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UNIVERSITY OF SOUTHAMPTON

Faculty of Engineering, Science and Mathematics

School of Electronics and Computer Science

Comparative Study of Different types of Coded Modulation

schemes using EXIT chart and BER Characteristics

by

Cherian Danny Joseph

23th September 2010

A dissertation submitted in partial fulfilment of the degree of

MSc in Wireless Communication

by examination and dissertation

SUPERVISOR: Dr. Robert G . Maunder


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Abstract

This report, extensively studies on the topic Coded Modulation(CM) a scheme which combines

both the coding as well as the modulation in together to get a bandwidth efficient scheme. This

project contributes a genuine comparative study on different Coded Modulation schemes such

as Trellis Coded Modulation (TCM), Bit interleaved Coded Modulation (BICM),Turbo Trel-

lis Coded Modulation (TTCM) and Bit Interleaved Coded Modulation with Iterative Decoding

(BICM-ID) in context of 8 level Phase Shift Keying(8PSK) over the Gaussian and uncorrelated

Rayleigh channels. Here, the comparison are done in terms of decoding complexity, the band-

width efficiency , the coding gain and the frame length by using the study tools such as EXtrinsic

Information Transfer Charts(EXIT charts) and BER curve characteristics.

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Acknowledgments

Firstly, I would like to thank my supervisor Dr. Robert. G. Maunder for his considerable

support and guidance all throughout the project. The tutorial section at the beginning of the

project was very useful in understanding the concepts of my project. He was tremendously

helpful when ever I faced any technical issues. The weekly meeting arranged helped me to

improve the quality of my project a lot. It was a real pleasure working under him.

I am also grateful Professor Lajos Hanzo and Dr Lie Liang Yang for the wonderful lectures

as well as the suggestions given during the course of Personal Multimedia Communication

which provided me a excellent base to start my Project.

I would like to thank all my class mates for their valuable supports and various helps through-

out the completion of MSc project. Meanwhile, I would also thank University of Southampton

who let me study this wonderful course.

Finally, I would like to dedicate this report to my parents for their unending love, care and

support that they have given me all throughout my life.

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Contents

Abstract i

Acknowledgement ii

List of Figures vi

List of Tables x

1 Introduction 1

1.1 A Historical Background on Coded Modulation . . . . . . . . . . . . . . . . . 2

1.2 Motivation and Organisation of this project . . . . . . . . . . . . . . . . . . . 3

1.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Chapter Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background Literature and Review 5

2.1 Modulation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Mapping Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 7-PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Convolutional Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 State and Trellis Diagram . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Viterbi Algorithm -(VA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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2.6.1 Virterbi Hard Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.2 Virterbi Soft Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Puncturing Convolution codes . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Trellis Coded Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.8.1 TCM Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.8.2 TCM encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Turbo Trellis coded Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9.1 TTCM Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9.2 TTCM Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.10 Bit-interleaved Coded modulation . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10.1 BICM Encoder and Decoder . . . . . . . . . . . . . . . . . . . . . . . 23

2.11 Bit-Interleaved Coded Modulation Using Iterative Decoding . . . . . . . . . . 26

2.11.1 BICM-ID Transmitter and Receiver . . . . . . . . . . . . . . . . . . . 27

2.12 BCJR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.12.1 Log-BCJR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.12.2 Symbol based MAP Algorithm . . . . . . . . . . . . . . . . . . . . . . 31

2.13 Exit Charts Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.13.1 EXIT chart Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.13.2 Area Properties of EXIT chart . . . . . . . . . . . . . . . . . . . . . . 35

2.13.3 EXIT band charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Results And Discussion 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Study of Individual Coded Modulation Schemes . . . . . . . . . . . . . . . . . 40

3.3.1 Uncoded QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 BICM Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 BICM-ID Simulation Results . . . . . . . . . . . . . . . . . . . . . . 44

3.3.4 TCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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3.3.5 TTCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Comparative Study of the Coded Modulation . . . . . . . . . . . . . . . . . . 57

3.4.1 Performance Over AWGN Channel . . . . . . . . . . . . . . . . . . . 57

3.4.2 Performance over Uncorrelated Narrowband Rayleigh Fading Channels 59

3.4.3 Effect of Block Length on Coded Modulation . . . . . . . . . . . . . . 59

3.4.4 Coding Gain versus Complexity . . . . . . . . . . . . . . . . . . . . . 60

3.4.5 Area/Capacity vs SNR . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Case Study:7 PSK over AWGN channel . . . . . . . . . . . . . . . . . . . . . 62

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Management and Planning 68

4.1 Initial Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Available Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Project Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Initial and Final Gantt chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Management Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Conclusions and Future Work 72

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A Publishable Paper 77

B Matlab codes 83

B.1 Original BCJR algorithm developed By Rob . . . . . . . . . . . . . . . . . . . 83

B.2 Symbol based BCJR algorithm for TCM decoders . . . . . . . . . . . . . . . . 86

B.3 2/3 rate Convolutional enocder . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B.4 BICM-ID modulation for 8 PSK using Natural Mapping . . . . . . . . . . . . 91

B.5 Soft Demodulation for 8 PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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B.6 EXIT band chart for inner code of BICM-ID(demapper) . . . . . . . . . . . . 93

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List of Figures

1.1 Factors affecting the design of Channel/Modulation Scheme [1] . . . . . . . . 2

2.1 Modulation and Demodulation of QPSK . . . . . . . . . . . . . . . . . . . . 6

2.2 Constellation Diagram for 4 and 8-PSK . . . . . . . . . . . . . . . . . . . . . 6

2.3 DCMC Channel Capacity for different Constellations under AWGN channel . . 7

2.4 Bit Mapping relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 7-PSK constellation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 7-PSK and 8-PSK capacity compariosn . . . . . . . . . . . . . . . . . . . . . 9

2.7 Convolutional Encoder of rate 1/2 . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8 State Diagram and One Stage of Trellis . . . . . . . . . . . . . . . . . . . . . 12

2.9 Constellation Diagram for 4 and 8-PSK . . . . . . . . . . . . . . . . . . . . . 16

2.10 8 PSK set partitioning [2],Ungerboeck . . . . . . . . . . . . . . . . . . . . . . 17

2.11 Encoder and trellis for the eight-state 8PSK . . . . . . . . . . . . . . . . . . . 18

2.12 RSC encoder and signal mapper forming the TCM encoder . . . . . . . . . . . 19

2.13 TTCM encoder Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.14 TTCM Decoder Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.15 Paaskes non systematic convolutional encoder, with bit based interleaves and a

8 PSK modulator forming a BICM encoder,employing the Gray Mapping . . . 24

2.16 BICM decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.17 Paaskes non systematic convolutional encoder . . . . . . . . . . . . . . . . . 25

2.18 Trellis Diagram for Paaskess eight state convolutional encoder . . . . . . . . . 25

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2.19 Reciver of BICM-ID with iteration . . . . . . . . . . . . . . . . . . . . . . . . 27

