Reduction of Peak-to-Average Power Ratio (PAPR) in ...

REDUCTION OF PEAK-TO-AVERAGE POWER RATIO

IN ORTHOGONAL FREQUENCY DIVISION

MULTIPLEXING

Farzana Rauf

Doctor of Philosophy

In

Electronic Engineering

MEHRAN UNIVERSITY OF ENGINEERING & TECHNOLOGY

JAMSHORO

October, 2016

i

REDUCTION OF PEAK-TO-AVERAGE POWER RATIO IN

ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING

A thesis submitted by

Farzana Rauf

In fulfillment of the requirements for the degree of

Doctor of Philosophy

In

Electronic Engineering

Department of Electronic Engineering

Institute of Information and Communication Technologies

Faculty of Electrical, Electronic and Computer System Engineering

Mehran University of Engineering and Technology

Jamshoro

October, 2016

ii

Dedicated to

my loving parents

who brought me on this earth,

my respected teachers, and

Worthy Vice Chancellor Dr. Usman Ali. G. Issani,

Iqra University Karachi,

who raised me up to sky.

iii

MEHRAN UNIVERSITY OF ENGINEERING & TECHNOLOGY

JAMSHORO

This thesis, written by Ms. Farzana Rauf Abro under the direction of her supervisors,

and approved by all the members of the thesis committee, has been presented to and

accepted by the Dean, Faculty of Electrical, Electronic and Computer Systems

Engineering, in fulfillment of the requirements of the degree of Doctor of Philosophy

in Electronic Engineering.

________________________ ___________________

Supervisor Co-supervisor

Prof. Dr. Manzoor Ahmed Hashmani Prof. Dr. Mukhtiar Ali Unar

____________________ ___________________

Internal Examiner External Examiner

____________________ ___________________

Co-Director IICT Director IICT

Prof. Dr. Zubair Ahmed Memon Prof. Dr. Mukhtiar Ali Unar

Mehran UET, Jamshoro Mehran UET, Jamshoro

__________________________________

Prof. Dr. Bhawani Shankar Chowdhury

(Dean Faculty of Electrical, Electronic and

Computer Systems Engineering)

Date: ______________________

iv

ACKNOWLEDGEMENTS

First of all I would like to thank Almighty Allah for giving me understanding,

intelligence and everything. I needed to finish up my research work with full

concentration and motivation.

Initially, with great respect, I would like to thank my supervisor Prof. Dr. Manzoor

Ahmed Hashmani, Director (Research and Publication), Faculty of Engineering,

Sciences, and Technology, Iqra University, Main Campus, Karachi who is first of all a

great man, a great engineer and a great team leader. I would have never dreamt of being

successful in this Ph.D. research work without his visionary guidance and mentoring.

He has been a continuous contributor of innovative ideas and knowledge throughout

my research work. He is no doubt the best resource; MUET has provided me for the

Ph.D. research.

I am also grateful to my respected co-supervisor, Prof. Dr. Mukhtiar Ali Unar,

Director Institute of Information and Communication Technologies, MUET, who has

always been a source of inspiration and who has critically judged and guided me during

my research. His expertise in this field and valuable observations has been very useful.

I have been fortunate to do Ph.D. in his kind Co-supervision.

I appreciate open heartedly cooperation and facilities provided by the Faculty of

Electrical, Electronic, and Computer Systems Engineering (FEECE) and the Institute

of Information and Communication Technologies (IICT). I am thankful heartedly to

Prof. Dr. Bhawani Shankar Chowdhry, Dean FEECE, Prof. Dr. Mukhtiar Ali

Unar, Director IICT and Prof. Dr. Zubair Ahmed Memon, Co-director IICT for their

phenomenal encouragement and immense cooperation. I would like to express sincere

v

gratitude to my beloved father Abdul Rauf Abro for his selfless support and guidance

during my Ph.D. research. His advices and guidelines had assisted me in many difficult

situation faced for completion of the thesis.

I especially appreciate the support and guidance provided by the Worthy Vice

Chancellor Dr. Usman Ali. G. Issani, Iqra University Karachi and Prof. Dr. Imtiaz

Hussain Kalwar for their special attention towards me during my Ph.D. Their

encouragement and guidance had always been source of motivation for me to complete

this Ph.D. research.

This research work is carried out at Mehran University of Engineering and Technology

(MUET) and I gratefully acknowledge MUET for providing me the opportunity to

study in a very interesting and challenging research area of security issues in wireless

sensor networks.

A portion of my research work has been carried out at University of Limerick, Ireland,

where I did my Ph.D. research for twelve months under the scholarship of Erasmus

Mundus Mobility for Life from 10th August 2013 to 11th September 2014. I am grateful

to my professors at University of Limerick, Ireland, Prof. Dr. Sean McGrath and

Prof. Dr. Elfed Lewis, Department of Electronic and Computer Systems Engineering,

for their guidance and support during this research.

vi

TABLE OF CONTENTS

Description Page

Acknowledgements (iv)

Table of Contents (vi)

List of Abbreviation (x)

List of Tables (xiii)

List of Figures (xiv)

Abstract (xvii

Chapter 1 INTRODUCTION 01

1.1 Rationale and Motivation 01

1.2 Research goal and Objectives 02

1.3 Structure of Dissertation 03

Chapter 2 ORTHOGONAL FREQUENCY DIVISION

MULTIPLEXING

06

2.1 Introduction to OFDM 06

2.2 Applications and Usage 07

2.2.1 ADSL 07

2.2.2 Wireless Local Area Networks (LAN), Metropolitan

Area Networks (MAN), and Personal Area Network (PAN)

07

2.2.3 Digital Video Broadcasting (DVB) 08

2.2.4 Digital Radio 08

2.2.5 Flash-OFDM (FOFDM) 09

2.3 Advantages and Disadvantages 09

2.3.1 Advantages 09

2.3.2 Disadvantages 10

2.4 Principles of Operation 10

2.4.1 Orthogonality 10

2.4.2 Implementation using FFT Algorithm 11

2.4.3 Guard Interval 12

2.4.4 Channel Coding and Time/Frequency Interleaving 13

2.4.5 Adaptive Transmission 14

2.4.6 OFDM with Multiple Access 14

2.4.7 Linear Transmitter Power Amplifier 15

vii

2.5 Efficiency Comparison between Single Carrier and Multi-

carrier OFDM Systems

16

2.6 System Model 18

2.6.1 Transmitter 18

2.6.2 Receiver 19

2.7 Mathematical Description 20

Chapter 3 REDUCTION OF PEAK-TO-AVERAGE POWER RATIO 22

3.1 Introduction 22

3.2 High Power Amplifier 23

3.2.1 Soft Limiter Power Amplifier 23

3.2.2 Solid State Power Amplifier 24

3.2.3 Travelling wave tube 24

3.3 PAPR Defined 24

3.4 Well Known PAPR Reduction Schemes 26

3.4.1 Signal Scrambling Schemes 26

3.4.1.1 Selective Mapping (SLM) 26

3.4.1.2 Partial Transmit Sequence (PTS) 28

3.4.1.3 Tone Reservation (TR) 29

3.4.1.4 Tone Injection (TI) 29

3.4.1.5 Interleaving Technique 30

3.4.2 Signal Distortion Schemes 31

3.4.2.1 Clipping and Filtering 31

3.4.2.2 Peak Reduction Carrier 32

3.5 Summary 33

Chapter 4 PERFORMANCE EVALUATION AND ANALYSIS

OF PAPR REDUCTION SCHEMES

35

4.1 Analysis of PAPR Reduction Schemes Through

Literature Survey

35

4.1.1 Clipping and Filtering Scheme 36

4.1.2 Coding Scheme 37

4.1.3 Peak Reduction Carriers Scheme 38

4.1.4 Envelope Scaling Scheme 39

4.1.5 PTS and SLM Schemes 39

4.1.6 Interleaving scheme 45

viii

4.1.7 Tone Reservation and Tone Injection Schemes 45

4.1.8 Active Constellation Extension Scheme 47

4.1.9 Comparison of PAPR Reduction Scheme 49

4.2 Performance Evaluation and Analysis of Selected

Mapping PAPR Reduction Schemes through Simulation

49

4.2.1 Motivation for Using Tone Reservation (TR) 51

4.2.2 Simulation Results on Performance of TR 52

4.2.3 PAPR Reduction Using Selective Mapping (SLM) 62

4.2.3.1 Motivation for Using Selective Mapping (SLM) 62

4.2.3.2 Selective Mapping (SLM) 62

4.2.3.3 Simulation Results and Discussion 64

4.2.3.4 Performance Analysis of SLM 65

4.3 Chapter Summary and Conclusion 69

Chapter 5 PROPOSAL OF A NOVEL PAPR REDUCTION

SCHEME BASED ON MNKB-RBF

70

5.1 Issue of SLM (Optimization Problem) 70

5.2 ANN (Artificial Neural Networks) and RBF (Radial Basis

Function)

70

5.3 Conventional RBF 73

5.4 A Recently Proposed “Novel Kernel Based RBF

(NKB-RBF)”

75

5.5 Issues of NKB-RBF 76

5.6 Proposed Solution: Modified NKB-RBF (MNKB-RBF) 78

5.7 Proposal of a Novel PAPR Reduction Scheme Based on

MNKB-RBF

79

5.8 Chapter Summary and conclusion 81

Chapter 6 PERFORMANCE EVALUATION OF PROPOSED PAPR

REDUCTION SCHEME

82

6.1 Performance Evaluation Environment 83

6.2 Simulation Environment and Test Cases 83

6.2.1 Regarding Datasets 83

6.2.2 Regarding Test Cases 84

6.3 Core Code of the Proposed Algorithm 84

6.4 Training Results 86

ix

6.4.1 Evaluation, Analysis, and Deductions 87

6.5 Testing Results 93

6.5.1 Evaluation, Analysis, and Deductions 94

6.5.2 Probability of Selecting Carrier of Low PAPR 100

6.6 Chapter Summary and Conclusion 102

Chapter 7 SUMMARY AND CONCLUSION 103

7.1 Summary 103

7.2 Conclusion 104

7.3 Future Work/Research Guideline 105

References 107

x

LIST OF ABRIVIATIONS

ACE = Active Constellation Extension

ADC = Analog to Digital Converter

ADSL = Advanced Digital Subscriber’s Line

AM = Amplitude Modulation

ANN = Artificial Neural Networks

BER = Bit Error Rate

BPSK = Binary Phase Shift Keying

BRAN = Broadcast Radio Access Network

CBC = Complement Block Coding

CC = Cyclic Coding

CCDF = Complementary Cumulative Distribution Function

CCI = Co-Channel Interference

CD = Cosine Distance

CDF = Cumulative Distribution Function

CDMA = Code Division Multiple Access

CF = Clipping and Filtering

CP = Cyclic prefix

COFDM = Coded Orthogonal Frequency Division Multiplexing

DAC = Digital-to-Analogue Converter

DMT = Discrete Multi-tone

DRM = Digital Radio Mondiale

DSL = Digital Subscriber’s Line

DVB = Digital Video Broadcasting

DS = Doppler Shift

ED = Euclidean Distance

ETSI = European Telecommunications Standards Institute

FEC = Forward Error Correction

FFT = Fast Fourier Transform

FOFDM = Flash Orthogonal Frequency Division Multiplexing

FPGA = Field Programmable Gate Array

FVs = Feature Vectors

xi

Giga Hertz = GHz

Hz = Hertz

IBO = Input Back-off

ICI = Inter Channel Interference

HPA = High Power Amplifier

IEEE = Institute of Electrical and Electronic Engineers

IFFT = Inverse Fast Fourier Transform

ISI = Inter-Symbol Interference

LAN = Local Area Network

LDPC = Low Density Parity Check

LTE = Long Term Evolution

MAN = Metropolitan Area Networks

MCBC = Modified Complementary Block Coding

MHz = Mega Hertz

MIPS = Million Instructions Per Second

MNKB-RBF = Modified Novel Kernel Based Radial Basis Function

NKB-RBF = Novel Kernel Based – Radial Basis Function

OFDM = Orthogonal Frequency Division Multiplexing

OFDMA = Orthogonal Frequency Division Multiple Access

PAN = Personal Area Networks

PAPR = Peak-to-Average Power Ratio

P/S = Parallel-to-Serial Conversion

PM = Phase Modulation

PRC = Peak Reduction Carrier

PSK = Phase-Shift Keying

PTS = Partial Transmit Sequence

QAM = Quadrature Amplitude Modulation

QoS = Quality of Service

QPSK = Quadrature Phase Shift Keying

RBF = Radial Basis Function

S/P = Serial-to-Parallel Conversion

SBC = Simple Block Coding

SII = Side Information Index

xii

SNDR = Signal-to-Noise-plus-Distortion Ratio

SLM = Selective Mapping

SNR = Signal-to-Noise Ratio

SL = Soft Limiter

SSPA = Solid State Power Amplifier

SVM = Support Vector Machine

TR = Tone Reservation

TWT = Travelling Wave Tube

UMB = Ultra-Mobile Broadband

UWB = Ultra-Wideband

VHF = Very High Frequency

VDSL = Very-high-speed Digital Subscriber Line

WRAN = Wireless Regional Area Network

WiMAX = Worldwide interoperability for Microwave Access

xiii

LIST OF TABLES

Description Page

Table 2.1 Performance Comparison between Single and Multicarrier 17

Table 3.1 Classification of Major PAPR Reduction Schemes 26

Table 3.2 Performance of PAPR Reduction Techniques 33

Table 4.1 Comparison of Different PAPR Reduction Schemes 49

Table 4.2 Different PRC Sum Variable Combinations 53

Table 4.3 PAPR Levels in dBs for OFDM Symbol Candidate Combination 65

Table 5.1 Criteria for Selection (Manual) of Mining Parameters α1 and α2 77

Table 6.1 Final Mean Square Error (MSE) Comparison 88

Table 6.2 Probability of Selecting Carrier of Low PAPR 101

xiv

LIST OF FIGURES

Description Page

Figure 2.1 OFDM Transmitter Model 18

Figure 2.2 OFDM Receiver Model 19

Figure 3.1 Block Diagram of SLM Scheme 27

Figure 3.2 Block Diagram of PTS Scheme 28

Figure 3.3 Block Diagram of Interleaving Scheme 31

Figure 4.1 PAPR distribution when “Clipping and Filtering Scheme” is

used

37

Figure 4.2 PTS Scheme Block Diagram 40

Figure 4.3(a) CCDF for PAPR of OFDM with and without PTS 41

Figure 4.3(b) CCDF for PAPR of OFDM with and without PTS 42

Figure 4.4 Block Diagram of SLM Scheme 43

Figure 4.5(a) CCDF for PAPR of OFDM with and without SLM 44

Figure 4.5(b) CCDF for PAPR of OFDM with and without SLM 44

Figure 4.6(a) PAPR of Tone Reservation, 12 Sub-Carriers, and 4 Peak

Cancellation Sub-Carriers

46

Figure 4.6(b) PAPR of Tone Reservation, 12 Sub-Carriers, and 4 Peak

Cancellation Sub-Carriers

47

Figure 4.7 The ACE Scheme for QPSK Modulation 48

Figure 4.8 TR-OFDM signal for -r1-r2-r3-r4 combination 54

Figure 4.9 TR-OFDM signal for -r1-r2-r3-r4 combination 54

Figure 4.10 TR-OFDM signal for +r1+r2-r3-r4 combination 55

Figure 4.11 TR-OFDM signal for -r1-r2+r3+r4 combination 55

Figure 4.12 TR-OFDM signal for -r1-r2-r3+r4 combination 56

Figure 4.13 TR-OFDM signal for -r1+r2+r3+r4 combination 56

Figure 4.14 TR-OFDM signal for +r1-r2-r3-r4 combination 57

Figure 4.15 TR-OFDM signal for +r1-r2-r3+r4 combination 57

Figure 4.16 TR-OFDM signal for +r1+r2+r3-r4 combination 58

Figure 4.17 TR-OFDM signal for -r1+r2-r3+r4 combination 58

Figure 4.18 TR-OFDM signal for +r1+r2+r3-r4 combination 59

Figure 4.19 TR-OFDM signal for +r1-r2+r3+r4 combination 59

Figure 4.20 TR-OFDM signal for +r1+r2-r3+r4 combination 60

xv

Figure 4.21 TR-OFDM signal for -r1+r2-r3-r4 combination 60

Figure 4.22 TR-OFDM signal for -r1-r2+r3-r4 combination 61

Figure 4.23 TR-OFDM signal for -r1+r2+r3-r4 combination 61

Figure 4.24 Block Diagram for SLM Scheme for OFDM System 63

Figure 4.25 CCDF graph for PAPR of OFDM with and without SLM 66






Figure 5.1 Architecture of the RBF based Neural Network 74

Figure 5.2 Block Diagram of the Proposed PAPR Reduction Scheme

Using MNKB-RBF

80

Figure 6.1 Core MatLab Code of the Proposed RBF (MNKB-RBF) 85

Figure 6.2 Training Cost Comparison between Novel RBF and Proposed

RBF (Phase-Rotated Sequences = 64, Modulation = 8-QAM)

89


RBF(Phase-Rotated Sequences = 64, Modulation = 16-QAM)

89



90



90



91



91



92



92



93

Figure 6.11 Testing Comparison between Novel RBF and Proposed RBF

(Phase-Rotated Sequences = 64, Modulation = 8-QAM)

96



96

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97



97



98



98



99



99



100

xvii

ABSRACT

It has been observed over the last several decades that greediness of applications in

terms of their demand for bandwidth never gets fulfilled. Hence, scientists, researchers,

and engineers keep working on new ways and means of providing higher bandwidth.

