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GENETIC ALGORITHM ASSISTED OPTIMUM DETECTION FOR BLOCK DATA

TRANSMISSION SYSTEMS

by

SHANMUGAIH PALANICHAMY KULANDAIVEL BABU

Thesis submitted in fulfillment of the requirements for the Degree of

Doctor of Philosophy

August 2010

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ACKNOWLEDGEMENTS

I would like to thank my supervisor Dr.Mohd. Fadzli Mohd. Salleh for his kind and dedication in supervising me, without whom, I would not have completed this research. Secondly I would like to thank my co-supervisor Prof. Farid Ghani, my mentor who gave me an opportunity to work under him in Universiti Sains Malaysia.

I would like to thank Dr. Laurel Kimura who corrected English for all my research papers and this thesis. I would like to thank Balan Sinnaih for his help in getting me around USM. I would also like to thank KDU College Penang Campus for allowing to take up part time research program in USM. Last but not least I would like to thank Malaysian government and people for their love and affection they have shown towards me during my stay.

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iii

TABLES OF CONTENTS

Acknowledgement Table of Contents List of Tables List of Figures List of Abbreviations List of Symbols Abstrak Abstract

ii iii ix x xix xx xxi xxiii

CHAPTER 1- INTRODUCTION

1.1 Preface 1

1.2 Problem Statement 2

1.3 Objectives of the Research 4

1.4 Summary of Original Contributions 4

1.5 Scope of the Work 6

1.6 Thesis Organization 6

CHAPTER 2 - IMPAIRMENTS IN DIGITAL WIRELESS COMMUNICATION

2.1 Introduction 8

2.2 Inter Symbol Interference and their Effects 9 2.2.1 Introduction to Digital Communication 9 2.2.2 General Block Diagram of Digital Communication 9

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iv Systems

2.2.3 Simplification and Approximation for Simulation 11 2.2.4 Communication Through Inter Symbol Interference

Channels

12

2.3 Modeling of a Wireless Channel 14

2.3.1 Channel 1: [1 0 0] 20

2.3.2 Channel 2: (1.5)-1/2 [0.5 1 0.5] 20 2.3.3 Channel 3: (19)-1/2[1 2 3 2 1] 22 2.3.4 Channel 4 (2)-1/2 [0.167 0.667 1 0.667 0.167] 23 2.3.5 Channel 5: (2)-1/2 [-0.167 0.667 1 0.667 -0.167] 24

2.3.6 Channel 6: (2)-1/2[1 1] 25

2.3.7 Channel 7: (2)-1/2[1 0 1] 27

2.3.8 Randomly varying Channel 28

2.4 Noise Considerations 30

2.4.1 Introduction: 30

2.4.2 Noise Consideration in the Receivers 33

2.5 Equalizers for ISI 34

2.6 Conclusion 36

CHAPTER 3 - BLOCK TRANSMISSION SYSTEM AND GENETIC ALGORITHMS

3.1 Introduction 38

3.2 Block Diagram of SC-BDTS 39

3.3 Data Format of SC-BDTS 39

3.4 Effect of Noise in Block Data Transmission Systems 41

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3.5 Equalizers for BDTS 42

3.5.1 Introduction 42

3.5.2 Block Linear Equalizers 43

3.5.3 Block Decision Feedback Equalizer 45 3.5.4 Maximum Likelihood Block Detectors 47 3.6 Performance Analysis of an Optimum BDTS 51 3.6.1 Bit Error Rate and Block Error Rate 51

3.6.2 Signal Space 51

3.6.3 Probability of Error 54

3.6.4 Implementation of MLBD 59

3.7 Orthogonal Frequency Division Multiplexing 59

3.7.1 Introduction 59

3.7.2 Block Based Transmission 60

3.7.3 OFDM Signal and Transmission Modeling 61

3.7.4. Model of a WLAN based OFDM 66

3.7.5 Equalization Techniques for OFDM 68

3.8 Genetic Algorithms 73

3.8.1 Introduction 73

3.8.2 Block Diagram representation of Genetic Algorithms 74 3.8.3 Components of Genetic Algorithms 78 3.8.4 Parameter Control in Genetic Algorithms 83 3.8.5 Hybridization of Genetic Algorithms 84

3.9 Micro-Genetic Algorithms 85

3.9.1 Introduction: 85

3.9.2 Working of µ-GA 86

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3.10 Hardware Implementation of GA 86

3.10.1 Introduction 86

3.10.2 Random Number Generation 88

3.10.3 Selection Unit 88

3.10.4 Crossover Unit 88

3.10.5 Efficient Implementation of Fitness Function 89

3.11 Conclusion 89

CHAPTER 4 - GENETIC ALGORITHM ASSISTED DETECTORS FOR BDTS

4.1 Introduction 91

4.2 General Simulation Model of a BDTS 93

4.3 Block Diagram of GA based SC-BDTS 94

4.4 Implementation methods for GA Parameters 96

4.4.1 Introduction 96

4.4.2 Coding of Solution Space 96

4.4.3 Objective Function 97

4.4.4 Stopping Criteria 100

4.5 Hybrid-GA Detection 102

4.6 Micro-GA based SC-BDTS 105

4.7 GA Based MC- BDTS 106

4.7.1 GA based ML detection in IEEE802.11a 107 4.7.2 Hybrid-µGA based ML Detector for OFDM 111 4.8 Procedure for Calculating the Complexity of MLBD Detectors 114 4.9 Hardware Architecture Design for Hybrid-µGA based MLBD 115

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4.10 Conclusion 118

CHAPTER 5 - RESULTS AND DISCUSSION

5.1 Introduction 119

5.2 Dependence of Objective Function Values on Channel Distortion

120

5.3 Parameter Setting in Genetic Algorithms 123 5.4 GA/Hybrid-GA based detectors for SC-BDTS 124

5.4.1 Initial Population Setting 125

5.4.2 BER Performance of SC-BDTS with Hybridization 126 5.5 Computational complexity of the Exhaustive Search and

Ordinary GA based MLBD for SC-BDTS

130

5.5.1 Complexity of Exhaustive Search versus GA based

Detectors 131

5.5.2 Effect of Noise on GA-Based Detector Complexity 132 5.5.3 Complexity of Hybrid GA-Based Detector versus Ordinary GA-Based Detector

134

5. 6 Performance Analysis of µGA based detectors for SC-BDTS 136

5.7 Computational advantages of µGA 140

5.8 Hybridization of µGA 141

5.9 GA based detectors for MC-BDTS 147

5.9.1 BER Performance 147

5.9.2 Complexity analysis of GA based Detector in MC-BDTS 149 5.10 Comparison of GA based and µGA Based Hardware 150

5.11 Conclusion 154

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CHAPTER 6 - CONCLUSION AND FURTHER STUDIES