2.20 A portion of Trellis diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.21 Exit chart of a 8-PSK De-mapper used in BICM-ID employing Natural Mapping 32

2.22 Scheme for generating the inner code EXIT charts . . . . . . . . . . . . . . . . 33

2.23 Scheme for generating the outer code EXIT charts . . . . . . . . . . . . . . . . 33

2.24 Convolutional Decoders EXIT chart for various memory and Generator Polyno-

mials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.25 Trajectories for iteractive decoding when Eb/N0 = 5 dB . . . . . . . . . . . . . 35

2.26 Mutal Infomration trajectories and EXIT band chart of TTCM , having a 1000

bit random Interleavers at Eb/N0 = 2.5 dB . . . . . . . . . . . . . . . . . . . . 36

3.1 System Overview for different coded Modulation Schemes [1] . . . . . . . . . 39

3.2 BER performance of uncoded QPSK for a Frame length of 1000 bits over AWGN

channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 BER performance of uncoded QPSK for a Frame length of 1000 bits over Rayleigh

channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 BER performance of BICM employing 8 PSK and for various Frame length over

AWGN channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 BER performance of BICM employing 8 PSK and for various Frame length over

Rayleigh channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 BER performance of BICM for various Constraint length . . . . . . . . . . . . 43

3.7 Externisic infomration transfer chart for the outer-code with different memeory 45

3.8 Externisic infomration transfer chart for the Demodulater/Demmaper for various

SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.9 BICM-ID performance using different number of iterations for a frame length

of 3000 information bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.10 EXIT band Chart trajectories for BICM-ID at a SNR =4.5 dB for various Frame

lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.11 BER performance comparison for different Frame length under AWGN channel 48

3.12 BER performance comparison for different Frame length under Rayleigh channel 49

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3.13 BER performance comparison for different code memory over AWGN channel 49

3.14 BER performance comparison for Different frame length under AWGN Channel 50

3.15 BER performance comparison for Different frame length under Rayleigh Channel 51

3.16 EXIT band chart for TCM decoder for various interleaving frame length at SNR

= 2.5 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.17 EXIT band chart for TTCM decoder for various interleaving frame length at

SNR = 2.5 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.18 TTCM BER performance using different iterations . . . . . . . . . . . . . . . 54

3.19 EXIT band chart trajectories for various Frame lengths at SNR=2.5 dB . . . . . 55

3.20 BER performance comparison TTCM for Different frame length under AWGN

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.21 BER performance comparison for TTCM for Different frame length under Rayleigh

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.22 BER performance of Coded modulation employing 8 PSK and using a Frame

length 2000 information bits . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.23 BER performance of Coded modulation employing 8 PSK and using a Frame

length 2000 information bits . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.24 Effects of block length on Coded Modulation . . . . . . . . . . . . . . . . . . 59

3.25 Coding gain at a BER of104 against the decoding complexity when compared

to the uncoded QPSK under AWGN channel . . . . . . . . . . . . . . . . . . . 61

3.26 Coding gain at a BER of104 decoding complexity when compared to the un-

coded QPSK under Rayleigh channel . . . . . . . . . . . . . . . . . . . . . . . 61

3.27 Area beneath EXIT function and DCMC Capacity plots in AWGN channel . . 62

3.28 BER comparison of BICM using 7-PSK and 8-PSK having a frame length 1000bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.29 EXIT Chart Comparison for de mapper of 7-PSK and 8-PSK at an SNR=0 dB . 63

3.30 BER comparison of BICM-ID using 7-PSK and 8-PSK having a frame length

1000 bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.31 EXIT Chart Comparison for de mapper of 7-PSK and 8-PSK at an SNR=2.5 dB 65

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3.32 BER comparison of TCM using 7-PSK and 8-PSK having a frame length 1500

bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.33 BER comparison of TTCM using 7-PSK and 8-PSK having a frame length 1500

bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Purposed Gantt chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Actual Gantt chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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Chapter 1Introduction

Channel Coding or Forward Error Correction(FEC) is the technique developed to combat the

effects of the channel impairment and help the receiver in its decision making process.The de-

sign of a good FEC depends on the various factors and can be illustrated in the Figure 1.1. It

is feasible to design a good coding schemes which are capable of reducing the Bit Error ra-

tio(BER) specifically for a given transmission channel. However, this implies there should be

further investments in terms of required complexity , the coding/interleaving delays and effec-

tive throughput. As time progressed there were different solutions developed for different codefeatures. For example, considering the case of wireless scenario where the power is a important

constrain, naturally the power reduction is an extremely important factor.

On the other hand, channel coding as well a modulation schemes can be joined together

to get a high rate channel coding schemes in collaboration with multilevel/phase modulation

schemes. In the project the main objective is to study such a scheme where coding gain can be

achieved without any bandwidth expansion and can be named as Coded Modulation. This

project studies a variety of coded modulation assisted system and will be investigating their

propagation in wireless environments.

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1.1. A HISTORICAL BACKGROUND ON CODED MODULATION 2

Implementationalcomplexity

Coding/interleavingdelay

Channel charactristics

Bandwidth

EffectiveThrougput Coding rate Coding Gain

Bit Error RateScheme

Coding/Modulation

Figure 1.1: Factors affecting the design of Channel/Modulation Scheme [1]

1.1 A Historical Background on Coded Modulation

The history of channel coding started with the pioneer works of Shannon [3] in the year 1948. He

predicted a reliable communication can be achieved with the aid of the channel coding by adding

redundant information bits to the transmitted signal. However, he didnt proposed any channel

coding scheme for real time implementation. This gave overall motivation for Hamming [4] andGolay [5], to come with first practical error control schemes known to be block codes. Later,

Convolution codes was introduced by Elias [6] in 1955 and a number of decoding algorithm

were developed by Fano [7] and Massey [8]. A breakthrough in convolutional decoding was the

invention of the maximum likelihood decoding sequence developed by Viterbi [9]. In 1970s

there were successful implementation of convolutional codes in deep space probes. In 1974 a

more complex Maximum A Posteriori(MAP) [10] algorithm was developed and was capable of

achieving minimum BER.

As time progressed and the evolutional of mobile communication system which has both

constrains in power and bandwidth-limited scenarios , in 1987 Urgerbock came with a new

bandwidth-efficient scheme which employed Set-Partitioning (SP) [11] for signal labelling and

this scheme came to be known as Trellis Coded Modulation [2].TCM combined both the con-

volutional encoder and multilevel/phase signal sets constituting a bandwidth efficient scheme


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1.2. MOTIVATION AND ORGANISATION OF THIS PROJECT 3

which is suitable for mobile/wireless communications [12]. Lately Zehavi [13] and Caire [14]came

up with powerful coded modulation scheme which utilised bit based interleaving in conjunction

with Gray signal labelling came to be know as Bit-Interleaved Coded Modulation. Another

milestone in history of error correcting codes was the invention of Turbo codes [15] by Berrou,

Glavieux and Thitimajshina in 1993 which used MAP [10] as its decoding algorithm and was

capable of approaching the Shannon limit. As a results this turbo codes were utilised in modern

third generation(3G) mobile phones as their Standardised error control scheme [16].