Not in the much distant past, a new modulation technique called Orthogonal Frequency

Division Multiplexing (OFDM) has been introduced which provides very high data

rates. The key feature of OFDM is the orthogonality of its carrier frequencies. Note that

in OFDM the high frequency input signal is modulated over a large number of low

frequency sub-carrier signals which are orthogonal to each other. This feature makes it

very robust against efficiency degradation at higher frequencies. That is the reason for

the OFDM to be a choice technology for modern high and ultra-high data rate

communication systems. However, it suffers from high levels of the peak power to the

average power also called Peak-to-Average Power Ratio (PAPR). Reducing PAPR in

OFDM is a hot research area. There are many schemes available which attempt to

reduce PAPR. Some are in fact able to reduce PAPR but not sufficient enough to make

these feasible. Others do reduce it but increase its complexity to an extent that these

become unfeasible to realize.

The main motivation behind this research effort is to find a mechanism which results

in minimum PAPR for OFDM based communication systems and has a reasonable level

of complexity so that it may be realizable.

This thesis has investigated and analyzed a number of PAPR reduction schemes

available in the literature. It has been identified that the Selective Mapping (SLM) is

xviii

better than others methods in terms of computational complexity for reduction of the

PAPR.

As an outcome of this research activity, a novel framework based on Artificial Neural

Networks (ANN) and SLM is proposed. The kernel being used by the ANN of the

proposed framework is called Modified Novel Kernel Based Radial Basis Function

(MNKB-RBF) and is a modified version of an already available kernel (NKB-RBF) in

the literature and is shown to be highly efficient. It has been shown through simulations

that the proposed kernel is more efficient than NKB-RBF and thus produces better

results in selection of low frequency sub-carrier with the lowest PAPR.

Keywords: Orthogonal Frequency Division Multiplexing, Peak-to-Average Power

Ratio, Selected Mapping, Artificial Neural Networks, Novel Kernel Base Radial Basis

Function, Modified Novel Kernel Base Radial Basis Function.

1

CHAPTER 1

INTRODUCTION

1.1 RATIONALE AND MOTIVATION

It has been seen over the years that as the data rates of communication systems increase

so does the demand by the applications. It has been observed that the current

applications demand very high data rates from the communication networks. Some

examples of these applications are; high definition TV, good quality video, on-line

gaming, vehicle navigation systems, etc.

One more demand of modern applications is an ability of the communication system to

sustain high data rates (without service disruption) to devices present in vehicles

moving at very high speed, i.e., fact moving cars, electric trains, airplanes, etc.

These demands of high date rates with seamless service require new and ingenious

techniques to be employed in new communication systems to provide ultra-high data

rates. One area of focus is the modulation techniques. It has been observed that efficient

modulation techniques can multiply the data rates that are made available by the

networks.

Not in distant past, a new modulation technique called Orthogonal Frequency Division

Multiplexing (OFDM) has been introduced which provides very high data rates and is

based on multiplexing of frequencies. The key feature of OFDM is the orthogonality of

its carrier frequencies. This feature makes it very robust against efficiency degradation

at higher frequencies (which results in higher data rates) unlike other modulation

techniques which suffer from substantial degradation of service at higher frequencies.

2

This is the reason for the OFDM to be the choice technology for the modern high and

ultra-high data rate communication systems.

However, like many of its contemporary modulation technologies, it has its share of

demerits. In particular, it suffers from high levels of the peak power to the average

power ratio also called Peak-to-Average Power Ratio (PAPR). In OFDM, PAPR is high

mainly due to the reason that the summation of peaks of many sub-carriers may result

in very high value. Note that in OFDM the high frequency input signal is modulated

over a large number of low frequency sub-carrier signals which are orthogonal to each

other. Though the average of these sub-carriers would be quite low, the peak power

which is the summation of all peak values may become very high in some cases.

Reducing PAPR in OFDM is a hot research area. There are many schemes available

which attempt to reduce PAPR. Some are in fact able to reduce PAPR but not sufficient

enough to make these feasible. Others do reduce it but increase its complexity to an

extent that these become unfeasible to realize. Sufficient details of these schemes, their

merits and demerits are provided in Chapter 3. To say it in brief, the issue is not settled

yet.

The motivation and rationale behind this research work is to study various PAPR

reduction techniques, for study their limitations and propose a novel approach to

overcome those limitations.

1.2 RESEARCH GOAL AND OBJECTIVES

The goal of the attempted research is:

To address and resolve (in a better way) the PAPR issue to improve the Quality

of Service (QOS) in OFDM based broadband communication systems.

3

In order to achieve the above stated goal, it is natural to fulfill/accomplish the

following objectives:

1. Literature Survey to Devise Strategy for PAPR Reduction

Literature survey has been conducted to comprehend already available

Methods/ scheme to reduce high PAPR.

2. Identify the Most Promising PAPR Reduction Scheme

Using means like literature survey and/or performance evaluation of

existing schemes, identify a scheme that has the most promise in order to

be used as a candidate for PAPR reduction.

3. Propose a Novel PAPR Reduction Scheme

Depending upon the outcome of 2nd objective, to propose a modified

scheme to achieve our stated goal of reducing PAPR.

4. Validation of Effectiveness of the Proposed Scheme

The last objective in order to fulfill our goal is to validate and verify the

effectiveness of the proposed scheme (outcome of the 3rd objective). The

proposed scheme has been validated through extensive simulations

studies.

1.3 STRUCTURE OF DISSERTATION

This dissertation consists of seven chapters in total.

Chapter 2: Provides the detailed technical knowhow about OFDM, its application,

usage, mathematical background, and its demerit (namely, high PAPR).

However, the detailed description of PAPR, its impact on system design

4

and performance, and an introduction of these schemes to handle PAPR

is given in Chapter 3.

Chapter 3: Chapter 3 provides an introduction to the schemes which are available in

the literature to reduce PAPR. A detailed account of a few prominent

schemes as per published literature as well as our own evaluation

(simulation based) is given in Chapter 4.

Chapter 4: This chapter focuses upon the following PAPR reduction schemes and

provides a critical review of these schemes based on the information

available in the literature. The schemes under focus are Clipping and

Filtering (CF), Tone Reservation (TR), Partial Transmit Sequence

(PTS), and Selective Mapping (SLM). Note that Chapter 2, Chapter 3,

and Chapter 4 are based primarily on the survey of the available

literature.

On the other hand, the subsequent chapters, i.e., Chapter 4 to Chapter 6 are based on

original work as part of this research activity and reflect upon the contribution of this

research work. Though performance evaluation of above stated PAPR reduction

schemes is available in literature, a comprehensive evaluation of these schemes under

uniform evaluation criteria is not available in the literature. Hence, such comprehensive

evaluation has been done and is reported in the later part of Chapter 4. This chapter

concludes that the most promising of all these PAPR reduction schemes as per our

evaluation is the SLM.

5

Chapter 5: In the first part of Chapter 5, a new Artificial Neural Network (ANN)

kernel is proposed called Modified Novel Kernel Base Radial Basis

Function (MNKB-RBF), which is a modified version of one of the most

efficient ANN kernels available to be used in optimization applications.

The later part of this chapter proposes a new framework/scheme based

primarily on the proposed MNKB-RBF to select for SLM a sequence of

sub-carriers from candidate sequences with minimum expected PAPR.

Chapter 6: Reports on the comprehensive performance evaluation of proposed kernel

(MNKB-RBF) and the proposed framework to minimize PAPR. The

results indicate that new proposed kernel and the framework both work as

anticipated, i.e., help SLM selects the sub-carriers sequence with

minimum PAPR in most cases.

Chapter 7: Provides a summary of this research work, states the contributions of this

activity, and also sums up the guidelines for those who would like to

continue from the point where this research activity reached.

6

CHAPTER 2

ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING

(OFDM)

2.1 INTRODUCTION TO OFDM

OFDM is a modulation technique which uses multiple low frequency sub-carrier

frequencies instead of a single high frequency carrier and is very popular these days to

transmit digital data. In particular, it is the most widely used encoding scheme for

wideband digital communication [1]. In a wideband system, the message bandwidth is

much greater than the channel’s coherence bandwidth. On the other hand, the

broadband communication system is a wideband data transmission system having

multiple transmissions simultaneously [2]. Applications of wideband systems are

numerous, for example, wireless networks, Digital Subscriber’s Line (DSL), digital

TV, and 4G mobile communications systems. As stated earlier, OFDM uses many sub-

carrier frequencies to modulate the original signal. The sub-carriers frequencies are

always spatially orthogonal to one other. The important fact is that the modulation used

on each sub-carrier is at a low symbol rate. However, overall symbol rate is quite high

because multiple sub-carriers are used. Note that for modulation, any of the

conventional modulation techniques methods such as QAM or Phase-Shift Keying

(PSK) is used.

We know that higher symbol rate over single carrier results in severe channel conditions

(attenuation, interference, fading, etc.). These conditions are reduced substantially due

to low symbol rate on each sub-carrier in OFDM. Nevertheless, to acquire high symbol

rate due to accumulation of low symbol rates of all sub-carriers. Moreover, due to low

7

symbol rate we do not need to use a large guard interval between symbols. This results

in elimination of Inter-Symbol Interference (ISI) and achievement of better Signal-to-

Noise Ratio (SNR).

2.2 APPLICATIONS AND USAGE

The OFDM based multiple access technology is being used in cable, wireless, many

4G cellular networks, and mobile broadband networks. Some of the main applications

and usage of OFDMA are given below.

2.2.1 ADSL

The Asymmetric Digital Subscriber’s Line (ADSL) connections use OFDM for

modulation. ADSL uses ANSI T1.413 and G.992.1 standards. DSL and ADSL provide

high speed data transmission over existing copper wires. The subsequent versions of

ADSL, i.e., ADSL2, ADSL2+, VDSL, and VDSL2, also use OFDM [3].

It is known that on copper wires in particular, the higher the frequency the higher the

attenuation. It means that the data rate is directly proportional to the frequency. Hence,

in order to achieve high data rate we need to use higher frequencies at the cost of higher

attenuation. But since OFDM uses many low frequency sub-carriers instead of a single

high frequency carrier, it is robust against attenuation. And thus it is a favorable choice

for use in DSL based technologies [3].

2.2.2 Wireless Local Area Networks (WLAN), Metropolitan Area Networks

(MAN) and Personal Area Networks (PAN)

Besides wired networks, OFDM is also widely used in all types of Wireless Local Area

Networks (LAN) and worldwide interoperability for Microwave Access (WiMAX).

https://en.wikipedia.org/wiki/WiMAX

8

Latest broadband wireless LANs (IEEE 802.11a/g/n) operate in very high frequency

bands, from around 2.4 GHz to around 5 GHz. Each stream in these standards may

provide data rates from 6 to 54 Mbps. In "HT mode" (with 802.11n), the data rate of

each stream is increased up 150 Mbps. These standards use four different modulation

schemes (BPSK, QPSK, 16-QAM, 64-QAM).

Similarly, PANs in the 3.1–10.6 GHz ultra-wideband spectrum are also using OFDM.

2.2.3 Digital Video Broadcasting (DVB)

Digital Video Broadcasting (DVB) is a well-known project of European Commission

for transmission of high quality digital video over large distances [4]. It is compulsory

for all television services in the whole European Community to use this standard. DVB

standard requires that Coded OFDM (COFDM) be used for modulation of television

data over carrier frequencies. The word “Coded” refers to the use of Forward Error

Correction (FEC), i.e., for error detection and correction.

2.2.4 Digital Radio

Like television transmission, COFDM is used in high quality radio transmission

standards as well. For example, COFDM is used in a European digital audio

broadcasting standard referred to as Digital Radio Mondiale (DRM) at VHF frequencies

[4]. The similar American standard, iBiquity also uses COFDM as lower layer

technology to broadcast audio signals in various frequency spectra [5].

Similarly wireless personal area network technology of ultra-wideband also utilizes

OFDM.

9

2.2.5 Flash-OFDM (FOFDM)

Flarion [6] developed a new standard based on OFDM and is called Fast Low Latency

Access with Seamless Handoff Orthogonal Frequency Division Multiplexing

(FOFDM) or Flash OFDM. It was later purchased by Qualcomm in January 2006 [7].

FOFDM was supposed to compete with GSM and 3G networks. For example, 450 MHz

spectrum previously used by NMT450 and CNetC450 is now being to FOFDM

operators.

USA based wireless carrier Nextel Communications also started using FOFDM based

wireless broadband networks in 2005. Sprint has deployed mobile version of WiMAX

which uses another variation of OFDM, namely Scalable Orthogonal Frequency

Division Multiple Access (SOFDMA) technology [8].

2.3 ADVANTAGES AND DISADVANTAGES

Listed below are the advantages and disadvantages of OFDM. These are discussed in

further detail in section 2.4 (Principles of Operation).

2.3.1 Advantages

Here are some of the main advantages of OFDM [14, 15].

Spectral efficiency of OFDM is much higher than the other advanced

modulation schemes such as spread spectrum, etc.[17]

OFDM does not need complex time domain equalization in order to handle

severe channel conditions.

It is highly robust against Co-Channel Interference (CCI).

At the same time OFDMA is robust against ISI and multipath related fading.

OFDM is not sensitive to time synchronization problems.

10

Fast Fourier Transform (FFT) is used to efficiently implemented OFDM.

2.3.2 Disadvantages

The disadvantages are listed below [9, 11].

High PAPR is one of the major disadvantages of OFDM based systems. A lot

of research is being conducted to find ways and means to reduce PAPR. This is

also the main focus of this research.

