california state university, northridge reducing peak

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

Reducing Peak to Average Power Ratio of OFDM by Using Selected Mapping

A graduate project submitted in partial fulfillment of the requirements

For the degree of Master of Science

In Electrical Engineering

By

Wanis Mohamed

May 2012

II

The graduate project of Wanis Mohamed is approved by:

Professor Mallard Benjamin Date

Professor Amini Ali Date

Professor Bekir Nagwa, Chair Date

California State University, Northridge

III

Acknowledgement

I would like to express my sincerest thanks to my advisor, Dr. Bekir Nagwa, for her guidance

and support. I would also like to thank my project committee, Dr. Amini Ali and Dr. Mallard

Benjamin, for their time in reviewing my work. I want to thank my family for their support.

IV

TABLE OF CONTENTS

Signature Page ……………………………………………………………………………………ii

Acknowledgement ………………………………………………………………….……………iii

List of Figures …………………………………………………………………….……………...vi

List of Tables ……………………………………………………………………..…………… viii

List of Abbreviations...…………………………………………………………..….……………ix

ABSTRACT.... ……………………………………………………………………………..........xi

CHAPTER 1 Introduction …………………………………………… . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2 OFDM System……………………………………………………………………...3

2.1 OFDM Concept . . . . . . . . . . . . . .. . . . . . . . . . . . . ………………………………3

2.2 OFDM Model…... . . . . . . . . . . . . . . . . . . . . ……………………………………. 6

2.3 Cyclic Prefix for OFDM ……………………. .. . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Advantages and Drawbacks OFDM……………………………………………..………..12

2.4.1 Advantages…………………………………………………………………12

2.4.2 Drawbacks.…………………………………………………………………12

CHAPTER 3 Peak to Average Power Ratio Reduction Techniques…………………………….13

3.1 Introduction to Peak to Average Power Ratio (PAPR). . . …… . . . . . ... . . . . . . 13

3.2 PAPR Reduction Techniques . . . . . . . ………………………... . . ... . . . . . . . . 15

3.2.1Signaldistortion techniques……………..……………….……..…………..16

3.2.1.1 Clipping……………………………………………………………16

3.2.1.2 Peak Windowing…… . ……………………….. . . . . . . . . . . . . . . . 17

3.2.1.3 Peak cancellation…………………………………………………...19

3.2.2 Coding Schemes…... ... . . ………………....... . . . . . . . . . . ……..... . … . . 20

V

3.2.3Symbol-scrambling techniques….………..………………….…………..…22

3.2.3.1 Partial Transmit Sequence..................................................................22

3.2.3.2 Selected Mapping...…….. . . . . . . . . . . . . … . . . . . . …… . ... . . . . . 24

CHAPTER 4 Selected Mapping…...…………………………………………………………….25

4.1 Introduction to Selected Mapping……..………………………………………25

4.2 Threshold Selected Mapping…………………………………………………..….26

4.3 Power Savings through Selected Mapping….………………………………….…29

CHAPTER 5 Computer Simulations…………………………………………………………… 33

5.1 OFDM Basic Model.………………………………………………………….. 33

5.1.1 OFDM Transmitter....………………………………………………………33

5.1.2 OFDM Receiver.………………………………………………………....... 35

5.1.3 The Performance Analysis …………………………………………………36

5.2 Threshold Selected Mapping… ………………………………………………...39

CHAPTER 6 Conclusions and Future Work.……………………………………………………45

6.1 Conclusion ……………………………………………………………………...45

6.2 Suggestions for Future Work....…………………………………………………46

REFERENCES…………………………………………………………………………………..47

APPENDIX………………………………………………………………………………………49

VI

List of Figures

Figure 2.1 Orthogonal Multicarrier versus Conventional Multicarrier………………………….. 4

Figure 2.2 Spectra of OFDM Individual Subcarrier...…………………………………………… 5

Figure 2.3 Spectra of OFDM Symbol……………………………………………………………. 6

Figure 2.4 OFDM Modulator……………………………………………………………………...7

Figure 2.5 OFDM Demodulator...………………………………………………………………...8

Figure 2.6 OFDM Block Diagram...………………………………………………………………9

Figure 2.7 OFDM Signal (a) without Cyclic Prefix at the Transmitter, (b) Without Cyclic Prefix

at The Receiver, (c) With Cyclic Prefix at The Transmitter, and (d) With Cyclic Prefix............ 11

Figure 3.1: Power Samples of One Symbol OFDM Signal..…………………………………… 14

Figure 3.2: Amplitude of Transmitted OFDM Symbol..……………………………………….. 17

Figure 3.3: Windowing an OFDM Time Signal...……………………………………………… 18

Figure 3.4: A Block Diagram of PAPR Reduction by Peak Cancelation ...……………………..19

Figure 3.5: (a) OFDM Symbol Envelop, and (b) Signal Envelope after Peak Cancellation…… 20

Figure 3.6: A Block Diagram of the PTS Technique………………………………………….....23

Figure 4.1 Block Diagram of SLM Technique………………………………………………......26

Figure 4.2 PAPR Reduction for SLM where N = 256 and U = 1, 2, 4, 8, 16…………………... 32

