place of the true U . Under normal operating conditions, theset of linear equations hHB(U) = 0T

NrNBP has h = 0L as unique solution.To overcome this problem, the equations are solved in the LS sense underan amplitude constraint ‖h‖ = 1. This leads to the following minimizationproblem

h = argmin‖h‖=1

hHB(U)BH(U)h

, (4.54)

where h represents a trial value of h. The solution is well known andis attained by choosing h as the unit-norm eigenvector associated to thesmallest eigenvalue of B(U)BH(U).

In conclusion, we can summarize the subspace-based procedure as fol-lows:

(1) observe a specified number KB of superblocks and compute the samplecorrelation matrix Rrr as indicated in Eq. (4.53);



(2) determine the noise subspace by computing the Nr smallest eigen-values of Rrr. Arrange the corresponding eigenvectors into a setU= u1, u2, . . . , uNr;

(3) use U to construct matrix B(U);(4) compute the smallest eigenvalue of B(U)BH(U) and take the corre-

sponding unit-norm eigenvector as an estimate h of the CIR vector.

For a given observation window, the accuracy of subspace-based meth-ods increases with the amount of redundancy introduced by the use of CPsand/or VCs. In particular, simulation results shown in [86] indicate thatenlarging the CP is more beneficial than increasing the number of VCs. Asmentioned previously, a major drawback of this class of schemes is repre-sented by the large number of blocks that are normally required to achievethe desired estimation accuracy.

4.3.2 EM-based channel estimation

In conventional OFDM systems with coherent detection, channel estimationand data decoding are normally kept as separate tasks. Albeit reasonableand easy to implement, this approach is not based over any optimality crite-rion. Better results are expected if the channel response and data symbolsare jointly estimated under a maximum likelihood (ML) framework. Unfor-tunately, using this strategy over an entire OFDM frame is computationallyunfeasible due to lack of efficient ways for maximizing the likelihood func-tion over all candidate data sequences. This problem is alleviated if thereceiver only exploits channel correlation in the frequency direction whileneglecting any time correlation over adjacent OFDM blocks. In this waythe equalization algorithm can operate on a block-by-block basis, with asubstantial reduction of the number of candidate sequences. However, evenwith the adoption of this simplified approach, joint ML estimation of chan-nel response and data symbols remains a challenging task as it is nowshown.

4.3.2.1 Likelihood function for joint data detection and channelestimation

In the following derivations we focus on a single OFDM block and neglectthe time index i for notational simplicity. The DFT output is given by

R(n) = H(n)c(n) + W (n), 0 ≤ n ≤ N − 1 (4.55)



where H(n) =∑L−1

`=0 h(`) e−j2πn`/N and h = [h(0), h(1), . . . , h(L− 1)]T

collects the CIR coefficients. Denoting R = [R(0), R(1), . . . , R(N − 1)]T

the observation vector, we may rewrite Eq. (4.55) in matrix form as

R = A(c)Fh + W , (4.56)

where c = [c(0), c(1), . . . , c(N − 1)]T is the transmitted data sequence, A(c)is a diagonal matrix with c along its main diagonal and F is an N×L matrixwith entries

[F ]n,` = e−j2πn`/N , 0 ≤ n ≤ N − 1, 0 ≤ ` ≤ L− 1. (4.57)

Vector W represents the noise contribution and is Gaussian distributedwith zero-mean and covariance matrix σ2

wIN .From Eq. (4.56), the likelihood function for the joint estimation of c

and h is found to be

Λ(c,h) =1

(πσ2w)N

exp− 1

σ2w

∥∥∥R−A(c)F h∥∥∥

2

, (4.58)

where c and h are trial values of c and h, respectively. The ML estimates ofthe unknown vectors are eventually obtained looking for the location whereΛ(c,h) achieves its global maximum, i.e.,

(c,h) = argmax(c,h)

Λ(c,h)

. (4.59)

4.3.2.2 Likelihood function maximization by EM algorithm

The maximum of Λ(c,h) in Eq. (4.58) can be found in two successive steps.First, we keep c fixed and maximize with respect to h. This produces

h(c) = [A(c)F ]†R, (4.60)

where [A(c)F ]† =[F HAH(c)A(c)F

]−1F HAH(c) is the Moore-Penrose

generalized inverse of A(c)F . After substituting Eq. (4.60) into Eq. (4.58)and letting c vary, we see that maximizing Eq. (4.58) is equivalent to max-imizing the following metric

g(c) = <e

RHA(c)F [A(c)F ]†R

. (4.61)

Inspection of Eqs. (4.60) and (4.61) indicates that the estimates of c

and h are decoupled in that the former can be computed first and is thenexploited to get the latter. However, maximizing g(c) in Eq. (4.61) appearsa formidable task. A certain simplification is possible if the data symbolsbelong to a PSK constellation. In this case we have AH(c)A(c) = IN , so



that [A(c)F ]† reduces to[F HF

]−1F HAH(c). Observing that F HF =N ·

IN , Eqs. (4.60) and (4.61) become

h(c) =1N

F HAH(c)R, (4.62)

g(c) =1N

∥∥RHA(c)F∥∥2

. (4.63)

Unfortunately, the direct maximization of g(c) in Eq. (4.63) is still in-tractable as it requires an exhaustive search over all possible data sequencesc, whose number grows exponentially with N . A possible way to overcomethis obstacle is the use of the EM algorithm. Under some mild condi-tions, the latter can locate the global maximum of the likelihood functionthrough an iterative procedure which is much simpler than the exhaustivesearch [34]. In the EM parlance, the observed measurements are replacedwith some complete data from which the original measurements are ob-tained through a many-to-one mapping. At each iteration, the algorithmcomputes the expectation of the log-likelihood function for the completedata (E-step), which is next maximized with respect to the unknown pa-rameters (M-step). Here, we follow the guidelines suggested in [102] andview the DFT output R as the incomplete data, whereas the complete dataset is defined as the pair R, h. Under these assumptions, during the j thiteration the EM algorithm proceeds as follows [102]:

EM-based joint channel estimation and data detection

• E-stepCompute

Q(c

∣∣∣c(j−1))

= Eh

p

(R

∣∣∣h, c(j−1))· ln p (R |h, c )

, (4.64)

where c(j−1) is the estimate of c at the (j − 1)th step, p(·) is theprobability density function (pdf) of the enclosed quantities andEh · indicates statistical expectation over the pdf of h.

• M-stepMaximize Q

(c

∣∣c(j−1))

over the set spanned by c to obtain datadecisions in the form

c(j) = arg maxc

Q

(c

∣∣∣c(j−1))

. (4.65)

Assuming that h is Gaussian distributed with zero-mean (Rayleigh fad-ing) and covariance matrix Ch =EhhH, after some manipulations it is



found that Eq. (4.65) can equivalently be rewritten as [102]

c(j) = arg maxc

<e

[RHA(c)F hMMSE(c(j−1))

], (4.66)

where

hMMSE(c(j−1)) = (N · IN + σ2wC−1

h )−1F HAH(c(j−1))R (4.67)

is the MMSE estimator of h as derived from the model Eq. (4.56) afterreplacing the true data vector c by its corresponding estimate c(j−1). De-noting HMMSE(n, c(j−1)) the N -point DFT of hMMSE(c(j−1)), we mayrewrite Eq. (4.66) in the following way

c(j) = arg maxc

N−1∑n=0

<e[R∗(n)c(n)HMMSE(n, c(j−1))

]. (4.68)

With uncoded transmissions, the above maximization is equivalent tomaximizing each individual term in the sum, i.e., making symbol-by-symboldecisions

c(j)(n) = argmaxc(n)

<e

[R∗(n)c(n)HMMSE(n, c(j−1))

], 0 ≤ n ≤ N − 1

(4.69)where c(j)(n) is the nth entry of c(j).

Inspection of Eq. (4.69) reveals the physical rationale behind the EMalgorithm. As is seen, at the jth iteration the estimate of c is computedthrough conventional frequency-domain detection/equalization techniques,where channel state information is achieved by means of the MMSE cri-terion using data decisions c(j−1) from the previous iteration. Clearly, aninitial estimate h(0) of the channel vector is needed to initialize the iterativeprocedure. One possibility is to insert some pilots within each OFDM blockand use them to compute h(0) according to Eq. (4.42). Alternatively, thechannel estimate obtained during the current OFDM block can be used inthe next block for initialization purposes.

As indicated in Eq. (4.67), the MMSE channel estimator requires knowl-edge of the channel statistics and noise power. These quantities can be es-timated on-time from the received samples as suggested in [102]. A simplersolution is found assuming high SNR values. In this case σ2

w is vanishinglysmall and hMMSE(c(j−1)) in Eq. (4.66) is thus replaced by the followingLS estimate

hLS(c(j−1)) =1N

F HAH(c(j−1))R. (4.70)

Albeit simple, this approach is expected to incur some performance penaltywith respect to the optimal solution Eq. (4.66).



4.4 Performance comparison

In this section we use computer simulations to compare the performance ofsome of the channel estimation techniques described throughout the chap-ter. In doing so we consider an OFDM system with N = 256 subcarriersand QPSK data symbols. The DAB/DVB pilot pattern of Fig. 4.3 (b) isemployed to multiplex 16 scattered pilots in each OFDM block. The trans-mission channel is characterized by Np = 4 multipath components. Thepath delays are kept fixed at τ1 = 0, τ2 = 1.4Ts, τ3 = 4.8Ts and τ4 = 9.7Ts,while the path gains αm(t) (m = 1, 2, 3, 4) are modeled as statistically in-dependent Gaussian random processes with zero-mean and autocorrelationfunction

Rm(τ) = σ2mJ0(2πfDτ). (4.71)

In the above equation, J0(x) denotes the zero-order Bessel function of thefirst kind, fD is the Doppler frequency and σ2

m =E|αm(t)|2 the statistical

power of αm(t). We assume an exponentially-decaying power delay profilewhere

σ2m = βe−m, m = 1, 2, 3, 4 (4.72)

and parameter β is chosen so as to normalize the received signal power tounity.

The channel taps hi(`) are expressed by

hi(`) =4∑

m=1

αm(iTB)g(`Ts − τm), ` = 0, 1, . . . , L− 1 (4.73)

where g(t) accounts for the signal shaping operated by the transmit andreceive filters, and has a raised-cosine Fourier transform with roll-off 0.22.The Doppler frequency is fD = 10−2/TB , while the channel length isL = 16. To prevent IBI, a CP of length Ng = 16 is appended to each block.

Figure 4.9 shows the BER performance as a function of Es/N0 for anuncoded QPSK transmission. The curve labeled “Ideal” refers to a sys-tem with perfect channel state information while the curves labeled “Two-cascaded 1D EQ” are obtained by performing zero-order or first-order 1Dpolynomial interpolation in the time-domain followed by 1D LS interpola-tion in the frequency-direction as indicated by Eq. (4.42). The EM-basedequalizer is initialized with channel estimates provided by the two-cascaded1D filters with first-order polynomial interpolation. We see that the first-order filter provides much better performance than zero-order interpolation



0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

Es/N

0 (dB)

BE

R

Two−cascaded 1D EQ (zero−order + LS interp.)Two−cascaded 1D EQ (first−order + LS interp.)EM−based EQ ( j = 1)EM−based EQ ( j = 2)Ideal

Fig. 4.9 BER comparison between two-cascaded 1D interpolation filters and EM-basedequalization as a function of Es/N0.

due to its enhanced tracking capability. The BER slightly improves if thechannel estimation and data detection tasks are coupled together by meansof the EM algorithm. Figure 4.9 indicates that in this way the error-rateperformance approaches that of the ideal system after only one iteration(j = 1), while marginal improvements are observed with more iterations.


Chapter 5

Joint Synchronization, ChannelEstimation and Data SymbolDetection in OFDMA Uplink

A frequency offset estimator based on the space-alternating generalizedexpectation-maximization (SAGE) algorithm has been presented in Chap-ter 3 for OFDMA uplink transmissions with generalized CAS. This schemecomputes estimates of all users’ carrier frequency offsets (CFOs) by ex-ploiting a training block transmitted at the beginning of the uplink frame.The frequency estimates are then employed during the payload section torestore orthogonality among the uplink signals by means of interferencecancellation or linear multiuser detection techniques.

In a high-mobility environment such as air traffic control and manage-ment [50], the users’ CFOs and channel responses may vary with time andtheir variations must continuously be tracked for reliable data detection.Hence, a robust scheme where data decisions are exploited in addition topilot symbols for the purpose of frequency and channel tracking is highlydesirable.

In this chapter we investigate the issue of joint frequency synchroniza-tion, channel estimation and data detection for all active users in the uplinkof a quasi-synchronous OFDMA system. As we shall see, the exact maxi-mum likelihood (ML) solution to this problem turns out to be too complexfor practical purposes as it involves a search over a multidimensional do-main. The complexity requirement is greatly reduced by resorting to theEM principle. This leads to an iterative scheme where the superimposedsignals arriving at the base station (BS) are first separated by means ofthe SAGE algorithm. The separated signals are subsequently passed to anexpectation-conditional maximization (ECM)-based processor, which up-dates frequency estimates while performing channel estimation and datadetection for each user. The resulting architecture is reminiscent of theparallel interference cancellation (PIC) receiver, where at each step inter-

135



ference is generated and removed from the received signal to improve thereliability of data decisions.

Simulations indicate that the joint synchronization, channel estimationand data detection scheme provides an effective means to track possiblefrequency variations that may occur in high-mobility applications. In par-ticular, it turns out that large CFOs can be corrected without incurringsevere performance degradation with respect to a perfectly synchronizedsystem where neither interchannel interference (ICI) nor multiple-accessinterference (MAI) is present. It is nevertheless fair to say that these ad-vantages come at the price of a higher computational load compared toother existing methods as those presented in [12,18,55,158].

5.1 Uncoded OFDMA uplink

5.1.1 Signal model

We consider the uplink of a quasi-synchronous OFDMA system in whichthe cyclic prefix (CP) is sufficiently long to accommodate both the chan-nel delay spreads and timing offsets of all active terminals. The chan-nel impulse responses (CIRs) are assumed static over one OFDMA block,even though they can vary from block to block. We denote hm,i =[hm,i(0), hm,i(1), . . . , hm,i(Lm − 1)]T the discrete-time CIR of the mth userduring the ith block and assume that the channel length Lm keeps constantover an entire frame. For convenience, we also define the mth extendedchannel vector as

h′m,i =[0T

θmhT

m,i 0TL−θm−Lm

]T, (5.1)

where θm is the mth timing error (normalized to the sampling interval Ts)and L = max

mLm + θm. As explained in Chapter 3, the fractional part

of the timing error can be absorbed into the CIR and, accordingly, is notconsidered in the following derivations.

At the BS receiver, the samples of the superimposed uplink signals thatfall within the ith DFT window are given by

ri(k) =M∑

m=1

rm,i(k) + wi(k), 0 ≤ k ≤ N − 1 (5.2)

in which M is the number of active terminals, wi(k) represents Gaussiannoise with zero-mean and power σ2

w and, finally, rm,i(k) is the signal from


Joint Synchronization, Channel Estimation and Data Detection 137

the mth user. Apart from an irrelevant phase shift that can be incorporatedas part of the channel response, from (3.70) we have

rm,i(k) =1√N

ej2πkεm,i/N∑

n∈Im

H ′m,i(n)cm,i(n) ej2πnk/N , 0 ≤ k ≤ N − 1

(5.3)where εm,i is the CFO of the mth user (possibly varying from block toblock), cm,i(n) are uncoded information symbols and H ′

m,i(n) denotesthe mth channel frequency response over the nth subcarrier, which reads

H ′m,i(n) =

L−1∑

`=0

h′m,i(`) e−j2πn`/N , 0 ≤ n ≤ N − 1. (5.4)

Without loss of generality, in the ensuing discussion we concen-trate on the ith received block and omit the time index i for nota-tional simplicity. Then, collecting the received samples into a vectorr = [r(0), r(1), . . . , r(N − 1)]T , after substituting Eqs. (5.3) and (5.4) intoEq. (5.2) we obtain

r =M∑

m=1

Γ(εm)F HD(cm)Uh′m + w, (5.5)

where

• Γ(εm) = diag1, ej2πεm/N , . . . , ej2π(N−1)εm/N

;

• F is the N -point DFT matrix with entries

[F ]p,q =1√N

exp (−j2πpq/N) , (5.6)

for 0 ≤ p, q ≤ N − 1;• cm is an N -dimensional vector with entries cm(n) for n ∈ Im and zero

otherwise;• D(cm) is a diagonal matrix with cm on its main diagonal;• U is an N × L matrix with elements [U ]p,q = exp (−j2πpq/N) for

0 ≤ p ≤ N − 1 and 0 ≤ q ≤ L − 1. In practice, the columns of U arescaled versions of the first L columns of F ;

• w is circularly symmetric white Gaussian noise with zero-mean andcovariance matrix σ2

wIN .


Since timing errors θm do not explicitly appear in the signal model Eq. (5.5),timing estimation is not strictly necessary in the considered system. Hence,



we only investigate the joint estimation of ε = [ε1, ε2, . . . , εM ]T , h′ =[h′

T

1 , h′T

2 , . . . , h′T

M ]T and c =[cT1 , cT

2 , . . . , cTM

]T based on received vectorr. In doing so, we follow an ML approach. Recalling that the entries of w

are independent Gaussian random variables with zero-mean and varianceσ2

w, the log-likelihood function for the unknown parameters ε, h′ and c

takes the form

Λ(ε, h′, c) = −N ln(πσ2

w

)− 1σ2

w

∥∥∥∥∥r −M∑

m=1

Γ(εm)F HD(cm)Uh′m

∥∥∥∥∥

2

, (5.7)

where the notation λ is used to indicate a trial value of an unknown pa-rameter λ.

The joint ML estimates of ε, h′ and c are found by searching for themaximum of Λ(ε, h′, c) with respect to ε, h′ and c. Unfortunately, thisoperation requires an exhaustive search over the multidimensional spacespanned by ε, h′ and c, which is prohibitively complex for practical imple-mentation. To circumvent this obstacle, we consider the iterative schemeproposed in [126] and depicted in Fig. 5.1. As is seen, a SAGE-based pro-cessor [45] is first used to extract the contribution of each user, say rp

(p = 1, 2, . . . , M), from the received vector r. Each rp is then exploited tojointly estimate εp, h′p and cp following an ECM approach [94].

r1

r2

rM(j)

(j)

(j)

r

ECM-BasedEstimator

Selector

p(j)

rData

Detection

ChannelEstimation

CFOEstimation

p(j-1)

p

(j-1)cp

(j-1)

cp(j)

cp(j)

p

(j)

p(j)

p(j)

p

(j)

SAGE-BasedSignal

Decompositionh'

ε

h'

h'

ε

ε

Fig. 5.1 Block diagram of the EM-based iterative receiver.



5.1.2.1 SAGE-based signal decomposition

In a variety of ML problems, direct maximization of the likelihood functionis analytically challenging. In such a case, the EM algorithm proves to beeffective as it achieves the same final result with a comparatively simpleriterative procedure. In the EM formulation, the observed measurementsare replaced with some complete data from which original measurementsare obtained through a many-to-one mapping [94]. At each iteration, theEM algorithm calculates the expectation of the log-likelihood function ofthe complete data set (E-step), which is then maximized with respect tothe unknown parameters (M-step). The process is terminated as soon asno significant changes are observed in the estimated parameters.

As mentioned in Chapter 3, the SAGE algorithm improves upon EM inthat it has a faster convergence rate. The reason is that maximization inthe EM algorithm is simultaneously performed with respect to all unknownparameters, which results in a slow process requiring searches over a spacewith many dimensions. In contrast, the maximization in the SAGE isperformed by updating a smaller group of parameters at a time. The SAGEalgorithm was first proposed in [45] and provides a practical solution toparameter estimation from superimposed signals [43]. In particular, it isnow exploited to decompose the maximization of Λ(ε, h′, c) in Eq. (5.7)into M simpler maximization problems.

For this purpose, we view the received vector r as the observed dataand take rm;m = 1, 2, . . . ,M as the complete data, where rm is thecontribution of the mth user to r in form of

rm = Γ(εm)F HD(cm)Uh′m + wm, m = 1, 2, . . . ,M. (5.8)

and wm (m = 1, 2, . . . ,M) are circularly symmetric and statistically inde-pendent Gaussian vectors satisfying the identity w =

∑Mm=1 wm [43].

The SAGE algorithm is applied in such a way that the parameters ofa single user are updated at a time. This leads to a procedure consistingof iterations and cycles, where M cycles make an iteration and each cycleupdates the parameters of a given user. To see how this comes about, we

call ε(j)m , h′

(j)

m and c(j)m estimates of εm, h′m and cm after the j th iteration,

respectively. Given initial estimates ε(0)m , h′

(0)

m and c(0)m , we compute

z(0)m = Γ(ε(0)

m )F HD(c(0)m )Uh′

(0)

m , m = 1, 2, . . . , M. (5.9)

Then, during the pth cycle of the j th iteration (with p = 1, 2, . . . , M),the SAGE proceeds as follows [45].



E-Step:

Compute

r(j)p = r −

p−1∑m=1

z(j)m −

M∑m=p+1

z(j−1)m (5.10)

where∑u

l is zero if u < l.

M-Step:

Compute[ε(j)

p , h′(j)

p , c(j)p

]= argmin

εp,hp,cp

∥∥∥r(j)p − Γ(εp)F HD(cp)Uh′p

∥∥∥2

, (5.11)

and then use updated parameters to obtain the following vector

z(j)p = Γ(ε(j)

p )F HD(c(j)p )Uh′

(j)

p . (5.12)

We see from Eq. (5.11) that the SAGE algorithm splits the maximizationof Λ(ε, h′, c) in Eq. (5.7) into a series of M simpler optimization problems.However, the multidimensional minimization in Eq. (5.11) still remains aformidable task. An iterative solution to this problem is presented in thenext subsection by resorting to the ECM algorithm.

5.1.2.2 ECM-based iterative estimator

Substituting Eq. (5.5) into Eq. (5.10) yields

r(j)p = Γ(εp)F HD(cp)Uh′p + η(j)

p , (5.13)

where

η(j)p = w +

p−1∑m=1

[zm − z(j)m ] +

M∑m=p+1

[zm − z(j−1)m ], (5.14)

and zm = Γ(εm)F HD(cm)Uh′m is the signal received from the mth user.Note that η

(j)p is a disturbance term that accounts for thermal noise and

residual MAI after the j th SAGE iteration, and is linearly related to thedata symbols of all interfering users. Then, assuming that these symbols areindependent and identically distributed with zero-mean, it follows from thecentral limit theorem that the entries of η

(j)p are nearly Gaussian distributed

with zero-mean and some variance σ2η(j). Under this assumption, it turns



out that the minimization problem in Eq. (5.11) is equivalent to the MLestimation of εp , h′p and cp starting from the observation of r

(j)p .

The ECM algorithm offers a practical solution to this problem. The onlydifference between this technique and the conventional EM algorithm is thatthe maximization step in the ECM algorithm is divided into several stages,where at each stage only one parameter is updated while all the othersare kept constant at their most updated values. This makes the ECMalgorithm suitable for multidimensional ML estimation problems, wherethe likelihood function has to be optimized over several parameters [94].

In the following, the ECM algorithm is employed to solve the optimiza-tion problem stated in Eq. (5.11). In doing so, we view r

(j)p as the ob-

served data and [ r(j)Tp h′Tp ]T as the complete set of data. Also, we denote

ξpdef= [cT

p εp ]T the parameters to be estimated and ξ(j,u)p = [ c(j,u)T

p ε(j,u)p ]T

the estimate of ξp at the uth ECM and j th SAGE iterations. Then, afterinitializing c

(j,0)p = c

(j−1)p and ε

(j,0)p = ε

(j−1)p , the ECM algorithm alternates

between an E-step and an M-step as follows.

E-Step:

We define

Υ(ξp

∣∣∣ξ(j,u)p

)= Eh′p

ln

[p

(r(j)

p

∣∣∣h′p, ξp

)]p

(r(j)

p

∣∣∣h′p, ξ(j,u)p

), (5.15)

where p(r

(j)p

∣∣∣h′p, ξp

)and p

(r

(j)p


)are conditional probability

density functions (pdf), Eh′p · denotes the statistical expectation overthe pdf of h′p and ξp = [ cT

p εp ]T is a trial value of ξp.

Function Υ defined in Eq. (5.15) can be rewritten as

Υ(ξp

∣∣∣ξ(j,u)p

)=

∫

Ω

ln[p(r(j)

p

∣∣∣h′p, ξp )]· p

(r(j)

p


)p(h′p) dh′p,

(5.16)where p(h′p) is the a-priori pdf of h′p.

To proceed further, we make the following assumptions:

(1) h′p is a circularly symmetric Gaussian vector with zero-mean (Rayleighfading) and covariance matrix Cp = Eh′ph′Hp ;

(2) the disturbance η(j)p in Eq. (5.13) is nearly Gaussian distributed with

zero-mean and covariance matrix σ2η(j)IN .