6.1 Introduction 155

6.2 Conclusion 157

6.3 Future work 158

REFERENCES 160

PUBLICATIONS 167

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LIST OF TABLES

Page

Table 2.1 Channel Impulse response of various channels 18 Table 3.1 Detection process for transmitted message block [-1 1] 49 Table 3.2 Chronology of important research activities on BDTS 50 Table 3.3 Minimum of distances to the decision boundaries 54 Table 3.4 The message blocks which have minimum distance

dmin/2 58

Table 3.5 Data rate and coding rate of IEEE 802.11a WLAN 67 Table 4.1 Coding of solution space for a message block of size 4

in binary format

97

Table 4.2 Few possible neighbors to BLE detected block [+1 +1 -

1 +1 -1 -1 +1 -1] 103

Table 5.1 Change in the objective function of 14th block with

respect to different channels 122

Table 5.2 Initial population variations for Ordinary and Hybrid- GA based detectors

126

Table 5.3 Block size versus Initial population 131 Table 5.4 Simulation results for µGA based BDTS 137 Table 5.5 Initial population and generations needed for Hybrid-

GA and Hybrid-µGA based detectors 151

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LIST OF FIGURES

Page Figure 2.1 Block diagram of a digital communication system 10 Figure 2.2 Simulation model of a baseband digital communication

system

12

Figure 2.3 QPSK constellation and effect of amplitude and phase

distortions 16

Figure 2.4 Magnitude and impulse response of Channel 1 20 Figure 2.5 Magnitude and phase response of Channel 2 21 Figure 2.6 Impulse response and Pole/Zero Plot of Channel 2 22 Figure 2.7 Magnitude and phase response of Channel 3 22 Figure 2.8 Impulse response and Pole/Zero plot of Channel 3 23 Figure 2.9 Magnitude and phase response of Channel 4 23 Figure 2.10 Impulse response and Pole/Zero plot of Channel 4 24 Figure 2.11 Magnitude and phase response of Channel 5 25 Figure 2.12 Impulse response and Pole/Zero plot of Channel 5 25 Figure 2.13 Magnitude and phase response of Channel 6 26 Figure 2.14 Impulse response and Pole/Zero plot of Channel 6 26 Figure 2.15 Magnitude and phase response of Channel 7 27 Figure 2.16 Impulse response and Pole/Zero plot of Channel 7 27 Figure 2.17 Impulse response of the channel for the 70th symbol 28 Figure 2.18 Frequency response of the channel for 70th symbol 28 Figure 2.19 Impulse response of the channel for 83rd symbol 29 Figure 2.20 Frequency response of the channel for 83rd symbol 30 Figure 2.21 Normalized Gaussian distribution with zero mean and

standard deviation of 1

31

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Figure 2.22 White noise power spectral density 32

Figure 2.23 Two symbol constellation 34

Figure 2.24 Baseband digital communication systems with Equalizer

35

Figure 2.25 BER performances of various equalizers for Channel 3 36 Figure 3.1 Model of a Block Data Transmission System 39 Figure 3.2 Structure of the transmitted symbols in a BDTS 40 Figure 3.3 Antipodal signaling of a message [1 0 1 1 0 1 1 0 1] in

BDTS

41

Figure 3.4 Block Linear Equalizers for BDTS 44

Figure 3.5 All possible received blocks for block size =2 52 Figure 3.6 Decision regions and decision boundaries for block

Size = 2 53

Figure 3.7 Theoretical Probability of Error for the channels shown in Table 3.4

58

Figure 3.8 Overlapping spectrums in OFDM (N=8 each allotted 16 Hz)

61

Figure 3.9 Block diagram of OFDM transceiver 62

Figure 3.10 OFDM symbol with cyclic prefix 62

Figure 3.11 OFDM Symbol in time domain 67

Figure 3.12 BER performance for GA based MLBD studied by

Sacchi et al. 73

Figure 3.13 Evolutionary cycle of a GA 75

Figure 3.14 Flow chart of a GA 77

Figure 4.1 Generic simulation model of a SC-BDTS system 93

Figure.4.2 GA based SC-BDTS 95

Figure 4.3 Euclidean distances of all possible blocks for channel 2 99 Figure 4.4 Pseudo code to plot the objective function 99

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Figure 4.5 Procedure to find the maximum generations for convergence

101

Figure 4.6 Hybrid-GA detector structure for Block Data

Transmission System 102

Figure 4.7 Pseudo code to generate the Initial population for Hybrid-GA based MLBD detectors

104

Figure 4.8 µGA based receiver for SC-BDTS 105

Figure 4.9 Hybrid-µGA based receiver 106

Figure 4.10 General block diagram of a GA based MLBD for an OFDM system

108

Figure 4.11 Methodology for simulation of GA based MLBD for

MC-BDTS 109

Figure 4.12 Methodology for simulation of µGA based MLBD for MC-BDTS

113

Figure 4.13 Data path of µGA based receiver 115 Figure 4.14 SIMD architecture for Ordinary GA. 116 Figure 4.15 SIMD based Hybrid-µGA detector showing the data

flow

117

Figure 5.1 Euclidean distances of all possible blocks for channel 1 120 Figure 5.2 Euclidean distances of all possible blocks for channel 2 121 Figure 5.3 Euclidean distances of all possible blocks for channel 3 121 Figure 5.4 Euclidean distances of all possible blocks for channel 4 122 Figure 5.5 Performance comparison of Selection, Crossover,

Initial population parameters in convergence 124 Figure 5.6 Bit Error Rate of BDTS for baseband antipodal

transmission through channel 1

127

Figure 5.7 Bit Error Rate of BTS for baseband antipodal

transmission through channel 2 128

Figure 5.8 Bit Error Rate of BTS for baseband antipodal

transmission through channel 3 129

Figure 5.9 Bit Error Rate of BDTS for baseband antipodal 130

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xiii transmission through channel 4

Figure 5.10 Computation requirements for GA based detector and Exhaustive Search based detector

132

Figure 5.11 Computation requirements in terms of additions for

GA-based detector in presence of noise 133 Figure 5.12 Computation requirements in terms of multiplications

for GA-based detector in presence of noise 134 Figure 5.13 Computation requirements in terms of additions for

GA-based and Hybrid-GA based detector

135

Figure 5.14 Computation requirements in terms of multiplications

for GA based and Hybrid-GA based detector 135 Figure 5.15 Bit Error Rate performance of µGA based BDTS for

Channel 1

138

Figure 5.16 Bit Error Rate performance of µGA based BDTS for

Channel 2 138

Figure 5.17 Bit Error Rate performance of µGA based BDTS for Channel 3

139

Figure 5.18 Bit Error Rate performance of µGA based BDTS for Channel 4

140

Figure 5.19 Complexity reduction in using µGA instead of GA in

the detector 141

Figure 5.20 Comparison of number of generations required for µGA and Hybrid-µGA for Channel 2