Figure 5.1 Simulink Model of OFDM Transmitter……………………………………………...33

Figure 5.2 Bernoulli Binary Generator Parameters……………………………………………...34

Figure 5.3 Rectangular QAM Modulator Baseband Parameters………………………………...34

Figure 5.4 Simulink Model of OFDM Receiver…………………………………………………35

Figure 5.5 Rectangular QAM Demodulator Baseband Parameters……………………………...35

Figure 5.6 Frequency Spectrum of Transmitted Signal…………………………………………36

VII

Figure 5.7 Frequency Spectrum of OFDM Signal………………………………………..……...37

Figure 5.8 OFDM Signal in Time Domain………………………………………………………37

Figure 5.9 Simulated BER vs. Theoretical BER…………………………………………………38

Figure 5.10 MATLAB Code……………………………………………………………………..39

Figure 5.11 PAPR Reduction for SLM where N = 64 and U = 1, 2, 4,8,16……………………. 40

Figure 5.12 PAPR Reduction for SLM where N = 128 and U = 1, 2, 4,8,16. …………….....….41

Figure 5.13 PAPR Reduction for SLM where N = 256 and U = 1, 2, 4,8,16………………..... ..41

Figure 5.14 PAPR Reduction for SLM where N = 512 and U = 1, 2, 4,8,16……………….... ...42

Figure 5.15 PAPR Reduction for SLM where N = 1024 and U = 1, 2, 4,8,16……………..… ...42

Figure 5.16 PAPR Reduction for SLM where α =2.8, N = 64, and U = 1, 2, 4,8,16……….……43

Figure 5.17 PAPR Reduction for SLM where α =2.8, N = 128, and U = 1, 2, 4,8,16……..…… 44

Figure 5.18 PAPR Reduction for SLM where α =2.8, N = 256, and U = 1, 2, 4,8,16…………..44

VIII

List of Table

Table 3.1 PAPR Reduction Comparison with Different Coding Schemes………………………22

Table 4.1 PAPR Reduction and Saving Gain Using SLM where N = 256 and U = 1, 2, 4,8,16...32

Table 5.1 PAPR Reduction Corresponding to Various Phase Sequences for Different Number of

Subcarriers ………………………………………………………………………………………43

IX

List of Abbreviations

4G Fourth Generation

ADSL Asymmetric Digital Subscriber Line

BRAN Broadband Radio Access Networks

Bc Coherence Bandwidth

CCDF Complementary Cumulative Distribution Function

CDF Cumulative Distribution Function

CP Cyclic Prefix

DAB Digital Audio Broadcasting

DFT Discrete Fourier Transform

DVB-T Digital Video Broadcasting-Terrestrial

FDM Frequency Division Multiplexing

FFT Fast Fourier Transform

HF High Frequency

ICI Inter Carrier Interference

IDFT Inverse Discrete Fourier Transform

IFFT Inverse Fast Fourier Transform

ISI Inter Symbol Interference

LOS Line of Sigh Path

MC Multicarrier Modulation

OFDM Orthogonal Frequency Division Multiplexing

PAPR Peak to Average Power Ratio

X

PSK Phase Shift Keying

PTS Partial Transmit Sequence

QAM Quadrature Amplitude Modulation

RF Radio Frequency

SLM Selected Mapping

XI

ABSTRACT

Reducing Peak to Average Power Ratio of OFDM by Using Selected Mapping

By

Wanis Mohamed

Master of Science in Electrical Engineering

Orthogonal frequency division multiplexing (OFDM) has become the most popular

modulation technique for high speed data transmission. However, high peak to average power

ratio (PAPR) is a major drawback of this modulation technique. Because high peak reduces the

power efficiency of the RF power amplifier at transmitter. This project gives an overview on the

popular OFDM reduction techniques, and indicates that selected mapping (SLM) is a promising

reduction technique.

1

CHAPTER 1

Introduction

This graduate project gives an overview on Orthogonal Frequency Division Multiplexing

(OFDM) technique and identifies popular Peak to Average Power Ratio (PAPR) reduction

schemes characteristically. In addition, it demonstrates that selected mapping (SLM) is a

promising reduction technique.

Wireless communications has many advantages, such as speed, simplicity, mobility and

flexibility, but in the same time it suffers from, inter-symbol interference (ISI) and multipath

propagation (frequency selective fading). Supporting high data rates channel of the conventional

single carrier system required various modulation techniques.

OFDM is the most popular one. The first OFDM scheme was proposed by Chang in 1966[1].

Even though the concept of OFDM has been around for several years, but it has not been

recognized as a great method for high speed bi-directional wireless data communication until

recent years. The fist applications of OFDM were in the military HF radio links. Today, the

OFDM technique is in many wirelesses and wired applications, such as broadband radio access

networks (BRAN), Digital Audio Broadcasting (DAB), Digital Video Broadcasting-Terrestrial

(DVB-T) and Asymmetric Digital Subscriber Line (ADSL). These days the OFDM technique is

considered as a strong candidate for the fourth generation (4G) of mobile communication

systems. OFDM has many advantages: such as, flexibility to the channel conditions without the

need of channel equalization, robustness to the fading, and resistance to multipath [1]. On the

other hand, OFDM suffers a high Peak to Average Power Ratio (PAPR). A high PAPR makes

the signal peaks move into the non-linear region of the RF power amplifier which causes signal

2

distortion. A large PAPR increases the complexity of the analog-to-digital and digital-to-analog

converters and reduces the efficiency of the RF power amplifier. Recently, researchers have

discovered many techniques on PAPR reduction, for instances, clipping, coding, and selected

mapping (SLM) [1].

This project is organized as follow. Chapter 2 reviews the basic concepts of OFDM, such as

transmitter, receiver and showing the difference between orthogonal multicarrier and

conventional multicarrier. In addition, it will present cyclic prefix which is a technique that is

used to resolve inter symbol interference (ISI) and inters carrier interference (ICI). Major

advantages and drawbacks of OFDM will be discussed as well.

Chapter 3 discusses the definition of PAPR and the high PAPR issues. And then represent

existing popular PAPR reduction schemes such as, Clipping, Interleaving, and Coding.

