Master Thesis Good

8/2/2019 Master Thesis Good

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Rateless Coding for OFDM-Based

Wireless Communications Systems

IQBAL HUSSAIN

Masters Degree Project

Stockholm, Sweden 2009


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To my Late brotherNawab Khan........


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Acknowledgments

First of all I would like to thank my supervising tutor Prof. Lars Kildehoj

Rasmussen at KTH for his abundant help and prolific suggestions. He made

himself readily available, had a patient ear, and always took the time to answer my

questions thoroughly sometimes not related to thesis. My deep appreciation and

heartfelt gratitude also goes to my thesis supervisor at LTH, Fredrik Tufvesson

for his support, patience and guidance. Both of my supervisors also reviewed the

whole thesis report very carefully for even the delicate specifics for which I am

very thankful to them. My special thanks go to Professor Mikael Skoglund for

allowing me to do my master thesis in communication Theory department KTH.

The atmosphere has always been a perfect source of motivation. My stay at

communication theory department was so nice that I included the possibility to

work in this department in my future plans. I have to mention nice time out

and discussions with communication theory department staff. I would also like

to thank Johannes Karlsson for his devotion to simulator computers which made

the simulation so flexible and extendible for me.

Last but not least, I would like to direct my warmest thanks to programme

secretary Pia Bruhn of master in wireless communications at Lund University

who was so generous in her support at various stages of my study in Sweden.

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Abstract

Performance of broadband wireless communication networks is limited by

available resources such as frequency bandwidth and transmission power. Also,

the time-varying features of wireless communication channels adversely affect

performance. Transmission schemes, adapting to instantaneous channel charac-

teristics can significantly improve performance. The block-fading channel is a

good model for OFDM-based wireless communications systems in which the fad-

ing occurs in block wise manner. Raptor code is new emerging rateless code whichhas shown amazing performance over variety of channels. There is a constraint

on the interleaving depth of OFDM-based system due to delay and maximum

packet size. This non-ideal interleaving affects the maximum achievable diversity

from the channel. We investigated the effect of correlation between fading blocks,

which relates to the limited interleaving possible between carriers in an OFDM

system, based on Raptor code. We investigated the performance of the Rap-

tor code over correlated slowly fading channels and compared it with half rate

standard (3,6) regular LDPC code, and to the ARQ systems using punctured

LDPC codes for short block length. We also compared the performance of the

Raptor code to the standard (3,6) regular LDPC code over Binary Erasure chan-

nel, Additive White Gaussian Noise channel, and fast fading Rayleigh chanel for

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Abstract

short block length. Enormous simulations are performed to get insight for future

research.

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Contents

Acknowledgments iii

Abstract v

Contents vii

List of Figures x

Acronyms xiii

1 Introduction 3

1.1 Background Discussion . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Channel Coding . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.4 Automatic Repeat Request (ARQ) . . . . . . . . . . . . . 6

1.1.5 Rate-Compatible (RC) Codes . . . . . . . . . . . . . . . . 7

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Linear Block Codes . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Codes on Graph . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.4 Rateless Codes . . . . . . . . . . . . . . . . . . . . . . . . 10

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Contents

1.2.5 Rateless code vs Fixed-rate code . . . . . . . . . . . . . . 12

1.2.6 Rateless code vs Automatic Repeat Request (ARQ) . . . . 13

1.3 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 System Model 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Channel Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Binary Erasure Channel . . . . . . . . . . . . . . . . . . . 24

2.2.2 Binary Symmetrical Channel . . . . . . . . . . . . . . . . 26

2.2.3 AWGN Channel . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.4 Fast Rayleigh Fading Channel . . . . . . . . . . . . . . . . 28

2.2.5 Block Rayleigh Fading Channel Model . . . . . . . . . . . 29

2.2.6 Correlated Block Rayleigh Fading Channel Model . . . . . 30

2.3 OFDM-based Wireless Communication systems . . . . . . . . . . 32

2.3.1 Multicarrier Modulation . . . . . . . . . . . . . . . . . . . 32

2.3.2 OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.3 Interleaving in OFDM System . . . . . . . . . . . . . . . 34

2.4 Interleaved-OFDM system as Block Fading Channel . . . . . . . . 36

3 Low Density Parity Check Codes 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Regular LDPC Codes . . . . . . . . . . . . . . . . . . . . . 40

3.1.2 Irregular LDPC Codes . . . . . . . . . . . . . . . . . . . . 41

3.1.3 Graphical Representation of LDPC Codes . . . . . . . . . 41

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Contents

3.1.4 Decoding Complexity . . . . . . . . . . . . . . . . . . . . . 43

3.2 Iterative Decoding of LDPC Codes . . . . . . . . . . . . . . . . . 43

3.3 LDPC Decoding for BEC . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Simulation Results and Conclusions . . . . . . . . . . . . . 45

3.4 LDPC Decoding for Binary Symmetrical Channel . . . . . . . . . 47

3.4.1 Soft Decision Message Passing Decoder for Binary Sym-

metrical Channel . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Soft Decision Iterative Decoder for BPSK AWGN Channel . . . . 52

3.5.1 Simulation Results and Conclusions for AWGN Channel . 58

3.6 Simulation Results and Conclusions for Rayleigh Fading Channel 60

3.6.1 Simulation Results and Conclusions for Block Fading Rayleigh

Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Digital Fountain Codes 63

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 LT Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 LT Encoding Process . . . . . . . . . . . . . . . . . . . . . 65

4.2.2 LT Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.3 Tanner Graph for LT Code . . . . . . . . . . . . . . . . . 68

4.3 Design of LT Degree Distribution . . . . . . . . . . . . . . . . . . 69

4.3.1 The Robust Soliton Distribution . . . . . . . . . . . . . . . 70

4.4 Simulations and Conclusions of LT code on BEC . . . . . . . . . 72

4.5 Raptor Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5.1 Tanner Graph and Construction of Raptor Code . . . . . . 75

4.6 BP Algorithm of Raptor code over Noisy Channels . . . . . . . . 78

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Contents

4.7 Capacity achieving Raptor Code for Noisy Channels . . . . . . . . 80

4.8 Simulations of Raptor code . . . . . . . . . . . . . . . . . . . . . 82

4.8.1 Simulations for AWGN Channel . . . . . . . . . . . . . . . 82

4.8.2 Simulations for Rayleigh Fading Channel . . . . . . . . . . 85

4.8.3 Simulations for 2-Block Fading Channel . . . . . . . . . . 86

5 Simulations Results 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Simulations of Raptor Code vs LDPC in BEC . . . . . . . . . . . 91

5.3 Raptor code vs LDPC code in AWGN Channel . . . . . . . . . . 94

5.4 Performance comparison of Raptor code and LDPC code in fast-

fading Rayleigh Channel . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Simulations and Conclusions of Raptor code for 2-Block Fading

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6 Simulations and Conclusions of Raptor code for correlated Block

Fading Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6.1 Raptor code vs LDPC in correlated Block Fading Channel 101

5.6.2 Raptor code vs punctured LDPC in correlated Block Fading

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Conclusions 105

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A Probability Calculations 109

Bibliography 111

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List of Figures

2.1.1 Communication System Model . . . . . . . . . . . . . . . . . . . . 22

2.2.1 The Binary Erasure Channel . . . . . . . . . . . . . . . . . . . . 25

2.2.2 The Binary Symetric Channel . . . . . . . . . . . . . . . . . . . . 26

2.2.3 The Additive White Gaussian Noise . . . . . . . . . . . . . . . . 27

2.2.4 Block Fading Channel code word representation . . . . . . . . . . 29

2.2.5 Model of Correlated Block Fading Channel . . . . . . . . . . . . . 31

2.3.1 Model of OFDM System . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Tanner Graph of LDPC Code . . . . . . . . . . . . . . . . . . . . 42

3.3.1 LDPC vs No LDPC in BEC . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Code word length comparison in BEC . . . . . . . . . . . . . . . . 47

3.4.1 General Message Updates . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Message updates during first iteration. . . . . . . . . . . . . . . . 54