142

Figure 5.21 Comparison of number of generations required for

µGA and Hybrid-µGA for Channel 3 143 Figure 5.22 Comparison of number of generations required for

µGA and Hybrid-µGA for Channel 4

143

Figure 5.23 Convergence of µGA and Hybrid-µGA based detectors

for a typical received block 144

Figure 5.24 Objective function evaluation for Exhaustive Search

detector and various GA based detectors 145 Figure 5.25 BER performance of various MLBD for channel

number 3

146

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Figure 5.26 Performance of OFDM based WLAN using various GA based detector (3-Tap channel)

148

Figure 5.27 Performance of OFDM based WLAN using various

GA based detectors (10-Tap channel) 148 Figure 5.28 Complexity analysis of various MLBD detectors for

OFDM based WLAN system in terms of objective function evaluation

150

Figure 5.29 Increase in hardware complexity for Hybrid-GA based

detectors with decrease in SNR 152

Figure 5.30 Increase in computational time for µGA based detector with decrease in SNR

153

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LIST OF ABBREVATIONS

AWGN Additive Gaussian White Noise

AP Access point

ASIC Application Specific Integrated Circuit BSS Base Service Sets

BlER Block Error Rate

BDTS Block Data Transmission Systems

BLE Block Linear Equalizers

BDFE Block Decision Feedback Equalizer

BGA Binary GA

BPSK Binary Phase Shift Keying BER Bit Error rate

CIR Channel Impulse Response

CDMA Code Division Multiple Access

CU Control Unit

CP Cyclic Prefix

dB Decibel

DFE Decision Feedback Equalizer DAB Digital Audio Broadcasting D/A Digital to Analog

DVB Digital Video Broadcasting DMT Discrete Multi Tone Modulation

DDB Distance to the decision boundary ETA Expanding Tree Algorithm

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xvi FFT Fast Fourier Transform

FPGA Field Programmable Gate Array FIR Finite Impulse Response

1G First generation

Gbps Giga Bits per Second

GA Genetic Algorithms

HDL Hardware Description Languages

IC Integrated Circuit

IBI Inter Block Interference ICI Inter Carrier Interference ISI Inter Symbol Interference iFFT Inverse Fast Fourier Transform

KHz Kilo Hertz

LOS Line of Sight

LE Linear Equalizer

LFSR Linear Feedback Shift Register LTE Linear Transversal Equalizer

MAC Medium Access Layer

MMSE Minim Mean Square Error ML Maximum Likelihood

MLSD Maximum Likelihood Sequence Detection MLBD Maximum Likelihood Block Detection µGA Micro Genetic Algorithms

MVA Modified Viterbi algorithm

MC Multi Carrier

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xvii MCM Multi Carrier Modulation N-LOS Non Line of Sight

NLBE Non-Linear Block equalizer

OFDM Orthogonal Frequency Division Multiplexing P/S Parallel to Serial

PHY Physical Layer

PE Processing Elements

QAM Quadrature Amplitude Modulation QPSK Quadrature Phase Shift Keying

QoS Quality-of-Services

RF Radio frequency

2G Second Generation

S/P Serial to Parallel SNR Signal To Noise Ration

SC Single Carrier

SC-FDMA Single Carrier - Frequency Division Multiple Access

SISO Single Input Single Output SIMD Single Instruction Multiple Data TDL Tapped delay line

3G Third generation

TSP Traveling Salesman Problem

VA Viterbi Algorithms

VC Virtual Channels

WLAN Wireless Local Area network

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WMAN Wireless Metropolitan Area Network ZF Zero Forcing

ZP Zero Padding

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LIST OF SYMBOLS

µ Micro

η Gaussian noise variable δ Delta function

∀ For all

Σ Summation

Λ Probability

σ Variance of AWGN

π Pi

® Registered trademark

∫ Integration

∞ Infinity

≤ Less than or equal to

≥ Greater than or equal to

≠ Not equal to

* Convolution

Tb Pulse duration

m Block size

h Channel Impulse Response g Known symbols size

) (f

θc Channel phase response

No Amplitude of Noise Spectral Density z Dimension of the signal space Prc Raised Cosine pulse

Rxx Autocorrelation Rxy Cross correlation

κ Efficiency of the Block Data Transmission System S Transmitted message blocks

bi Uninterrupted bit stream

H Channel matrix

N Sample values of the statistically independent Gaussian

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random variables representation with zero mean and with a common variance σ2

n Received block

⊂ Belongs to (Used in description of sets) dmin Minimum distance

PBER Probability of bit error PblER Probability of block error Nmin Number of blocks having dmin

mg Cyclic Prefix length in OFDM e Euclidean distance

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PENGESAN OPTIMUM DENGAN BANTUAN GENETIK ALGORITMA UNTUK SISTEM PENGHANTARAN BLOK

DATA

ABSTRAK

Dunia sedang memperlihatkan peningkatan dalam keperluan untuk transmisi data berkeupayaan tinggi di bidang applikasi multimedia tanpa wayar. Masalah penyuraian saluran berasaskan masa bukannya sesuatu yang baru di dalam sistem transimisi data berkeupayan tinggi tanpa wayar. Teknologi-teknologi baru yang mempergunakan kaedah transmisi data berbentuk “block” di mana data ditukarkan ke bentuk “block” yang diasingkan oleh simbol yang telah diketahui terlebih dahulu, telah diperkenalkan untuk megatasi masalah ketidaksepasangan disebabkan oleh penyuraian saluran. Pengaturan kaedah sebegini menyekat “Inter Symbol Interference” di dalam “block” tertentu dan membenarkan pemprosesan data secara

“block” demi “block”. System transmisi data berbentuk blok (BDTS) merupakan salah suatu teknik yang berpromisi dan didapati di dalam generasi keempat teknologi tanpa wayar seperti “Orthogonal Frequency Division Multiplexing (OFDM)”,

“Single Carrier Frequency Division Multiple Access (SC-FDMA)” dan “Discrete Multi Tone (DMT) Modulation Scheme”. Di dalam BDTS, besar kemungkinan pemprosesan pengesanan adalah optimum tetapi pelaksanaannya memerlukan penghitungan yang intensif. Semasa saluran tanpa wayar mengalami kehilangan yang mendalam atau memorinya berlebihan berbanding simbol yang dikenali terlebih dahulu, kadar ralat bit (BER) memberi kesan kepada “Quality of Service (QOS)”. Di dalam kes-kes sebegini, pengesanan ML menjadi mandatori. Kaedah

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untuk pelaksanaan pengesan ML adalah diperlukan untuk bersearah dengan kemajuan dalam teknologi “microchip”. Di dalam penyelidikan ini, “Genetic Algorithm” (GA) digunakan sebagai kaedah untuk melaksanakan pengesan ML.

