wideband beamforming concepts and...

WIDEBANDBEAMFORMINGCONCEPTS ANDTECHNIQUES

Wei LiuUniversity of Sheffield, UK

Stephan WeissUniversity of Strathclyde, UK

A John Wiley and Sons, Ltd., Publication

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WIDEBANDBEAMFORMING

Wiley Series on Wireless Communications and Mobile Computing

Series Editors: Dr Xuemin (Sherman) Shen, University of Waterloo, CanadaDr Yi Pan, Georgia State University, USA

The “Wiley Series on Wireless Communications and Mobile Computing” is a series ofcomprehensive, practical and timely books on wireless communication and networksystems. The series focuses on topics ranging from wireless communication and codingtheory to wireless applications and pervasive computing. The books provide engineersand other technical professionals, researchers, educators, and advanced students in thesefields with invaluable insights into the latest developments and cutting-edge research.

Other titles in the series:

Misic and Misic: Wireless Personal Area Networks: Performance, Interconnection, andSecurity with IEEE 802.15.4, January 2008, 978-0-470-51847-2

Takagi and Walke: Spectrum Requirement Planning in Wireless Communications: Modeland Methodology for IMT-Advanced, April 2008, 978-0-470-98647-9

Pérez-Fontán and Espiñeira: Modeling the Wireless Propagation Channel: A simulationapproach with MATLAB, August 2008, 978-0-470-72785-0

Ippolito: Satellite Communications Systems Engineering: Atmospheric Effects, SatelliteLink Design and System Performance, August 2008, 978-0-470-72527-6

Lin and Sou: Charging for Mobile All-IP Telecommunications, September 2008,978-0-470-77565-3

Myung and Goodman: Single Carrier FDMA: A New Air Interface for Long TermEvolution, October 2008, 978-0-470-72449-1

Wang, Kondi, Luthra and Ci: 4G Wireless Video Communications, April 2009,978-0-470-77307-9

Cai, Shen and Mark: Multimedia Services in Wireless Internet: Modeling and Analysis,June 2009, 978-0-470-77065-8

Stojmenovic: Wireless Sensor and Actuator Networks: Algorithms and Protocols forScalable Coordination and Data Communication, February 2010, 978-0-470-17082-3

Liu and Weiss, Wideband Beamforming – Concepts and Techniques, March 2010,978-0-470-71392-1

Riccharia and Westbrook, Satellite Systems for Personal Applications: Concepts andTechnology, July 2010, 978-0-470-71428-7

Hart, Tao and Zhou: Mobile Multi-hop WiMAX: From Protocol to Performance, October2010, 978-0-470-99399-6

Qian, Muller and Chen: Security in Wireless Networks and Systems, November 2010,978-0-470-512128

WIDEBANDBEAMFORMINGCONCEPTS ANDTECHNIQUES


Stephan WeissUniversity of Strathclyde, UK

A John Wiley and Sons, Ltd., Publication

This edition first published 2010 2010 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Liu, Wei, 1974-Wideband beamforming : concepts and techniques / Wei Liu, Stephan Weiss.

p. cm.Includes bibliographical references and index.ISBN 978-0-470-71392-1 (cloth)1. Beamforming. 2. Antenna radiation patterns. 3. Adaptive antennas. 4. Adaptive signal processing.

5. Adaptive filters. 6. Broadband communication systems. I. Weiss, Stephan, 1968- II. Title.TK7871.67.A33L58 2010621.382′2 – dc22

2009052109

A catalogue record for this book is available from the British Library.

ISBN 9780470713921 (H/B)

Typeset in 10/12 Times by Laserwords Private Limited, Chennai, India.Printed and bound in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire.

www.wiley.com

In memory of my motherWei Liu

About the Series EditorsXuemin (Sherman) Shen (M’97–SM’02) received the B.Scdegree in electrical engineering from Dalian Maritime Univer-sity, China in 1982, and the M.Sc. and Ph.D. degrees (both inelectrical engineering) from Rutgers University, New Jersey,USA, in 1987 and 1990 respectively. He is a Professor and Uni-versity Research Chair, and the Associate Chair for GraduateStudies, Department of Electrical and Computer Engineering,University of Waterloo, Canada. His research focuses on mobil-ity and resource management in interconnected wireless/wirednetworks, UWB wireless communications systems, wirelesssecurity, and ad hoc and sensor networks. He is a co-author

of three books, and has published more than 300 papers and book chapters in wirelesscommunications and networks, control and filtering. Dr Shen serves as a Founding AreaEditor for IEEE Transactions on Wireless Communications; Editor-in-Chief for Peer-to-Peer Networking and Application; Associate Editor for IEEE Transactions on VehicularTechnology; KICS/IEEE Journal of Communications and Networks, Computer Networks;ACM/Wireless Networks; and Wireless Communications and Mobile Computing (Wiley),etc. He has also served as Guest Editor for IEEE JSAC, IEEE Wireless Communica-tions, and IEEE Communications Magazine. Dr Shen received the Excellent GraduateSupervision Award in 2006, and the Outstanding Performance Award in 2004 from theUniversity of Waterloo, the Premier’s Research Excellence Award (PREA) in 2003 fromthe Province of Ontario, Canada, and the Distinguished Performance Award in 2002 fromthe Faculty of Engineering, University of Waterloo. Dr Shen is a registered ProfessionalEngineer of Ontario, Canada.

