Iterative Decoding of Concatenated Codes.

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Iterative Decoding of Concatenated

Codes

By Kjetil Fagervik

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Thesis submitted to the University of Surrey

for the degree of

Doctor of Philosophy

Centre for Communications Systems Research

School of Electronic Engineering, Information Technology and

Mathematics

University of Surrey

Guildford, Surrey

United Kingdom

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Acknowledgements

I w ould like to direct som e deep-felt gratitude to a w ide range of people and in stitu tio n s w ho have m ade th e w ork described in this thesis possible. Firstly I w ould like to th a n k Professor Barry Evans, Professor A hm et Kondoz and my advisor Tony Jeans a t th e C entre for Com m unications Systems Research (CCSR) a t th e University of Surrey, as w ell as CCSR itself, for providing funding and facilities for this PhD study, for th e ir continuous faith in th is research project, and for allowing m e th e freedom to define and carry o u t th e research in th e w ay I th o u g h t m ost suitable. I w ould particularly like to direct g ratitu d e to m y advisor Tony Jeans for fruitful discussions a t tim es w hen th e future direction of th e project app eared diffuse. I also th an k Dr. Roger Seebold for his friendship, and for first suggesting to m e th e possibility of undertaking a PhD.

I w ould also like to th a n k th e Norwegian Research Council (Norges Forskningsrad) for funding th e la tte r 1.5 years of this project, enabling financial freedom to fully pursue th e research.

I w ould also like to th a n k o th er researchers and PhD stu d e n ts a t CCSR for great friend ship, invaluable discussions and critical com m ents on th e research. In particular, I th a n k Stephan W esem eyer for m eticulously reading and com m enting u p o n this thesis, H ai Pang Ho for discussions on hard-decision decoding algorithm s of block codes, John Paffett of Surrey Satellite Technology Ltd. for discussions concerned w ith practicality issues of Turbo Codes and Philipos Psilionis for providing H-263 coded video sequences for 'rea l- life' testin g of Turbo Codes.

Finally, I w ould like to th a n k m y wife Liz for all her s u p p o rt and encouragem ent th ro u g h o u t th e w ork on this PhD research project.

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Summary of Thesis

This th esis is concerned w ith th e area of decoding techniques of concatenated, erro r cor recting codes using various s o ft-in /so ft-o u t decoding algorithm s, as well as w ith th e construction of th ese codes.

Initially, we consider in som e detail th e th eo ry behind a com m unications system s, w hereby th e tra n sm itte r, channel and receiver are quantified and analytically defined. We use th ese definitions to define soft-decision decoding algorithm s of erro r correcting trellis codes, after having considered th e th eo ry behind th e construction of such codes, w here b o th of th e com m on classes of convolutional and block codes are tre a te d . We th e n m ove onto concatenated coding schem es, considering b o th trad itio n al, serially con catenated coding schem es w hereby th e o u ter code is decoded by m eans of hard-decision decoding m ethods, as well as new soft-decision decoding schem es. We tre a t in som e detail th e construction of parallel concatenated codes, decoded by m eans of iterativ e d e coding algorithm s, also denoted Turbo Codes. We th e n extend th e principles introduced in th is p a rt to also apply to serially concatenated codes of Reed-Solom on and convolu tional codes. Finally, w e consider spectrally efficient coded m odulation techniques using iterativ e decoding techniques.

The m ain research achievem ents resulting from this w ork include:

• The reduction in complexity and decoding delay latency of iterativ e decoding schem es involving trad itional, parallel concatenated System atic Recursive Convo lutional (RSC) codes, as well as several novel code and decoder configurations using th ese codes.

• The developm ent and application of s o ft-in /so ft-o u t decoding algorithm s of se ri ally concatenated convolutional and Reed-Solom on codes. N on-iterative and ite ra tive decoding algorithm s are investigated and presented in this thesis, for b o th th e AWGN and Rayleigh fading channels. A m ajor finding of th is research is th a t serially concatenated coding schem es appear m ore suitable for system s in which very low Bit Error Rates are required th a n do parallel concatenated schem es. These results apply for b o th AWGN and Rayleigh fading channels.

• The proposal and investigation of spectrally efficient coded m odulation schem es in volving binary BCH and non-binary Reed-Solom on codes for which very high spec tra l efficiency m ay be obtained, even w hen used w ith m odulation schem es w ith a sm all alp h ab et. This w ork has also resulted in a novel, low com plexity d em o d u latio n algorithm for giving soft outputs a t th e bit level for non-binary m o dulation schem es.

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List of Figures

2.1 Elements of communications system. Filtering of the signal is inherent in both m odulator and dem odulator...

2.2 M-PSK signal g e n e ra tio n ...

2.3 Frequency response of the VRC filter and the RC filter...

2 .4 The pdf of the Rayleigh fading envelope re(t) and its accumulated distribution . .

3.1 Illu stratio n of code p a r a m e t e r s ...

3.2 Illustration of decoder p a r a m e t e r s ...

3.3 Illustration of convolutional c o d e ...

3 .4 Convolutional encoder of code with [G(E>)] = [7 5] ...

3.5 State Diagram of code with [G(£>)] = [7 5] ...

3.6 Block diagram of R = 2/3 convolutional code. [G(D)] as defined in equation (3.35).

3.7 Trellis derived from th e state diagram of Figure 3.5 of th e [7 5] convolutional code

3.8 Trellis derived from the state diagram of the R = 2/3 convolutional code depicted in Figure 3.6 . . ...

3.9 Block diagram of the R S C (7,5) c o d e ...

3.10 BCH encoder with g(D) = 1 3 ... ...

3.11 Block diagram of polynomial division circuit ...

3.12 Block diagram of encoder of systematic cyclic c o d e s ...

3.13 Equivalent systematic encoder of the code drawn in Figure 3 . 1 3 ...

3 .1 4 RS encoder constructed over G F(23) ...

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LIST OF FIGURES x

3.16 State diagram of BCH code with g(D) = 1 3 ... 52

3.17 Trellis of BCH code shown in Figure 3 . 1 0 ... 53

3.18 State diagram of systematic BCH code with G(D) = 1 3 ... 53

3.19 Trellis of BCH code shown in Figure 3 . 1 3 ... 54

4.1 Illustration of code p a r a m e te r s ... 55

4 .2 Trellis of the considered c o d e ... 56

4.3 Trellis of code to be d e c o d e d ... 70

4 .4 Branch metrics as seen in the tr e l l i s ... 72

4.5 Values associated with each state after forward recursion ... 74

4 .6 Values associated with each state after backward r e c u r s i o n ... 75

4 .7 Path through trellis of the decoded s e q u e n c e ... 76

4 .8 Example of notation in trellis. Decoding depth is given by the param eter S. . . . 78

4 .9 Trellis of code to be decoded using S O V A ... 81

4 .1 0 Trellis of code after receipt of the first 2 channel s y m b o l s ... 82

4.11 Trellis of code at tim e i = 2... 83

4 .1 2 Trellis of code at tim e i = 3... 84

4.13 Trellis of code at tim e *' = 4... 85

4 .1 4 Trellis of code at tim e * = 5... 86

4.15 Trellis of code at tim e *' = 5 with decoded sequence h i g h lig h te d ... 87

4 .1 6 Trellis configuration when the mean of the reliability value, A(-) = AEcb... 88