3.5.2 Update message for variable nodes . . . . . . . . . . . . . . . . . 56

3.5.3 System Block diagram for simulation. . . . . . . . . . . . . . . . . 58

3.5.4 LDPC in AWGN with different code word length . . . . . . . . . 59

3.6.1 LDPC in Rayleigh Fading Channel . . . . . . . . . . . . . . . . . 61

3.6.2 LDPC in Block Fading Channel . . . . . . . . . . . . . . . . . . . 62

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List of Figures

4.2.1 Bipartite Graph of LT code . . . . . . . . . . . . . . . . . . . . . 66

4.2.2 Tanner Graph of (a) LDPC code(b) LT code . . . . . . . . . . . 68

4.4.1 LT code in BEC channel . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 Tanner Graph of Raptor Code . . . . . . . . . . . . . . . . . . . 76

4.5.2 Graphical representation of Raptor code . . . . . . . . . . . . . . 77

4.8.1 System Block Diagram of Raptor code . . . . . . . . . . . . . . . 83

4.8.2 Raptor code in AWGN . . . . . . . . . . . . . . . . . . . . . . . . 84

4.8.3 Raptor code in Rayleigh Fading Channel . . . . . . . . . . . . . . 86

4.8.4 Raptor code in Block Fading Channel . . . . . . . . . . . . . . . . 87

5.2.1 Raptor code vs LDPC in BEC . . . . . . . . . . . . . . . . . . . . 92

5.3.1 Raptor code vs LDPC in AWGN Channel . . . . . . . . . . . . . 93

5.4.1 Raptor code vs LDPC in Rayleigh Fading Channel . . . . . . . . 95

5.5.1 Raptor code vs LDPC in BFC using AWGN deg. dest. . . . . . . 96

5.5.2 Raptor code vs LDPC in BFC using BEC Deg. Dist. . . . . . . . 97

5.6.1 Raptor code in correlated block Fading Channel . . . . . . . . . . 99

5.6.2 Effect of correlation on rate of Raptor code . . . . . . . . . . . . . 100

5.6.3 Raptor code vs LDPC in correlated BF Channel . . . . . . . . . . 101

5.6.4 Raptor code vs Punctured LDPC code . . . . . . . . . . . . . . . 102

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Acronyms

DMC Discrete Memoryless ChannelISI Inter Symbol Interference

BEC Binary Erasure Channel

BSC Binary Symmetrical Channel

LDPC Low Density Parity Check

FEC Forward Error Correction

BPSK Binary Phase Shift Keying

AWGN Additive White Guassian Noise

LLR Log-Likelihood Ratio

LT Luby TransformBIAWGN Binary Input Additive White Gaussian Noise

BIMSC Binary Input Memoryless Symmetrical Channel

SNR Signal to Noise Ratio

BER Bit Error Rate

i.i.d Independent Identically Distributed

BIAWGN Binary Input Additive White Gaussian Noise

ARQ Automatic Repeat Request

RMS Root Mean Square

HARQ Hybrid ARQ

RC Rate Compatible

DVB Digital Video Broadcasting

OFDM Orthogonal Frequency Division Multiplexing

WLAN Wireless Local Area Network

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Chapter 1

Introduction

1.1 Background Discussion

1.1.1 Overview

Digital communication has had promising impacts on our lives from the last

few decades. Its practical applications in satellite, military, internet, sea and

space communications, digital audio and video broadcasting and mobile commu-nications have brought a revolution in our society. The reliable transmission of

information over noisy channels is one of the basic requirements of digital infor-

mation and communication systems. Reliable transmission over communication

channel has been the subject of much research for many years. Channel coding is

viable method to ensure reliable communication by introducing redundancy to the

information to be transmitted. Channel coding transforms signal constellation

points to a higher dimensional signalling space. Due to this higher dimensional

space, the distance between constellation points increases and hence enhances bit

error detection and correction. Channel coding can be classified in two categories

as automatic repeat request (ARQ) scheme and forward-error correction (FEC)

scheme. ARQ combines error detection and retransmission strategies to ensure

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1. Introduction

that data is delivered accurately despite the occurrence of errors during transmis-

sion. On the other hand, FEC tries to correct errors at the receiver end. ARQ

schemes require to send a small number of redundant information along with user

information, which can be used to detect errors during transmission, while FEC

schemes require redundant information to detect and correct errors. To minimize

the number of redundant information bits and achieve high error-correction ca-

pability, on can use powerful powerful error-correcting codes. LDPC codes were

introduced by Gallager [7], which have shown better performance over a variety of

channels. Finite-length LDPC codes have also been shown to outperform turbo

codes. Rateless codes such as LT codes [11] and Raptor codes [14] are the class

of digital fountain codes [10], having capacity achieving property. This chapter

introduces the basic background and the literature review on the works related

to the thesis. Furthermore, the main objectives of the thesis are also presented

1.1.2 Channel Coding

The shannon channel theorem [20], states the maximum rate at which infor-

mation can be transmitted reliably over a given communication channel with a

specified bandwidth. The maximum rate is called channel capacity. The capacity

of a bandlimited AWGN channel with bandwidth B, is given by

C = B log 2(1 + Es/N0), bits per second (bps)

The Shannon theorem shows that if information is transmitted with rate equal

to capacity or less, then there exists a coding technique which allows the proba-

bility of error at the receiver to be made arbitrarily small. The converse is also

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Background Discussion 1.1

important that if the information rate is greater than the capacity of the chan-

nel, then there exist no coding technique which makes probability close to zero.

However Shannon theorem did not give any clue about the construction of the

coding scheme.

Forward Error Correction code uses redundancy to correct transmission errors

at the receiver and hence no feedback channel is required. FEC offers constant bit

throughput and varying reliability depending on channel condition. Conventional

FEC codes reduce the required transmit power for a given BER at the expense

of increased signal bandwidth or a reduced data rate. In ARQ schemes a small

amount redundant bits are added to detect transmission errors. Retransmission

will take place if and only if errors are detected at the receiver end, hence needs

a feedback channel. As contrast to FEC, ARQ offers constant reliability and

varying bit throughput.

There are two main types of conventional FEC codes which are named as block

codes and convolutional codes. Block codes accept k information bits and produce

n encoded bits by adding (n-k) redundant bits. On the other hand, convolutional

codes transform k information bits to n encoded bits in serial manner and this

transformation depends on the current as well as L last information bits, where L

is the constraint length of the code.Trellis codes combine channel code design and

modulation to reduce the BER without bandwidth expansion or rate reduction.

Recent advancements in coding technology, such as LDPC codes [7] and Ratless

codes [11, 14] offer performance that approaches the channel capacity of AWGN

and fading channels for larger block length.

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1. Introduction

1.1.3 OFDM Systems

Orthogonal Frequency Division Multiplexing is based on multi-carrier modu-

lation technique which is more immune to frequency selective fading than single

carrier systems. In OFDM, the data stream is distributed over a number of lower

rate streams and these streams are modulated over different carriers. Lower data

rate of each stream in OFDM system as compared to the single carrier system

increases the symbol duration and hence reduces the effects of multipath propa-

gation. Inter symbol interference can be removed by using cyclic prefix which is

copying the last part of a symbol at the start of a symbol. By using orthogonal

transmit signal and cyclic prefix of at least equal to maximum delay spread avoids

not only inter symbol interference but also inter-carrier interference. As channel

is divided in parallel subchannels in OFDM system, so channel equalization is

simpler than the adaptive equalization used in single carrier system. Drawbacks

of OFDM system as compared to the single carrier system is its sensitivity to fre-

quency offset and phase noise. Also there is possible limited interleaving present

among the subcarrier of OFDM system which can severely affect the performance.

1.1.4 Automatic Repeat Request (ARQ)

The Automatic Repeat Request (ARQ) is a control mechanism which com-

bines error detection and retransmission schemes to ensure reliable transmission.