Dr Yi Pan is the Chair and a Professor in the Departmentof Computer Science at Georgia State University, USA. DrPan received his B.Eng. and M.Eng. degrees in computer engi-neering from Tsinghua University, China, in 1982 and 1984,respectively, and his Ph.D. degree in computer science from theUniversity of Pittsburgh, USA, in 1991. Dr Pan’s researchinterests include parallel and distributed computing, opticalnetworks, wireless networks, and bioinformatics. Dr Pan haspublished more than 100 journal papers with over 30 paperspublished in various IEEE journals. In addition, he has pub-lished over 130 papers in refereed conferences (including

viii About the Series Editors

IPDPS, ICPP, ICDCS, INFOCOM, and GLOBECOM). He has also co-edited over 30books. Dr Pan has served as an Editor-in-Chief or an editorial board member for 15 jour-nals including five IEEE Transactions and has organized many international conferencesand workshops. Dr Pan has delivered over 10 keynote speeches at many internationalconferences. Dr Pan is an IEEE Distinguished Speaker (2000–2002), a Yamacraw Distin-guished Speaker (2002), and a Shell Oil Colloquium Speaker (2002). He is listed in Menof Achievement, Who’s Who in America, Who’s Who in American Education, Who’sWho in Computational Science and Engineering, and Who’s Who of Asian Americans.

Contents

About the Series Editors vii

Preface xiii

1 Introduction 11.1 Array Signal Processing 11.2 Narrowband Beamforming 41.3 Wideband Beamforming 71.4 Wideband Beam Steering 11

1.4.1 Beam Steering for Narrowband Arrays 121.4.2 Beam Steering for Wideband Arrays 131.4.3 A Unified Interpretation 17

1.5 Summary 18

2 Adaptive Wideband Beamforming 192.1 Reference Signal-Based Beamformer 19

2.1.1 Least Mean Square Algorithm 202.1.2 Normalized Least Mean Square Algorithm 222.1.3 Recursive Least Squares Algorithm 232.1.4 Comparison of Computational Complexities 242.1.5 Frequency-Domain and Subband Adaptive Algorithms 262.1.6 Simulations 26

2.2 Linearly Constrained Minimum Variance Beamforming 282.2.1 A Simple Formulation of Constraints 292.2.2 Optimum Solution to the LCMV Problem 302.2.3 Frost’s Algorithm for LCMV Beamforming 312.2.4 Simulations 31

2.3 Constraints Design for LCMV Beamforming 332.3.1 Eigenvector Constraint Design 332.3.2 Design Example 352.3.3 Application to Wideband DOA Estimation 36

2.4 Generalized Sidelobe Canceller 382.4.1 GSC Structure 382.4.2 GSC with Tapped Delay-Lines 422.4.3 Blocking Matrix Design 462.4.4 Simulations 48

x Contents

2.5 Other Minimum Variance Beamformers 482.5.1 Soft Constrained Minimum Variance Beamformer 492.5.2 Correlation Constrained Minimum Variance Beamformer 51

2.6 Robust Adaptive Beamforming 522.6.1 Spatially Extended Constraints 522.6.2 Norm-Restrained Approaches 57

2.7 Summary 60

3 Subband Adaptive Beamforming 613.1 Fundamentals of Filter Banks 61

3.1.1 Basic Multirate Operations 623.1.2 Perfect Reconstruction Condition for Filter Banks 663.1.3 Oversampled Modulated Filter Banks 68

3.2 Subband Adaptive Filtering 703.3 General Subband Adaptive Beamforming 74

3.3.1 Reference Signal Based Beamformer 753.3.2 Generalized Sidelobe Canceller 763.3.3 Reconstruction of the Fullband Beamformer 793.3.4 Simulations 79

3.4 Subband Adaptive GSC 823.4.1 Structure 823.4.2 Analysis of the Computational Complexity 823.4.3 Reconstruction of the Fullband Beamformer 833.4.4 Simulations 83

3.5 Temporally/Spatially Subband-Selective Beamforming 843.5.1 Partially Adaptive GSC 853.5.2 Temporally/Spatially Subband-Selective Blocking Matrix 873.5.3 Temporally/Spatially Subband-Selective Transformation Matrix 953.5.4 Application to Subband Adaptive GSC 983.5.5 Extension to the General Subband Adaptive Beamforming Structure 1003.5.6 Simulations 103

3.6 Frequency-Domain Adaptive Beamforming 1053.6.1 Frequency-Domain Formulation 1063.6.2 Constrained Frequency-Domain Adaptive Algorithm 1083.6.3 Frequency-Domain GSC 1093.6.4 Simulations 111

3.7 Transform-Domain Adaptive Beamforming 1123.7.1 Transform-Domain GSC 1133.7.2 Subband-Selective Transform-Domain GSC 1153.7.3 Simulations 115

3.8 Summary 118

4 Design of Fixed Wideband Beamformers 1194.1 Iterative Optimization 119

4.1.1 Traditional Methods 1194.1.2 Convex Optimization 120

Contents xi

4.2 The Least Squares Approach 1264.2.1 Standard Formulation 1264.2.2 Constrained Least Squares 128

4.3 The Eigenfilter Approach 1314.3.1 Standard Approach 1324.3.2 Maximum Energy 1374.3.3 Total Least Squares 139

4.4 Summary 142

5 Frequency Invariant Beamforming 1435.1 Introduction 1435.2 Design Based on Multi-Dimensional Inverse Fourier Transform 144

5.2.1 Continuous Sensor and Signals 1445.2.2 Discrete Sensors and Signals 1515.2.3 Design Examples 1555.2.4 Further Generalization to the FIB Design 163

5.3 Subband Design of Frequency Invariant Beamformers 1675.3.1 First Implementation 1695.3.2 Second Implementation–Scaled Aperture 1735.3.3 Design Examples 175

