SUPERPOSITION CODED MODULATION

TONG JUN

DOCTOR OF PHILOSOPHY CITY UNIVERSITY OF HONG KONG MAY 2009

CITY UNIVERSITY OF HONG KONG 香港城市大學

Superposition Coded Modulation 疊加編碼調製

Submitted to Department of Electronic Engineering 電子工程系 in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 哲學博士學位

by

Tong Jun 童軍

May 2009 二零零九年五月

Abstract Coded modulation (CM) is an eective high-rate transmission technique that has attracted tremendous attention. The revolutionary trellis coded modulation (TCM) scheme can achieve high spectrum- and power-eciencies based on the joint design of coding and modulation. However, it requires a specially-tailored code for each particular transmission rate. This issue can be tackled by multi-level codes (MLC) and bit-interleaved coded modulation (BICM) where the on-shelf binary codes can be directly applied. All these CM schemes can yield signicant coding gains. However, they still suer from capacity loss, high decoding and design complexities. Superposition coded modulation (SCM) is an alternative CM scheme recently proposed. Compared with conventional schemes which usually employ non-linear bit-to-symbol mapping, SCM produces transmit signals by superimposing independent binary coded sequences (each referred to as a layer). By properly choosing the weighting factors, the same binary component code can be used at all layers. This greatly simplies the search of good component codes. Furthermore, thanks to the linearity involved, a successive-interference cancelation (SIC)-type receiver can be applied to reduce complexity. SCM has other advantages such as capability of achieving shaping gain, diversity gain and unequal error protection. However, as a newly proposed CM scheme, SCM is still limitedly explored. In this thesis, we make a comprehensive study on the theoretical and practical aspects of SCM. In the rst contribution, the analysis and design of SCM over memoryless channels is investigated. The basic features of SCM are described, followed by the information-theoretic analysis. Dierent encoding/decoding strategies are compared from the capacity point of view, which also provides insights into the design of capacity-approaching schemes. Error-bounding analysis is then discussed to predict the asymptotic performance with low-complexity codes. In order to examine the convergence property of iterative decoding, a mutual-information-based (MI) evolution technique is proposed subsequently. These analysis techniques provide convenient tools for performance evaluation and optimization. In the second contribution, we study the power eciency of SCM with high data rates. In particular, we consider the peak-to-average power ratio (PAPR) problem. In SCM with very large constellations, the transmit signal exhibits a high PAPR, which may degrade the eciency of the radio frequency power amplier. We apply a straightforward but eective clipping technique to handle this

ii problem. We show that the capacity loss is marginal for clipping depth of practical interests if the optimal receiver is used. We also devise a low-cost, iterative soft compensation receiver for real implementations. Numerical examples are provided to demonstrate that SCM with clipping can still provide ecient high-rate transmissions. In the third contribution, we consider the application of SCM in orthogonal frequency-division multiplexing (OFDM) systems. As well known, OFDM also suers from high PAPR. We apply clipping to solve this problem for general coded OFDM schemes and devise a novel soft compensation method that outperforms conventional approaches. Our focus is on the comparison of the performance with dierent signaling schemes. The major nding is that SCM-OFDM schemes are more robust to the clipping eect than other alternatives such as those based on conventional BICM. This is veried by both theoretical proofs and simulation studies. We also extend the MI evolution method to clipped SCM-OFDM systems with soft compensation for performance analysis and optimization. In the fourth contribution, we discuss SCM in communication systems suering from interferences, such as the inter-symbol interference (ISI) in multi-path channels and the cross-antenna interference (CAI) in multiple-input multiple-output (MIMO) channels. A generic linear vector channel is used to model such systems. We rst outline an iterative linear minimum mean square error (LMMSE) receiver to provide a unied solution. We then study a semi-analytical evolution technique to provide quick prediction of the system performance. We show that SCM has a signicant advantage in canceling interferences and can greatly improve the overall performance when compared with other conventional CM schemes. In the nal contribution, we consider the applications of SCM in multi-user communications. In particular, we incorporate SCM into the OFDM interleavedivision multiple-access (OFDM-IDMA) scheme. Analysis and design issues are discussed. We show that SCM-based OFDM-IDMA schemes are more robust in fading channels than other alternative multiple-access schemes. In particular, it provides a simple and eective means of achieving multi-user gain with low PAPR. In summary, this thesis presents a comprehensive study of SCM over a variety of channels including memoryless and multi-path/multi-user/multi-antenna channels. Both single- and multi-carrier transmissions are examined. The theoretical and simulation studies show that SCM oers a very attractive option for high-rate transmissions.

Acknowledgments First of all, I am deeply grateful to my supervisor Prof. Li Ping for his patient guidance. His constructive suggestions and constant encouragements are invaluable to my life and study. I also thank him for helping me improve my presentation skills. Many thanks to Prof. Xiao Ma, Dr. Raymond Leung, Dr. Lihai Liu, Dr. Keying Wu, Mr. Peng Wang, Dr. Qinghua Guo, Dr. Xiaojun Yuan, Mr. Hao Wu, Mr. Yueqian Li, Dr. Dengsheng Lin, Dr. Sheng Tong, Dr. Shuling Che, Dr. Haitao Li, Mr. Hao Wang, Mr. Chongbin Xu, Miss Nian Geng and Mr. Zhonghao Zhang for helpful discussions. I also thank those stas of Digital and Mobile Communications LAB, IT LAB, general oce of Department of Electronic Engineering and Chow Yei Ching School of Graduate Studies who have given me lots of help. I am grateful to Dr. Shenghui Song and Dr. Qian Huang for their kindness during the tough time. I cherish those nice memories. Last but not least, I am also greatly indebted to my family for their love and support during all the time.

Contents List of gures

viii

List of tables

xiii

Abbreviations

xiv

Notations

xvi

1 Introduction 1.1

1

High-Rate Transmission Schemes . . . . . . . . . . . . . . . . . . .

1

1.1.1

Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Coded Modulation (CM) . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Trellis Coded Modulation (TCM) . . . . . . . . . . . . . . .

3

1.1.4

Multi-Level Codes (MLC) . . . . . . . . . . . . . . . . . . .

3

1.1.5

Bit-Interleaved Coded Modulation (BICM) . . . . . . . . . .

4

1.1.6

Coded Modulation With Iterative Decoding . . . . . . . . .

5

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Limitations of Conventional Coded Modulation Schemes . .

5

1.2.2

Superposition Coded Modulation (SCM) . . . . . . . . . . .

6

1.3

Research Contributions and Thesis Outlines . . . . . . . . . . . . .