Pengesan berasaskan GA direka untuk system komunikasi “Single Carrier” (SC) dan juga “Multi Carrier” (MC). Di dalam sistem komunikasi SC, saluran dengan kehilangan yang mendalam akan digunakan. Manakala di dalam sistem komunikasi MC, saluran yang bermemori lebih daripada “Cyclic Prefix” (CP) yang menghasilkan BER rendah akan dipertimbangkan. Kerumitan penghitungan dikurangkan melalui pengurangan kecil di dalam prestasi BER daripada pengesan optimum dengan menggunakan GA di dalam pengesan ML, contohnya sistem pencarian “Exhausitive”. Analisis masalah di dalam pengesan ML yang lebih lanjut, memperkenalkan GA yang berlainan jenis, contohnya “Micro-GA” (µGA) untuk membantu proces pengesanan. µGA menggunakan populasi yang kecil dan bantuan penghitungan minima berbanding dengan GA. Seni bina perkakasan untuk pelaksanaan “Hybrid - µGA” dicadangkan dengan menggunakan seni bina selari berjenis “Single Instruction Multiple Data”. Prestasi BER di dalam µGA adalah sebanding dengan sistem pengesan optimum dan menawarkan pengurangan sebanyak 50% di dalam penghitungan berbanding dengan GA. Process pengacukan di antara GA dan µGA dengan meghasilkan populasi awal daripada output pengesan linear “suboptimal” telah memperbaiki pemusatan GA berasaskan pengesan ML. Ini telah megurangkan lagi 50% kerumitan penghitungan.

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GENETIC ALGORITHM ASSISTED OPTIMUM DETECTION FOR BLOCK DATA TRANSMISSION SYSTEMS

ABSTRACT

Recent years have witnessed an increase in demand for higher transmission data rates for wireless multimedia communications applications. Wireless data transmission systems with high data rates always face problems over time dispersive channels. New technologies that employ block data transmission where the data is converted into blocks separated by known symbols have been introduced to combat the impairments due to dispersive channels. This arrangement confines the Inter Symbol Interference (ISI) to a particular block and allows Block-by-Block data processing. Block Data Transmission Systems (BDTS) is one of the promising techniques for high data rate data transmission and found in the fourth generation wireless technologies such as Orthogonal Frequency Division Multiplexing (OFDM), Single Carrier Frequency Division Multiple Access (SC-FDMA) and Discrete Multi Tone (DMT) Modulation schemes etc. In BDTS, Maximum Likelihood Block-by-Block detection processing is optimum but computationally intensive to implement. When wireless channels experience deep fades or when the channel memory is longer than the known symbols, Bit Error Rate (BER) affects the Quality of Service (QOS). In these cases ML detection becomes mandatory. The methods for implementation of ML detectors are necessary with today’s advances in microchip technologies. In this research Genetic Algorithm (GA) is used as a method to implement ML detectors. The GA based detectors are designed for Single Carrier (SC) as well as Multi Carrier (MC) communication systems. In SC communication systems, channels with deep fades are used. In MC communication

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systems, channels whose memories are longer than Cyclic Prefix (CP) that produce low BER are considered. By using GA for ML detection, computational complexity is reduced with small reduction in terms of BER performance from the optimum detector, i.e., the Exhaustive search system. Further analysis of the problems in ML detectors has introduced a variant of GA, i.e., Micro-GA (µGA) to assist the detection process. The µGA uses a smaller population and less computational recourses as compared to GA. The hardware architecture for implementation of the Hybrid- µGA is proposed, using a Single Instruction Multiple Data type of parallel architecture. The BER performance of µGA is comparable with the optimum detector system and offers 50% reduction in computation as compared to GA. The hybridization of GA and µGA by generating the Initial population from the output from a suboptimal linear detector has improved the convergence of GA based ML detectors, which led to another 50% percent reduction in computational complexity.

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1

CHAPTER 1 INTRODUCTION

1.1 Preface

The demand for high data rate mobile services with spectral, power efficiency and stringent Quality-of-Services (QoS) is expected. The challenges posed in the fourth generation (4G) mobile communication systems will be enormous. The users will demand a particular service with particular QoS. For example, the data rate of such systems is expected to have more than 1Giga bits per second (Gbps). One of a few hurdles to achieve high data rate is Inter Symbol Interference (ISI), which is due to the underlying channel characteristics in a mobile environment. With the help of advances in semiconductor technologies and advanced signal processing algorithms, we can construct complex communication systems that will meet the demands of the future.

A Block-by-Block transmission scheme found in the present and proposed future communication technologies are designed to reduce ISI. In this system blocks of data symbols are separated by zero symbols/known symbols, which confine the ISI to a block. Orthogonal Frequency Division Multiplexing (OFDM), Single Carrier Frequency Division Multiple Access (SC-FDMA), Discrete Multi Tone Modulation (DMT) schemes, to name a few, operate on a Block-by-Block basis. Certain multiple antenna systems operate in this fashion, and Block-by-Block detection is also found in some multiuser Code Division Multiple Access (CDMA) systems (Xu, et al., 2006).

Multipath fading channels cause severe ISI (Sklar, 1997). Shortwave  ionospheric radio, tropospheric scatter radio and land  mobile radio are a few

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examples of such channels. In Single Carrier Block Data Transmission Systems (SC- BDTS) when there is a null in the channel frequency response or in a Multi Carrier (MC) system like OFDM having channel length in excess of the Cyclic Prefix (CP) length or when there is clipping noise, the effects of ISI is high. This effect leads to a high Bit Error Rate (BER) that cannot be efficiently dealt by a Zero Forcing (ZF) equalizer or a Decision Feedback Equalizer (DFE). The use of optimum detectors like Maximum Likelihood Sequence Detection (MLSD) becomes mandatory in this case (Proakis & Sallehi, 2002). Even though Viterbi Algorithms (VA) can be used to implement optimal detectors, their computational complexity increases with an increase in signal sets and channel memory. In BDTS the number of zero symbols/known symbols per data block is dependent on the channel memory. Short data blocks based BDTS can have VA type optimum detectors with reduced efficiency (Kaleh, 1995). A large data blocks based BDTS is preferred, but optimum detection in such a system leads to high complexity. In such cases global optimization techniques like Genetic Algorithms (GA) are the alternate choice to implement such optimal detectors.

1.2 Problem Statement

It is found that when ISI is severe, the MLSD is superior to other equalizer methods. The performance can be further improved by converting symbols as blocks, separated by known or zero symbols. This blocking of symbols restricts the effect of ISI to be confined to a single block and allows block based processing. In BDTS, Block-by-Block processing is optimum and symbol based processing is suboptimum.