Thus, bearing in mind Eq. (5.13), we may write

p(h′p) =1

πL det(Cp)exp−h′Hp C−1

p h′p, (5.17)

p(r(j)

p


)≈ 1

[πσ2η(j)]N

exp− 1

σ2η(j)

∥∥∥r(j)p − z(j,u)

p

∥∥∥2

, (5.18)

ln[p(r(j)

p

∣∣∣h′p, ξp )]≈ −N ln[πσ2

η(j)]− 1σ2

η(j)

∥∥∥r(j)p − zp

∥∥∥2

, (5.19)

with

z(j,u)p = Γ(ε(j,u)

p )F HD(c(j,u)p )Uh′p, (5.20)

and

zp = Γ(εp)F HD(cp)Uh′p. (5.21)

Substituting Eqs. (5.17)-(5.19) into Eq. (5.16) and skipping additive andmultiplicative terms independent of ξp, we may replace Υ

(ξp

∣∣∣ξ(j,u)p

)with

the equivalent function

Φ(ξp

∣∣∣ξ(j,u)p

)= −

∥∥∥r(j)p − Γ(εp)F HD(cp)Uh′p,MMSE(ξ(j,u)

p )∥∥∥

2

−

σ2η(j) · trD(cp)U [P (c(j,u)

p )]−1

UHDH(cp), (5.22)

where

h′p,MMSE(ξ(j,u)p ) = [P (c(j,u)

p )]−1

UHDH(c(j,u)p )FΓH(ε(j,u)

p )r(j)p (5.23)

is the MMSE estimate of h′p obtained with ξp = ξ(j,u)p , while

P (c(j,u)p ) = UHEp(c(j,u)

p )U + σ2η(j)C−1

p (5.24)

with

Ep(c(j,u)p ) = diag

∣∣∣c(j,u)p (n)

∣∣∣2

; n = 0, 1, . . . , N − 1

. (5.25)

We see from Eqs. (5.22)-(5.24) that evaluating Φ(ξp

∣∣∣ξ(j,u)p

)requires

knowledge of Cp and σ2η(j). Thus, suitable schemes must be devised to

estimate these parameters. A practical solution to this problem is foundby assuming high SNR values. In this case we expect that σ2

η(j) becomes

vanishingly small and Φ(ξp

∣∣∣ξ(j,u)p

)can reasonably be approximated by

Φ(ξp

∣∣∣ξ(j,u)p

)= −

∥∥∥r(j)p − Γ(εp)F HD(cp)Uh′p,LS(ξ(j,u)

p )∥∥∥

2

, (5.26)

where h′p,LS(ξ(j,u)p ) is the least-squares (LS) estimate of h′p and takes the

form

h′p,LS(ξ(j,u)p ) = [UHEp(c(j,u)

p )U ]−1UHDH(c(j,u)p )FΓH(ε(j,u)

p )r(j)p . (5.27)



In the sequel, function Φ(ξp

∣∣∣ξ(j,u)p

)is used in place of Φ

(ξp

∣∣∣ξ(j,u)p

).

Although this approach may entail some performance penalty at low andmedium SNRs, it has the advantage of being practically implementable,while computing Φ

(ξp

∣∣∣ξ(j,u)p

)seems hardly viable in practice.

M-Step:

The M-step aims at maximizing the right-hand-side of Eq. (5.26) with re-spect to ξp. This goal is achieved using a two-stage procedure. Followingthe notation of [94], we denote ξ

(j,u+g/2)p the estimate of ξp at the gth

stage of the uth ECM iteration, where g = 1, 2. Then, the maximum ofΦ

(ξp

∣∣∣ξ(j,u)p

)is found as follows.

• Step 1:

ξ(j,u+1/2)p =

[(c(j,u)

p )T ε(j,u+1)p

]T

, (5.28)

where

ε(j,u+1)p = arg max

εp

−

∥∥∥r(j)p − Γ(εp)F HD(c(j,u)

p )Uh′p,LS(ξ(j,u)p )

∥∥∥2

.

(5.29)

Note that the quantity∥∥∥Γ(εp)F HD(c(j,u)

p )Uh′p,LS(ξ(j,u)p )

∥∥∥2

is indepen-

dent of εp since ΓH(εp)Γ(εp) = IN . Thus, Eq. (5.29) can equivalentlybe replaced by

ε(j,u+1)p = arg max

εp

<e

[r(j)H

p Γ(εp)F HD(c(j,u)p )Uh′p,LS(ξ(j,u)

p )]

.

(5.30)• Step 2:

ξ(j,u+1)p =

[(c(j,u+1)

p )T ε(j,u+1)p

]T

, (5.31)

where

c(j,u+1)p = argmin

cp

N−1∑n=0

∣∣∣R(j)p (n, ε(j,u+1)

p )− cp(n)H ′(j,u)p,LS (n)

∣∣∣2

,

(5.32)with R(j)

p (n, ε(j,u+1)p ); n = 0, 1, . . . , N − 1 and H ′(j,u)

p,LS (n);n =

0, 1, . . . , N − 1 being the N -point DFTs of ΓH(ε(j,u+1)p )r(j)

p andh′p,LS(ξ(j,u)

p ), respectively.



An approximation of the CFO estimate in Eq. (5.30) can be obtainedin closed-form after replacing Γ(εp) with its Taylor series expansiontruncated to the second order term and using ε

(j,u)p as starting point,

i.e.,

Γ (εp) ≈ Γ(ε(j,u)p ) + j(εp − ε(j,u)

p )Γ′(ε(j,u)p )− 1

2(εp − ε(j,u)

p )2Γ′′(ε(j,u)p ),

(5.33)where Γ′(ε(j,u)

p ) = ΨΓ(ε(j,u)p ), Γ′′(ε(j,u)

p ) = Ψ2Γ(ε(j,u)p ) and Ψ =

(2π/N) · diag 0, 1, . . . , N − 1. Substituting Eq. (5.33) into Eq. (5.30)and setting the derivative with respect to εp to zero yields

ε(j,u+1)p = ε(j,u)

p +=m

r

(j)Hp Γ′(ε(j,u)

p )F HD(c(j,u)p )Uh′p,LS(ξ(j,u)

p )

<e

r(j)Hp Γ′′(ε(j,u)

p )F HD(c(j,u)p )Uh′p,LS(ξ(j,u)

p ) .

(5.34)After a specified number NU of iterations, we terminate the ECM pro-cess and replace Eq. (5.11) with

[ε(j)p , h′

(j)

p , c(j)p ] = [ε(j,NU )

p , h′p,LS(ξ(j,NU )p ), c(j,NU )

p ]. (5.35)

In the sequel, the iterative scheme relying on Eqs. (5.27), (5.32) and(5.34) is referred to as the EM-based receiver (EMBR).

5.1.3 Practical adjustments

The following guidelines may be helpful for a practical implementation ofEMBR:

(1) It is well known that a good initialization is essential for EM-typealgorithms. Hence, the problem arises of how to obtain initial estimates

ε(0)m , h′

(0)

m and c(0)m to start the SAGE procedure. If ε and h′ vary slowly

in time, frequency and channel estimates obtained in a given block canbe used to initialize the iterative process in the next block. Estimatesfor the first data block may be obtained in a data-aided fashion byexploiting a training sequence placed at the beginning of the uplinkframe [124,127].The initial CFO estimates ε(0) = [ε(0)

1 , ε(0)2 , . . . , ε

(0)M ]T are next exploited

to accomplish frequency correction using one of the methods discussedin [12,18,55,158]. This operation aims at restoring orthogonality amongsubcarriers and produces the following N -dimensional vectors (one for



each user)ψm = D (cm) Uh′m + γm, m = 1, 2, . . . ,M. (5.36)

where γm is a disturbance term that accounts for thermal noise andresidual MAI caused by imperfect separation of the users’ signals. Fi-nally, initial data decisions are obtained as in conventional OFDMtransmission, i.e.,

c(0)m = arg min

cm

N−1∑n=0

∣∣∣ψm(n)− cm(n)H ′(0)m (n)

∣∣∣2

, (5.37)

where ψm(n) is the nth entry of ψm and H ′(0)m (n); n = 0, 1, . . . , N−1

is the N -point DFT of h′(0)

m .In applications characterized by high user mobility, initializing theSAGE iterations with channel estimates from the previous block mayresult in poor performance due to fast fading. In these circumstances,a possible solution is to insert scattered pilots in each OFDMA block

and compute h′(0)

m through conventional pilot-aided estimation tech-niques [101]. Albeit robust against rapidly varying channels, this ap-proach inevitably results into a reduction of the overall data throughputdue to the increased overhead.

(2) For PSK transmissions, matrix Ep(c(j,u)p ) defined in Eq. (5.25) becomes

independent of c(j,u)p since

∣∣∣c(j,u)p

∣∣∣2

is either unitary or zero dependingon whether the nth subcarrier is assigned to the mth user or not. Insuch a case, evaluating h′p,LS(ξ(j,u)

p ) in Eq. (5.27) does not require any

on-line matrix inversion since [UHEp(c(j,u)p )U ]−1 can be pre-computed

and stored in the receiver. A further simplification is possible if thesubcarriers of the pth user are uniformly distributed over the signalbandwidth with separation interval N/P , where P is the number ofsubcarriers in each subchannel. In this hypothesis, UHEp(c

(j,u)p )U

reduces to P · IL and Eq. (5.27)becomes

h′p,LS(ξ(j,u)p ) =

1P

UHDH(c(j,u)p )FΓH(ε(j,u)

p )r(j)p . (5.38)

(3) Intuitively speaking, the SAGE procedure should be stopped when nosignificant variations are observed in the log-likelihood function, i.e.,

Λ(ε(j), h′(j), c(j))− Λ(ε(j−1), h′(j−1)

, c(j−1)) < λth,

for some threshold λth. A simpler stopping criterion is to terminatethe SAGE procedure after a preassigned number of iterations.



5.1.4 Performance assessment

The performance of EMBR has been assessed by computer simulation inan OFDMA scenario inspired by the IEEE 802.16 standard for WirelessMetropolitan Area Networks [177]. Without loss of generality, we onlyprovide results for user #1.

The simulated system has N = 128 subcarriers and a signal bandwidthof 1.429 MHz, which corresponds to a sampling period of Ts = 0.7 µs.The useful part of each OFDMA block has length T = NTs = 89.6 µs

while the subcarrier spacing is 1/T = 11.16 kHz. We consider an inter-leaved CAS where each user is provided with a set of P = 32 subcarriersuniformly spaced over the signal bandwidth. In this way, the maximumnumber of active users in each OFDMA block is R = 4. We assume afully-loaded system with M = 4 active terminals and let the users’ CFOsbe ε = ρ · [1,−1, 1,−1]T , where the attenuation factor ρ is modeled as adeterministic parameter belonging to interval [0, 0.5] [55]. Information bitsare mapped onto uncoded QPSK symbols using a Gray map. The channelresponses hm,i have length L = 5 while the timing errors θm are indepen-dently generated at the beginning of each frame and take values in the set0, 1, 2, 3. A CP of length Ng = 8 is used to avoid interblock interference(IBI). In this way, the duration of the extended OFDMA block (includingthe CP) is TB = (N + Ng)Ts = 95.2 µs.

The channel taps hm,i(`) are modeled as statistically independentnarrow-band Gaussian processes with zero-mean and autocorrelation func-tion

Ehm,i(`)h∗m,i+n(`)

= σ2

` J0 (2πnfDTB) , ` = 0, 1, 2, 3, 4 (5.39)

where fD is the Doppler bandwidth, J0(x) is the zero-order Bessel functionof the first kind and

σ2` = E|hm,i(`)|2 = βm · exp(−`). (5.40)

In Eq. (5.40), β1 is chosen such that the signal power of user #1 isnormalized to unity, i.e., E‖h1‖2 = 1, while parameters βm (m ≥ 2)affect the signal-to-interference ratio. The Doppler bandwidth is relatedto the carrier frequency fc and mobile velocity v by fD = fcv/c. Lettingfc = 2 GHz and v = 60 km/h, we obtain fD ≈ 110 Hz, corresponding to1% of subcarrier spacing.

The uplink frame is composed by 10 OFDMA blocks. Frequency andchannel estimates obtained in a given block are used to initialize the it-erative process in the next block, while initialization for the first block is



achieved using a training sequence placed at the beginning of the frame[127]. For each block, initial CFO estimates ε(0) are employed to restoreorthogonality among subcarriers by resorting to the scheme proposed byCao, Tureli, Yao and Honan (CTYH) in [12], where a linear transformationis applied to the DFT output to obtain vectors ψm (m = 1, 2, . . . ,M) in

Eq. (5.36). The latter are exploited to get initial channel estimates h′(0)

m .For this task we employ the pilot-aided estimator described in [101] andassume that 8 pilots are uniformly placed in each subchannel. Initial datadecisions are eventually obtained according to Eq. (5.37).

The number NU of ECM iterations is set to 1 while the number Ni ofSAGE iterations is varied throughout simulations to assess its impact onthe system performance.

Performance with ideal frequency and channel information

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

ρ

BE

R

IdealEMBR (N

i=1)

EMBR (Ni=5)

HL (Ni=5)

CTYH

Fig. 5.2 BER performance vs. ρ for uncoded QPSK with Eb/N0 = 20 dB and perfectknowledge of the CFOs and channel responses.



Figure 5.2 shows the BER performance as a function of ρ in case ofperfect knowledge of CFOs and channel responses, i.e., εm = εm andh′m,LS = h′m for m = 1, 2, 3, 4. This scenario was also considered in[12,18,55,158] and is used here to assess the ability of the system to mitigateICI and MAI produced by frequency offsets. Users have equal power withEb/N0 = 20 dB. Comparisons are made with both CTYH [12] and the iter-ative scheme proposed by Huang and Letaief (HL) in [55], where frequencycorrection is accomplished at the output of the receive DFT by means ofinterference cancellation techniques and windowing functions. Five itera-tions are employed with HL while the number of SAGE iterations is eitherNi = 1 or 5.

The curve labeled “ideal” is obtained by assuming that all CFOs haveperfectly been corrected at the mobile terminals (MTs), i.e., εm = 0 form = 1, 2, 3, 4. This provides a benchmark for the BER performance sincein this case users’ signals at the DFT output are orthogonal and no inter-ference is thus present. As expected, the BER of all considered schemesdegrades with ρ due to the increased amount of ICI and MAI. Interestingly,EMBR provides similar results with either Ni = 1 or Ni = 5, meaning thatconvergence is achieved after one single iteration. Also, this scheme largelyoutperforms the other methods. A possible explanation is that CTYH oper-ates similarly to a linear multiuser detector where interference is mitigatedat the price of non-negligible noise enhancement. As to the HL scheme, thewindowing functions applied to the DFT output may lead to a significantloss of signal energy in the presence of relatively large CFOs.

Performance with estimated frequency offsets and channel

responses

We now assess the performance of EMBR when the frequency and channelestimation tasks are coupled with the decision making process. Figure 5.3shows the BER of the considered schemes as a function of Eb/N0 withρ = 0.3. Users have equal power and the number of iterations is Ni = 5 withboth EMBR and HL. For comparison, we also illustrate the performanceof the ideal system with perfect frequency and channel information, whereall CFOs have been corrected at the MTs. Again, the best performance isachieved by EMBR. In particular, at an error rate of 10−2, the gain overCTYH is approximately 4 dB while a loss of 3 dB is incurred with respectto the ideal system. As for HL, it performs poorly and exhibits an errorfloor at high SNRs.



0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

IdealCTYHEMBR (N

i = 5)

HL (Ni = 5)

Fig. 5.3 BER performance vs. Eb/N0 for uncoded QPSK and ρ = 0.3.

Resistance to near-far effect

In practical systems, power control is employed to mitigate the near-farproblem arising from the different path losses incurred by uplink signals.However, power control cannot be assumed when a new user is entering thesystem as its power level is still to be measured. Therefore, it is of interestto assess the performance of the considered schemes in the presence of astrong interferer. For this purpose, we consider a scenario in which thepower of user #2 is larger than that of the others by a factor α ≥ 1. Thiscondition is obtained setting β2 =

√α · β1 in Eq. (5.40), while keeping

βm = β1 for m = 3, 4. Simulation results illustrating the BER of user #1are shown in Fig. 5.4 as a function of α (expressed in dB) for ρ = 0.3 andEb/N0 = 20 dB. As expected, the system performance degrades with α.In particular, the BER of EMBR and CTYH increases by a factor of twowhen α passes from 0 to 5 dB, while larger degradations occur with HL.



0 1 2 3 4 510

−3

10−2

10−1

100

α (dB)

BE

R

IdealCTYHEMBR (N

i=5)

HL (Ni=5)

Fig. 5.4 BER performance in the presence of a strong interferer for uncoded QPSKwith Eb/N0 = 20 dB and ρ = 0.3.

5.2 Trellis-coded OFDMA uplink

The receiver structures discussed in the previous subsection are specificallydesigned for uncoded transmissions. On the other hand, we know thatchannel coding is a fundamental part of any multicarrier system as it pro-vides a natural way for exploiting the frequency diversity offered by themultipath channel. For this reason, it is of practical interest to extend theEMBR to coded systems.

5.2.1 Signal model for coded transmissions

Figure 5.5 illustrates the basic block diagram of the mth MT transmitterin a coded OFDMA uplink. Here, a block of binary information data am

is trellis-encoded into a vector bm of coded bits. The latter are then fedto a block interleaver, which helps to break up error bursts. After dividingthe interleaved bits xm into adjacent segments of length ϑ, each segment



Convolutional encoder

Block interleaver

Mapper OFDM modulator

a m b m x m c To the channel

m

Fig. 5.5 Block diagram of the mth MT transmitter in a coded OFDMA system.

is mapped onto a modulation symbol taken from a constellation with 2ϑ

points. This produces a vector cm of N symbols which is finally passed tothe OFDM modulator and launched over the channel. At the BS receiver,the observation vector r is still expressed as in Eq. (5.5), where the entriesof cm are now coded symbols obtained as illustrated in Fig. 5.5.

Hard-decisionDecoder

cp(j)

r Hard-decisionEM-baseddetector

Re-encodingand

symbol mapping

ap

(j)

hp(j)

p(j)ε

c p

(j)

Fig. 5.6 Block diagram of an EM-based receiver employing a hard-decoding strategy.

One possible way for applying the EMBR to a coded OFDMA systemis depicted in Fig. 5.6. As is seen, at each iteration the EM-based de-tector provides decisions about the coded symbols of all users, which arethen passed to the hard-decoding unit. The retrieved information bits arere-encoded and re-mapped before being returned to the EM detector forthe next iteration. This approach is relatively simple, but cannot provideoptimum performance as it does not exploit any information regarding thelikelihood of the detected symbols (also referred to as soft information).Inspired by the turbo decoding principle, a number of turbo processingtechniques have recently been developed to improve the channel estima-tion [116] or interference suppression tasks [47] by taking advantage of thesoft information associated with the decoded data. In the ensuing discus-sion, the turbo principle is applied to a coded OFDMA uplink. In particu-lar, we exploit soft-decision feedback from a maximum a posteriori (MAP)



decoder to jointly perform frequency synchronization, channel estimationand interference cancellation.


with coded transmissions

Hard-detected symbols

Lessreliable

Morereliable

Fig. 5.7 Hard-decision detection in a QPSK transmission.

Figure 5.7 shows the classical concept of hard data detection of QPSKsymbols. The noisy points in the I/Q diagram represent the output of thechannel equalizer and are classified into one out of four possible constel-lation symbols. Although some of these points may be more reliable thanothers, the hard-decision process masks out this reliability since points lyingin the same decision region are treated exactly in the same way, regardlessof their distances from the corresponding constellation symbol. In codedsystems, reliability information can be exploited by representing the ten-tative decoded symbols through their statistical expectation. In this way

June

15,2007

10:2

World

Scien

tific

Book

-9in

x6in

book

Join

tSynch

roniza

tion,C

hannel

Estim

atio

nand

Data

Detectio

n153

DFTBlock

De-Interleaver

MAP

Decoder

CFO & Channel

Estimation

Block

Interleaver

a

pr( j)

Soft

Symbol Estimator

pEc (n)

CFO

Compensation

Data

Detector

LLR(R |b )p

( j)

p

d

cp( j+1)

h'p

(j+1)

LLR(a |R )p

( j)

LLR(b |R )( j)d

p

p

Rp

( j)

p(j+1)

(n) (n) (n)LLR(R |x )p

( j) d

p(n) (n)

p p(n)LLR(x |R )( j)

d

p p(n)

Fig. 5.8 Block diagram of the ECM-based MAP decoder.



the system performance is greatly improved as compared to hard-decisiondecoding.

We follow the same approach employed with uncoded transmissions andconsider an iterative receiver structure in which a SAGE-based processor isfirst used to extract the contribution of each user, say rp (p = 1, 2, . . . , M),from the received vector r. Each rp is then exploited to estimate εp, h′p andcp in a joint fashion according to the ECM principle. The overall receiverarchitecture is depicted in Fig. 5.8. The main difference with respect to theuncoded case is that now the receiver can effectively exploit informationabout the reliability of the detected symbols.

The SAGE algorithm is applied in the same way as in uncoded sys-tems. In particular, during the pth cycle of the j th iteration (withp = 1, 2, . . . , M), the contribution of the pth user to the received vectorr is estimated as

r(j)p = r −

p−1∑m=1

z(j)m −

M∑m=p+1

z(j−1)m , (5.41)

where z(j)m is given in Eq. (5.12) and represents an estimate of the signal

zm = Γ(εm)F HD(cm)Uh′m received from the mth user.Following the same steps outlined in Sec. 5.1.2.2, we substitute Eq. (5.5)

into Eq. (5.41) and obtain

r(j)p = Γ(εp)F HD(cp)Uh′p + η(j)

p , (5.42)

where η(j)p is defined in Eq. (5.14).

The ML estimates of εp, h′p and cp are derived from r(j)p using the

ECM algorithm. After initializing c(j,0)p = c

(j−1)p and ε

(j,0)p = ε

(j−1)p , the

uth iteration of the ECM-based MAP decoder proceeds in the following way[116]. The estimated CFO ε

(j,u)p is first used to compute the N -dimensional

vector

R(j)p = FΓH(ε(j,u)

p )r(j)p , (5.43)

with entries R(j)p (n) for n = 0, 1, . . . , N − 1.

Next, we callxd

p(n); d = 0, 1, . . . , ϑ− 1

the nth segment of ϑ inter-

leaved bits that are mapped onto cp(n). Recalling that η(j)p is nearly Gaus-

sian distributed, the log-likelihood ratio (LLR) of R(j)p (n) conditioned on

xdp(n) is given by



LLR(R(j)

p (n)∣∣xd

p(n))

= logPr

(R

(j)p (n)

∣∣xdp(n) = +1

)

Pr(R

(j)p (n)

∣∣xdp(n) = −1

)

= log

∑

cp(n)∈Sd+1

exp− |R(j)

p (n)−H′(j,u)p,LS (n) cp(n)|2σ2

η(j)

∑

cp(n)∈Sd−1

exp− |R(j)

p (n)−H′(j,u)p,LS (n) cp(n)|2σ2

η(j)

,

(5.44)

where Sdα (with α = ±1) is the set of constellation symbols for which

xd = α, while H′(j,u)p,LS (n) represents the nth entry of H ′

p,LS(ξ(j,u)p ). The

latter is the LS estimate of the channel frequency response for a givenξ

(j,u)p = [ c(j,u)T

p ε(j,u)p ]T , and reads

H ′p,LS(ξ(j,u)

p ) = Uh′p,LS(ξ(j,u)p ), (5.45)

where h′p,LS(ξ(j,u)p ) is defined in Eq. (5.27).

In an attempt of reducing the computational complexity, one can usethe max-log approximation in Eq. (5.44) to obtain [116]

LLR(R(j)

p (n)∣∣xd

p(n))≈ max

cp(n)∈Sd+1

−|R(j)

p (n)− H′(j,u)p,LS cp(n)|2

− maxcp(n)∈Sd

−1

−|R(j)


,

(5.46)

where the quantity σ2η(j) has been dropped since the frequent re-

normalization process during MAP decoding removes in practice the effectof any common factors.

The sequence

LLR(R

(j)p (n)

∣∣xdp(n)

)at the output of the data de-

tector is then de-interleaved to yield

LLR(R

(j)p (n)

∣∣bdp(n

). These

quantities are employed by the MAP decoder to generate the sequenceLLR

(bdp(n)

∣∣∣R(j)p

)and LLR

(ap

∣∣∣R(j)p

)using the BCJR algorithm [5].

Readers are referred to [79] and references therein for a formal treatmentof the BCJR algorithm. Finally, the stream

LLR

(bdp(n)

∣∣∣R(j)p

)is inter-

leaved and employed to evaluate the expected values of the coded channelsymbols cp.



Letting c(j,u+1)p (n) = E cp(n) and assuming for simplicity a QPSK

constellation (d = 0, 1), it can be shown that [116]

c(j,u+1)p (n) =

1√2

[eLLR(x0

p(n)|R(j)p ) − 1

eLLR

(x0

p(n)∣∣∣R(j)

p

)+ 1

+ jeLLR(x1

p(n)|R(j)p ) − 1

eLLR

(x1

p(n)∣∣∣R(j)

p

)+ 1

].