8

1.4

List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2

2 Analysis and Design of SCM

12

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

2.4

2.2.1

SL- and ML-SCM . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2

Receiver Principles . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3

Demapping Algorithms . . . . . . . . . . . . . . . . . . . . . 16

Capacity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1

Maximum Achievable Rates . . . . . . . . . . . . . . . . . . 19

2.3.2

Optimization of ML-SCM . . . . . . . . . . . . . . . . . . . 24

Error-Bound Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1

Analysis of SL-SCM . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2

Analysis of ML-SCM . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3

Impact of SCM Parameters . . . . . . . . . . . . . . . . . . 30

Contents 2.5

v

Analysis of Iterative Decoding

. . . . . . . . . . . . . . . . . . . . 33

2.5.1

Analysis of SL-SCM . . . . . . . . . . . . . . . . . . . . . . 34

2.5.2

Analysis of ML-SCM . . . . . . . . . . . . . . . . . . . . . . 38

2.5.3

Simplications in AWGN Channel . . . . . . . . . . . . . . . 38

2.5.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6

Design of SCM Systems . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7

Two-Dimensional (2-D) SCM

2.8

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.9

Appendix: Equivalence of the LMMSE and GA Approaches . . . . 47

. . . . . . . . . . . . . . . . . . . . . 43

3 SCM with Clipping and Iterative Decoding

49

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3

3.4

3.5

3.6

3.2.1

Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2

Peak-to-Average Power Ratio . . . . . . . . . . . . . . . . . 51

3.2.3

Received Signal Model . . . . . . . . . . . . . . . . . . . . . 51

Eect of Clipping on the Achievable Rates . . . . . . . . . . . . . . 52 3.3.1

Continuous Input Signal . . . . . . . . . . . . . . . . . . . . 53

3.3.2

Discrete Input Signal . . . . . . . . . . . . . . . . . . . . . . 55

Iterative Decoding of Clipped SCM . . . . . . . . . . . . . . . . . . 57 3.4.1

Overall Iterative Detection Principle . . . . . . . . . . . . . 58

3.4.2

Optimal Realization of the ESE

3.4.3

ESE Based on GA for Unclipped SCM . . . . . . . . . . . . 59

3.4.4

Modied GA Method (MGA) for Clipped SCM . . . . . . . 61

3.4.5

Soft Compensation for Clipped SCM . . . . . . . . . . . . . 61

3.4.6

Evaluation of the Conditional Means and Variances . . . . . 62

3.4.7

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

. . . . . . . . . . . . . . . 58

Comparison with BICM . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5.1

Comparison in AWGN Channels . . . . . . . . . . . . . . . . 68

3.5.2

Comparison in Fading Channels . . . . . . . . . . . . . . . . 69

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 SCM-OFDM With Clipping and Soft Compensation

72

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3

Iterative Soft Compensation . . . . . . . . . . . . . . . . . . . . . . 75 4.3.1

Overall Iterative Decoding Principle . . . . . . . . . . . . . . 76

4.3.2

Soft Compensation . . . . . . . . . . . . . . . . . . . . . . . 77

Contents 4.3.3 4.4

4.5

4.6

vi Proposed Clipping Noise Estimation Method . . . . . . . . . 78

Optimal Signaling for Soft Compensation . . . . . . . . . . . . . . . 79 4.4.1

Residual Clipping Noise . . . . . . . . . . . . . . . . . . . . 79

4.4.2

Signaling Schemes . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.3

Local and Global Statistics . . . . . . . . . . . . . . . . . . . 83

4.4.4

Minimum Global Variance . . . . . . . . . . . . . . . . . . . 83

4.4.5

Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.6

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Analysis and Design of SCM-Based OFDM . . . . . . . . . . . . . . 90 4.5.1

Multi-level SCM (ML-SCM) Revisited . . . . . . . . . . . . 91

4.5.2

Global Variance in ML-SCM . . . . . . . . . . . . . . . . . . 91

4.5.3

Performance Prediction . . . . . . . . . . . . . . . . . . . . . 92

4.5.4

Symbol Variance Minimization . . . . . . . . . . . . . . . . . 97

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.6.1

Residual Clipping Noise and Global Variance . . . . . . . . . 97

4.6.2

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 99

4.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.8

Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping

. . . . . . . . . . . . . . . . . . . . . . . . 104

4.8.1

Iterative Bayesian Method . . . . . . . . . . . . . . . . . . . 105

4.8.2

Iterative Soft Compensation Method . . . . . . . . . . . . . 107

4.8.3

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 SCM over Linear Vector Channels

110

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2

Iterative LMMSE Detection . . . . . . . . . . . . . . . . . . . . . . 111

5.3

5.4

5.2.1

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2.2

Overall Iterative Detection Principles . . . . . . . . . . . . . 111

5.2.3

ESE Function . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.4

Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . 114

Evolution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3.1

Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . 115

5.3.2

Evolution Analysis . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.3

Impact of Signaling Schemes . . . . . . . . . . . . . . . . . . 118

5.3.4

Extension to MIMO Channels . . . . . . . . . . . . . . . . . 119

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4.1

Single-User Multi-path Channels . . . . . . . . . . . . . . . 121

Contents 5.4.2 5.5

vii MIMO Channels . . . . . . . . . . . . . . . . . . . . . . . . 123

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 OFDM-IDMA

126

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.4

6.3.1

Transmitter Principles . . . . . . . . . . . . . . . . . . . . . 129

6.3.2

Receiver Principles . . . . . . . . . . . . . . . . . . . . . . . 130

Performance Analysis and Optimization

. . . . . . . . . . . . . . . 132

6.4.1

Performance Prediction . . . . . . . . . . . . . . . . . . . . . 133

6.4.2

Spreading and SCM

. . . . . . . . . . . . . . . . . . . . . . 135

6.5

Numerical Results and Discussions . . . . . . . . . . . . . . . . . . 136

6.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Conclusions and Future Work

139

7.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.2

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography

142

List of Figures 1.1

Memoryless communication channel. . . . . . . . . . . . . . . . . .

2

1.2

Constrained capacities achieved by various input signals. . . . . . .

2

1.3

Original coded multi-ary schemes. . . . . . . . . . . . . . . . . . . .

3

1.4

Trellis coded modulation (TCM). . . . . . . . . . . . . . . . . . . .

3

1.5

Multilevel codes (MLC). . . . . . . . . . . . . . . . . . . . . . . . .

4

1.6

Bit-interleaved coded modulation (BICM). . . . . . . . . . . . . . .

4

2.1

Transmitter structure of SL- and ML-SCM. S/P, ENC, Π are serialto-parallel converter, encoder and interleaver, respectively. . . . . . 14

2.2

Constrained capacities achieved by one-dimensional (1-D) SCM with equal-power allocations. . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3

Capacities achieved by SL- and ML-SCM over AWGN channels.