Chapter 4 discusses selected mapping (SLM) technique which is a well known technique to

reduce the peak-to-average power ratio (PAPR).Chapter 5 shows results of computer simulations

on the performance of SLM technique. And finally, the conclusions and the recommendations

for future work will be given in Chapter 6.

3

CHAPTER 2

Orthogonal Frequency Division Multiplexing System

2.1 Orthogonal Frequency Division Multiplexing (OFDM) Concept

In the communication system, the transmitted signals may not reach the receiver antenna

directly because of diffraction, reflection, and scattering, which caused by buildings, mountains,

and that resulting in blocking the line-of-sight path (LOS). In case of blocking LOS the received

signals will come from different directions and this effect is known as multipath propagation

(Frequency Selective Channel).

The frequency selective channel has big effects on the transmitted data, and there are many

techniques used to decrease the effect of the frequency selective channel such as the Viterbi

algorithm and equalization. Orthogonal Frequency Division Multiplexing (OFDM) is a technique

that can also be used to alleviate frequency selective channels.

In the OFDM system the channel bandwidths (W) divided into N lower rate data stream and

then transmit them simultaneously over number of subcarriers. The individual bandwidth (W/N)

of the subcarrier is smaller than the coherence bandwidth (Bc ), which is the maximum

bandwidth over which two frequencies of a signal experience correlated amplitude fading . Since

the individual bandwidth smaller than the coherence bandwidth the channel is said to be a flat

fading channel. As it can be seen from figure 2.1, subcarriers in the OFDM system allowed to be

overlapping because the orthogonality makes the separation of the subcarriers will be end at the

receiver and that is the fundamental of the OFDM system which saving bandwidth[2].

4

In the normal frequency division multiplexing (FDM) system, on the other hand, the total

frequency spectrum is divided into N non overlapping frequency sub channels. If the bandwidth

of the subcarrier is small, it can be considered to be a flat fading channel. And the subcarriers are

spaced apart in such a way that they do not interfere with each other. In the normal frequency

division multiplexing the received signals can be received at the receiver by using conventional

filters and demodulators. Figure 2.1 shows the difference between the frequency division

multiplex system (conventional multicarrier) and the overlapping multicarrier (such as OFDM)

[2].

Figure 2.1 Orthogonal Multicarriers versus Conventional Multicarrier [2]

5

Figure 2.2 illustrates the spectrum of an individual data subcarrier. In the indivdual spectra

the OFDM signal equal to the bandwidth of each subcarrier. Figure 2.3 shows the spectrum of

the OFDM symbol. And it can be seen that at the center frequency of each subcarrier there is no

cross talk.

Figure 2.2 Spectra of OFDM individual subcarrier [2]

6

Figure 2.3 Spectra of OFDM symbol

2.2 OFDM Model

The OFDM signal is a sum of subcarriers that they are modulated individually by using either

quadrature amplitude modulation (QAM) or phase shift keying (PSK) and then they are

simultaneously transmitted as data stream. Figure 2.4 shows the OFDM modulator and it can be

represented very efficiently by the inverse fast Fourier transform (IFFT) as in the following

equation [2]:

X (t) =

(2.1)

Where

= (2.2)

= 0 otherwise

And

, (2.3)

7

is the Kth subcarrier frequency, with being the lowest, is the symbol duration, N is the

number of OFDM subcarrier and is the symbol transmitted during nth timing interval using

Kth subcarrier.

dn,0

:

dn,N-1

Figure 2.4 OFDM modulator

Σ x(t)

8

Figure 2.5 is a simplified block diagram of the OFDM demodulator and it is based on the

orthogonality of the subcarrier and it can be introduced as the following equation [2]

(2.4)

And because of the orthogonality relationship of the subcarriers the demodulator can be

represented digitally, the inverse fast Fourier transform (IFFT) to the fast Fourier transform

(FFT), modulation to demodulation of the OFDM signal and the equation can be implemented as

shown in the figure 2.5 by using the fast Fourier transform (FFT) [2]:

Td

dn,0

x (t) :

Td

dn,N-1

Figure 2.5 OFDM demodulator

dn,N-1

9

Figure2.6 below shows the block diagram of the OFDM system (transmitter, receiver and

fading channel), and as it can be seen there is cyclic prefix (CP) block which will be discussed

next.

Transmitter

Receiver

Figure 2.6 OFDM block diagram

10

2.3 Cyclic Prefix for OFDM

To explain the cyclic prefix for OFDM let us start by assuming two OFDM symbols that they

have experienced a delay spread td and channel dispersion. Figure 2.7a illustrates slow subcarrier

(slow delay spread at td ) and fast subcarrier (fast delay spread at td ) inside each OFDM symbol

on the transmitted signal. Figure 2.7b shows slow subcarrier delayed by td against fast subcarrier

on the received signal. As it can be seen from the figure 2.7b that the slow subcarrier in the

OFDM symbol interfere with another OFDM symbol and that is called inter symbol interference

(ISI). Moreover, the OFDM waveform in the discrete Fourier transform (DFT) window is

incomplete so that the orthogonality condition for the subcarrier is lost which result in inters

carrier interference (ICI).

Cyclic prefix is a technique that is used to resolve ISI and ICI. Figure 2.7c shows a cyclic

prefix of the OFDM symbol into the guard interval ∆G. And the waveform in the guard interval is

a copy of that in the DFT window with shift ts. The OFDM signal with guard interval on the

received signal is shown in figure 2.7d. As it can be seen from the figure 2.7d the OFDM symbol

of the slow subcarrier is in the DFT window because the cyclic prefix has moved into the DFT

window to replace the signal that has shifted out of this OFDM symbol [3].