OFDM is sensitive to problems related to frequency synchronization.

It is also sensitive to Doppler shift (DS) which is caused due to motion of

communicating entities [9].

Some loss of efficiency caused due to guard interval.

2.4 PRINCIPLES OF OPERATION

The following sub-sections describe the main characteristics and principles of

operation of OFDM.

2.4.1 Orthogonality

As stated earlier, OFDM is a frequency division multiplexing scheme with an additional

requirement that the sub-carrier frequencies must be orthogonal to one another.

Because of the orthogonality of sub-carriers, crosstalk between sub-carriers is

eliminated. Moreover, inter-carrier guard-bands are not required. This results in great

amplification in designs of both the transmitter and the receiver. Note that a separate

filter for each subcarrier is not required in OFDM which would not be the case if

conventional FDM were used [10, 11, 12, 13].

11

The condition of orthogonality makes it mandatory to separate sub-carriers in frequency

by a factor of ∆f = k

TU Hertz. Here TU is the size in seconds of the receiver’s window,

and k is a positive integer generally equal to 1. Hence, if N sub-carriers are used in an

OFDM based communication, the total bandwidth will be B ≈ N ∆f Hertz.

Because of orthogonality, we can utilize almost all of the allocated bandwidth resulting

in very high spectral efficiency. OFDM generally gives benign electromagnetic

properties which result in very low interference between co-channel users.

However, retaining orthogonality requires careful frequency synchronization between

the transmitter and the receiver. If a frequency deviation occurs, the sub-carriers will

not remain orthogonal which consequently result in Inter Channel Interference (ICI),

ISI and other severe channel conditions. Frequency deviations usually occur due to

reasons such as Doppler Shift or mismatched transmitter and receiver oscillators.

DS is caused by the relative movement of the transmitter and/or receiver. DS can easily

be handled by the adjustment (addition or subtraction) of frequency at the receiver’s

end, however, when this gets combined with multipath phenomenon, it becomes very

difficult to correct. And as the speed increases so does its worsening effect. This factor

limits the usage of OFDM in high-speed vehicles [9].

2.4.2 Implementation using FFT Algorithm

The implementation of modulation and demodulation can be efficiently done at both

the sender and the receiver side due to the orthogonality feature of sub-carriers. Note

that inverse FFT algorithm is applied on the transmitter side and forward FFT algorithm

on the receiver side. The recent substantial cost reduction in the digital signal

12

processing components that can efficiently calculate the FFT has boosted the use of

OFDM in wideband communication systems.

In order to be used, the time required to compute the FFT (or inverse FFT) has to be

less than the time for each symbol. For example for Digital Video Broadcast which

uses FFT 8k puts a compulsion on computation time to be less than or equal to 896 μs.

For an 8192-point FFT the computational requirement can be calculated to be 428 MIPS

as shown below:

MIPS = Computational Complexity

TSymbolx 1.3 x10−6 (2.1)

= 147456 x 2

896 x10−6x 1.3 x10−6 = 428

The computational demand increases approximately linearly (not exponentially) with

respect to FFT which implies that if the size of FFT is doubled, the amount of time

would be approximately double. This feature is very important for OFDM to be used in

all cases. If it were an exponential increase, the use of OFDM in higher frequency

communication would have been cost ineffective.

2.4.3 Guard Interval

In OFDM, many low frequency sub-carrier signals are used instead of one high

frequency signal. Note that both would provide the same high data rate because data

rates of all sub-carriers have to be added. However, having many low frequency sub-

carriers is very advantageous (some were described in preceding sub-sections).

The rate at which symbols are modulated over each sub-carrier is kept low. Hence, each

modulated symbol in terms of time is relatively long as compared to time characteristics

13

of the channel. Longer symbols suffer less from ISI caused by multipath phenomena.

Note that when the duration of each symbol is long, it becomes feasible to insert guard

interval between the OFDM symbols which eliminates ISI. The guard intervals reduce

the sensitivity of OFDM to time synchronization issues and simultaneously eliminates

the requirement of pulse-shaping filter.

For an illustration consider the following example. On a conventional single carrier

wireless channel one million symbols are modulated in one second which implies that

each symbol is approximately one microsecond long. Shorter symbols (in time) make

it very difficult to synchronize in time and hence a mechanism to handle the multipath

problem has to be put in place. If the same symbol rate, i.e., one million symbols per

second is distributed evenly among one thousand sub-carriers, the symbol rate on each

sub-carrier would be one thousand symbols per second. This implies that the duration

of each symbol on each carrier would be approximately one thousand times longer, i.e.,

one millisecond. Assume that we insert a guard interval of 1/8 of the symbol length,

i.e., 125 microseconds after each symbol. It has been seen that ISI can be eliminated if

the multipath time-spread is shorter than the guard interval indicating that a maximum

difference of 37.5 kilometers between all paths can be tolerated which is generally more

than enough.

2.4.4 Channel Coding and Time/Frequency Interleaving

Generally speaking, OFDM is used together with FEC also called channel coding as

well as time and/or frequency interleaving.

The benefit of frequency interleaving of sub-carriers is that it increases robustness

against channel fading. Frequency interleaving ensures that when one part of the

14

bandwidth being used fades, then the bit errors of that part are distributed into the

unaffected part of the bandwidth. Similarly, time interleaving distributes bit errors far

away in time. As evident, the purpose of frequency and time interleaving is to eliminate

concentration of errors. This attribute is very important because FEC’s ability of error

detection and correction becomes highly reduced or in some cases impossible to detect

and correct errors when errors are concentrated. Hence, interleaving is almost always

used on OFDM systems, so that FEC decoders could detect and correct more evenly

distributed bit errors.

2.4.5 Adaptive Transmission

In order to mitigate severe channel conditions, OFDM uses the technique of “Adaptive

Transmission”. In adaptive transmission, information about the channel condition is

sent over a return channel. This information is then used to adapt modulation, coding

scheme and power allocation for all or some sub-carriers as the need may be to optimize

transmission. Some sub-carriers can even be disabled if these are suffering from high

interference or attenuation.

For example, ADSL and VDSL uses Discrete Multi-tone (DMT) modulation which is

an adaptive OFDM based communication system. The granularity of the adaptation is

a single subcarrier.

2.4.6 OFDM with Multiple Access

As stated in previous sections, OFDM is a digital modulation technique which is

primarily used to modulate one original stream of data on a single communication

channel. However, it can be used to provide multiple access (access to multiple users

15

over a channel) using well known techniques of dividing time, frequency, or code. In

this form it is termed Orthogonal Frequency Division Multiple Access (OFDMA).

In OFDMA, multiple access is provided by assigning different sub-channels to different

users. Different Quality of services (QOS) for different users is achieved by assigning

different number of sub-channels effectively by increasing or decreasing the assigned

bandwidth to a user.

OFDMA is used in a variety of modern high-speed communication systems. Some are

mentioned below:

WiMAX IEEE 802.16 based Mobile Wireless MAN standard.

MWBA – IEEE 802.20 based Mobile Wireless MAN standard

Long Term Evolution (LTE) – Fourth generation mobile broadband standard.

Ultra-Mobile Broadband (UMB) – Qualcomm/3GPP2 project, intended as a

successor of CDMA2000, but replaced by LTE.

Wireless Regional Area Networks (WRAN) – IEEE 802.22 based network

which will potentially use OFDMA.

2.4.7 Linear Transmitter Power Amplifier

One of the main demerits of OFDM is a high PAPR. The average power of each sub-

carrier is quite low. However, because of the independent phases of sub-carriers of

OFDM, the peak power is determined by accumulation of powers of sub-carriers and

may become quite high. Handling high PAPR is a hot research area. Some ways to

handle PAPR are given below.

A high resolution Digital-to-Analogue (DAC) Converter is placed in the

transmitter.

16

A high resolution Analogue-to-Digital (ADC) Converter in the receiver.

Ensuring a linear signal chain.

If linearity in the signal chain is not ensured, it will cause inter-modulation distortion

which in turn may cause the following.

Raise the noise floor.

Produce Inter-Carrier Interference (ICI).

Generate out-of-band radiation.

Ensuring linearity is a very complex issue. In particular it is very demanding for the

transmitter because the amplifiers used in the transmitter are deliberately made non-

linear so as to minimize power consumption. One way to handle this is to allow a small

amount of clipping to limit PAPR. However, the transmitter output filter has the effect

of restoring peak levels which were previously clipped. Hence, clipping indeed is not a

suitable solution to reduce PAPR.

Since PAPR is in fact the main area of focus of this research activity, a detailed report

and discussion on PAPR and the schemes that attempt to reduce PAPR are given in

subsequent chapters.

2.5 EFFICIENCY COMPARISON BETWEEN SINGLE CARRIER AND

MULTI-CARRIER OFDM SYSTEMS

A communication system’s performance is generally measured using two very

important parameters, i.e., the power efficiency and the bandwidth efficiency. Power

efficiency is defined as the ability of a communication system to transmit signal with

low Bit Error Rate (BER) at low power levels. On the other hand bandwidth efficiency

of a communication system is its ability to utilize the available bandwidth to the

17

maximum, i.e., to achieve highest possible date rate per Hertz. For example, the

bandwidth efficiency of an OFDM system on fiber using multiple sub-carriers can be

calculated from the following formula.

η = 2 RS

BOFDM (2.2)

Here, RS is the symbol rate in giga symbols per second (Gsps),

and BOFDM is the bandwidth of OFDM signal.

The bandwidth efficiency of a multicarrier system such as OFDM is higher as compared

to a single carrier system. Table 2.1 shows comparison of performance of OFDM

systems having single carrier and multi-carrier modulation. The last column in this table

(titled “Bandwidth Efficiency”) indicates a performance gain of 76.7% at the cost of 1

dBm increase in receiver power as indicated in the 2nd last column (titled “Power at

Receiver”).

Table 2.1: Performance Comparison between Single Carrier and

Multicarrier OFDM [10]

S. No. Type of

Transmission QAM

Sub

carriers Bit Rate

Fiber

Length

Power

at

Receiver

Bandwidth

Efficiency

1 Single Carrier 64 1 10 Gbps 20 km -37.3

dBm 6.0000

2 Multi Carrier 64 128 10 Gbps 20 km -36.3

dBm 10.6022

18

2.6 SYSTEM MODEL

The subsequent sub-sections describe OFDM system models for the transmitter and the

receiver side.

2.6.1 Transmitter

In OFDM, the carrier signal is the summation of multiple sub-carriers which are

orthogonal to one another. Some conventional modulation scheme, such as QAM,

Phase Shift Keying (PSK), etc., is used to modulate given data on each sub-carrier.

As shown in Figure 2.1, s[n] is a stream of digital data. This digital data stream is first

of all de-multiplexed into N parallel streams. Using a set of symbols, now each sub-

stream of digital data is modulated using a conventional modulation scheme (QAM,

PSK, etc.). Since each stream is modulated independently of the other, some streams

may have higher symbol/data rate than the others.

Figure 2.1: OFDM Transmitter model

Source: Wikipedia, the free encyclopedia

X

X

X0

s[n]

FFT-1

X1

XN-2

XN-1

DAC

DAC

Constellation Mapping

𝕽𝒆

fe

𝟗𝟎𝒐 Serial to Parallel

s(t)

𝕽𝒎

19

After modulation, inverse FFT is applied on each sub-stream which produces a set of

complex time-domain samples for that sub-stream. As shown in Figure 2.1, the next

stage which consists of DACs converts real and imaginary components into analogue

domain. The analogue signals are then used to modulate at the carrier frequency, fc,

respectively. These modulated signals are then summed up to give the signal s(t) which

is to be transmitted.

2.6.2 Receiver

The corresponding receiver model of an OFDM system is shown in Figure 2.2. It can

observe in this figure that the receiver receives the signal r(t) and quadrature-mixes it

to baseband frequency. The resultant baseband analogue signal is converted into digital

stream in the next stage which consists of ADCs. Note that this signal is still in time

domain. FFT is applied on this signal to convert it into frequency domain. This produces

N parallel sub-stream of modulated symbols. Each stream is demodulated to produce

sub-streams of digital data. These sub-streams are combined into one stream of digital

stream S[n] which is approximately equal to the stream of digital data s[n] transmitted

by the transmitter.

Figure 2.2: OFDM Receiver model. Source: Wikipedia, the free encyclopedia

FFT

X Y0

Y1

YN-2

YN-1

Symbol Detection

fc

𝟗𝟎𝒐

Parallel to Serial

r(t)

X ADC

ADC

𝕽𝒎

𝕽𝒆

𝐬∧[𝐧]

20

2.7 MATHEMATICAL DESCRIPTION

In an OFDM system, if N sub-carriers are used, and each sub-carrier is modulated using

M symbols, then the set symbol of this OFDM system consists of MN elements.

Mathematically, low-pass OFDM signal is expressed as:

v(t) = ∑ Xkej2πkt/T , 0 ≤ t < T

N−1

k=0

(2.3)

Where, Xk denotes the set of data symbols used, N is the number of sub-carriers and T

is the time of an OFDM symbol. The orthogonality of the sub-carrier is ensured by

spacing them by a 1/T. This property is expressed as:

1

T∫(ej2πk1t/T)

∗(ej2πk2t/T) dt

T

0

(2.4)

= 1

T∫(ej2π(k2−k1)t/T) dt = δk1k2

T

0

(2.5)

Where * operates returns the complex conjugate and δ is the Kronecker delta.

A guard interval of length Tg is inserted before each OFDM block in order to eliminate

Inter Symbol Interference (ISI). During this interval, a cyclic prefix is transmitted so

that the signal in the interval −Tg ≤ t < 0 is the same as the signal in the interval

(T − Tg) ≤ t < T. Therefore, the OFDM signal with cyclic prefix can be represented

by the following equation.

v(t) = ∑ Xkej2πkt/T , −Tg ≤ t < T

N−1

k=0

(2.6)

21

The above stated signals can be either real or complex. Real valued signals are generally

transmitted at baseband wired applications such as DSL use this approach. The complex

valued signals are generally transmitted after conversion to a carrier frequency, wireless

application use this approach.

In general, the transmitted signal is represented as:

s(t) = ℜ{v(t)ej2πfct} = ∑ |Xk|cos(2π[fc + k/T]t + arg[Xk])

N−1

k=0

(2.7)

22

CHAPTER 3

REDUCTION OF PEAK-TO-AVERAGE POWER RATIO

As put forward in chapter 2, OFDM is a choice technology for high speed multi-user

communication systems. However, one major problem with OFDM hinders its

widespread use. The problem is that of “high level of PAPR”. This chapter presents an

introduction and explanation of this problem besides a brief of the schemes/methods

used to palliate this problem up to a degree of success. Note that “Reduction of PAPR”

is still a hot research area which is evident from the fact that a lot of research papers are

being published on this topic currently.

3.1 INTRODUCTION

One of the emerging technologies in the field of communications is the wireless

technology. It offers effective data transmission and a growing concept of 4G and 5G

communications. The concept of OFDM system states that it is a kind of modulation

scheme which accommodate multiple users simultaneously. As stated earlier, OFDM

suffers from the drawback of high PAPR [14, 15]. Numerous techniques, for example,

Selective Mapping [16], Partial Transmit Sequence [17], Clipping and Filtering [18, 19,

20], Tone Reservation [21], Companding [22], etc. are available which can be

employed to reduce PAPR effect in OFDM systems [14, 15, 16]. Different parameters

such as distortion rate, data rate, power, etc. are analyzed by the study of different PAPR

reduction schemes. OFDM communication systems find its applications in digital

television and audio broadcasting, DSL internet access, wireless networks, and 4G

mobile communications [23].