β1 = 3/2, β2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4

Maximum achievable rates for the mismatch-decoded layer (layer1) in a 2-layer SCM with APP and GA demapping. The number included in the legend is the value of ρ = β1 /β2 . . . . . . . . . . . . 24

2.5

Performance of turbo-coded 2- and 3-layer ML-SCM with dierent design rules and APP demapping.

2.6

Performance of turbo-coded 2- and 3-layer ML-SCM with dierent design rules and GA demapping.

2.7

. . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . 28

Error bounds for an SL-SCM with β2 /β1 = 2 over AWGN channels with APP demapping. R = 1 bit/dim. . . . . . . . . . . . . . . . . 29

2.8

EF bounds of ML-SCM over AWGN channels with GA demapping.

R = 1 bit/dim. {βk } = {1, 1} and {1, 2} are compared. 2.9

BER and variance

σc2

. . . . . . 31

v.s. input mutual information Iλ for the

decoder of (5, 7)8 coded BPSK schemes over AWGN and fullyinterleaved Rayleigh fading channels. . . . . . . . . . . . . . . . . . 35 2.10 Simulation (solid lines) versus MI evolution results (dashed lines) for SL- and ML-SCM schemes with K = 8 and {βk } = {1 × 8} over AWGN channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.11 Simulation results for SL- and ML-SCM with {βk } = {1×6, 1.4×2} over fully-interleaved Rayleigh fading channels.

. . . . . . . . . . . 40

2.12 Simulation, MI evolution, and EF bound results for ML-SCM over fully-interleaved Rayleigh fading channels. . . . . . . . . . . . . . . 41

List of Figures

ix

2.13 Simulation results for ML-SCM with dierent component codes and block lengths over AWGN channels. R = 1 bit/dim. K = 2,

β1 = β2 = 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.14 QPSK with (a) Gray and (b) anti-Gray mapping. . . . . . . . . . . 42 2.15 SCM-1. (α1 = α2 = 1. PAPR = 3.01 dB.) . . . . . . . . . . . . . . 43 2.16 SCM-2. (α1 = 1, α2 = 2. PAPR = 2.55 dB.) . . . . . . . . . . . . . 43 2.17 SCM-3. (α1 = 1, α2 = 2eiπ/4 . PAPR = 1.947 dB.) . . . . . . . . . . 44 2.18 SCM-4. (α1 = 1, α2 = 3/2. PAPR = 2.87 dB.)

. . . . . . . . . . . 44

2.19 Performance of SL-SCM with dierent 2-D signaling schemes and APP demapping with (It = 10) and without (It = 1) iterative decoding over AWGN channels. . . . . . . . . . . . . . . . . . . . . 45 2.20 Capacities of ML-SCM based on QPSK with dierent mapping rules over AWGN channels.

. . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1

Encoder of a two-dimensional ML-SCM system. . . . . . . . . . . . 50

3.2

Capacities of clipped Gaussian input signals over (a) AWGN and (b) Rayleigh fading channels.

. . . . . . . . . . . . . . . . . . . . . 53

3.3

SCM constellations (a) without and (b) with clipping. M = 5,

3.4

{|αm |} = {1, 1.4565, 2.1218, 3.0912, 4.5031} and {]αm = mπ/10, ∀m}. 55 Capacities of clipped SCM schemes over (a) AWGN and (b) Rayleigh fading channels. (The optimal ML decoding is assumed.) . . . . . . 56

3.5

Block diagram of the iterative decoding/detection algorithm. . . . . 58

3.6

Performance of the clipped ML-SCM with a doped code and different detection methods over AWGN channels. (a) M = 4, (b)

M = 5. J = 105 . For MGA w/ SC, the MGA method is rst used for ItM = 1 iteration and then the SC method is used for ItS = 5 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7

Performance of SCM at R = 2 bits per two dimensions over fullyinterleaved Rayleigh fading channels. J = 2048. The number of iterations is 10. For the clipped cases with SC, we set ItM = 6, and

ItS = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8

Performance of turbo-coded ((23, 35)8 ) ML-SCM and BICM at R =

2 bits per two dimensions over AWGN channels. . . . . . . . . . . . 69

List of Figures 3.9

x

Performance of ML-SCM and BICM-ID over fully-interleaved Rayleigh fading channels. J = 2048. For the ML-SCM without clipping, the number of iterations is 12 for R = 2 and 3, and 16 for R = 5. For the clipped ML-SCM, we set ItM = 6, ItS = 6 for R = 2 and 3, and

ItM = 6, ItS = 10 for R = 5. The number of iterations is 10 for all the BICM-ID results. . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1

Block diagram of coded OFDM systems with clipping, where Π denotes interleaver and Π−1 de-interleaver. . . . . . . . . . . . . . . 74

4.2

CCDF of the PAPR in clipped OFDM systems. The oversampling factor L = 4, the number of sub-carriers N = 256 and the clipping ratio CR = 0, 1, · · · , 6 dB (from left to right). . . . . . . . . . . . . 75

4.3

σd2 versus σx2 . L = 4. N = 256. The average signal power E[|X[n]|2 ] is normalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4

Examples of 16-ary signaling schemes.

. . . . . . . . . . . . . . . . 82

4.5

Examples of signaling schemes. . . . . . . . . . . . . . . . . . . . . 88

4.6

Block diagram of the Monte Carlo simulation for generating TMI (·, ·). 95

4.7

SRCNR achieved by dierent detection methods for a BICM-IDOFDM scheme. Clipping with L = 4, N = 256, and CR = 0 dB is assumed.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.8

Impact of signaling schemes on σx2 . . . . . . . . . . . . . . . . . . . 99

4.9

Comparison of dierent detection methods for a clipped BICM-IDOFDM scheme with SCM signaling (K = 4, {|βk |} = {1 × 4}).

L = 4, N = 256, CR = 0 dB.

. . . . . . . . . . . . . . . . . . . . . 100

4.10 Comparison of BICM-ID-OFDM schemes with dierent signaling schemes when clipping and the soft compensation are used. L =

4, N = 256 and CR = 0 dB. AWGN channels are assumed. . . . . . 101 4.11 EXIT charts for the clipped BICM-ID-OFDM schemes in Fig. 4.10 at Eb /N0 = 4 dB.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.12 Comparisons of turbo-coded ML-SCM-OFDM and BICM-ID-OFDM with clipping and soft compensation over AWGN channels. L = 4,

N = 256. The frame length is 32768. R = 2 bits per two dimensions. The number of iterations between the ESE and DEC is 6. The number of the iterations in the turbo decoder is 6. For the ML-SCM, {|βk |} = {1 × 2, 1.5 × 2}.