(5.47)

The detected symbols c(j,u+1)p (n) are grouped to form a vector c

(j,u+1)p

defined asc(j,u+1)

pdef= [c(j,u+1)

p (0), c(j,u+1)p (1), . . . , c(j,u+1)

p (N − 1)]T , (5.48)which is next employed to update the CFO estimate according to Eq. (5.34).

Finally, ε(j,u+1)p and c

(j,u+1)p are substituted into Eq. (5.45) to update

the channel estimates. After NU iterations, we terminate the ECM processand update the SAGE processor with

[ε(j)p , h′

(j)

p , c(j)p ] = [ε(j,NU )

p , h′p,LS(ξ(j,NU )p ), c(j,NU )

p ]. (5.49)In summary, during the pth cycle of the j th iteration (with p =

1, 2, . . . , M), the iterative algorithm proceeds as follows.

E-Step:

Compute r(j)p according to Eq. (5.41);

M-Step:

• Update H ′p,LS(ξ(j,u)

p ) based on Eq. (5.45) and compute

LLR(R(j)

p (n)∣∣xd

p(n))≈ max

cp(n)∈Sd+1

−|R(j)


− maxcp(n)∈Sd

−1

−|R(j)


.

(5.50)

• Generate

LLR(bdp(n)

∣∣∣R(j)p

)and LLR

(ap

∣∣∣R(j)p

)by exploiting

LLR(R

(j)p (n)

∣∣bdp(n

)using the BCJR algorithm;

• Update c(j,u+1)p and the estimation parameters based on Eqs. (5.47)

and (5.49), respectively;• Finally, use updated parameters to obtain the following vector

z(j)p = Γ(ε(j)

p )F HD(c(j)p )Uh′

(j)

p . (5.51)



5.2.3 Performance assessment

The performance of EMBR when applied to a coded OFDMA uplink isassessed by computer simulations under the same operating conditions ofFig. 5.3. The only difference is that the information bits are now encodedby a rate-1/2 convolutional encoder with generator polynomials (5, 7) (inoctal) and an 8×8 block interleaver is employed to scramble the coded bitswithin the OFDM block. The interleaved bits are then mapped onto QPSKsymbols using a Gray map. The number NU of ECM iterations is set to 3while the number of SAGE iterations is Ni = 5. The CTYH scheme is usedto initialize the EMBR. Again, results are only provided for user #1.

0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

Cod

ed B

ER

IdealCTYHEMBR (N

i=5)

HDEMBR (Ni=5)

Fig. 5.9 BER performance vs. Eb/N0 for a coded QPSK transmission.

Figure 5.9 illustrates BER results as a function of Eb/N0 in case ofusers with equal average power and ρ = 0.3. The curve labeled “ideal”corresponds to perfect knowledge of CFOs and channel responses and pro-vides a benchmark for the BER performance. At an error rate of 10−3,the gain of EMBR with respect to CTYH is nearly 6 dB after five itera-



tions, while a loss of 4 dB is incurred with respect to the ideal system. Forcomparison, we also show the performance of a hard-decision EM-basedreceiver (HDEMBR) which operates as illustrated in Fig. 5.6 using a hard-decoding Viterbi processor. As is seen, HDEMBR performs poorly sincehard-decoding does not allow to exploit any reliability information.


Chapter 6

Dynamic Resource Allocation

One attractive feature of multicarrier transmissions is the possibility ofdynamically allocating system resources according to the changing envi-ronmental conditions. Many studies have demonstrated that significantperformance improvement is achieved in single-user OFDM systems if trans-mission power and data rate are properly adjusted over each subcarrier totake advantage of the channel frequency selectivity. This idea is usually re-ferred to as adaptive modulation while the set of algorithms and protocolsgoverning it is known as link adaptation [13,75].

The goal of any link adaptation algorithm is to ensure that the most ef-ficient set of modulation parameters (or transmission mode) is always usedover varying channel conditions. Different mode selection criteria can beenvisaged depending on whether the system is attempting to maximize theoverall data throughput under a total power constraint or to minimize theoverall transmit power given a fixed throughput. In any case, the adapta-tion algorithm tends to allocate more information bits onto better qualitysubcarriers, i.e., those exhibiting the highest signal-to-noise ratios (SNRs),whereas small-size constellations are normally employed over severely fadedsubcarriers in order to increase their robustness against thermal noise. Insome extreme situations a number of subcarriers may even be left unusedif the corresponding SNR is too poor for reliable data transmission. In therelated literature, the problem of efficiently mapping information bits overthe available carriers is referred to as bit loading.

The concept of link adaptation has also been extended to OFDMAsystems. In this case the base station (BS) not only has the opportunityof optimally allocating power and data rate over different subchannels, butcan also exploit instantaneous channel state information for dynamicallydistributing subcarriers to the active users. The adoption of a dynamic

159



carrier assignment scheme allows a more effective use of the available systemresources, even though it complicates the link adaptation problem to a largeextent as compared to point-to-point communications.

The aim of this chapter is to present the basic concept of link adapta-tion in multicarrier systems. Section I investigates adaptive bit and powerloading in single-user OFDM applications. Here, we revisit the classicalwater-filling power allocation policy and formulate the rate-maximizationand margin-maximization problems. Practical bit loading schemes based ongreedy techniques are illustrated for either uniform or non-uniform powerallocation. We also present the concept of subband adaptation and discusssome signaling schemes enabling exchange of side information between thetransmit and receive ends of an adaptive modulation system.

Section II is devoted to link adaptation in a multiuser OFDM network.After discussing the multiaccess water-filling principle, we extend the rate-maximization and margin-maximization concepts to a typical OFDMAdownlink scenario. As we shall see, in such a case optimum assignmentof system resources results into a multidimensional optimization problemwhich does not lend itself to any practical solution. To overcome this dif-ficulty, we present some suboptimum schemes in which the subcarrier allo-cation and bit loading tasks are performed separately and with affordablecomplexity.

6.1 Resource allocation in single-user OFDM systems

The research on resource allocation in multicarrier systems was fueled bythe success of the asymmetric digital subscriber line (ADSL) service in theearly nineties [1, 8]. This technology employs a Digital Multitone (DMT)modulation for high-speed wireline data transmissions. Due to crosstalkfrom adjacent copper twisted pairs, the ADSL channel is characterized byremarkable frequency-selectivity. The latter can usefully be exploited as asource of diversity by applying suitable link adaptation techniques.

In this Section we review the main concepts behind bit and power load-ing in point-to-point OFDM transmissions. Although originally devised forADSL applications, the investigated methods apply to multicarrier wire-less services as well. The only requirement is that the fading rate is nottoo fast, as dynamic resource allocation is hardly usable in the presence ofrapidly-varying transmission channels.


Dynamic Resource Allocation 161

6.1.1 Classic water-filling principle

We start discussing the water-filling power allocation principle, which allowsone to achieve the theoretical capacity offered by a frequency-selective chan-nel. Capacity is operationally defined as the maximum data rate that thechannel can support with an arbitrarily low error-rate probability. Froman information theoretic perspective, it represents the maximum mutualinformation between the transmitted data symbols and the received signalvector, where maximization is performed over the probability density func-tion (pdf) of the transmitted data [27]. In the ensuing discussion, theseconcepts are applied to an OFDM communication system.

Assuming perfect timing and frequency synchronization, the outputfrom the receive DFT is expressed by

R(n) = H(n)S(n) + W (n), 0 ≤ n ≤ N − 1. (6.1)

where H(n) is the channel frequency response over the nth subcarrier, S(n)the corresponding input symbol with power Pn =E|S(n)|2 and W (n)is white Gaussian noise with zero-mean and variance σ2

w. Inspection ofEq. (6.1) indicates that the OFDM channel can be viewed as a collectionof parallel independent AWGN subchannels, one for each subcarrier.

In a practical system, the transmitted power is normally constrained tosome value Pbudget. Mathematically, this amounts to setting

N−1∑n=0

Pn ≤ Pbudget, (6.2)

with Pn ≥ 0 for n = 0, 1, . . . , N − 1. It is known that among all input vec-tors S = [S(0), S(1), . . . , S(N − 1)]T satisfying the overall power constraintEq. (6.2), the mutual information I(S, R) between S and the observationvector R = [R(0), R(1), . . . , R(N − 1)]T is maximized when the data sym-bols S(n) are statistically independent and Gaussian distributed withzero-mean [105]. In this case we have

I(S, R) =N−1∑n=0

log2

(1 +

Pn |H(n)|2σ2

w

). (6.3)

The channel capacity C is obtained by maximizing the right-hand-side ofEq. (6.3) with respect to P = [P0, P1, . . . , PN−1]

T , i.e.,

C = maxP

N−1∑n=0

log2

(1 +

Pn |H(n)|2σ2

w

). (6.4)



Since the objective function in Eq. (6.4) is convex in the variables Pn,the optimum power allocation under the convex constraints Eq. (6.2) canbe found using Lagrangian methods. For this purpose, we consider theaugmented cost function

J =N−1∑n=0

log2

(1 +

Pn |H(n)|2σ2

w

)+ λ

(Pbudget −

N−1∑n=0

Pn

), (6.5)

where λ is the Lagrangian multiplier. The Kuhn–Tucker (KT) optimalityconditions are given by

KT conditions:

∂J∂Pn

= 0 if Pn > 0

∂J∂Pn

≤ 0 if Pn = 0(6.6)

where ∂J/∂Pn is the derivative of J with respect to Pn, which reads

∂J

∂Pn=

1[Pn + σ2

w/ |H(n)|2]ln 2

− λ. (6.7)

The optimum power allocation satisfying the KT conditions is found to be

P (opt)n =

(µ− 1

γn

)+

, (6.8)

where (x)+ = max x, 0, γn = |H(n)|2 /σ2w is the so-called channel SNR

and µ = 1/(λ ln 2) is a parameter that must be chosen so as to meet thetotal transmit power constraint

N−1∑n=0

(µ− 1

γn

)+

= Pbudget. (6.9)

This solution lends itself to an interesting physical interpretation. As de-picted in Fig. 6.1, the quantities 1/γn can be thought of as the bottom of avessel in which the transmit power Pbudget is poured similarly to water. Inparticular, the quantity µ represents the height of the water surface, whileP

(opt)n is the depth of the water at subcarrier n. Since the power alloca-

tion process resembles the way by which water distributes itself in a vessel,this optimal strategy is referred to as water-filling or water-pouring. It isworth noting that the bottom level may occasionally become higher thanthe water surface. When this happens, no power is allocated over the cor-responding subcarriers since the latter are too faded for supporting reliabledata transmission. In general, the water-filling strategy takes advantageof the channel frequency-selectivity by giving more power to high-quality



subcarriers while those characterized by the worst channel SNRs are usedto a lesser extent or avoided altogether. Once the power has been optimallydistributed over the signal spectrum according to Eq. (6.8), specific codingtechniques should be employed over each subcarrier to attain the data ratepromised by the channel capacity.

subcarrier n

n1/ γ

water level, µ

0 21

unused subcarriers

nP(opt )

N _ 1

Fig. 6.1 Water-filling power allocation over the available subcarriers.

Inspection of Eq. (6.9) reveals that the water level µ is related to thequantities 1/γn and Pbudget but, unfortunately, the presence of the non-linear operator (·)+ prevents the possibility of computing it in closed-form.As a consequence, the optimum power allocation specified by Eq. (6.8)can only be found through iterative procedures. Two prominent schemeshave been suggested in the literature. In the first one, a tentative level µ

is re-calculated at each new iteration after discarding the subcarrier thatexhibits the lowest channel SNR. Specifically, denote N (i) the set of subcar-rier indices that are considered for power allocation during the ith iteration,where N (0) = 0, 1, 2, . . . , N − 1 is used for initialization purposes. Then,the water level is first computed from Eq. (6.9) as

µ(i) =1

cardN (i)

Pbudget +

∑

n∈N (i)

1/γn

, (6.10)

where card· represents the cardinality of the enclosed set. This value isnext inserted into Eq. (6.8) to obtain the tentative power allocated over the



nth subchannel in the form

P (i)n =

µ(i) − 1/γn, if n ∈ N (i),

0 otherwise.(6.11)

At the end of each iteration, if the subcarrier with the lowest channel gainhas a negative power assignment (i.e., P

(i)n < 0), we discard this subcarrier

from the iterative process by setting the corresponding power level to zeroand removing its index from N (i). The remaining subcarriers are thenused to form the set N (i+1) which is employed in the next iteration. Thealgorithm stops as soon as all power assignments are non-negative. Inthe sequel, this method is referred to as the iterative subcarrier-removalalgorithm.

An alternative scheme to solve the non-linear Eq. (6.8) with respect toµ relies on the use of the well-known bisection algorithm. To explain thismethod, we denote

P (µ) =N−1∑n=0

(µ− 1

γn

)+

(6.12)

the total required power for a given water level µ, and assume that duringthe ith iteration the desired water level µ lies in a coarsely estimated intervalI(i) = [µ(i)

` , µ(i)u ]. Then, we take the middle point of I(i) as a rough estimate

of µ, say µ(i) = (µ(i)` + µ

(i)u )/2, and evaluate the corresponding required

power P (µ(i)) based on Eq. (6.12). A refined estimate of µ is thus obtainedby comparing P (µ(i)) with Pbudget. Specifically, if P (µ(i)) < Pbudget theinterval I(i+1) = [µ(i+1)

` , µ(i+1)u ] to be used in the next iteration is such

that µ(i+1)` = µ(i) and µ

(i+1)u = µ

(i)u , otherwise we set µ

(i+1)` = µ

(i)` and

µ(i+1)u = µ(i). In this way the interval width is halved at each new iteration,

thereby improving the accuracy of the estimated water level. The algorithmis stopped as soon as µ

(i)u −µ

(i)` < ε, where ε is a specified positive parameter.

Clearly, smaller values of ε result into more accurate estimates of µ.

Example 6.1 For illustration purposes, in this example we consider anOFDM system with only eight subcarriers. The channel is frequency-selective and characterized by the SNR values given in Table 6.1.

The goal is to distribute an overall power Pbudget = 1 over the avail-able subcarriers using either the iterative subcarrier-removal method orthe bisection algorithm. The latter is initialized with µ

(0)` = 0.1 and

µ(0)u = 0.6, while the stopping criterion is µ

(i)u −µ

(i)` < 10−4. Although both

schemes achieve the same final power distribution depicted in Fig. 6.2, the



Table 6.1 Channel SNRs in Example 6.1.

Subcarrier index, n Channel SNR, γn (dB)

1 -0.77912 6.10633 19.72394 36.88005 41.31906 23.16187 31.46328 26.6705

subcarrier-removal method stops after just one iteration whereas it takes13 iterations for the bisection algorithm to reach the same result. Clearly,the convergence speed of the bisection procedure is largely determined bythe width of the initialization interval I(0). As a final remark, we observethat the first two subcarriers in Fig. 6.2 are left unused due to their poorchannel quality.

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

subchannel indices

Allocated power, P

n

Inverse SNR, γn−1

Fig. 6.2 Water-filling power distribution in Example 6.1.



6.1.2 Rate maximization and margin maximization

Although the water-filling solution represents the optimal power assignmentstrategy for maximizing the data rate, its practical relevance is limited bythe fact that it does not provide any clear indication about the kind ofsignaling and coding schemes that must be used over each subcarrier toapproach the theoretical channel capacity. In addition, it tacitly assumesan arbitrarily low error-rate probability, whereas practical communicationsystems are normally designed for a non-zero target error-rate which isspecified by the requested quality-of-service. These inherent drawbacksof the water-filling principle have motivated an intense research activitytoward the development of efficient bit and power loading schemes operatingunder a variety of error probability constraints. For instance, in [10,20,71]transmission power and data rate are assigned such that the bit-error-rate(BER) across tones does not exceed a given threshold pe,max. This resultsinto the following uniform BER constraint

pe,n ≤ pe,max, n ∈ N (6.13)

where pe,n is the BER over the nth subcarrier and N the set of modulatedsubcarriers. A less stringent requirement is adopted in [171] and [173] byspecifying the average error probability over the entire OFDM block. If bn

is the number of bits allocated over the nth subcarrier, the correspondingconstraint is stated as

pe =∑N−1

n=0 bn pe,n∑N−1n=0 bn

≤ pe,max, (6.14)

and results into a non-uniform error probability across subcarriers.Whatever the adopted BER constraint, practical loading algorithms

are normally derived on the basis of two main optimization criterions. Afirst possibility is to distribute a given amount of power Pbudget over theavailable subcarriers such that the number of bits per transmitted blockis maximized. This results into the following rate-maximization concept(RMC)

maximize Rb =N−1∑n=0

bn (6.15)

subject toN−1∑n=0

Pn = Pbudget, with bn, Pn ≥ 0 (6.16)



where Pn is the power allocated over the nth subcarrier.The second approach is known as the margin-maximization concept

(MMC), which aims at minimizing the overall transmission power for agiven target data rate Rtarget. Mathematically, we have

minimize PT =N−1∑n=0

Pn (6.17)

subject toN−1∑n=0

bn = Rtarget, with bn, Pn ≥ 0. (6.18)

Although RMC and MMC represent the most popular approaches forthe design of loading algorithms, in some applications there might be thedesire to employ a given power Pbudget to transmit at a target data ratewith the lowest possible error probability. A practical scheme based on thisconcept is found in [3].

6.1.3 Rate-power function

The uniform BER constraint Eq. (6.13) establishes a strict relationshipbetween the number bn of bits allocated over the nth subcarrier and thecorresponding transmission power Pn. The functional dependence betweenthese quantities is dictated by the specified BER pe,n and by the availablecoding and modulation schemes. For instance, with an uncoded BPSKtransmission (bn = 1) we have [123]

pe,n = Q(√

2Pnγn

), (6.19)

where γn = |H(n)|2 /σ2w is the channel SNR over the nth subcarrier while

the Q-function is defined as

Q(x) =1√2π

∫ ∞

x

e−t2/2dt. (6.20)

For QPSK (bn = 2), 16-QAM (bn = 4) and 64-QAM (bn = 6) constellationswith Gray mapping the uncoded BER is reasonably approximated as [123]

pe,n ≈ 4bn

(1− 1

2bn/2

)Q

(√3Pnγn

2bn − 1

). (6.21)

In some works [10,114] the gap-approximation analysis is adopted to estab-lish a more general relationship between Pn and bn in the form [23]

bn = log2

(1 +

Pnγn

Γn

), (6.22)



where Γn is the so-called SNR gap, which is calculated on the basis ofthe target BER, the selected coding scheme and the system performancemargin. Unfortunately, the gap approximation provides accurate resultsonly when the size of the employed constellation is adequately large, asituation that is typical of ADSL applications but rarely occurs in wirelesscommunications. Some useful comments on the validity of Eq. (6.22) aregiven in [3].

Solving Eqs. (6.19), (6.21) or (6.22) with respect to Pnγn yieldsPnγn = f(bn, pe,n), (6.23)

where f(b, p) is referred to as the rate-power function. The latter is nor-mally viewed as a function of the variable b with p as a parameter. Inpractice, it represents the received SNR that is required on a given sub-carrier for reception of b information bits at a target BER p. Figure 6.3illustrates f(b, p) vs. b for p = 10−5 and some popular coding and mod-ulation schemes. The continuous function approximation is derived fromEq. (6.22) and is expressed by

f(b, p) = Γ(2b − 1

), (6.24)

where Γ is selected so as to fit the points corresponding to the consideredcoding/modulation schemes in a least-squares sense.

6.1.4 Optimal power allocation and bit loading under BER

constraint

The optimal solutions to the RMC and MMC problems are not availablein closed-form and can only be approached through iterative methods. Tosee how this comes about, in what follows we restrict our attention to theRMC criterion (similar reasonings also apply to the MMC case). We beginby considering an average error rate constraint and state the optimizationproblem as


bn (6.25)

with respect to b = [b0, b1, . . . , bN−1]T and P = [P0, P1, . . . , PN−1]

T , sub-ject to

N−1∑n=0

Pn = Pbudget, (6.26)

pe(b, P ) =∑N−1

n=0 bn pe,n(bn, γnPn)∑N−1n=0 bn

= pe,max, (6.27)



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

6

8

10

12

14

16

18

20

22

24

Number of bits, b

f(b,

p) (

dB)

LS fitting

BPSK (R = 1/2)

QPSK (R = 1/2)

QPSK (R = 2/3)

16QAM (R = 1/2)

16QAM (R = 3/4)

16QAM (R = 2/3)

64QAM (R = 2/3)

64QAM (R = 3/4)

Fig. 6.3 Rate-power function.

with bn, Pn ≥ 0. Here, we treat each bn as a continuous variable andassume that the functional dependence of pe,n on the quantities bn andγnPn is specified in some way using the rate-power function.

The Lagrangian function for the constrained maximization problemEq. (6.25) is defined as

L(b, P ,λ) =N−1∑n=0

bn + λ1

[Pbudget −

N−1∑n=0

Pn

]+ λ2

[pe,max − pe(b, P )

],

(6.28)where λ= [λ1, λ2]

T is the set of Lagrangian multipliers. Conditions foroptimum bit and power loading are derived by setting to zero the derivativesof L(b,P ,λ) with respect to b and P . This produces the following set of2N equations

∂L∂bn

= 1− λ2∂pe

∂bn= 0,

∂L∂Pn

= −λ1 − λ2∂pe

∂Pn= 0,

(6.29)

for n = 0, 1, . . . , N − 1.



After appropriate definition of the constant terms ξ1 and ξ2, Eq. (6.29)can also be rewritten as

∂pe

∂bn= ξ1,

∂pe

∂Pn= ξ2,

(6.30)

for n = 0, 1, . . . , N − 1.Unfortunately, there is no explicit solution to the conditions Eq. (6.30).

An iterative algorithm for approaching the optimal vectors b and P is pro-posed in [8] using convex simplex techniques. This scheme requires a searchover a multidimensional parameter space and exhibits a long convergencetime which makes it unsuited for practical implementation.

A certain reduction of complexity is possible if we replace the averageerror probability constraint in Eq. (6.27) with a uniform BER constraint inwhich the same BER pe,max is imposed over all subcarriers, i.e.,

pe,n(bn, γnPn) = pe,max, (6.31)for n = 0, 1, . . . , N − 1.

In this way the optimization process has only to be performed withrespect to b rather than over the set (b, P ) since the power Pn is univocallydetermined by the constraint Eq. (6.31) once bn has been specified. Indeed,using the rate-power function defined in Eq. (6.23), we have

Pn =f(bn, pe,max)

γn. (6.32)

The cost function for the new optimization problem takes the form

L(b,λ) =N−1∑n=0

bn + λ

[Pbudget −

N−1∑n=0

f(bn, pe,max)γn

], (6.33)

and conditions for optimal bit allocation are found by setting to zero thederivative of L(b,λ) with respect to b. This yields

∂f(bn, pe,max)∂bn

= γn/λ, (6.34)

for n = 0, 1, . . . , N − 1, from which it follows that the data rate is maxi-mized when each subcarrier operates at a slope γn/λ over the rate-powerfunction. An iterative algorithm to approach the solution Eq. (6.34) hasbeen proposed by Campello in [10]. Compared to [171], this scheme is muchsimpler to implement and also exhibits faster convergence thanks to the re-duced number of optimization parameters. The price for these advantagesis a slight reduction of the achievable data rate as a consequence of theuniform BER constraint in Eq. (6.31). The latter is more stringent thanthe average constraint in Eq. (6.27) and inevitably reduces the number ofdegrees of freedom that are exploited by the optimization process.



6.1.5 Greedy algorithm for power allocation and bit loading

The RMC and MMC problems as stated in the previous subsections assumea constellation size with infinite granularity and their optimum solution willinvariably lead to noninteger bit allocation across tones. A more practicalapproach is to specify a finite set of allowable PSK or QAM constellations,which are then selected on a subcarrier-by-subcarrier basis according to therelevant channel gains. Hence, it is of interest to look for efficient bit andpower loading schemes that result into the assignment of an integer numberof bits over each subcarrier. For this purpose, we still concentrate on theRMC problem which is now restated as


bn (6.35)

with respect to b = [b0, b1, . . . , bN−1]T and P = [P0, P1, . . . , PN−1]

T undereither a uniform or average BER constraint and subject to

N−1∑n=0

Pn ≤ Pbudget, (6.36)

bn ∈ 0, 1, . . . , bmax , (6.37)

where Pn ≥ 0 and 2bmax is the maximum size of the employed constellations.The optimization problem formulated in Eqs. (6.35)-(6.37) has been

extensively studied by many authors (see for example, [20,56,82,123]). Itssolution is found through iterative greedy techniques in which bit loadingacross tones is performed incrementally or decrementally one bit at a time.From an operational point of view, we distinguish between bit-filling andbit-removal schemes. In the former case we start from an initial all-zero bitallocation and add one bit at a time to the subcarrier requiring the leastadditional power to meet the specified BER constraint. Vice versa, the bit-removal approach starts with an initial maximum bit allocation bn = bmax

for n = 0, 1, . . . , N − 1 and removes one bit at a time from the subcarrierthat guarantees the maximum power saving for operation at the targetBER. Both algorithms are stopped as soon as the required transmissionpower PT approaches the maximum admissible value Pbudget.