23

3.2 HIGH-POWER AMPLIFIERS

The purpose of PAPR reduction is to counteract the nonlinear effect of the High

Power Amplifier (HPA). Usually, HPAs are characterized as memory-less nonlinear

amplifiers in accordance with

g(x(t)) = F(|x(t)|)ej(∅(t)+Ф(|x(t)|)) (3.1)

Where g(x(t)) is the output of the HPA; x(t) = |x(t)|ej∅(t) is the time domain signal

input to the HPA; F(|x(t)|) and Ф(|x(t)|) are, respectively, the AM/AM and the

AM/PM distortion functions respectively, where AM denotes the Amplitude

Modulation, and PM denotes the Phase Modulation. Usually HPAs can be partitioned

into three categories: the Soft Limiter (SL), the Solid State Power Amplifier (SSPA),

and the Traveling-Wave Tube (TWT). Their characteristics can be described as follows.

3.2.1 Soft Limiter Power Amplifier

The Soft Limiter (SL) [24] is the simplest model of the HPA. It introduces no distortion

in the phase of the input signal and simply clips the signal magnitude when it exceeds

a threshold.

Therefore, the output of the SL can be written as

g(x(t)) = {Aej∅(t), |x(t)|A,

x(t), other wise ( 3.2)

Where A > 0 represents the threshold of the Soft Limiter.

24

3.2.2 Solid State Power Amplifier

The Solid State Power Amplifier is the most commonly used amplifier in wireless

communications. The output of SSPA can be written as [3].

g(x(t)) =|x(t)|

(1 +|x(t)|

A

2p

)

12p

ej∅(t); (3.3)

i.e., it introduces no distortion in the signal phase. When p → ∞, the SSPA becomes

the SL. Usually, p = 3 for a practical SSPA.

3.2.3 Traveling-Wave Tube

Travelling Wave Tubes (TWTs) are wideband amplifiers widely used in satellite

communications [25, 26]. The AM/AM and AM/PM functions of TWT can be written

as [27].

F(|x(t)|) =|x(t)|

1 + ( |x(t)|

2A )2 ,

or ∅(|x(t)|) =π

3

|x(t)|2

3|x(t)2| + 4A2 . (3.4)

3.3 PAPR DEFINED

In OFDM System Model, it can be noticed that the input channel signals are modulated

first using either Phase Shift Keying (PSK) or QAM and then undergo Inverse Fast

Fourier Transform (IFFT) operation at the transmitter end [28, 29]. This creates low-

frequency sub-carriers (orthogonal to one another) at the transmitter side [30]. These

transmitted signals can deliver high peak values in the time domain and these high peak

values when get summed up due to alignment produce high ratio of peak power to the

average power. The high PAPR is a consequence of the summing up of sonic waves

25

and non-constant envelope [31]. The injurious effect of high PAPR is that it brings

down the performance of power amplifier. Therefore, RF power amplifiers need to be

controlled in a very large linear region, otherwise the signal peaks will enter into a non-

linear region and will cause deformation. Though there are many schemes which reduce

PAPR, the efficiency of any PAPR reduction scheme is measured through Cumulative

Distribution Function (CDF). PAPR of a signal is calculated by the following equation

[14].

PAPR[x(t)] =Pmax

Pav=

max0≤t≤NT

[|x(t)|2]

E{|x(t)2|} (3.5)

Where Pav is the average power of signal x(t) and is calculated in the frequency

domain, since IDFT is a (scaled) unitary transform and E{⋅} represents the value

operator. The nonlinear distortion in the HPA occurs in the analog domain. But, the

majority of the signal processing to reduce PAPR OFDM signal is used in the digital

domain. In general, the PAPR in the digital domain is not necessarily the same as PAPR

in the analog domain.

The PAPR (in dB) of the transmitted OFDM signal can be written as in equation (3.6)

CFRmax = 10log(N)dB (3.6)

It is a random variable, because PAPR is a function of input data. The crest factor is

generally characterized by the square root of the PAPR.

Crest Factor, C. F = √PAPR or PAPR = (C. F)2 (3.7)

The clipping ratio (CR) is define as follows:

CR =Amax

Aave (3.8)

26

Where Amax is the maximum amplitude of the output signal after clipping, and Aave is

the average amplitude of input signal before clipping.

3.4 WELL KNOWN PAPR REDUCTION SCHEMES

This section describes various PAPR reduction schemes and discusses their

performance as given in the literature.

As shown in Table 3.1, the schemes to reduce PAPR are primarily divided into two

classes, i.e., Signal Scrambling Schemes and Signal Distortion Schemes.

In section, 3.4.1, described some popular signal scrambling schemes, and in section

3.4.2, similarly some major signal distortion schemes for PAPR reduction are

elaborated.

Table 3.1: Classification of Major PAPR Reduction Schemes [17]

Signal Scrambling Schemes Signal Distortion Schemes

Selective Mapping (SLM) Clipping and Filtering (C and F)

Partial Transmit Sequence (PTS) Peak Windowing

Tone Reservation (TR) Peak Reduction Carrier

Tone Injection (TI) Companding

Interleaving

3.4.1 Signal Scrambling Schemes

3.4.1.1 Selective Mapping (SLM)

The research that introduced the “Selective Mapping Technique” was penned down by

Bamul, Fischer and Huber in 1996. SLM [16, 32, 33] is one of the favorable PAPR

27

reduction techniques as it does not introduce distortion and effectively reduces PAPR.

In this technique the input data blocks are multiplied by each of the given phase-rotated

sequences to generate alternative input symbol sequences. Part of the alternative

sequences is processed further under IFFT and their PAPR is determined. Then the

signal with lowest PAPR is selected for transmission [16, 32, 33]. SLM is a technique

which is utilized to lessen the PAPR effect in OFDM Systems. It is a type of Phase

Rotation Method. As shown in the block diagram of SLM (Figure 3.1), the Side

information index (SII) should be transferred to appropriate retrieval of data cube at the

recipient position [34, 35].

Figure 3.1: Block diagram of SLM scheme [17]

In SLM, input data is partitioned into smaller data blocks of length N (Figure 3.1).

These smaller data blocks result into the parallel data streams obtained through serial

Serial- to-

Parallel

Converter

Choose the

one with

minimum

PAPR

IDFT

IDFT

IDFT

X

X

X

S

S(1)

S(2)

S(U)

s(1)

s(2)

s(U)

SK

Side

Information

S

B(1)

B(U)

B(2)

28

to parallel converter. Each element in each data bock of parallel streams is multiplied

with a phase-rotated sequence [15], [52].

After data block is phase rotated, the rotated OFDM data blocks represent similar

information, multiplied with known phase sequence. The fundamental idea that lies in

this technique is that it helps to select the signal with lowest PAPR value from a pool

of phase-rotated sequences.

3.4.1.2 Partial Transmit Sequence (PTS)

Partial Transmit Sequence technique [17] is popularly used technique for PAPR

reduction and the concept of PTS Scheme can be clearly seen in its block diagram as

shown in Figure 3.2. PTS is based on the concept of addition of phase rotation to

develop a candidate signal and to select one signal with low PAPR [29, 30, 36, 37]. The

statistics of a multicarrier signal gets enhanced by practicing this technique.

Figure 3.2: Block diagram of PTS scheme [17]

Data

Source

Serial to

Parallel

Partition

in to

Clusters

Selection optimal

combination with

lowest PARP

Parallel

To Serial

X

X

X

b1

U

N-point

IFFT

N-point

IFFT

N-point

IFFT

u(1)

u(2)

u(Z-1)

b2

b(Z-1)

Phase OptimizationSide Information

If necessary

29

The important idea that lies behind PTS scheme is to divide the original OFDM carrier

sequence into several sequences. Distinct weights are multiplied with each sequence

until the best solution is accomplished. This idea is visible in block diagram of PTS

(Figure 3.2) [17]. The portioning into sub-carriers of a single carrier is a major drag on

performance and must be taken into consideration. The three ways of sub-block

portioning schemes are adjacent, interleaved and pseudo-random portioning [29]. PTS

scheme has high level of computational complexity and it also needs to handle SII as

that in SLM scheme.

3.4.1.3 Tone Reservation (TR)

In this scheme, N sub-carriers (tones) are divided into two parts, i.e., data tones and

peak reduction tones. The central thought behind this scheme is that a humble set of

sub-carrier frequencies (tones) are created for PAPR reduction. This method is applied

to minimize the high peak values. At the transmitter side, the computation of time

domain signal can be made. The effectiveness for PAPR reduction in tone reservation

approach depends on the tones that are reserved. TR is a low complexity scheme [15].

This technique demonstrates that by reserving even a modest fraction of tones leads to

big reductions in PAPR value. Moreover, no additional data is required to be handled

at the recipient side. However, reserving tones is not only an optimization issue but

leads to inefficient use of spectrum.

3.4.1.4 Tone Injection (TI)

The Tone Injection (TI) approach was given by Seung. H., and Jae. H. Lee in [15]. The

basis of the tone injection scheme is a general additive method which provides a desired

PAPR reduction. By the execution of the additive method of the multicarrier signal, the

30

PAPR reduction is accomplished. A circle of equivalent constellation points is being

used by TI scheme for original constellation points to minimize the effect of PAPR.

The demerits of the TI are that this access requires additional information for

deciphering the signal at the receiver and causes extra IFFT operations which results in

complex circuitry.

3.4.1.5 Interleaving Technique

This is a technique which utilizes a set of Interleavers for PAPR reduction and is thus

called Interleaving technique. In this scheme, the high value of PAPR is reduced by

applying a set of Interleavers, but not by using the set of phase sequences as was the

case with both PTS and SLM techniques. A long correlation pattern is worn down to

melt off the elevated values of correlated data structures [15]. By the use of this

Interleaving technique, higher code rate without expansion in bandwidth is obtained as

compared to conventional OFDM systems, without an increased number of sub-

carriers. The Interleaving technique is moderately complex in nature.

As shown in Figure 3.3, Interleavers are used to produce permuted data blocks from the

original data block. PAPR is then computed for the original and the permuted data

blocks. The data block with the lowest PAPR is then selected for transmission. These

computations are comprehensive in nature and make this scheme moderate complex in

terms of computational complexity. The block diagram of interleaving scheme is shown

in Figure 3.3.

The performance (in terms of reduction in PAPR) of the Interleaving scheme depends

on the number and design of the Interleavers.

31

Figure 3.3: Block diagram of Interleaving scheme

3.4.2 Signal Distortion Schemes

3.4.2.1 Clipping and Filtering

The technique is one of the simplest techniques and is being mostly used for getting

reduced value of PAPR. The functioning of clipping and filtering technique is clear

from its name itself, i.e., it clips the part of the signals which are not allowed to enter

the specified region. The operation of clipping and filtering technique can be realized

by using the HPA with the saturation region below the signal span will automatically

induce the signal to be clipped [18.19,20]. The amplitude clipping is mathematically

defined as follows:

C(x) = {x, x ≤ AA, x > A

(3.9)

Clipping is generally speaking at the transmitter side. The receiver is supposed to

estimate the clipping performed by the sender and compensates accordingly. Since, at

Select One

With

Minimum

PAPR

32

most one clipping is done by the sender per OFDM symbol, thus the receiver has to

estimate two parameters, i.e., location and size of the clip. It is difficult for the receiver

to estimate this data. Hence, clipping introduces both in-band distortion and out-of-

band radiation into OFDM signals. This degrades system performance including bit-

error-rate and effective use of the available spectrum. Though filtering (pruning) can

reduce out-of-band radiation, it can not reduce in-band distortion. However, on the

other hand filtering may cause regrowth of some peaks so that the signal after clipping

and filtering will go past the clipping level at some stages [18].

3.4.2.2 Peak Reduction Carrier

Peak Reduction Carrier technique was proposed by Muller and Huber [38]. This

technique is demonstrated to have the capability to reduce the elevated value of PAPR

by using data bearing Peak Reduction Carrier in OFDM systems. This technique is

associated with the use of higher order modulation scheme to represent a lower order

modulation symbol [15]. The phase shift keying modulation scheme is suited for Peak

Reduction Carrier as in this envelope of all the sub-carriers are the same. The

implementation of QAM scheme will result in serious BER degradation. When the

higher order modulation schemes are used by this technique to represent lower order

modulation scheme data then, there is increased probability of error and hence the

overall BER performance gets degraded.

33

Table 3.2: Performance of PAPR Reduction Techniques [17]

S. No. PAPR Reduction

Techniques Performance

1. Selective Mapping

(SLM) Reduces Distortion

No Power Raise

Selects sub-carriers with the lowest

PAPR value

2. Partial Transmit

Sequence (PTS) Reduces Distortion

No Power Raise

High Computational Complexity

3. Tone Reservation (TR) Reduces Distortion

Power Gets Raised

Less Complex as Compared with PTS

4. Tone Injection (TI) Reduces Distortion

Power Gets Raised

PAPR Reduction without Data Rate

Reduction

5. Clipping and Filtering Introduces Distortion

No Power Raise

One of the Simplest

3.5 SUMMARY

OFDM is a kind of multicarrier and multiuser modulation technique. Wireless

communication is emerging technology in the present times and OFDM systems are in

use because of its advantages such as providing High Spectral Efficiency, Increased

Bandwidth Power and its Robustness against Multipath Interference. But the OFDM

system suffers from the demerit of high values of PAPR. In this chapter, several

techniques for PAPR reduction are reviewed and discussed. These techniques are

separated into two classes. 1. Signal Scrambling Techniques and 2. Signal Distortion

Techniques.

34

A number of techniques such as Selective Mapping, Partial Transmit Sequence, Tone

Reservation, Tone Injection, Interleaving, Clipping and Filtering and Peak Reduction

Carrier techniques are talked about in this chapter. The analysis of PAPR reduction

techniques as given in the literature on various parameters is done. It can be said that

different PAPR reduction techniques are able to reduce the PAPR effectively but each

having its own demerits. Hence, the jury is still out and the research work is still being

done to optimize these schemes.

35

CHAPTER 4

PERFORMANCE EVALUATION AND ANALYSIS OF PAPR

REDUCTION SCHEME

This chapter consists of two major parts. First part is given in section 4.1 and its sub-

sections which produce a detailed account of various PAPR reduction schemes based

on the analysis done on the documents available on these schemes in the literature.

Based on this analysis of various schemes, two PAPR reduction schemes are selected

as the schemes having more promise and are to be considered for further improvement.

Second part of this chapter (section 4.2 and its sub-sections), describe the performance

evaluation done by us of the selected schemes. The objective of this performance

evaluation is two-fold, i.e., to verify the performance of these schemes against the

performance given in the literature, and to understand their workings for the purpose of

further enhancement.

4.1 ANALYSIS OF PAPR REDUCTION SCHEMES THROUGH

LITERATURE SURVEY

This section provides a detailed account of the following PAPR reduction schemes

based on the analysis done on the documents available in the literature related to these

schemes. The details on each scheme are given in the subsequent sub-sections of this

section.

Clipping and Filtering Scheme

Coding Scheme

Peak Reduction Carriers Scheme

36

Envelope Scaling Scheme

PTS and SLM Scheme

Tone Reservation and Tone Injection Schemes

Active Constellation Extension(ACE) Scheme

4.1.1 Clipping and Filtering Scheme

This is one of the simplest schemes to reduce PAPR. This scheme can be implemented

without any significant computational overhead to the OFDM modulator [39, 40]. In

this scheme, a power threshold is first of all fixed. The signal power is compared with

the threshold power. If the power of the signal is less than the threshold then it is

transmitted as it is but if the power of the signal power is greater, then instead of the

signal power, the threshold power is used to send the signal. This simplicity, however,

has a cost involved. Since, a part of OFDM signal is clipped altogether, it causes loss

of information. This loss of information in turn increases the BER. Higher BER

severely affects some critical application like real-time video transmission etc. [41].