Equalizers for this block based data transmission system are the Block Linear Equalizer (BLE) and the Block Decision Feedback Equalizer (BDFE), which are

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symbol based, and the Maximum Likelihood Block Detection (MLBD), which is block based. In the MLBD, if a symbol is represented by one bit then, a block (message) of size m, when received should be compared with all possible 2m blocks in the receiver. When this comparison is based on Euclidean distance, then it is equivalent to the implementation of an optimum detection procedure (Proakis &

Salehi, 2002). For small block sizes, all possible transmitted blocks can be searched.

This is called an Exhaustive Search, and it becomes quite impossible when the block size is increased for efficiency as the number of known symbols is fixed irrespective of block size. A global optimization process like GA can replace the Exhaustive Search. In this research, the optimum detection problem is implemented using GA, a global search technique.

This implementation has to be analyzed in terms of performance and complexity. The performance in terms of BER is done using MATLAB® based simulation. The GA has various parameters and they are problem dependent. For this problem these parameters are analyzed and the best are to be taken. New variants of GA like Micro-GA (µGA) are also to be considered. Hybrid systems consisting of a simple conventional system assisted by a complex intelligent system are proven to be more efficient than using an intelligent system alone. An intelligent system like using GA based optimum detectors which is an intelligent system can be integrated with suboptimal linear equalizers, which are simpler in terms of implementation. All the above systems are to be analyzed in terms of their BER performance, complexity and hardware implementation issues.

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1.3 Objectives of the Research

The present research centers on the BDTS, which has many advantages and applications, as discussed in the previous section. The key objectives of this research include:

1. To develop a GA assisted optimal BDTS receiver for Single Carrier and Multi Carrier communication systems.

2. To develop a reduced complexity GA using Hybrid techniques (Hybrid-GA) assisted optimum detector for Single Carrier and Multi Carrier communication systems.

3. To use Micro GA and hybridize the same to further reduce the complexity of the optimum detector for Single Carrier and Multi Carrier communication systems.

1.4 Summary of Original Contributions

The original contributions of the research can be generally categorized into two different areas namely: GA based BDTS for Single Carrier communication systems and GA based BDTS for Multi Carrier communication. The contributions are summarized below:

1. GA assisted MLBD for SC-BDTS

Extensive studies in BDTS equalizers did not concentrate on the MLBD due to the complexity of the Exhaustive Search procedure involved. In this research the suitability of GA for SC-BDTS is proposed and the performance is compared with the Exhaustive Search process.

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5 2. Hybridization of the GA process.

GA starts with a set of possible solutions. The solution in this case is a set of possible messages that would have been transmitted. In this research to speed up convergence, the starting space, which has probable solutions, will have the output of BLE and its neighbors. The neighbors are the blocks which differ by one, two or three bits from the output of the BLE. By doing this, computational complexity is reduced by a substantial amount.

3. Micro-GA (µGA) used instead of GA.

The µGA uses a smaller population size than GA and its fast convergence, when compared to GA is reported for many applications. This technique is tried with BDTS and its BER performance is comparable to GA based BDTS with a reduced complexity.

4. Development of the Hybrid-µGA based optimal detectors in IEEE 802.11a based WLAN

The algorithms and techniques developed for the SC-BDTS are applied for Wireless Local Area Networks (WLAN) as defined in the IEEE 802.11a system. In the literature, GA based optimum detection was used in OFDM systems under nonlinear distortion. The other possible scenario where MLBD is mandatory is when channel length is higher than Cyclic Prefix (CP) length. For this scenario, in this research, µGA and Hybrid-µGA are used along with Ordinary GA as an optimum detector. This detector has shown similar performance as Maximum Likelihood (ML) detectors in terms of BER but with a reduction in complexity of about fifty percent (50%) and above.

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1.5 Scope of the Work

In this research, the SC-BDTS uses zero symbols as guard symbols.

Therefore, no channel estimation is performed. Instead, the Channel Impulse Response (CIR) is assumed as the known quantity in advance and time invariant.

This arrangement allows the performance analysis of various equalizers and detectors under different levels of deep fades and channel lengths. In Multi Carrier BDTS (MC-BDTS) consideration, IEEE 802.11a is used instead of the recent IEEE 802.11n, which is better in terms of performance and capacity. The reason for choosing IEEE 802.11a is due to the simplicity in simulation.

The existing Single Instruction Multiple Data (SIMD) architecture for GA is extended for µGA and the performance of µGA in SIMD architecture is compared.

No Field Programmable Gate Array (FPGA) or Application Specific Integrated Circuits (ASIC) implementation is done.

1.6 Thesis Organization

The remainder of the thesis is organized into six chapters. Chapter 2 serves as a background for the reader. In this chapter, for a continuous transmission digital communication system, the effect of noise, ISI and their combination is detailed.

Various methods to mitigate those effects are also shown in detail. The chapter also includes the topic of channel modeling and noise consideration for deriving probability of error. Chapter 3 presents the literature review of the BDTS, its signal format and its impairments. Part of this chapter is dedicated to the earlier remedy of using equalizers to mitigate the impairments encountered. A visit to the concept and applications of GA and its variants is presented. The earlier proposal of GA for

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detection in an OFDM system under clipping noise along with a few applications of GA for a CDMA system is shown in detail.

In Chapter 4, the methodology opted for implementing the GA and its variants assisted optimum detector for SC-BDTS and MC-BDTS is presented. The procedure for calculating the computational complexity of the GA, GA variants and Exhaustive Search based detectors are also shown. The chapter also deals with design procedure used to study the performance of GA and µGA on parallel architectures.

Chapter 5 provides the results, which mainly include the BER performance analysis of GA and its variants based optimum detectors for both SC and MC communication systems. For SC systems, channels with spectral nulls and for MC systems channel length higher than CP length are taken into consideration. The results were compared with the Exhaustive Search method, which is an optimum detector implementation and the BLE which is the simplest method. In Chapter 6 the conclusions and future research direction of the thesis is presented.

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CHAPTER 2

IMPAIRMENTS IN DIGITAL WIRELESS COMMUNICATION

2.1 Introduction

Many applications in wireless multimedia communication such as video telephony, wireless Internet, Digital Audio Broadcasting (DAB), Digital Video Broadcasting (DVB), etc., require a large volume of data transmission over a limited amount of resources. The high data rate transmissions especially through the time dispersive communications channels, introduce Inter Symbol Interference (ISI). This interference is the major limiting factor in such communication systems. It is well known that equalizers are used to combat ISI (Proakis & Salehi, 2002). They come in different types of design complexity and offer different performances.