Thus, the main idea of this technique is to replicate part of the OFDM waveform from the

back to the front to develop a guard period. And at the receiver, certain position within the cyclic

prefix is picked as the sampling starting point, which satisfies the condition td < ∆G where td is the

delay spread and ∆G the guard interval. As soon as the above condition is satisfied, there will be

no channel dispersion such as ISI and ICI [3].

11

Figure 2.7 OFDM signal (a) without cyclic prefix at the transmitter, (b) without cyclic

prefix at the receiver, (c) with cyclic prefix at the transmitter, and (d) with cyclic prefix at

the receiver [3]

12

2.4 Advantages and Drawbacks of OFDM

2.4.1 Advantages

By using the parallel multicarrier transmission, OFDM converts frequency selective

fading channels to non-selective fading subchannels (flat fading).

OFDM is good for broadcasting applications because it allows single frequency

networks to be used.

OFDM flexible to the channel conditions without the need of channel equalization

algorithms and it is also easy in meeting various design requirements, such as

complexity.

OFDM is robust to inter symbol interference (ISI ) and inters carrier interference (ICI)

by using Cyclic prefix technique

2.4.2 Drawbacks

The OFDM system is sensitive to the carrier frequency offset and Doppler shift.

The OFDM system is more complicated than single-carrier modulation.

High peak to average power ratio reduces the power efficiency of the RF power

amplifier and because of that makes the design of RF amplifier becomes more difficult.

The next chapter will discuss some popular techniques that use to reduce the effect of high peak

to average power ratio.

13

Chapter 3

Peak to Average Power Ratio Reduction Techniques

3.1 Introduction to Peak to Average Power Ratio (PAPR)

In the orthogonal frequency division multiplexing (OFDM) the peak power might be much

larger than the average power, due to adding up subcarriers coherently which resulting in large

peak-to-average power ratio (PAPR). PAPR is a very important situation in the communication

system because it has big effects on the transmitted signal. Low PAPR makes the transmit

power amplifier works efficiently, on the other hand, the high PAPR makes the signal peaks

move into the non-linear region of the RF power amplifier which reduces the efficiency of the

RF power amplifier. In addition, high PAPR requires a high-resolution digital- to- analog

converter (DAC) at the transmitter, high-resolution analog -to -digital converter (ADC) at the

receiver and a linear signal. Any non-linearity in the signal will cause distortion such as inter-

carrier interference (ICI) and inter symbol interference (ISI). The PAPR effect is shown in

figure. 3.1. And it can be seen that the peak power is about 17 times the average power [4].

The peak to average power ratio (PAPR) of a continuous time signal is given by [4],

PAPR

(3.1)

And for the discrete time signal PAPR

(3.2)

14

Figure 3.1: Power samples of one symbol OFDM signal

The input signal to the amplifier in the OFDM system is an analog signal and the time domain

samples of the output from the inverse fast Fourier transform (IFFT) is [4]:

, 0 (3.3)

If the number of subcarriers (N) is large are zero mean Gaussian random variables. And for

complex Gaussian the OFDM signal is Rayleigh distributed with variance , and the

phase of the signal is uniform. The peak value of the signal that has Rayleigh distribution will

15

exceed any values with nonzero probability. Thus the probability of the PAPR of the discrete

signal exceeds a threshold =

is given by [4]:

(3.4)

Let us show how PAPR increases by increasing the number of subcarriers N. Assume N

Gaussian independent and identically distribute (i.i.d) random variables , 0 with

zero mean and unit power. The average signal power = is then

=

=1 (3.5)

The maximum value occurs when all the add coherently, which is

max[

=[

= N (3.6)

Thus, the maximum PAPR is N for N subcarriers.

3.2 PAPR Reduction Techniques

One of the major disadvantages of OFDM systems is that the OFDM signal has high Peak to

Average Power Ratio (PAPR), and to deal with this problem many typical techniques have been

proposed. Each one is different from others in complexity and performance, and can be divided

into three major categories:

16

. Signal distortion techniques

.Signal Clipping

. Peak windowing

. Peak cancellation

. Coding Schemes

. Symbol-scrambling techniques

. Partial Transmit Sequences

. Selected Mapping

3.2.1Signal distortion techniques

3.2.1.1 Clipping

Clipping is the simplest technique that is used to reduce PAPR in OFDM system. The basic

idea of this technique is to clip the parts of the signals that have high peak outside of the allowed

region. The following equation shows the amplitude clipping [5],

(3.7)

Where A is a positive real number and it presents the clipping level.

Since the clipping is always performed at the transmitter, signals at the receiver have to

estimate the clipping that has occurred at the receiver. In general, one clipping occurs per

OFDM symbol, and the receiver has to calculate two important parameters: location and size of

17

the clipping signals. Clipping method is a nonlinear process and may cause in or out distortion

into the OFDM system, which may affect the bit error performance (BER), besides, it may cause

peak regrowth. Peak regrowth happens when clipping exceed the clipping level. And by

repeating clipping and filtering process again the effect of this distortion can be eliminated [5].

Figure 3.2: Amplitude of transmitted OFDM symbol [1]

3.2.1.2 Peak windowing

Another method used to reduce the PAPR in OFDM system is Peak windowing. The main

idea of this method is to multiply large signal peaks by a Gaussian shaped window to reduce the

out of band radiation. As the matter of fact, any window could be used to minimize out band

radiation. The window has to be as narrowband as it needed, however, it should not be too long

in time domain because many signals might be affected which will result in increasing bit error

rate (BER). Appropriate windows that offer good result in reduction PAPR in OFDM are

18

Kaiser, Cosine and Hamming functions. Figure 3.3 shows an example of reducing PAPR by

using peak windowing and indicates that how by increasing the window level the distortion will

decrease [6].