The results shown in Figure 4.1, illustrate that as we increase the number of clips done

on the data, so does the reduction in PAPR. For example, it can be seen that the

Complementary Cumulative Distribution Function (CCDF) is the highest for the

original signal (without clipping). CCDF after first clipping is lower than the original

clipping. Same pattern can be seen in Figure 4.1 as the number of clips are further

increased. However, it may increase BER, which may cause distortion.

37

Figure 4.1: PAPR distribution when “Clipping and Filtering Scheme” is used

4.1.2 Coding Scheme

In this method FEC codes are used for PAPR reduction. The FEC codes help scramble

the data so that the OFDM symbol created has a smaller PAPR. The advantage of PAPR

reduction using Coding is that this method not only provides error correction but also

PAPR reduction. The disadvantage is that coding reduces the information rate (bit rate)

since redundant information needs to be transmitted. However, since FEC are typically

used in communication systems, therefore the idea of using FEC for PAPR reduction

is quite attractive. To develop a Complement Block Coding (CBC) scheme, this helps

to reduce the PAPR of the OFDM signal. A reduction of about 3dBs is achieved using

this method. The convolutional codes are also used to reduce the PAPR of OFDM

signals. The motivation is that convolutional codes are already used for various

38

communication systems such as UMTS, LTE, WiMAX etc., so using this scheme for

PAPR reduction will be efficient. The use of Low Density Parity Check (LDPC) codes

for PAPR reduction show that the use of LDPC can reduce the PAPR of OFDM by

about 60%. Also, the combination of LDPC codes and PTS for PAPR reduction of

OFDM shows that such schemes can reduce the PAPR by about 3.7 dBs for 8 partitions.

In [14], the authors have compared the PAPR reduction capabilities of Cyclic Coding

(CC), Simple Block Coding (SBC), Complement Blocking Coding (CBC) and

Modified Complementary Block Coding (MCBC). They propose the use of CBC and

MCBC codes since they offer high coding rates and provide flexibility between coding

rate choices and complexity.

4.1.3 Peak Reduction Carriers Scheme

This scheme uses bearings to reduce the Peak Reduction Carriers (PRCs) data to reduce

the effective PAPR [15]. It includes the use of a higher order modulation scheme to

represent a lower order modulation symbol. The amplitude and phase of the PRC is

positioned within the constellation region symbolizing the data symbol to be

transmitted. This scheme is suitable for PSK modulation. Note that in PSK the

envelopes of all sub-carriers are the same. When QAM modulation is implemented in

the OFDM systems, the carrier envelope scaling will result in serious BER degradation.

BER degradation can be reduced but at the cost of substantial increase in the side

information to be transmitted by the sender.

39

4.1.4 Envelope Scaling Scheme

Envelope scaling technology is proposed in [42]. Based on this technique, many

algorithms are available to reduce the peak signal before the expected input envelope

by scaling some part of the subcarriers after these are sent to the IFFT [42]. Generally,

256 subcarriers are used with Quadratic Phase Shift Keying (QPSK) modulation

schemes to make the envelope subcarriers equivalent. The core of this scheme states

that a bundle of only some sub-carriers is to be made which reduces PAPR. Hence, the

receiver does not require further additional information to decode the sequence. A

reduction of 4dB has been reported in literature for this scheme [42].

4.1.5 PTS and SLM Schemes

PTS and SLM which use multiple signal representation represent another class of PAPR

reduction schemes. Both PTS and SLM are shown not to degrade BER performance.

However, their computational cost is much higher [43, 44, 45].

PTS is one of the most popular distortion less PAPR reduction schemes in that it does

not increase the BER [44, 45]. However PTS is computationally complex and is thus

comparatively slow. This is a serious disadvantage when compared to other available

techniques. During OFDM transmission, it is critical that the side information is

received without errors. The performance of PTS depends upon two factors, i.e., (1) the

phase factors and (2) the segmentation method. In practice the computational load of

PTS is reduced by fixing the phase factors, segmentation method, and the number of

segments [36, 37].

For PTS, the data is divided into non-overlapping parts and the subcarriers within each

part are applied a phase shift so as to reduce the overall PAPR as much as possible as

40

shown in Figure 4.2. This figure is same as figure 3.2 in Chapter 3(section 3.4, and its

sub-sections 3. 4.1.4).

Figure 4.2: PTS scheme block diagram [17]

The above block diagram shows the functional steps of PTS [36, 37].

First start with a serial sequence of incoming data.

In the next step, the incoming data having N subcarriers is divided into non-

overlapping parts by the serial to parallel converter.

This is followed by IFFT being performed on the data blocks.

The individual signals now need to be assigned phase factors having a complex

value. This is achieved by means of the PTS method to determine the best

possible phase shifts for each data part so as to reduce the overall PAPR to the

minimum value.

On the receiver end, the reverse operation is performed.

41

Figures 4.3(a) and 4.3(b) compare the performance of PTS with the Original signal

(without PTS) in terms of their PAPR reduction capability.

Figure 4.3(a) shows the performance of PTS which uses 4 different phase rotations and

16 different segmentations (number of sub-band is 64, the oversampling factor L is 8).

We can see in this figure that the above stated combination of parameters reduces PAPR

by about 2dB.

Similarly Figure 4.3(b) shows the performance of PTS when the number of different

phase sequences is 4 and the number of different segmentation combinations is 256.

This figure shows that this combination can reduce PAPR of an OFDM signal by about

5.8 dB.

Figure 4.3(a): CCDF for PAPR of OFDM with and without PTS [17]

42

Figure 4.3(b): CCDF for PAPR of OFDM with and without PTS [17]

Out of many PAPR reduction schemes, it can be found SLM is one of the most

promising schemes. In SLM statistically independent data blocks are generated from

the given OFDM data blocks. These data blocks are generated using a set of phase-

rotated sequences and one with the lowest PAPR is selected for transmission. The goal

is to perform a change of phase on the modulated symbols (before IFFT) in order to

reduce the probability of constructive interference between the subcarriers (after IFFT)

has been computed. The block diagram of SLM is shown in Figure 4.4, in this figure it

is cleared that each block of data is multiplied with a phase rotator Z for different length

N.

Bz = [bz, 0, bZ, 1, … , bZ, N − 1]T, Z = 1,2, … , Z − 1

43

Resulting in 𝑍 different data blocks, so the zth phase sequence after multiplying is given

as:

Figure 4.4: Block diagram of SLM scheme [17]

UZ = [U1bZ,1, U2bZ,2, … , UN−1bZ,N−1]2

, (z = 1, 2, … , Z − 1). (4.1)

Therefore, OFDM signal becomes as given in equation 4.2.

uz(t) =1

√N∑ Unb

z,nej2πfnt

N−1

n=0

where 0 ≤ t ≤ NT, = 1,2, … , U (4.2)

From the data blocks U(z)(z = 0, 1, … , Z − 1)the one with smallest PAPR is selected

for transmission. After that the data is transmitted along with the phase sequences as

side information. At the receiver, reverse operation is performed to recover the original

data. This methodology is calculable with a wide range of modulation and with a

number of sub-carriers.

Data

Source

Partition

In to

Clusters

Serial to

Parallel

Select

one

With

Minimum

PAPR

Parallel

To Serial

N-point

IFFT

N-point

IFFT

N-point

IFFT

X

X

X

U

U(1)

U(2)

U(Z-1)

u(1)

u(2)

u(Z-1)

B(1)

B(2)

B(Z-1)

44

Figure 4.5(a): CCDF for PAPR of OFDM with and without SLM

Figure 4.5(b): CCDF for PAPR of OFDM with and without SLM

45

By analysing the graph shown in Figure 4.5(a), it is deducted that SLM reduces PAPR

by as much as 3dB (quite significant) when 04 different phase-rotated sequences are

used.

Similarly, Figure 4.5(b) shows that when in SLM, the number of OFDM symbol

candidates C=8, different phase sequences is 4, number of sub bands is 64, and the

oversampling factor L= 4, PAPR of an OFDM signal is reduced by about 3.7dB.

Under the above observation, it is clear that both SLM and PTS schemes are very

similar in nature. The only difference is the provision of phase-rotated sub-sequences

in SLM before the IFFT operation and in PTS scheme after the IFFT operation.

4.1.6 Interleaving Scheme

The interleaving scheme is very similar in its nature to SLM scheme. As we have seen

that the sub-sequences are separated by doing phase-rotation in SLM, however in

Interleaving, intervals are used to separate sub-sequences. The interleaver generates the

modified data blocks which are actually permuted to the data blocks of the original one.

Finally, the data black with the least PAPR is selected for transmission. The

performance of this scheme in terms of PAPR reduction heavily depends on the amount

and design of intervals [14].

4.1.7 Tone Reservation and Tone Injection Schemes

After thorough literature survey it is found that like some other schemes, both Tone

Reservation and Tone Injection are categorized as efficient schemes too (in terms of

PAPR reduction).

In contrast to the other PAPR reduction methods discussed above, TR seems to be

efficient in terms of both complexity and BER requirements [46, 47]. Since TR does

46

not manipulate data subcarriers therefore it does not cause any errors to it. TR utilizes

a set of reserved subcarriers for PAPR reduction. The task is to optimize the signal of

non-data bearing subcarriers, while keeping the data subcarriers unchanged [48].

In TR, the receiver does not need to share the complexity of this PAPR reduction

algorithm. That is the processing load for the selection of TR carriers needs to be

performed by the transmitter only [21, 49].

It can be observed from Figure 4.6 (a) and 4.6 (b) that the PAPR reduction is inversely

related to the number of PRCs. Therefore, the lower the PRC sum, the more the

reduction in the PAPR.

Figure 4.6(a): PAPR of Tone Reservation, 12 sub-carriers, and 4 Peak

Cancellation sub-carriers

47

Figure 4.6(b): PAPR of Tone Reservation, 12 sub-carriers, and 4 Peak

Cancellation sub-carriers

TR has many advantages. In TR the data rate does not get reduced substantially.

Moreover, side information is also not needed to be transmitted. However, TI suffers

from some disadvantages too. The most important disadvantage is that exhaustive

search for the best constellation from a large number of constellations is required. This

implies that the computational complexity of TI is quite high.

TI is more complex than TR [50]. This is because in TI both the injected signal and

information signal occupy the same frequency bands.

4.1.8 Active Constellation Extension Scheme

The method of Active Constellation Extension is also used to minimize the PAPR and

falls in to the same line of methods as TI. This is a non-linear method utilized for

reduction of the PAPR of OFDM signal in addition to using Companding and

48

Clipping. ACE has a major edge over other techniques in that it improves the BER

without affecting the rate of data exchange which is a very important requirement for

wireless communication. Nevertheless, the advantage of good BER comes at the cost

of computational complexity as it is intensive to determine the best possible

constellation which is an iterative process [50, 51].

Figure 4.7: The ACE scheme for QPSK modulation [17], [36]

lm

Re

49

4.1.9 Comparison of PAPR Reduction Schemes

Table 4.1: Comparison of Different PAPR Reduction Schemes [15], [17]

PAPR

Reduction

Schemes

Metric

Power

Increase

Implementation

Complexity

Bandwidth

Expansion

BER

Degradation

Required

Processing

Clipping /

Filtering No Low No Yes

Tx: Amplitude Clipping

Rx: None

Coding No Low Yes No Tx: Encoding

Rx: Decoding

TR Yes High Yes No

Tx: IDFTs, Final value

PRCs

Rx: Ignore non-data

carriers

ACE Yes High Yes No

Tx: IDFTs, projection on

“shaded area”

Rx: None

PTS No High Yes No

Tx: M IDFT, Wm-1

complex vector sums

Rx: Side information

extraction, Inverse PTS

SLM No High Yes No

Tx: U-IDFTs


extraction, Inverse SLM

Interleaving No Low Yes No

Tx: K-IDFTs, K-1

interleavings


extraction, Inverse

interleaving

TI Yes High Yes No

Tx: IDFT, search for

maximum point in time,

tones to be modified

Rx: Modulo-D operation

4.2 PERFORMANCE EVALUATION AND ANALYSIS OF SELECTED

PAPR REDUCTION SCHEMES THROUGH SIMULATIONS

Wireless digital communications is an ever increasing phenomenon thereby requiring

structures that are dependable and extremely efficient. OFDM allows efficient use of

the available spectrum, tolerance in multipath delay, and robustness to fading. It has

50

therefore been used for high speed communication as well as it is a part of several

standards.

As a result it has been chosen for high data rate communications, DVB, Digital Video

Broadcasting Terrestrial (DVB-T) and mobile Worldwide interoperability for

Microwave Access (WiMAX) based on OFDM access technology (Jiang., et al., 2007)

Digital Signal Processing has helped develop recent interest in this technique and it has

found use in many international standards such as IEEE 802.11, IEEE 802.16, IEEE

802.20, European Telecommunications Standards Institute (ETSI) Broadcast Radio

Access Network (BRAN) committees, and high-speed digital subscriber lines (HDSL,

ADSL, and VDSL) [52].

Even with its many advantages, OFDM suffers from power issues i.e. the largest power

value during an OFDM transmission can be equal N times the average power of the

signal (N being the number of carriers used in the signal). This is a major drawback for

its use as it results in distortions in the output. To circumvent this, one solution is to

reduce the power being transmitted thus reducing the PAPR of the signal. Having a

lower PAPR allows for a higher average power to be sent for a fixed value of peak

signal power. This increases the signal to noise ratio.

To achieve this, a number of methods have been proposed [52]. These techniques are

Clipping [53] Clipping and Filtering [40, 41, 54], coding [55, 56] Tone Reservation

[47, 48] and Active Constellation Extension [50]. Other methods having discrete

solutions such as Tone Injection [49] and multiple signal representation techniques such

as Partial Transmit Sequence [57, 58, 59] Selected Mapping [44, 45, 60], and Inter

Leaving [50] have also been proposed.

51

4.2.1 Motivation for Using Tone Reservation (TR)

There have been many methods proposed for decreasing the PAPR values during an

OFDM transmission. Of all the presented techniques, TR is considered to be an

effective technique to achieve this task. Each method of TR has its own benefit and

suitable for use in a certain scenario.

All the approaches used here to apply the tone reservation technique have their own

benefits, so that each can be used in different condition. If the complexity is not an

issue then Signal to Clipping Noise Ratio (SCR) Gradient TR iterative algorithm or

Adaptive Scaling TR algorithm can be used that give reasonable PAPR reduction. On

other hand if system cannot be offered much complexity, then other algorithm like

Gaussian pulse based TR is suitable. Active set TR Algorithm can be used where there

is a need to attack only high peaks, the complexity is bit higher but techniques is

reasonable as not all the symbols samples are checking.

The simplest of the methods for the reduction of PAPR are Clipping and Companding,

which rely on literally clipping the amplitude of the multicarrier signal. These

procedures have a number of shortfalls such as in-band distortions and noise

amplification which result in BER degradation. Another class of PAPR reduction

techniques includes multiple signal representation methods such as PTS and SLM.

These techniques do not degrade the BER performance but are computationally quite

expensive. BER performance is improved due to TR. Some subsets of subcarriers are

exploited by the TR for controlling PAPR.