. . . . . . . . . . . . . . . . . 103

List of Figures

xi

4.13 Performance of convolutional-coded ML-SCM-OFDM and BICMID-OFDM with clipping and soft compensation over STVFS channels. L = 4, N = 256, and CR = 2 dB. R = 4. The frame length is 4096. It = 10. For the SCM scheme, K = 32, {|βk |} =

{1 × 12, 1.58 × 4, 2.10 × 6, 2.49 × 2, 2.73 × 2, 3.58 × 4, 3.93 × 2}. . . 104 4.14 Simulation versus evolution results for the ML-SCM-OFDM with

R = 4 bits per two dimensions over fully-interleaved Rayleigh fading channels. The frame length is 16384. In the clipped case, L = 4,

N = 256, and CR = 2 dB. It = 12. . . . . . . . . . . . . . . . . . . 105 4.15 Comparison of dierent clipping eect mitigation methods for Nyquistrate sampled Cartesian clipping at CR = 4 dB. R = 2; {βk } =

{1, 1.05, i, 1.05i}. The number of iterations is 10 for all curves. Other parameters are the same as those in Fig. 4.9. 5.1

. . . . . . . . 108

Transmitter and receiver structure of a coded multi-ary system over linear channels. Π denotes interleaver and Π−1 de-interleaver.

. . . 112

5.2

LMMSE approach to the ESE. . . . . . . . . . . . . . . . . . . . . . 112

5.3

Evolution model of the iterative LMMSE receiver.

5.4

Comparison of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICM-ID

. . . . . . . . . 115

with 16-QAM MSP signaling over Proakis B channel. The dashed and solid curves represent the evolution and simulation results, respectively. The number of iterations It = 10. 5.5

. . . . . . . . . . . . 119

Performance of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICMID with 16-QAM signaling schemes (Gray, Mixed and MSP) over random multi-path channels. It = 10. Information block length is 8192.

5.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

FER of single-user MIMO systems with SL-SCM and BICM-MSP over single-path channels. The receiver has nr = 4 antennas. 16QAM is always assumed. (For SL-SCM, {βk } = {1, 2, i, 2i}.) Information block length = 2048. It= 10. System throughputs R = 2 for nt = 1 and R = 8 for nt = 4.

5.7

. . . . . . . . . . . . . . . . . . . 121

BER of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICM-ID with 16-QAM MSP mapping over quasi-static MIMO multi-path channels. R = 4. It = 20. Information block length is 8192.

. . . . . . 122

List of Figures 5.8

xii

FER for Q = 2-user MIMO systems with SL-SCM and BICM-ID over multi-path channels. For SL-SCM, {βk } = {1, 2, i, 2i}. For BICM-ID, 16-QAM is used. Information block length = 1024 for each user. It = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.9

FER of Q = 2-user MIMO systems employing dierent SL-SCM schemes for R = 8 and R = 12. For curves marked by (1, 2) ,

{βk } = {1, i, 2, 2i}. The cases of other curves are similar. It = 10. Information block length is 8192 and 12288 respectively for R = 8 and R = 12. 6.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Average MTSP versus system sum rate R for dierent multiple access schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2

Transmitter/receiver structure for OFDM-IDMA. The QPSK modulation, cyclic prex insertion and removal for OFDM are not shown for simplicity. ENC and DEC denote encoder and decoder, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.3

Transmitter structure of OFDM-IDMA with SCM, where SP represents spreader.

6.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Performance of OFDM-IDMA with and without clipping.

The

spreading length is S = 8. The rate of each user is 1/8. Q = 16.

R = 2. The information block length for each user is 512. All users have the same transmit power. 6.5

. . . . . . . . . . . . . . . . . . . . 136

Performance of two-user OFDM-IDMA schemes with dierent spreading factors S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.6

Performance of OFDM-IDMA and OFDMA with clipping in an uplink scenario.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

List of Tables 2.1

Weighting Factors for 2- and 3-layer SCM Schemes with the Turbo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2

Weighing Factors for SCM Schemes With the Convolutional Code . 41

3.1

Parameters of the ML-SCM Schemes in Fig. 3.6 . . . . . . . . . . . 63

3.2

Parameters of the ML-SCM Schemes at R = 2 bits per two dimensions in Fig. 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3

Parameters of the ML-SCM and BICM-ID Schemes in Fig. 3.9 . . . 70

6.1

Received Power Prole of the OFDM-IDMA Systems in Fig. 6.6. . . 137

Abbreviations APP

A Posteriori Probability

AWGN

Additive White Gaussian Noise

BC

Broadcasting Channel

BER

Bit-Error-Rate

BICM

Bit-Interleaved Coded Modulation

BICM-ID

BICM with Iterative Decoding

CAI

Cross-Antenna Interference

CDMA

Code Division Multiple Access

CM

Coded Modulation

EF

Error Floor

ESE

Elementary Signal Estimator

EXIT

Extrinsic Information Transfer

FEC

Forward Error Control

GA

Gaussian Approximation

IBI

Inter-Block Interference

IDMA

Interleave Division Multiple Access

i.i.d.

Independent and Identically Distributed

ISI

Inter-Symbol Interference

LLR

Log-Likelihood Ratio

LMMSE

Linear Minimum Mean Square Error

MAC

Multiple-Access Channel

MAP

Maximum A Posteriori

MAI

Multiple-Access Interference

MI

Mutual Information

MIMO

Multiple-Input Multiple-Output

ML

Maximum Likelihood

MLC

Multi-Level Codes

ML-SCM

Multi-Level Superposition Coded Modulation

MSD

Multi-Stage Decoding

MSP

Modied Set-Partitioning

MTSP

Minimum Transmitted-Sum-Power

MUD

Multi-User Detection

MUG

Multi-User Gain

List of Tables

xv

OFDM

Orthogonal Frequency-Division Multiplexing

OFDMA

Orthogonal Frequency-Division Multiple-Access

PAPR

Peak-to-Average Power Ratio

PAM

Pulse Amplitude Modulation

PID

Parallel Independent Decoding

PSK

Phase-Shift Keying

QAM

Quadrature Amplitude Modulation

SCM

Superposition Coded Modulation

SIC

Successive Interference Cancelation

SISO

Soft Input Soft Output

SL-SCM

Single-Level Superposition Coded Modulation

SINR

Signal-to-Interference-Noise Ratio

SNR

Signal-to-Noise Ratio

SP

Set Partitioning

STVFS

Slow Time-Varying Frequency Selective

Notations (·)∗ (·)>

Conjugate of complex numbers

(·)† A ck C CX

Conjugate transpose of a matrix

d d˜ Eb E[·]

Transpose of a matrix Clipping threshold A coded bit at the k th bit position Capacity of a channel Constrained capacity with a given constellation X Time-domain clipping noise Estimate of d Average energy per information bit Mean of a random variable

f (d, β) Pairwise error probability at Hamming distance d in SCM with weighting vector β F Discrete fourier transform matrix g(·) Rule of clipping h[n] Channel coecient for the nth transmit symbol H Iγ Iλ K