However, Turbo codes had a disadvantage of having a low coding rate and hence the re-

quirement for high bandwidth is inseparable. Therefore, there was many researches going on to

find a better efficient method how the bandwidth could be saved. A higher spectral efficient bi-

nary turbo codes codes knows as BICM-bases Turbo Coded Modulation [17] was introduced in

1994. A more recent and better Bandwidth efficient Turbo Trellis coded modulation(TTCM) [1]

scheme was introduced which has same properties as of Turbo codes but, by adding new punc-

turing mechanism the bandwidth utilisation is reduced here. In 1998 , Lie [18] proposed a new

iterative joint decoding and demodulation assisted BICM known to be Bit Interleaved Coded

Modulation with Iterative decoding(BICM-ID) which uses Signal Partitioning as its signal la-

belling scheme. The main focus of this project is to study and compare the performance of TCM,

BICM, TTCM and BICM-ID schemes with the uncoded QPSK using Gray mapping scheme.

1.2 Motivation and Organisation of this project

1.2.1 Motivation

The radio spectrum available is a scarce resource and they are extremely costly. Therefore, how

efficient the bandwidth could be exploited to accommodate the ever-increasing traffic demands

is the main question. The coded modulation is capable of achieving the substantial coding gain

by expanding the multi-point in the symbol mapping keeping the bandwidth the same. The

fundamental objective of the project is to study this novel schemes and trying to evaluate the

performance in terms of decoding complexity, the bandwidth efficiency , the coding gain and

the frame length for all the four coded modulation by the aid of the BER characteristics and


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1.2. MOTIVATION AND ORGANISATION OF THIS PROJECT 4

EXIT chart techniques. Its interesting to note that the EXIT chart have been invented since all

of the coded modulation were. So we can use this techniques to to make a good comparison that

were not featured in the original coded modulation papers.

1.2.2 Chapter Organisation

Chapter 2: Four different coded modulation such as TCM, BICM, TTCM and BICM-

ID are introduced. Each of these coded modulation is studied separately in terms of the

overall coding structure, signal labelling types both Gray and Set-Partitioning, interleaving

and mainly the BCJR Log based decoding philosophy. The BCJR algorithm for both the

bit based and symbols based algorithm are also highlighted.

Chapter 3: The above said coded modulation performance are studied separately when

communicating over the AWGN and the narrowband channels with aid of the EXIT charts

and BER curves. Extra care has been taken to make the comparative study genuine in

terms of the complexity of structure, interteaver length and block length.

Chapter 4: This section explains the management and planning section. The main task

explained here is the Initial project scope, the available system resource for example soft-

wares used for the simulation, the project tasks which describes the how difficult the given

task where, what all Risks that was encountered during these project, the Initial and Final

Gantt chart and Finally the management techniques used for the successful completion of

this project.

Chapter 5: The major findings are summarised here.

Appendix A: A publishable paper in the format of IEEE for this project had been devel-

oped and shown here.

Appendix B: Important Matlab codes are given here.

Having presented an overview of the project, lets discuss on detailed discourse on coded

modulation in the coming chapters.


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Chapter 2Background Literature and Review

2.1 Modulation Scheme

During transmission of the signal, modulation is a process that is used to convey the information.

Modulation can be done either in digital or in the analog domain. Modulation are done with the

aid of the carrier wave, which is usually high frequency wave to convey the information of

much lower frequency input signal.The main types of Digital modulation are Binary Phase Shift

keying (BPSK), Frequency shift keying(FSK), Quadrature amplitude Shit keying(QAM) etc.The modulation and demodulation can be expressed in complex notation i. e) in-phase and

Quadrature components which are real and imaginary parts of the signals. The complex notation

are convenient to represent since in phase and quadrature component parts of the signal behave in

the same way as the real and the imaginary parts of the complex numbers. Let the the transmitted

signal be x(t) be expressed as shown below.

x(t) = xi(t) + j.xq(t)

The transmitted signal y(t) is obtained by just taking the real part of complex carrier part

(ct)) of the signal is:

y(t) = Re{x(t). exp(jct)} = xi(t).cos(ct) + xq(t).sin(ct)

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2.1. MODULATION SCHEME 6

a) QPSK Modulation b) QPSK demodulation

90 90

xi(t)

xq(t)

N

N

CarrierGeneration y(t)

xi(t)

xq(t)

LOy(t)

N

N

Figure 2.1: Modulation and Demodulation of QPSK

The Figure 2.1 represents the generation and detection of the QPSK modulation and these sig-

nals are converted to discrete signal level and are mapped to the in-phase and quadrature phase

components by assigning them to the particular points and are known to be constellation digram

or constellation patterns.

I

Q

dd

d

d

I

Q

d

d

d

d

QPSK 8-PSK

Figure 2.2: Constellation Diagram for 4 and 8-PSK

In this project the fundamental schemes used are of Quadrature Phase Shift Keying (QPSK)

and that of 8-Phase Shift Keying. The constellation diagram of the following modulation schemes

are given in the Figure 2.2

QPSK can be considered as the special case of the M ary PSK where the phase of the car-rier takes on one of the available values, namely i = 2(i1)/M where i = 1, 2, 3, 4 , M.Signalling interval duration for one of the M possible signals are given as


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2.2. CAPACITY 7

si(t) =

2E

Tcos

2fct +

2

M(i 1)

i = 1, 2, 3, 4 , M (2.0)

where E is the energy per symbol and fc is the carrier frequency.

2.2 Capacity

Capacity of the given channel is an upper bound of the achievable throughput which can be

express as the Shannons capacity formulae given as

Bmax =C

Bp = log2(1 + SN R)bits/channeluse (2.0)

Here Equation 2.2 refers to the Gaussian distributed input into the channel and cab be as-

sumed as Continuous-input Continuous-output Memoryless Channel (CCMC) capacity.

10 5 0 5 10 15 200

1

2

3

4

5

6

7

SNR(dB)

bits/Channelu

se

AWGN channel Capacity

Shannon

BPSK

QPSK

8PSK

16QAM

Figure 2.3: DCMC Channel Capacity for different Constellations under AWGN channel

The Discrete-input Continuous-output Memoryless Channel (DCMC) considers the main

effect by limiting the channel by log2M (this shows how many bits per symbols are used for

respective modulations) when the channel SNR is increasing where M is the number of points

in the constellation diagram.The Channel capacity for different modulation under AWGN can

be illustrated in the Figure 2.3.


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2.3. MAPPING SCHEMES 8

2.3 Mapping Schemes

The bit mapping is a procedure which shows, how the bit are mapped to corresponding symbol

before the transmission. There are two major classification of bit-mapping methods, they are

Gray Mapping and Natural Mapping( set partitioning). In Gray Mapping the bit mapping is done

in such a way that the Hamming Distance between each of the adjacent constellation point is one.