4 .1 7 Trellis configuration with A(-) = Y l E cb, but yielding wrong d e c i s i o n ... 90

4 .1 8 State Diagram of [G(£>)] = [7 5]s code... 91

4 .1 9 The theoretically obtained estim ates of th e pdf’s of the reliability values before trace-back a t Eb/No = OdB. Solid line: Wrongly decoded. Dotted line: Correctly decoded b its... 92

4 .2 0 The theoretically obtained estim ates of the pdf’s of the reliability values before trace-back a t Eb/No = 3dB. Solid line: Wrongly decoded. Dotted line: Correctly decoded b its... 93

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LIST OF FIGURES xi

4.21 The pdf’s of the reliability values before trace-back at E b / N 0 = O d B ... 94

4.22 The pdf's of the reliability values before trace-back at E b/N o = 3 d B ... 95

4.23 Trellis configuration showing trace-back operation of S O V A ... 96

4 .2 4 The pdf's of the reliability values after trace-back at E b / N 0 = O d B ... 97

4.25 The pdf’s of the reliability values after trace-back at Eb/No = 3 d B ... 98

4 .2 6 The theoretically obtained estimates of the pdf’s of the reliability values after trace- back at E b /N o = OdB. Solid line: Wrongly decoded. Dotted line: Correctly decoded b its... 99

4 .2 7 The theoretically obtained estim ates of the pdf’s of the reliability values after trace- back a t E b /N o = 3dB. Solid line: Wrongly decoded. Dotted line: Correctly decoded b its... 99

5.1 Block diagram of serially concatenated channel coding scheme... 101

5.2 Block diagram of non-systematic, v = 6 convolutional c o d e ... 103

5.3 BER performance of non-systematic, v = 6 convolutional c o d e ... 104

5 .4 BER performance of RS (255,223) and RS (204,188) c o d e s ... 105

5.5 8 x M Block I n t e r l e a v e r ... 106

5.6 Configuration of the Forney convolutional in te r le a v e r ... 107

5 .7 BER performance of DVB-S system using the un-punctured, v = 6 convolutional code.108 5.8 RS encoder constructed over G F(23) ... 109

5.9 Performance of SOVA and BCJR algorithms in concatenated schemes... 110

5.10 Performance of bit and symbol interleaving in the concatenated coding schemes.. I l l 5.11 Performance of concatenated v = 6 convolutional code and RS (7,5,3) code over the Rayleigh fading channel... 112

5.12 Performance of concatenated v — 6 convolutional code and RS (15,13,3) code over th e Rayleigh fading channel... 113

5.13 Performance of concatenated convolutional codes using soft-in/soft-out decoding 114 5 .1 4 Performance comparison of concatenated convolutional codes with and w ithout in terleaving... 115

5.15 Block diagram of effective code formed by concatenation of Ri — 1/2 and R n = 2/3 code with no interleaver... 116

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L IST OF FIGURES xii

5.16 Block diagram of P codes in parallel concatenation. ... 117

6.1 Block diagram of 2 parallel concatenated RSC c o d e s ... 120

6.2 Block diagram of decoding algorithm for parallel concatenated RSC codes.... 121

6.3 Block diagram of iterative decoding algorithm of two parallel concatenated convo lutional codes. Al(di) denotes the soft outputs of the first decoding stage of the p th iteration, whereas A£(<&) denotes the soft outputs of the second decoding stage of the same iteration ... 121

6 .4 Block diagram of p iterations of a pipe-lined iterative decoding algorithm of two parallel concatenated c o d e s ... 122

6.5 Performance comparison of Turbo Codes with different interleavers (2 iterations) 125

6.6 Performance comparison of Turbo Codes with different interleavers (2 iterations) over the fully interleaved Rayleigh fading c h a n n e l... 126

6 .7 State diagram of RSC code with generator polynomials g(D)RSc(7,5) = [1 5/7]s . 128

6.8 Performance comparison of R = 1/2 Turbo Codes with differing number of decoder iterations. The constituent encoders both had generator polynomials

9 (D) r s c (3 7,2 1) = [1 21/37]s - Interleaving: Convolutional w ith N = 13 and J = 3. . 130 6.9 Performance comparison of R = 1/2 Turbo Codes with encoders of differing com

plexity. Interleaving: Convolutional with N = 13 and J = 3. 131

6.10 Performance comparison of R = 1/2 Turbo Codes of the conventional scheme with th a t of a scheme using two different codes... 132

6.11 Performance comparison of Turbo Codes w ith different interleaver puncturing rates. The constituent encoders had generator polynomials gi(D) r s c (3 7,2 1) = [1 21/37]s

and 9 2(D) r s c (3 1,2 7) = [1 27/31]s... 133

6.12 Performance comparison of Turbo Codes with differing multiplication factors. All curves are for a decoder consisting of 12 iterations... 134

6.13 Performance comparison of Turbo Codes with different interleaver sizes. The con stituent encoders were gi(D)RS0(3 7,2 1) and 9 2(D)Rsc(3i,\27) ... 135

6 .1 4 Performance comparison of Turbo Codes w ith SOVA, gi(D)RSc(3 7,2i) = [1 21 /37]s

and 9 2( D ) r s c (3 1,2 7) = [1 27/31]s... 136

6.15 Simple iterative decoder using S O V A ... 137

6.16 N on-stationary encoder of RSC code g(D)RSc(7,5) = [1 5/7]s and non-systematic

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LIST OF FIGURES

6.17 Performance of trellis term inated Turbo Codes

xiii

140

6.18 Performance of hybrid concatenated coding schemes. Interleaver for inner Turbo Code: Convolutional with J = 53, N = 33. Interleaver between outer and inner

code: Convolutional symbol interleaver with J = 17, N = 12... 141

6.19 Iterative decoding algorithm of serially concatenated c o d e s ... 144

6.20 Performance of iterative decoder of RS (7,5,3) and RSC[37,2i]8 codes... 145

6.21 Performance comparison of parallel and serially concatenated schemes, decoded iteratively. ... 146

6.22 Performance comparison of parallel and serial concatenated schemes, decoded it eratively. The inner RSC code is punctured to have rate R n = 2/3... 147

6.23 Performance comparison of iterative decoders of parallel and serially concatenated schemes over the Rayleigh fading channel... 148

6 .2 4 Performance comparison of Turbo Codes over the Rayleigh fading channel with CSI available ... 149