To better illustrate these iterative procedures, we assume a uniformBER constraint across subcarriers. This allows us to use the rate-powerfunction f(b, pe,max) defined in Eq. (6.23), where pe,max is the maximumBER that can be tolerated by the system. Then, the bit-filling and bit-removal algorithms are summarized as follows:



Bit-filling algorithm

• Initialization1) let bn = 0 and PT = 0;2) ∆P+

n = f(1, pe,max)/γn for each n ∈ N = 0, 1, . . . , N − 1;

• Bit assignment iterations:repeat the following procedure:1) n = arg min

n∈N∆P+

n ;2) PT = PT + ∆P+

n

3) if PT > Pbudget then stop the algorithm;4) bn = bn + 1;5) ∆P+

n = [f(bn + 1, pe,max)− f(bn, pe,max)] /γn;6) if bn = bmax, then remove n from N ; end.

Bit-removal algorithm

• Initialization:1) let bn = bmax and2) initialize ∆P−n for each n ∈ N = 0, 1, . . . , N − 1 as follows

∆P−n = [f(bmax, pe,max)− f(bmax − 1, pe,max)] /γn;

3) let PT =∑N−1

n=0 f(bmax, pe,max)/γn.

• Bit removal iterations:repeat the following procedure until PT ≤ Pbudget:1) n = arg max

n∈N∆P−n ;

2) bn = bn − 1;3) PT = PT −∆P−n ;4) If bn = 0, then remove n from N , otherwise compute

∆P−n = [f(bn, pe,max)− f(bn − 1, pe,max)] /γn;

end.

For the bit-filling algorithm, during initialization, the power neededto transmit one bit is calculated for each subcarrier. At each iteration,



the subcarrier requiring the minimum additional power ∆P+n is assigned

one more bit and the new additional power for that subcarrier is updatedtogether with the overall transmission power PT . If the number of bits hasachieved its maximum allowable value bmax, then the selected subcarrier isexcluded from any further assignment by removing its index from N . Thestopping criterion is governed by PT , which cannot overcome the assignedpower Pbudget.

On the other hand, the initialization for the bit-removal algorithm isperformed by allocating the maximum number of bits over all subcarriers.At each iteration, one single bit is subtracted from the subcarrier thatprovides the maximum power saving ∆P−n for operation at the target BER,and the transmit power PT is correspondingly updated. If no more bits areleft on the selected subcarrier, the latter is excluded from further iterations,otherwise the new amount of power saving is calculated. The optimum bitallocation is obtained as soon as PT becomes smaller than or equal toPbudget.

Although bit-filling and bit-removal procedures converge to the samebit allocation across tones, the computational load involved with these al-gorithms is typically different and depends on the achieved data rate Rb.In particular, bit-removal is to be preferred when Rb > Nbmax/2 since inthis case the convergence is faster than with bit-filling. It is also importantto note that the resulting bit allocation is optimal only in relation to theconsidered function f(b, p). Actually, the selection of different modulationschemes as possible transmission modes will lead to the consideration ofdifferent rate-power functions, which may result into possibly different bitallocations for the same set of channel SNRs.

6.1.6 Bit loading with uniform power allocation

Greedy techniques based on bit-filling or bit-removal strategies provide opti-mum joint distribution of power and data rate in practical situations wherefinite-granularity constellations have to be employed. The main difficultyof these methods is the extensive requirement of sorting and searching op-erations, which may prevent their applicability when the number of bits perOFDM block is relatively large. A simpler approach relies on the observa-tion that in general only negligible throughput penalties occur if the optimalpower assignment is replaced by a uniform allocation of power across sub-carriers [180]. This simplified strategy has the advantage of reducing thedimensionality of the optimization problem in that the quantities Pn are



kept fixed at some specified value P and only bit loading is performed adap-tively. A scheme based on this suboptimal approach is derived in [9] underan average BER constraint. In this case the RMC problem is reformulatedas


bn (6.38)

under a uniform power allocation and subject to

pe(b) =∑N−1

n=0 bn pe,n(bn, γnP )∑N−1n=0 bn

≤ pe,max, (6.39)

and

bn ∈ 0, 1, . . . , bmax , (6.40)

where the BER pe,n(bn, γnP ) over the nth subcarrier is univocally deter-mined by the number bn of allocated bits and by the received SNR γnP .Note that the maximization of the objective function Rb is only performedwith respect to b = [b0, b1, . . . , bN−1]

T since the available power Pbudget isnow uniformly distributed over the modulated subcarriers.

The corresponding solution is found iteratively by means of the followingbit-removal algorithm with uniform power allocation (BRA-UniPower):

The suboptimum BRA-UniPower algorithm

• Initialization:1) let bn = bmax;2) set Pn = Pbudget/N for n ∈ N = 0, 1, . . . , N − 1 andcompute pe(b).

• Bit removal iterations:repeat the following procedure until pe(b) ≤ pe,max:1) n = arg max

n∈Npe,n(bn, γnPn);

2) bn = bn − 1;3) if bn = 0, then remove n from N and reassign the power so thatPn = Pbudget/cardN for n ∈ N ;4) recompute pe(b) for the current bit allocation and power distri-bution;end.



During initialization, the maximum number of bits is tentatively allo-cated over each subcarrier under a uniform power assignment. At eachiteration, the algorithm searches for the subcarrier n exhibiting the worstBER performance and reduces the corresponding data rate by one singlebit. If bn = 0, the index n is removed from N so as to exclude the se-lected subcarrier from transmission and the power Pbudget is redistributeduniformly over the remaining subcarriers. The average BER pe(b) is nextcomputed for the current bit assignment and compared with its maximumadmissible value pe,max. The process is stopped as soon as pe(b) ≤ pe,max.

This algorithm allows a certain computational saving with respect toa system in which data rate and transmission power are jointly adjustedaccording to some specified optimality criterion. However, the need forrecomputing the average BER pe(b) at each new iteration still represents aserious drawback for practical implementation.

A further reduction of complexity is possible if we adopt a uniformBER constraint pe,n(bn, γnP ) ≤ pe,max instead of specifying the averageerror rate as in Eq. (6.40). In such a case, bn is explicitly determined bysolving the equation pe,n(b′n, γnP ) = pe,max with respect to b′n and takingthe integer part of the corresponding solution. This yields

bn = min bmax, int(b′n) , (6.41)where we have also borne in mind that bn cannot exceed a prefixed valuebmax. In this way, bit and power allocation is performed through the fol-lowing iterative process, which is referred to as uniform-BER and uniform-power loading algorithm (UniBER-UniPower) :

The suboptimum UniBER-UniPower algorithm

• Initialization:1) let Pn = Pbudget/N and2) set bn = min bmax, int(b′n) for n ∈ N = 0, 1, . . . , N − 1.

• subcarrier removal iterations:repeat the following procedure until bn > 0 for all n ∈ N ;1) if one or more bn’s are zero, then let n = arg min

n∈Nγn and

remove n from N ;2) reassign the power so that Pn = Pbudget/cardN for n ∈ N ;3) recompute bn = min bmax, int (b′n) for n ∈ N according to thenew power distribution;end.



As is seen, a preliminary bit distribution is derived from Eq. (6.41) as-suming Pn = Pbudget/N as a tentative power assignment. If some bn’s turnout to be zero, the algorithm iterates by removing the worst quality sub-carrier from the set N and redistributing the overall power Pbudget acrossthe remaining tones. Bit loading is then recomputed according to the newpower distribution. The algorithm is stopped as soon as bn > 0 for alln ∈ N .

The most demanding task in the described procedure is the need forrecomputing the bit allocation each time a subcarrier is excluded fromtransmission. A simpler yet suboptimal solution is obtained by replacingthe subcarrier removal iterations with a single cancellation stage in whichall subcarriers presenting an initial zero-bit assignment are simultaneouslydiscarded. This approach results into a significant reduction of complexitysince now the final bit assignment is directly derived from Eq. (6.41) afterassuming Pn = Pbudget/N for n ∈ 0, 1, . . . , N − 1, thereby dispensingfrom any iteration [31]. The final power allocation is eventually obtainedby distributing Pbudget over the modulated subcarriers (i.e., those char-acterized by a positive bit assignment). In general, this strategy incurssome throughput penalty compared to a system in which the power is re-distributed each time a subcarrier is removed from N . The reason is thatpower redistribution may allow some subcarriers to pass from an initialzero-bit assignment to some positive allocation bn > 0 as a consequence ofthe increased power level. The suboptimal algorithm excludes these subcar-riers from data transmission, even though they could actually be exploitedto convey some minimum information with the required reliability.

6.1.7 Performance comparison

In this Section we use computer simulations to compare the performanceof the discussed bit-loading schemes in terms of achievable data through-put. For this purpose, we assume that a power budget of 10 dBm is avail-able in an uncoded OFDM system with N = 256 subcarriers. The signalbandwidth is 10 MHz while the noise power spectral density is −80 dBm.The transmission mode is selected from a set of four possible modulationschemes, namely BPSK, QPSK, 16-QAM and 64-QAM. As a result, thequantities bn take values in the set 1, 2, 4, 6 for n = 0, 1, . . . , 255. Thechannel model is the same employed in Sec. 4.4, and comprises four mul-tipath components with fixed path delays and an exponentially decayingpower delay profile. A total of 200 snapshots are generated in order to



average the simulation results over the channel statistics.

10−4

10−3

10−2

10−1

20

40

60

80

100

120

140

160

180

200

220

Target BER

Num

ber

of b

its p

er O

FD

M b

lock

, Rb

Water−fillingGreedy BFA/BRABRA−UniPowerUniBER−UniPower

Fig. 6.4 Number of allocated bits as a function of the target BER.

Figure 6.4 illustrates the total bit rate Rb =∑255

n=0 bn achieved by theloading algorithms as a function of the target BER. For comparison, we alsoshow the data throughput provided by the classical water-filling solution.As expected, the greedy bit-filling/bit-removal algorithms (BFA/BRA) out-perform their suboptimal BRA-UniPower and UniBER-UniPower versionsat the price of a higher computational load. On the other hand, the dif-ference between the two suboptimal schemes with uniform power alloca-tion is quite negligible, particularly at low error probabilities. In the lowtarget-BER region we see that the water-filling policy achieves a significantadvantage over the other algorithms due to its implicit assumption of aninfinite granularity constellation. As the target-BER grows large, however,this advantage reduces to such a point that greedy BFA/BRA become theleading schemes at BER> 3×10−2. This fact can be explained by recallingthat the water-filling solution has been derived under the assumption of anarbitrarily small BER, whereas the considered greedy-based techniques can



trade data throughput against error-rate probability. This means that afair comparison between the water-filling policy and other loading schemescan only be made in the low BER region.

6.1.8 Subband adaptation

The adaptive techniques illustrated so far operate on a subcarrier basisin that the optimum constellation size and/or power level are individuallydetermined for any subcarrier according to instantaneous channel state in-formation. This approach offers a large amount of flexibility on one hand,but on the other it may entail a prohibitive signaling overhead since thereceiver has to be informed as to which modulation parameters are em-ployed over each subcarrier. To alleviate this drawback, system resourcescan be allocated in a blockwise fashion following a subband adaptation cri-terion [49,74]. The basic idea behind this approach is to divide the availablespectrum into several groups of adjacent subcarriers which are referred toas subbands, and use the same set of modulation parameters (constellationsize, code rate, power level) over all subcarriers in the same subband. Inthis way the signaling task is substantially simplified at the price of some-what reduced flexibility in resource assignment. Roughly speaking, thepenalty incurred by subband adaptation in terms of achievable throughputis determined by the extent of channel variations over each subband. If thesubband width is smaller than the channel coherence bandwidth, the chan-nel appears as nearly flat across the subband and no significant penalty isincurred with respect to a system that operates at subcarrier level.

In those applications where system complexity is a critical issue, sub-band loading can be used in conjunction with uniform power distributionover the signal spectrum. In such a case, letting M = M1,M2, . . . ,MJbe the set of possible transmission modes (each characterized by a givenconstellation size, code rate and other possible modulation parameters), theproblem is to select the best mode over each subband so as to obtain thehighest throughput at some specified target BER. Again, the optimizationcan be performed under either a uniform or average error rate constraint.In the former case the BER over each subcarrier is kept smaller than agiven value pe,max, while in the latter an upper limit pe,max is imposed tothe average error probability

pe(Mj) =1

Ns

∑n

pe,n(Mj), (6.42)

where Ns is the number of subcarriers in the subband, pe,n(Mj) is the



BER over the nth subcarrier for a given transmission mode Mj and thesummation is extended to all subcarriers in the considered subband.

In case of a uniform BER constraint, a mode Mj can be activated on acertain subcarrier only if the instantaneous SNR exceeds a given thresholdρj which depends on the adopted modulation parameters and target BER.For example, with an uncoded BPSK transmission operating at an errorrate of 10−3 we have ρ = 6.8 dB, while ρ = 9.8 dB is requested for anuncoded QPSK. On the other hand, since the channel quality varies acrosssubcarriers and a single mode must be employed in each subband, thetransmission parameters in the considered subband are conservatively se-lected on the basis of the subcarrier which exhibits the lowest SNR. Clearly,this approach results into some performance loss with respect to a systemin which the available resources are assigned on a subcarrier basis. Thereason is that in each subband the transmission mode and the associateddata throughput are exclusively dictated by the most faded subcarrier eventhough other subcarriers with better channel quality could safely supporthigher data rates. This problem can be mitigated by a proper design of thesubband width, which should be made adequately smaller than the channelcoherence bandwidth. In this way all relevant subcarriers undergo similarchannel impairments and, in consequence, the selected transmission modeis likely to be optimal over the entire subband.

As anticipated, subband adaptation can also be performed under anaverage error-rate constraint. In such a case, the average BER pe(Mj) inEq. (6.42) is computed for all available modes Mj , and in each subbandthe mode M exhibiting the highest data rate and satisfying the conditionpe(M) ≤ pe,max is selected for transmission. This adaptation strategy isexpected to mitigate to some extent the throughput penalty associatedwith the uniform BER constraint. The reason is that in each subbandall subcarriers contribute to the average error rate and, in consequence,the transmission mode is not exclusively selected on the basis of the worstquality subcarrier.

6.1.9 Open-loop and closed-loop adaptation

Any link adaptation technique exploits instantaneous channel state infor-mation to determine the best set of modulation parameters to be employedin the next transmission. One main assumption behind this approach isthat the fading rate is not too rapid since otherwise channel predictionmay be obsolete at the time of transmission, thereby resulting into a wrong



selection of the modulation parameters.Roughly speaking, we distinguish between two different classes of adap-

tation techniques. The former class is suitable for time-division-duplex(TDD) systems, where the same frequency band is used for both uplink anddownlink transmissions and the communication channel can reasonably beconsidered as reciprocal. In this case the receiving station estimates thechannel quality during the downlink phase and exploits this estimation toselect the best mode for the next uplink transmission. We refer to thisoperating method as open-loop adaptation since the local transmitter ad-justs the modulation parameters by only relying on channel measurementsacquired during the previous slot and without exploiting any feedback fromthe remote receiver.

On the other hand, if the communication link is not reciprocal as infrequency-division-duplex (FDD) systems, channel state information de-rived from the received OFDM blocks cannot be used to determine themodulation parameters for the next transmission stage because of the dif-ferent propagating conditions encountered in the two communication links.In this case adaptive modulation can be established on condition that theremote receiver performs channel estimation and instructs the transmit-ter as to which parameters are the best to be used. This policy is knownas closed-loop adaptation since the transmission mode is activated on thebasis of a specific feedback from the remote receiver rather than being au-tonomously selected by the transmitter. Although closed-loop adaptationis expected to be intrinsically robust against interference and other non-reciprocal effects, it suffers from an inherent feedback delay which mightresult into outdated information. This makes the Doppler fading rate arather critical parameter in closed-loop adaptive modulation systems.

6.1.10 Signaling for modulation parameters

Signaling plays a major role in the design of an adaptive communicationlink. In an open-loop system where channel estimation and parameteradaptation are performed by the local transmitter, the remote receiver mustbe informed as to which transmission mode is currently in use. Vice versa,in a closed loop scenario the modulation parameters are decided by thereceiver itself, which therefore has to communicate its choice to the remotetransmitter. In any case, it is important that signaling information beexchanged with a high level of reliability since otherwise the receiver mightbe induced to adopt a wrong detection strategy and would be unable to



successfully decode the information data.One popular signaling scheme is based on the insertion of one or more

dedicated subcarriers in each subband to convey information about the setof employed modulation parameters. If NM is the number of possible trans-mission modes, a single NM -PSK symbol would in principle be sufficientfor this purpose. However, in order to reduce the probability that signalinginformation may be corrupted by channel impairments, multiple dedicatedsymbols can be placed across the subband to take advantage of the channelfrequency diversity. A drawback of this signaling method is the throughputpenalty that results from the use of dedicated subcarriers.

An alternative approach is based on blind detection algorithms. Theseschemes try to estimate the currently employed transmission mode fromthe received signal without requiring any extra overhead. An example ofblind algorithm is presented in [73] for systems employing subband adap-tation. Let Mj ; j = 1, 2, . . . , J be the set of possible transmission modesand denote Y (n) = R(n)/H(n) the nth DFT output divided by the cor-responding channel estimate H(n). Using (4.6), we can interpret Y (n) asan estimate of the data symbol c(n) transmitted over the nth subcarrierand embedded in additive noise. Then, inside the constellation associatedto the transmission mode Mj we select the symbol cj(n) that is closest toY (n) and compute the following error signal

ej =∑

n

|Y (n)− cj(n)|2 , j = 1, 2, . . . , J. (6.43)

where the summation is extended to all subcarriers in the considered sub-band. Clearly, ej is a measure of the Euclidean distance between the re-ceived symbols Y (n) and the constellation points associated to Mj . Tosee how the quantities e1, e2, . . . , eJ can be used to decide which trans-mission mode is currently in use, we temporarily neglect the noise contri-bution and assume perfect channel estimation. In this ideal setting we haveY (n) = c(n) and, in consequence, the error signal associated to the actuallyemployed transmission mode turns out to be zero due to a perfect agreementbetween the received symbols and the corresponding constellation points.Although in the presence of thermal noise and channel estimation inaccu-racies this error signal may not be exactly zero, under normal operatingconditions it is expected to be relatively small. Hence, it makes sense toargue that the transmission mode employed over the considered subband isthe one associated to the minimum error signal. In other words, we decide



that Mj is currently in use if

j = arg minjej . (6.44)

Compared to signaling schemes that make use of dedicated subcarriers,this blind method has the advantage of dispensing from any overhead, eventhough a larger SNR is required to achieve the same level of reliability.In particular, it is found in [73] that the system performance is largelydictated by the number of subcarriers in each subband and by the numberof allowable transmission modes, which in practice cannot be greater thanfour.

6.2 Resource allocation in multiuser OFDM systems

In a typical multiple-access system, users’ signals undergo independent fad-ing attenuations because of the different spatial positions occupied by re-mote terminals. As a consequence, a subcarrier that appears in a deep fadeto one terminal may exhibit a much higher channel gain for other users. Totake advantage of this multiuser diversity effect [78], the available subcar-riers should be dynamically assigned to users on the basis of instantaneouschannel state information. Compared to conventional OFDMA systemswith non-adaptive resource allocation, this approach allows a more efficientuse of the system resources. The net result is an increased data throughputsince a given subcarrier will be left unused only if it appears in a deep fadeto all terminals, a situation that rarely occurs due to the mutual indepen-dence of the users’ channel responses.

From the above discussion it follows that optimum resource allocation ina multiuser scenario requires the adoption of a dynamic carrier assignmentpolicy in addition to adaptive bit and power loading. This makes the linkadaptation task much more challenging than in single-user systems. Asusers cannot share the same subcarrier, the allocation process results intoa combinatorial optimization problem for which no optimal greedy solutionexists. This fact has recently stimulated an intense research activity towardthe development of suboptimum resource assignment schemes characterizedby good performance and affordable complexity. The common idea behindthese methods is to consider carrier allocation and bit loading as separatetasks to be performed independently rather than jointly.

The concept of dynamic resource allocation in an OFDMA downlinktransmission is illustrated in Fig. 6.5. At the BS, information about the



OFDM MODULATOR

User 1 data

User 2 data Subcarrier allocation

and bit-loading

N -point IDFT

Add CP and D/A

Frequency- domain samples

User M data

Channel state information

Fig. 6.5 OFDMA downlink transmission with adaptive resource allocation at the BStransmitter.

users’ channel responses are passed to the subcarrier allocation and bitloading unit, which maps the users’ data over the selected subcarriers us-ing the more appropriate transmission mode (coding and/or modulationscheme). In order to guarantee a specified error rate probability, the powerlevel over each subcarrier is properly adjusted on the basis of the employedtransmission mode. The resulting frequency-domain samples are finally fedto an OFDM modulator and transmitted over the channel.

At the mth mobile terminal, the received signal is demodulated and therecovered frequency-domain samples are passed to the subchannel selector,which only retains information from subcarriers assigned to the mth userwhile discarding all the others. The selected samples are then fed to the de-coding unit, which provides final bit decisions using the appropriate detec-tion strategy. Clearly, the BS must inform the users’ terminals as to whichsubcarriers and transmission modes have been assigned to them, otherwisethe subchannel selector and data decoding unit cannot properly be config-ured. This requires the exchange of side information with a correspondingpenalty in data throughput due to the transmission overhead. The amountof side information is somewhat reduced by adopting a subband allocationpolicy where users are given blocks of contiguous subcarriers with similarfading characteristics.



Bit decisions

OFDM DEMODULATOR

Subchannel selector

Information feedback from the BS subcarrier allocation

and bit-loading unit

A/D and

remove CP

N -point DFT

User m decoder

Fig. 6.6 Block diagram of the mth receiving terminal in an OFDMA downlink trans-mission with adaptive resource allocation.

6.2.1 Multiaccess water-filling principle

The extension of the water-filling principle to a multiuser scenario is notstraightforward except for the unrealistic case where all users are char-acterized by the same channel response. The first pioneering results inthis area were presented by Cheng and Verdu in their excellent paper [17].They derived the capacity region and the optimal power allocation for afrequency-selective Gaussian multiaccess channel, where two or more userswith independent power constraints transmit data to a common BS receiver.In what follows, the results of [17] are applied to the uplink transmissionof a multicarrier system accommodating M simultaneously active users.

Assuming perfect timing and frequency synchronization, the DFT out-put at the BS receiver takes the form

R(n) =M∑

m=1

Hm(n)Sm(n) + W (n), 0 ≤ n ≤ N − 1. (6.45)

where Hm(n) is the channel frequency response of the mth user overthe nth subcarrier, Sm(n) is the corresponding input symbol with powerPm,n =E|Sm(n)|2 and W (n) is white Gaussian noise with zero-mean andvariance σ2

w. In this uplink scenario, the power constraints are stated as

N−1∑n=0

Pm,n ≤ Pm,budget, (6.46)



for m = 1, 2, . . . ,M , where Pm,budget represents the amount of availablepower for the mth user and Pm,n ≥ 0 for n = 0, 1, . . . , N − 1.

Unlike the single-user case, the multiaccess channel is characterizedby a M -dimensional capacity region CR ∈ RM

+ (we denote RM+ the set

of M -tuples with non-negative real-valued entries). Each point R =(R1, R2, . . . , RM ) in this region represents a combination of rates at whichusers can send information with an arbitrarily low error-rate probability.For the sake of simplicity, in the following we limit our attention to a two-user scenario. In this case CR is a convex set in the positive quadrant ofthe (R1, R2)-plane which can be written as [17]

CR = ∪P1,P2

(R1, R2) :

0 ≤ R1 ≤N−1∑n=0

log2 (1 + P1,nγ1,n)

0 ≤ R2 ≤N−1∑n=0

log2 (1 + P2,nγ2,n)

R1 + R2 ≤N−1∑n=0

log2 (1 + P1,nγ1,n + P2,nγ2,n)

,

(6.47)where Pm = [Pm,0, Pm,1, . . . , Pm,N−1]

T (m = 1, 2) are power vectors satis-fying the constraint Eq. (6.46) while γm,n = |Hm(n)|2 /σ2

w is the channelSNR of the mth user over the nth subcarrier. From the above equationwe see that CR is the union of an infinite number of rate regions, eachcorresponding to a different pair (P1,P2) and representing a pentagon inthe (R1, R2)-plane.

A possible example of capacity region is depicted in Fig. 6.7. The ab-scissa of the corner point A indicates the maximum rate at which user 1can reliably send information over the channel (single-user capacity) whenuser 2 is not transmitting (R2 = 0). This point is achieved by optimallyallocating the power P1,budget over the channel H1(n) according to the clas-sical single-user water-filling principle. The converse is true for the cornerpoint B, which is attained by applying the water-filling policy to H2(n)assuming that user 1 is turned off. Any other point on the boundary curveconnecting A and B is achieved by an appropriate choice of (P1, P2) and isoptimal in that it maximizes a linear combination of the users’ rates, say

R(α) = αR1 + (1− α)R2, (6.48)

with α ∈ [0, 1]. This can readily be seen by considering the family of parallelstraight lines in the (R1, R2)-plane over which R(α) keeps constant. Theselines have a common slope α/(α − 1) and, due to the convexity of CR,



A1R

B

Q

Slope = _ 1

Sum-ratemaximization

pointCAPACITY

REGION

RC

2R

Fig. 6.7 Example of capacity region in a two-user scenario.

only one of them is tangent to the boundary curve in some point Q(α).The coordinates of Q(α) provide the values R1 and R2 that maximize R(α)over the capacity region.