5.4 Frequency Invariant Beamforming for Circular Arrays 1765.4.1 Phase Mode Processing 1775.4.2 FIB Design 1815.4.3 Design Example 181

5.5 Direct Optimization for Frequency Invariant Beamforming 1825.5.1 Convex Optimization 1825.5.2 Least Squares 1855.5.3 Eigenfilter 186

5.6 Beamspace Adaptive Wideband Beamforming 1885.6.1 Structure 1885.6.2 Analysis of the Beamspace Adaptive Method 1905.6.3 Design of Independent FIBs 1925.6.4 Simulations 193

5.7 Summary 197

6 Blind Wideband Beamforming 1996.1 Blind Source Separation 199

6.1.1 Introduction 1996.1.2 A Blind Source Extraction Example 201

6.2 Blind Wideband Beamforming 2046.3 Blind Beamforming Based on Frequency Invariant Transformation 206

6.3.1 Structure 2076.3.2 The Algorithm 2086.3.3 Simulations 208

6.4 Summary 211

xii Contents

7 Wideband Beamforming with Sensor Delay-Lines 2137.1 Sensor Delay-Line Based Structures 213

7.1.1 Introduction 2137.1.2 Wideband Response of the SDL-Based Structure 217

7.2 Frequency Invariant Beamforming 2187.2.1 2-D Arrays 2207.2.2 3-D Arrays 224

7.3 Adaptive Beamforming 2287.3.1 Reference Signal Based Beamformer 2297.3.2 Linearly Constrained Minimum Variance Beamformer 2307.3.3 Discussions 2327.3.4 Simulations 233

7.4 Beamspace Adaptive Beamforming 2357.4.1 Structure 2357.4.2 Simulations 236

7.5 Summary 238

8 Wideband Beamforming for Multipath Signals 2398.1 The Wideband Multipath Problem 2408.2 Approach Based on a Narrowband Beamformer 241

8.2.1 Structure 2418.2.2 Simulations 243

8.3 Approach Based on Blind Source Separation 2468.3.1 Structure 2468.3.2 Simulations 247

8.4 MIMO System 2498.4.1 Evolution to a MIMO System 2508.4.2 MIMO Beamforming and Equalization 252

8.5 Summary 254

Appendix A: Matrix Approximation 255

Appendix B: Differentiation with Respect to a Vector 259

Appendix C: Genetic Algorithm 261C.1 The Principle 261

C.1.1 Chromosome Representation 261C.1.2 Parent Selection 262C.1.3 Genetic Operation 262C.1.4 Fitness Evaluation 263C.1.5 Initialization 263C.1.6 Termination 263

C.2 Design Example in Section 3.5.2 264

Bibliography 267

Index 283

PrefaceBeamforming is a spatial filtering technique for receiving signals illuminating an arrayof sensors from some specific directions, whilst attenuating signals from other directions.Depending on the signal bandwidth, it can be divided into two categories: narrowbandbeamforming and wideband beamforming. For narrowband beamforming, it is achievedby an instantaneous linear combination of the received array signals. However, when theinvolved signals are wideband, we have to employ an additional processing dimensionfor effective operation, such as tapped delay-lines (or FIR/IIR filters), or the recentlyproposed sensor delay-lines, which lead to a wideband beamforming system.

Wideband beamforming has been studied extensively in the past due to its applica-tions in various areas ranging from radar, sonar, microphone arrays, radio astronomy,seismology, medical diagnosis and treatment, to communications. In particular, sincespeech/sound is a natural source of wideband signals, much of the research and develop-ment in wideband beamforming has been focused on the area of microphone arrays.

Traditionally, beamforming is considered as part of the wider area of array signalprocessing and chapters relating to beamforming can be found in many books on arraysignal processing. Recently, due to its importance in the wireless communications area,there have been some books dedicated to beamforming in the form of smart antennatechniques.

However, since in many current wireless communication applications the signal band-width is still relatively narrow, almost all of the books within the smart antenna literatureare focused on narrowband beamforming and the topic of wideband beamforming is byand large ignored. With the introduction of ultra-wideband systems, one or two chapters onwideband beamforming have recently appeared in books about ultra-wideband communi-cations and wideband radar, etc. With the increasing importance of wideband beamformingand recent advances in this area, it appears timely to have a book dedicated to this topicfor the benefit of the wireless communications community. However, the concepts andtechniques presented in this book for wideband beamforming are general and not limitedto the wireless communications area, or any other specific applications.

There has been a huge amount of work going on in the past half a century in the areaof wideband beamfoming and it is impossible to cover all of them in the first attempt ofproducing a single book dedicated to this area. Although we have tried our best to givean extensive review of this topic in Chapters 1, 2 and 4 about both fixed and adaptivebeamforming techniques, the remaining part of the book is mostly based on our ownresearch over the past ten years in this area. Our primary goal is to give a systematicintroduction to the various concepts and techniques in wideband beamforming in the

xiv Preface

form of a self-contained monograph and also present some of the most recent researchand development in this area.

The contents of the book are organized into eight chapters.Chapter 1 is a brief introduction to the general area of array signal processing, includ-

ing both narrowband and wideband beamforming, with a detailed analysis for the beamsteering process for both cases. It will be shown that unlike the narrowband case, wherethe steered beam response is a circularly shifted version of the original one given a halfwavelength spacing, a more complicated relationship exists for a wideband beamformer.