6.25 Performance comparison of concatenated RS and RSC code over the Rayleigh fading channel with CSI available... 150

7.1 Encoder and signal m apper of pragmatic Turbo Coded m o d u la tio n ... 151

7.2 Decoder of pragmatic Turbo Coded m o d u la tio n ... 152

7.3 Signal Constellation Diagram for Gray Coded 8 P S K ... 158

7 .4 Performance Comparison of the a lg o rith m s ... 160

7.5 RS codes in parallel c o n c a te n a tio n ... 161

7.6 Iterative decoding algorithm of parallel concatenated RS codes ... 162

7.7 Performance of some selected Turbo Coding schemes with QPSK m odulation. . . 164

7.8 Performance of coding schemes with high spectral efficiency using 8PSK modulation. 165 7.9 Performance of coded 8PSK systems in Rayleigh fading. The num ber before the # denotes the num ber of iterations used... 166

A .l Block diagram of DVB-S transm itter . . ... 175

A .2 DVB-S framing structure after MPEG-2 and RS encoding... 176

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LIST OF FIGURES

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List of Tables

4.1 BCJR decoding example: Channel O u t p u t s ... 71

4.2 BCJR decoding example: Branch M e t r i c s ... 71

4.3 Transfer f u n c t i o n s ... 89

A .l Inner convolutional code r a t e s ... 180

A. 2 Perform ance R e q u i r e m e n t s ... 181

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Chapter 1 Preface

The invention of Turbo Codes, th e description of which was first published in [1], m arked a significant step forw ard in th e area of Forw ard Error C orrection coding and inform a tio n th eo ry . P rom inent researchers have even gone so far as to say th a t th e ad v e n t of Turbo Codes is th e m ost im p o rta n t event since th e publications by Claude Shannon of th e founding papers of th e area of inform ation theory, e.g. references [2], [3] and [4 ]. The initial reaction to Turbo Codes was one of scepticism and disbelief, m aking Turbo Codes prone to criticism based on practicality issues. It is a fact th a t th e schem e p resen ted in [1] im posed su b stan tial processing and m em ory requirem ents on th e decoders, yielding a very high overall delay in th e decoding process. How ever, soon after th is initial p a p e r on Turbo Codes, articles sta rte d appearing in conference proceedings and journals on a global scale, in which th e results of [1] w ere validated, often w ith less complex decoders. This stu d y in th e area of Turbo Codes was initiated in th e beginning of 1995, and to our know ledge th e re w ere only a handful of papers available on Turbo Codes a t th a t tim e, com prising [1], [5], [6], [7], [8] and [9]. How ever, th e concept of Turbo Codes sparked off nothing less th a n an explosion in term s of research u n d ertak en in this area, and som e 3 years afterw ards th e re are literally hundreds, possibly th o u san d s, of papers available on ite rativ e decoding and related subjects. At th e tim e of w riting, th e re is a w ell-k ep t datab ase of papers related to this area a t th e In te rn et site of th e Com m unications, Con tro ls and Signal Processing Laboratory (CCSP), University of Virginia, w ith th e WWW address h ttp ://w w w .e e .v irg in ia .e d u /C S L /tu rb o _ co d e s/.

The invention of Turbo Codes was app aren tly a gradual and experim ental process. To qu o te B attail [10], who worked closely w ith th e te a m who invented Turbo Codes:

'The invention of Turbo Codes has been an unprecedented event in th e field of com m unication [1]. The design of th ese codes did n o t indeed consist of optim ising som e given criterion, as usual, b u t was th e resu lt of an experi

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CHAPTER 1. PREFACE ₂

m ental process w here sim ulation was used in order to jointly adjust several param eters so as to optim ise th e final targ et, namely, th e b it-e rro r rate (BER). [ . . . ] .

Since Turbo Codes did n o t actually result from applying a preexisting theory, m ost of th e ir outstanding features rem ain to be explained.

This la tte r sta te m e n t rem ains tru e , although ground-breaking, theoretical w ork was u n d ertak en by H agenauer e t al., and presented in [11] and B enedetto and M ontorsi, which was p resen ted in [12] and o th er papers by th e sam e authors, perhaps th e m o st im p o r ta n t of th ese being references [13], [14], [15], [16] and [17], and [18]. Researchers, notably Divsalar and M ontorsi, have also contributed im m ensely in th e field, som e of th e m ore im p o rta n t publications being [19], [20], [21] and also [18].

All th ese publications have resulted in an im proved understanding of Turbo Codes. H ow ever, several aspects of Turbo Codes rem ain unclear, and m ost schem es a p p e ar to have been designed heuristically, w ith incom plete th eo ry to back up th e perform ance, which is in general obtained experim entally through sim ulations. In this w ork w e have, in gen eral term s, tak en th e sam e approach. W henever theoretical developm ents are m ade, we will also use sim ulation results to back up this theory.

In th is study, w e will aim for a general approach w hen treatin g th e subject of iterativ e decoding, or Turbo Coding. We will n ot aim to cover all th e different variants of coding schem es which have been invented - if such tre a tm e n ts are desired, it will be of value to refer to reference [22] and th e JSAC journal on concatenated coding techniques, ref erence [23]. How ever, th e aim is to place iterative decoding into th e context of already know n th e o ry by showing th a t iterative decoding algorithm s are a sim ple extension of al gorithm s th a t have b een know n for alm ost 30 years. Having tak e n th is general approach, we are th u s in a position w here iterative decoding m ay be u n d ertak en for literally any concatenated coding schem e, and several such schem es will be presen ted in th is thesis.

1.1 Thesis Outline

The th esis is organised as follows:

In C hapter 2 w e define and quantify th e tra n sm itte r and receiver of th e communications system s th a t w e have used in th e modelling and sim ulations of la te r chapters. H ere, the channels over which w e have obtained sim ulation results are defined, as well as th e issues of soft decision channel inform ation. This analysis forms th e basis for fu rth er analysis u n d ertak en in C hapters 3 and 4.

C hapter 3 tre a ts conventional erro r correcting codes using a general approach. Initially, th e th e o ry behind th e construction of th ese codes is considered, and th e re a fte r we focus

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CHAPTER 1. PREFACE 3

on th e trellis and finite s ta te m achine structures of th ese codes, which is crucial for th e decoding o p eratio n . This will be defined in C hapter 4. Both block and convolutional codes will be considered in C hapter 3, as we will use b oth th ese classes of codes in su b se q u en t chapters.

In C hapter 4, th e corresponding (probabilistic) decoding algorithm s of th e codes d e scribed in C hapter 3 are considered, again from a general point of view. We will focus on soft o u tp u t algorithm s, as this class of algorithm s is crucial to th e o p eratio n of iterativ e decoding schem es. First, th e V iterbi algorithm is briefly explained, paving th e w ay for th e d erivation th e optim um , soft o u tp u t, sym bol by sym bol decoding algorithm , d en o ted th e BCJ R algorithm after th e authors of [24]. Then w e will show how th e V iterbi algo rith m m ay also be m odified to deliver soft o u tputs. This algorithm was first introduced in [25], and was d en o ted th e Soft O utput Viterbi Algorithm (SOVA). H ere, we will consider this algorithm in considerable detail, and a com plete derivation of th e algorithm will be given. Decoding examples will be given for b o th th e BCJR and th e SOVA, and for th e SOVA w e will also derive approxim ate expressions for th e pdf's of wrongly decoded and correctly decoded d a ta bits.