No internal separation method is applied in TR to eventually calculate effective

PAPR. All of the measurements are computed at Transmitter side without

involvement of the receiver. Recent counterpart communication systems comprising

52

3G and 4G standards like LTE-Advanced its predecessor, and Wireless

Interoperability for Microwave Access (WiMAX) utilize TR for this purpose.

Distinguished sided quality of TR is industrious BER enhancement i.e. since TR does

not control information subcarriers accordingly it doesn't outcome in any omission to

it.

4.2.2 Simulation Results on Performance of TR

In order to validate the proposed scheme in this research few simulation experiments

are conducted having different no of carriers for PRC observations. To check flat

signal observation 12 subcarriers are considered, consequently. we decide +r1-r2+r3

and+r4 as a PRC and transmit on x1: x2, that results x1: x2 summation to create a peak,

while r4 creates an anti- peak, which concludes that the exact ant peak or results output

to be a flat signal from which data cannot be retrieved.

Similarly (Figure 4.8 through 4.23) compares the TR OFDM signal with 12 subcarrier

and 4 peak cancellation simulation results. The blue line represents the original value

of PAPR in dB, whereas red line indicates the result after PAPR reduction in dB.

Table 4.2 presents the summary of the results presented in Figures 4.8 to 4.23,

depicting the significant reduction in PAPR.

Table 4.2 eventually provides use of TR method results to accomplish that PRC-4

with combination +r1+r2+r3 and r4 provides the greatest reduction of PAPR to be

2.45 dB, from all experimented combinations.

53

Table 4.2: Different PRC Sum Variable Combinations [46]

Figures. Nos

Peak

Cancellation

Sum

Variable

Carrier Sum

Calculation

PAPR

Level (dB)

Figure 4.8 sum1 -r1-r2-r3-r4 2.45

Figure 4.9 sum2 -r1-r2-r3-r4 1.4

Figure 4.10 sum3 +r1+r2-r3-r4 1.1

Figure 4.11 sum4 -r1-r2+r3+r4 0.7

Figure 4.12 sum5 -r1-r2-r3+r4 0.7

Figure 4.13 sum6 -r1+r2+r3+r4 0.8

Figure 4.14 sum7 +r1-r2-r3-r4 0.4

Figure 4.15 sum8 +r1-r2-r3+r4 0.7

Figure 4.16 sum9 +r1+r2+r3-r4 1.1

Figure 4.17 sum10 -r1+r2-r3+r4 0.9

Figure 4.18 sum11 +r1+r2+r3-r4 1.1

Figure 4.19 sum12 +r1+r2+r3-r4 1.3

Figure 4.20 sum13 +r1+r2-r3+r4 1.0

Figure 4.21 sum14 -r1+r2-r3-r4 0.25

Figure 4.22 sum15 -r1-r2+r3-r4 0.2

Figure 4.23 sum16 -r1+r2+r3-r4 0.3

54

Figure 4.8: TR-OFDM signal for -r1-r2-r3-r4 combination

Figure 4.9: TR-OFDM signal for -r1-r2-r3-r4combination

55

Figure 4.10: TR-OFDM signal for +r1+r2-r3-r4 combination

Figure 4.11: TR-OFDM signal for -r1-r2+r3+r4 combination

56

Figure 4.12: TR-OFDM signal for -r1-r2-r3+r4 combination

Figure 4.13: TR-OFDM signal for -r1+r2+r3+r4 combination

57

Figure 4.14: TR-OFDM signal for +r1-r2-r3-r4 combination

Figure 4.15: TR-OFDM signal for +r1-r2-r3+r4 combination

58

Figure 4.16: TR-OFDM signal for +r1+r2+r3-r4 combination

Figure 4.17: TR-OFDM signal for -r1+r2-r3+r4 combination

59

Figure 4.18: TR-OFDM signal for +r1+r2+r3-r4 combination

Figure 4.19: TR-OFDM signal for +r1-r2+r3+r4 combination

60

Figure 4.20: TR-OFDM signal for +r1+r2-r3+r4 combination

Figure 4.21: TR-OFDM signal for -r1+r2-r3-r4 combination

61

Figure 4.22: TR-OFDM signal for -r1-r2+r3-r4 combination

Figure 4.23: TR-OFDM signal for -r1+r2+r3-r4 combination

62

4.2.3 PAPR Reduction Using Selective Mapping (SLM)

4.2.3.1 Motivation for Using Selective Mapping (SLM)

SLM is arguably the most popular PAPR reduction scheme which can be used for

OFDM. The main benefit of SLM is the guarantee of achieving a smaller PAPR

providing that the mapping sequences were pre-selected in an optimal manner. Another

important motivation for using SLM is that it does not cause distortion, which means

that the SNR of the transmitted signal remains the same. It must be mentioned here that

SLM does require the knowledge of scrambling sequences at the receiver end, but this

can be communicated via log2(Z) bits, where Z represents the number of SLM

sequences used.

4.2.3.2 Selective Mapping (SLM)

A promising scheme for reducing PAPR in OFDM is selected mapping. Within this

scheme, data blocks which are statistically independent are generated by means of

different phase sequences with the lower one being utilized and transmitted. This

scheme has the potential to reduce the probability of a major PMEPR (Peak-to-Mean

Envelope Power Ratio) for multicarrier transmission scheme. The goal is to perform a

change of phase on the modulated symbols (before IFFT) in order to reduce the

probability of constructive interference between the subcarriers (after IFFT) has been

computed. A block diagram of SLM is represented in (Figure. 4.24).

63

Figure 4.24: Block Diagram for SLM Scheme for OFDM System [15] [17]

Each block of data is multiplied with phase rotator Z for different length N.

Bz = [bz, 0, bZ, 1, … , bZ, N − 1]T, Z = 1,2, … , Z − 1

Resulting in 𝑍 different data blocks, so the zth phase sequence after multiplying is given

as:

UZ = [U1bZ,1, U2bZ,2, … , UN−1bZ,N−1]2

, (z = 1, 2, … , Z − 1).

Therefore, OFDM signal becomes as given in equation (4.3).

uz(t) =1

√N∑ Unb

z,nej2πfnt

N−1

n=0

where 0 ≤ t ≤ NT, = 1,2, … , U. (4.3)

Among the data blocks U(z)(z = 0, 1, … , Z − 1) only one of the modified data block,

that has the lowest PAPR value is chosen to be sent. After that the data is transmitted

Data

Source

Partition

In to

Clusters

Serial to

Parallel

Select

one

With

Minimum

PAPR

Parallel

To Serial

N-point

IFFT

N-point

IFFT

N-point

IFFT

X

X

X

U

U(1)

U(2)

U(Z-1)

u(1)

u(2)

u(Z-1)

B(1)

B(2)

B(Z-1)

64

along with the phase sequences as side information. The opposite operation is

performed at the receiver to recover the original signal. The SLM technique requires Z

IFFT operations to be performed to use it in OFDM signals and the quantity of bits used

as side data is ⌈log2(Z)⌉ for each block of information. This methodology is calculable

with a wide range of modulation and the number of sub carriers. It is cleared that the

performance of SLM in terms of PAPR reduction is dependent on the number of phase

factors Z and the design of the phase components.

4.2.3.3 Simulation Results and Discussion

By closely observing the graphical analysis, it is inferred that the PAPR reduction in

terms of dB in the CCDF for OFDM system is shown with dotted (- - - -) line and

without SLM solid (______) line. The graphical analysis and simulation results of PAPR

reduction SLM methodology can be utilized for OFDM based system. From Figure

4.25 to Figure 4.30. SLM technique uses different number of OFDM Symbol candidates

(C), with 4 phase sequences, 64 number of sub-band, and L = 4 (Oversampling factor).

This combination can reduce the PAPR of an OFDM signal at different level in dB.

But, Figure 4.30 shows the reduction approximately about 5.5 dB by using number of

OFDM symbol candidates C=256, 4 different phase sequences, 64 number of sub band,

L=4 (Oversampling factor). Table 4.3 summarizes the results presented in figures 4.25

to 4.30.

It is observed that SLM with increasing number of OFDM symbol candidate improves

the PAPR performance and no BER increases. This also decreases the system

complexity.

65

4.2.3.4 Performance Analysis of SLM

The major aim of this sub-section is to conduct a performance analysis of SLM

technique for PAPR reduction. The proposed methodology can be utilized for the

OFDM based system. One workable solution is SLM which is an easy way to mitigate

the effect of PAPR by means of undistorted processing. This is done by increase the

number of OFDM symbol candidate with 4 different phase sequences, 64 number of

sub band, L=4 (Oversampling factor), combinations can reduce the PAPR of an OFDM

signal by about 5.5 dB. This is quite enough. Furthermore it makes no BER degradation,

no interference to different subcarriers and does not require any extra processing at the

receiver. The potential of the research is quite evident from the simulation results

presented. Further work can be carried out to reduce the complexity. It can be seen that

the SLM scheme is very useful and offers advantages over the other techniques to

reduce the PAPR efficiently.

Table 4.3: PAPR Levels in dBs for OFDM Symbol Candidate Combinations [34]

Figure Nos

OFDM

Symbol

Candidates

Phase

Sequences

Over

Sampling

Factors

Sub-band

PAPR

Level

(dB)

Figure. 4.25 C=8 4 4 64 3.7

Figure. 4.26 C=16 4 4 64 4.3

Figure. 4.27 C=32 4 4 64 4.7

Figure. 4.28 C=64 4 4 64 5.0

Figure. 4.29 C=128 4 4 64 5.3

Figure.4.30 C=256 4 4 64 5.5

66

Figure 4.25: CCDF graph for PAPR of OFDM with and without SLM


4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

number of OFDM symbol candidates=8

PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100


PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

67



4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100


PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100


PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

68



4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100


PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100


PAPRa[dB]

CC

DF (Pr(

PA

PR

>PA

PR

a))

Original

SLM

69

4.3 CHAPTER SUMMARY AND CONCLUSION

This chapter consists of two parts, i.e., section 4.1 and section 4.2. Section 4.1 provides

details and performance analysis of various PAPR reduction schemes as seen in the

literature. It is concluded from the analysis presented in section 4.1 is learnt that though

there are many schemes available to reduce PAPR, two are the most promising and can

be focused for further optimization. These are (1) Tone Reservation, and (2) Selective

Mapping.

In order for us to fully understand their functionality and to do an independent

evaluation, simulation-based performance evaluation of both of these schemes is given

in section 4.2. The results agree with the results already available in the literature and

show that both the schemes can substantially reduce PAPR without distortion.

However, SLM performs marginally better than TR. It is also noted that the

performance of SLM mainly depends upon the selection of appropriate sequence of

sub-carriers from a set of sub-carrier sequences. If selected appropriately, we can have

a large gain in PAPR reduction in SLM. However, if an optimal sequence of sub-

carriers is not selected, it may even lead to worsening of performance.

In short, SLM has great promise but needs further investigation and tuning to perform

optimal selection of sub-carrier sequences from a set of available sub-carrier sequences.

70

CHAPTER 5

PROPOSAL OF A NOVEL PAPR REDUCTION SCHEME

BASED ON MNKB-RBF

5.1 ISSUE OF SLM (OPTIMIZATION PROBLEM)

As stated in the conclusion of Chapter 04, SLM is an efficient, distortion-less PAPR

reduction scheme. It reduces PAPR substantially and has good promise for further

improvement as well.

SLM scheme is based on the core principle of selecting one sequence of sub-carriers

from a set of available sequences of phase-rotated sub-carriers. Selection of a particular

sequence is a major issue and determines the performance of SLM in terms of the

magnitude by which the PAPR is reduced. Hence, selection of an appropriate phase-

rotated sequence of sub-carriers is essentially an optimization problem.

In order to enhance performance of SLM, we intend to solve the above stated problem

by using a framework which would use ANN which is one of the best ways to solve

optimization problems.

5.2 ARTIFICIAL NEURAL NETWORKS (ANNs) AND RADIAL BASIS

FUNCTION (RBF)

Computational model for ANNs was first proposed by McCulloch and Pitts [62]. Since

then, ANNs have been recognized as a decision making tool by many researches [63,

64, 65]. ANN is particularly very useful in solving optimization problems. Optimization

problem is either difficult or almost impossible to solve with the help of conventional

rule-based programming [66].

71

Simple but yet powerful generalization capability of ANN had drawn the attention of

numerous past and present researchers [63, 64, 66, 67]. It all started with Rosenblatt

when he created the perceptron [68], a pattern recognition algorithm for supervised

classification. However, Rosenblatt’s idea could not be translated into a computer

program until the development of back propagation algorithm which has so far been the

most popularly used algorithm in ANN paradigm [70]. Thereafter, immense research

was done in this field, and in last 50 years or so there has been extraordinary growth in

this domain and the result is invention of several sophisticated algorithms [67, 71, 72].

An RBF network [73] is an ANN and whose activation functions are radial basis

functions. It was first introduced by Broom head and Lowe [71] and since then it has

become a very popular methodology to solve optimization problems that suit ANN

paradigm [72, 73, 74, 75]. The main advantage of RBF when compared with other

algorithms based on ANN paradigm is the simplicity of the computation of network

parameters [73]. Another very important feature of RBF based ANNs is to be able to

perform complex nonlinear mappings that allow a fast linear and robust learning

mechanism [66]. Originally, RBF networks were developed for data interpolation in

high dimensional space [73]. Nonetheless, RBF networks have been used in diverse

optimization domains, including pattern classification [68], time series prediction [74],

systems and control [75], and function approximation [76].

Some of the most commonly used basis functions are Gaussian functions [73],

multiquadrics functions [73], thin plate spline function [73], inverse multiquadrics

functions [73], and so forth. There is no general rule, but the choice of a radial basis

function is highly problem specific. Also, most applications using RBF make use of a

free shape parameter that plays pivotal role in the accuracy of the method and is

72

commonly chosen with the help of cross-validation [77] technique. It is a standard

practice [73] to learn three sets of parameters for RBF network: locations, widths, and

weight factors of RBF kernel. Enormous amount of work [78, 79, 80] has already been

done to select those parameters optimally.

In the conventional RBF kernel, mostly Gaussian of the Euclidean distance (ED)

between feature vector and neuron’s center is used [78]. However, there can be

scenarios where Euclidean distance is not the dominant measure to find separation

among the features, for example, if two feature vectors are separated by equal distance

from a center but separated from the center via unequal angles. In that case, the cosine

of the angle can play a vital role in differentiating the feature vectors.

There are some existing works in the literature that had discussed usage of cosine

measure with RBF kernels [81, 82, 83, 84, 85, 86, 87]. Karayiannis and Randolph-Gips

[88] have proposed a novel RBF which is a normalized version of the multiquadratic

radial basis function, where the cosine represents the angle between the transformed

vectors rather than the original vectors. Liu et al. [89] have used cosine similarity

measure to achieve high performance of classification by selecting meaningful features.

They compute the cosine similarity among the kernels rather than the original vectors.

By doing so, they transformed all the vectors to the same length, whereas we do not

perturb the feature space. Moreover, these cosine kernels are developed for Support

Vector Machine (SVM). Cho and Saul [90] have used arc cosine of the angles between

inputs in their kernel.

73

5.3 CONVENTIONAL RADIAL BASIS FUNCTION (RBF)

RBF networks in their general form consist of three layers. These layers are called an

input layer, a hidden layer, and a linear output layer. The hidden layer is one where

nonlinear activation functions operate. The layout is shown in Figure 5.1. Generally,

the input is a real vector, x ∈ Rn. The network output maps the input vector to a

scalar, y: Rn → R, which is achieved by employing the following equation:

yi = ∑ ωiφi(‖x − ci‖) +

N

i=1

bj ∀j = 1,2 … , No (5.1)

Where N and No are the number of hidden and output layer neurons, respectively, ci ∈

Rn is the center for ith neuron, ωi is output layer weight for ithneuron, bj is the bias

term for the jth output neuron, and φi is the basis function associated with ith hidden

neuron.