Figure 3.3: Windowing an OFDM time signal [2].

19

3.2.1.3 Peak Cancellation

The basic idea of peak cancelation is to reduce the amplitude of the data samples when the

magnitude exceeds a certain threshold. A comparator can be used to check whether the OFDM

symbol exceeds the threshold or not. In case the amplitude is above the threshold, the peak and

the side lobes are scaled in way so that they maintain the certain threshold. Figure 3.4

demonstrates the block diagram of an OFDM transmitter with peak cancelation which is located

after the cyclic prefix (CP). And an example is shown in figure 3.5 which indicates the peak

amplitude is reduced to 3 dB corresponding to the peak cancellation.

Figure 3.4: A Block diagram of PAPR reduction by Peak Cancelation [1].

20

Figure 3.5: (a) OFDM symbol envelop, and (b) signal envelope after peak cancellation [2]

3.2.2 Coding Schemes

As it described in previous section when N signals that have the same phase added together

resulting the peak power which is N times the average power. Since not all code words result in a

bad PAPR, the good PAPR can be obtained when measures are taken to reduce the occurrence

probability of the same phase of the N signals which is the main idea of coding schemes. The

first block coding scheme was simple and it introduced by Jones et al. The main idea of this

21

technique is by adding a Simple Odd Block Code (SOBC) at the last bit across the channels to

map 3 bits into 4 bits codeword. Later, Cyclic Coding (CC) was introduced by Wulich to reduce

the PAPR. More advanced Simple Block Code (SBC) was introduced in 1998, by Fragiacomo to

reduce the PAPR in the OFDM system, but, this code was ineffective when the frame size is

large. Later, new coding schemes were introduced to reduce the PAPR in the OFDM system;

they are Complement Block Coding (CBC) and Modified Complement Block Coding (MCBC)

schemes. CBC and MCBC are attractive coding scheme because they are effective in the large

size frame and they are flexible in choosing the coding rate. CBC and MCBC are working by

adding bits to the original data bits to reduce the occurrence probability of the peak signals [6].

Table 3.1 compares results of the PAPR reduction with various coding schemes, where N is

the number of subcarriers, n is the number of bits, and R is coding rate. When coding rate R is

equal to 3/4, MCBC code scheme obtains more PAPR reduction when it compares to other

coding schemes, at any length of frame size. In CBC coding scheme almost 3-dB PAPR

reduction could be obtained when R > (N-2)/N.

Thus, due to the flexibility in choosing the coding rate and low complexity, CBC and MCBC

are attractive coding schemes for OFDM systems with long frame sizes.

22

TABLE 3.1

PAPR Reduction Comparison with Different Coding Schemes [5]

N n R PARR Reduction (dB)

CBC SBC MCBC SOPC CC

4 1 3/4 3.56 3.56 - 3.56 3.56

8 1 7/8 2.52 2.52 - 2.52 (R=7/8)

3.66 (R=3/4) 2 3/4 2.67 3.72 2.81

16 1 15/16 2.74 1.16 - 1.18

(R=15/16)

3.74 (R=3/4) 2 7/8 2.74 2.52 -

3 13/16 2.74 - -

4 3/4 2.74 2.98 3.46

32 1 31/32 1.16 0.55 - 0.58 (R=31/32)

-

2 15/16 1.16 1.16 -

3 29/32 2.75 - -

4 7/8 2.50 2.51 -

5 27/32 2.75 - -

8 3/4 2.75 3.00 3.45

3.2.3 Symbol-scrambling techniques

3.2.3.1 Partial Transmit Sequences (PTS)

Partial transmit sequences (PTS) is one of the most important methods that is used to reduce

PAPR in the OFDM system. And it can be presented in two main steps. First, by dividing the

original OFDM signal into a number of sub-blocks. Secondly, adding the phase rotated sub-

blocks to develop a number of candidate signals to pick the one with smallest PAPR for

transmission. There is another way that it can also be used to express PTS method by

multiplying the original OFDM signal with a number of phase sequences [7].

23

Let us assume that X = {Xk}, where (k = 0,… ,N -1) , is the frequency domain (FD) data of an

OFDM signal xn = inverse discrete Fourier transform (IDFT) { Xk} (n = 0,…, N -1) , where N is

the number of subcarriers. It can be reduced the PAPR of signal x = {xn} by using PTS method

in the following steps [7],

Make M is the frequency domain (FD) data sequences, ( = 1,…,M ), by multiplying the

phase sequences = { }(k = 0,1, …, N-1) with X elements , it can get the following result,

= [ X0,

X1,…,

XN-1 ] , = 1,…,M (3.8)

Where = exp (

),

is uniformly distributed in [0, 2 ).

Get M candidates time domain (TD) via IDFTs

= IDFT { }, = 1,…, M (3.9)

All the candidates have the same information x, but different PAPRs. The one with the smallest

PAPR in is selected for transmission. Figure 3.6 shows an example of PTS technique.

Figure 3.6: A block diagram of the PTS technique.

24

3.2.3.2 Selected Mapping

Selected mapping (SLM) is a promising PAPR reduction technique of OFDM system. The

main idea of SLM technique is to generate a number of OFDM symbols as candidates and then

select the one with the lowest PAPR for actual transmission. This technique will be discussed in

details in the next chapter.

25

CHAPTER 4

Selected Mapping

4.1 Introduction to Selected Mapping (SLM)

Selected Mapping (SLM) technique is the most promising reduction technique to reduce Peak

to Average Power Ratio (PAPR) of Orthogonal Frequency Division Multiplexing (OFDM)

system. The first SLM scheme was introduced by Bauml, Fischer and Huber in 1996 [8].The

basic idea of this technique is based on the phase rotation. The lowest PAPR signal will be

selected for transmission from a number of different data blocks (independent phase sequences)

that have the same information at the transmitter. Figure 4.1 shows a block diagram of SLM

scheme [9].