A back-channel is required for acknowledgement and non-acknowledgement mes-

sages for lost/wrong data frames at the receiver. ARQ works well for many

one-to-one protocols such as TCP/IP protocol, but its performance seriously de-

grades for broadcast applications. In broadcast applications, data frames may be

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Background Discussion 1.1

retransmitted even if they are received by many receivers. In these cases, data

receivers may ask for retransmitting the data which are already received by other

receivers. Consequently, the data source needs to retransmit most of the data

and hence inefficiently uses valued bandwidth. Furthermore in ARQ, most of the

time source will be idle if the distance between source and destination is too long.

In Hybrid ARQ, ARQ and error-correction schemes are combined to reduce

retransmission. When the packet arrives at the receiver, it is first decoded by

the FEC decoder and then checked for errors. If errors are detected, a retrans-

mission request for the corresponding data frame is sent back to the transmitter.

In applications with fluctuating channel conditions such as satellite packet trans-

mission and mobile communication, incremental redundancy (IR) HARQ schemes

demonstrate higher throughput efficiency by adopting their error correcting code

redundancy to different channel conditions.

1.1.5 Rate-Compatible (RC) Codes

Adaptive code rate is always desirable especially in the varying channel en-

vironment. By using puncturing and extending properties to the conventional

forward error-correction coding schemes, we can achieve flexible code rates. Punc-

tured codes can be achieved by deleting parity bits to get wide range of higher

rates than the code rate of mother code. Higher bandwidth efficiency can be

achieved in punctured codes at the expense of degraded performance. Extended

codes can be achieved by appending more parity bits to the mother code. By

extending a code we can get only lower code rate than the code rate of the

mother code. Extending leads to codes of increased minimum distance and im-

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1. Introduction

proved performance.The addition of rate-compatibility property to puncturing

and extending further enhances the performance of Rate-compatible codes in

time varying channels. The limitation in rate-compatible code is that the coded

bits of a high-rate punctured code are also used by the lower-rate codes. In other

words, the high-rate codes are embedded into the lower-rate codes of the family.

If the higher rate codes are not sufficiently powerful to decode channel errors,

only small amount of extra bits which were previously punctured (deleted) have

to be transmitted in order to improve the code performance.

1.2 Literature Survey

1.2.1 Linear Block Codes

A binary block code generates a block of n coded bits from k information

bits. The number of redundant bits added to every k information bits is (n k).A code is linear if the addition of any two valid codewords results in another

valid codeword. A systematic block code is also specified by its generator matrix

G = [IkP], where Ik is (k k) identity matrix and P is a k (n k) matrix that

determines the (n k) parity check bits. The systematic block code can also be

specified by a parity-check matrix H of the form H = [PTInk], where PT is the

transpose of the matrix P. Obviously, n is larger than k in general and the code

efficiency is usually evaluated by k/n, which is called the code rate of the coding

algorithm. A (n, k) parity-check code is a linear block code whose codewords

satisfy a set of (n k) linear parity-check equations. It is traditionally defined

by its (n k) n parity-check matrix H, whose (n k) rows specify the (n k)

equations.

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Literature Survey 1.2

1.2.2 Codes on Graph

The main purpose of coding theory has been to design the codes which can

achieve Shannon capacity [20]. Convolution codes were designed to approach the

Shannon limit within a gap of few decibels with reasonable decoding complexity.

However, reducing this gap required impractical complexity until the discovery

of the iterative message-passing algorithms. Using an iterative message-passing

decoder, low density parity check and turbo codes have provided excellent perfor-

mance and a small gap to the Shannon limit with a practical decoding complexity.

This better performance of message-passing algorithms based codes drew a lot of

attention to this field of study, which soon extended to a more general class of

codes called codes defined on graphs.

Graphs are used to visualize the constraints of the codes. The advantage

of codes defined on graphs is that they can be decoded using message-passing

algorithms. Degree of the vertices determine decoding complexity while girth

and diameter provides the qualitative analysis of the message-passing algorithms

[4]. Other communication components can also be modelled in the graph of codes.

Since the rediscovery of LDPC codes, there has been a lot of research activities

and improvements in the area of codes defined on graphs [3]. Research on LDPC

codes has played a major role in this field, as many of the new classes of codes

which are defined on graphs are influenced by the structure of LDPC codes.

1.2.3 LDPC Codes

LDPC Code, which is based on bi-partite graph, was first proposed by Gallager

[7] in the early 1960s, but it did not get proper attentions until years later.

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1. Introduction

LDPC codes are also called as Gallager codes, in honor of Robert G.Gallager,

who proposed the concept of LDPC in his phD thesis. The bi-partite graph

contains two sets of nodes which are called as variable nodes and check nodes.

The bi-partite graph is build in such a way that for each check node, the module-

2 sum of the values of its incident variable nodes is equal to zero. There are

various methods for LDPC code, having encoding algorithm which can run in

linear time. The most efficient decoding algorithm for LDPC codes is the belief-

propagation (BP) algorithm. During each iteration, BP algorithm updates the

probability that a variable node is zero based on the information obtained from

the check nodes in the previous round. The time complexity of the decoding is

proportional to the number of edges in the bipartite graph. LDPC code requires

a small reception overhead at the receivers to reconstruct the message symbols.

It has been proved that the performance of LDPC codes which is obtained from

an appropriate highly irregular bipartite graph rather than a regular graph is

very close to the Shannon bounds. LDPC codes are also widely used in many

applications, e.g. in the DVB technology. The message-passing algorithm for

LDPC codes in Binary Erasure Channel has been studied in [5]. Similarly based

on [8], the results of LDPC codes in Additive White Gaussian Noise and correlated

and uncorrelated fast Rayleigh fading channels have been demonstrated in [6].

LDPC codes are used in DVB-S2, WiMax and 10GBase-T Ethernet.

1.2.4 Rateless Codes

The traditional block codes have several problems in the real communication

systems. For example, in the systems where the channel error rate is unknown

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by the encoder or the decoder before coding, the traditional codes which have

fixed code rates, have to estimate the loss rate and choose the closest code rate

to adapt the channel conditions. However, these coding algorithms are nearly

useless if the estimated rate cannot be estimated correctly or can be changed

during the transmission. Communication methods used to communicate over

such channel usually employ feedback from receiving end to transmitter. These

feedback messages are either acknowledgments for missing packets or for every

packet. In case of acknowledgments for missing packets, transmitter only retrans-

mit the missing packets while in other case transmitter retransmit packets with

negative acknowledgment. However, from Shannon theory [2, 20], it can be easily

concluded that these protocols are inefficient and wasteful of bandwidth because

if the channel has erasure probability, then its capacity will be (1 ), whether

we will use feedback or not. This inefficient utilization of bandwidth becomes

more predominant when a data source wants to make broadcasting on a network

which consists of many channels with different loss rates. When using the fixed

rate codes, sender is forced to generate code words based on the worst code rate

to ensure reliable transmission. Thus, a new family of FEC codes which is robust

are proposed to address these problems.

M.Luby in [11] presented the first practical realization of the rateless code

and performance was further improved in [14] by Amin Shokrollahi. Digital

Fountain Codes are also called universal erasure code because it can be used

independently of channel loss rate and showed awe-inspiring performance over

every erasure channel [11]. As LT code has excellent performance in Binary

Erasure channel [11] but exhibits error floor in fading channels and impractical

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1. Introduction

decoding complexity for long block codes [12]. Raptor code [14] is concatenation

of the outer code with LT code to combat the problem of error floor in fading

channels and to provide linear time encoding and decoding complexity. Raptor

code has demonstrated the capacity approaching property for the binary erasure

channel [14]. Raptor code has not only beat LT code but also has shown near

optimal performance on wide variety of channels. Raptor code has also provided

amazing performance in AWGN channel [12, 15], and Rayleigh fading channels

[19] for larger block length.

1.2.5 Rateless code vs Fixed-rate code

Digital Fountain Code is considered as rateless, which means, unlike the

traditional block codes such as LDPC codes and RS codes, Digital Fountain

Code does not have a fixed code rate and rate is determined by the number of

transmitted codeword symbols required before the decoder is able to decode. The

rate is then not known a priori as it is in traditional fixed-rate block codes. It

can generate as many codeword symbols as needed to recover all the message

bits regardless of the channel performances. The first practical realization of the

rateless codes was presented by M.Luby in [11] and was further improved in [14].