Inspection of Eq. (6.48) provides a useful interpretation of α as a pa-rameter that determines the relative users’ priorities. Specifically, as α

approaches unity the priority given to user 1 increases and the point Q(α)moves on the boundary curve toward A. When α = 1/2 both users are giventhe same priorities. In this case the corresponding boundary point resultsin the maximization of the sum-rate R1 +R2 and is graphically determinedby considering the tangent line with slope −1 as illustrated in Fig. 6.7.

From the above discussion it appears that in a two-user scenario differentusers’ priorities result into different optimum operating points, each locatedon the boundary of the capacity region. Hence, the task is to find, for anygiven value of α, the optimum pair (P1, P2) that allows one to achieve theboundary point where R(α) is maximum. A geometrical solution to this



problem has been presented in [17] and consists of two fundamental steps.In the first step, an equivalent transfer function H(eq)(n) is computed fromH1(n) and H2(n), and the classical water-filling principle is then applied toH(eq)(n). This provides the optimum allocation of the total available powerP1,budget + P2,budget in the frequency-domain. The second step determineshow the total power Pn = P1,n +P2,n allocated over each subcarrier shouldbe optimally split among the active users. The result is that in general eachsubcarrier has to be shared by both users, who therefore interfere with eachother. In this case, the successive decoding idea suggests that the user withthe lowest priority (say user 1) should be decoded first while treating theother user’s signal as noise. The receiver then regenerates the signal ofuser 1 and subtracts it from the received waveform. This results into anexpurgated signal which is eventually employed to detect the informationsent by user 2.

An interesting situation occurs when both users are given the samepriority. As mentioned earlier, in this case the optimum power assignmentmaximizes the sum-rate R1 + R2 over the capacity region and achieves theboundary point Q depicted in Fig. 6.7. A prominent result of [17] is thatthe optimum power split among equal-priority users corresponds to theclassical OFDMA concept in which subcarriers are grouped into disjointclusters that are exclusively assigned to users. This means that OFDMAis capable of achieving the sum-rate capacity promised by the Gaussianmultiaccess channel.

In case of only two users with equal priorities, the optimum power as-signment (P1, P2) is found through a geometrical procedure which is rem-iniscent of the water-filling argument. The basic idea behind this methodis to properly scale the water-filling diagrams associated with the channelresponses H1(n) and H2(n) such that they present the same water leveland can thus be combined into a single diagram. More specifically, lettingρ1 and ρ2 be the scaling coefficients, we arbitrarily fix the water level tounity and plot the curves ρ1/γ1,n and ρ2/γ2,n as a function of n on thesame diagram. As indicated in Fig. 6.8, we treat the minimum of the twocurves as the bottom of the vessel where water is poured, and adjust ρ1

and ρ2 such that: 1) the total amount of water is ρ1P1,budget + ρ2P2,budget;2) the amount of water in the region where ρ1/γ1,n ≤ ρ2/γ2,n is equal toρ1P1,budget.

In general, the coefficients ρ1 and ρ2 can only be obtained graphicallyor numerically as they depend on the channel transfer functions and powerconstraints in a rather complicated fashion which makes their analytical



subcarriers

waterlevel

1ρ γ/ 1,n 2

ρ γ/2,n

P1,budget1

ρ P2,budget2

ρ

Fig. 6.8 The water-filling principle in a two-user scenario.

derivation a rather difficult task. Anyway, assuming that these parametershave been derived in some manner, the optimum power assignment for thetwo users is eventually found after scaling the shaded regions in Fig. 6.8by ρ1 and ρ2. As anticipated, different users are given different subcarriersaccording to the OFDMA principle. In particular, the frequency bandwhere ρ1/γ1,n ≤ ρ2/γ2,n is assigned to user 1 while the remaining part isavailable for user 2. Clearly, if min ρ1/γ1,n, ρ2/γ2,n exceeds the watersurface for some n, the corresponding subcarriers are left unused as theycannot support reliable data transmission.

6.2.2 Multiuser rate maximization

Although relevant from an information theoretic perspective, the multiuserwater-filling policy turns out to be too complex for practical purposes dueto lack of efficient methods for determining the scaling coefficient of eachindividual channel response. As in the single-user case, a more convenient



approach for dynamic resource allocation is based on the rate-maximizationconcept (RMC). This strategy aims at maximizing the aggregate data rateof all active users under fixed constraints in terms of total transmissionpower and error-rate performance.

To see how the RMC can be extended to a typical OFDMA downlinkscenario with M active users, we denote bm,n the number of bits of themth user that are allocated over the nth subcarrier. We also assume thatbm,n ∈ 0, 1, . . . , bmax, where bmax is determined by the maximum allow-able constellation size. Since each subcarrier cannot be shared by morethan one user, for any index n only one single m ∈ 1, 2, . . . ,M may ex-ist for which bm,n 6= 0. The performance requirement of the mth user isspecified by the maximum tolerable BER pm,max. In order to maintain thedesired quality of service, the power allocated to the mth user over thenth subcarrier must equal Pm,n = f(bm,n, pm,max)/γm,n, where f(b, p) isthe rate-power function indicating the minimum SNR that is required todetect b information bits at a target BER p. Note that in this way the sameerror probability pm,max is maintained over all subcarriers assigned to themth user (uniform BER constraint).

Under the above assumptions and statements, the multiuser RMC prob-lem is mathematically formulated as

maximize Rb =M∑

m=1

N−1∑n=0

bm,n (6.49)

with respect to the bit assignments bm,n, where maximization is subjectto

PT =M∑

m=1

N−1∑n=0

f(bm,n, pm,max)γm,n

≤ Pbudget, (6.50)

and

if bm′,n 6= 0 , then bm,n = 0 for all m 6= m′. (6.51)

The constraint Eq. (6.50) specifies an upper limit Pbudget to the total trans-mission power while Eq. (6.51) ensures that each subcarrier is exclusivelyassigned to only one user, as demanded by the OFDMA concept.

From Eqs. (6.49)-(6.51) we see that extending the RMC criterion toa multiuser scenario results into a combinatorial maximization problemfor which no practical solution is available. Things become easier if allusers are characterized by a common BER constraint pm,max = pmax form = 1, 2, . . . , M . This particular situation is considered in [78], where the



optimum solution to the RMC problem is found in two successive steps. Inthe first step each subcarrier is exclusively assigned to the user exhibitingthe highest channel SNR over it. More precisely, the m′th user is given thenth subcarrier on condition that

m′ = arg max1≤m≤M

γm,n . (6.52)

In the second step, the number of bits allocated over any assigned subcarrieris determined so as to maximize the objective function Rb in Eq. (6.49) un-der the power constraint Eq. (6.50). This task is accomplished in the sameway as in single-user OFDM transmissions. Indeed, after all subcarriershave been assigned, the OFDMA downlink can be viewed as an equivalentsingle-user system with channel SNRs given by γ

(eq)n = max

1≤m≤Mγm,n for

n = 0, 1, . . . , N − 1 and with a data rate that equals the aggregate datarate of the original multiuser scenario. Optimum bit assignment is thusachieved by means of RMC-based greedy techniques as those discussed inSec. 6.1.5.

Numerical results illustrated in [78] indicate that for a given power con-sumption PT the achievable sum-rate Rb increases with the number of usersdue to multiuser diversity effects [65]. However, a fundamental drawbackof the RMC criterion as stated in Eqs. (6.49)-(6.51) is that it does not pro-vide any guarantees on the minimum achievable data rate of each individualuser. Actually, in some extreme situations maximizing the aggregate datarate may result into the assignment of all available subcarriers to only asubset of users exhibiting good channel quality, thereby excluding all otherusers from transmission.

6.2.3 Max-min multiuser rate maximization

One possible approach to overcome the inherent limitations associated withthe sum-rate maximization criterion is described in [130]. The idea is to dis-tribute system resources so as to maximize the minimum data rate amongstall users for a fixed transmission power and assigned error probabilities. Theresulting strategy is called the max-min rate-maximization concept and ismathematically formulated as

maximize R(min)b = min

1≤m≤M

N−1∑n=0

bm,n

(6.53)

with respect to the bit assignments bm,n and subject to the constraintsEqs. (6.50), (6.51). The rationale behind the “max-min” operation in



Eq. (6.53) is to assign more power to users exhibiting poor channel condi-tions so that they can achieve a data rate comparable to that of other userswith better channel quality.

Unfortunately, the problem stated in Eq. (6.53) is not convex and canonly be solved through a numerical search over all admissible bit assign-ments satisfying Eqs. (6.50) and (6.51). In practical applications this searchturns out to be prohibitively complex due to the large number of possiblecandidate assignments. A way out is offered by the use of Lagrangian re-laxation (LR) techniques, where the Lagrange method of optimization isapplied to an integer parameter which is relaxed to take on noninteger val-ues. The LR approach is adopted in [130] to transform Eq. (6.53) intoa similar but more tractable optimization problem. In particular, the re-quirement bm,n ∈ 0, 1, . . . , bmax is relaxed by allowing bm,n to take onany noninteger value within the interval [0, bmax]. In addition, a new set ofvariables αm,n is introduced to indicate the percentage of times each sub-carrier is shared by a given user. This amounts to considering a very largenumber of OFDM blocks (say JB) where users are allowed to time-sharethe available subcarriers. In this respect, αm,n represents the ratio betweenthe number of blocks where the nth subcarrier is assigned to the mth userand the total number of blocks JB . Clearly, the assumption behind thisapproach is that the users’ channel responses do not change significantlyover a timing interval spanning JB blocks.

After scaling both the transmit power and data rate by the correspond-ing time-sharing factor αm,n, the new optimization problem is stated as

maximize min1≤m≤M

N−1∑n=0

αm,nbm,n

(6.54)

with respect to bm,n and αm,n, where maximization is subject toM∑

m=1

N−1∑n=0

αm,nf(bm,n, pm,max)

γm,n≤ Pbudget, (6.55)

andM∑

m=1

αm,n = 1, (6.56)

for n = 0, 1, . . . , N−1, with bm,n ∈ [0, bmax] and αm,n ∈ [0, 1]. As indicatedin [130], the solution to the above problem is found iteratively by means ofstandard optimization software as long as the rate-power function f(b, p)is convex with respect to b. However, this solution cannot directly be used



for a couple of reasons. A first difficulty is that in general the number bm,n

of allocated bits is noninteger and may not correspond to any practicalmodulation/coding scheme. In addition, some of the quantities αm,n maybe within (0, 1), thereby indicating a time-sharing allocation policy. Thisrepresents a potential problem in most wireless communication systemssince the channel responses are typically time-varying and do not keepunchanged long enough to make time-sharing a feasible solution.

6.2.4 Multiuser margin maximization

In real-time multimedia communications, the users’ bit rates are generallydictated by the employed data compression algorithms. In such a case thesystem resources cannot be assigned according to the RMC criterion asthere is no guarantee that each user can meet its individual rate require-ment. When a specified throughput must be retained for each user, themargin maximization concept (MMC) turns out to be the most appropriateapproach for adaptive resource allocation. This strategy aims at minimizingthe total power consumption under fixed constraints in terms of individualbit rates and error probabilities. This feature makes it particularly suitedfor applications where different classes of services must simultaneously besupported.

To fix the ideas, we denote Rm the number of information bits of themth user that must be conveyed by each OFDM block and call pm,max themaximum admissible BER. Then, recalling that the power allocated to themth user over the nth subcarrier is given by Pm,n = f(bm,n, pm,max)/γm,n,we state the multiuser MMC optimization problem as

minimize PT =M∑

m=1

N−1∑n=0

f(bm,n, pm,max)γm,n

, (6.57)

with respect to the bit assignments bm,n, where bm,n ∈ 0, 1, . . . , bmaxand subject to

N−1∑n=0

bm,n ≥ Rm, (6.58)

for m = 1, 2, . . . , M , and

bm,n = 0, (6.59)

if bm′,n 6= 0 for all m′ 6= m.The constraints Eq. (6.58) specify the users’ rate requirements while

Eq. (6.59) avoids that a given subcarrier is shared by more than one user.



It is worth noting that in some works related to DSL applications theindividual rate requirements in Eq. (6.58) are replaced by a single sum-rateconstraint [83–85]. Although this approach has the advantage of increasingthe number of degrees of freedom for the minimization of PT , it has thefundamental drawback of not considering fairness among users.

Similarly to the RMC policy, the multiuser MMC criterion results intoa combinatorial optimization problem whose solution requires an exhaus-tive search over all possible bit assignments. The complexity associatedwith the exhaustive search turns out to be prohibitive for practical imple-mentation. Again, the use of Lagrangian relaxation techniques proves tobe useful as it provides a computationally manageable (yet suboptimum)solution. Following this approach, we still allow users to time-share eachsubcarrier over a number JB of OFDM blocks and assume that bm,n cantake any noninteger value within the interval [0, bmax]. Then, calling αm,n

(m = 1, 2, . . . ,M) the time-sharing factors for the nth subcarrier, we for-mulate a modified MMC-based optimization problem as

minimize PT =M∑

m=1

N−1∑n=0

αm,nf(bm,n, pm,max)

γm,n(6.60)

with respect to bm,n and αm,n, subject toN−1∑n=0

αm,nbm,n = Rm, for m = 1, 2, . . . , M. (6.61)

andM∑

m=1

αm,n = 1, for n = 0, 1, . . . , N − 1. (6.62)

where αm,n ∈ [0, 1] and bm,n ∈ [0, bmax]. A numerical solution to the aboveproblem is found in [172] using convex optimization techniques. The onlyrequirements are that f(b, p) is convex with respect to b and the aggre-gate data rate is less than Nbmax (which is the maximum number of bitsthat one OFDM block can convey). As mentioned previously, however, atime-sharing allocation policy is hardly usable in a wireless scenario as aconsequence of the time-varying nature of the channel responses. Further-more, the fact that bm,n can take any value within [0, bmax] poses somedifficulties in the selection of a practical modulation scheme that may at-tain the required bit rate. Note that simply quantizing bm,n and αm,n doesnot provide a feasible solution since the resulting bit allocation is not guar-anteed to satisfy the individual rate requirements specified in Eq. (6.58).



One possible approach to overcome these problems is based on a two-step suboptimal procedure in which subcarrier assignment and bit loadingare performed separately instead of jointly. This strategy has been sug-gested in many works, including [78] and [172]. In particular, in [172] theavailable subcarriers are exclusively allocated to users on the basis of theoptimum time-sharing factors αm,n satisfying Eqs. (6.60)-(6.62). The al-location criterion is that any subcarrier must be assigned to the user whoexhibits the largest time-sharing factor over it. After subcarrier allocation,bit loading is independently performed for each user over the assigned sub-carriers. Any conventional greedy algorithm based on the MMC criterionmay be used for this purpose.

6.2.5 Subcarrier assignment through average channel

signal-to-noise ratio

As mentioned previously, a suboptimum yet practical approach for adap-tive resource allocation in OFDMA systems is based on a strict separationbetween the subcarrier assignment and bit loading tasks. Even in thiscase, however, allocating the available subcarriers to the active users onthe basis of some optimality criterion remains a difficult problem. Therelaxation-based solution described in [172] requires knowledge of the opti-mum time-sharing factors αm,n, which can only be determined iterativelyby means of convex optimization methods. A potential drawback of thisapproach is the large number of iterations that may be required to achieveconvergence.

A simpler scheme suggested in [77] divides the subcarrier assignmenttask in two successive steps. The first step, known as bandwidth allocation,determines the number of subcarriers that each user will get on the basis ofthe individual rate requirements and average channel SNRs. In the secondstage, full channel state information is exploited to properly allocate thesubcarriers to each user. By solving these subproblems separately, a goodassignment of system resources is possible with affordable complexity.

The bandwidth allocation step operates in accordance to the MMCprinciple of minimizing the total power consumption under individual con-straints in terms of data rate and error probability. From a mathematicalviewpoint, the problem is that of determining the number Nm of subcar-riers that must be reserved to the mth user (m = 1, 2, . . . , M) for reliabletransmission of Rm bits per OFDMA block. To simplify the derivation, wetemporarily assume that each user signal undergoes flat-fading distortion



and experiences the same channel SNR over each subcarrier. The latter isset equal to the average SNR across the signal bandwidth and reads

γm =1

Nσ2w

N−1∑n=0

|Hm(n)|2 , for m = 1, 2, . . . , M. (6.63)

In the above hypothesis, the optimal loading strategy results into a uniformbit distribution, which amounts to transmitting bm(Nm) = Rm/Nm bitsover each allocated subcarrier. The total transmission power associated tothe mth user is thus given by

Pm(Nm) =Nm

γm

f(Rm/Nm, pm,max), (6.64)

where f(b, p) is the rate-power function and pm,max denotes the maximumtolerable BER.

Note that Pm(Nm) decreases with Nm if f(b, p) is strictly convex anduniformly increasing as illustrated in Fig. 6.3. Under the above assump-tions, the objective of the bandwidth allocation process is to find the set ofintegers N1, N2, . . . , NM that solves the following optimization problem:

minimize PT =M∑

m=1

Nm

γm

f(Rm/Nm, pm,max) (6.65)

subject to

M∑m=1

Nm = N, (6.66)

and

Nm ∈⌈

Rm

bmax

⌉, . . . , N

, (6.67)

where bmax is the maximum number of bits that can be allocated over anysubcarrier and the notation dxe indicates the smallest integer greater thanor equal to x. The constraint Eq. (6.66) indicates that no more than N

subcarriers are available for all active users, while Eq. (6.67) specifies thata minimum of dRm/bmaxe subcarriers is needed for the mth user to satisfya rate requirement of Rm bits per OFDMA block.

The solution to the above problem is found through the following iter-ative procedure:



Bandwidth allocation based on average SNR (BABS) algo-rithm

• Initialization:1) let Nm = dRm/bmaxe and2) let ∆Pm = Pm(Nm) − Pm(Nm + 1) for each m ∈ M =1, 2, . . . , M.

• Resource allocation iterations: repeat the following procedure:1) if

∑Mm=1 Nm = N then stop the algorithm;

2) m = arg maxm∈M

∆Pm;3) Nm = Nm + 1;4) ∆Pm = Pm(Nm)− Pm(Nm + 1);end.

As is seen, in the initialization stage each user is given the minimumnumber of subcarriers that is needed to satisfy its rate requirement. Thepower saving ∆Pm resulting from the assignment of one additional subcar-rier is also computed for all users. Assuming that there is enough bandwidthto satisfy all individual rate requirements, after initialization a total of

N −M∑

m=1

⌈Rm

bmax

⌉(6.68)

subcarriers are still available for further assignment. Then, at each iterationone additional subcarrier is given to the user m that allows the maximumpower saving and the new saving ∆Pm is evaluated for the selected user.The procedure terminates as soon as the number of allocated subcarriers isequal to N .

It is worth noting that the BABS algorithm only determines the numberof subcarriers that must be reserved to each user. After its application,the next step is to specify which subcarriers are actually to be assigned.This task is accomplished by exploiting knowledge of the users’ channelresponses across the transmission bandwidth. One feasible solution basedon heuristic arguments is presented in [77]. This scheme is known as theamplitude craving greedy (ACG) algorithm as each subcarrier is assignedto the user exhibiting the highest channel gain over it. Clearly, once a userhas obtained the number of subcarriers specified by the BABS algorithm,it cannot bid for any more.



Let Im be the set of subcarrier indices assigned to the mth user anddenote card· the cardinality of the enclosed set. Then, the ACG proceedsas follows:

Amplitude craving greedy (ACG) algorithm

• Initialization:1) let Im = ∅ for each m ∈M = 1, 2, . . . , M.

• Subcarrier assignment iterations: repeat the following procedurefor each subcarrier n ∈ 0, 1, . . . , N − 1:1) m = arg max

m∈M

|Hm(n)|2

;

2) Im = Im ∪ n;3) if cardIm = Nm, then remove m from M ;end.

After initializing all sets Im to ∅, at each iteration a subcarrier is as-signed to that user m exhibiting the maximum channel gain in the set M.If the selected user has obtained the desired number Nm of subcarriers, itsindex is removed from M so as to exclude the user from any further as-signment. To counteract the effect of channel correlation between adjacentsubcarriers, it is recommended that the latter be processed in some randomorder rather than in the natural order n = 0, 1, . . . , N − 1. In addition, theusers’ channel responses should be normalized to a common average energybefore starting the assignment process so that weak users may have a fairchance when bidding against more powerful users.

Simulations indicate that BABS and ACG algorithms perform well un-der realistic channel and data traffic scenarios, thereby providing a com-putationally efficient method for subcarrier allocation in OFDMA systems.After this operation has been completed, bit and power loading is inde-pendently performed for each user over the corresponding set of assignedsubcarriers. Again, greedy techniques based on the MMC criterion can beresorted to if the objective is to guarantee a target throughput under aspecified BER constraint.

6.3 Dynamic resource allocation for MIMO-OFDMA

In recent years, the multiple-input multiple-output (MIMO) technologywith multiple antennas deployed at both the transmit and receive ends



has been shown capable of achieving much higher spectral efficiency thanconventional single-input single-output (SISO) transmission schemes [152].This fact has inspired considerable research interest on dynamic resourceallocation for MIMO-OFDMA. In these applications users are still sepa-rated on a subcarrier basis, but each subcarrier is now characterized bya channel matrix of dimensions NR × NT , with NT and NR denoting thenumber of transmit and receive antennas, respectively. After diagonaliz-ing this channel matrix by means of singular-value-decomposition (SVD),each subcarrier is converted into a set of parallel flat-fading SISO subchan-nels which are commonly referred to as eigenchannels or eigenmodes. Thismeans that we can view a MIMO channel as a source of spatial diversity.The latter can be exploited to improve reliability and coverage by means ofspace-time coding techniques [151] and/or to increase the data rate throughspatial multiplexing [46]. In particular, the presence of several eigenmodesfor each subcarrier offers the opportunity of simultaneously transmittingparallel data streams over the same frequency band, thereby increasing theachievable data throughput to a large extent.

As mentioned previously, in MIMO-OFDMA each subcarrier is exclu-sively assigned to only one user, who can therefore access all the associatedeigenchannels. One possible drawback of this approach is that if some ofthese eigenchannels are deeply faded, they are definitively wasted as noother user is allowed to exploit them. An alternative strategy relies on thepossibility of separating users in the spatial domain so that all of them canaccess the same set of subcarriers. This technique is commonly known asspace division multiple-access (SDMA), and is characterized by increasedspectral efficiency due to the opportunity of frequency reuse. In practice,SDMA is implemented by adopting a beamforming approach where mul-tiple antennas deployed at the BS are used to transmit information overorthogonal spatial channels. The combination of SDMA and OFDMA re-sults in a new technology called SDMA-OFDMA [157]. The research ondynamic resource allocation for SDMA-OFDMA was first pioneered byKoutsopoluos [80] and later investigated in [181] under the constraint ofinstantaneous QoS provisioning.

Recent advances on resource allocation for MIMO multicarrier trans-missions have motivated further investigations on the performance penaltyinduced by imperfect channel state information (CSI). In TDD systems,the BS can exploit the reciprocity between alternative downlink and up-link transmissions to get information about the downlink channel, whereasin a FDD network CSI must be fed back by the mobile terminals on a



dedicated control channel. In MIMO multicarrier systems, the number ofspatial eigenchannels increases linearly with the number of antennas so thatthe amount of CSI that must be returned to the BS is much greater than inSISO transmissions and may far exceed the capacity of the control channel.As a result, in most cases only imperfect or partial channel information isavailable at the BS, which may greatly degrade the performance of existingresource allocation schemes. A few methods have recently been proposedto cope with imperfect channel knowledge in adaptive MIMO multiusersystems. Several sources of degradation have been considered, includingoutdated information due to feedback delay, channel estimation inaccura-cies induced by Gaussian noise [174] and quantized CSI for bandwidth-constrained control channels [175].

6.4 Cross-layer design

The research on dynamic resource allocation is closely related to some re-cent developments in the field of cross-layer design for wireless networks.In a conventional communication system the network protocol is dividedinto several layers, each of which is designed independently of the othersto accomplish some specific tasks. While such an approach reduces thecomplexity involved in the design of a complicated network, it ignores anypossible interaction among different layers. For instance, in a conventionalnetwork protocol the channel estimation process is performed in the phys-ical (PHY) layer while subcarrier assignment is handled by the multiple-access control (MAC) layer without exploiting the interdependence of thesetwo tasks. As discussed throughout this chapter, however, the system per-formance is greatly improved if subcarriers are dynamically allocated to theactive users on the basis of instantaneous channel state information. Somepioneering works in the field of cross-layer design have recently appearedin the literature [81,147].