Channel coecient matrix

L M N N0

Oversampling factor in OFDM systems

Pb p(y|x) Pr(x) Q R

Bit error rate

R V[·] V x(X)

Covariance matrix of the received signal vector

Mutual information between the decoder outputs and coded bits Mutual information between the decoder inputs and coded bits Number of coded bits per symbol Number of layers in SCM schemes employing QPSK at each layer Number of subcarriers in OFDM systems Noise spectral density Conditional probability density function of y given x Probability of x Number of users Total rate of system Variance of a random variable Covariance matrix of transmit signal vector x Time (frequency)-domain transmit signal vector

List of Tables xk (Xk ) xRe (xIm ) y(Y ) Y [n]

xvii

Time (frequency)-domain transmit signal vector of layer-k Real (imaginary) part of a complex number Time (frequency)-domain received signal vector Frequency-domain received signal

αm βk γk Γ[n] κ

Weighting factor for the mth QPSK layer in SCM

λk σ2 σd2 σc2

LLR for coded bit ck from the demapper

σx2 ξ[n] ζk I(X; Y ) X

Average variance of the transmit symbols

Xkb EX [·]

Subset of constellation points whose k th bit has value b

Weighting factor for the k th bit in SCM LLR for coded bit ck from the decoder Signal-to-interference-plus-noise ratio w.r.t. nth transmit symbol Attenuation factor in Price's theorem for non-linear devices Variance of channel noise Average power of the residual clipping noise Average variance of (−1)c Interference-plus-noise in the LMMSE estimates Interference w.r.t. layer-k Mutual information between random variables X and Y Constellation of the transmit signal Mathematic expectation w.r.t. X

Chapter 1

Introduction 1.1 High-Rate Transmission Schemes 1.1.1 Channel Capacity The groundbreaking paper of Claude E. Shannon A Mathematical Theory of Communication in 1948 [89] establishes the fundamental limit on the data rates of reliable communications over a discrete memoryless channel as described by Fig. 1.1. Shannon names this limit the channel capacity and derives its expression as the maximum mutual information between the channel inputs and outputs

C = max I(X; Y ) p(x)

(1.1)

where the maximization is taken over all possible input distributions p(x) of x. In particular, when the channel is characterized by an additive white Gaussian noise (AWGN) with average power σ 2 and the transmit power is constrained to P , Shannon shows that the channel capacity (hereafter also referred to as Shannon limit) is computed as

µ C = log2

where

P σ2

P 1+ 2 σ

¶ (1.2)

is the signal-to-noise ratio (SNR). It is proved that the capacity can

be achieved by Gaussian input signals, random coding and maximum likelihood (ML) decoding [27, 28]. In most practical communication systems, digital signals uniformly drawn from discrete constellations are widely used. Typical examples include phase-shift keying (PSK), pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM). In such cases, the maximum achievable rate, which is usually referred to as the constellation constrained capacity 1 , is computed as · µ ¶¸ p(Y |X) CX = I(X; Y ) = log2 (|X |) + EX,Y log2 P Z∈X p(Y |Z)

(1.3)

where X represents the signal constellation of size |X |, p(y|x) the channel tran1 The

constrained capacity is also referred to as the coded modulation capacity in [19].

1.1. High-Rate Transmission Schemes

2

Figure 1.1: Memoryless communication channel.

Figure 1.2: Constrained capacities achieved by various input signals. sition probability and the expectation EX,Y [·] is over the joint distribution of X and Y . When discrete signaling schemes are used, the constrained capacities are lower than the Shannon limits. This is illustrated in Fig. 1.2 where several twodimensional, discrete constellations are compared with the continuous signaling schemes. Note that the continuous uniform signaling is a limiting case of square QAM with innite signal points.

1.1.2 Coded Modulation (CM) Shannon's celebrated ndings motivate tremendous work to design practical coding schemes to approach the channel capacity. In the rst three decades from 1948, the research eorts are mainly devoted to the low-SNR regime, i.e., the

power-limited regime. In this regime, the transmission rate is low and binary coding together with antipodal signaling (e.g., binary phase-shift keying (BPSK)) is sucient to approach the Shannon limit. (The loss is within 0.2 dB for C < 0.5 bit per dimension.) The focus is to increase the power eciency, or equivalently, to reduce the SNR required for a desired reliability of communications at a given rate. Code design and optimization can be performed in the Hamming space. The Hamming code, Golay code, convolutional code, Bose-Chaudhuri-Hocquenghem (BCH)

1.1. High-Rate Transmission Schemes

3

Figure 1.3: Original coded multi-ary schemes.

Figure 1.4: Trellis coded modulation (TCM). and Reed-Solomon (RS) codes, etc., mark the major breakthroughs in those ages. Until late 1970s, little progress had been made in the high-SNR regime, namely, the bandwidth-limited regime. In this regime, multi-ary signaling is necessary to ensure high spectrum eciency. Original approaches were based on a direct, serial concatenation of binary codes and multi-ary modulation schemes, as illustrated in Fig. 1.3. Massey was the rst to point out that [69] joint design of coding and modulation can potentially improve the spectrum and power eciency. The key is to maximize the Euclidean distance between transmitted signal sequences by optimizing the code in the signal space. This gave birth to the coded modulation

(CM) that is now recognized as one of the most eective approaches to improving power and bandwidth eciencies in the high-SNR regime.

1.1.3 Trellis Coded Modulation (TCM) Ungeboeck's trellis coded modulation (TCM) scheme (Fig. 1.4) is perhaps the rst breakthrough in designing practical CM schemes. At the transmitter, specially tailored trellis codes are ingeniously combined with expanded constellations and set-partitioning mapping [101] to rene the Euclidean distance properties of the transmitted symbol sequences. At the receiver, demodulation and decoding are jointly performed based on the minimum Euclidean-distance rule and Viterbi algorithm. TCM can improve power eciency signicantly without sacricing spectrum eciency and was widely used in modems.

1.1.4 Multi-Level Codes (MLC) In the meantime, Imai showed that comparable gains can also be attained by multilevel codes (MLC) [53]. In MLC (Fig. 1.5), each bit position of a constellation label is protected by an individual binary component code at a rate carefully

1.1. High-Rate Transmission Schemes

4

Figure 1.5: Multilevel codes (MLC).