Gray Mapping was a successful mapping scheme for non-iterative schemes but when an iterative

schemes are used then Gray mapping cannot minimise the number of bit error. Therefore when

an iterative schemes are used we mainly uses Natural mapping schemes. The illustration of

mapping schemes are shown in the Figure 2.4 . Much more detailed analysis of natural Mapping

scheme can be seen in the section 2.8.1.

bits Symbolsbits Symbols

00

01

10

11

00

11

Gray Mapping

10

01

Natural Mapping

0

1

2

3

0

1

2

3

Figure 2.4: Bit Mapping relation

2.4 7-PSK

The 7-PSK is much similar toward the 8 PSK modulation system except that the 7 -PSK system

has a 7 constellation points and a point at the origin. The modulated signal for a 7 PSK can be

expressed as follows:


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2.4. 7-PSK 9

2 1 0 1 22

1

0

1

2

Inphase (AU)

Quadrature(AU)

Constellation points of 7 PSK

Figure 2.5: 7-PSK constellation diagram

10 5 0 5 10 15 200

0.5

1

1.5

2

2.5

3

SNR(dB)

Bits/channeluse

Comparison of Capacity Curve between 7 and 8 PSK schemes

8PSK

7PSK

Figure 2.6: 7-PSK and 8-PSK capacity compariosn


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2.5. CONVOLUTIONAL CODE 10

si(t) =

2ET

cos(

2fct +27

(i 1)) (1 i 7)0 (m = 8)

where E is the energy in each respective waveforms when 1 i 7) of the transmitted

signal.

The Constellation points for the 7 PSK is shown FIgure 2.5 here we can see that the conven-

tional difference between 8 PSK as here there will a point in centre of the constellation digram.

The 7-PSK can achieve a capacity much higher than that of the 8-PSK and can be be capac-

itive efficient modulation scheme. The capacity comparison of 8-PSK and 7-PSK is shown in

the figure 2.6

S0 S1

V

U

V(1)

V(2)

S0 S1S+0 S

+1

Figure 2.7: Convolutional Encoder of rate 1/2

2.5 Convolutional Code

In general , a rate k/n convolution encoder has M-element shift register, k per input information

bit and n output coded bits which are given by the linear combination of the content of the

register and the input information bits.

Generally a convolution encoder of rate 1/n is used, one of the most widely used are the

binary convolutional codes .The n generator polynomial is described by the specific connections

to the register stage. Upon clocking the shift register output moves to next state and so and so.

The generator polynomial are constituted by a binary pattern, indicating the presence or absence


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of a specific link from a particular shift register stage. For example by referring the figure 2.7 ,

the generator polynomial is constituted by:

g1 = [[1 0 1]] , g2 = [[1 1 1]]2.5.1 State and Trellis Diagram

The main technique for characterising the operation of the state machine, as the Convolution

encoder (CC) encoder is just to refer the state diagram of the Figure 2.8 . The diagram fully

describes the operation of the encoder , how the state is been transferred from one state to the

other state and which all path it can choose when its is going from one path to the other. Now,considering the case of the above said convolutional encoder there are two bits in the two shift

registers at any time, there are four possible states and these state transition are governed by

the incoming bit U. The constraint imposed by the respective encoder restrict them to only two

legitimate state transition depending on the nature of the input.

Another simple way of representing the encoder is to portray its trellis diagram, which is

given in Figure 2.8. On the left side there are four state are portrayed.State transition are gov-

erned by the incoming bits and a state transition due to the logical zero is indicated in the figure

2.8

Let the input to the convolutional encoder be denoted as U,next state be S+0 , S+1 and Output

be V, then the State Transition equation can be written as:

S+0 = U S+1 = S0

V1 = U S1 V2 = U S0 S1

Since here the total number of memory elements(m) are 2 there are 2m = 22 = 4 states(S0, S1).

These states will change accordingly to the input information word U to get the output(Vi) and

the next states(S+0 , S1) respectively as demonstrated in the Table 2.1


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Initial State Information Final State OutputS0S1 U S+0 S

+1 V1V2]

00 0 00 00

00 1 10 11

01 0 00 11

01 1 10 00

10 0 01 10

10 1 11 01

11 0 01 01

11 1 11 10

Table 2.1: State Transition Table

0

1

00

10

01

11

Figure 2.8: State Diagram and One Stage of Trellis


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2.6. VITERBI ALGORITHM -(VA) 13

2.6 Viterbi Algorithm -(VA)

Viterbi Algorithm was developed by Andrew Viterbi [9], it is established that the the algorithm

calculates the maximum likelihood code sequence from the received data. The Viterbi algo-

rithm has found its application in number of application such as GSM phones, space probes etc.

The Virtebi algorithm can be done either by hard or soft decoding which is explained in later

subsections

2.6.1 Virterbi Hard Decoding

The state si(k) denotes a state in trellis in the stage i,Each state Si(k) in the trellis is associated in

a state metric or branch metric M(si(k)) and apath in the trellis yk. So in short Viterbi algorithm

states At time i, the most likely paths per state yki ( the closest to received sequence) will be

eventually coincide at the some time (i-l) . The beginning of Virterbi, the algorithm operates

from the zero state, after that will compare the output of the received signal with respect to the

encoded sequence of the trellis on the basis of the Hamming distance(HD), its nothing but the

number of different bit position between two binary sequence. Now, if the received symbol

10, the associated hamming distance is one with respect to both 00 and 11 of the encoded

sequence. During this stage decoder is unable to express any preference to the whether it was

00 or 11 was more. These Hamming distance are known as the context of the Viterbi decoding

and known to be the branch metric. Now proceeding to next symbol it will agin compute the

hamming distance of all possible four legitimate paths and the received signal. These distance

will yield to the new branch metric associated with with second trellis stage. By now the

encoded symbol of two original input bits have been reside. Now the obtained branch metric

is added to previous branch metric to obtain the path metric . A low Hamming distance canindicates a high similarity between the received sequence and the encoded sequence concerned,

which is characteristic of the most likely encoded sequence, since the probability of a high

number of error is exponentially decreasing with numbers of error.


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2.7. PUNCTURING CONVOLUTION CODES 14

2.6.2 Virterbi Soft Decoding

In the hard decision Viterbi decoding, based on the location of the received coded symbol, the

coded bit was estimated if the received symbol is greater than zero, the received coded bit is 1;if the received symbol is less than or equal to zero, the received coded bit is 0.

In Soft decision decoding, rather than estimating the coded bit and finding the Hamming

distance, the distance between the received symbol and the probable transmitted symbol is found

out. This is done as eight level confidence for example, scale +4 indicates the highest possible

confidence concurring the demodulators decision for a binary 1 and -4 for the lowest possible

confidence. In fact, if the demodulator output -4 , the low confidence of in a logical implies a

high probability of a binary zero. Bearing this eight level confidence scale in min, the received

bits can be decoded.