Let us assume that the original input data X [X0, X1,…,XN-1 ]T multiplied with independent

phase sequences =

( =0,1, U-1) , where U is the number of phase

sequences. Both the input data and phase sequences have the same length N ( = 0, 1…, U-1).

After multiplication, inverse fast Fourier transform (IFFT) will be applied on each sequence to

convert the signal from frequency domain to the time domain. The result from multiplication will

generate the data block of an OFDM system that has different time domain signals, with length

of U, and different PAPR values, =

. The last step is comparing

the PAPR among the independent data blocks and the candidate with the lowest PAPR will be

selected for transmission. The following equation expresses the optimal candidate that has the

lowest PAPR and selected for transmission [9],

26

= [PAPR ( ] (4.1)

Figure 4.1 Block diagram of SLM technique

4.2 Threshold Selected Mapping

As it described in Chapter 2 the complex baseband of an OFDM signal that has N subcarriers

with Nyquist sample rate can be expressed as [10],

, 0 (4.2)

27

Where are the modulation symbols. The central limit theorem shows that, if the number of

subcarriers N is large, are zero mean Gaussian random variables. And for complex

Gaussian the OFDM signal is Rayleigh distributed with a variance of 0.5, and the phase of the

signal is uniform. The peak value of the signals that have Rayleigh distribution will exceed any

values with nonzero probability. Let us assume that the average power of is equal to 1, and

is the independently and identically distributed (i.i.d) Rayleigh random variables. The

probability density function of is given by [10],

( , = 0, 1, 2…, N-1 (4.3)

The maximum value of is equivalent to PAPR. If then the

cumulative distribution function (CDF) of and the probability of peak to average power

ratio (PAPR) below threshold are given by [10],

= …..

(4.4)

The complementary cumulative distribution function (CCDF) is used when PAPR value exceeds

the threshold. To find the probability that PAPR of an OFDM signal exceeds the threshold ,

assume the following complementary cumulative distribution function (CCDF) [10],

= 1 -

= (z)

= 1- (4.5)

28

In SLM technique each data block will create U times phase sequences, if each mapping

considered statistically independent, then CCDF of the Peak to Average Power Ratio (PAPR) in

Selected Mapping (SLM) will be,

= (1- )U

(4.6)

Where U is the number of phase sequences, N is the number of subcarriers, and z is threshold.

As it can be seen from equations (4.5) and (4.6), they derived when the number of subcarriers

N is large and the samples are independent with Nyquist sampling rate. But, both equations don’t

mention the oversampled and band limited. It is because the fact that the sampled signal does not

need to have the maximum point of the original signal. On other hand, it is important to

oversamples OFDM signals by oversampling factor L to obtain better value of PAPR. Tellado

indicates that an oversampling of four is adequate to reach the real PAPR values [11]. And it is

quite difficult to derive the solution of the peak power distribution; therefore, Nee and Prasad

show an approximation to explain the probability of PAPR by approximated N subcarriers and

oversampling distribution by α· N subcarriers without oversampling, and they mention that

when α =2.8 is the best value to reach better PAPR when subcarriers N 64. The

approximation is shown below,

(4.7)

When PAPR value exceeds the threshold z, the probability of PAPR for oversampling case can

be written as,

29

(4.8)

(4.9)

4.3 Power Savings through Selected Mapping

The average input power in the OFDM system need to be adjusted to decrease the affect of

the distortion in the peak of the signals, to do so, an input backoff (IBO) needs to be applied.

IBO is the measurement of how much reduction of the input power is needed, so that the desired

output power can be achieved. The amount of IBO applied is related to peak to average power

PAPR and the efficiency , high PAPR result in increasing IBO and decreasing . IBO is

equivalent to PAPR in certain probability. The efficiency of the power amplifier that is used in

OFDM system can be given as [13],

(4.10)

Class A amplifiers, for instance, are inefficient amplifiers, the efficiency range is between 10-

25%, and they can increase their efficiency to 50% which is the maximum. Thus, an ideal linear

power amplifier should be used to maintain the saturation point. This ideal power amplifier has

the following condition,

(4.11)

30

Power savings can be defined as the related power consumption to the efficiency ,

(4.12)

Now, by substituting (4.11) into (4.12), it can point out this result,

Therefore, the power saving from efficiency to another can be written as follows,

=2 ( ) (4.13)

To calculate savings gain , let us indicate that the saving gain as the ratio of savings power

to the output power,

(4.14)

From equation (4.13) into (4.14), it can infer to the result that,

Thus, the savings gain as the result of Peak to Average Power Ratio can be expressed as,

(4.15)

31

Figure 4.2 shows the performance of peak to average power ratio (PAPR) reduction of OFDM

symbol by using selected mapping (SLM) schemes. That was achieved by using equation (4.6),

where number of subcarriers N is set to 256 , with different values of phase sequences U (1, 2, 4,

8, and16). It is clear from the figure that by increasing the number of phase sequences U (SLM

scheme) large PAPR reduction can be obtained. The main focus on here is the saving power

through selected mapping, as it mentioned in the previous section the saving gain is the

difference in peak to average power ratio . Table 4.1 gives an overview

of several PAPR reduction performances corresponding to probability of clipping where N

equal to 256. All values in the table are corresponding to the curves. It is clear from the table

that, by increasing the phase sequences (SLM phase sequences) savings in gain is increased as

well. Equation (4.14) infers that by increasing in saving gain, the power saving will increase.