Digital Fountain Codes also called as universal erasure code because they can be

used independently of channel loss rate and having good performance over every

erasure channel [11].

Our motivation is to use rateless codes as compared to the fixed-rate code

due its advantages in different requirements. In fixed-rate codes the rate is fixed

and independent of realizations of the channel, so there is a tradeoff between

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efficiency and reliability. Fixed-rate codes achieve high rate transmission at the

expense of reliability and vice versa, while in rateless codes, codeword length is

determined by the channel realization offering higher efficiency and reliability as

compared to fixed-rate codes. To achieve best possible code rate, fixed-rate codes

required channel information at the transmitter which is some time difficult to

achieve due to limited bandwidth of feedback channel, while in rateless codes

efficient transmission is achieved irrespective of the channel information at the

transmitter. For large relay networks the feedback information of fixed-rate codes

contributes significantly as compared to the rateless codes, so rateless code is a

good candidate for the relay networks.

1.2.6 Rateless code vs Automatic Repeat Request (ARQ)

Rateless code has become a natural candidate for the Hybrid ARQ. In con-

ventional Incremental Redundancy IR-HARQ protocol initially the transmitter

sends only as many codeword symbols as necessary to ensure a high probability

of successful transmission. If the decoding fails, the receiver sends a negative

acknowledgement and the channel information to the transmitter. Taking into

account the channel information of the past transmission(s), the transmitter sends

only as many additional codeword symbols as necessary to insure a high proba-

bility of successful transmission. Similarly in rateless codes transmitter send code

word symbols to the receiver unless it get a feedback message from the receiver

about successful transmission. Both rateless codes and codes based on IR-HARQ

schemes adjust its transmission according to the channel conditions.

Encoding and Decoding complexities of rateless schemes are higher than IR-

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1. Introduction

HARQ schemes. However, when Raptor codes are used, we only need to encode

as many parity bits as we need to send in the initial transmission or in subsequent

re-transmission(s). On the other hand, for punctured codes based on IR-HARQ

schemes, we need to encode all parity bits, even though we may need to send only

a small portion of them. In conventional IR-HARQ schemes a feedback channel

is required for frequent and significant feedback messages, while for rateless this

is not required for one time single bit feedback message for a given number of

information bits.

1.3 Thesis Motivation

Adaptive modulation and coding enables robust and bandwidth-efficient trans-

mission over time-varying channels. Modulation and coding techniques that do

not adapt to fading conditions need an acceptable reliability when the channel

quality is not so good. Thus, these fixed -rate systems are effectively designed

for the worst-case channel conditions. Adapting to the channel fading can in-

crease average throughput, reduce required transmit power, or reduce average

probability of bit error by taking advantage of good quality channel conditions.

In rateless coding techniques transmission of information will take place at high

efficiency and reliability as compared to fixed-rate coding.

OFDM is finding use in practical applications in Digital Audio Broadcasting

(DAB) and Terrestrial Digital Video Broadcasting (DVB-T) in Europe, wireless

networking and broad band internet access. Wireless Local Area Networks use

OFDM as their physical layer transmission technique. The European standard

is ETSI HiperLAN/2, American standard is IEEE 802.11a/g , HiSWAN is high-

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Thesis Motivation 1.3

speed wireless local area network standard set by Association of Radio industries

and Business Japan ; all of which has similar physical layer specifications based

on OFDM. OFDM is also a strong candidate for IEEE Wireless Personal Area

Network (WPAN) standard and for fourth generation (4G) cellular systems.

Real-time applications actually need short block length which is less than a

few thousands. This limitation may be due to memory and buffer size and delay

in transmission. As GSM frame duration is 4.615 ms which corresponds to the 8

user data and data rate on radio link interface is 270.83 Kbps which corresponds

to the frame length of 1250 bits. The standard of the multimedia broadcast and

multiuser services of 3G wireless networks works for small code word length up to

500 bits [25]. WiMAX has quite flexible physical layer frame duration varies from

2 ms to 20 ms, however 5 ms is more conveniently used. WiMAX using channel

bandwidth 1.25 MHz and QPSK modulation having aggregate uplink data rate

154 kpbs with 5 ms frame length and 128 OFDM symbols which corresponds to

770 bits frame length. The basic transmission element in the FDD physical layer

of UMTS is a radio frame. A radio frame has duration of 10 ms, and it is broken

down into 15 time slots of 0.667 ms each. Each time slot contains 2,560 chips

which corresponds from 10 to 640 bits depending on spreading factor. Frame

length of Digital Audio Broadcasting based on MPEG-1 is 24 ms and data rates

are between 32 and 192 kpbs for a single channel. In this case Digital Audio

Broadcasting has a frame length between 768 and 4068 bits.So codes with small

frame length have significant role in present and future wireless communication

technologies.

Rateless codes are used at transport layer or network layer but many of the

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1. Introduction

existing system uses fairly small block length at physical layer. So our motivation

is to investigate the performance of Raptor code for shorter block length because

for larger block length, rateless code has already outperformed different codes

over variety of channels.

Performance of broadband wireless communication networks is limited by

available resources such as frequency bandwidth and transmission power. Various

phenomena such as multipath propagation, terminal mobility and users interfer-

ence, result in channels with time-varying parameters. These time-varying fea-

tures of wireless communication channels severely affect performance. So there

is a need to develop practical adaptive transmission schemes for OFDM-based

wireless communications systems. The block-fading channel is a good model for

such systems, so we want to investigate the use of rateless codes [11, 14] which

is new code family and ideally suitable for the multi-channel data transmission

environment [22] over such channels. We decided to investigate the effect of cor-

relation between fading blocks, which relates to the limited interleaving possible

between carriers in an OFDM system.

Interleaving is used to mitigate the the burst error effect of channel in com-

munication over a fading channel. For large coherence time or equivalently low

doppler spread of the fading, high interleaving is required to break the memory

of the channel, which is not affordable in some communication system due to

some constraints. OFDM system recommended by IEEE.802.11, or in other ap-

plications like GPRS, there is a constraint on the interleaving depth due to the

maximum allowable packet size or processing delay.

Thesis Objective:

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Thesis Contributions 1.4

The overall objective is to investigate the error rate performance of Raptor

code with short block length over uncorrelated and correlated slowly fading chan-

nels and compare it with fixed rate standard (3,6) regular LDPC code.

In order to be able to reach the proposed objective, we used the following proce-

dure:

We investigated (3,6) rate 1/2 regular LDPC code over the binary erasure

channel (BEC).

We analyzed (3,6) rate 1/2 regular LDPC code over the additive white

Gaussian noise channel (AWGN).

We compared the performance of the (3,6) rate 1/2 regular LDPC code

over the Rayleigh block-fading channel with additive white Gaussian noise

(BF-AWGN) and AWGN only channel.

We thoroughly investigated the Luby Transform (LT) codes over the BEC.

By simulations we investigated Raptor codes over the BEC.

Finally Investigation of Raptor codes over the AWGN and over BF-AWGN

was carried out.

1.4 Thesis Contributions

This thesis has contributed to field of rateless coding in different ways. Along

with LDPC codes, rateless codes are natural candidate for use in HARQ schemes

used in applications with fluctuating channel conditions. Raptor code is the class

for Fountain codes, designed for reliable transmission of data over an erasure

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1. Introduction

channel with unknown erasure probability. The main contribution of this master

thesis can be summarized as follows:

We compared the performance of Raptor code and half rate standard (3,6)

regular LDPC code over binary erasure channel for short block length and

by simulations showed that Raptor code outperforms the corresponding

LDPC code.

We showed by simulations that degree distribution optimized for binary

erasure channel outperforms the degree distribution optimized for AWGN

channel over fast fading Rayleigh, uncorrelated and correlated block fading

channels for short block length.

We demonstrated the comparison of Raptor code and half rate standard

(3,6) LDPC code over AWGN channel for short block length.

We compared the performance of Raptor code and half rate standard (3,6)

LDPC code over fast fading Rayleigh channel and block fading channel.