2.7 Puncturing Convolution codes

Puncturing is a process by systematically deleting or not sending, some output bits of a high-

rate-encoder. Since, the trellis structure of the low rate ermines the same, the rate of information

bits per symbol sequence does not change. As a result, the putout sequence belong to the higher

rate punctured convolution codes (PC) codes.

One of the goals of puncturing is that the same decoder can be used for a verity of high

rate codes. One way to achieve decoding of a PC code is using the viterbi algorithm described

earlier by the insertion of the deleted symbols in the position that were not send. This process

is known to be depuncturing. The deleted symbols are marked by some kind of special flag.

The Viterbi Algorithm as mentioned previously works on the maximum likelihood sequence

of received data may not guarantee the minimised symbol error therefore, much more decoding

algorithm known as BCJR algorithm can be used which will to find the individual probability of

the incoming bit and can be proved much better than VA. The detailed algorithm can seen in the

section 2.12.


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2.8. TRELLIS CODED MODULATION 15

2.8 Trellis Coded Modulation

The basic idea of Trellis Coded Modulation [2] is that, instead of sending the symbol m formed

after the respective modulation is done, a extra parity bit is introduced by doubling the number of

constellation points while trying to maintain the same effective throughput. For example if there

are two information bits for 4-level Phase Shift Keying(PSK), a parity bit is being introduced

by scaling the original constellation points to eight, i.e by making it to 8 PSK. As a result the

redundant bit can be absorbed by the expanded constellation diagram, instead of increasing the

signalling rate of the system(bandwidth).

Ungerboeck [2], in his paper fully describes how to employ the TCM schemes in redun-

dant non-binary modulation(symbol based) with the combination of a finite state Forward Error

Correction(FEC) encoder, which selects the coded signal sequence. The extra bits formed by

corresponding convolution encoder will restrict the possible state transformation among the con-

secutive phasor to a certain legitimate constellation. The receiver tries to decode the incoming

noisy signal by a trellis based soft-decision maximum-lilkelihod detector and tries to map it to

the each of the legitimate phasor sequence by the restrictions imposed by the convolution en-

coder. The term Trellis is used to describe this scheme is because the overall operation can

be described by a corresponding state transition diagram similar to that of binary convolution

encoder. The only difference in TCM is that, here trellis branches are labelled with respect to

the redundant non-binary modulated phasors.

2.8.1 TCM Principle

The illustration of TCM principle is done by using the example of a eight-state trellis code for 8

PSK, since this is almost simple and will help in better understanding.When a transmission of two bit/symbol by a coded 8 PSK is used, the suggestion is to use a

2/3 rate convolution coder with a Maximum free Hamming Distance(HD) for a respective con-

straint length [19] and a Gray coding as the mapping scheme. Yet there are problems when this

approach is used. As Ungerboeck [2] noted in his paper encoder should be design to maximising

the free Euclidean Distance(ED) but, when Gray code mapping is used it does not translate the


27/106


large HD into larger ED and the permeation of binary output may have significant influence on

the ED.

This is the good time to explain the concepts of Euclidean Distance(ED) and Hamming dis-

tance(HD). Euclidian distance in a signal constellation is the distance between different points

in the constellation diagram with respect to reference point. In Figure 2.9 do, d1, and d2 repre-

sents the minimum squared Euclidian Distance dmin. Just like real numbers have the concept of

distance, so do the binary numbers. Lets compare two binary numbers, say 1011 and 0100 , the

hamming distance between this numbers is 4. Thus, the Hamming distance is distance obtained

by comparing the respective binary number and adding them.0 00 01 11 1 010101 01 010101 01 0101 010 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 10000000000000000000111111111111111111123 0

1

d0 =

2

d0 =

2

000(0)

001(1)011(3)

101(5)

110(6)

010(2)

111(7)

(B2,B1,B0)

d1 =

2

d2 = 2

d0 = 2sin(/8)d0

d2

d1

100(4)

4-PSK Signal Set8-PSK Signal Set

Figure 2.9: Constellation Diagram for 4 and 8-PSK

Ungerboeck [2], in his paper proposed an new approach know as set partitioning which

aims more directly at maximum free ED. The mapping follows a special partitioning of the

signal set into subset with increasing the minimum distance 0 1 2 between thesignals. This can be illustrated in the Figure 2.10.

In short designing the encoder, Ungerboeck summarised the following rules that were to be

applied to the assigned channel signals

1. Transmission originating, or merging into any of the same state should receive signals

from subset BO or B1 or should have distance of at-least 1=

2 between them.

2. Parallel Transmission should receive the signals form the subset C0 or C1 or C2 or C3

3. ALL 8 PSK signals are used in trellis diagram with equal probability.


28/106


0 = 2sin(/8) = 0.765

1 = 1.414

2 = 2

SUBSET B1B0

C0 C1 C2 SUBSET C3

0 1

0 01 1

Figure 2.10: 8 PSK set partitioning [2],Ungerboeck

In the Figure 2.11 illustrates encoder and trellis state diagram for 8-state 8PSK. Owing to the

limitation imposed by the encoder, there are only a limited set of state transitions associated

with the certain phasor sequence is only possible. Now, the above mentioned rules will be much

clearer with this example. For example, let the correct path be all zero path and what will be

the shortest distance between the two paths which diverges and then remerges, that is given by

minimum squared distance free Euclidian distance of the code as seen form trellis in the Figure

2.11

d2free = d21 + d20 + d21 (2.-2)

where dfree is the minimum distance between any sequence in the trellis and known to be

free Euclidian distance.


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0101010101010101

0101010101010101

0101010101010101

0101010101010101

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 01 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 01 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 01 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 01 1 1 1 1 1 1 10 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 01 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 4 2 61 5 3 74 0 6 25 1 7 32 6 0 4

3 7 1 5

6 2 4 0

7 3 5 1

6

6

7

Ungerbock eight State PSK Code

s2 s1 s0

u2

u1

s2 s+

1s1

c2

c1

c0s0s

+

0s+

2

Figure 2.11: Encoder and trellis for the eight-state 8PSK

2.8.2 TCM encoder

TCM encoder is a specific encoder that is been selected from the family of Recursive Systematic

codes [2], which only attaches one parity bit to that of the information bits i.e only m out ofm

bits are RSC encoded. Hence, the branches that can diverge from and merge from is 2m, the

total number of parallel transmission associated with each of 2m branches are m < m, 2mm.

The Constraint length (K 1) will define the numbers of shift registers in the encoder. Figure2.12 shows the generalised 8-PSK TCM encoder, which has a been designed to achieve a high

FED over the AWGN channel. Its a systematic encoder which attaches a extra parity bit for theoriginal 2 bit information word. The resultant 3-bit word are mapped to one of the 23=8 possible

point of an 8PSK modulator.