C hapter 5 is concerned w ith concatenation of codes. We consider b o th conventional con catenated coding schem es, w hereby th e inner code is a soft-decision decoded convolu tional code and th e o u ter code is a hard-decision decoded Reed-Solom on code. M ore im p ortantly, som e novel schem es will be introduced, w hereby th e decoder of th e o u te r R eed-Solom on code perform s soft-decision decoding. We will show th a t su b sta n tia l cod ing gains m ay be achieved using this approach.

C hapter 6 and C hapter 7 contain th e m ajor results of this study. C hapter 6 initially consid ers conventional Turbo Codes, and w e subsequently d em o n strate perform ance v a ria tio n of th ese schem es if certain code param eters are varied, as for instance th e in terleav er size of th e codes, th e decoding algorithm , or th e code complexity. We will p rese n t novel solutions in o rd er to low er th e error-floor effect in Turbo Codes. We will show how th e perform ance of th ese schem es m ay be im proved by increasing th e com plexity of th e codes. In C hapter 6 w e also design entirely new iterative decoding schem es of serially concatenated convolutional and Reed-Solom on codes. We will show th a t th e se schem es m ay be favourably com pared to conventional Turbo Codes.

In C hapter 7, w e investigate coding and m odulation schem es w here th e in te n tio n has been to design th e se schem es w ith high spectral efficiency in m ind, trad in g th is off w ith th e synchronisation problem s associated w ith m odulation schem es w ith a large sym bol a lp h ab et. We will argue th e case th a t it m ay be of in te rest to use a low a lp h a b e t m o d u la tio n schem e, b u t to increase th e rate of th e codes, th ere b y concluding th a t high ra te block codes m ay be a suitable choice for th e constituent encoders in iterativ e coding schem es. A w ide range of novel coding schem es will be considered, including th e use of high ra te Reed-Solom on and binary BCH codes in parallel concatenation, th e concatenation e n abling th e use of iterativ e decoding algorithm s. The results of th ese schem es will be

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CHAPTER 1. PREFACE ₄

com pared w ith th a t of conventional Turbo Coded m odulation.

In C hapter 8 we draw som e conclusions ab o u t this study, and we indicate som e new directions in which future research w ould be beneficial.

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Chapter 2 System Definitions

In this chapter w e will define and quantify th e p aram eters w ithin a com m unications sys te m which will directly affect th e perform ance and characteristics of th e channel coding system . This im plies th a t th e channel encoder, th e m o d u lato r and d em o d u lato r as well as th e decoder will be described in th e following sections. We will quantify w h at is m ea n t by a ‘soft in p u t' or 'so ft o u tp u t’, enabling us to give a com plete description of som e code decoding algorithm s in an o th e r chapter.

The block diagram of th e elem ents in a com m unications system th a t will be tre a te d here are show n in Figure 2.1. It is assum ed th a t th e source is binary, o u tp u ttin g a sequence of 0's and l 's w ith equal probability. This sequence is th e n applied to a channel e n coder, w hose ta sk it is to tran sfo rm th e source d a ta into a new binary sequence w ith a o n e -to -o n e relationship betw een th e input and o u tp u t sequence. The purpose of this tran sfo rm atio n is to enable th e signals to b e tte r w ithstand th e effects of channel im p air m ents, such as noise, fading and jam m ing. The original inform ation can th e n be decoded in th e receiver by m eans of a decoder. Usually, th e aim of channel coding is to reduce th e probability of bit error, o r to reduce th e required E b/N 0 for a given Bit Error Rate (BER), w here E t is th e tra n sm itte d energy per bit of inform ation and N 0 is th e noise pow er spec tra l density. The price to be paid for this is usually an increase in th e required system bandw idth. Channel coding m ay be divided into tw o m ain classes, nam ely w aveform coding, which transform s signal waveform s into m ore ro b u st w aveform s, and stru ctu red

sequences, which tran sfo rm data sequences into m ore ro b u st sequences by adding re 

dundancy bits (parity bits) which will be used to correct or d e te c t errors in th e received sequence. It is th e last class of channel coding which form s th e area of this research, and th e te rm used for this ty p e of coding is norm ally Forward Error Correction Coding (FEC).

The ta sk of th e m o d u lato r is to m ap th e encoder o u tp u t sequence into a s e t of analogue channel sym bols w ith characteristics th a t enable propagation th ro u g h th e com m unica tions m edia. This m apping m ay take a variety of different form s, b u t in Section 2.1 and

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CHAPTER 2. S Y S T E M DEFI NI TI ONS 6 Channel Decoder Sink Channel Encoder Modulator Source Demodulator

F igure 2.1: Elements of communications system. Filtering of the signal is inherent in both m od ulator and demodulator.

2.3 w e will seek to generalise this m apping, as well as to give examples from specific types of m apping. In practical system s, th e class of m apping used will depend on th e characteristics of th e m edium through which th e signal propagates. In Section 2.2 w e will consider tw o types of channels, nam ely th a t of th e additive w hite Gaussian noise (AWGN) channel and th a t of th e m em oryless Rayleigh fading channel. The following analysis m ay be found in a range of standard text books, including [26] and [2 7 ].

2.1 The Modulator

In th e m odulator, th e encoder o u tp u t sequence is m apped into a s e t of analogue channel sym bols. This m eans th a t, a t any tim e i, a sequence of m binary sym bols is tran sfo rm ed into one of a s e t of M = 2m channel w aveform s. We denote this se t of channel w aveform s

{si(t)}, I e { 1 . .. M } . We shall assum e th a t this m apping is m em o ryless in th e following, i.e. th a t th e m apping of inform ation sequence into a channel w aveform does n o t depend on previously tra n sm itte d waveform s.

A channel w aveform is characterised by being a sinusoidal signal able to p ropagate through th e radio channel. In ord er to rep resent one of M possible, tim e-d isc re te bi nary inform ation sequences, one or m ore param eters of this sinusoid has to be varied according to a decision rule which depends solely on th e input sequence. This p a ra m ete r could be frequency, so th a t we have frequency m odulation, it could be am plitude, im plying am plitude m odulation, or th e phase of th e signal could be varied, im plying phase m odulation. The choice of m odulation schem e would depend heavily on th e channel

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CHAPTER 2. S Y S T E M DEFI NI TI ONS 7

characteristics. In o rd er to lim it th e am ount of analysis, we will concentrate on M -ary Phase Shift Keying (M-PSK) system s. These m odulation schem es are attractiv e in m any applications, e.g. G eo-stationary satellite transm ission [28], as all channel sym bols have equal energy, th ere b y reducing im perfections associated w ith n on-linearity in th e s a te l lite am plifiers.