We presented the performance of the Raptor code in correlated block fading

channel and showed by simulations that correlation causes degradation in

performance and degradation is more sever at lower rates.

We compared the performance of the Raptor code to half rate standard (3,6)

regular LDPC code in correlated block fading channel based on the degree

distribution optimized for BEC and showed by simulations that Raptor

code experience more degradation specially at low correlation values.

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Thesis Organization 1.5

We presented the performance comparison of the Raptor code and punc-

tured LDPC in correlated block fading channel for short block length.

1.5 Thesis Organization

The rest of this thesis will be organized as follow.

In chapter 2 we presents the channel models, which are used in our project

for the performance comparison of the Raptor code and half rate standard (3,6)

LDPC code.

Chapter 3 discusses necessary background and structure of LDPC codes and

their decoding algorithms. Simulations results were provided at the end of LDPC

over Binary erasure, AWGN and Rayleigh fading channels for small code word

block length.

In chapter 4 , the concept of digital fountain codes [10] was introduced. More-over, the performance of the LT and Raptor codes were demonstrated for short

block length codes over binary erasure , AWGN , fast fading Rayleigh and uncor-

related block fading channels.

Chapter 5 presents simulation results to gauge difference in performance of

the Raptor code and half rate regular (3,6) LDPC code over binary erasure,

AWGN, fast fading channel and uncorrelated block fading channel. Chapter 5

also discusses the performance of the Raptor code in correlated block fading

channel and compare it with half rate standard(3,6) LDPC code for short block

length. In chapter 5 we also introduced the performance comparison of the Raptor

code and punctured LDPC code in correlated block fading channel for short block

length.

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1. Introduction

Finally, in Chapter 6 we present a summary of our work, conclusions, and

future directions for the continuation of this thesis.

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Chapter 2

System Model

2.1 Introduction

Shannon ground-breaking approach in his landmark paper [20] demonstrated

that, if the information or entropy rate is below the capacity of the channel,

then proper methods are available to encode information messages and it canbe be received with out errors even if the channel distorts the message during

transmission. Recent developments in coding theory, have design codes which

have performance very close to the channel capacity. The utilization of error

control coding has become an integral part of the modern communication system.

A typical Digital communication model is represented by block diagram as shown

in Figure 2.1.1. This model is suitable from coding theory and signal processing

point of view. Information is generated by source which may be human speech,

data source, video or a computer. This information is then transformed to electric

signals by source coder which are suitable for digital communication system.

To ensure reliable transmission over communication channel encoder introduce

redundancy to the user information.The modulator is a system component which

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2. System Model

Figure 2.1.1. Communication System Model

transforms the message to signal suitable for the transmission over channel.

The physical medium through which information is transmitted is called com-

munication channel. For example telephone lines in wired system and environ-

ment between the transmitter and receiver in wireless system. The channel model

is a mathematical model of the channel characteristics. During transmission, in-

formation is transmitted from source to sink through communication channel.

Error may arise from the channel noise, so encoder and decoder blocks must be

design to minimize the errors introduced by channel. The messages from the

source are binary sequences of length k bits. The encoder performs mapping of

the messages to the encoded words. so the code words are binary sequence of

length n bits but not all combinations of n bits are code words. There are 2k

code words out of 2n possible binary sequences.

After the signal has passed the channel it is distorted due to channel noise.

So the decoder must be designed in such a way to minimize the error between the

received code word and the transmitted code word. The design parameters of the

decoder are complexity, delay and rate. As in the absence of decoder, the receiver

takes decisions independently on each bit but as now with the introduction of

decoder, the receiver takes decision on n bits, so it needs a more complex system,

but in todays VLSI technology, the complexity is not a major issue. Also to

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Channel Modelling 2.2

wait for n bits introduces delay in the system which is very critical. As n is

greater than k, so this redundancy in the coded system is very important to

take into consideration because now the system attempts channel n times to

communicate k bits as compared to the un-coded system in which case the system

will communicate k bits in k attempts of the channel. But with the introduction

of coding we can reduced the transmit power for the same error rate as compared

to the un-coded system, or equivalently we can reduce the bit error rate for the

same transmit power. As high power radio frequency devices are more expensive

than the low power radio frequency devices which further enhance the importance

of the coding techniques.

2.2 Channel Modelling

The goal of wireless channel modeling is to find useful analytical models forthe variations in the channel. The most prominent draw back of the wireless com-

munications is channel fading. Various properties such as multipath propagation,

terminal mobility and user interference, result in channel with time-varying pa-

rameters. Fading of the wireless channel can be classified into large-scale and

small-scale fading. Large-scale fading involve the variation of the mean of the

received signal power over large distances relative to the signal wavelength. On

the other hand, small-scale fading involve the fluctuations of the received signal

power over distances comparable with the wavelength. Models for the large scale

variations are useful in cellular capacity-coverage optimization and analysis, and

in radio resource management such as handoff, admission control, and power con-

trol. Models for the small scale variations are more useful in the design of digital

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2. System Model

modulation and demodulation schemes that are robust to these variations. We

hence focus on the small scale variations in this class. Reflection, diffraction and

scattering in the communication channel causes rapid variations in the received

signal. The reflected signals arrive at different delays which cause random am-

plitude and phase of the received signals. This phenomenon is called multipath

fading. If the product of the root mean square (RMS) delay spread which is stan-

dard deviation of the delay spread and the signal bandwidth is much less than

unity, the channel is said to suffer from the flat fading.The relative motion be-

tween the transmitter and the receiver (or vice versa) causes the frequency of the

received signal to be shifted relative to that of the transmitted signal. The fre-

quency shift, or Doppler frequency, is proportional to the velocity of the receiver

and the frequency of the transmitted signal . A signal undergoes slow fading when

the bandwidth of the signal is much larger than the Doppler spread (defined as

a measure of the spectral broadening caused by the Doppler frequency). The

combination of the multipath fading with its time variations causes the received

signal to degrade severely. This degradation of the quality of the received signal

caused by fading needs to be compensated by various techniques such as diversity

and channel coding. In the forthcoming subsections we will briefly discuss a few

of standard channel models which we will frequently use in our simulations.

2.2.1 Binary Erasure Channel

A Discrete Memoryless Channel (DMC) is a class of channel for which both the

input and output letters belong to finite alphabet.The most simple and popular

example of the DMC channel models is Binary Erasure Channel (BEC), which

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0

1

1-

1-0

1

Erasure

Input Output

Figure 2.2.1. The Binary Erasure Channel

is characterized by transition probability and shown in Figure 2.2.1. To model

a BEC, two inputs are needed and they are either 0 or 1. The output consists

of 0, 1 and additional element called erasure. The bits are either transmitted

correctly with probability (1 ) or erased with probability . The capacity of

this channel is given by (1 ) [2]. Binary erasure channel can be used to model

internet system where packets can be either forwarded accurately or dropped due

to congestion or other disturbance in network.

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Figure 2.2.3. The Additive White Gaussian Noise

having constant spectral density.The AWGN model does not take account for

the phenomena of fading, interference dispersion in time and frequency. The

source of Gaussian noise are thermal vibrations of atoms in antennas, shot noise,

black body radiation etc.The AWGN channel is a good model for many satellite

and deep space communication links.The model for AWGN channel is illustrated

in Figure 2.2.3 and described by the additive white Gaussian noise term Z. In

particular, the capacity of a bandlimited AWGN channel with bandwidth B, is

given by

C = B log 2(1 + S/N), bits per second (bps)

where B is bandwidth of the system, S is the signal power of the input signal

X and N is the noise power of noise signal Z which are expressed by follwoing

relations:

S = EX2

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2. System Model

N = E

Z2

So the channel capacity depends on the signal to noise ratio. The signal-to-noise

ratio of the real-valued output R of the matched filter is then given by

S

N=

EbN0/2

with bit energy Eb and noise power spectral density N0. In [23], comparison

of capacities of the AWGN channel and BSC has been carried out. In order

to achieve the same channel capacity binary symmetrical channel required more

signal-to-noise ratio as compared to AWGN channel. This gain also translates to

the coding gain which can be achieved by soft-decision decoding as compared to

hard-decision decoding of channel codes.