Figure 1.6: Bit-interleaved coded modulation (BICM). chosen according to the protection capability of that position. This is in contrast to TCM where all bits are protected by an overall trellis code. Since binary codes have been well studied, the system design problem is simplied. At the receiver, multi-stage decoding (MSD) in a layer-by-layer manner can be applied to achieve the constrained capacity, which has lower complexity than the joint decoder in TCM. A more comprehensive study of MLC is made by Wachsmann et al in [104] where the information-theoretic foundations, error bounds and design rules are studied. Both TCM and MLC can be optimized in the Euclidean space to rene the distance properties of coded sequences.

1.1.5 Bit-Interleaved Coded Modulation (BICM) Bit-interleaved coded modulation (BICM) (Fig. 1.6), rst proposed by Zehavi [114] and later thoroughly studied by Caire et al [19, 33], provides an even simpler option. BICM has a transmitter structure similar to the original multi-ary scheme described in Fig. 1.3 with separated coding and modulation. However, bit-level interleaving permutation is introduced between the two stages. This improves the diversity order in fading channels and also facilitates iterative decoding/demodulation. The BICM receiver is based on symbol-wise soft demapping together with bit-level decoding, which is demonstrated [19] to be capacity approaching when proper bit-to-symbol mapping (e.g., Gray mapping) is employed at the transmitter. By simply concatenating an on-shelf binary code, an interleaver and a signal mapper (with Gray rule), BICM can achieve comparable gains as the more complicated TCM and MLC schemes. Thanks to its good performancecomplexity tradeo, BICM has been recognized as one pragmatic approach to CM and has been accepted by various communication standards [33].

1.2. Motivation

5

1.1.6 Coded Modulation With Iterative Decoding Since 1993, the advent of turbo codes [11] and the re-discovery of the low-density parity check (LDPC) codes [82] have triggered remarkable developments of CM. The basic principles in these random-like codes, such as concatenation, random interleaving and iterative decoding, have been incorporated into CM in a variety of ways. In [39], the turbo code is directly applied as the component code of BICM, yielding near-capacity performance in the high-SNR regime. In [83] and [9], the concatenation structure is combined with the set-partitioning rules originating from TCM. Furthermore, in a coded modulation scheme that can be decomposed into serial or parallel concatenated functional blocks, iterative decoding can be performed based on exchanges of probabilistic information between blocks. For example, in BICM, the demapper and decoder can work in an iterative manner, which leads to the powerful BICM-ID (BICM with iterative decoding) scheme [60]. In each case, signicant coding gains over conventional CM schemes can be achieved, pushing the performance much closer to the Shannon limit.

1.2 Motivation 1.2.1 Limitations of Conventional Coded Modulation Schemes Thanks to its distinguished performance, CM has been adopted in various applications, such as satellite communications, wireless communications and digital subscriber loops. However, the above CM schemes still have restrictions in certain aspects as follows:

• Most conventional CM schemes are based on uniform constellations where signal points are equispaced and equiprobable. According to Section 1.1.1, such schemes are suboptimal from the viewpoint of capacity. In an AWGN channel, there is an asymptotic gap of πe/6 = 1.53 dB (the so-called ultimate shaping gap [36]) between the achievable performance of TCM (and other schemes based on uniform signaling [53, 76]) and the channel capacity [104]. This gap cannot be bridged by using coding techniques. For modern communication systems where capacity-approaching codes are widely adopted, it is of interest to design schemes to close the shaping gap. Conventional approaches to solve this issue are based on signal shaping techniques such as trellis shaping" [36]. The performance can be improved noticeably but the complexity increases since additional shaping operations are required.

1.2. Motivation

6

• CM achieves high coding gains at the expense of signal constellation expansion. The demapping complexity can be a concern for high-rate applications. For example, consider a BICM scheme with a 2K -ary constellation and non-linear bit-to-symbol mapping such as Gray mapping. The demapping complexity is O(2K ) that grows exponentially with K . This constitutes an obstacle to the application of high-order modulation for high transmission rates. On the other hand, it is well known that the diversity gains in fading channels can be increased by using large constellations. Conventional CM schemes are disadvantageous in this aspect as the constellation size is restricted due to the complexity reasons mentioned above.

• For conventional MLC scheme, the rate of the component code at each level should be carefully chosen to ensure good performance. Usually, unequal, irregular rates need to be assigned to dierent layers. This increases the design and implementation complexities since dierent codes/decoders are needed. They are also not very suitable for adaptive modulation since the scheme should be specically designed for each transmission rate.

• The conventional CM schemes are mainly designed and optimized for relatively simple channels such as AWGN and Rayleigh fading channels. Their performance in more adverse channels, e.g., channels with severe interference or non-linear distortion, may degrade seriously. The above also motivates the research reported in this thesis.

1.2.2 Superposition Coded Modulation (SCM) Superposition coded modulation (SCM) [23, 32, 38, 66, 92, 113] refers to a coded modulation scheme where the transmit signal is a linear superposition of binary symbols. With SCM, K coded sequences {x1 , x2 , · · · , xK } are weighted summed to form the transmit signal

x=

K X

βk xk

(1.4)

k=1

where each xk is a sequence of coded symbols and {βk } are constant weighting factors. SCM was originally studied by Duan et al. [32] as an alternative scheme to achieve shaping gains in single-user AWGN channels and later studied by Gadkari, Sun et al. [38, 92] for broadcasting applications. Ma et al. further investigated the capacity limits and design of SCM [66]. More recently, Chen and Wu et al

1.2. Motivation

7

have extended SCM to more complicated channels such as wideband and multiantenna channels [23, 113]. SCM has attracted increasing attention due to its low complexity, high exibility and attractive performance.

• Since the SCM signal is a linear superposition of independent signals, the transmit signal exhibits a Gaussian-like distribution according to the central limit theorem. This leads to a straightforward approach to achieving shaping gain that is necessary to achieve the Shannon limit in the high-SNR regime. The complexity is lower than that in conventional equispaced constellationbased schemes since no additional shaping operation is required.

• SCM is highly connected to the superposition coding concept that has been widely discussed for multi-user channels [28, 99, 106]. In particular, it can be treated as a perfectly coordinated multiple-access (MAC) system. Hence, a variety of low-complexity detection techniques [14, 64, 103, 107] developed for MAC can be applied to SCM. In particular, interference-cancelation-type detection can be used, whose complexity increases only logarithmically with constellation size, rather than linearly as in conventional CM schemes. This enables the usage of very large constellations that can lead to signicant diversity gains in fading channels [58].

• SCM is suitable for adaptive modulation through adjusting constellations (and so rate) according to channel conditions [85]. With SCM, rate change can be made smoothly by using a low-rate component code and dierent number of layers. The transmitter and receiver structures remain roughly the same for dierent K . This is more exible than traditional approaches, such as switching among, say, TCM using 8-PSK, 16-QAM, 32-QAM etc, for channel adaptation [41, 42]. The latter has the drawbacks of abrupt rate change and high receiver cost due to the need of many dierent TCM decoders.