2.1.1 Representation of M-ary Phase Shift Keying (M-PSK) Signals

We will in th e following consider a 'o n e -sh o t' scenario, i.e. th a t only one sym bol is being tra n sm itte d . This is done in o rder to simplify th e calculations. In M -PSK system s, th e

M channel w aveform s are represented by [27]

w here g ( t ) is th e signal pulse shape and th e tra n sm itte d phase 8t = ^ ( l - 1) + 0, I e

( 1 . . . M }, w here 0 is a phase bias, e.g. th e DVB-S stan d ard [28] which defines Q uad ratu re Phase Shift Keying (QPSK) m odulation, i.e. M = 4, w ith 0 = 7t/4. f c is th e carrier frequency, Ts is th e d u ratio n of one channel sym bol and

s i ( t ) J{27T(l-l)/M+4>) j 2 i r f c t

(2 .1)

I, = cos + (2.2)

and

Q* = s in t j(1~ 1) + <^ (2.3)

We notice from th e above th a t all of th e M possible channel sym bols have equal energy, nam ely

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CHAPTER 2. S Y S T E M D EFI NI TI ONS 8

Assuming for sim plicity th a t / 0T" g(t)2dt = 1 , th e signal energy is given by E s = m E cb.

Also, equ atio n (2.1) represents a tw o dim ensional signal as defined by tw o o rthonorm al signal w aveform s f i ( t ) and f 2(t), so th a t

si(t) = s n f i ( t ) + s i2f 2(t), (2.6) w here

fi {t ) = 5 ( ^ y |- c o s ( 2 7 r / ct), (2.7) and

f2(t) = - g ( t ) i j ^ s m ( 2 7 r f ct), (2 .8)

w here th e orthogonality arises because

t5

J

f i ( t ) ■f 2{t)dt = 0. (2.9) o

and th e orth o n o rm ality arises from (in addition to equation (2.9))

t3 t s

J

=

J

{f2{t))2dt = 1. (2.10)

0 0

We also observe th a t s n and s/2 are given by

s n = V m E cbIi (2.11) and

si2 = \ j mEcbQi (2 .12)

Figure 2.2 show s how an M -PSK signal m ay be generated in a straightforw ard m anner. The binary inform ation sequence first enters th e m apper, which m aps th e in p u t sequence into a sym bol, defined by th e C artesian coordinates // and Qi, which are th e n scaled, filtered and m odulated independently.

The pulse shaping filter

The role of th e pulse shaping filter is tw o-fold: Task one is to lim it th e frequency range over which th e filter input signal exists. Secondly, th e role of th e filter is to sh ape th e signal so th a t Inter-Sym bol Interference (ISI) is m inim ised. To avoid excessive ISI, th e pulse shaping filters should be of such a natu re th a t a t th e optim um sam pling p o in t of one pulse, all signal pulses apart from the current should have value zero [27, 26]. As ‘brick-wall' filters w ith th e theoretical sinc(^-) im pulse response are n o t viable in practice, th e

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CHAPTER 2. S Y S T E M D EFI NI TI ONS Jt ? c o s( 2tt/ c ( ) 9 g(t) 2 ^ ^ ^ cos [2nfct 4- j j ( l — 1) In p u t d ata Symbol M apper / / i C . a = 0.35 sin (27r f ct)

F igure 2.2: M-PSK signal generation

raised cosine, R C , pulse shape is w idely used. The frequency response of such a filter is

given by

f 1 : | / | < f N ( 1 - a )

Hr cU ) = < | + : f N ( l - a ) < |/ | < / w(l + a ) , (2.13)

[ 0 : | / | > / j v ( l + a)

w here f x = 1/2T S is th e N yquist frequency, and a is th e roll-off, or excess b andw idth factor. This is th e stan d ard R C form at [27, 28, 2 6 ]. Our aim is th a t th e signal should undergo this frequency response in th e e n d -to -en d transm ission, i.e. from source to sink. O ften it is assum ed th a t th e channel itself has un it gain for all frequencies, i.e. a pure AWGN, or a m em ory-less fading channel, and hence th e frequency response of th e

R C m ay be sp lit into th e tra n s m itte r filter and th e receiver filter only. Thus w e have

H r c V ) = H t ( f ) ■ H r ( f ) , (2.14) w here H t ( f) is th e tra n sm itte r filter and H r ( f ) is th e receiver filter. The o th er condition w e w ould like to im pose on th e filter is th a t th e receiver filter should be m atched to th e pulse shaped signal. Hence,

H r { f ) = (2.15)

w here th e * denotes complex conjugacy. As in our case, th e I and Q channels are filtered in d ep en d en tly as purely real signals, we can sim ply e q u ate H r ( f ) and H t ( f ) . This im plies th a t th e filters in b o th tra n sm itte r and receiver should be

H r ( f ) = H t ( S) = v ' H n c ( f ) - (2.16)

We will d e n o te this filter th e y/R C filter. Figure 2.3 show s th e frequency response of b o th th e R C and th e V R C filters for an excess bandw idth figure a = 0.35, as defined in th e ETSI DVB-S stan d ard [2 8 ], as well as th e responses w hen a = 1.0. It is noticeable from th e Figure, or by inspection of equation (2.13), th a t th e higher th e a , th e m ore

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bandw idth is used. The advantage of using a higher a is th a t th e synchronisation issue becom es considerably easier [29]. A high value of a implies th a t th e im pulse response of th e y/R C filters decays quickly, th u s reducing th e IS1 if th e receiver sam pling clock is slightly o u t of synchronisation. 0.8 c 0.6 'co (D 0.4 _{RG (1;,0) i} RRC (1.0) RG (Q.35) RRC.(0.35.) 0.2 0.0 -0.5/T 0.0/T 0.5/T 1.0/T

N orm alised Frequency

Figure 2.3: Frequency response of the y/RC filter and the RC filter.

2 . 2

The Channel

In th is Section, we will define tw o types of channels, nam ely th a t of th e A dditive W hite Gaussian Noise (AWGN) Channel, and th a t of th e Rayleigh fading channel. Both th e se ty p es of channels are widely used in th e evaluation of com m unications system s.

2.2.1 The Additive White Gaussian Noise Channel

As th e nam e im plies, th e AWGN channel causes w hite noise w ith a Gaussian am p litu d e probability density to be added to th e signal. The noise arises from th e ever p rese n t th e r m al noise in th e tra n sm itte r and receiver equipm ent. The te rm ‘W hite’ is given because th e noise is un-filtered, and hence it has a frequency spectrum which, w hen averaged

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over tim e, has a constant pow er over all frequencies. The probability density function of a Gaussian ran d o m variable n x is given by

PAWGnOs) = ^ ^ f e"(X‘"/iz)2/2<72’ (2'17)

w here n x is th e m ean and a 2 is th e variance of th e Gaussian random variable, x is th e dim ension along which n x m ay vary; in this case this w ould be th e am p litude of th e noise. The received signal will th u s be given by

r ( t ) = s(t) + n(t), (2.18) w here s(t) is th e band-lim ited tra n sm itte d signal, typically defined by th e sq u a re -ro o t of th e filter definition given in eq u atio n (2.13). The noise in th e channel theoretically exists over an infinite range of frequencies, and th e to ta l noise pow er is th u s given by

im plying th a t th e un-filtered noise has in fact infinite pow er, assum ing th e noise pow er spectral density N 0 ^ 0. H ow ever, this is n ot so in th e receiver. The signal and noise will be filtered, and in term s of am plitude response vs. frequency, th e overall filtering of th e signal is th a t of R C filtering, w hereas th e noise is only filtered w ith a y/R C filter. At th e o u tp u t of this filter, th e noise is n o t 'w hite' w ithin th e given b andw idth of th e signal any longer, which is to say th a t it will have a roll-off region as defined by th e filter. It is convenient to define an equivalent noise bandw idth B neq over which th e noise sp ectru m is w hite, and which gives th e sam e noise pow er as th e actual system . It is straightforw ard to find B neq w ith knowledge of th e receiver filter characteristics. The receiver filter am p litu d e response is given by H r ( f ) and th e w hite noise a t th e in p u t to th e filter has a pow er spectral density of N 0/2 over all frequencies. The m ean sq u a re noise pow er a t th e o u tp u t of th e filter is th u s given by

+oo

K =

j

^ \ H r(f)\2df

(

2

.