In Chapter 2, we will study a range of basic approaches to adaptive wideband beam-forming. The latter can be achieved by a standard adaptive filtering structure when areference signal is available. When we know the direction of arrival (DOA) angle of thesignal of interest, a linearly constrained minimum variance (LCMV) beamformer can beconstructed and realized by either a constrained adaptive algorithm or an unconstrainedone through the structure of a generalized sidelobe canceller (GSC). In addition to thestandard LCMV beamformer, two other minimum variance beamformers will also bestudied, including the soft-constrained beamformer and the correlation constrained beam-former. To improve the robustness of the beamformer in the presence of steering vectorerrors, the topic of robust adaptive beamforming is addressed at the end of this chapter.

Chapter 3 is focused on various subband techniques and structures for adaptive wide-band beamforming, which can normally achieve a higher convergence rate and a lowercomputational complexity. Since the discrete Fourier transform (DFT) and inverse DFT(IDFT) pair can be considered as a simple maximally decimated filter banks system,frequency-domain adaptation techniques are also studied in this chapter.

Chapters 4 and 5 are devoted to the fixed wideband beamformer design problem, withChapter 4 for a general design using the iterative optimisation method, the least squaresmethod and the eigenfilter method, and Chapter 5 for a special class of fixed widebandbeamformers – the frequency invariant beamformer. The design of a frequency invariantbeamformer can be achieved by many different methods and at the end of Chapter 5, anapplication of the frequency invariant beamforming technique to the adaptive widebandbeamforming problem is also studied, which leads to a beamspace adaptive widebandbeamformer.

Chapter 6 is focused on a different class of adaptive beamformers: the blind adaptivewideband beamformer, which is based on the concept of blind source separation. For thisclass of beamformers, neither a reference signal nor the DOA information of the desiredsignal is needed and only some assumptions on the statistical properties of the sourcesignals are required.

In Chapter 7, we will introduce a totally different approach to wideband beamformingbased on the recently proposed sensor delay-line system. A special property of the resultantwideband beamforming structure is that there is not any form of temporal processingrequired, such as tapped delay-lines or FIR/IIR filters. Therefore it can be considered asa wideband beamforming structure with spatial-only information. Most of the techniquesdeveloped for the traditional wideband beamformers can be applied to this new structuredirectly. However, further studies are needed in the future to fully exploit its potential forwideband beamforming in various signal environments and applications.

In the last chapter, Chapter 8, we will study the wideband beamforming problem ina multipath environment. For the case with a small number of multhpath signals, two

Preface xv

solutions will be provided employing the wideband beamspace adaptive beamformingstructure studied in previous chapters. When a large number of multipath signals is presentto the beamformer, we will have a generalized signal mixing problem independent of thearray geometry and the original array system can be considered as a general multipleinput multiple output (MIMO) system. A brief introduction to the MIMO system is thenprovided from the viewpoint of beamforming at the end.

Acknowledgements

I would like to thank Professor S. C. Chan and Dr K. L. Ho for introducing me to theresearch area of signal processing and in particular filter banks and wavelets when mybackground was still mainly in space physics. I am very grateful to my Ph.D. supervisorsDr S. Weiss and Professor L. Hanzo. With their support I moved to the area of widebandbeamforming which has defined a major part of my research life today. Many thanks aredue to Professor S. Chen who has been supporting me in various ways since the verystart of my studies at the University of Southampton, UK and Dr D. P. Mandic who ledme into the area of blind source separation during my postdoctoral research work, whichopened up a new horizon for my work in wideband beamforming. I would also like tothank Professors P. A. Houston, B. Chambers and R. J. Langley in the Department ofElectronic and Electrical Engineering at the University of Sheffield, UK for their supportand help, especially during my ‘starting years’ of my academic career at the Universityof Sheffield. Special thanks must go to Professors I. K. Proudler, J. G. McWhirter, A.Cichocki and R. Wu, Dr D. C. McLernon and Professor M. Ghogho. Much of the researchpresented here was conducted in collaboration with them. Thanks also must go to Dr Y.Zhang and Professors R. W. Stewart and J. Li for their valuable comments and suggestionsduring the preparation of this book.


1Introduction

1.1 Array Signal Processing

Array signal processing is one of the major areas of signal processing and has beenstudied extensively in the past due to its wide applications in various areas ranging fromradar, sonar, microphone arrays, radio astronomy, seismology, medical diagnosis andtreatment, to communications (Allen and Ghavami, 2005; Brandstein and Ward, 2001;Fourikis, 2000; Haykin, 1985; Hudson, 1981; Johnson and Dudgeon, 1993; Monzingoand Miller, 2004; Van Trees, 2002). It involves multiple sensors (microphones, antennas,etc.) placed at different positions in space to process the received signals arriving fromdifferent directions. An example for a simple array system consisting of four sensors withtwo impinging signals is shown in Figure 1.1 for illustrative purposes, where the directionof arrival (DOA) of the signals is characterized by two parameters: an elevation angle θand an azimuth angle φ.

We normally assume the array sensors have the same characteristics and they areomnidirectional (or isotropic), i.e. their responses to an impinging signal are independentof their DOA angles. According to the relative locations of the sensors, arrays can bedivided into three classes (Van Trees, 2002):

• one-dimensional (1-D) arrays or linear arrays;• two-dimensional (2-D) arrays or planar arrays;• three-dimensional (3-D) arrays or volumetric arrays.Each of them can be further divided into two categories:

• regular spacing, including uniform and nonuniform spacings;• irregular or random spacing.Our study in this book will be based on arrays with regular spacings.

For the impinging signals, we always assume that they are plane waves, i.e. the arrayis located in the far field of the sources generating the waves and the received signalshave a planar wavefront.