• It is easy to optimize SCM, since it involves only K weighting factors for a 2K -ary constellation. In contrast, the optimization for a general BICM over a 2K -ary constellation has complexity O(2K !) with exhaustive search.

• Multi-level SCM schemes with unequal power allocation among the layers naturally provide unequal error protection [3, 109]. This is advantageous for some applications such as in multimedia broadcasting where dierent types of information have dierent requirement of error probability. In fact, it

1.3. Research Contributions and Thesis Outlines

8

has been shown in [38] that SCM can signicantly outperform time-division schemes for broadcasting. However, as a young CM scheme, SCM is still limitedly understood and is faced with challenges for real applications.

• The analysis and design of SCM are still limitedly explored. Previous results are mainly based on simulation and numerical studies. It is of interest to nd more convincing analytical tools.

• The Gaussian-like transmit signal of SCM exhibits a high peak-to-average power ratio (PAPR), which requires D/A converters and power ampliers with high dynamic ranges. This increases the device cost and reduces the power eciency. The same issue exists in orthogonal frequency-division multiplexing (OFDM) systems employing SCM or other CM schemes.

• Modern communication systems should be able to deliver reliable transmissions over a variety of complicated channels such as multi-path channels and multiple-input multiple-output (MIMO) channels. It is necessary to nd eective tools to analyze and optimize the performance of SCM in such applications.

• Allowing multi-user concurrent transmissions has been indispensable for standard communication systems. What are the pros and cons of using SCM in the multi-user scenario?

1.3 Research Contributions and Thesis Outlines In light of the above, this thesis makes a comprehensive study of SCM. We will consider the analysis and design of SCM schemes in a variety of scenarios. Methods to improve performance and reduce complexity and their performance analyses are the focuses of the investigations. Our major contributions are as follows. Chapter 2 presents a comprehensive study of SCM over memoryless channels. Two types of SCM schemes, i.e. the single-level SCM (SL-SCM) and multi-level SCM (ML-SCM), are analyzed and compared. The information-theoretic aspect of several low-complexity, suboptimal decoding methods is analyzed, based on which a new design rule is proposed for ML-SCM. The error bounding techniques are applied to examine the asymptotic performance of SCM with practical component codes. A semi-analytical mutual information (MI)-based evolution technique is proposed to characterize the convergence behavior of iterative decoding.

1.3. Research Contributions and Thesis Outlines

9

It is shown that ML-SCM outperforms SL-SCM in achievable rates and convergence speed of iterative decoding but suers from a higher error oor. Numerous examples demonstrate that SCM provides a exible, low-complexity solution to high-rate communications. In Chapter 3, we apply clipping to reduce the PAPR of SCM. The impact on performance is rst investigated by evaluating the mutual information driven by the clipped input signals. It is shown that the rate loss is marginal for moderate clipping thresholds if optimal encoding/decoding is used. This fact is conrmed by examples with capacity-approaching component codes and a posteriori probability (APP) demapping. In order to reduce the detection complexity of SCM with a very large number of layers, we develop a suboptimal soft compensation (SC) method. A variety of simulation results for AWGN and fading channels are presented. It is shown that with the proposed SC method, the eect of clipping can be eciently compensated and a good tradeo between PAPR and bit-error-rate (BER) can be achieved. Comparisons with BICM demonstrate that SCM oers signicant advantages for high-rate transmissions over fading channels. Chapter 4 deals with the analysis and design of SCM with clipped OFDM modulation. An iterative compensation scheme is proposed to mitigate the clippinginduced distortion, which can outperform conventional approaches. The focus is on the impact of signaling schemes on the iterative compensation performance. By symbol variance analysis, it is proven that SCM can minimize the residual clipping noise power. This indicates that OFDM systems employing SCM are more robust to clipping eect than other alternatives when clipping and iterative compensation are applied. Several analysis and design issues are discussed subsequently. Numerical results show that SCM-based schemes can provide signicantly better performance in high-rate transmissions with clipping. Chapter 5 is concerned with SCM over linear vector channels like MIMO channels. We apply a low-cost iterative receiver based on linear minimum-mean-squareerror (LMMSE) estimation to jointly mitigate the inter-symbol interference (ISI), cross-antenna interference (CAI) and multiple-access interference (MAI). We then propose an evolution technique to predict the performance and study the impact of signaling schemes. The tradeo between the abilities to combat ISI/CAI/MAI and channel noise are studied for dierent signaling schemes. We show that SCM is advantageous for the LMMSE estimation and can lead to better error-rate performance. Chapter 6 deals with the analysis and design of orthogonal frequency-division multiplexing interleave-division multiple-access (OFDM-IDMA) that can be seen

1.4. List of Publications

10

as an extension of SCM in multi-user communications. We begin with the analysis of the information-theoretic advantages of non-orthogonal transmission schemes in fading multiple-access channels. We then turn attention to practical design issues. A signal-to-noise ratio (SNR) evolution technique is developed to predict the BER performance and to optimize system parameters. Numerical examples show that OFDM-IDMA can (i) alleviate the PAPR problem; (ii) deliver signicant multi-user gain (MUG); (iii) provide robust communications in frequency-selective channels; and (iv) support high single-user throughput. We will summarize our major results and discuss possible directions for future work in Chapter 7.

1.4 List of Publications This thesis is partly based on the following papers. Journal Articles, 1. Lihai Liu, Jun Tong, and Li Ping, Analysis and optimization of CDMA systems with chip-level interleavers, IEEE J. Select. Areas Commun., vol. 24, no. 1, pp. 141150, Jan. 2006. 2. Li Ping, Qinghua Guo, and Jun Tong, The OFDM-IDMA approach to wireless communication systems, IEEE Wireless Commun. Mag., pp. 18 24, June 2007. 3. Jun Tong, Li Ping, and Xiao Ma, Superposition coded modulation with peak-power limitation, IEEE Trans. Inform. Theory., vol. 55, no. 6, pp. 25622576, June 2009. 4. Jun Tong, Qinghua Guo, and Li Ping, Analysis and design of OFDMIDMA systems, European Trans.Telecomm., vol. 19, no. 5, pp. 561569, Aug. 2008. 5. Li Ping, Jun Tong, Xiaojun Yuan, and Qinghua Guo, Superposition coded modulation and iterative linear MMSE detection, to appear in IEEE J.