20

)

—oo

We w an t to define th e bandw idth B neq as th e frequency range over which th e noise can be th o u g h t of as having th e noise spectral density N 0/2 , b u t still give th e sam e m ean sq u are pow er as defined in equ atio n (2.20) [26]. The pow er of th is noise w ould be equal to

K = B neq ■ ^ ■ \Hr(fo)\2, (2.21)

w here \Hr ( f 0)\2 is th e square of th e gain of th e filter a t th e centre frequency / 0 of th e filter. For a baseband, sym m etric, double sided filter, th is im plies th a t H r ( f 0) = H r (0).

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CHAPTER 2. S Y S T E M DEFI NI TI ONS 12 Thus, +00

/

Bneq + 0 0 / I “ CO (2 .22) | t f r ( / 0 ) | 2

It is of in te rest to define th e equivalent noise bandw idth of th e system w hen th e receiver filter is th e square ro o t of th e filter defined in equation (2.13). This filter gives zero gain a t / > /w ( 1 + a), and |iTr (0)|2 = 1 for this filter, implying th a t

f N ( l - a ) / j v ( l + a )

= 2 <

I df+

/

Bneq 1 1 . 2 + 2 Sm 0 f N ( l - C t ) = 2/jv(l — o) + /at( 1 + o;) — / / / ( l — a) + 2 a/iv * _ ( In - f %fN a df + -7r [c° s ( _ ^ a ) ) - c o s( ^ Q)] = 2 /at (2.23)

This m eans th a t th e (double sided) equivalent noise bandw idth of th e y/R C filter is equal to th e sym bol rate, independently of th e excess bandw idth Figures. As th e filter is sym m etric around its centre frequency, th e equivalent noise frequencies are in th e range "257 < / < afc- The to ta l noise pow er is thus equal to

p - = i ^

n Tc 2 (2.24)

The to ta l pow er of zero-m ean Gaussian noise is also given by a2. From this, w e get th e relationship

No

2 _XJneq

= Tsa 2 W/Hz. (2.25)

For th e case of m odelling th e com m unications system , it is convenient to assum e th e sym bol period Ts = 1, so th a t iVo = 2cr2.

2.2.2 The Fading Channel

In this section a s h o rt tre a tm e n t of th e Rayleigh fading channel will be given, which will su it th e purpose of defining th e m odels used in sim ulations of codes and decoders, results of which will be presented in su b seq u en t chapters. For a m ore general ap proach to th e issue of fading com m unications m edia, references [30] and [31] could be referred to .

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CHAPT ER 2. S Y S T E M D EFI NI TI ONS 13

Also, reference [27] tre a ts th e subject of fading com m unication m edia in considerable detail.

The use of n atural, i.e. n o t m an-m ade m edia for radio com m unications im plies unavoid able involvem ent w ith th e random fluctuations which often accom pany n atural ph en o m ena. Thus, th e a tte n u a tio n experienced in propagation m ay fluctuate, or th e propagation p a th length m ay change. M oreover, several different transm ission paths m ay exist and it m ay be unavoidable to excite th e several different paths sim ultaneously. N ature m ay n o t be th e only source for th e occurrence of such difficulties; e.g. reflections in buildings, aeroplanes or even satellites will cause considerable fluctuations in th e com m unications m edia.

V ariations in th e channel which tak e place in a tim e interval m uch sh o rte r th a n th e s h o rt e st d u ratio n of in te re st to th e com m unications application, i.e. much s h o rte r th a n th e sym bol d u ratio n in digital transm ission, will, depending on th e front end of th e receiver, usually be evidenced in som e averaged form . At th e o th er extrem e, fluctuations m ay also tak e place relatively slowly. These changes could occur in fractions of an hour, daily, m onthly o r seasonally. The com m unications system will evidently have to be designed to cope w ith th e w o rst case scenario, b u t it is unavoidable th a t th e se fluctuations im pose varying signal to noise ratios.

Of p articular in te re st in th e design of th e com m unications system are th e fluctuations w hich occur in tim e intervals in -betw een th e tw o extrem es described above. This ty p e of fading is dom inated by m u lti-p a th fading. The m eaning of m u lti-p a th is im plicit in its nam e: The propagation m edium contains several distinguishable p aths, or beam p a t te rn s, so th a t som e fraction of th e to ta l received energy unavoidably arrives over each p a th . For th e p u rpose of sim plicity, it is b est to view th is dispersiveness of th e beam p a tte rn s as a s e t of ray path s along which th e electro-m agnetic energy propagates. The d ifferentiation b etw een several paths implies th a t th ese path s are resolvable, i.e. th e ir lengths are sufficiently different th a t signals sta rtin g o u t sim ultaneously on each ray can be distinguished as arriving sequentially. Accordingly, th e com m unications receiver is faced w ith superpositions of ‘echoes' of current and previous sym bols, which is w h a t is m ea n t by th e te rm m u lti-p a th fading. The super-positioning of th e sym bols m ay add constructively or destructively, depending on th e values of th e echoes and th e tru e sym bols.

If such changes occur random ly and continually, th e observed resu lta n t carrier will corre spondingly change random ly in th e envelope and phase relative to som e fixed reference phase. A statistical m odel of th ese random fluctuations is th e Rayleigh fading m odel, and will be d ealt w ith in th e following Sections.