Thus, power saving can be achieved through selected mapping.

32

Figure 4.2 PAPR Reduction for SLM where N = 256 and U = 1, 2, 4,8,16.

Table 4.1

PAPR Reduction and Saving Gain Using SLM where N = 256 and U = 1, 2, 4,8,16

U (Phase Sequences ) PAPR (Peak to Average Ratio) ( Savings Gain )

1 (No SLM) 10.9 0

2 9.6 2.6

4 8.6 4.6

8 7.9 6

16 7.4 7

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

PAPR Reduction

33

CHAPTER 5

Computer Simulations

This chapter discusses the performance of selected mapping (SLM) with different values of

subcarriers N and phase sequences U. it can be seen from the simulations results that it is

possible for SLM scheme to reduce peak to average power ratio (PAPR).

5.1 OFDM Basic Model

5.1.1 OFDM Transmitter

The basic Simulink model of OFDM transmitter is shown in figure 5.1. it is clear that OFDM

transmitter consists on Bernoulli Binary Generator, Rectangular 16-QAM , Select Rows,

Complex to Real imag, Math Function, Matrix Concatenate, IFFT, and Selector . Figure 5.2 and

5.3 respectively show the parameter of Bernoulli Binary Generator and Rectangular QAM

Modulator. The M-ary number parameter is set to 16 and the sample time is set to 0.001/256 sec.

Figure 5.1 Simulink Model of OFDM Transmitter

34

Figure 5.2 Bernoulli Binary Generator Parameters

Figure 5.3 Rectangular QAM Modulator Baseband Parameters

35

5.1.2 OFDM Receiver

The basic Simulink model of OFDM receiver is shown in figure 5.4.As it can be seen OFDM

receiver consists on Selector, FFT, Select Rows, Complex to Imag, and Real-Imag to Complex,

matrix Concatenate, Frame Conversion, and Rectangular QAM Demodulator. Figure 5.5 shows

the parameters of Rectangular QAM Demodulator. M-ary number parameter is set to 16.

Figure 5.4 Simulink Model of OFDM Receiver

Figure 5.5 Rectangular QAM Demodulator Baseband Parameters

36

5.1.3 The Performance Analysis

The frequency spectrum of QAM signal and frequency spectrum of OFDM signal are

shown in figures 5.6, and 5.7 respectively. Figure 5.8 shows time domain of modulated OFDM

signal with 256 subcarriers and M-ary is set to 16. As it can be seen from the figure that OFDM

signal consists on high peak to average power, due to adding up subcarriers coherently with the

same phase. This high PAPR is very important situation because it is reducing the efficiency of

RF power amplifier, and resulting in inter carrier interference. Thus, SLM scheme is the

promising technique in reducing the effect of PAPR, and the performance of this technique will

be discussed in the next section.

Figure 5.6 Frequency Spectrum of Transmitted Signal

37

Figure 5.7 Frequency Spectrum of OFDM Signal

.

Figure 5.8 OFDM Signal in Time Domain

38

Figure 5.9 shows the results of probability of bit error (BER) performance for both the

simulation and theoretical. The MATLAB code in figure 5.10 is used to compare simulated BER

with theoretical BER. The point of this comparison between the simulation and theoretical

results is to make sure that the Simulink model gives the correct results. It is clear from the

figure 5.9 that simulation curve is close to the theoretical curve.

Figure 5.9 Simulated BER vs. Theoretical BER

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

100

Eb/No (dB)

Pro

babili

ty o

f B

it E

rror

Theory

Simulation

39

Figure 5.10 MATLAB Code

5.2 Threshold Selected Mapping

This section discusses PAPR reduction and it indicates that large PAPR reduction is possible

with selected mapping scheme. Figures 5.11, 5.12, 5.13, 5.14, and 5.15 respectively show the

performance of peak to average power ratio (PAPR) reduction of threshold selected mapping

(SLM) schemes by using equation (4.6) for different values of phase sequences U and

subcarriers N. It is clear from the figures that by increasing the number of phase sequences U

better PAPR reduction can be obtained. For instance, figure (5.11) is a plot of PAPR reduction

curves for OFDM symbol where N=62. From the figure it can be seen that when there is no

SLM which is at U=1 threshold needed to get good PAPR reduction performance is 10.5, while

for U = 16, only 6.2 is needed to get good PAPR reduction performance , by assuming that

close all; clear all;

BERVec=[];

t=10001;

EbNoVec=[0:2:10];

for n=1:length(EbNoVec);

EbNo=EbNoVec(n);

sim('ofdmmodel');

BERVec(n,:)=BER(t,1);

end

BERtheory=berawgn(EbNoVec,'qam',16);

figure

semilogy(EbNoVec,BERtheory,'b',EbNoVec,BERVec(:,1),'r');grid on

legend('Theory ','Simulation')

ylabel('Probability of Bit Error')

xlabel('Eb/No (dB)')

40

probability of clipping is for both cases. Table 5.1 gives an overview of several PAPR

reduction performances corresponding to probability of clipping for different values of

phase sequences U and subcarriers N.

Figure 5.11 PAPR Reduction for SLM where N = 64 and U = 1, 2, 4,8,16

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

41



4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

42



4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

43

Table 5.1 PAPR Reduction Corresponding to Various Phase Sequences for Different

Number of Subcarriers

PAPR

N 64 128 256 512 1024

D=1 ( No SLM) 10.5 10.9 10.9 11.2 11.4

D=2 8.8 9.2 9.6 9.9 10.2

D=4 7.7 8.2 8.6 8.9 9.4

D=8 6.8 7.4 7.9 8.4 8.7

D=16 6.2 6.8 7.4 7.9 8.3

Now, let us figure out the effect of oversampling on SLM that is by using equation (4.9) for over

sampling case where α =2.8. Figures 5.16,5.17,and5.18 respectively show PAPR curves for

sampled OFDM symbol for different values of phase sequences U and subcarriers N. it is clear

from the figures that all probability level are almost the same as on Nyquist samples (previous

curves). Thus, by applying oversampling on SLM nothing is going to change.