Now consider a plane wave with a frequency f propagating in the direction of thez-axis of the Cartesian coordinate system as shown in Figure 1.2. At the plane defined

Wideband Beamforming Wei Liu and Stephan Weiss 2010 John Wiley & Sons, Ltd

2 Wideband Beamforming

signal #1

q

x

z

yf

sensor #4

sensor #1

sensor #2

sensor #3

signal #2

Figure 1.1 An illustrative array example with four sensors and two impinging signals

plane z = constant

y

z

rk

x

Figure 1.2 A plane wave propagating in the direction of the z-axis of the Cartesian coordinatesystem

by z = constant, the phase of the signal can be expressed as:φ(t, z) = 2πf t − kz (1.1)

where t is time and the parameter k is referred to as the wavenumber and defined as(Crawford, 1968):

k = ωc= 2π

λ(1.2)

where ω is the (temporal) angular frequency, c denotes the speed of propagation in thespecific medium and λ is the wavelength. Similar to ω, which means that in a temporalinterval t the phase of the signal accumulates to the value ωt , the interpretation of kis that over a distance z, measured in the propagation direction, the phase of the signalaccumulates to kz radians. As a result, k can be referred to as the spatial frequency of asignal.

Different from the temporal frequency ω, which is one-dimensional, the spatial fre-quency k is three-dimensional and its direction is opposite to the propagating direction ofthe signal. In a Cartesian coordinate system, it can be denoted by a three-element vector:

k = [kx, ky, kz]T (1.3)

Introduction 3

with a length of:

k =√

k2x + k2y + k2z (1.4)This vector is referred to as the wavenumber vector. In the case shown in Figure 1.2, wehave kx = ky = 0 and kz = −k. Let ẑ = [0, 0, 1]T denote the unit vector along the z-axisdirection, then we have k = −kẑ.

These two quantities are not independent of each other and as shown in Equation (1.2),they are related by the following equation:

k = 2πfc

(1.5)

Any point in a 3-D space can be represented by a vector r = [rx, ry, rz]T , where rx , ry andrz are the coordinates of this point in the Cartesian coordinate system. With the definitionof the wavenumber vector k, the phase function φ(t, r) of a plane wave can be expressedin a general form:

φ(t, r) = 2πf t + kT r (1.6)For the case in Figure 1.2, we have:

kT r = −k(ẑT r) = −krz (1.7)Therefore, as long as the points have the same coordinate rz in the z-axis direction, theyhave the same phase value at a fixed time instant t .

For the general case, where the signal impinges upon the array from an elevation angleθ and an azimuth angle φ, as shown in Figure 1.1, the wavenumber vector k is given by:

k = kxky

kz

= k

sin θ cos φsin θ sin φ

cos θ

(1.8)

Then the time independent phase term kT r changes to:

kT r = k(rx sin θ cos φ + ry sin θ sin φ + rz cos θ) (1.9)The wavefront of the signal is still represented by the plane perpendicular to its propaga-tion direction.

There are three major research areas for array signal processing:

1. Detecting the presence of an impinging signal and determine the signal numbers.

2. Finding the DOA angles of the impinging signals.

3. Enhancing the signal of interest coming from some known/unknown directions andsuppress the interfering signals (if present) at the same time.

The third research area is the task of beamforming, which can be divided into nar-rowband beamforming and wideband beamforming depending on the bandwidth of theimpinging signals, and wideband beamforming will be the focus of this book. In the nextsections, we will first introduce the idea of narrowband beamforming and then extend itto the wideband case.


1.2 Narrowband Beamforming

In beamforming, we estimate the signal of interest arriving from some specific directionsin the presence of noise and interfering signals with the aid of an array of sensors. Thesesensors are located at different spatial positions and sample the propagating waves inspace. The collected spatial samples are then processed to attenuate/null out the interferingsignals and spatially extract the desired signal. As a result, a specific spatial response ofthe array system is achieved with ‘beams’ pointing to the desired signals and ‘nulls’towards the interfering ones.

Figure 1.3 shows a simple beamforming structure based on a linear array, whereM sensors sample the wave field spatially and the output y(t) at time t is given byan instantaneous linear combination of these spatial samples xm(t), m = 0, 1, . . . ,M − 1, as:

y(t) =M−1∑m=0

xm(t)w∗m (1.10)

where ∗ denotes the complex conjugate.The beamformer associated with this structure is only useful for sinusoidal or narrow-

band signals, where the term ‘narrowband’ means that the bandwidth of the impingingsignal should be narrow enough to make sure that the signals received by the oppositeends of the array are still correlated with each other (Compton, 1988b), and hence it istermed a narrowband beamformer.

We now analyse the array’s response to an impinging complex plane wave ejωt with anangular frequency ω and a DOA angle θ , where θ ∈ [−π/2 π/2] is measured with respectto the broadside of the linear array, as shown in Figure 1.3. For convenience, we assumethe phase of the signal is zero at the first sensor. Then the signal received by the firstsensor is x0(t) = ejωt and by the mth sensor is xm(t) = ejω(t−τm), m = 1, 2, . . . , M − 1,where τm is the propagation delay for the signal from sensor 0 to sensor m and is afunction of θ . Then the beamformer output is:

y(t) = ejωtM−1∑m=0

e−ωτmw∗m (1.11)

signal

*

*

*

q

tM−1

t1 1

0w

w

wx1(t)

xM−1

x0(t)

y(t)

(t)

M−1

Figure 1.3 A general structure for narrowband beamforming

Introduction 5

with τ0 = 0. The response of this beamformer is given by:

P(ω, θ) =M−1∑m=0

e−jωτmw∗m = wH d(ω, θ) (1.12)

where the weight vector w holds the M complex conjugate coefficients of the sensors,given by:

w = [w0 w1 . . . wM−1]T (1.13)and the vector d(ω, θ) is given by:

d(ω, θ) = [1 e−jωτ1 . . . e−jωτM−1]T (1.14)We refer to d(θ, ω) as the array response vector, which is also known as the steeringvector or direction vector (Van Veen and Buckley, 1988). We will use the term ‘steeringvector’ to avoid confusion with the response vector used in linearly constrained minimumvariance beamforming introduced later in Section 2.2 of Chapter 2.