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2.2.3 Characterisation of the Multi-Path Fading Channel

Using a complex envelope approach, a tran sm itted signal Si(t) a t som e carrier frequency

f c can be expressed as [27]

8i(t) = Vt [ui(t)ej2nfct], (2.26) w here Ui(t) denotes th e envelope of th e signal. Ignoring any AWGN, th e received b a n d  pass signal will th e n tak e th e form

s (t ) = Y , a p(t)si(t - t 0 - t p ) , (2.27)

p

w here t 0 is som e conveniently chosen representative value of th e average propagation tim e, rp is th e additional relative delay on th e pth p ath and th e real num ber a p is th e p a th a tte n u a tio n factor for th e pth p ath . Substituting equation (2.26) into eq u atio n 2.27 yields th e following expression for s (t ):

s(t) =

Y

a p(t)ui(t - Tp ) e j 2 n f c ^ Tp) (2.28) . p

w here th e average transm ission tim e t 0 has been ignored. The equivalent low -pass re ceived signal r(t) is given by

r W = Y f a P ^ Ui^ ~ Tp)e~j2wfcTp• (2.29)

p

If Ui(t) is an im pulse 8{r,t), r(t) denotes th e tim e variant (varying w ith r ) im pulse re sponse h(r, t) of th e channel, i.e.

h (r ,t ) = Y / a p(t)fi{r ~ Tp)e~i2nf cTp. (2.30)

p

If w e s e t Ui(t) = 1, i.e. th e tra n sm itte d signal is a sinusoid w ith am plitude 1 and frequency

f c, r(t) becom es r(t) = Y * P(t)e~j2nfcTp- (2.31) i.e. p(t) cos($p) - j ^ a :p ( £ ) s in ( $ p ) , (2.32) p p w here th e phase $ p = 2 n fcTk

It is evident th a t th e phase is a random variable which will have a uniform d istrib u tio n in th e interval { —7r.. .7r}, as it is assum ed th a t th e r k are uniform ly d istrib u ted aro u n d

t 0, and th e frequency f c is a constant and f c ^ 0.

We use th e n o tatio n X = a p(t) cos($p) and Y = Z)Pa pW sin(^p)» a n d in th e lim it

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central lim it th eo re m , so th a t th e sum s in expression 2.32 becom e tw o Gaussian random variables w ith zero m ean. Hence, X and Y are ind ep en d en t Gaussian random variables. W hen this is th e case, and also w hen th e X and Y are indep en d en t from sam ple to sam ple, th e channel th ro u g h which th e signal propagates is denoted th e fully in terleaved Rayleigh

fading channel. The envelope of th e signal a t any one tim e in sta n t will be given by

r,(«) = V x 1 + r 2,

and th e phase of th e received signal a t any tim e in sta n t is given by

(2.33) ta n ' ( £ ) ta n -1 (y) + f s g n ( y ) 0 ifX > 0 ifX < 0 ifX = 0, r ^ 0 i f x = o, r = o (2.34)

The d istrib u tio n of r e(t) is related to th e central x 2 d istrib u tio n w ith 2 degrees of freedom , and has been nam ed th e Rayleigh distribution.

If X and Y have non-zero m ean values, th e d istrib u tio n of r e{t) is related to th e n o n central x 2 d istrib u tio n w ith 2 degrees of freedom . The resu lta n t signal will th e n be biased in a certain direction, which will often be th e case in a real com m unications system , since it is likely th a t one particular propagation p a th will dom inate th e scenario. In th is case, th e d istrib u tio n of r e(t) is th a t of th e Rice distribution. We will n o t be concerned w ith this in th is w ork, instead concentrating on th e lim iting cases of th e pure AWGN and th e fully interleaved Rayleigh fading channels, as th ese tw o channel m odels will indicate th e b e st and w o rst case scenarios respectively.

The Rayleigh Probability Density Function

Starting w ith equations (2.33) and (2.34), we se t R = r e(t) and 0 = 6(t). Because X and

Y are in d ep en d en t Gaussian random variables, th e jo in t d istrib u tio n of X and Y is

P x r ( x , y ) = (2-35)

w here X and Y have th e sam e variance a 2. To find th e d istrib u tio n of th e envelope of th e signal, w e have to tran sfo rm th e pdf’s from th e C artesian dom ain involving X and

Y to th e polar dom ain and th e n re-express th e density in this new co-ordinate sy stem

involving R and 0 , (R > 0, 0 < 0 < 2tt), w ith th e transform ations given in equ atio n s

(2.33) and (2.34). The inverse transform ation is given by

X = R cos(0) and

Y = jRsin(0), (2.36)

The jo in t density of R and 0 is

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and it m ay easily be shown, e.g. [32], th a t

P{r < R < (r 4- dr), 6 < 0 < (6 + dd)) = p x y ( r cos(6),rsin(9))rdrd9 (2.38) and, com bining w ith equation (2.37),

Pr o(r,9) = rpx y(r cos (9), r sin (9)). (2.39)

Thus,

PR&(r,e) =

r e -( r2)/2- 2 (2.40)

2ira2

The d istrib u tio n of r is found by averaging PR&( r, 9) over all values of 9 from 0 to 2n. Thus,

27T

Pr{t) =

J

P R e (r ,9 ) d9 = ^ e -7"2/ 2^ , (2.41)

o

which describes th e Rayleigh distribution. The variance of th e d istrib u tio n is a* = (2 - 7r/2)or2 [33]. In o rd er to preserve th e to ta l tra n sm itte d pow er of th e signal, a 2, th e variance of b o th X and Y m ust be equal to 1/2.

The d istrib u tio n and th e cum ulative distribution m ay be seen in Figure 2.4. As m ay be deduced from th e figure, th e expected value of r, E[r], is n o t equal to 1. This im plies

th a t in a Rayleigh fading channel, th e average signal to noise ratio E cb/N 0 in th e receiver is n o t given by th e tra n sm itte d energy per bit over th e noise spectral density - rath e r, one has to tak e into account this fading to m easure th e signal to noise ratio p er bit. In th e sim ulations to be presented in su b sequent chapters, w e have first m easured th e signal energy p er bit (tim e averaged) and se t th e noise level in accordance to th e desired

received E cb/N 0.

In reference [34] m any of th ese issues are addressed, including th e op tim u m design of Turbo Codes over th e Rayleigh fading channel.

2.3 The Optimum Receiver In The Memoryless Channel

In th is section th e design of th e optim um receiver will be considered w hen th e chan nel th ro u g h which th e signal has travelled is m em oryless, which is th e case for b o th th e AWGN channel and th e fully interleaved Rayleigh fading channel tre a te d in th e preced ing Sections. We will assum e optim um m atched filter receivers. Then, th e subject of m axim um likelihood and m axim um a posteriori decision rules will be tre a te d , th e re b y defining a soft decision o u tp u t. The tre a tm e n t will be restricted to concern only th e d e te c tio n of (I Q) m odulated signals, as is th e case for th e general M -PSK signal defined in eq u atio n (2.1), although in th e initial section we will consider th e general case.

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C HAPT ER 2. S Y S T E M DEFI NI TI ONS 17 1.0 0.8 Rayleigh distribution Cumulative distribution 0.6 0.4 0.2 0.0 0 1 2 3 r

Figure 2.4: The pdf of the Rayleigh fading envelope re(t) and its accumulated distribution

2.3.1 The Optimum Detector

T reatm ents of th e optim um d etecto r is given in b o th references [26] and [27]. H ere w e will use th e m ain results of these.