Figure 5.16 PAPR Reduction for SLM where α =2.8, N = 64, and U = 1, 2, 4,8,16

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

44



4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

z

P(P

AP

R>

z)

U=1

U=2

U=4

U=8

U=16

45

CHAPTER 6

Conclusions and Future Work

6.1 Conclusion

OFDM system has been discussed in this project. It indicated that OFDM is a popular

communication system due to the advantages this system has. For instance, the ability of the

system in converting frequency selective fading channels to flat fading channels. Also, the

robustness to inter symbol interference and inters carrier interference. In addition, the flexibility

of this system to the channel conditions, and easiness in meeting design requirement.

Peak to average power ratio issue was also discussed, showing how it affected the transmitted

signal. There were many reduction techniques presented to solve high peak to average power

ratio such as, Signal distortion techniques, Coding Schemes, and Symbol-scrambling techniques.

Selected mapping (SLM) technique was the main focus of the project. SLM explained in

details and showed that SLM is the most promising reduction technique. It was also mentioned

that power saving could be achieved through selected mapping.

This project also showed the simulation results of OFDM symbol with and without SLM. The

simulation results indicated that large PAPR reduction is possible with selected mapping scheme,

and showed how by increasing the number of phase sequences U large PAPR reduction can be

obtained.

46

6.2 Suggestions for Future Work

The following are suggestions of interesting topics that can be pursued as extensions of this

project

. Improving power efficiency by using selected mapping.

. Monomial phase sequences for selected mapping detection.

47

REFERENCES

1- Adarsh B. Narasimhamurthy, Mahesh K. Banavar, and Cihan Tepedelenlio˘glu, “OFDM

Systems for Wireless Communications”,2010, ISBN: 9781598297010

2- Jha, Uma Shanker, “OFDM Towards Broadband Wireless Access”, Artech House Books,

Norwood, 2007,ISBN: 9781580538091

3- Shieh, William; Djordjevic, Ivan, “OFDM for Optical Communications”, Academic Press,

Burlington, 2009,ISBN= 8790080952062

4- Andrea,Goldsmith, “ Wireless Communications”, Cambridge University,2005, ISBN 978-0-

521-83716-3

5- Tao Jiang, Member IEEE, and Yiyan Wu, Fellow, IEEE, “An Overview: Peak-to-Average

Power Ratio Reduction Techniques for OFDM Signals” VOL. 54, NO. 2, 2008

6- A. Zolghadrasli and M. H. Ghamat,” An Overview of PAPR Reduction Techniques for

Multicarrier Transmission and Propose of New Techniques for PAPR Reduction” Iranian Journal

of Electrical and Computer Engineering, VOL. 7, NO. 2, 2008.

7- Guangyue Lu1, Ping Wu and Catharina Carlemalm-Logothetis,” Partial Transmit

Sequences Method for Reduction of PAPR in Real -Valued OFDM Systems “Signals and

Systems Division, Dept. of Engineering Sciences, Uppsala University Uppsala, Sweden

8- Bauml, R., Fischer, R., and Huber, J,”R Guangyue Lu1, Ping Wu and Catharina Carlemalm-

Logothetis,” Reducing the peak-to-average power ratio of multicarrier modulation by selected

mapping" IEE Electronics Letters, vol. 32,pp. 2056-2057, 1996.

9- Pankaj Kumar Sharma,” Power Efficiency Improvement in OFDM System using SLM

with Adaptive Nonlinear Estimator” World Applied Sciences Journal 7 (Special Issue of

Computer & IT): 145-151, 2009, ISSN 1818.4952

10- Cho, Yong Soo; Kim, Jaekwon; Yang, Won Young; Kang, Chung G. ” MIMO-OFDM

Wireless Communications with MATLAB “Wiley, Hoboken, 2010,ISBN: 9780470825624

11- J. Tellado,” Multicarrier Modulation with Low PAR:Applications to DSL and Wireless”

Norwell, MA:Kluwer, 2000.

48

12- Richard van Nee, Ramjee Prasad” OFDM for wireless multimedia communications”. Artech

house, 2000. ISBN 0-98006-530-6

13-R. J.Baxley and G.T.Zhou “ Power Saving Analysis of Peak to Average Power Ratio

Reduction in OFDM” IEEE,Vol.50,No.3,2004

49

APPENDIX

MATLAB m-file code for figure (4.2)

clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=256;

p=ones (length (z), 12);

for j=1:length(U);

p(:,j)= 1-((1-exp(-(z))).^N).^U(j);

semilogy(zdb,p); grid on ;legend('U=1','U=2','U=4','U=8','U=16')

axis ([4 12 10^-4 1]);

hold on

ylabel('P(PAPR>z) ')

xlabel('z')

end


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=64;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^N).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end

50


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=128;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^N).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=256;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^N).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end

51


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=512;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^N).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=1024;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^N).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end

52


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=64;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^(2.8*N)).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=128;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^(2.8*N)).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end

53


clear all;close all

zdb= [0:1:12] ;

z= 10.^(zdb/10);

U = [1, 2, 4, 8, 16]; N=256;


for j=1:length(U);

p(:,j)= (1-(1-exp(-(z))).^(2.8*N)).^U(j);


axis ([4 12 10^-4 1]);

hold on


xlabel('z')

end

california state university, northridge reducing peak

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