In our notation, we generally use lowercase bold letters for vector valued quantities,while uppercase bold letters denote matrices. The operators {·}T and {·}H represent trans-pose and Hermitian transpose operations, respectively.

Based on the steering vector, we briefly discuss the spatial aliasing problem encounteredin array processing. In analogue to digital conversion, we sample the continuous-timesignal temporally and convert it into a discrete-time sequence. In this temporal samplingprocess, aliasing is referred to as the phenomenon that signals with different frequencieshave the same discrete sample series, which occurs when the signal is sampled at arate lower than the Nyquist sampling rate, i.e. twice the highest frequency of the signal(Oppenheim and Schafer, 1975). With temporal aliasing, we will not be able to recoverthe original continuous-time signal from their samples. In array processing, the sensorssample the impinging signals spatially and if the signals from different spatial locationsare not sampled by the array sensors densely enough, i.e. the inter-element spacing of thearray is too large, then sources at different locations will have the same array steeringvector and we cannot uniquely determine their locations based on the received arraysignals. Similar to the temporal sampling case, now we have a spatial aliasing problem,due to the ambiguity in the directions of arrival of source signals.

For signals having the same angular frequency ω and the corresponding wavelengthλ, but different DOAs θ1 and θ2 satisfying the condition (θ1, θ2) ∈ [−π/2 π/2], aliasingimplies that we have d(θ1, ω) = d(θ2, ω), namely:

e−jωτm(θ1) = e−jωτm(θ2) (1.15)For a uniformly spaced linear array with an inter-element spacing d , we have τm =

mτ1 = m(d sin θ)/c and ωτm = m(2πd sin θ)/λ. Then Equation (1.15) changes to:e−jm(2πd sin θ1)/λ = e−jm(2πd sin θ2)/λ (1.16)

In order to avoid aliasing, the condition |2π(sin θ)d/λ|θ=θ1,θ2 < π has to be satisfied.Then we have |d/λ sin θ | < 1/2. Since | sin θ | ≤ 1, this requires that the array distanced should be less than λ/2.


In the following, we will always set d = λ/2, unless otherwise specified, then ωτm =mπ sin θ and the response of the uniformly spaced narrowband beamformer is given by:

P(ω, θ) =M−1∑m=0

e−jmπ sin θw∗m (1.17)

Note for an FIR (finite impulse response) filter with the same set of coefficients (Oppen-heim and Schafer, 1975), its frequency response is given by:

P(�) =M−1∑m=0

e−jm�w∗m (1.18)

with � ∈ [−π π] being the normalized frequency. For the response of the beamformergiven by Equation (1.17), when θ changes from −π/2(−90◦) to π/2(90◦), π sin θchanges from −π to π accordingly, which is in the same range as � in Equation (1.18).With this correspondence, the design of uniformly spaced linear arrays can be achievedby the existing FIR filter design approaches directly.

As a simple example, if we want to form a flat beam response pointing to the direc-tions θ ∈ [−π/6 π/6]([−30◦ 30◦]), while suppressing signals from directions θ ∈ [−π/2−π/4] and [π/4 π/2], then it is equivalent to designing an FIR filter with a passband of� ∈ [−0.5π 0.5π] and a stopband of � ∈ [−π − 0.71π] and [0.71π π] (sin π/6 = 0.5and sin π/4 = 0.71). We can use the MATLAB function remez to design such a filter(Mat, 2001), and then use the result directly as the coefficients of the desired beamformer.One of the design results is given by (M = 10):

wH = [0.0422 0.0402 − 0.1212 0.0640 0.51320.5132 0.0640 − 0.1212 0.0402 0.0422] (1.19)

Substituting this result into Equation (1.17), we can draw the resultant amplituderesponse |P(θ, ω)| of the beamformer with respect to the DOA angle θ . |P(θ, ω)| iscalled the beam pattern of the beamformer to describe the sensitivity of the beamformerwith respect to signals arriving from different directions and with different frequencies.Figure 1.4 shows the beam pattern (BP) in dB, which is defined as follows:

BP = 20 log10|P(θ, ω)|

max |P(θ, ω)| (1.20)

For the general case of d = αλ/2, α ≤ 1, the response of the beamformer given byEquation (1.17), will change to:

P(ω, θ) =M−1∑m=0

e−jmαπ sin θw∗m (1.21)

Its design can be obtained in a similar way as above and the only difference is that theFIR filter can have an arbitrary response over the regions � ∈ [−π − απ] and [απ π]without affecting that of the narrowband beamformer.

Introduction 7

−80 −60 −40 −20 0 20 40 60 80−100

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0

Direction of arrival of the signal, q

Mag

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Figure 1.4 The beam pattern of the resultant narrowband beamformer with M = 10 sensors

1.3 Wideband Beamforming

The beamforming structure introduced in the last section works effectively only for nar-rowband signals. When the signal bandwidth increases, its performance will degradesignificantly. This can be explained as follows.