RBF solves a problem by mapping it into a high dimensional space in a nonlinear

manner and then applies linear decision boundary. The concept of transformation to

high dimensional space is justified by Cover’s theorem, according to which

classification via linear separation becomes easier by translating the features from low

dimension to high dimension [91].

The significance of adding bias to the output is to improve the approximation quality

by shifting the decision boundary. The weights of the network govern the position of

the decision boundary in the feature space. However, during the adaptive weight update,

if bias is not used, then the hyper-plane is forced to pass through the origin of the feature

space defined by the inputs or feature vectors. Although it is valid for some problems,

in many others this separation boundary is desired to be located somewhere else.

74

Figure 5.1: Architecture of the RBF based Neural Network [66]

As a general rule, all inputs are connected to each hidden neuron. The domain of

activation function is a norm which is typically taken to be the Euclidean distance

between input and the centers of every neuron. Most commonly used RBF kernels are

as follows [66].

Multiquadrics are given as:

φi(‖x − ci‖) = (‖x − ci‖2 + τ2)1/2 (5.2)

Inverse multiquadrics are given as:

φi(‖x − ci‖) = 1

(‖x − ci‖2 + τ2)1/2 (5.3)

And Gaussian as:

φi(‖x − ci‖) = e−‖x−ci‖2/β2 (5.4)

ϕ1

Σ ϕ2

ϕn

ωi

X1

X2

Xn

y

Bias

Input layer Hidden layer Output layer

75

Where τ > 0 a constant and β is spread parameter. The sensitivity of a hidden neuron

towards a data point varies in proportion with the distance of the data point from its

center. For example, in case of a conventional ED based RBF network that uses

Gaussian in its kernel, this sensitivity can be fine-tuned by adjusting β; if β is large, it

implies less sensitivity and vice versa.

5.4 A RECENTLY PROPOSED “NOVEL KERNEL BASED RBF (NKB-

RBF)”

Motivated by the observation that “in many scenarios Euclidean distance is not the

dominant measure to find the separation among features”, Aftab et.al, propose in [92]

a novel RBF kernel which consists of a linear combination of Gaussian and cosine RBF

kernels. The cosine RBF kernel computes the cosine of the angle between supplied

feature vector and the center vector associated with that neuron.

Aftab et.al, in [92] state that intuition suggests that ED is not the only measure to

contrast the FVs. For example, in the case when FVs are equally separated in distance,

then the ED will be no more effective. To deal with this issue, they proposed a

generalized RBF kernel by linearly combining the conventional ED based RBF kernel

and a cosine based RBF kernel which is formulated as follows:

φi(x, ci) = α1φi1(x. ci) + α2φi2(‖x − ci‖) (5.5)

Where α1, α2 are weightage parameters for cosine and Euclidean kernels, respectively,

which can acquire values in this range: 0 ≤ α1, α2 ≤ 1. Moreover, φi1(x. ci) and

φi2(‖x − ci‖) are the cosine and the Euclidean kernels, respectively, for 𝑖𝑡ℎ neuron.

These are further defined as follows:

76

φi1(x. ci) ≜ cos(θi) = x. ci

‖x‖. ‖ci‖ (5.6)

φi2(‖x − ci‖) = e−‖x−ci‖2/β2 (5.7)

Where x. c𝑖 represent the dot product between the two vectors.

By observing Equation (5.6), we can notice that the kernel φi1(x. ci) computes the

cosine of angle between x and ci. Hence, φi1(x. ci) may attain the values in the range

[−1, +1]. If it returns to 1, it implies that the x is aligned with ci, whereas its 0 return

value corresponds to the scenario when x is perfectly orthogonal to ci: and the return

value of −1 indicates that x and ci are aligned in opposite directions.

5.5 ISSUES OF NKB-RBF

Recently proposed Novel Kernel Based RBF (NKB-RBF) is shown in [92] to perform

well with different problems as it utilizes the complimentary property of two kernels

that are based on the Euclidean (distance) and Cosine (angle) or correlation measure.

However the NKB- RBF suffers from the manual selection of the mixing parameter.

We can see in Equation (5.5) that α1 and α2 are the mixing parameters of Cosine and

Euclidean kernels. The manual selection (as suggested by MNKB-RBF) of the mixing

parameters α1 and α2 is a critical issue particularly in the situations with no

generalization of the problem. For the selection of these mixing parameters one needs

prior information about the problem. In cases where cosine is the good measure of

similarity or in other words the angle is the discriminating element we will have to

choose higher value for α1 (close to 1) and lower value for α2 (close to 0) to have

optimal performance. The reserve must be selected for these mixing parameters, i.e.,

α1 (close to 0) and lower value for α2 (close to 1), for the problem where angle is not

77

the optimal discriminating element. If this guide line is not followed, we will face

degradation in performance instead of improvement. Table 5.1 shows the criteria for

the manual selection of mixing parameters α1 and α2.

Table 5.1: Criteria for Selection (Manual) of Mining Parameters 𝛂𝟏 and 𝛂𝟐 [92]

α1 α2

Weightage to the Cosine

Distance (CD) should be high

in case where CD is the

distinguishing element.

Should be low if the CD is the

confusion factor

NKB-RBF allows to choose

α1 manually which is not

possible in the dynamic

scenario of SLM

Weightage to the Euclidean

Distance (ED) should be high

in case where ED is the

distinguishing element.

Should be low if the ED is the

confusion factor

NKB-RBF allows to choose α2

manually which is not possible

in the dynamic scenario of

SLM

To get the best performance from the NKB-RBF one needs to select the optimal values

of the mixing parameters. The manual selection of mixing parameters as suggested by

NKB-RBF requires prior knowledge of the system. This restricts the application of

NKB-RBF to only the problems where prior information about the system is known. In

the case of SLM, we need as adaptive optimization algorithm that can select the optimal

weights of the individual kernels to harness the complimentary properties of the two

kernels without the prior knowledge of the incoming signal type.

78

5.6 PROPOSED SOLUTION: MODIFIED NKB-RBF (MNKB-RBF)

The critical issue of NKB-RBF is the manual tuning of the mixing parameters α1

and α2. In order to use it for optimization of selection of one phase rotated sub-

sequence of signals among many candidate in SLM (with minimum PAPR), we need

to make the tuning of α1 and α2 dynamically adaptive. Since the core objective of the

proposed adaptive algorithm is to minimize the overall error of the system, we propose

to use the error energy besides distance for tuning of α1 and α2. We call our new

algorithm as Modified NKB-RBF or MNKB-RBF in short.

In order to incorporate the error energy, we replace α1 and α2 with the η(n) and 1-η(n)

in Eq. (5.5), to make them time varying. We can rewrite the kernel equation as:

φi(x, ci) = η(n)φi1(x. ci) + (1 − η(n))φi2(‖x − ci‖) (5.8)

Where,

η(n) = weight of Cosine Distance (CD)

1- η(n) = weight of Euclidean Distance (ED)

The MNKB-RBF algorithm uses the update rule of a Robust Variable Step-Size Least

Mean Square (RVS-SLMS) algorithm [93] for its learning rate where the update is

obtained by an estimate of the autocorrelation between current error e(n) and past error

e(n-1).

If ρ(n) is the final output error, the error energy of the MNKB-RBF algorithm is defined

as follows:

ρ(n) = β ρ(n) + (1 − β)e(n)e(n − 1) where 0 < β < 1 (5.9)

Note that ρ(n) is the error energy at nth instant.

79

In the proposed method the weight of the cosine kernel can be calculated as follows:

η(n + 1) = τ ∗ η(n) + σρ(n) where 0 < τ < 1, 0 < σ < 1 (5.10)

Here η(n+1) is the weight of the Cosine Distance for next iteration and τ, σ and β are

the momentum coefficients.

η(n + 1) = {

1, η(n + 1) > 1

η(n + 1), 0 < η(n + 1) < 1

0, η(n + 1) < 0

(5.11)

5.7 PROPOSAL OF A NOVEL PAPR REDUCTION SCHEME BASED ON

MNKB-RBF

In section 5.6, we proposed a novel technique for the autonomous selection of weights

of mixing parameters of the NKB-RBF algorithm and named it MNKB-RBF. The

proposed kernel can be used in the dynamic environments where little or no prior

information about the discriminating measure in known. As in the case of Selective

Mapping we want to make our system adaptive and suitable for dynamically selecting

an optimal sub-sequence (with lowest PAPR) from a set of available sub-sequences of

the frequencies with unknown effects on PAPR. The proposed kernel is expected to

perform well in this scenario because it autonomously selects weights based on the error

energy.

80

Figure 5.2: Block diagram of the Proposed PAPR Reduction Scheme

Using MNKB-RBF

The framework of the proposed technique (based on MNKB-RBF and SLM) has been

shown in Figure 5.2, which selects the sub-sequence with the lowest PAPR from the

given sequences. From this figure it can be seen that the selection of the optimal phase

rotation is performed by advanced technique of optimized weighted kernel. In the

proposed system the signal will first pass through the frequency transformation block

and then the SLM block, which will select the appropriate carrier signal for the given

signal.

The SLM block in modified to indicate the selection of the best carrier sub-sequence

based on the intelligent decision of the dynamic method of MNKB-RBF to minimize

S/P IFFT

SLM Encoder

MNKB-RBF phase rotation selector

Input Data

.

.

.

.

.

.

MNKB-RBF based SLM Module

Modified Novel Kernel Based-Radial Basis Function Neural Network Phase Rotation Selection System

utilizes the intelligence of Artificial Neural Networks to select the optimal sequence set for PAPR reduction

81

the chances of the rise in PAPR. The argument is supported by extensive

simulations/experiments performed and discussed in Chapter 6.


In this chapter, we stated that in SLM, the selection of an appropriate phase-rotated

sequence of sub-carriers with minimum PAPR is essentially an optimization problem.

And that the issue can be optimally addressed by using ANNs. We showed that the

performance of ANN largely depends upon the kernel (Radial Basis Function) being

used. We introduced an RBF most recently proposed (NKB-RBF) and is shown to

perform well in many optimization applications. The performance of NKB-RBF largely

depends upon the tuning parameters α1 and α2. Moreover, NKB-RBF suggests manual

selection of weights of these tuning parameters. In many applications in general and in

our case in particular manual selection of weights of these parameters is simply not

possible due to dynamic nature (real-time selection) of our application.

Hence, we proposed in this chapter a modified version of NKB-RBF and termed it

MNKB-RBF. The key feature of MNKB-RBF is its ability to automatically adjust the

weights of the tuning parameters taking into account the error energy.

Also described in this chapter is the framework proposed by us for reduction of PAPR

which is based on SLM and our proposed MNKB-RBF.

82

CHAPTER 6

PERFORMANCE EVALUATION OF PROPOSED PAPR

REDUCTION SCHEME

In chapter 5, it is stated that selection of a phase-rotated sequence of signals with the

lowest PAPR from a set of given sequences is an optimization issue. And that the

problem can be solved using Artificial Neural Networks. However, the performance of

an ANN depends upon the RBF kernel being used. Though many conventional kernels

exist but a recently introduced kernel that is referred to as NKB-RBF has shown better

performance. It is also described in Chapter 5, that the performance of NKB-RBF in

turn depends upon careful manual selection of two tuning parameters α1 and α2. Since,

in proposed case manual selection of these parameters is not possible, so a modified

kernel is proposed and referred to as MNKB-RBF. MNKB-RBF selects optimal values

of α1 and α2 automatically instead of manually (for details please refer to chapter 5).

The performance of the proposed scheme (chapter 5) for selection of a phase-rotated

sequence of signals from a set of available sequences depends on MNKB-RBF and

SLM. Since SLM is already a known and evaluated scheme, hence the performance of

the proposed scheme solely depends upon the performance of MNKB-RBF. In this

chapter, the MNKB-RBF is presented with thorough evaluation, which shows that

MNKB-RBF performs better than NKB-RBF.

83

6.1 PERFORMANCE EVALUATION ENVIRONMENT

We have performed a comprehensive evaluation of Modified Novel Kernel Based RBF

(MNKB-RBF) against Novel Kernel Based RBF (NKB-RBF) by using simulations in

MATLAB of ANNs which use these kernels.

In order to have a good level of confidence in the deduction that we make after

comparison of these two kernels. Nine different datasets have been generated on which

the performance of both of these kernels is tested and compared. The details of the

simulation environment, test cases, and the nature of the datasets are being given in the

following section.

6.2 SIMULATION ENVIRONMENT AND TEST CASES

In order to train and test the ANNs which use the Novel RBF and the proposed RBF

(MNKB-RBF), first of all nine different datasets of one hundred randomly generated

messages are generated. In each dataset, the initial fifty messages are for the purpose of

training the ANN and the later fifty messages are to be used for testing the performance

of the ANN.

6.2.1 Regarding Datasets

Regarding dataset 1, 2, and 3, the carrier is to be selected from a pool of 64 sequences

of phase-rotated carriers. However, 8- QAM, 16-QAM, and 32-QAM are used to

modulate each message in dataset 1, dataset2, and dataset 3 respectively. This is done

in order to see whether or not changing modulation changes the pattern of results or

leads to the same conclusion. In our graphic or numeric results it indicates the nature of

dataset by the following scheme [64 x 8]. In this scheme, the first numeric value

indicates the number of phase-rotated sequences in the pool and the second numeric

84

value representing the number of symbols used in QAM for modulation. For example

in [64 x 8], the pool has 64 sequences and 8 symbols are used to modulate the messages.

Similarly, in dataset 4, 5, and 6, the pool has 128 phase-rotated sequences with

messages modulated using 8-QAM, 16-QAM, and 32-QAM respectively.

Lastly, in dataset 7, 8, and 9, there are 256 phase rotated sequences and messages are

modulated using 8-QAM, 16-QAM, and 32-QAM respectively.

6.2.2 Regarding Test Cases

Multiple iterations of simulations are performed for both “Training Phase” as well as

“Testing Phase” on all of the datasets as specified in the above section. Since, a large

number of iterations are performed, the deductions derived from the analysis of these

results (in section 6.4 and 6.5) have higher level of confidence and their reliability is

thought to be better too.

6.3 CORE CODE OF THE PROPOSED ALGORITHM

The proposed kernel has been implemented in MATLAB. Produced below is the core

portion of the code representing the main part of the proposed algorithm. The phase-

rotated sequence of the carrier selected by this code is fed to the “SLM Encoder” for

encoding and later transmission to the destination. The performance of new proposed

algorithm is decided by the fact that it should select a carrier with minimum PAPR.

The code given in Figure 6.1 implements the proposed MNKB-RBF algorithm. It is

clear from lines 1, 5, and 6 that many iterations are done in order for the error energy

to become minimum. In line number 7 and 8, Euclidean Distance (φi1) and Cosine

Distance (φi2) are calculated for the ith iteration (refer to Equation 5.8). In line number

85

9, overall Distance (φi) is calculated for the ith iteration incorporating both φi1 and φi2

weighted by the error energy e(n) (Equation 5.8).

Figure 6.1: Core MATLAB code of the Proposed RBF (MNKB-RBF)

The loop from line number 6 to 10 determines final output error energy ρ(n) for the nth

instant (Equation 5.9) incorporating the momentum controlling coefficient β. Finally,

the weighted sum of the error energy e(n) and the final error energy ρ(n) is calculated

in line number 16, which in turn becomes error energy for the next instant, i.e., (n+1)th

instant (Equation 5.10).