The connections between the incoming bit and the modulo-2 adder is what is known to be

generator polynomial. The coefficient of these generator polynomial in general is defined as:


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000

001

010

011

s2 s1 s0

u2 c2

u1c1

c0

100

101 111

110

8 PSK Mapper

Tx Symbols

Figure 2.12: RSC encoder and signal mapper forming the TCM encoder

gj(D) = gjK.Dk + gjK1.D

K+ + gj1.D + gj0 (2.-2)

where D represents delay of each one register stages. The coefficient gji takes the value

1 when there is connection while, it will be zero when there is no connection at the specific

encoder stage. Hence, the generator polynomial for the Figure 2.12 in the binary format is:

g0(D) = 1001

g1(D) = 0100

g2(D) = 0010

Here g0 represents the feedback polynomial and gj(D) forj 1 is the generator polynomial

associated with the jth information bit. The generator polynomial in octal representation is:

g(D) =[

11 08 04]

(2.-2)

Let (S2, S1, S0) denotes the each state of the Convolutional code is shown in the Figure2.11

and the respective cord words(C2, C1, C0) generated can be seen from the table 2.2 and the

Trellis diagram for the eight sate is given Figure 2.11.

The conventional TCM decoders are demodulated with the aid of the modified Viterbi Algo-

rithm (VA) [20]. The fundamental principle of VA is maximum likelihood sequence algorithm

which may not guarantee the minimised symbol error. Hence, the decoding of TCM will be


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2.9. TURBO TRELLIS CODED MODULATION 20

States Information Word

(S2, S2, S1 00 01 10 11000 000 010 100 110

001 001 011 101 111

010 000 010 100 110011 001 011 101 111

100 000 010 100 110

101 001 011 101 111

110 000 010 100 110

111 001 011 101 111

CodeWords=(C2, C1, C0)

Table 2.2: Codeword Table for the 2/3 rate TCM encoder

using a modified MAP algorithm known as Symbol based MAP algorithm this will discussed

later in the section 2.12.

2.9 Turbo Trellis coded Modulation

Turbo codes [15] was a major milestone in the forward error correction codes which can even

achieve an excellent bit error rates at low SNR. The original proposal was for the BPSK schemebut were soon successful with multilevel coded as well. Robertson [21] soon introduced the con-

cept of the Turbo Trellis Coded Modulation employing two TCM codes as parallel concate-

nation of two recursive TCM encoder, and adapted puncturing mechanism to avoid the obvious

disadvantage of the rate loss.

2.9.1 TTCM Encoder

TTCM encoder are the parallel concatenation of two TCM encoders made of Ungerboeck en-

coder and a signal mapper as shown in the figure 2.13. The first TCM encoder normally op-

erates with the original bit sequence (a1, a2), while the second encoder works with the inter-

leaved version of the input bit sequence (b1,b2). Ungerboeck encoder is the same encoder

explained in the previous section of TCM, having a code rate of 2/3 and a generator matrix of

g(d) = [11 08 04] in octal representation. The 8 -P SK signal Mapper translates the encoded


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TCM Encoder

TCM Encoder

Tx Signal

a2

a1

b2

d

c

b1

Figure 2.13: TTCM encoder Structure

bits to the complex symbols by aid of the natural Mapping method (SP). Theses complex signal

are then alternatively selected, an effective puncturing mechanism that punctures the parity bits.

Here,lets explain how selector selects the complex signal coming from the each of the encoder.

Let first encoder produces the symbols c = (0, 2, 4, 5, 1, 6) while, the second encoder output is

d = (0, 3, 6, 4, 0, 7) now the Tx signal will be of the symbols (0, 3, 4, 4, 1, 7) i.e each symbols is

selected alternatively from each encoders.

2.9.2 TTCM Decoder

TTCM Decoder is much similar to that of binary turbo codes [15], except the difference in the

nature of the information passed from one decoder to other decoder respectively and the treat-

ment of the very first decoding step and schematic of the decoder is shown in the Figure 2.14

Here, the main concern is how the symbol-based non binary TTCM scheme is being done. In

symbol-based non binary scheme the systematic bit as well as the parity bits are transmittedtogether as in the form of complex enveloped symbol and cannot be separated from the extrinsic

components, since the noise and the fading that effect the parity components will also affects the

corresponding systematic components. Therefore , here in this case the symbol-based informa-

tion can be spilt into two components:

1. the a-priori component of the non binary symbol provided by the alternative decoders.


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1

BCJR Symbol

Decoder-1Based

Decoder-2

Based

Input

cc

dc

b1e

b2e

a2e

b2a

b1a

a1e

a2a a1

a

y2p

y1p

z1p

BCJR Symbolz2p

y1a

y2a

z1a

z1e

z2e

y1e

y2e

z2a

Figure 2.14: TTCM Decoder Schematic

2. the inseparable extrinsic information as well as the systematic components of the non-

binary symbol

Now, we can concentrate the working of the TTCM decoder. The received symbols are

separated into two different symbol to make sure that upper decoder receives only the sym-

bols encoded by the upper encoder vice versa for the seconder decoder as well and this can

be described as the first step in the decoding. After this, each decoder produces its symbol

based probabilities and generates the a-priori and extrinsic information based on Log-Based

BCJR algorithms The decoders then provides the corresponding a posteriori (yp,zp) which is

subtracted with incoming a-priori (aa,ba) information to make sure that each of the decoder

doesnt receive the same information more than once. The extrinsic information are then inter-

leaved/deinterleaved by the random inderleavers to become the a-priori information and made

to iterate between them. During the final decoding the a posteriori information are de inter-

leaved from the decoder-2 and uses the Hard decision for selecting the maximum a- posteriori

probability associated with the information word.


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2.10. BIT-INTERLEAVED CODED MODULATION 23

2.10 Bit-interleaved Coded modulation

Bit-interleaved Coded Modulation (BICM) was the idea proposed by Zehavi [13] in order to

improve the diversity of the code in Rayleigh channel. The design of the coded modulation

schemes are affected by several factors such as High Free Euclidian Distance which is desired

for the AWGN channel, while its was interested to note that a high Effective Code Length and a

high minimum product distance were the main factors effecting the fading channel. The diversity

of the code can be defined as length of the shortest error path [22] and one should be aware

that the shortest error distance are not necessarily be the minimum distance error. Unfortunately

there was no TCM codes available which can compensate the above said difficulties. In order to

solve the problem Zehavi came with an idea to render the codes diversity equal to that smallest

number of different bits, employing the bit-based interleaving.In short, the bit based interleaves

main purpose was to:

To maximise the diversity and to disperse the bursty error introduced by the fading channel

To render the bit with respect to the Transmitted symbol uncorrelated or independent with

each other

2.10.1 BICM Encoder and Decoder

The BICM encoder uses non-systematic eight state code of a rate 2/3 using a free bit-based

hamming distance of four [1].The purposed model is illustrated in the figure 2.15. Let a be

the encoding sequence, before the encoding of the sequence starts all three shift register are

initialised as zero. After the bits are encoded, the each encoded bits will be interleaved by three

individual parallel random interleavers of the length equal to each incoming coded bits resulting

in a binary vector ofC2,C1,C0. These group of three bits are then mapped to the 8-PSK signal

set according to that of Gray Mapping. These mapping signals points are just digitally pulse

shaped signals then transmitted through the channel. The BICM decoder is implemented as

shown in the figure 2.16. At the receiver, the faded noisy signal will be demodulated into six bit

metric associated with the three bit position, from each received symbols.