In order to study the effect of ISI towards transmission data, it is very important to analyze the properties of the channels. Much research has been done to study the channel properties which led to channel models. Although channel modeling is not part of the objectives of this research, it is important to study how the properties of the channels contribute to ISI which is impairment to the communication systems. The Additive White Gaussian Noise (AWGN) is impairment due to the electron movement in a certain device that affects the transmitted signal. Thus, in a way, it limits the performance of the communication system. In this chapter the channel models, along with their properties used in the research, are studied. Section 2.2 provides the basic discussion about ISI in digital communications; Section 2.3 introduces the multipath fading effects along with channel modeling. In Section 2.4, the effect of noise on digital wireless transmission is studied. Section 2.5 presents the equalizers, and in Section 2.6 the Maximum

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Likelihood Sequence Detectors (MLSD) are studied. Section 2.7 compares various equalization methods, and Section 2.8 concludes the chapter.

2.2 Inter Symbol Interference and their Effects

2.2.1 Introduction to Digital Communication

Digital communication systems, which use a predefined set of electrical signals to represent digital messages, are the preferred choice due to immunity to noise, cheap hardware, error control and security. Noise is inherent in any communication system. The messages, that can be voice, music, image or video, can be represented in digital format as bits. These bits are either directly or grouped together and mapped to electrical signals, which are then transmitted over the analog communication channels. The bits, which can be as individuals or as groups, are called symbols. In real world communication scenarios, we come across different types of channels, source destination pairs, signals, QoS, number of users, traffic over communication links, etc. In the case of digital communication systems, the design of a communication system for a specific scenario is unique. However the impairments seen in almost any communication system are the same.

2.2.2 General Block Diagram of Digital Communication Systems

A complete digital communication system used in the real world can be shown as a block diagram. The block diagrammatic approach is a system level representation that gives the overall function of each block. Each block, in turn, can be decomposed into smaller blocks until the need for decomposition vanishes.

The general block diagram of a digital communication system is shown in Figure 2.1. It contains three parts, namely the transmitter, channel and receiver, along with related functional blocks. In the transmitter, the source encoder removes the

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10

redundancy, and the channel encoder adds controlled redundancy for error detection and correction. The baseband modulator maps the bits into suitable signals for Radio Frequency (RF) modulation. The transmit filter restricts the bandwidth of the source.

The communication channel in the research is assumed to be a wireless channel where noise and other interferences are added to the modulated signal.

Figure 2.1 Block diagram of a digital communication system (Haykin, 1997)

In digital communication, good waveforms can be selected to match the channel impairments, thereby increasing the reliability when compared to an analog system, where such a possibility is not available (Anderson, 1997). In the receiver, the reverse process of the transmitter is performed so that the messages are retrieved from the changes it went through during transmission. The additional system blocks like carrier recovery and timing recovery are used to aid the receiver to process the

Information Source

Source Encoder

Channel Encoder

Baseband modulator and Transmit filter

RF Modulator Communication Channel

Baseband Demodulator Equalizer

Channel Decoder

Receiver Filter RF Demodulator

Source Decoder

Information Sink

Carrier Recovery

Timing Recovery Noise

Interference from other sources

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received signals with more accuracy. Due to efficiency, cost reduction, and reliability, digital communication has replaced the analog communication system.

2.2.3 Simplification and Approximation for Simulation

In general research design of any new or better digital communication systems, the objective of research plays an important role in dealing with the complexity of design and analysis. In Section 2.2.2, the general description of the digital communication systems clearly indicates that designing and testing the whole system is very complex and time consuming. Therefore a high level model as shown in Figure 2.2 is more suitable, as it has few parameters and reduces the computational burden if simulation is to be performed (Tranter et al., 2006).

Simulation is used to design and analyze a communication system without prototyping it in the laboratory. It has the advantage of reducing the development time and cost of the research. The results of such simulations are also accurate to a level of prototyping and testing. Approximations and assumptions are used to simplify simulation (Tranter et al., 2004). However if a part of the system is studied by itself, this can be done at a low level, while keeping other blocks at a high level.

As seen in the abstract of this research, mainly the detection process is taken into consideration and the Bit Error Rate (BER) performance is one of the main goals. A much-simplified model which can be taken into consideration is shown in Figure 2.2.

In this model the source and channel encoder/receiver is removed along with RF modulation. The BER of a digital communication system in RF or baseband transmission is the same. This leads to the removal of RF modulation/demodulation along with the carrier recovery, as it is also assumed that timing is unaltered in the simulation process.

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Figure 2.2 Simulation model of a baseband digital communication system

A general baseband digital communication system is shown in Figure 2.2. It consists of a transmit filter to restrict the bandwidth of the message signals and a bandlimited linear time invariant channel corrupted by AWGN. The linearity and time invariance is an approximation when the analysis is done for a short time. In the receiver there is another filter to restrict the out-of-band noise, and a decision device used as a detector to make a decision on the received signal as it is corrupted by noise. If the blocks are considered as linear, the overall response of the cascaded or parallel blocks can be added or multiplied, thereby greatly reducing the complexity and simulation time.

2.2.4 Communication Through Inter Symbol Interference Channels

In Figure 2.2 we found that the messagesXk, when communicated over a

channel, are corrupted by the noise. The received messages X^k may be the same as transmitted or different due to the addition of noise. When they are different, then an error has occurred. The other source of error in a digital communication system is ISI. In this section, the problem of ISI is illustrated. Let the messages be binary and let these messages be called as symbols. A symbol may have one or more bits. When

Transmit Filter

g(t)

Channel h(t)

Receive Filter

c(t)

Decision Device Σ

Messages

Xk AWGN n(t)

Transmitter Receiver Channel

y(t) Messages

^

Xk

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13

a symbol having a single bit is converted to an antipodal rectangular pulse mk , the amplitude of each bit is defined as

If g(t), h(t) and c(t) are impulse responses of the transmit filter, channel and the receive filter, respectively, the transmitted signal s(t) is defined as

=

k

b

kg t kT

m t

s() ( ) (2.2) The output of the receive filter is written as

+

=

k

b

kpt kT t

m t

y( ) α ( ) η( ) (2.3) The pulse p(t) is the rectangular pulse, which is used to represent “1” or “-1”. The received pulse is due to the double convolution

) (

* ) (

* ) ( )

(t g t ht ct

p =

α (2.4) where “*” represents convolution and for a normalized pulse p(0) = 1, α accounts for the scaling in the amplitude, and η(t) is assumed to be AWGN.

The receiver samples the output in synchronization with the transmission and compares the sampled value to a threshold. If the value of the sample is higher than the threshold, it decides as “1” and otherwise as “0”.

In deciding the reception of the ith pulse the received signal y(t) is sampled at time ti = iTb (with i taking on integer values and Tb the pulse rate).