The need for a cross-layer design approach has been further driven bythe success of wireless networks. In contrast to wired systems, wirelessnetworks are characterized by time-varying channel transfer functions. Asa result, a close collaboration between the PHY layer and upper layers ishighly required in order to more efficiently distribute the available systemresources among users. Some novel approaches have been proposed forgeneral communication systems in which channel information is exploitedto improve either the carrier-sense multiple-access (CSMA) scheme used



in the MAC layer [14] or the transmission scheduling of multiple usersin the network layer [165]. However, most of these techniques have onlybeen devised for single-carrier modulations. Their extension to multicarriersystems is still largely unexplored and needs further investigations.


Chapter 7

Peak-to-Average Power Ratio(PAPR) Reduction

One of the major obstacles to the practical implementation of a multicarriersystem is represented by the relatively high peak-to-average power ratio(PAPR) of the transmitted waveform. Recalling that the OFDM signal is asuperposition of N sinusoids modulated by possibly coded data symbols, thepeak power can theoretically be up to N times larger than the average powerlevel. This fact poses two different problems. The first one is related to theA/D and D/A converters, which must be equipped with a sufficient numberof bits to cover a potentially broad dynamic range. The second difficultyis that the transmitted signal may suffer significant spectral spreading andin-band distortion as a consequence of intermodulation effects induced bya non-linear power amplifier (PA). One possible method to circumvent thisproblem is the use of a large power backoff which allows the amplifier tooperate in its linear region. However, this results into considerable powerefficiency penalty, which translates into expensive transmitter equipmentsand reduced battery lifetime at the user’s terminal. It is thus of interestto look for some efficient schemes that can reduce the occurrence of largesignal peaks at the input of the PA so as to minimize the detrimental effectsof non-linear distortions without sacrificing the power efficiency.

In this chapter we present basic material related to the PAPR mitigationproblem in OFDM transmissions. After defining the PAPR and analyzingits statistical properties, some of the most representative PAPR-reductiontechniques available in the literature are reviewed in detail. We also showhow large amplitude fluctuations of the received signal may affect the designof the automatic gain control (AGC) unit at the receive side.

201



7.1 PAPR definitions

The continuous-time baseband representation of an OFDM signal with N

subcarriers is given by

s(t) =1√N

N−1∑n=0

c(n) ej2πnfcst, 0 ≤ t < T, (7.1)

where c(n) is the data symbol transmitted onto the nth subchannel, fcs

denotes the subcarrier spacing and T = 1/fcs is the data block duration(excluded the cyclic prefix). As indicated in Eq. (7.1), s(t) is the superpo-sition of N modulated complex sinusoidal waveforms, each correspondingto a given subcarrier. In the extreme situation where all sinusoids interfereconstructively, their sum will result into a large signal peak that greatlyexceeds the average power level. Furthermore, assuming that N is ade-quately large, we can reasonably approximate s(t) as a Gaussian randomprocess by virtue of the central limit theorem (CLT). As shown later, thisassumption plays an important role in the statistical characterization of thesignal amplitude.

After baseband processing, s(t) is up-converted to a higher carrier fre-quency fc. The resulting RF waveform is expressed by

sRF (t) = <es(t)ej2πfct

, (7.2)

which represents the actual input to the PA. Thus, strictly speaking thePAPR should be defined over sRF (t) rather than over s(t). However, sincethis approach would lead to some mathematical complications, it is a com-mon practice to measure the PAPR at baseband. This procedure providesaccurate results as long as fc À 1/T , a condition that is always met in allpractical systems.

With the above assumption, the continuous-time PAPR is defined as

γcdef=

max0≤t<T

|s(t)|2

E|s(t)|2 , (7.3)

and is sometimes referred to as the peak-to-mean envelope power (PMEPR)[144, 150]. Without loss of generality, one can normalize s(t) such thatE|s(t)|2 = 1. In this case γc reduces to

γc = max0≤t<T

|s(t)|2 . (7.4)

In principle, the maximum of |s(t)|2 can be computed by setting itsderivative to zero. Unfortunately, this operation is not trivial since the


Peak-to-Average Power Ratio (PAPR) Reduction 203

derivative is a sum of sinusoidal functions and its roots cannot easily befound. To overcome this difficulty, it is expedient to replace the continuous-time waveform s(t) by its samples s(L)

k taken at some rate L/Ts, whereTs = T/N while L is a suitable integer which is commonly referred to asoversampling factor. This leads to the definition of the following discrete-time PAPR

γddef= max

0≤k≤LN−1

∣∣∣s(L)k

∣∣∣2

, (7.5)

where s(L)k is obtained after setting t = kTs/L into Eq. (7.1), i.e.,

s(L)k =

1√N

N−1∑n=0

c(n) ej2πnk/LN , 0 ≤ k ≤ LN − 1. (7.6)

Inspection of Eq. (7.5) reveals that the discrete-time PAPR is computed

through a numerical search over the set∣∣∣s(L)

k

∣∣∣2

; k = 0, 1, . . . , NL− 1

,

thereby avoiding the need for solving highly non-linear equations. More-over, comparing Eqs. (7.4) and (7.5) it is easily seen that γd approaches γc

as L grows large. For this reason, γd is normally employed as a practicalmetric for evaluating the performance of PAPR-reduction techniques. Aninteresting question is how large the oversampling factor must be chosento make γd a sufficiently accurate approximation of the continuous-timePAPR. This issue was first assessed by Tellambura in [155], and representsthe subject of the next section.

7.2 Continuous-time and discrete-time PAPR

For notational simplicity, in the ensuing discussion we denote Pa(t) = |s(t)|2the instantaneous envelope power of the transmitted signal. Then, fromEq. (7.1) we have

Pa(t) =1N

N−1∑n=0

N−1∑

`=0

c(n)c∗(`) ej2π(n−`)t/T , 0 ≤ t < T (7.7)

which can also be rewritten as

Pa(t) =1N

N−1∑n=0

|c(n)|2 +2N

N−1∑m=1

N−1−m∑

`=0

<e

c(`)c∗(` + m) e−j2πmt/T

.

(7.8)



A necessary condition for Pa(t) to achieve its maximum at t = t∗ is that

∂Pa(t)∂t

∣∣∣∣t=t∗

= 0. (7.9)

Thus, the global maximum of Pa(t) is found by evaluating the roots of∂Pa(t)/∂t = 0 and comparing the values taken by Pa(t) over these roots.As mentioned previously, solving the equation ∂Pa(t)/∂t = 0 is in general aformidable task due to the non-linear nature of the trigonometric functionsin Eq. (7.8). One possible method to circumvent this obstacle is proposedin [155] by transforming Pa(t) into a linear sum of Chebyshev polynomi-als. Unfortunately, this procedure can only be used in conjunction withreal-valued data symbols and does not apply to general PSK or QAM con-stellations. For this reason, in the sequel we limit our attention to a BPSKmodulation where c(n) ∈ ±1.

We begin by rewriting Eq. (7.8) as

Pa(t) =N−1∑m=0

βm cos(2πmt/T ), (7.10)

with

βm =

1 m = 0

2N

N−1−m∑

`=0

c(`)c(` + m) , m = 1, 2, . . . , N − 1.(7.11)

Then, we recall that the mth order Chebyshev polynomial is defined asTm(t) = cos(m cos−1 t) and can be computed through the following recur-sion [122]

Tm(t) = 2t · Tm−1(t)− Tm−2(t), m = 2, 3, . . . (7.12)

with T0(t) = 1 and T1(t) = t. Hence, using the identity Tm(cos θ) =cos (mθ), we rewrite Eq. (7.10) as

Pa(t) =N−1∑m=0

βmTm [cos(2πt/T )] , (7.13)

which provides an expression of Pa(t) as a combination of Chebyshev poly-nomials in the variable ξ = cos(2πt/T ). Computing the derivative ofEq. (7.13) with respect to t yields

∂Pa(t)∂t

= −2π

Tsin(2πt/T ) ·Q [cos(2πt/T )] , (7.14)



where Q(ξ) is the following polynomial of order N − 2

Q(ξ) =N−1∑m=0

βm∂Tm (ξ)

∂ξ. (7.15)

Inspection of Eq. (7.14) reveals that the stationary points of Pa(t) satisfyeither sin(2πt/T ) = 0 or Q [cos(2πt/T )] = 0. Recalling that 0 ≤ t < T , theformer equation is solved by t/T = 0 or 1/2. As for the latter equation, itssolutions are given by

tiT

=cos−1(ξi)

2π, i = 1, 2, . . . , I. (7.16)

where ξi; i = 1, 2, . . . , I are all real-valued roots of Q(ξ) lying in theinterval [−1, +1]. Clearly, I ≤ N − 2 since Q(ξ) is a polynomial of degreeN − 2 and has a total of N − 2 roots. The latter can be computed numer-ically using one of the many software packages that provide the roots of apolynomial equation.

Once the stationary points of Pa(t) have been found, they are collectedinto a set

Λ =

0, 1/2,cos−1 (ξ1)

2π,cos−1 (ξ2)

2π, · · · ,

cos−1 (ξI)2π

. (7.17)

The continuous-time PAPR γc is eventually computed by evaluatingPa(t) over the entries of Λ and picking up the maximum, i.e.,

γc = maxλ∈Λ

Pa(λ) . (7.18)

Figure 7.1 illustrates the complementary cumulative distribution func-tion (CCDF) of γc and γd for a BPSK-OFDM signal with N = 32 sub-carriers. The quantity γc is computed from Eq. (7.18) using the discussedprocedure based on Chebyshev polynomials, while γd is obtained by look-

ing for the maximum in the set∣∣∣s(L)

k

∣∣∣2

; k = 0, 1, . . . , NL− 1

with either

L = 1, 2 or 4. In any case, the curves represent the probability that themeasured PAPR exceeds the threshold γ indicated on the horizontal axis.We see that the discrete-time PAPR obtained without oversampling (L = 1)is only a rough estimate of γc, meaning that some caution must be takenwhen γd is used as a measure of the PAPR. As expected, the differencebetween γd and γc reduces as L increases and becomes negligible whenL = 4. These results validate the rule-of-thumb idea that the discrete-timePAPR with four-time oversampling provides an accurate approximation ofthe continuous-time PAPR [52].



0 2 4 6 8 10 1210

−5

10−4

10−3

10−2

10−1

100

γ (dB)

Pro

b (P

AP

R >γ

)

γc

γd ( L=1)

γd ( L=2)

γd ( L=4)

Fig. 7.1 CCDF of γc and γd for a BPSK-OFDM signal with different oversamplingfactors.

It is worth pointing out that the results reported in [155] has only beenverified on BPSK systems. Whether the same conclusions are valid withhigher-order modulations is still an open question which is worth for furtherinvestigations. Nevertheless, the pioneering work of [155] has laid down thefundamental guidelines for the identification of a practical PAPR metric. Inthe sequel, the quantity γd obtained from the oversampled time-domain se-quence s(L)

k is used as a measure of the true PAPR and exploited to assessthe performance of the PAPR-reduction techniques considered throughoutthis chapter.

7.3 Statistical properties of PAPR

The statistical properties of the PAPR are normally given in terms of thecorresponding CCDF. From the central limit theorem we know that the realand imaginary parts of the time-domain samples s(L)

k can reasonably beapproximated as statistically independent Gaussian random variables with



zero-mean and variance σ2 = 1/2 (recall that the signal has been normalized

such that E|s(t)|2 = 1). This means that∣∣∣s(L)

k

∣∣∣2

follows a central chi-square distribution with two degrees of freedom [115], and its cumulativedistribution function (CDF) is thus given by

Pr∣∣∣s(L)

k

∣∣∣2

≤ γ

= 1− e−γ , for γ ≥ 0. (7.19)

The CDF of γd is easily computed when L = 1 [109] since in this casethe N samples s(L)

k are mutually independent and we can write

Pr γd ≤ γ =N−1∏

k=0

Pr∣∣∣s(L)

k

∣∣∣2

≤ γ

. (7.20)

Substituting Eq. (7.19) into Eq. (7.20) provides the CCDF of γd in theform

F (γ) = 1− Pr γd ≤ γ= 1− (1− e−γ)N

.(7.21)

Unfortunately, results obtained with L = 1 have a rather scarce practicalrelevance since in this case the quantity γd is only a rough approximationof the true PAPR. The use of an oversampling factor L ≥ 2 introduces astatistical correlation among neighboring samples, which makes Eq. (7.21)a poor approximation of the true CCDF. Many attempts have been made toderive more accurate expressions of the CCDF in situations where the signalsamples are statistically dependent. For instance, in [104] the oversamplingeffect is taken into account by introducing an ad-hoc parameter α in theexponent of Eq. (7.21), yielding

F (γ) = 1− (1− e−γ

)αN. (7.22)

Numerical results indicate that α = 2.8 is a good choice when N is ade-quately large and L ≥ 4.

Figure 7.2 compares the analytical result Eq. (7.22) with the simulatedCCDF of γd for a QPSK-OFDM signal. The oversampling factor is L = 8and the number of subcarriers is N = 64, 256 or 1024. As expected, theprobability that the signal power exceeds a given threshold increases withN . Furthermore, we see that Eq. (7.22) represents a reasonable approxi-mation of the true CCDF, especially for large values of N .

Since the occurrence of large peaks in the envelope of OFDM signals isan event with non-negligible probability, effective PAPR-mitigation tech-niques are essential to enable the use of efficient non-linear power ampli-fiers without incurring severe spectral spreading and/or in-band distortion.Some popular methods for PAPR reduction are described in the subsequentsections.



4 5 6 7 8 9 10 11 1210

−6

10−5

10−4

10−3

10−2

10−1

100

γ (dB)

Pro

b (P

AP

R >γ

)

SimulatedAnalysis

N = 64

N = 256

N = 1024

Fig. 7.2 CCDF of γd for a QPSK-OFDM signal with oversampling factor L = 8 anddifferent number of subcarriers.

7.4 Amplitude clipping

The simplest approach to limit the amplitude peaks in a multicarrier wave-form is to deliberately clip the signal before amplification [111]. This oper-ation is normally accomplished at baseband using a soft envelope limiter. Ifclipping is directly applied to the analog signal s(t) (i.e., after D/A conver-sion), the output y(t) of the limiter appears as indicated in Fig. 7.3, whereA is the maximum permissible amplitude over which the signal is clipped.

The distortion caused by the clipping process is mathematically ex-pressed as

d(t) = y(t)− s(t), (7.23)

and is viewed as an additional source of noise. Since the derivative of d(t)exhibits discontinuities at the clipping instants, its bandwidth is theoreti-cally infinite. This means that in general amplitude clipping gives rise toin-band distortion as well as out-of-band emission. The former degradesthe bit-error-rate (BER) performance while the latter reduces the spectral



-A

A

s ( t ) y ( t )

Unclipped signal Clipped signal

-A

A

t t

Fig. 7.3 The effect of clipping on the transmitted signal.

efficiency of the communication system. Filtering after clipping can reduceout-of-band radiation to a large extent, but may also produce some peakregrowth in the filtered signal [88].

7.4.1 Clipping and filtering of oversampled signals

In practical applications clipping and filtering is performed digitally (i.e.,before D/A conversion) on an oversampled version of the OFDM signal [52].Letting J ≥ 1 be the employed oversampling factor, we denote

s(k) =1√N

N−1∑n=0

c(n) ej2πnk/JN , 0 ≤ k ≤ JN − 1. (7.24)

the samples of s(t) corresponding to a given block of data c =[c(0), c(1), . . . , c(N − 1)]T . Note that J should not be confused with pa-rameter L defined in Sec. 7.1. Indeed, the former is the oversampling factorthat is actually used in the OFDM transmitter to execute clipping and fil-tering operations, while the latter is just a parameter employed in computersimulations for PAPR measurements.

As illustrated in Fig. 7.4, the oversampled data sequence, s =[s(0), s(1), . . . , s(JN − 1)]T , can be efficiently generated as the IDFT ofthe zero-padded data block c(ZP ), which is obtained by extending c with



c

CLIPPING & FILTERING

Zero padding

(ZP) c JN -point IDFT

s Soft limiter

s LPF D/A

y y ( t )

Fig. 7.4 Clipping and filtering operations on the oversampled OFDM signal.

(J − 1)N zeros

c(ZP ) = [c(0), c(1), . . . , c(N − 1), 0, 0, . . . , 0︸︷︷︸(J−1)N zeros

]T . (7.25)

Each sample s(k) is then clipped by a soft envelope limiter. Lettingρkejϕk be the representation of s(k) in polar coordinates, the output fromthe limiter is given by

s(k) =

s(k),Aejϕk ,

if ρk ≤ A

if ρk > A.(7.26)

It is a common practice to normalize the clipping level A to the root-mean-square (rms) value of the input signal. This results into the followingclipping ratio (CR)

µ =A√Pin

, (7.27)

where Pin =E|s(k)|2 is the average power of the unclipped samples.As is intuitively clear, the clipping process leads to a certain reduction of

the output power. If the OFDM signal can be modeled as a zero-mean cir-cularly symmetric complex Gaussian process, the amplitude ρk is Rayleighdistributed and the average power of the clipped samples turns out to be

Pout = (1− e−µ2)Pin. (7.28)

Note that the difference between Pout and Pin reduces as µ grows largeand becomes zero when µ = ∞, which corresponds to an ideal systemwithout clipping.

As mentioned earlier, in general the power spectral density (PSD) of thenon-linear distortion introduced by the amplitude limiter has a theoreticallyinfinite bandwidth. Hence, aliasing will occur if clipping is carried outon the samples s(k) rather than on the continuous-time signal s(t). In



particular, when clipping is done at the Nyquist rate (J = 1), the spectrumof the resulting distortion is folded back into the signal bandwidth. Thisgives rise to considerable in-band distortion, with ensuing limitations of theerror-rate performance. Furthermore, extensive simulations indicate thatthe PAPR reduction capability of Nyquist-rate clipping is not so significantdue to considerable peak regrowth after D/A conversion [108, 110]. As aresult, clipping is normally performed on an oversampled version of theOFDM signal (J > 1).

The oversampled approach has the advantage of reducing in-banddistortion and peak regrowth to some extent, but inevitably generatesout-of-band radiation that must be removed in some way. The con-ventional solution to this problem is to pass the clipped samples s =[s(0), s(1), . . . , s(JN − 1)]T through a low-pass filter (LPF) as indicatedin Fig. 7.4. This produces a vector y = [y(0), y(1), . . . , y(N − 1)]T of time-domain samples, which are extended by the cyclic prefix (CP) and fed tothe D/A converter. The resulting baseband waveform is then upconvertedand passed to the power amplifier before being launched over the channel.

LPF

JN -point DFT

Out-of-band removal

s y c ' c N -point IDFT

Fig. 7.5 Filtering process to remove out-of-band radiation.

The filtering process is outlined in Fig. 7.5. The sequence s is trans-formed in the frequency domain through a DFT operation which producesthe following vector of length JN

c′ = [c(0), c(1), . . . , c(N − 1)︸︷︷︸in-band components

, c(N), c(N + 1), . . . , c(JN − 1)︸︷︷︸out-of-band components

]T , (7.29)

with entries

c(n) =1

J√

N

JN−1∑

k=0

s(k)e−j2πnk/JN . 0 ≤ n ≤ JN − 1. (7.30)



Next, out-of-band radiation is suppressed by discarding the last (J−1)Nelements of c′ (out-of-band components) while leaving the first N ele-ments unaltered (in-band components). This yields a vector of N modifiedfrequency-domain samples c = [c(0), c(1), . . . , c(N − 1)]T , which is in fact adistorted version of the original data block c. Vector c is then transformedback in the time domain through an N -point IDFT, which yields the se-quence y of N modified time-domain samples. After D/A conversion, theanalog signal y(t) can be expressed in terms of the modified symbols c(n)as

y(t) =1√N

N−1∑n=0

c(n)ej2πnfcst, 0 ≤ t < T. (7.31)

It is worth noting that the filtering procedure sketched in Fig. 7.5 isequivalent to an ideal brick-wall low-pass filter which totally eliminatesout-of-band radiation regardless of the oversampling factor J . Clearly, theentire filtering process becomes useless when clipping is performed at theNyquist rate. The reason is that in this case there are no out-of-bandcomponents to be suppressed in c′ and, in consequence, the architecturein Fig. 7.5 reduces to a pair of N -points DFT/IDFT units, which simplyprovides y = s.

Albeit necessary for suppressing out-of-band emission, the filtering-after-clipping approach results into some peak regrowth. A consequence ofthis fact is that the analog signal y(t) may occasionally exceed the clippinglevel A at some instants. As reported in many works, however, filtering theoversampled and clipped version of the OFDM signal produces much lesspeak regrowth than clipping at Nyquist rate. This conclusion is also sup-ported by the simulation results shown in Fig. 7.6, illustrating the CCDFof the PAPR for a clipped QPSK-OFDM signal with N = 256 subcarriers.The clipping-ratio is set to µ = 1 while the oversampling factor is J = 1, 2or 4. The curve pertaining to the unclipped signal (µ = ∞) is also shownfor comparison. The PAPR of the analog signal y(t) is measured as

γd =max

0≤k≤LN−1

∣∣∣y(L)k

∣∣∣2

P, (7.32)

where y(L)k are samples of y(t) taken at rate L/Ts while P is the power

of the current OFDM block after clipping and filtering, which is given by

P =1N

N−1∑n=0

|c(n)|2 . (7.33)



As discussed in Sec. 7.2, the quantity γd provides an accurate measureof the PAPR as long as parameter L is properly designed. The value L = 8is adopted throughout simulations.

2 3 4 5 6 7 8 9 10 11 12

10−4

10−3

10−2

10−1

100

γ (dB)

Pro

b (P

AP

R >γ

)

µ = ∞

J = 1

J = 2

J = 4

Clipping with oversampling

Fig. 7.6 PAPR CCDF for a clipped and filtered OFDM signal with oversampling.

Inspection of Fig. 7.6 reveals that clipping at Nyquist rate consider-ably reduces the PAPR of the transmitted signal as compared to a systemwithout clipping. However, much better results are obtained if clipping isexecuted on the oversampled waveform. In particular, a PAPR reductionof approximately 2 dB is achieved when J is increased from 1 to 4. Clearly,this advantage comes at the expense of a higher computational complex-ity due to the larger dimension of the IDFT unit in Fig. 7.4 and the needfor filtering the signal after clipping. Theoretical analysis [145] and com-puter simulations [88] indicate that in many cases a good trade-off betweenperformance and complexity is obtained with an oversampling factor of4. Repeated clipping and filtering operations can also be used to furtherreduce the overall peak regrowth after D/A conversion [2].



7.4.2 Signal-to-clipping noise ratio

The in-band distortion affecting the clipped signal is normally measured interms of signal-to-clipping noise ratio (SCNR) [2]. This quantity is definedas the ratio of the average received signal power to the average power ofthe clipping distortion, and can be computed by resorting to the Bussgangs’theorem [134]. To see how this comes about, we consider a conventionalOFDM receiver in which the incoming waveform is low-pass filtered andsampled at Nyquist rate. After discarding the CP, the remaining samplesare passed to an N -point DFT unit to retrieve the information symbols. Incase of ideal timing and frequency synchronization, the DFT output takesthe form

R(n) = H(n)c(n) + W (n), 0 ≤ n ≤ N − 1. (7.34)

where H(n) is the channel response over the nth subcarrier, W (n) accountsfor thermal noise and c(n) is a distorted version of the original symbol c(n).The relationship between c(n) and the clipped sequence s(k) is providedby Eq. (7.30). A more useful expression for c(n) is found by observing thats(k) is the output of a memoryless non-linearity driven by the unclippedsignal s(k), as indicated in Eq. (7.26). If the number of subcarriers isadequately large, from the central limit theorem we know that the sequences(k) is approximately Gaussian distributed with zero-mean. Hence, byapplying the Bussgangs’ theory, we can write the output of the non-linearityas [134]

s(k) = ηs(k) + d(k), 0 ≤ k ≤ JN − 1. (7.35)

where d(k) is a zero-mean distortion term uncorrelated with s(k), while η

is an attenuation factor given by

η =E s(k)s∗(k)

E|s(k)|2 . (7.36)

For a soft envelope limiter characterized by a clipping ratio µ, it can beshown that [108]

η = 1− e−µ2+√

πµ

2erfc(µ), (7.37)

with

erfc(x) =2√π

∫ ∞

x

e−t2dt. (7.38)

Substituting Eq. (7.35) into Eq. (7.30) and bearing in mind Eq. (7.24),yields

c(n) = ηc(n) + D(n), 0 ≤ n ≤ N − 1. (7.39)



where

D(n) =1

J√

N

JN−1∑

k=0

d(k) e−j2πnk/JN , (7.40)

represents the in-band distortion over the nth subcarrier. It is worth notingthat, although the probability density function of d(k) is in general verynon-Gaussian due to the presence of a large peak at zero corresponding tothe unclipped samples, the distribution of D(n) is approximately Gaussianas long as the number of clips occurring in each OFDM block is adequatelylarge. The reason is that in the latter case D(n) is the sum of several non-zero random variables d(k) as indicated in Eq. (7.40), and the central limittheorem can thus be applied.

Inspection of Eq. (7.39) reveals that in general the clipping processresults into a shrinking of the signal constellation plus an added noise-likeeffect. Calling C2 =E|c(n)|2 the power of the original data symbols, theSCNR over the nth subcarrier is found to be

SCNRn =η2C2

PD,n, (7.41)

where PD,n =E|D(n)|2 is the PSD of d(k). Obviously, SCNRn is inde-pendent of the channel frequency response since clipping noise is introducedat the transmitter side and, in consequence, it fades along with the signal.