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8 PSK Mapper

Interleaver-1

Interleaver-3

Tx Signal

a2

a1

b2

Interleaver-2b1

b0

c2

c0

c1

S2

S0

S1

Figure 2.15: Paaskes non systematic convolutional encoder, with bit based interleaves and a 8 PSK

modulator forming a BICM encoder,employing the Gray Mapping

8 PSK DeMapperBCJR Decoder Deinterleaver-2

Deinterleaver-3

Rx Signal

Deinterleaver-1a2

p

a1p

c2e

c1e

c0e

b2a

b1a

b0a

Figure 2.16: BICM decoder

These bit metric are then de-interleaved by the three independent bit de-interleavers to form

the estimate code words. Then the BCJR decoder is invoked for selecting the best possible

estimate of the original information bits

To understand the how the BICM coding is done, we can use the Paaskes eight state con-

volutional code is used. The generator polynomial for a rate -k/n code there are k generator

polynomials, each having n coefficients i.e) gi =(g0, g1, , gn), i k will be the generator

polynomial for the ith information bits. The generator polynomial for encoder shown in the

figure 2.17 in octal representation is:

g1 = [4 2 6]g2 = [1 4 7] (2.-2)A two-bit information namely a2, a1 are encoded in each cycle in order to form a code word

bi where i (0,1 2). The encoder has three shift register and namely S0, S1 and S2 as given inthe figure. Let S+0 , S

+1 and S

+2 represents the next states of the three memory bits. The encoded

code word for the convolutional code is given as :


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b2

b1

b0

a2

a1S2 S1

S0

S0

S+1 S1

S+0

S+2 S2

Figure 2.17: Paaskes non systematic convolutional encoder

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

01

10

11

00

Figure 2.18: Trellis Diagram for Paaskess eight state convolutional encoder

S+0 = a2 (2.-1)

S+1 = S2 (2.0)

S+2 = a1 (2.1)


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2.11. BIT-INTERLEAVED CODED MODULATION USING ITERATIVE DECODING 26

States Information words

(S2, S2, S1 00 01 10 11000 000 110 101 011

001 110 000 011 101

010 101 011 000 110011 011 101 110 000

100 100 001 010 111

101 010 100 111 001

110 001 111 100 010

111 111 001 010 100

Codeword=(C2, C1, C0)

Table 2.3: Codeword Table (b2,b1, b0)for the Paaskess eight state convolutional encoder

b2 = a2 S1 (2.2)b1 = a1 S0 (2.3)b0 = a1 S0 S1 S2 a2 (2.4)

The eight possible states corresponding to state0 to State7 is shown in the Figure 2.18. Thetrellis diagram gives the all possible transition of the encoder for the Figure2.17. The Table 2.3

illustrates the all possible combination of the code word for the given set of the information

word a2,a1 with help of the the trellis diagram to denote the next states.

2.11 Bit-Interleaved Coded Modulation Using Iterative De-

coding

As noted previously Bit-Interleaved Modulation was purposed to increase the diversity of the

Ungerboeck TCM scheme under the Rayleigh channel. Li [23, 18] suggested an new scheme

of Bit-Interleaved Coded Modulation using Iterative Decoding which employed Set-Partitioning

signal labelling system as that of Ungerboeck. Introduction of soft-decision feedback from

the decoders output to the demapper/demodulator input to iterate between them is advanta-


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2.11. BIT-INTERLEAVED CODED MODULATION USING ITERATIVE DECODING 27

geous of the fact that, it improves the reliability of the soft information passed to the demap-

per/demodulator.

2.11.1 BICM-ID Transmitter and Receiver

A BICM-IDs Transmitter is a serial concatenation of convolution encoder, parallel bit in-

teraleaver and a high order modulator. Here, for the sake of making the comparative study more

realistic we are using a rate 2/3 code and of a 8 PSK Modulation. The BICM-IDs transmitter is

similar to that transmitter explained in the section 2.10.1.

BICM-IDs receiver is almost similar to that of BICMs receiver except the fact, the iterative

process is used in order to achieve global optimum through a step-by step local search. Iterative

decoding is recent success in Forward error correcting codes and can achieve a rate equivalent

to that Shannon capacity.

Rx Signal

BCJR DEMAPPER

1

1

2

2

3

1

3

1

1

c2e

c2a

c0e

c0a

c1e

c1

a

b1a

b0a

b0e

b1

e

b2e

b2a

a2

a1

Figure 2.19: Reciver of BICM-ID with iteration

The Figure 2.19 shows the receiver of the BICM-ID, the receiver uses a sub optimal, iterative

method to calculate the decoded bits. At the initial step , received signal is demodulated and can

use equal likelihood assumption to generate bit metrics cei {0, 1, 2} which is interleaved bycorresponding de-interleavers to become the a priori information bai {0, 1, 2} to the Log-based BCJR decoders to generate the a posteriori bit probablities for the information and the

coded word.

On the second pass the extrinsic, a posteriori vectors bei {0, 1, 2} are interleaved as the apriori information to the demodulators assuming that all the bits are independent of each other


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2.12. BCJR ALGORITHM 28

(by a design of a good interleaver) and will again iterate the above said steps until the final step is

reached. The finally api {0, 1} are decoded by using the hard decision to decode the incomingmessage sequence.

2.12 BCJR Algorithm

The BCJR algorithm was developed by L.Bahi, J.Coocke, F Jelinek and J.Raviv in 1974. The

algorithm is suitable for exitimating bit/symbol probabilities for a finite-state Markov source

transmitted through a discrete memoryless channel. The further details of the BCJR algorithm

can be seen from the paper [10].

2.12.1 Log-BCJR algorithm

Lets first find how the soft information can be expressed by Logarithmic Likelihood Ratios(LLRs).

The LLR of the received data say b is defined as the log ratio of the probabilities of the bit taking

two possible values 0 and 1and denoted as L(b). The LLR can be represented as follows:

L(b) = ln

P(b = 0

|yk)

P(b = 1|yk) (2.4)

where yk is the received signal. As from the equation 2.12.1, if the LLR is positive, that means

the probability of logic zero can be higher than the probability of logic zero while, when LLR

is negative the probability of logic one is higher. The Log based BCJR algorithm is selected

due to following advantages, one is that since the calculation are done in the log domain it can

avoid unnecessary numerical overflows and second is that multiplication and division in the log

domain is addition and substation by the properties of the logarithm.The ultimate aim of the Log-BCJR algorithm is to calculate the extrinsic LLR from the

corresponding decoded sequence. The calculation of extrinsic LLR say ye lead to the calculation

of the three internal variable such as , and .

The (T) value represents the conditional probabilities that corresponds to each transition

in trellis, here in the case of coded modulation schemes, the can be sub grouped into


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two, a priori transition probabilities y and the channel transition probablities c.