−∞

=

+

=

k

i b k

i m p i k T t

t

y( ) α [( ) ] η( ) (2.5)

−∞

=

+

− +

=

i kk

i b k

i

i m m p i k T t

t

y( ) α α [( ) ] η( ) (2.6) mk = +1 if symbol Xk is 1

-1 if symbol Xk is 0

(2.1)

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14

In Equation 2.6 the first term is due to the ith pulse and the second term is due to the residual effect of all other pulses (Proakis & Salehi, 2002). These residues, from the before and after pulses of the ith pulse, is called ISI. The channels that produce these residues are called dispersive channels. An example of a dispersive channel is land mobile channel whose characteristics are randomly changing and have multipath propagation (Stüber, 2002).

If the pulse shape is controlled such that

⎩⎨

= =

i k

k T i

k i

p b

, 0

, ) 1

( (2.7)

then the receiver output given in Equation 2.6 gives the perfect reconstruction in the absence of noise, with only the scaling factor. The result is shown in Equation 2.8.

i

i m

t

y( )=α (2.8)

Pulse p(t), which satisfies the condition given in Equation 2.7, are Sinc and Raised Cosine pulses (Haykin, 1997).

One of the solutions to deal with the ISI is the equalizer. It can be categorized generally as linear or nonlinear. Examples of linear equalizers are Zero Forcing (ZF) equalizer and Minimum Mean Square Error (MMSE) equalizer. Examples of nonlinear equalizers are the Decision Feedback Equalizer (DFE) and the Maximum Likelihood Sequence Detection (MLSD). Out of the above four, the MLSD is the optimum and outperforms the others with the disadvantage of high complexity (Forney, 1972, Proakis & Salehi, 2002).

2.3 Modeling of a Wireless Channel

A communication channel is the medium in which a signal passes from a transmitter to a receiver. The channel can be a copper wire, twisted pair wire, optical

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cable, free space or underwater. In a wireless communication system, the received signal at the antenna of a receiver is the superimposed signal that comprises the signals arriving from different directions and time. Each path behaves like an individual fading path. This fading is due to reflection, delay and scattering, etc.

(Proakis & Salehi, 2002; Sklar, 1997). This fading process is characterized as a Rayleigh distribution for a Non Line-of-Sight (N-LOS) path and a Rician distribution for a Line-of-Sight (LOS) path. If the number of paths is just one, then it is called frequency flat fading, and if more, as frequency selective fading (Marvin & Alouini, 2000; Tranter et al., 2004).

The frequency selective fading in time domain is called as delay spread and measured as the overall time span of the path delays from the first path to the last path. This is also called as the length of the channel. Due to delay spread there is a time dispersion of the signal. This time dispersion causes ISI. The channels, which have the above characteristics, are found in different communication systems such as Mobile Radio, Ionosphere High frequency (HF) channel, underwater channel and telephone channels (Fonta & Espineira, 2008).

The signal distortions, which occur in the channels, are mainly at the amplitude and phase. A channel with impulse response h(t) can be considered as a bandlimited linear filter. This impulse response can be used to design, plan or develop any communication system. This can be done by a measurement system that uses the unique properties of the pseudo random numbers transmission and extraction via a channel, where the magnitude and phase response are obtained from the received sequence (Benvenuto & Cherubini, 2002). The Fourier transform of the impulse response is given as H(f)= Y(f)ejθc(f),where h(t) and H(f)are Fourier transform pairs, H(f) is the channel’s amplitude response, and θc(f) is the

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c l o d f t s n I e

v a a

channel’s ph linear functi occurs. If th distortion o frequency. S the variation some freque null can be w If )θc(f is effect is pha The visualized u and phase di a Quadrature

Figur

hase respons on of freque he term H(f ccurs. Amp Some channe

n are minim encies are ca

wide or narr not a linear ase distortion

amplitude using the con

istortions. T e Phase Shif

re 2.3 QPSK

se. The term ency. If both

)

f is not co plitude respo

els exhibit s mal. The cha alled channel row with res function of n.

and phase nstellation po This figure sh

ft Keying (Q

K constellatio

16 m H(f) mu h do not follo onstant and v onse will g severe ampli annels that h

ls with spect pect to the s frequency w

distortions oints in Figu hows the am QPSK) conste

on and effec

ust be consta ow the above varies with f give the det itude variatio

have very lo tral null. The signal bandw within the ch

of a seque ure 2.3 due t mplitude and ellation diag

ct of amplitu

ant and (θc e condition t frequency, th

ails of amp on and for s ow amplitud e frequency width and are

annel bandw

ence transm to the effect phase distor gram.

de and phase )

f must be then distortio hen amplitud plitude versu some channe

de (80dB) f range of suc e unpredicte width, then th

mitted can b t of amplitud rtion effect o

e distortions e a

on de us els for ch ed.

he

be de on

s

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17

The amplitude distortion is noted when the constellation points move towards the center and the phase distortion leads towards the rotational spread of the constellation points (Marvin & Aloiuni, 2000; Tranter et al., 2004; Maurice, 2006).

The channel characteristics are analyzed in the following aspects:

1. Impulse response 2. Pole/Zero locations 3. Phase response 4. Magnitude response

The wireless channel of a communication system can be modeled so that it can be used in simulation for finding the system’s performance. A well known technique is to use the impulse response measurement. A linear baseband channel utilized in this research is modeled as a continuous time channel. The overall impulse response h(t) includes all components of the transmitter and receiver filters. These filters are used for pulse shaping, linear modulation and demodulation as shown in Figure 2.2 (Ghani, 2004). The transmitted signal is modeled as:

=

i

i t kT

s t

s() δ( ) (2.9) where si’sare the transmitted symbols, i is any integer and δ(t) is the standard unit impulse function. The continuous channel model can be discretized when one sample per symbol period T is considered. Thus, this channel can be efficiently modeled as a Finite Impulse Response (FIR) filter (Proakis & Salehi, 2002).

In this analysis two scenarios are taken into consideration given as the following (Choi, 2006):

1. The gain of a path remains constant throughout the simulation. The worst case scenario happens when the frequency response of the channel shows spectral nulls. This spectral null produces a high BER.

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2. The gains of the delayed paths vary. This variation is made to exhibit a Rayleigh distributed pattern to simulate the real world scenario. Thus, this scenario produces a random frequency response.

In this research, the time invariant model of the channel is considered where the channel is typically varying slowly with time. In addition, in this work, it is assumed that the channel is known and fixed for the duration of a detecting a block (Crozier, 1989).

The Channel Impulse Response (CIR) h is defined in Equation 2.10.

hi = (h0, h1, …… hg) (2.10) In this research seven different channels are chosen which are used widely in various literatures (Porat & Friedlander, 1991; Kaleh, 1995; Declercq et al., 2001;

Zhang, 2002; Frédérique & Sari, 2005; Choi, 2006) for simulation of Single Carrier systems. Each CIR has its own properties. The channels are shown in Table 2.1.