1. for k=1:epoch 2. I(k)=0; 3. ind=randperm(m); 4. 5. for n=1:m 6. for i=1:n1 7. ED(i)=exp((-(norm(P(ind(n),:)-c(i,:))^2)))/beeta^2; 8. CD(i)=abs(P(ind(n),:)*c(i,:)')/(norm(P(ind(n),:))*norm(c(i,:))+1e-50); 9. phi(n,i)=eta(n)*ED(i)+(1-eta(n))*CD(i); 10. end 11. y(n,:)=w*phi(n,:)'; 12. d(n,:)=f(ind(n),:); 13. e(n+1,:)=d(n,:)-y(n,:); 14. 15. p(n+1)=(p(n)*beeta)+((1-beeta)*(e(n,:)*e(n+1,:)')); 16. eta(n+1)=(tau*eta(n))+(sigma*p(n)); 17. 18. I(k)=I(k)+e(n+1,:)*e(n+1,:)'; %%% Objective Function 19. 20. w=(w+eta*e(n+1,:)'*phi(n,:)); 21. end 22. e(1,:)=e(end,:); 23. End

86

6.4 TRAINING RESULTS

The training of both kernels, i.e., NKB-RBF and MNKB-RBF is done using three

different training cases. Each test case consists of tests done on three datasets of similar

nature as listed below.

Training Case I: Dataset having 64 sequences of phase-rotated carriers.

Dataset 1 – [ 64 x 8]

Dataset 2 – [ 64 x 16]

Dataset 3 – [ 64 x 32]

Training Case II: Dataset having 128 sequences of phase-rotated carriers.

Dataset 4 – [128 x 8]

Dataset 5 – [128 x 16]

Dataset 6 – [128 x 32]

Training Case III: Dataset having 256 sequences of phase-rotated carriers.

Dataset 7 – [256 x 8]

Dataset 8 – [256 x 16]

Dataset 9 – [256 x 32]

The results of the training phase in graphical form are shown in Figures 6.2 to 6.10.

The results shown in Figure 6.2, 6.3, and 6.4 correspond to “Training Case I”. Similarly

Figure 6.5, 6.6, and 6.7 to “Training Case II” and Figure 6.8, 6.9, and 6.10 relate to

“Training Case III”.

Evaluation and analysis of training phase results are given in the following sub-section.

87

6.4.1 Evaluation, Analysis, and Deductions

The x-axis in these graphs (Figures 6.2 to 6.10) represents the number of epochs for

which the simulation was run. The y-axis gives the magnitude of the Mean of the

Squared Error (MSE). Each graph has two curves, i.e., the dashed-line curve (red

colored curve) for the Novel RBF (NKB-RBF) and the solid-line curve (blue curve) for

the Proposed RBF (MNKB-RBF).

Let us first of all analyze graphs of the “Training Case I” (64 sequences of carriers),

i.e., graphs shown in Figure 6.2, 6.3, and 6.4. It can be seen that in Figure 6.2, 8 symbols

are used for modulation, that for only the first epoch, the MSE is higher for MNKB-

RBF. Whereas, from 2nd epoch and onwards, MSE for MNKB-RBF is lower than NKB-

RBF. This indicates much early successful training of ANN which uses the proposed

kernel. After around only 7th epoch, MSE for MNKB-RBF reaches its minimum which

is 2 and is an excellent result showing that less time is needed to train MNKB-RBF

based ANN.

It can be seen that curves in Figure 6.2, 6.3, and 6.4 do not show substantial change in

their pattern even after increase in modulation symbols from 8 to 64. This leads us to

deduce that MNKB-RBF is robust against variation in modulation symbol rate. This is

important for MNKB-RBF to be indeed practically used in not only PAPR reduction

schemes but also in other optimization applications.

The results of “Training Case II and III” which are shown in Figures 6.5 to 6.10 are

focused now, some interesting observations can be made. For one, the performance of

MNKB-RBF does not change even though the pool of candidate phase-rotated

sequences is increased from 64 to 256. In all of these graphs it can be seen that MSE

exponentially reduces (solid blue curve) and minimizes after around 7 epochs. This

88

analysis leads us to the deduction that MNKB-RBF is robust against variation in the

size of the pool of phase-rotated carrier sequences as well.

On the other hand the performance of NKB-RBF has degraded with increase in the pool

size of carrier sequences. MSE for NKB-RBF is not reducing fast enough which is

evident from the less steepness of the dashed-line red curves for NKB-RBF in Figures

6.2 to 6.10.

Now look at the final value of MSE for which both NKB-RBF and MNKB-RBF

stabilize. It is visible in these graphs that the final MSE value for NKB-RBF is around

5, whereas for MNKB-RBF, MSE keeps on decreasing as we increase the size of the

pool of sequences and becomes almost zero. The final MSE values (approximated) are

summarized against size of pool of sequences in the following table for MNKB-RBF.

Hence, it is concluded that MNKB-RBF performs much better than NKB-RBF in terms

of training of ANN.

Table 6.1: Final Mean Square Error (MSE) Comparison

Size of Pool of

Carrier Sequences

MSE for

Novel RBF

MSE for

Proposed RBF

64 4.0 2.0

128 3.5 1.0

256 3.0 0.1

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Figure 6.2: Training cost comparison between Novel RBF and Proposed RBF

(Phase-rotated sequences = 64, Modulation = 8-QAM)

Figure 6.3: Training Cost Comparison between Novel RBF and Proposed RBF


90





91





92





93



6.5 TESTING RESULTS

The second most important phase of performance evaluation of primarily MNKB-RBF

is done and is generally termed as “Testing Phase” in the ANN community. However,

in order to evaluate MNKB-RBF, we need to do its comparison with other similar

algorithm. Hence, like the “Training Phase” performance of MNKB-RBF is compared

with NKB-RBF and additionally with simple SLM as well.

Like the “Training Phase”, in the “Testing Phase” too three basic test cases are

performed. Each testing case in turn consists of tests done on three datasets of similar

nature as listed below:

94

Testing Case I: Dataset having 64 sequences of phase-rotated carriers.

Dataset 1 – [ 64 x 8]

Dataset 2 – [ 64 x 16]

Dataset 3 – [ 64 x 32]

Testing Case II: Dataset having 128 sequences of phase-rotated carriers.

Dataset 4 – [128 x 8]

Dataset 5 – [128 x 16]

Dataset 6 – [128 x 32]

Testing Case III: Dataset having 256 sequences of phase-rotated carriers.

Dataset 7 – [256 x 8]

Dataset 8 – [256 x 16]

Dataset 9 – [256 x 32]

The results of the testing phase in graphical form are shown in Figures 6.11 to 6.19.

The results shown in Figure 6.11, 6.12, and 6.13 correspond to “Testing Case I”.

Similarly Figure 6.14, 6.15, and 6.16 to “Testing Case II” and Figure 6.17, 6.18, and

6.19 relate to “Testing Case III” respectively.

Evaluation and analysis of training phase results are given in the following sub-section.

6.5.1 Evaluation, Analysis, and Deductions

The x-axis in these graphs (Figures 6.11 to 6.19) represents the “Message Number”

under focus. Recall that a total of 50 messages are to be sent as separate transmissions

95

selecting a phase-rotated sequence from a pool of given sequences. The sequence is to

be selected in a manner to minimize PAPR for that message. Hence, it can be seen that

there are 50 messages on x-axis with PAPR for each message shown on y-axis.

Each graph has three curves black solid-line curve representing our proposed RBF

(MNKB-RBF), blue dashed-line curve for the Novel RBF (NKB-RBF) and red solid-

line curve for simple SLM.

First of all let us analyze the graphs for the “Testing Case I” shown in Figures 6.11,

6.12, and 6.13. At first glance it seems that the results are random. But a careful look at

these graphs reveals the following.

For majority of the messages, the red solid-live curve is below the other

two curves indicating high values of PAPR for simple SLM and thus it

is knocked out of the competition with NKB-RBF and MNKB-RBF

Comparing and carefully analyzing the solid-line black curve (MNKB-

RBF) with the dashed-line blue curve (NKB-RBF). It can be seen that

for majority of the messages PAPR for MNKB-RBF is lower than NKB-

RBF. For only a few messages, PAPR for MNKB-RBF is higher than

NKB-RBF.

As a whole it is concluded that MNKB-RBF outperforms both Simple

SLM and NKB-RBF in terms of reduction in PAPR levels.

The same pattern, i.e., curve for MNKB-RBF is below other two curves for most of the

messages, is observable in results of Test Case II and III as shown in the graphs of

Figures 6.14 to 6.19. Hence, these results concur with the conclusion we drove from

the results of Test Case I.

96

Figure 6.11: Testing Comparison between Novel RBF and Proposed RBF




97





98





99





100



6.5.2 Probability of Selecting Carrier of Low PAPR

In order to be clearer about the deduction drawn in the previous sub-section, the results

of the previous sub-section are reprocessed to find the probability of selection of a

phase-rotated carrier sequence with the lowest PAPR from a given pool of phase-

rotated sequence. The probability is calculated for all three schemes, i.e., Simple SLM,

Novel RBF, and Proposed RBF is shown in Table 6.2.

The 1st column of this table (from left) gives the test number. The details regarding the

size of the pool of sequences and the number of symbols used for modulation are given

in the 2nd column. Whereas, the 3rd, 4th, and 5th column show the probability calculated

for the Simple SLM, Novel RBF, and the Proposed RBF. Now note the values in these

columns give the probability of the selection of the lowest PAPR which implies that the

101

scheme with the highest probability is the best one. Hence, for each row the highest

values are underlined and produced in bold font.

One can easily notice that except for Test Case 1-3 and 3-2, the performance of the

Proposed RBF is the best of the three. Therefore, it is concluded with a higher level of

confidence that the performance of the Proposed RBF is better than the other two

contenders.

Table 6.2: Probability of Selecting Carrier of Low PAPR

Test Case

Number

Test Case Details

(Carriers x Symbols) SLM

Novel

RBF

Proposed

RBF

Test Case 1-1 256 x 32 0.54 0.82 0.88

Test Case 1-2 256 x 16 0.48 0.74 0.80

Test Case 1-3 256 x 8 0.44 0.80 0.76

Test Case 2-1 128 x 32 0.46 0.58 0.60

Test Case 2-2 128 x 16 0.40 0.52 0.66

Test Case 2-3 128 x 8 0.40 0.66 0.68

Test Case 3-1 64 x 32 0.58 0.58 0.62

Test Case 3-2 64 x 16 0.42 0.60 0.40

Test Case 3-3 64 x 8 0.40 0.56 0.58

102


This chapter has elaborated upon the extensive performance evaluation of the proposed

PAPR reduction framework which is based upon SLM and the proposed MNKB-RBF.

Both the training and the testing results indicate that the proposed framework performs

better than simple SLM and when SLM is used in conjunction with NKB-RBF.

103

CHAPTER 7

SUMMARY AND CONCLUSION

7.1 SUMMARY

In this dissertation, first of all described the reasons for conducting this research

activity. The argument goes like this. Lately most of the broadband systems use OFDM

as the main modulation technique. ODFM is used due to its main feature of using a

large number of low frequency sub-carrier signals (orthogonal to each other) instead of

one high-frequency carrier to achieve high data rate. The orthogonality of the sub-

carriers and being of low frequency ensure robustness against many kinds of

interferences. However, OFDM also suffers from high level of PAPR. A lot of research

activity is being done by the research community in this area to device frameworks and

algorithms which may ensure low PAPR values.

Many schemes and frameworks do exist in the literature which claim to produce low

PAPR values, for example, Clipping and Filtering, Peak Reduction Carriers, Envelop

Scaling, Coding, Partial Transmit Sequence, Selective Mapping, Tone Reservation,

Tone Injection, etc. A literature-survey has been performed based performance of

evaluation of some of the promising among these in order for us to select one to two

schemes to focus upon for performance improvement. However, it was felt to conduct

our own simulation-based performance evaluation for two reasons, i.e., firstly, to

understand in a better way the working of these schemes, and secondly to be absolutely

sure which scheme has the potential for further improvement. It was found that SLM

fulfills research criteria.

104

Since the core idea of SLM scheme is to choose one sequence of low frequency sub-

carriers from a pool of given sub-carriers in such a way that PAPR gets reduced, hence

it effectively is an optimization issue. Therefore, a new framework is proposed which

uses Artificial Neural Networks (ANN) in conjunction with SLM. The job of the ANN

is to optimize the selection of sub-carrier sequence such that the PAPR is the lowest for

that sequence. Because this scheme is to be used in real life and real time environments,

the performance of the ANN is critical issue. The performance of the ANN in turn

depends upon the RBF Kernel used in the ANN.

Recently, a new kernel has been proposed and termed to as Novel Kernel Based RBF

(NKB-RBF) and is shown to have better performance. However, the performance of

NKB-RBF depends upon values selected for the weightage parameters α1 and α2. In

NKB-RBF, the weights of α1 and α2 are selected manually which is not feasible in our

case of usage in real time environment. Hence, proposed a modified version of this

kernel and call it as Modified NKB-RBF (MNKB-RBF). The proposed framework is

evaluated which uses SLM and MNKB-RBF based ANN, which optimally selects a

sequence of phase-rotated sub-carriers to be better and more efficient than Simple SLM

and NKB-RBF based SLM as well.

7.2 CONCLUSION

As stated in the previous section, the focus of this research activity is to propose a

mechanism/framework which minimizes PAPR in majority of transmissions. Such a

mechanism is indeed proposed and described in this dissertation. The core of the

proposed scheme consists of both SLM scheme and ANN which uses a modified

105

version (MNKB-RBF) of already available kernel. The conclusions of this research

activity are enumerated below.

SLM, one of the PAPR reduction schemes, has the potential for further

improvement in performance and can lead to optimal selection a sequence of

phase-rotated sub-carrier from a pool of available sequences such that its PAPR

is the lowest.

SLM used in conjunction with ANN has the potential to lead to optimal

selection.

A recently proposed ANN kernel (NKB-RBF) is shown to perform better but

suffer from the manual selection of weights of the tuning parameters α1 and α2.

The proposed kernel (MNKB-RBF) which uses error-energy to automatically

adjust weights of α1 and α2 is shown in this dissertation through simulation

results to perform better than NKB-RBF.

From analysis of simulation results (comprehensive set of scenarios) it is

concluded that the probability of selection of a sequence of phase-rotated sub-

carriers from a pool of given sequences in the proposed framework is the highest

among all contenders (Simple SLM, NKB-RBF based SLM).

7.3 FUTURE WORK/RESEARCH GUIDELINE

There are multiple ways in which this research activity can be taken to the next level.

A few of these ways are listed below.

Though simulation based evaluation does provide a pretty good idea on

performance, hardware based implementation and experimentation may open

up new issue and avenues regarding realizability of MNKB-RBF. It would be

106

interesting to implement MNKB-RBF in the hardware and study its usability in

the real environment.

It would be very interesting to compare the performance of the Proposed RBF

when it is implemented in both ordinary hardware and in Field Programmable

Gate Array (FPGA). In this aspect the selection the performance comparison

parameters may require a lot of brain storming. Note that FPGA provides a lot

of flexibility in terms of change management. However, the time taken by both

implementations to decide the phase-rotated sequence of sub-carrier may or

may not be substantially different.

In this project error energy has been used to automatically adjust the weights of

the tuning parameters α1 and α2. There may be other and better choices

available. It is worth exploring these choices.

107

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