Suppose there are in total M impinging signals sm(t), m = 0, 1, . . . , M − 1, from direc-tions of θm, m = 0, 1, . . . , M − 1, respectively. The first one s0(t) is the signal of interestand the others are interferences. Then the array’s steering vector dm for these signals isgiven by:

dm(ω, θ) =[1 e−jωτ1(θm) . . . e−jωτM−1(θm)

]T(1.22)

Ideally, for beamforming, we aim to form a fixed response to the signal of interest andzero response to the interfering signals. Note for simplicity, we do not consider the effectof noise here. This requirement can be expressed as the following matrix equation:

1 e−jωτ1(θ0) . . . e−jωτM−1(θ0)1 e−jωτ1(θ1) . . . e−jωτM−1(θ1)...

.... . .

...

1 e−jωτ1(θM−1) . . . e−jωτM−1(θM−1)

w∗0w∗1...

w∗M−1

=

constant0...

0

(1.23)

Obviously, as long as the matrix on the left has full rank, we can always find a set ofarray weights to cancel the M − 1 interfering signals and the exact value of the weightsfor complete cancellation of the interfering signals is dependent on the signal frequency(certainly also on their directions of arrival).

For wideband signals, since each of them consists of infinite number of different fre-quency components, the value of the weights should be different for different frequencies


and we can write the weight vector in the following form:

w(ω) = [w0(ω) w1(ω) . . . , wM−1(ω)]T (1.24)This is why the narrowband beamforming structure with a single constant coefficient foreach received sensor signal will not work effectively in a wideband environment.

The frequency dependent weights can be achieved by sensor delay-lines (SDLs), whichwere proposed only recently and will be studied in Chapter 7. Traditionally, an easy wayto form such a set of frequency dependent weights is to use a series of tapped delay-lines(TDLs), or FIR/IIR filters in its discrete form (Compton, 1988a; Frost, 1972; Mayhanet al., 1981; Monzingo and Miller, 2004; Rodgers and Compton, 1979; Van Veen andBuckley, 1988; Vook and Compton, 1992).

Both TDLs and FIR/IIR filters perform a temporal filtering process to form a frequencydependent response for each of the received wideband sensor signals to compensatethe phase difference for different frequency components. Such a structure is shown inFigure 1.5. The beamformer obeying this architecture samples the propagating wave fieldin both space and time. The output of such a wideband beamformer can be expressed as:

y(t) =M−1∑m=0

J−1∑i=0

xm(t − iTs)× w∗m,i (1.25)

where J − 1 is the number of delay elements associated with each of the M sensorchannels in Figure 1.5 and Ts is the delay between adjacent taps of the TDLs.

array sensors

*

*

*

delays

**

*

*

*

** * *

M−1,0 M−1,1 M−1,2 M−1,J−1

y (t)

x0(t)

x1(t)

xM−1(t)

w0,1 w0,2

w1,1 w1,2

w w w

w

w1,J−1

0,J−1

w1,0

w0,0

w

Figure 1.5 A general structure for wideband beamforming

Introduction 9

In vector form, Equation (1.25) can be rewritten as:

y(t) = wH x(t) (1.26)The weight vector w holds all MJ sensor coefficients with:

w =

w0w1...

wJ−1

(1.27)

where each vector wi , i = 0, 1, · · · , J − 1, contains the M complex conjugate coefficientsfound at the ith tap position of the M TDLs, and is expressed as:

wi = [w0,i w1,i · · · wM−1,i]T (1.28)Similarly, the input data are also accumulated in a vector form x as follows:

x =

x0(t)

x1(t − Ts)...

xJ−1(t − (J − 1)Ts]

(1.29)

where xi (t − iTs), i = 0, 1, . . . , J − 1, holds the ith data slice corresponding to the ithcoefficient vector wi :

x(t − iTs) =[x0(t − iTs) x1(t − iTs) · · · xM−1(t − iTs)

]T(1.30)

Note that this notation incorporates the narrowband beamformer with the special case ofJ = 1.

Now, for an impinging complex plane wave signal ejωt , assume x0(t) = ejωt . Then wehave:

xm(t − iTs) = ejω(t−(τm+iTs )) (1.31)with m = 0, 1, . . . ,M−1, i = 0, . . . , J − 1. The array output is given by:

y(t) = ejωtM−1∑m=0

J−1∑i=0

e−jω(τm+iTs ) · w∗m,i

= ejωt × P(θ, ω) (1.32)where P(θ, ω) is the beamformer’s angle and frequency dependent response. It can beexpressed in vector form as:

P(θ, ω) = wH d(θ, ω) (1.33)


where d(θ, ω) is the steering vector for this new wideband beamformer and its elementscorrespond to the complex exponentials e−jω(τm+iTs ):

d(θ, ω) = [e−jωτ0 . . . e−jωτM−1 e−jω(τ0+Ts) . . . e−jω(τM−1+Ts ). . . e−jω(τ0+(J−1)Ts ) . . . e−jω(τM−1+(J−1)Ts )]T (1.34)

For J = 1, it is reduced to the steering vector introduced for the narrowband beamformerin Equation (1.14).

For an equally spaced linear array with an inter-element spacing d , we have τm =mτ1 and ωτm = m(2πd sin θ)/λ for m = 0, 1, . . . , M − 1. To avoid aliasing, d < λmin/2,where λmin is the wavelength of the signal component with the highest frequency ωmax.Assume the operating frequency of the array is ω ∈ [ωmin ωmax] and d = αλmin/2 withα ≤ 1. In its discrete form, Ts is the temporal sampling period of the system and shouldbe no more than half the period Tmin of the signal component with the highest frequencyaccording to the Nyquist sampling theorem (Oppenheim and Schafer, 1975), i.e. Ts ≤Tmin/2.