The values are the forward recursion corresponding to each state in each step in the

respective trellis.

The values are the backward recursion corresponding to each state in each step in the

respective trellis.

0

1

2

3

Q1

sk1 sk

Figure 2.20: A portion of Trellis diagram

Finally, the above said three variables can used to calculate probability of specific transition

in the trellis and can be used to denote as . The calculation of extrinsic information is y is

the joint probability of the corresponding to , and .

Now, its time to deal how the calculation of above said variables are dealt with with the help

of the Figure 2.20 and diveded into for parts.

1. calcuation: With the aid of the Figure 2.20,the transition probability t(p,q) or the

branch metric for the branch Sk1 = p to Sk = q can be calculated as:

t(p,q) = P(Sk = q|Sk1 = p) = P(yt = x(p,q)) (2.4)

where P(yt = x(p,q)) is the apriori probability of the message symbol yt.


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2. calculation: The value of probabilities are calculating starting from the beginning

of the trellis and proceed forward through the trellis. The t(p) for all given states p (0,

Q

1) and calculated as :

k(q) =

Q1p=0

k1(p)(p,q) (2.4)

3. Calculation: The value of probabilities are calculated starting from the end and work-

ing backward through the trellis. The backward recursion can be done in the similar way

as that of.

k1(p) =

Q1

q=0

k(p)(p,q) (2.4)

4. y calucation: The value ofy can be calculated as:

y =

(p,q)Sx

k(q).t(p,q).k1(q) (2.4)

5. Finally, The extrinsic LLRs of the uncoded bits are calculated with aid of Jacobin loga-

rithm.

The Jacobian logarithm can be defined as:

f(1, 2) = ln(e1 + e2) (2.5)

= max{1, 2} + ln(1 + e|12|) (2.6)= max(1, 2) (2.7)

In actual practice ln(1 + e|12|) is implemented by the a aid of Look-Up Table. This

version of Log Bases BCJR which is realised by looking the Look-Up-Table is known to be

Approx-Log-Bases BCJR algorithm.


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2.13. EXIT CHARTS ANALYSIS 31

2.12.2 Symbol based MAP Algorithm

Till, now we have just seen how binary based MAP algorithm works. This section will explain

working of non-binary based MAP algorithm or symbol bases MAP decoding algorithm [21].The algorithm works just same as that of binary MAP algorithm except in the face that it

differs its calculation of (T). As noted previously, can be sub grouped into two a priori

transition probabilities y and channel transition probablities c. Here, the main difference in

calculation of non-binary based MAP is the calculation of the channel transition probabilities,

the calculation of the c is as follows:

Lets consider the received noisy signal be yk and transmitted noiseless signal be xk then the

channel transition probablities c is given as:

c(T) = |yk sk|2

22n(2.7)

where 2 = N0/2 is the noise variance , and N0 is the noises Power Spectral Density(PSD)

Thepriori transition probabilities y are added with obtained channel transition probablities

c in order to get the final (T). One thing we should note here is that the information symbols

in most cases are independent and eqi probable. However, if the information symbols have some

priori knowledge about the incoming symbols, this can used as the a priori probabilities. These

a priori probabilities used to improve after each iteration until they converge.

Rest all the calculation of, beta and are much similar to binary MAP algorithm.

The non-binary MAP algorithm can also be evaluated in the same logarithmic domain to

reduce the multiplicative complexity especially the overflow and to mitigate the computational

complexity. The transformation of non-binary MAP algorithm are same as explained above.

2.13 Exit Charts Analysis

EXtrinsic Information Transfer(EXIT) [24] chart analysis is powerful that is used to check the

convergence of the iterated decoders. BER chart is one of most powerful tool to analysis the

performance how good the decoder is, but it was not able to explain in detail how the decoder


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converges when an iterative decoding is done. EXIT chart measures the Mutual Information

(MI) that is exchanged between the constituent decoder in a iterative process. The EXIT chart

is expressed in terms of the Log Likelihood ratio of both apriori information Ia and extrinsic

information Ie.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ia

Ie

Exit Chart Generation For BICMID Demapper

SNR =6 dB

SNR =5 dB

SNR =4 dB

SNR =3 dB

SNR=2 dBSNR=1dB

SNR =0 dB

Increase inSNR

Figure 2.21: Exit chart of a 8-PSK De-mapper used in BICM-ID employing Natural Mapping

To elaborate the exit chart lets, consider the BICM-IDs demaper EXIT curve plot. EXIT

curve are obtained by as a function ofIa and Ib for a various ranges of SNR. Note that when Ia

= 0 it correspond to the first iteration of decoding when all the a priori values are equal to zero

the extrinsic information are almost less reliable an can results in high probability of decoding

errors. In contrast, as the a priori increases the extrinsic information are getting more reliable

and confident. Another important aspect is that as SNR values are getting strong, however

irrespective ofa priori information, the extrinsic are getting more and more reliable.

2.13.1 EXIT chart Generation

Figure 2.22 and 2.23 explains schematics by how the EXIT chart simulation are carried out. Fig-

ure 2.22 shows how the EXIT charts are being obtained for inner code/demmaper. As mentioned

above BICM-ID is using an serial Concatenated codes, which is having an outer convolutional

encoder and inner 8-PSK demmaper. Figure 2.22 demonstrates how the EXIT chart curve for

the inner code.


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Mapper

Inner Code/

Demapper

Inner Decoder/

Random BitsGenerate N

Channel

Generate LLRs

Measure MIIe

Ia

a b

bc

ae

aa

Figure 2.22: Scheme for generating the inner code EXIT charts

Outer Decoder

Outer Code

Generate LLRs

Generate N Bits

Measure MI

bc

Ia

a

Ie

ba

be

Figure 2.23: Scheme for generating the outer code EXIT charts


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Usually first step in EXIT generation is to generate N random bits and to form vector of

binary input say a. After that it will be passed though either inner code or outer code which

is needed accordingly. If we have to genearte the EXIT chart for Inner code then output of

the inner code b are passed through the channel having a fixed SNR. The incoming coded

signal are decoded and made to iterate for a known predefined a priori information ba and

will be measuring the mutual information for the respective extrinsic information. In the similar

manner we will be generating the EXIT chart for the outer code , but the difference is that the

here encoded bits will not send through the channel rather it will be producing the a priori

informations from the encoded outer code itself . As in the case of Inner code this will iterated

for the known values of the a priori and measures the mutual information . The figure 2.21

shows the EXIT chart curve for the de-mapper, the overall performance can be explained by

the two values: one is Ia=0 known as no -priori information and other is known to be perfect

a-priori information (Ia=1). Different Eb/N0(dB) values causes the line to shift up and down.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ia

Ie

Comparison of Effect of Constraint Length

K=7 G=[103,30,66]

K=4 G=[11, 02, 04]

K=5 G=[23, 04 ,16]

K=6 G=[45, 26, 34]

Figure 2.24: Convolutional Decoders EXIT chart for various memory and Generator Polynomials

The outer codes EXIT function is

dis sera tat ion

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