2.2.4 Fast Rayleigh Fading Channel

For most practical channels, where signal propagation takes place in the at-

mosphere and near the ground, the freespace propagation model is inadequate to

describe the channel and predict system performance. When there is no line-of-

sight present between transmitter and receiver then the multipath is produced

only from reflections of the objects presents in the environment. This form of

scattering is purely diffuse and can be assumed to form a continuum of paths,

with no one path dominating the others in strength.When the channel gain fol-

lows a Rayleigh probability density function (pdf), the channel is said to be a

Rayleigh fading channel. Received signal can be modeled as y = te + n. The

is the normalized Rayleigh fading factor and related to the fading coefficient

of the channel ht through = |ht|, where the real and imaginary components of

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Figure 2.2.4. Block Fading Channel code word representation

ht are Gaussian random variables. If sufficient channel interleaving is introduce,

then fading coefficients ofht are independent. The conditional probability density

function (pdf) of the normalized Rayleigh fading factor is given by p() = 2e2

, n is the Gaussian random variables with zero mean and variance 2 and

phase is independent random variable being uniform on [0, 2]. Rayleigh fading

is viewed as a reasonable model for urban environments on radio signals.

2.2.5 Block Rayleigh Fading Channel Model

The block fading model is more common representation of the slowly varying

fading channels. In block fading channel the random channel gain or normal-

ized Rayleigh fading factor remains constant over a certain block of the symbols

transmitted through the channel. Code word representation of the Block fading

channel is shown in Figure 2.2.4. So the code designed which works better in

fast fading channel, may not behave very well in the Block fading channel. As

the block fading channel is nonergodic channel [17], so we cannot use the channel

capacity as we did in the case of fast fading channel. For information-theoretical

rate limit we rather use the outage probability instead of capacity [18].

If we want to transmit N symbols over m-block fading channel where m the

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2. System Model

number of normalized independent fading factor. The number of symbols which

are affected by each fading block is given by l N/m. The received symbol yi is

given by

yi = j ti + ni

where i = 1, . . . N , and j = 1 + [(i 1)/l], where [x] represent the integer part

of a real number x. The j is defined as for Rayleigh fading channel for fading

block j, where j = 1, . . . , m and ni is an i.i.d AWGN sample with zero mean and

variance 2 = N0/2, where N0/2 is the two-sided noise power spectral density.

For our simulation we will use 2-Block fading channel such that we divide our

code word length in two blocks and each block has independent fading factor.

The simulated model can be explained as

y1 = 1te1 + n1

y2 = 2te2 + n2

Where 1 and 2 are independent normalized Rayleigh fading factor.1 is for the

first half of block length bits te1 and 2 is for the second half blocks length bits te2

independent of each other, n1 and n2 are Additive Gaussian random noise with

zero mean and variance 2 .

2.2.6 Correlated Block Rayleigh Fading Channel Model

In BF model, transmitted sequence is divided into blocks and all the symbols

belonging to the same block experience the same fading. In some cases, the fading

blocks are assumed to be independent from each other, but in some applications

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Figure 2.2.5. Model of Correlated Block Fading Channel

such as the OFDM system, there is a considerable correlation exist among the

fading blocks due imperfect interleaving. The non-ideal interleaving is due tothe constraints on such as maximum allowable packet size and processing delay

requirement.

The coded system of correlated Block fading channel is shown Figure 2.2.5.

The k information bits encoded to code word of n bits. The coded bits are inter-

leaved and mapped to L symbol vectors xi. Each symbol vector xi is transmitted

through a subchannel of block fading channel each having normalized rayleigh

fading factor i. Fading coefficients are correlated by following correlation ma-

trix.

C = E[h], = (1, . . . , L)

t

where L is the number of fading blocks.

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2. System Model

2.3 OFDM-based Wireless Communication sys-

tems

2.3.1 Multicarrier Modulation

Single carrier modulation scheme is limited by the delay spread of the chan-

nel. The simple idea of multicarrier transmission to overcome this limitation is

to split the data stream into a number of sub-streams of lower data rate and

to transmit these data sub-streams on adjacent sub-carriers. In a single carrier

modulation, data is sent serially over the channel by modulating a single carrier.

In multipath fading channel, the time dispersion can be significant as compared

to symbol period, which result as inter symbol interference (ISI). In such situation

a more complex equalizer will be required to combat the channel distortion. In

multicarrier modulation, the available bandwidth is divided in a number of sub-bands, called subcarriers. The data is divided into several parallel data streams

or channels, one for each sub-carrier. Each sub-carrier is modulated with a con-

ventional modulation scheme at a low symbol rate, maintaining total data rates

similar to conventional single-carrier modulation schemes in the same bandwidth.

The symbol duration can be made greater than the channel maximum delay by

selecting more sub-carrier. On the other hand, bandwidth of the each sub-carrier

must be kept small as compared to the coherence bandwidth of the channel so

that each sub-carrier experience flat fading and hence simple equalization will

be required at the receiver end. Increasing the number of sub-carrier increases

the symbol time which significantly reduce ISI and hence simplifies equalization.

However, the performance of long symbol time signals is degraded in time variant

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OFDM-based Wireless Communication systems 2.3

Figure 2.3.1. Model of OFDM System

channels. If the symbol time is greater than the coherence time of the channel

then transmission is heavily affected during a single symbol transmission and the

overall performance is degraded.

2.3.2 OFDM

To achieve higher spectral efficiency in multicarrier system, the sub-carriers

must have overlapping transmit spectra but at the same time they need to be

orthogonal to avoid complex separation and processing at the receiving end. Mul-

ticarrier modulation schemes that fulfil above mentioned conditions are called

orthogonal frequency division multiplex (OFDM) systems. Instead of baseband

modulator and bank of matched filters Inverse Fast Fourier Transform (IFFT)

and Fast Fourier Transform (FFT) is efficient method of OFDM system imple-

mentation as shown in Figure 2.3.1 because it is cheap and does not suffer from

inaccuracies in analogue oscillators.

Inter symbol interference occurs when the signal passes through the time-

dispersive channel. In an OFDM system, it is also possible that orthogonality

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2. System Model

of the subscribers may be lost, resulting and inter carrier interference. OFDM

system uses cyclic prefix (CP) to overcome these problems. A cyclic prefix is

the copy of the last part of the OFDM symbol to the beginning of transmitted

symbol and removed at the receiver before demodulation. The cyclic prefix should

be at least as long as the length of impulse response. The use of prefix has

two advantages that are it serves as guard space between successive symbols

to avoid ISI and it converts linear convolution with channel impulse response to

circular convolution. As circular convolution in time domain translates into scalar

multiplication in frequency domain, the subcarrier remains orthogonal there is no

ICI in addition with memory and time saving in measurement. In Figure 2.3.1, L

coded vector xi are generated by proper coding, interleaving and mapping. After

adding cyclic prefix, OFDM signal is passed through multipath channel. At the

receiver the cyclic prefix is removed and received signal is passed through FFT

block to get L received vectors yi,where vk,t are zero mean Gaussian noise with

variance N0/2 of k-th sample of the t-th OFDM symbol. N0 is the noise power,

k = (1, 2, . . . , N FFT 1) and t = (1, 2, . . . , M ), where M is the number of OFDM

symbols and NFFT is the size of FFT.

2.3.3 Interleaving in OFDM System

Diversity is a technique, used to mitigate the multipath fading. Diversity can

be achieved when there are multiple independent fading channel exist between

transmitter and receiver. From channel coding point of view, closely related

transmitted coded bits must separate in time or frequency to experience inde-

pendent fading of channel. Bits are closely related in block code if they are part

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OFDM-based Wireless Communication systems 2.3

of the same code while in convolution code if not so many constraint lengths

present between them. In time interleaving bits must be separated in time while

in frequency interleaving bits are physically separated in frequency domain. Time

interleaving introduces decoding delay because the receiver has to wait until all

closely related bits are received for decoding. In time interleaving ,closely related

bits of the code experience independent fading, when the time separation between

the bits are greater than coherence time of the channel. Similarly the bandwidth

of the system should be greater the coherence bandwidth of the channel to get

efficient frequency interleaving.