With the normalized frequency � = ωTs , ω(mτ1 + iTs) changes to mµ� sin θ + i�with µ = d/(cTs), then the steering vector d(θ, ω) changes to:

d(θ, ω) = [1 . . . e−j (M−1)µ� sin θ e−j� . . . e−j�(µ sin θ(M−1)+1). . . e−j (J−1)� . . . e−j�(µ sin θ(M−1)+J−1)]T (1.35)

and we have:

P(θ, ω) =M−1∑m=0

J−1∑i=0

e−j�(mµ sin θ+i) × w∗m,i

=M−1∑m=0

e−jmµ� sin θJ−1∑i=0

e−j i� × w∗m,i

=M−1∑m=0

e−jmµ� sin θ ×Wm(ej�) (1.36)

where Wm(ej�) =∑J−1

i=0 e−j i� × w∗m,i is the Fourier transform of the TDL coefficients

attached to the mth sensor. For the case where α = 1 and Ts = Tmin/2, we have µ = 1.Now given the coefficients of the wideband beamformer, we can draw its 3-D beam

pattern |P(θ, ω)| with respect to frequency and DOA angle, according to Equation (1.36).To calculate the beam pattern for Nθ number of discrete DOA values and N� numberof discrete temporal frequencies, an Nθ ×N� matrix is obtained holding the responsesamples on the defined DOA/frequency grid.

As an example, consider an array with M = 5 sensors and a TDL length J = 3. Supposethe weight vector is given as:

W = [0 0 0 0 0 0.2 0.2 0.2 0.2 0.2 0 0 0 0 0]T (1.37)The beam pattern of such an array is shown in Figure 1.6 for N� = 50 and Nθ = 60,where the gain is displayed in dB as defined in Equation (1.20).

Introduction 11

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Figure 1.6 A 3-D wideband beam pattern example based on an equally spaced linear array withM = 5, J = 3 and µ = 1

In the above example, the values of the weight coefficients are fixed and the resultantbeamformer will maintain a fixed response independent of the signal/interference sce-narios. In statistically optimum beamforming, the weight coefficients need to be updatedbased on the statistics of the array data. When the data statistics are unknown or timevarying, adaptive optimization is required (Haykin, 1996), where according to differentsignal environments and application requirements, different beamforming techniques maybe employed. Both kinds of beamformers will be studied later in this book.

1.4 Wideband Beam Steering

For a narrowband beamformer, we can steer its main beam to a desired direction by addingappropriate steering delays or phase shifts (Johnson and Dudgeon, 1993; Van Trees, 2002).The relationship between the steered response and the original one is simple for a halfwavelength spaced linear array: the former one is a circularly shifted version of the latterone, i.e. the sidelobe shifted out from one side is simply shifted back from the other side.

Intuitively, we may think that adding steering delays for wideband beamformers hasthe same effect as in the narrowband case. However, this is not true and in general thereis not a one-to-one correspondence between the original beam response and the steeredone (Liu and Weiss, 2008c, 2009a).

In this section we will give a detailed analysis about this relationship. We will seethat after adding steering delays to the originally received wideband array signals, themain beam will be shifted to the desired direction; however for the sidelobe region, forone side, it is shifted out of the visible area and for the other side, it is not a simpleshifted-back of those shifted out, but exhibits a very complicated pattern.


1.4.1 Beam Steering for Narrowband Arrays

For a uniformly spaced narrowband linear array, its response can be expressed as:

P(sin θ) =M−1∑m=0

w∗me−jmµ� sin θ (1.38)

which is a special case (J = 1) of Equation (1.36).Suppose the set of coefficients w∗m, m = 0, 1, . . . , M − 1, forms a main beam pointing

to the broadside of the array (θ = 0). In order to steer the beam to the direction θ0,we can add a delay of (M − 1)(d sin θ0)/c to the first received array signal, a delay of(M − 2)(d sin θ0)/c to the second received array signal, and so on. Then the new responsewith a main beam pointing to θ0 is given by:

P(sin θ − sin θ0) = e−j (M−1)µ� sin θ0M−1∑m=0

w∗me−jmµ�(sin θ−sin θ0) (1.39)

where the term e−j (M−1)µ� sin θ0 represents a constant delay for all signals and will beignored in the following equations and discussions.

To avoid spatial aliasing, d = λ/2, where λ is the signal wavelength. We also assumethe sampling frequency is twice that of the signal frequency. Then, we have µ = 1 and� = π . As a result, Equation (1.39) changes to:

P(sin θ − sin θ0) =M−1∑m=0

w∗me−jmπ(sin θ−sin θ0) (1.40)

Since the function e−jmπx is periodic with a period of 2, compared to Equation (1.38),the response given by Equation (1.40) is simply a circularly shifted version of the responsein Equation (1.38) for one period sin θ ∈ [−1 1]. As an example, suppose we have abroadside main beam response P(sin θ) with a maximum response at sin θ = 0 for sin θ ∈[−1 1], as shown in Figure 1.7. Then after shifting it by sin θ0 < 0, the new responsewill be given by Figure 1.8.

Now we consider the effect as a function of θ . For the remaining part of Section 1.4,without loss of generality, we always assume θ0 < 0. For −1 < sin θ ≤ (1+ sin θ0), wehave:

−1 < −1− sin θ0 < sin θ − sin θ0 ≤ 1 (1.41)

P(sin q )

sin q−1 1

Figure 1.7 A broadside main beam example for an equally spaced narrowband linear array

wideband beamforming concepts and...

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