Substituting Eq. (7.39) into Eq. (7.34) yields

R(n) = H(n) [ηc(n) + D(n)] + W (n), 0 ≤ n ≤ N − 1. (7.42)

which represents the equivalent model of a clipped OFDM transmissionchannel as depicted in Fig. 7.7.

Clipping and filtering

D(n)η

R(n)c(n)c(n)

Multipath channel

H(n) W(n)

Fig. 7.7 Equivalent model of a clipped OFDM multipath channel.



Although some attempts have been made in the literature to derive the-oretical expressions of PD,n, computer simulations are normally employedfor SCNR measurements. In Fig. 7.8 the SCNR is shown as a functionof the clipping ratio µ for an OFDM signal with 256 subcarriers. Theoversampling factor is J = 4 and data symbols are taken from a QPSKconstellation with unit power, i.e., C2 = 1. The results are numericallyobtained by averaging the right-hand-side of Eq. (7.41) over the availablesubcarriers. As is seen, SCNR increases with µ and remains quite largeeven in the presence of severe clipping.

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Clipping ratio, µ

SCN

R (

dB)

Fig. 7.8 SCNR as a function of µ for a QPSK-OFDM signal with 256 subcarriers andoversampling factor J = 4.

The impact of clipping noise on the error-rate performance is shown inFig. 7.9. Here, the BER obtained with several values of µ over an AWGNchannel is illustrated as a function of Eb/N0, where Eb is the average energyper bit after clipping and filtering while N0/2 is the two-sided noise PSD.Compared to the unclipped signal (µ = ∞), the SNR penalty incurredwith µ = 1.0 is 3.5 dB at a target BER of 10−4. The degradation reducesto approximately 0.5 dB when µ ≥ 1.5, while an irreducible error floor is



observed with µ ≤ 0.5. As shown in [110], the distortion caused by theclipping process can be alleviated by means of suitable coding techniques.

0 1 2 3 4 5 6 7 8 9 10 11 1210

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

µ = 0.1

µ = 0.5

µ = 1.0

µ = 1.5

µ = 2.0

µ = ∞

Fig. 7.9 The impact of clipping noise on the error-rate performance of a QPSK-OFDMtransmission over an AWGN channel.

7.4.3 Clipping noise mitigation

Several methods have been proposed in the literature to mitigate the harm-ful effects of clipping noise in OFDM systems. Some of them attempt toretrieve the original amplitude of clipped samples by interpolating the re-ceived oversampled signal [137]. However, correct interpolation requiressome out-of-band emission at the transmit side, thereby leading to a reduc-tion of the spectral efficiency. An alternative scheme that does not requireany bandwidth expansion is derived in [16] making use of iterative inter-ference cancellation techniques. This method operates in the frequencydomain and employs detected data to regenerate the clipping noise dis-tortion. The latter is then subtracted from the DFT output at each newiteration.



To better illustrate this approach, we assume that the receiver has per-fect knowledge of the channel frequency response and collect data decisionstaken at the jth iteration into a vector c(j) = [c(j)(0), c(j)(1), . . . , c(j)(N −1)]T . Then, the clipping-noise canceler proceeds as follows:

(1) The detected symbols c(j)(n) undergo the same clipping and filteringoperations as those performed at the transmitter (see Fig. 7.10). Thisproduces the sequence of N samples c(j) = [c(j)(0), c(j)(1), . . . , c(j)(N−1)]T which, similarly to Eq. (7.39), can be represented as

c(j)(n) = η c(j)(n) + D(j)(n), 0 ≤ n ≤ N − 1. (7.43)

where η is given in Eq. (7.37).(2) The clipping noise terms D(j)(n) are derived from Eq. (7.43) in the

form

D(j)(n) = c(j)(n)− η c(j)(n), (7.44)

and are subtracted from the DFT output so as to obtain a refinedobservation sequence

R(j)(n) = R(n)−H(n)D(j)(n). (7.45)

Substituting Eqs. (7.42) and (7.44) into Eq. (7.45) yields

R(j)(n) = ηH(n)c(n) + H(n)[D(n)− D(j)(n)

]+ W (n), (7.46)

where D(n) − D(j)(n) is the residual clipping noise over the nth sub-carrier.

(3) The refined DFT output R(j) = [R(j)(0), R(j)(1), . . . , R(j)(N − 1)]T isfed to the channel equalization and data detection unit, which deliversnew data decisions c(j+1)(n) (n = 0, 1, . . . , N − 1) to be employed inthe next iteration.

Zero padding

JN -point IDFT

Soft limiter

y Out-of-band

removal

JN -point DFT

c ( j ) c ( j )

Fig. 7.10 Regeneration of the clipped and filtered signal at the receiver.

Simulation results reported in [16] indicate that the accuracy of theestimated clipping noise component D(j)(n) increases with the number ofiterations, thereby improving the error-rate performance. From Fig. 7.10 it



turns out that the crux in the computation is represented by the JN -pointIDFT and DFT pair, which must be performed at each iteration. However,in many cases the required complexity is moderate since incremental gainsdiminish after the first iteration and a couple of iterations are often sufficientto restore the system performance.

7.5 Selected mapping (SLM) technique

One possible approach for PAPR control in multicarrier systems is basedon the idea of mapping the data block c = [c(0), c(1), . . . , c(N − 1)]T intoa set of adequately different signals and then choosing the most favorableone for transmission. This technique is called selected mapping (SLM) andits main concept is shown in Fig. 7.11.

c Q

c 2

(ZP ) c 1

(ZP ) c 2

(ZP ) c Q

s 1

( L ) s 2

( L ) s Q

c

Zero padding

LN -point IDFT

Zero padding

LN -point IDFT

Zero padding

LN -point IDFT

Generate candidate

blocks SELECTOR

1 c ( L )

s ( L ) q

Fig. 7.11 Block diagram of the SLM technique.

As is seen, the transmitter generates a number Q of candidate datablocks cq = [cq(0), cq(1), . . . , cq(N − 1)]T (q = 1, 2, . . . , Q) using some suit-able algorithm. Each block has length N and conveys the same informationas the original data sequence c. The latter is normally included into theset of candidate blocks by letting c1 = c. After transforming all blockscq in the time-domain, the one exhibiting the lowest PAPR is selected fortransmission.

Since the PAPR of the continuous-time waveform cannot precisely becomputed from its Nyquist-rate samples, each candidate block is paddedwith (L − 1)N zeros and fed to a LN -point IDFT unit. This provides Q



oversampled sequences s(L)q (q = 1, 2, . . . , Q) with entries

s(L)q,k =

1√N

N−1∑n=0

cq(n) ej2πnk/LN , 0 ≤ k ≤ NL− 1. (7.47)

and characterized by the following discrete-time PAPRs

γq =max

0≤k≤LN−1

∣∣∣s(L)q,k

∣∣∣2

Pq

, q = 1, 2, . . . , Q. (7.48)

with

Pq =1N

N−1∑n=0

|cq(n)|2 . (7.49)

As mentioned in Sec. 7.2, setting L = 4 is sufficient to capture the peaks ofthe continuous-time waveform.

The selector in Fig. 7.11 computes the quantities γq and chooses thesequence s

(L)q such that

q = arg min1≤q≤Q

γq . (7.50)

The selected sequence is then passed to the D/A converter and the corre-sponding waveform is finally launched over the channel after up-conversionand power amplification.

To better illustrate the PAPR-reduction capability of the SLM tech-nique, we denote Fq(γ) = Pr γq ≥ γ the CCDF of γq and observe that

Fq(γ) = Pr

Q⋂

q=1

(γq ≥ γ)

, (7.51)

since γq is the minimum of the set γq. If the candidate sequences s(L)q

are sufficiently “different”, the random variables γq may be considered asnearly independent and Eq. (7.51) reduces to

Fq(γ) =Q∏

q=1

Fq(γ). (7.52)

Figure 7.12 illustrates function Fq(γ) for N = 256 and some values of Q.The results are derived analytically under the simplifying assumption thateach factor Fq(γ) in Eq. (7.52) can be expressed as indicated in Eq. (7.22).In this case we have

Fq(γ) =[1− (

1− e−γ)αN

]Q

, (7.53)



4 5 6 7 8 9 10 11 1210

−6

10−5

10−4

10−3

10−2

10−1

100

γ (dB)

Pro

b (P

AP

R >γ

)

Q = 1Q = 2Q = 4Q = 8Q = 16

Fig. 7.12 Function Fq(γ) for different values of Q.

with α = 2.8. As expected, the amount of PAPR reduction depends on thenumber Q of candidate sequences. We see that significant gains are achievedin passing from Q = 1 to Q = 4, while only marginal improvements areobserved with higher values of Q.

Unfortunately, the result Eq. (7.53) is only an approximation of theCCDF of γq. The reason is that in practice the quantities γq are not trulyindependent as they are derived from sequences s

(L)q that convey the same

information c. Thus, the question arises as to how candidate blocks cq thatresult into adequately different sequences s

(L)q can be generated.

The solution suggested in [66] employs a set of Q pseudo-random in-terleavers to get permuted versions of the original data block c. In such acase, the entries of cq are given by

cq(n) = c(πq(n)), n = 0, 1, . . . , N − 1. (7.54)

where n → πq(n) is a one-to-one mapping, with πq(n) ∈ 0, 1, . . . , N − 1for all n.

An alternative approach is sketched in Fig. 7.13, where the candidate



Q

c

1

1 b

b 2

b Q

c

2 c

c

Fig. 7.13 Generation of candidate sequences through pseudo-random phase shifts.

blocks are obtained through an element-wise multiplication of c by Q differ-ent pseudo-random phase sequences bq = [ejϕq(0), ejϕq(1), . . . , ejϕq(N−1)]T

[7]. This produces the following modified symbols

cq(n) = c(n)ejϕq(n), n = 0, 1, . . . , N − 1. (7.55)

To reduce the system complexity, the phase shifts ϕq(n) are normally chosenas multiples of π/2. In this way cq(n) is obtained from c(n) by means ofsimple sign inversions, thereby dispensing from any multiplication.

The computational requirement of the SLM technique is mainly relatedto the generation of the sequences s

(L)q . Since this operation involves Q

oversampled IDFTs for each OFDM block, in practice parameter Q mustbe carefully designed so as to guarantee a reasonable trade-off in terms ofsystem complexity and PAPR-reduction capability. Compared with ampli-tude clipping, SLM has the considerable advantage of being distortionlessas it does not produce any inter-modulation among subcarriers nor unde-sired out-of-band emission. Clearly, in order to recover the original datasymbols, the receiver must be informed as to which interleaver or phasesequence has been employed at the transmitter to generate the selectedblock cq. Since both the transmitter and the receiver can store the per-mutation indices πq(n) or phase vectors bq in memory, the integer q

represents the minimum side information that must be sent to the receiver



for each OFDM block. This operation requires log2 Q dedicated bits thatmust carefully be protected against channel impairments since an error inthe reception of q would entail the loss of the entire data block. An SLMtechnique that eliminates the need for any exchange of side information isdiscussed in [9].

7.6 Partial transmit sequence (PTS) technique

In the SLM technique, the data block is mapped into different sequencesof frequency-domain samples. As indicated in Fig. 7.11, in such a casea dedicated IDFT operation is required to measure the PAPR associatedwith each candidate sequence. In applications where system complexityis a critical issue, this approach limits the number of possible candidatewaveforms to only a few units, with a corresponding decrease of the PAPRreduction capability. To circumvent this problem, the partial transmit se-quence (PTS) technique generates candidate sequences in the time-domainrather than in the frequency-domain. In this way, a large set of candidatesis obtained with only a few IDFT operations as it is now explained.

c

cM

c2

1c

1b

2b

Mb

(b)s(L)

(L)v1

(L)v2

(L)vM

LN-pointIDFT

LN-pointIDFT

LN-pointIDFT

Σ

PAPR optimization

Subblockpartitioning

Zeropadding

Zeropadding

Zeropadding

Fig. 7.14 Block diagram of the PTS technique.

Figure 7.14 illustrates the basic idea behind the PTS approach. Theinput data vector c = [c(0), c(1), . . . , c(N − 1)]T is partitioned into M dis-joint subblocks cm = [cm(0), cm(1), . . . , cm(N − 1)]T (m = 1, 2, . . . , M)



with entries

cm(n) =

c(n),0,

if n ∈ Jm

otherwise.(7.56)

The sets Jm collect the indices of subcarriers assigned to the varioussubblocks and satisfy the identities

M⋃m=1

Jm = 0, 1, . . . , N − 1 , (7.57)

and

Jm ∩ J` = ∅, for m 6= `. (7.58)

Hence, from Eq. (7.56) we have

c =M∑

m=1

cm. (7.59)

Three different strategies can be adopted for generating the M subblocks.In the subband design the subcarriers of any subblock occupy adjacentpositions in the signal spectrum, while in the interleaved design they areuniformly spaced over the signal bandwidth. A more versatile approach isbased on a pseudo-random design, where subcarriers are randomly parti-tioned into M clusters. In any case, subblocks of equal size are normallyemployed even though in principle an arbitrary number of subcarriers mightbe included in each subblock.

Returning to Fig. 7.14, we see that vectors cm are concatenated with(L − 1)N zeros and transformed in the time-domain through a bank ofM separate and parallel IDFT units. This operation provides a set ofoversampled vectors v(L)

m ;m = 1, 2, . . . ,M which are referred to as partialtransmit sequences (PTSs). The latter are next combined using M complexrotating factors b = [b1, b2, . . . , bM ]T , with bm = ejϕm . After combining,the time-domain samples

s(L)(b) =M∑

m=1

bmv(L)m (7.60)

are fed to the D/A converter and transmitted over the channel. The objec-tive of the PAPR optimization block is to find the set of phase shifts ϕmthat minimize the PAPR of the transmitted sequence s(L)(b).

To reduce the complexity associated with the optimization problem, thephase shifts are normally constrained to vary in a finite set of W elements.In this case the optimum weighting vector b is computed as

b = arg minb

PAPR[s(L)(b)]

, (7.61)



where bm = ejϕm and ϕm ∈ 2π`/W ; ` = 0, 1, . . . ,W − 1. It is worthnoting that in practice the number of phase shifts that must be optimized isM−1 since we can arbitrarily set b1 = 1 without incurring any performancepenalty. Hence, a total of WM−1 permissible vectors b is to be tested inEq. (7.61), with a complexity that increases exponentially with the numberM of PTSs.

Various techniques have been suggested to reduce the complexity of theoptimization problem stated in Eq. (7.61) [22,51,156]. In the iterative flip-ping algorithm [22], the weighting factors bmare determined one by onein M−1 steps following the natural order m = 2, 3, . . . , M . For illustrationpurposes, we assume W = 2 so that bm ∈ ±1 and recall that b1 is arbi-trarily set to unity without any loss of performance. Then, after initializingbm = 1 for m = 1, 2, . . . ,M , during the first step the algorithm flips the signof b2 and evaluates the PAPRs of the two signals generated with weightingfactors [1, 1, 1, . . . , 1]T and [1,−1, 1, . . . , 1]T . The value b2 that yields thelowest PAPR is then retained and used in the next step, where signals ob-tained with [1, b2, 1, . . . , 1]T and [1, b2,−1, . . . , 1]T are tested to find b3. Theiterative process continues in this fashion until all factors b2, b3, . . . , bMhave been determined.

The flipping algorithm can easily be generalized to any value of W . Inthis case the rotating factors are taken from the set P = ej2π`/W ; ` =0, 1, . . . , W − 1 and W different alternatives are explored at each step.The search complexity associated with the flipping procedure is thus pro-portional to W (M − 1), which translates into considerable computationalsaving with respect to the ordinary PTS technique. The price for this ad-vantage is a certain degradation of the system performance in terms ofPAPR reduction. Better results are obtained by allowing r > 1 weightingfactors to be simultaneously flipped at each new iteration [51]. In general,a suitable design of r allows one to achieve a reasonable trade-off betweenperformance and complexity.

As is intuitively clear, the PAPR reduction capability of the PTS tech-nique improves with M and W due to the increased number of candidatesequences s(L)(b). In order to keep the system complexity to a tolerablelevel, in practice the number of PTSs cannot exceed a few units, while W

is normally set to 4 since in this case ϕm is a multiple of π/2 and no multi-plication is required when rotating and combining the PTSs in Eq. (7.60).Another factor that may considerably affect the system performance is theparticular strategy adopted for generating the M subblocks. Although nu-



merical simulations indicate that the pseudo-random criterion representsthe best choice in terms of PAPR minimization, the subband design isnormally preferred for its simplicity. It is worth noting that when M is apower of two and an interleaved design is adopted for subblock partitioning,a computationally efficient implementation of the IDFT algorithms is pos-sible by taking into account that the majority of elements in each subblockis zero.

r

1 b

2 b

M b

N -point DFT

Subblock partitioning

R

R M

R 2

1 R

X M

X 2

1 X

*

*

*

to channel equalization

and data detection

Fig. 7.15 Coherent receiver for an OFDM system employing the PTS technique.

Figure 7.15 illustrates the block diagram of an OFDM receiver for asystem employing the PTS technique. The received samples r are trans-formed in the frequency-domain through an N -point DFT operation andthe resulting vector R is partitioned into M subblocks R1,R2, . . . , RMusing the same partitioning policy employed at the transmitter. The entriesof Rm are given by

Rm(n) =

H(n)c(n)bm + W (n),

0,

if n ∈ Jm,

otherwise.(7.62)

where H(n) is the channel frequency response over the nth subcarrier whileW (n) represents thermal noise.

The subblocks are then rotated back so as to generate M vectorsX1, X2, . . . , XM, with Xm = b∗mRm. Recalling that bm = ejϕm , from



Eq. (7.62) it follows that Xm has entries

Xm(n) =

H(n)c(n) + W ′(n),0,

if n ∈ Jm,

otherwise.(7.63)

with W ′(n) = b∗mW (n). The non-zero elements of Xm are then passedto the channel equalization and data detection unit, which provides finaldecisions on the information symbols conveyed by the mth subblock. Fromthe above discussion it turns out that, similarly to SLM, the PTS is adistortionless technique in which the receiver must be informed about thespecific set of rotation factors that have been employed at the transmitterto generate the time-domain samples. An unambiguous representation ofb has thus to be sent to the receiver as side information. Since b is takenfrom a set of WM−1 admissible vectors, a total of (M − 1) log2 W bits isrequired to represent this side information.

An interesting alternative to the coherent receiver architecture ofFig. 7.15 is represented by a differential decoding approach which, how-ever, can only be used on condition that a subband strategy is adopted forgenerating the M subblocks. Since the entries of any given subblock arerotated by the same angle, the phase relations among subcarriers remain un-changed in each subblock. Hence, if the transmitted information is mappedas phase differences between adjacent subcarriers, differential decoding canbe applied on a subblock-by-subblock basis without requiring knowledge ofthe rotation vector b. Clearly, in this case one additional carrier must beinserted in each subblock to provide the necessary phase reference. Thiscalls for a total of M redundant subcarriers, with a corresponding overheadthat is independent of W .

Figure 7.16 illustrates the performance of the SLM and PTS techniquesin terms of CCDF of the corresponding PAPR levels. The OFDM systemhas N = 256 QPSK modulated subcarriers and the candidate transmitsignals in the SLM algorithm are obtained as depicted in Fig. 7.13 usingQ = 8 different phase vectors. To make comparisons with the same num-ber of IDFT units, M = 8 subblocks are generated in the PTS scheme. Asubband design criterion has been adopted with clusters of 32 adjacent sub-carriers assigned to any subblock. For simplicity, only binary phase shiftsare employed in both SLM and PTS, meaning that bm ∈ ±1. The over-sampling factor is L = 4, which results into 1024-points IDFT operations.In addition to the ordinary PTS scheme, the possibility of reducing thesearch complexity by means of the flipping algorithm is also investigated.We see that the ordinary PTS performs remarkably better than the SLM



4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

100

γ (dB)

Pro

b (P

AP

R >γ

)

Conventional OFDMSLMPTSIterative flipping PTS

Fig. 7.16 Comparison between SLM and PTS in terms of PAPR reduction.

technique. The reason is that the former minimizes the PAPR by exploringamong WM−1 = 27 candidate signals while in the latter the number ofalternative waveforms is limited to Q = 23. However, both schemes ensureconsiderable PAPR reduction as compared to a conventional system wherenothing is done to control amplitude fluctuations. Although the use of theflipping algorithm can significantly reduce the system complexity, a penaltyof approximately 1 dB is incurred with respect to the ordinary PTS.

7.7 Coding

It is a well recognized fact that the frequency diversity offered by the mul-tipath channel cannot fully be exploited in OFDM systems without em-ploying some form of channel coding. A natural question is whether theredundancy introduced by channel coding can be exploited not only forerror correction purposes, but also as a means for minimizing the PAPRof the transmitted waveform. The possibility of using block coding for



Table 7.1 PAPR γd of BPSK-modulated codewords with N = 4.

Code words BPSK symbols PAPR (dB)

b(0) b(1) b(2) b(3) c(0) c(1) c(2) c(3) γd

0 0 0 0 1 1 1 1 6.021 0 0 0 -1 1 1 1 2.320 1 0 0 1 -1 1 1 2.321 1 0 0 -1 -1 1 1 3.720 0 1 0 1 1 -1 1 2.321 0 1 0 -1 1 -1 1 6.020 1 1 0 1 -1 -1 1 3.721 1 1 0 -1 -1 -1 1 2.320 0 0 1 1 1 1 -1 2.321 0 0 1 -1 1 1 -1 3.720 1 0 1 1 -1 1 -1 6.021 1 0 1 -1 -1 1 -1 2.320 0 1 1 1 1 -1 -1 3.721 0 1 1 -1 1 -1 -1 2.320 1 1 1 1 -1 -1 -1 2.321 1 1 1 -1 -1 -1 -1 6.02

PAPR reduction was originally proposed in the seminal work [69], whereonly codewords exhibiting the lowest PAPR are selected for transmissionwhile discarding all the others. Table 7.1 illustrates the highly-cited exam-ple given in [69], where the discrete-time PAPR is listed for all possible datablocks in a BPSK-OFDM system with N = 4 subcarriers and oversamplingfactor L = 4.

We see that four data blocks are characterized by a maximum PAPRof 6.02 dB and another set of four blocks results into a PAPR of 3.72dB. Clearly, using a suitable coding scheme that avoids transmitting thesesequences helps to reduce the PAPR of the transmitted signal. In theparticular example shown in Table 7.1, this goal is achieved with an oddparity check code of rate 3/4 where the first three elements b(0), b(1), b(2)in each codeword represent the information bits while the fourth element iscomputed as b(3) = b(0)⊕ b(1)⊕ b(2)⊕ 1, with ⊕ denoting the arithmeticaddition in the binary Galois field. In this way the PAPR becomes 2.32 dBfor all codewords, thereby leading to a reduction of 3.70 dB with respect tothe uncoded system. It is shown in [69] that higher gains of 4.58 and 6.02dB are possible in case of N = 8 subcarriers using coding schemes with rates7/8 and 3/4, respectively. Clearly, these benefits are achieved at the price ofsome penalty in terms of spectral efficiency due to the inherent redundancy



introduced in the transmitted signal. Note that the latter is only exploitedfor PAPR reduction purposes rather than to protect information againstchannel impairments. In addition, the method in [69] becomes impracticalfor large values of N since the best codes can only be found through anexhaustive search and prohibitively large look-up tables are required for theencoding and decoding operations.

a Encoder Mapper to OFDM

modulator

c b ( w ) c ' Phase shifter

Fig. 7.17 Coding and phase rotation for simultaneous error control and PAPR reduc-tion.

A more sophisticated approach proposed by Jones and Wilkinson in [68]relies on the design of combined coding schemes for simultaneous errorcontrol and PAPR reduction. This solution employs conventional linearblock codes to achieve the desired level of error protection and the coderedundancy is subsequently exploited to minimize the PAPR. The basicidea behind this method is sketched in Fig. 7.17. Let ϑ be the numberof points in the employed constellation and assume that a (Nϑ, k) binaryblock code has been chosen for its correction property. As is seen, a block a

of k information bits is first transformed into a vector b of Nϑ coded bits.The latter is next divided into N adjacent segments of length ϑ, whereeach segment is independently mapped onto a modulation symbol c(n).This produces a codeword c = c(0), c(1), . . . , c(N − 1) of length N foreach block of k information bits. We denote C =cm; m = 1, 2, 3, . . . , 2kthe set of all possible codewords. Then, in an attempt of reducing thePAPR, the codewords are element-wise multiplied by a same rotating vectorw =

[ejψ(0), ejψ(1), . . . , ejψ(N−1)

]T, where the phase shifts ψ(n) vary in

the compact set [0, 2π]× [0, 2π]× · · · × [0, 2π]. The rotated version of cm isdenoted c′m(w) and reads

c′m(w) = cm(0)ejψ(0), cm(1)ejψ(1), . . . , cm(N − 1)ejψ(N−1). (7.64)

Since distances among codewords remain unchanged after rotation, the newcode C′(w) =

c′m(w); m = 1, 2, 3, . . . , 2k

has the same error correction

capability as the original code C. However, it may exhibit a lower PAPR ifthe phase shifts are suitably chosen. Hence, for a given code C, the problem



is to find an optimal vector w = [ejψ(0), ejψ(1), . . . , ejψ(N−1)]T such that

w = arg minwPAPR [C′(w)] , (7.65)

where PAPR[C′(w)] is defined as

PAPR [C′(w)] = maxc′m(w)∈C′(w)

PAPR [c′m(w)] , (7.66)

with PAPR[c′m(w)] denoting the PAPR of the waveform associated to themth rotated codeword c′m(w).