Different techniques are available to implement interleaving. Block interleaver

takes a block of some symbols and then randomly permute the order of the sym-

bols in the block. At the receiver permutation is reversed to get the original

order. Convolution interleavers perform in sequential manner. Block interleavers

are best choice for frequency interleavers due to processing of block of sub-carrier

used, while convolution interleavers suited to the time interleavers due its ex-

cellent decoding properties. In certain situations, time interleaving did not give

sufficient diversity so time and frequency interleaving may be used in combina-

tion.

OFDM system using interleaving (either time or frequency) depending on

different applications. Digital Audio Broadcasting (DAB) is typically OFDM

based system .From [24], for DAB to be working in 225 MHz , vehicle speed is

48 Km/h turn to Doppler frequency 10 Hz, and using time interleaving results

in decoding delay of several seconds, which is not suitable for practical system.

OFDM has the property to combine the time and frequency interleaving to achieve

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2. System Model

best interleaving.

2.4 Interleaved-OFDM system as Block Fading

Channel

As we live in the world of reality, therefore in practical multicarrier sys-

tem such as OFDM, correlation exists among the sub-carriers due to non-ideal

interleaving[24]. One of many reasons for the lost of the orthogonality between

the sub-carriers of the OFDM system is higher values of Doppler frequency. From

practical point of view these correlation must be take in to consideration.

In OFDM system information bits are distributed over a number of sub-

carriers and transmitted in parallel channels. Each subcarrier experiences flat

fading channel but as due to limited interleaving correlation exist among sub-

carriers. Similarly in block fading channel, channel gain remains constant over

a certain block of the symbols transmitted through the channel. Therefore to

achieve the performance analysis of OFDM system we can use the generalized

model of the correlated block fading channel [26]. So the whole blocks, from

IFFT to FFT in Figure 2.3.1, can viewed as correlated block fading channel of

Figure 2.2.5.

As an example, the limitations on depth of interleaving in transmission of

OFDM system recommended by IEEE.802.11 are maximum allowable packet size

and processing delay requirements. This results in a non-ideal interleaving which

effectively limits the maximum achievable diversity from the channel. In such a

channel, fading gain occurs in block wise fashion, i.e., all transmitted symbols

of the same block sense the same fading. Also there is considerable correlation

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Interleaved-OFDM system as Block Fading Channel 2.4

among the fading blocks. We call such a channel as Correlated Block Fading

(CBF) channel. In some situations, the block fading channel can be assumed

independent but in systems based on OFDM, there exist some correlation. So

limited interleaved OFDM -based system can be properly modelled by correlated

block fading channel.

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Chapter 3

Low Density Parity Check Codes

3.1 Introduction

A linear code with sparse parity-check matrix is said to be ( Low Density

Parity Check) LDPC code. LDPC code was first proposed by Gallager [7] in the

early 1960s along with its elegant iterative decoding property, but it did not get

proper attention until years later. Due to heavy computations, LDPC codes were

ignored but got an amazing comeback in the last few years. The name of LDPC

code is used in relation with their Parity-check matrix which have low density

of 1,s compared the number of 0,s. In contrast to other coding scheme , LDPC

codes offer a better performance with low decoding complexity. LDPC codes are

also called the capacity approaching codes.

In parity-check matrix representation of a LDPC, the code word c = (c1, . . . , cn)

can be expressed as simple linear algebraic equation HcT = 0T, where H is parity-

check matrix. The elements of the parity-check matrix are 0,s and 1,s and all

arithmetics are modulo 2, that is, multiplication ofc by a row ofH means taking

the XOR of the bits in c corresponding to the 1,s in the row of H. The number

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3. Low Density Parity Check Codes

of 1,s in each row of the parity-check matrix is called the weight of each row

of H, while the number of 0,s in each column is the weight of each column the

parity-check matrix. If the length of the block code is n then the number of rows

of parity-check matrix can be calculated as n wc/wr which has to be integer

and where wc represents the weight of column and wr is the row weight ofH. The

rate of the LDPC code is R (1 wcwr

).

3.1.1 Regular LDPC Codes

A LDPC code is regular if the number of 1,s in column wc and number of 1,s

in row wr are constant for a given parity-check matrix. LDPC codes proposed by

Gallager in [7] are regular for which parity-check matrix can be defined as

H =

H1

H2

...

Hwc

where the submatrices Hs has a special structure. For any integer and wr greater

than 1, each submatrix Hs has a row weight equal to wr and a column weight 1

and of size wr. The submatrix H1 has the special form of for i = 1, . . . , ,ith row contain all 1,s of wr in column (i 1)wr to iwr.The other submatrices

are just the column permutation of H1. Gallager [7] showed that the ensemble of

such codes has excellent distance properties provided that wc 1 and wr > wc.

Mackay in [21] proposed independently different methods for generating sparse

parity-check matrix H.

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Introduction 3.1

3.1.2 Irregular LDPC Codes

A LDPC code is regular if the number of 1,s in columns and rows are not

constant for a given parity-check matrix. Irregular LDPC codes can be parame-

terized by the degree polynomials (x) and (x), which can be defined as

(x) =

dli=2

ixi1 and (x) =

dri=2

ixi1

where i(x) and i(x) are the fractions of edges belonging to degree-i variable

and check nodes, and dl and dr are the maximum variable and check node degrees

respectively. The optimization of the (x) and (x) is found by combination of

density evolution and optimization algorithm.

3.1.3 Graphical Representation of LDPC Codes

In coding theory, codes connected with graphs have been defined in variety of

ways. Tanner graph is the best way to represent the LDPC codes as this graph

not only gives good visualization but also describes the decoding algorithm as

well. Tanner graphs of LDPC codes are bipartite graphs with variable or bit

nodes on one side and constraint or check nodes on the other. Each variable node

corresponds to a bit, and each parity-check node corresponds to the constraint on

the bits of the code word. Edges are used to connect the corresponding variable

and check nodes. The tanner graph representation of the LDPC codes is closely

analogous to the more standard parity-check matrix representation of a code.

Such parity-check matrix can easily be generated [7, 21] and it consists of check

nodes equal to the number of parity and variable nodes equal to the number of

bits in a codeword. The entry (i, j) is 1 if and only if the ith check node is

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Figure 3.1.1. Tanner Graph of LDPC Code

connected to the jth variable node in the graph. LDPC codes are defined by the

graph and they consist of a set of vector c of block length n such that H.cT = 0T.

However, not every binary linear code has a representation by a sparse bipartite

graph, if it does, then the code is called a low-density parity-check (LDPC) code

[2]. Graphical representation of the half-rate regular (3, 6) LDPC code with code

word length 10 for the following parity-check matrix is shown in Figure 3.1.1.

H =

1 1 1 1 0 1 1 0 0 0

0 0 1 1 1 1 1 1 0 0

0 1 0 1 0 1 0 1 1 1

1 0 1 0 1 0 0 1 1 1

1 1 0 0 1 0 1 0 1 1

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Iterative Decoding of LDPC Codes 3.2

3.1.4 Decoding Complexity

LDPC codes have achieved outstanding performances over additive white

Gaussian noise channel and Rayleigh fading channel [6]. It is evident from simula-

tion results that LDPC codes perform near to the Shannon limit for larger block

lengths.For example, LDPC codes perform within the 0.04 dB of the Shannon

limit at bit error rate of 106 with a block length of 107 [1].

Simplicity of LDPC codes have been exploited by recent design improvements,

advancement in computational technology and hardware realization and hence

produced such coding systems that outperform turbo codes with lower complexity.

LDPC codes have the advantage of the controlled sparseness of the code which

causes a specified and small number of 1,s in each row and column. The small

number one 1,s in each column not only reduces the number of equations to be

solved by the decoder, but also makes it feasible for practical decoding by means

of iterative decoding.