Table 2.1 Channel Impulse response of various channels

Channel No. Channel Impulse Response (hi)

Normalized Channel Impulse Response (hn)

1 [1 0 0 ] 1 0 0

2(Proakis 1998) (1.5)-1/2 [0.5 1 0.5] 0.4082 0.8165 0.4082 3(Choi , 2007) (19)-1/2[1 2 3 2 1] 0.2294 0.4588 0.6882

0.4588 0.2294 4(Porat , 1991;

Choi, 2007) (2)-1/2 [0.167 0.667 1 0.667

0.167] 0.1197 0.4782 0.7169

0.4782 0.1197 5(Porat , 1991;

Choi, 2007)

(2)-1/2 [-0.167 0.667 1 0.667 - 0.167]

-0.1197 0.4782 0.7169 0.4782 -0.1197

6(Kumaran, 2005) (2)-1/2[1 1] 0.7071 0.7071

7(Kumaran, 2005) (2)-1/2[1 0 1] 0.7071 0 0.7071

Table 2.1 is a collection of channels which may have spectral nulls of various widths to reflect the deep fades. These channels are considered to show the effect of

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ISI. The above table also has channels with various lengths, which helps in demonstrating the fact that ISI and its effects do not depend on the channel length but on the characteristics. Channel 1 is an ISI free channel and it is used for comparison purposes. One can model any fading channel in a similar way.

The characteristics of each channel can be analyzed in terms of magnitude, phase, impulse response or by plotting Pole/Zero locations. Each of the above analysis techniques is used in the design considerations. For example, in Pole/Zero plot of a channel characterization analysis, the location of zeros acts as the constraint in the MLSD scheme which utilizes Viterbi Algorithm (Lee & Messerschimitt, 1994) and it can also check for the physical realizability of the channel model. The impulse response technique of channel characterization shows the extent of propagation delay, path loss and Doppler effects. In the magnitude response technique of channel characterization, the number of signal replicas and their arrival times can be determined from the shape of the magnitude and phase response (Asrar, 2003).

Each channel is to be normalized as explained below. Normalization of the channel is done before simulation and it is done by dividing each element in the CIR by the sum of squares of all the elements. The normalization procedure is given as Equation 2.11 (Anderson, 1997):

=

= n

i i in i

h h h

1 2

(2.11)

By normalizing, the energy of the channel is placed on the average symbol energy and the ISI property of the channel separately in h(t) (Anderson, 1997). By this arrangement the energy received is the same as transmitted when the filter (channel) energy is normalized. The ISI loss is contained in the distance to the decision boundary between the symbols which are having separate boundaries.

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20 2.3.1 Channel 1: [1 0 0]

This Channel 1 is typically a non-ISI channel. It does not introduce ISI and has a flat magnitude response. The magnitude and impulse responses are plotted in Figure 2.4. These findings are well explained in Proakis & Salehi (2002) and Ghani (2004).

Figure 2.4 Magnitude and impulse response of Channel 1

The phase response is also linear. The symbols, which pass through this channel do not get affected and remain unaltered. The main limitation of this channel is the AWGN alone. The performance of a communication system for this channel gives the upper limit (Haykin, 1997).

2.3.2 Channel 2: (1.5)-1/2 [0.5 1 0.5]

The magnitude response seen from Figure 2.5 of Channel 2 shows the severe amplitude distortion.

0 0.2 0.4 0.6 0.8

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Magnitude Response (dB)

0 1 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Samples

Amplitude

Impulse Response

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21

Figure 2.5 Magnitude and phase response of Channel 2

There is a spectral null and the severity of the amplitude distortion at the spectral null is also very high (see Figure 2.5). The implementation of a linear equalizer in the communication system is not suitable for this type of channel. The phase response as shown in Figure 2.5 is linear and this shows that there is no phase distortion introduced by this channel. This channel is used often in many literatures, for example see Ghani (2004) and Proakis & Salehi (2002).

The impulse response plotted as Figure 2.6 shows that the span of ISI is small. However, this does not relate to the severity of the distortion (Proakis &

Salehi, 2002). The coefficients of the impulse response also give an idea about the amplitude distortion. The locations of poles and zeros are shown in Figure 2.6. There are two zeros which lie on the unit circle.

0 0.2 0.4 0.6 0.8

-60 -50 -40 -30 -20 -10 0 10

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Magnitude Response (dB)

0 0.2 0.4 0.6 0.8

-3 -2.5 -2 -1.5 -1 -0.5 0

Normalized Frequency (×π rad/sample)

Phase (radians)

Phase Response

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22

Figure 2.6 Impulse response and Pole/Zero Plot of Channel 2

2.3.3 Channel 3: (19)-1/2[1 2 3 2 1]

The magnitude and phase response of Channel 3 are shown in Figure 2.7. The magnitude response shows a severe spectral null. The amplitude distortion is also very high. These findings are well explained in Proakis & Salehi (2002) and Ghani (2004). However, the phase response is linear and the channel does not introduce phase distortion.

Figure 2.7 Magnitude and phase response of Channel 3

The impulse response and Pole/Zero plots are shown in Figure 2.8. The impulse response shows that the span of the ISI is wide. The coefficient of the

1 2 3

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Samples

Amplitude

Impulse Response

-1 -0.5 0 0.5 1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Real Part

Imaginary Part

2

Pole/Zero Plot

0 0.2 0.4 0.6 0.8

-60 -50 -40 -30 -20 -10 0 10

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Magnitude Response (dB)

0 0.2 0.4 0.6 0.8

-6 -5 -4 -3 -2 -1 0

Normalized Frequency (×π rad/sample)

Phase (radians)

Phase Response

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impulse response shows the magnitude of the distortions. The Pole/Zero plot shows that there are two zeros located on the unit circle.

Figure 2.8 Impulse response and Pole/Zero plot of Channel 3

2.3.4 Channel 4: (2)-1/2 [0.167 0.667 1 0.667 0.167]

The magnitude and phase response of Channel 4 is shown in Figure 2.9.

These findings are well explained in Proakis & Salehi (2002) and Ghani (2004). The magnitude response shows a spectral null. The amplitude distortion is severe for this channel. The phase response is linear and the channel does not introduce phase distortion.

Figure 2.9 Magnitude and phase response of Channel 4

0 1 2 3 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Samples

Amplitude

Impulse Response

-1 -0.5 0 0.5 1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Real Part

Imaginary Part

2

2 4

Pole/Zero Plot

Zero Pole

0 0.2 0.4 0.6 0.8

-70 -60 -50 -40 -30 -20 -10 0

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Magnitude Response (dB)

0 0.2 0.4 0.6 0.8

-6 -5 -4 -3 -2 -1 0

Normalized Frequency (×π rad/sample)

Phase (radians)

Phase Response

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