It is worth noting that in this way PAPR reduction comes for free since,as mentioned previously, both C and C′(w) are perfectly equivalent in termsof error rate performance and decoding complexity. At the receive side, thephase shifts introduced by w can easily be compensated for by appropriatecounter-rotation of the DFT output. For this purpose, w must be knownto the receiver.

The main drawback of the described approach is the heavy computa-tional load that is required to solve the optimization problem Eq. (7.65).An algorithm for finding the optimum rotation vector is discussed in[68] under the assumption that the phase shifts belong to a finite setΨ = 2π`/W ; ` = 0, 1, . . . , W − 1. Unfortunately, this method is onlyapplicable to relatively short codes because of the huge complexity involvedin computing the PAPR of all phase-shifted codewords. A computation-ally efficient solution to this problem is outlined in [150], where a simpli-fied method is proposed to identify codewords characterized by the highestPAPR and a gradient-based iterative minimization technique is next usedto search for the optimum rotation vector. A possible shortcoming is thatin general the objective function in Eq. (7.65) presents various local minimawhich may attract the gradient algorithm toward spurious locks.

A third approach for the design of low-PAPR coding schemes was mo-tivated by the observation that the PAPR of an OFDM signal is at most3 dB if the modulation sequence is constrained to be a member of a Go-lay complementary pair [48, 119]. For a long time these sequences werenot recognized to possess sufficient structure to form a practical codingscheme until a theoretical connection has been established between themand the first- and second-order Reed-Muller codes [32]. This connectionoffers the opportunity to combine the error correcting capability of classi-cal Reed-Muller codes with the attractive PAPR control property of Go-lay complementary sequences. Further improvements to this approach arefound in [33], where a range of flexible coding schemes using binary, qua-ternary and higher order modulations has been designed to achieve desired



tradeoffs in terms of PAPR control, spectral efficiency and error-correctingcapability. Computationally efficient decoding algorithms have also beendeveloped based on the fast Hadamard transform (FHT). A unified theorylinking Golay complementary sets of polyphase sequences and Reed-Mullercodes has been presented by Paterson in [117] and exploited to design abroad range of coding options employing high-order modulations. Unfortu-nately, the usefulness of all these techniques is somewhat limited by the factthat they can only be applied to multicarrier systems with a small numberof subcarriers in order to keep the computational complexity to a tolerablelevel. One possible advantage is that no side information is required at thereceiver to recover the transmitted data symbols.

7.8 Tone reservation and injection techniques

An efficient family of PAPR reduction methods is based on the idea ofadding a data-dependent vector e = [e(0), e(1), . . . , e(N − 1)]T in the fre-quency domain to the original data block c = [c(0), c(1), . . . , c(N − 1)]T soas to reduce the peaks of the resulting OFDM signal. The most representa-tive examples in this family are the tone reservation (TR) and tone injection(TI) techniques [153] which are discussed below. Both schemes have theremarkable advantage of being distortionless, as the added vector can easilybe canceled out at the receiver without incurring any performance loss.

7.8.1 Tone reservation (TR)

c c

e PTRs

generation

N -point IDFT

to D/A conversion

Fig. 7.18 Block diagram of the TR technique.

In the TR approach, the transmitter does not send information overa small set of Q subcarriers which are reserved for PAPR control. These



subcarriers are referred to as peak reduction tones (PRTs) and are normallydistributed in a pseudo-random fashion across the signal bandwidth. Forillustration purposes, we denote Jc = i1, i2, . . . , iQ the set collecting theindices of the PRTs while data-bearing subchannels have indices in the setJ = 0, 1, . . . , N − 1−Jc. As shown in Fig. 7.18, at the transmitter vectorse and c are summed up to form a block c = [c(0), c(1), . . . , c(N − 1)]T offrequency-domain samples with entries

c(n) =

c(n),e(n),

n ∈ J,

n ∈ Jc.(7.67)

The sequence c(n) is then transformed in the time-domain through anN -point IDFT unit and passed to the D/A converter, which provides thecontinuous-time signal

s(t) =1√N

N−1∑n=0

c(n) ej2πnfcst, 0 ≤ t < T. (7.68)

Since e and c are constrained to lie into disjoint frequency subspaces, atthe receiver the information symbols are simply recovered by selecting theoutputs of the DFT with indices in the set J . Clearly, this requires thatthe receiver be informed as to which subcarriers are reserved to the PRTs.

Collecting the non-zero entries of e into a Q-dimensional vector e =[e(i1), e(i2), . . . , e(iQ)]T , the goal of the TR scheme is to find the optimale that minimizes the PAPR of s(t). As we know, a practical approach toaccomplish this task is to replace s(t) by its samples s

(L)k (k = 0, 1, . . . , NL−

1) taken with oversampling factor L ≥ 4. The TR optimization problemcan thus be cast as a constrained quadratic program

minimize γ (7.69)

with respect to e ∈ E and subject to∣∣∣s(L)

k

∣∣∣2

≤ γ, for all k = 0, 1, . . . , NL− 1. (7.70)

where E is the multidimensional space of admissible vectors e and s(L)k is

given by

s(L)k =

1√N

∑

n∈J

c(n)ej2πnk/LN+1√N

∑

n∈Jc

e(n)ej2πnk/LN , 0 ≤ k ≤ NL−1.

(7.71)Finding the exact solution to the above problem is in general a computa-tionally expensive task. However, since we are minimizing a linear function



under quadratic constraints, the problem is also convex. This property maybe exploited to obtain a good, yet suboptimal, solution. For instance, anefficient method to iteratively approach the optimum e has been suggestedin [38] using the sub-gradient algorithm.

Increasing the number Q of PRTs provides the optimization processwith more degrees of freedom. In this way, the PAPR reduction capabilityof the TR technique is improved at the price of a throughput penalty dueto the reduced number of data-bearing subcarriers. In general, a tradeoffbetween these conflicting requirements is sought through a careful design ofparameter Q. Computer simulations indicate that gains of approximately3 dB and 6 dB in terms of PAPR reduction can be achieved with a loss indata rate of less than 0.2% and 5%, respectively.

Another factor that remarkably affects the system performance is theset Jc of PRT positions. Finding the optimal Jc that minimizes the PAPRresults into a combinatorial optimization problem which cannot be solvedwith affordable complexity. However, experimental results indicate thata good selection is obtained by generating a sufficiently large number ofpseudo-random sets and choosing the best one.

In wireline DSL applications, the throughput penalty associated withthe TR technique is partly alleviated by placing the PRTs over frequencysubchannels that would go otherwise unused because of their relatively poorSNRs. Unfortunately, a similar approach cannot be pursued in wirelesssystems since in these applications no fast channel state feedback is availableto adaptively decide which subcarriers should be used to send informationand which others should be reserved to PRTs.

7.8.2 Tone injection (TI)

Tone injection can be viewed as an improvement of the TR technique inthat it aims at reducing the PAPR without sacrificing the spectral efficiency.The basic idea is to send information over all subcarriers using an expandednon-bijective constellation set, where each data symbol is mapped into asubset of equivalent points. The signal peaks are then reduced throughappropriate selection of the constellation point within the subset.

To explain the TI principle, we consider a conventional M -ary QAMconstellation A where 2d is the minimum distance between neighboringpoints. In this case, the real and imaginary parts of each symbol take valuesin the set ±d,±3d, . . . ,±(

√M − 1)d, with

√M denoting the number of

levels per dimension. In the expanded constellation set A, any symbol c



of the original constellation is mapped into one of several equivalent pointsc = c + pD + jqD, where p and q are suitable integers while D is a positivereal number known at the receiver. In the ensuing discussion, we refer toS(c) = c + pD + jqD; p, q ∈ I ⊂ Z as the subset associated to c. Theintegers p and q provide extra degrees of freedom that are exploited toreduce the PAPR of the transmitted signal. Clearly, to ensure that theinformation symbol c can be recovered from c without any ambiguity, it isnecessary that different points of A are mapped onto disjoint subsets of A.As explained in [154], this condition requires a careful design of parameterD. In particular, setting D = 2ρd

√M with ρ ≥ 1 yields disjoint subsets and

results into approximately the same error-rate probability as a conventionalOFDM system without TI.

Subset associated to c =1+ j

Subset associated to c = 1+ j _

Subset associated to c = 1 j _ _

Subset associated to c =1 j _

d

Original constellation

Fig. 7.19 The expanded constellation set in the TI technique.



Figure 7.19 depicts the expanded constellation A in case of ρ = 1 andQPSK symbols (M = 4). For illustration purposes, the integers p and q

are constrained to the set I = −1, 0, 1 . As is seen, A is obtained byreplicating the original constellation A through known translation vectors.Four subsets are present, each containing nine symbols and correspondingto a different symbol of A. Note that the original symbol c can perfectlybe recovered from anyone of the points c ∈ S(c) by simply using a modulooperator that acts independently over the real and imaginary parts of itsinput according to the following rule

MOD(x) = x− 2ρd√

M

⌊x + ρd

√M

2ρd√

M

⌋, (7.72)

where the notation bzc represents the smallest integer not exceeding z. Inpractice, MOD(x) performs a periodic mapping of the complex plane intothe square region xR + jxI

∣∣∣xR, xI ∈ (−ρd√

M, ρd√

M ] with side length

2ρd√

M .Denoting c(n) = c(n) + pnD + jqnD the selected point in the subset

S(c(n)) associated with c(n), the oversampled sequence of time-domainsamples can be written as

s(L)k =

1√N

N−1∑n=0

c(n) ej2πnk/LN , 0 ≤ k ≤ NL− 1. (7.73)

Note that the vector c = [c(0), c(1), . . . , c(N − 1)]T of frequency-domainsamples is obtained as shown in Fig. 7.18 after defining e(n) = pnD+ jqnD

for n = 0, 1, . . . , N − 1.Since it is desirable to reduce the peaks of the transmitted signal as

much as possible, we look for the integers p = p0, p1, . . . , pN−1 andq = q0, q1, . . . , qN−1 that minimize the PAPR of the sequence s

(L)k . This

results into an integer programming problem whose complexity grows ex-ponentially with the number N of available subcarriers. Fortunately, goodapproximations to the optimum solution can be obtained through efficientiterative methods that dispense one from exploring all candidate vectorsp and q. A further reduction of complexity is possible if the expandedconstellation set is only employed over a small fraction of the available sub-carriers. Clearly, this approach reduces the number of candidate vectorsto be explored at the price of some performance loss in terms of PAPRreduction.

Inspection of Fig. 7.19 reveals that the modified symbol c(n) has moreenergy than c(n) whenever pn 6= 0 or qn 6= 0. This means that the TI



technique reduces the PAPR at the expense of a certain increase of thetotal transmission power. However, no loss in data rate is incurred since,contrarily to the TR scheme, all subcarriers are employed to transmit data.At the receiver side, the original symbols c(n) are recovered by pass-ing the decoded sequence c(n) through the modulo operator Eq. (7.72),thereby avoiding the need for any exchange of side information between thetransmitter and receiver.

7.9 PAPR reduction for OFDMA

In OFDMA systems, the available subcarriers are divided into mutuallyexclusive subchannels that are assigned to distinct users for simultaneoustransmission. As illustrated in Fig. 3.14, three different strategies can beadopted to accomplish the subcarrier assignment task. In the subband CASeach subchannel is composed by a set of adjacent subcarriers while in theinterleaved CAS the subcarriers of each user are uniformly spaced over thesignal bandwidth to take advantage of the channel frequency diversity. Themore flexible strategy is represented by the generalized CAS, where usersare provided with the best quality subcarriers that are currently available.

From a physical layer perspective, the OFDMA downlink is essentiallyequivalent to an OFDM system. The only difference is that in OFDMAeach block conveys simultaneous information for multiple subscribers whilein OFDM the transmitted data are intended for a single specific user. Thissuggests that statistical PAPR characterization as well as PAPR reductionmethods devised for single-user OFDM systems can be extended to theOFDMA downlink in a straightforward fashion. A rather different situationoccurs in the OFDMA uplink. Here, each signal employs only a fraction ofthe available subcarriers and the underlying subcarrier assignment schemeis expected to play a major role in determining the PAPR of the transmit-ted waveform. A theoretical analysis presented in [166] indicates that, onaverage, the generalized CAS results into higher signal peaks than the sub-band or interleaved CAS. In any case, the PAPR problem in the OFDMAuplink is not as serious as in the downlink because of the relatively smallnumber of modulated subcarriers. This explains why the topic of PAPRreduction in uplink transmissions has remained largely unexplored up tonow.

In what follows we revisit some of the PAPR control methods describedthroughout this chapter and show how they can be extended to an OFDMA



downlink.

7.9.1 SLM for OFDMA

The SLM technique applies to the OFDMA downlink without any substan-tial modification with respect to the single-user case. The only differenceis that in OFDMA the candidate signals are exclusively generated by shift-ing the phase of the original data symbols while in OFDM they can alsobe obtained through pseudo-random permutations. The latter approach isnot suited for OFDMA as it would result into a modification of the sub-carrier allocation scheme, which is clearly unfeasible in systems employingrigid subband or interleaved CAS. Information about the employed phasesequence is broadcasted to all active terminals using some dedicated sub-carriers. This information is exploited by each user to retrieve its owndata.

7.9.2 PTS for OFDMA

The PTS technique employed in OFDM systems can easily be modified forOFDMA downlink transmissions. In such a case the subcarriers of eachuser are grouped into one or more subblocks, and PTSs are next obtainedby transforming these subblocks in the time-domain. One subcarrier persubblock is reserved to provide information about the phase factor employedover that subblock. At the receiving terminal this subcarrier is extractedand used as a phase reference for data detection over the correspondingsubblock.

7.9.3 TR for OFDMA

The TR approach is applied to the OFDMA downlink exactly in the sameway as in single-user OFDM systems. As suggested in [166], however, acertain reduction of complexity is possible if a set of PRTs is exclusivelyassigned to each user and optimized for the data sequence of that useronly. This results into a suboptimal optimization process in which data ofdifferent users are processed independently at the transmit side for PAPRmitigation. To further reduce the computational load, the amplitude of thePRTs may be optimized over a finite set of values and stored in a look-uptable for every possible information sequence [182]. In this way, there isno need to recompute the optimum PRT values at each new transmitted



block since the latter are simply obtained from the look-up table with theinformation sequence serving as a memory address.

7.10 Design of AGC unit

The presence of large amplitude fluctuations in the OFDM signal requires acareful design of the automatic gain control (AGC) unit and A/D converterat the receiver side. Figure 7.20 illustrates the front-end of a conventionaltwo-branch receiver for digital transmissions.

r (t)

rI (t)

rQ(t)

Powermeasurement

zQ(t) zQ(k) yQ,q

(k)

1/β

yQ(k)

zI(t) zI(k)Sampler Limiter Quantizer

yI (k)

Sampler Limiter Quantizer

A/D converterAGC

I/Qdemodulator

I,q(k)y

RF

Fig. 7.20 Front-end of a typical two-branch receiver.

After I/Q demodulation, the baseband signals rI(t) and rQ(t) are passedto the AGC unit, where they are scaled by a factor 1/β. The resulting sig-nals zI(t) and zQ(t) are next fed to the A/D converter, which consists ofa sampling device plus a quantizer operating over a finite dynamic range[−A, A]. As is intuitively clear, the scaling factor β must properly be de-signed so as to minimize the distortions introduced by the quantizationprocess. For this purpose, it is convenient to model the overall quanti-zation unit as the cascade of a limiter with cutting level A followed by aquantizer with infinite dynamic range. This approach offers the opportunityof separately assessing the impact of clipping distortions and quantizationerrors on the system performance.

Without loss of generality, in the ensuing discussion we let A = 1 anddenote Nb the number of bits reserved to the A/D conversion. The design ofNb depends on many parameters, including the computational requirementas well as the accuracy needed for a given constellation size. In practice,Nb = 10 is commonly adopted for a 64-QAM constellation, while smallervalues of Nb are used with lower order modulations. The AGC gain is



adaptively adjusted on the basis of appropriate power measurements in anattempt of achieving a balanced trade-off between two conflicting require-ments. On one hand, small values of β enlarge the dynamic range of thesignal at the input of the A/D converter, thereby reducing the effects ofquantization errors. On the other hand, a too large signal dynamic is unde-sirable as it increases the occurrence of clipping events. In what follows, welook for the optimum AGC gain that maximizes the SNR at the quantizeroutput. In doing so we limit our attention to the I branch in Fig. 7.20 andneglect the index I for notational simplicity.

If the number of modulated subcarriers is adequately large, we knowthat r(t) can be approximated as a zero-mean Gaussian random processwith some power σ2

r . In this case, samples z(k) at the input of thelimiter are Gaussian distributed with probability density function

pZ(z) =1

σz

√2π

e−z2/(2σ2z), (7.74)

where σz = σr/β is the rms value of z(k). The output of the limiter ismathematically described as

y(k) =

1,

z(k),−1,

if z(k) ≥ 1,

if − 1 < z(k) < 1,

if z(k) ≤ −1.

(7.75)

In practice, we can view y(k) as the sum of the useful signal z(k) plusa clipping noise term wc(k), i.e.,

y(k) = z(k) + wc(k), (7.76)

where wc(k) is obtained after substituting Eq. (7.75) into Eq. (7.76), andreads

wc(k) =

1− z(k),0,

−1− z(k),

if z(k) ≥ 1,

if − 1 < z(k) < 1,

if z(k) ≤ −1.

(7.77)

To proceed further, we define the clipping noise power asPc =E|wc(k)|2. Then, from Eq. (7.77) it follows that

Pc =

−1∫

−∞(1 + z)2pZ(z)dz +

∞∫

1

(1− z)2pZ(z)dz, (7.78)

or, equivalently,

Pc = 2

∞∫

1

(1− z)2pZ(z)dz. (7.79)



Substituting Eq. (7.74) into Eq. (7.79) and performing standard com-putations, yields

Pc =(

1µ2

+ 1)

erfc(

µ√2

)− 1

µ

√2π

e−µ2/2, (7.80)

where erfc(x) is the complementary error function given in Eq. (7.38) whileµ = 1/σz is the clipping crest factor, which is defined as the ratio betweenthe maximum allowable amplitude A = 1 and the rms of z(k). Recallingthat σz = σr/β, we also have µ = β/σr.

Next, we consider the quantization error eq(k) = y(k)−yq(k) introducedby the Nb-bit quantizer. Letting ∆ = 2/2Nb be the quantization step-size,we can approximate eq(k) as a random variable with uniform distributionin the interval [−∆/2, ∆/2) [123]. The power of the quantization noise isthus given by

Pq =∆2

12=

13 · 22Nb

. (7.81)

Neglecting for simplicity the effect of thermal noise, the SNR at theoutput of the A/D converter is found to be

γA/D =PZ

Pc + Pq, (7.82)

where PZ = σ2z is the power of z(k).

Bearing in mind that σ2z = 1/µ2, after substituting Eqs. (7.80) and

(7.81) into Eq. (7.82) we obtain

γA/D =

[(µ2 + 1

)erfc

(µ√2

)− µ

√2π

e−µ2/2 +µ2

3 · 22Nb

]−1

. (7.83)

Figure 7.21 illustrates γA/D as a function of µ for Nb = 8 and 10. Asexpected, at low values of µ clipping noise dominates the system perfor-mance and γA/D increases with µ. However, when the crest factor goesbeyond its optimal value µopt, the SNR starts to decrease since in this casethe quantization error becomes the most critical impairment to the systemperformance. Inspection of Fig. 7.21 reveals that µopt is close to 4 witheither Nb = 8 or 10. These results indicate that optimum performance isachieved when the I/Q components of the received waveform are scaled suchthat their rms is approximately four times smaller than the clipping levelA. This is a consequence of the large amplitude fluctuations characterizingthe OFDM signal.



2 2.5 3 3.5 4 4.5 5 5.5 615

20

25

30

35

40

45

50

55

Crest factor, µ

γ A/D

(dB

)

Nb = 8

Nb = 10

Fig. 7.21 Output SNR vs. µ for an A/D converter with Nb bits.

Recalling that µ = β/σr, we can use the quantities µopt and σr todetermine the optimum AGC coefficient in the form

βopt = µoptσr. (7.84)

While µopt can be inferred from the theoretical curves of Fig. 7.21, anestimate of σr is normally obtained by measuring the average power of thereceived signal as indicated in Fig. 7.20.


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Index

Access point (AP), 9

Automatic gain control (AGC), 239

Bluetooth, 8

Carrier assignment schemes (CAS),77

generalized, 78

interleaved, 78

subband, 78

Channel equalization, 108

Channel estimation

decision-directed channelestimation, 129

pilot-aided channel estimation, 111

2D Wiener interpolation, 115

two-cascaded 1D interpolationfilters, 117

subspace-based channel estimation,126

Channel impulse response (CIR), 53

Clipping ratio (CR), 210

Code-division multiple-access(CDMA), 7

Code-division multiple-access-2000(CDMA-2000), 7

Coherence bandwidth, 23

Coherence time, 26

Complementary cumulativedistribution function (CCDF), 207

Cyclic prefix (CP), 39

Digital Audio Broadcasting (DAB),14

Digital subscriber line (DSL), 11

Digital video broadcasting-terrestrial(DVB-T), 2

Doppler spread, 24

Equalization

maximum-ratio-combining (MRC),111

minimum-mean-square-error(MMSE), 34, 108

zero-forcing (ZF), 34, 109

Excess delay, 20

Root-mean-squared (RMS) delayspread, 22

Expectation-maximization algorithm(EM), 90

Fading

large-scale fading, 19

small-scale fading, 19

Fading channels, 27

frequency and time-selective fadingchannels, 33

frequency-nonselective andslowly-fading channels, 28

frequency-selective fading channels,29

time-selective fading channels, 31

Frequency attenuation factor, 105

Frequency-division multiple-access

255



(FDMA), 5

Greedy power allocation techniques,171bit-filling algorithm, 172bit-removal algorithm, 172uniform-BER and uniform-power

allocation bit loadingalgorithm, 175

uniform-power allocation bitloading algorithm, 174

Group Special Mobile (GSM), 6

Hard-decision decoding, 154High performance LAN

(HiperLAN2), 10

IEEE 802.11 family, 10IEEE 802.15, 8Interchannel interference (ICI), 56

Joint data detection and channelestimation, 129

Link adaptationbit adaptation, 166open/closed-loop adaptation, 179subband adaptation, 178

Log-likelihood ratio (LLR), 154

Multi-user detection (MUD), 8Multimedia mobile access

communication (MMAC), 10Multiple-access interference (MAI), 8Multiple-input multiple-output

(MIMO), 13Multiuser power allocation

margin-maximization concept, 192max-min rate-maximization

concept, 190rate-maximization concept, 188

OFDM-based multiple-accessschemes, 46MC-CDMA, 46OFDM-TDMA, 46

OFDMA, 48Orthogonal frequency division

multiplexing (OFDM), 37

PAPRContinuous-time PAPR, 202discrete-time PAPR, 203peak-to-mean envelope power

(PMEPR), 202PAPR reduction techniques

amplitude clipping, 208coding, 228partial transmit sequence (PTS),

223selected mapping (SLM), 219tone injection (TI), 234tone reservation (TR), 232

Path loss, 19Personal area networks (PANs), 8Power allocation

margin-maximization concept(MMC), 167

rate-maximization concept (RMC),166, 188

Power delay profile (PDP), 21

Rate-power function, 167Rayleigh fading

Jake’s model, 32power spectral density, 32

Reference blocks for synchronization,61

Scattered pilot patterns, 112Signal-to-clipping noise ratio

(SCNR), 214Soft information, 151Software Defined Radio (SDR), 13Space-alternating projection

expectation-maximizationalgorithm (SAGE), 90

Subscriber Station (SS), 11Synchronization

frequency synchronization, 51frequency acquisition, 69frequency tracking, 72


Bibliography 257

Morelli and Mengali scheme,71

Schmidl and Cox scheme, 69SNR loss, 59

sampling clock synchronization, 51timing synchronization, 51

fine timing tracking, 67Schmidl and Cox scheme, 63Shi and Serpedin scheme, 65SNR loss, 56timing acquisition, 62

Synchronization impairmentsfrequency offset, 58timing offset, 54

Time-division multiple-access(TDMA), 6

Virtual carriers (VC), 37

Water-filling principlemultiple users, 184single user, 161

Wireless local area networks(WLANs), 1

Wireless metropolitan area networks(MANs), 1

Worldwide Interoperability forMicrowave Access Forum (WiMax),12

broadband wireless communications

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