3.2 Iterative Decoding of LDPC Codes

The tanner graph exploits the dependency structure of the various bits re-

ceived from the channel. The iterative or a message passing algorithm discussed

in [7] and which can be easily explained by a tanner graph. As the name specifies,

in each round the messages are passed from the variable nodes to check nodes and

from check nodes to variable nodes. The message passed from the variable node

depends upon the received value of bit from the channel and values received from

its neighbor check nodes except the check node to which it will send the message.

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Similarly, the message sent from the check node depends on its neighbors vari-

able nodes except the variable node to which it will send the message. The most

popular iterative algorithm is the belief propagation algorithm in which the mes-

sages passed from the variable nodes to the check nodes and vice versa are not the

hard decision values but probabilities, conditional likelihoods or log-likelihoods.

It is also assumed that the message passed between variable and check nodes are

independent of each other.

If the messages which are to be exchanged between the variable and check

nodes are independent, then the corresponding node can accurately calculate

the log-likelihood based on the received beliefs or messages. This independent

assumption is not fulfilled in practice, but in bipartite graph it is applicable for l

first iterations if the neighborhood of a variable node up to depth l is a tree [2].

3.3 LDPC Decoding for BEC

The binary erasure channel (BEC) is perhaps the simplest channel model as

described in section 2.2.1. The message passing algorithm of LDPC has been

generalized in [5] using hard decision rule for Binary erasure channel.Variable

nodes receive bits information from the channel and some of them are erased

depending on the channel erasure probability. Then based on the received bits

information, variable nodes will send erasure, 0 or 1 to the check nodes. The

messages sent by variable nodes are received by the check nodes and hence cal-

culate the messages to be send to the corresponding variable nodes. From check

nodes perspective if any incoming message except the variable node to which it

will send this outgoing message is erasure then outgoing message will be erasure.

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Figure 3.3.1. Performance comparison of LDPC vs No LDPC Decoding inBEC for code word length 512.

increases beyond 0.5 value then both the erasure probabilities with and without

iterative LDPC decoding become linear. So up to threshold value of erasure

probability with iterative LDPC decoding outperforms the un-coded system but

after that threshold value both are behaving in the same manner.

In order to exploit the difference in performance of different code word block

length codes, we simulated 256 and 512 code word block length in BEC channel.

LDPC code with block length 512 outperform 256 code word block length But

again this better performance is subject to some threshold value and after that

all codes with different code block lengths behave in the same fashion. Similarly

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LDPC Decoding for Binary Symmetrical Channel 3.4

Figure 3.3.2. Performance comparison of different code word length.

if we increase the length of block code, then after a certain limit further increase

in the code word length cant result in a better performance. In our simulation

result this threshold is around 0.445 channel erasure probability and is shown in

Figure 3.3.2.

3.4 LDPC Decoding for Binary Symmetrical

Channel

Message passing or belief propagation decoding algorithm for binary sym-

metrical channel is extensively discussed in [7]. Similar to the binary erasure

channel the message passing decoder consists of a number of rounds. The incom-

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ing messages at the check node except the corresponding variable node and after

calculation at the check node forward the outgoing message to the corresponding

variable node. In a similar fashion the incoming message at the variable node,

except the corresponding check node are processed and forward the outgoing

message to the check node. Here, too, the independent message assumption is

made.

In round 0 of Gallager,s A algorithm [7] using message passing decoder and

hard decision rule,variable node sends its received information from the channel

to the check node. But after round 0, if all the incoming message except the

concerned check node are of the same value, then variable node sends that infor-

mation to the check node; otherwise, it sends its received value to the check node.

On the other hand, check node sends module 2 additions of all incoming messages

except from the message of the the corresponding variable node to which it will

send this message.

Gallager, s B algorithm [7] is similar to the Gallager, s A algorithm but more

flexible and fancy than it. In this algorithm for a certain degree of edge and a

particular round there is a threshold such that if the variable node receives the

same messages at least equal to that threshold excluding the corresponding check

node, then it sends that information to the check node; otherwise, it sends its

received bit information from channel to the check node.

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3.4.1 Soft Decision Message Passing Decoder for BinarySymmetrical Channel

This decoder is similar to the hard decision decoder. However, the update

messages exchanged between two sets of nodes are not the hard decision but it is

some sort of belief calculated based on the incoming messages.

At iteration zero each variable node calculates the prior probability that the

received bit is 1. LetPi be the conditional probability such that the received code

bit is 1 and given as

Pi = pr(ci = 1|yi)

The message from variable node i to the check node j that the received

bit by variable node i is 1 is given by qij(1) = Pi , equivalently the message

delivered from the variable node ito the check node j that the given bit

received by variable node i is zero is qij(0) = 1 Pi. The check nodes update

their equations according to [1, 7] as following:

The message from the check node jto the variable node i given that this

node believes that the received bit is zero from the incoming message excluding

the corresponding variable node i to which this message has to be send is given

by following equation

rji(0) =1

2+

1

2

iVj\i

(qij(0) qij(1))

where Vj represent the neighbor of check node j. Neighbor of the check node can

be defined as the number of variable nodes connected to it. This is the probability

that the whole sequence contain an even number of 1,s and hence rji(0) is the

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probability that message to be send by check node j to the variable node i

is zero. Similarly the probability that message sent from the check node j to

the variable node i is 1 is given by the following equation,

rji(1) = 1 rji(0)

The variable nodes calculate their update messages from the incoming messages

of check nodes except the corresponding check node as follows [1, 7].

qij(0) = K0

jCi\j

rji(0)

and so we can get

qij(1) = K1

jCi\j

rji(1)

where K0 = Kij(1 pi) ,K1 = Kijpi and Ci represent the neighbors of variable

node i. Kij,K0 and K1 should be selected in such a way that the sum ofprobability calculated for the update equation should be 1. The method of update

messages calculations is shown in Figure 3.4.1.

This completes one of the iteration and the variable nodes take decision of

the received bits as follows [7].

Qij(0) = K00

jCirji(0)

and

Qij(1) = K11

jCi

rji(1)

where K00 = Ki(1 pi) and K11 = Kipi.

Again K00,K11 and Ki should be selected in such a way that the sum of the

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Figure 3.4.1. General message update (a) from check node to variable node(b) from variable node to check node..

calculated probabilities should be equal to 1. The decision rule can be taken as

ci =

0 if Qi(0) > Qi(1)

1 otherwise

The maximum number of iterations are either explicitly mentioned or the iter-

ations should be continued until the code word satisfies the parity-check matrix

equation i.e. HcT = 0. Due to large number of multiplications there is a sta-

bility problem such that the intermediate result may become zero and may give

some erroneous final result so the best way to convert this algorithm into the

log-domain.

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3.5 Soft Decision Iterative Decoder for BPSK

AWGN Channel

Soft-decision message passing decoder is the approximation of the bitwise-

MAP decoder. In other words the soft-decision message passing decoder approx-

imately computes the probability that the given bit is either zero or one given

the entire received vector. Let x = (x1, x2,

, xn) be the message bits, the

BPSK method map the signal from {0, 1} to 1 where +1 represent 1 and

-1 represent 0 respectively. After transmitting over an AWGN channel, white

Gaussian noise is added to the original signal and the received analogue signal

vector r = (r1, r2, ...rn) will have a value around -1 and +1. In a hard-

decision decoder, the BPSK demodulator estimates the received binary vector

y = (y1, y

2,...y

n) based on the sign of the elements of vector r so that

yi =

1 if ri > 0

0 otherwise

Let us consider the probability that the received bit is zero for given received

entire vector is give by the following equation.

pr(ci = 0 | r)

Or actually it calculate the likelihood ratio given by

li =pr(ci = 0 | r)pr(ci = 1 | r)

and the log likelihood ratio can then be represented by LLR = loge li.

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Soft Decision Iterative Decoder for BPSK AWGN Channel 3.5

This soft-decision message passing decoder is similar to the Gallager A algo-

rithm in which there are iterations and in each iteration there are two steps. In

first step messages were passed to check nodes from variable nodes and in the

next step messages have been delivered from check nodes to variable nodes.

If the bits are equally likelihood then the channel likelihood ratio for eac

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