At tim e t = Ts, th e o u tp u ts from th e m atched filters are

ri = VsiiTs) + y ni(Ts), (2.42) w here y Si(Ts) represents th e signal com ponent and y ni(Ts) represents th e noise com po n e n t of r i. E quation (2.42) is th e o u tp u t th a t maximises th e o u tp u t S / N ratio. The o u tp u t

S / N ratio is defined as

So_ _ _ f s (Ts) ,2 43x

No E[yl(Ts)Y

w here E[*] denotes th e expectation o perator. Thus, since th e noise is assum ed to have zero m ean, E[y2(Ts)] is sim ply th e variance of th e noise, a 2, w here a 2 = N 0/2 , if w e assu m e u n it sym bol duration. We d enote th e sym bol signal energy to be E s , or in term s of th e previous definitions of th e M -ary PSK signal, E s = mEb. Thus, in line w ith previous results, th e m axim um signal to noise ratio obtainable a t th e o u tp u t of th e d e m o d u la to r

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(assum ing no am p litude fading, i.e. th a t p = 1 for all signals) is given by

k = ( 2 M )

2 E s N 0 ’

w here E s is th e energy of th e original signal in th e interval 0 < t < Ts

(2.45)

We shall define th e o u tp u t from th e channel, as given in equation (2.42), th e so ft decision

channel outputs.

A m atched filter dem o d u lato r produces a vector r = {ri r 2 . .. rn} in a way th a t m aximises

th e o u tp u t S / N of th e d em o dulator [27]. In this section we will consider th e optim um decision rule based on th e observation of this vector r. Our aim is to design a signal detector th a t m akes a decision on th e tran sm itted signal in each signal interval based on th e observation of th e vector r in each interval, so th a t th e probability of a correct decision is m axim ised. Thus, a decision rule based on th e com putation of th e p o sterior probabilities is ap p ro p riate, w here th e posterior probability is defined as

P (signal si was tran sm itted | r), I = 1,2 , . . . , M, (2.46) which is norm ally abbreviated as P(si | r). Using Bayes’ rule, th e p o sterio r probabilities m ay be expanded into th e full form of th e M axim um a Posteriori (MAP) decision rule.

p { s i , r) = g f r L f f i W , (2.47) w here p(r | si) is th e conditional pdf of th e observed vector given sj, and P (sj) is th e a priori probability of th e Ith signal being tra n sm itte d . The MAP algorithm th u s w orks on choosing th e s e t of signals th a t m aximises th e probability of eq u atio n (2.47).

The d en o m inator of equ atio n (2.47) m ay be expanded into

M

P(r) = ^ P ( r I si)p (*i)- (2-48)

i=i

This is essentially a norm alising factor, so th a t th e sum m ation of all th e probabilities

E p (s' i r) = 1- (2 -49)

i

m ay be o m itted from th e calculations, as it is com m on for all I.

Thus, th e decision rule from equation (2.47) may be re-expressed as choosing th e signal se t si which maximises

P ( s, I r) = p(r | si)P(si). (2.50)

We notice th a t th e evaluation of th e posterior probabilities P (s/ | r) requires know ledge of th e a priori probabilities P(si) and th e conditional pdf p(r | si) for all /. The conditional

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C HAPTER 2. S Y S T E M D EFI NI TI ONS 19

pdf p(ri | su ) depends on th e characteristics of th e channel. As sta te d previously, w e will consider th e m em oryless channel only, and we will assum e th a t th e am p litude fading coefficients are constant over an entire sym bol period. Thus, p (n | su) m ay be expressed as

p{n

I

su

) =

V2

7T<7 exp

i n - psu

)21

2 O-2 (2.51)

w here a 2 is th e variance of th e noise, and a 2 = \ N 0 in th e case of norm alised bandw idth. Also, since N P(r I si) = Y [ p ( r i | s u ) , Z = l,2 ,...,A f 2 = 1 th e n I \ p (si) P (r I si) = N exp

V2

7T<T N

-E

1 = 1 (n - p s u f 2 a 2 / = 1 , 2 , . . . , M (2.52) (2.53)

The am p litu d e fading coefficients will be th e sam e for all /, as this is a uniquely defined variable, which only changes w ith tim e. We recall th a t th e sum m ation p a ra m ete r N is th e nu m b er of basis functions.

W hen th e {st} are equally likely, th e probability P(si) is of no im portance as far as th e decision rule is concerned, and m ay th u s be om itted in equation (2.53). The decision rule w e end up w ith, is th a t of th e M axim um Likelihood algorithm , nam ely th a t by choosing th e signal th a t maximises p (r | si), we have also chosen th e signal th a t m aximises p {r | s*). The m axim um likelihood rule is th u s given by choosing th e signal se t si th a t m aximises

N p(T | si) = x/2 - N exp 7T<7

-E

i= 1 i n - psuy 2cr2 (2.54)

It is easier to w ork w ith th e natural logarithm of equ atio n (2.53), and as th e n a tu ra l logarithm is m onotonically increasing w ith its argum ent, this will n o t affect th e decision rule. By taking th e natural logarithm of equation (2.53), w e end up w ith

1 1 N

ln p (r | s t ) = - - N l n 2 n a 2 - — - p s u ) 2 + ln P (s /), Z = 1 ,2 , . . . , M (2.55)

2 = 1

Since th e first te rm of eq u atio n (2.55) has no effect on th e decision, i.e. it is a constant, th e decision rule for maximising p {r | s{) becomes one of choosing th e signal th a t m inim ises th e sq u a re of th e Euclidean distance w ith th e n atu ral logarithm of th e a priori p robability s u b tra c te d from it, i.e.

N

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CHAPTER 2. S Y S T E M D EFI NI TI ONS ₂₀

This decision rule is d enoted m in im u m distance detection. Equation (2.56) m ay be ex panded further, th ere b y achieving additional reduction in complexity:

N N N

D (r,si) - 2 p 'Y ^ r isii + ^ 2 p 2s2ii - \n P { s i ) , l = (2.57)

i= 1 i=l i= 1

N

Only th e th re e last term s of equation (2.57) have any effect on th e decision, since X) rl

2 = 1

will be equal for all /. Therefore our decision rule becomes th a t of th e correlation d e te c 

tion, i.e. we w an t to m inim ise

N N

D (t,si) = - 2 p J 2 r i S u + J 2 p 2sli - ln P ( » i ) . (2.58)

2 = 1 2 = 1

In th e case w hen all st have equal energy, as is th e case for an M -PSK signal, th e m iddle te rm m ay also be o m itted, and reversing th e signs of equation (2.58), th e decision rule for th ese m odulation schem es becomes one of maximising

N

D (r ,s t) = 2 p 'Y ^ r isu + ln P (s/). (2.59)

i= 1

Also, for th e M axim um Likelihood case, w hen all th e P(si) are equal, th e final decision rule m ay be fu rth er sim plified to be

N

D (r,si) = 2 p ^ n s n . (2.60)

i—1

In this w ork w e shall generally assum e th a t this is th e case in th e channel, i.e. th a t all si are equally probable.

2.4 Conclusion

In this chapter, im p o rta n t param eters in a com m unications system have been identified, and quantitatively and qualitatively described. We have considered th e general case of m em oryless channels, including th e AWGN and th e Rayleigh fading channel. We have also considered th e optim um d etector on th e basis of th ese system p aram eters.

Iterative Decoding of Concatenated Codes.