Select. Areas Commun, Special Issue on Capacity Approaching Codes. Aug. 2009. 6. Jun Tong and Li Ping, Performance analysis of superposition coded modulation, submitted to Physical Communications. Journal Articles in Preparation,

1.4. List of Publications

11

1. Jun Tong, Li Ping, and Vijay K Bhargava, Iterative soft compensation for OFDM systems with clipping and superposition coded modulation, to be

submitted to IEEE Trans. Commun.. Conference Publications, 1. Jun Tong and Li Ping, Iterative decoding of superposition coding, in

Proc. 4th Int. Symp. Turbo Codes & Related Topics, Munich, Germany, 3-7 April 2006. 2. Jun Tong, Li Ping, and Xiao Ma, Superposition coding with peak-power limitation, in Proc. IEEE Int. Conf. Commun., ICC'06, Istanbul, Turkey, 11-15 June 2006. 3. Jun Tong, Qinghua Guo, and Li Ping, Performance analysis of OFDMIDMA systems with peak-power limitation, in Proc. ISSSTA'2008, Bologna, Italy, 25-28 Aug. 2008. 4. Li Ping, Jun Tong, Xiaojun Yuan, and Qinghua Guo, Performance analysis of multi-ary systems with iterative linear minimum-mean-square-error detection, in Proc. 5th Int. Symp. Turbo Codes & Related Topics, Lausanne, Switzerland, 1-5 Sept. 2008. 5. Jun Tong and Li Ping, Iterative detection techniques for clipped OFDM systems, in Proc. IEEE Globecom 2008, New Orleans, 30 Nov.-4 Dec. 2008. 6. Li Ping, Jun Tong, Xiaojun Yuan, and Qinghua Guo, Impact of signaling schemes on iterative minimum-mean-square-error detection, in Proc. IEEE

Globecom 2008, New Orleans, 30 Nov.-4 Dec. 2008.

Chapter 2

Analysis and Design of SCM 2.1 Introduction SCM is a coded modulation scheme in which the transmit signal x is a weighted sum of, say, K coded symbols {x1 , x2 , · · · , xK } where each xk is drawn from a BPSK constellation [23, 32, 38, 66, 67, 98]. According to the encoding strategies, SCM schemes can be classied into single-level SCM (SL-SCM) and multi-level SCM (ML-SCM) where {xk } are from one common codeword and K independent codewords, respectively (c.f. Fig. 2.1 ). Due to the similarity between SCM and multi-access systems (with BPSK signal for each user), the successive interference cancelation (SIC) principles can be applied to decode SCM, which can lead to very attractive performance-complexity tradeo for various types of channels. SL-SCM is a special BICM scheme [19, 114] where the bit-to-symbol mapping is realized by superimposing K independent, antipodal symbols. When the SIC detection is used, it has lower complexity than conventional BICM schemes. Due to this fact, SL-SCM has been explored as an ecient means of achieving signal space diversity [98]. The optimization of SL-SCM is also relatively easy: Only

K weighting factors need to be considered, in contrast to 2K signal points for conventional BICM with the same constellation size. ML-SCM is a special MLC scheme [53, 104] with superposition mapping. By properly choosing the weighting factors, the same component code can be used at all layers, in contrast to conventional MLC schemes where dierent codes should be used at dierent layers. In [23, 32, 66, 113], it is shown that SCM provides a simple and ecient approach to achieving the capacity of both scalar and vector channels. SCM can be applied in dierent scenarios. For example, in broadcasting channels (BC) [28], it is desirable to serve dierent data with dierent qualities according to their importance. This can be easily realized via ML-SCM: rst classify the data into dierent importance classes (each corresponding to a layer) and then apply ML-SCM with un-equal power allocation among layers. More important classes are assigned more transmit power. This can achieve unequal error protection and high adaptability of the system. In this thesis, we will show that SCM

2.2. System Model

13

can also provide more robust performance in systems with non-linear distortion and interference. SCM has attracted increasing attention due to its low complexity, high exibility and performance. However, most previous work has relied on Monte-Carlo simulations to evaluate the performance and the understanding of SCM is still limited. In light of these facts, we make a comprehensive study of SCM in this thesis. The primary objective is to provide (semi-) analytical tools (other than simulation) for performance prediction and useful guidelines for system design. In this chapter, we focus on two memoryless channels: the AWGN channel and the fully-interleaved at fading channel. (For such channels, the fading eect of channel can be averaged out.) We leave more complicated channels to the remainder of the thesis. The organization of this chapter is as follows:

• Section 2.2 describes the basic features of SCM. • Section 2.3 examines the theoretical limits of SCM performance. In particular, we derive the maximum achievable rate of the SIC receiver with a suboptimal demapping algorithm. Based on the capacity analysis, a simple and eective design rule is developed for ML-SCM.

• Section 2.4 analyzes the error rate performance of SCM by using bounding techniques, based on which some useful design guidelines of SCM are discussed. We also show that SL-SCM can achieve a lower error oor than ML-SCM.

• Section 2.5 analyzes the convergence property of the iterative decoding algorithms. A semi-analytical mutual information (MI)-based evolution technique is proposed to evaluate overall system performance.

• Section 2.6 briey discuss the design of SCM with dierent component codes. • Finally, Section 2.7 discusses the design of SCM based on two-dimensional signals.

2.2 System Model In this section, we rst address the basic features of SCM and then outline the decoding algorithms for SCM schemes.

2.2. System Model

14

Figure 2.1: Transmitter structure of SL- and ML-SCM. S/P, ENC, Π are serialto-parallel converter, encoder and interleaver, respectively.

2.2.1 SL- and ML-SCM Consider a coded modulation system employing binary component codes. The information bits are rst encoded. The resultant coded bits are then interleaved and packed into binary K -tuples {c = (c1 , c2 , · · · , cK )} with each ck ∈ {0, 1}. The signal mapper maps each c to a symbol x ∈ X to transmit, where X is the signal constellation of size |X | = 2K . With SCM [23, 32, 38, 66], x is generated by superposition mapping as

x = µ(c) =

K X

βk (−1)ck

(2.1)

k=1

where {βk } are constant weighting factors. Note that each (−1)ck ∈ {+1, −1} is a BPSK symbol. We distinguish the following two types of SCM schemes as illustrated in Fig. 2.1:

• SL-SCM: {ck } for dierent k are from a common encoder, as in BICM [19, 24].

• ML-SCM: Each ck is from an independent encoder, as in MLC [53, 104]. From (2.1), {βk } are the crucial design parameters of SCM. They can be chosen arbitrarily to produce constellations with dierent shapes and performance. Each |βk |2 can be seen as the power allocated to a bit position k . The most straightforward design is probably to assign the same value to all {βk }, which is referred to as an equal-power scheme in [23]. This scheme usually leads to equispaced constellations with overlapped signal points. The transmit signal follows a binomial distribution over the resultant constellations. Another choice is to set

2.2. System Model

15

{βk } = {1, 2, 4, · · · } which results in the conventional 2K -ary PAM constellations with equispaced, equiprobable signal points. The information-theoretic comparison of such SCM constellations can be found in [23, 66]. In this chapter, we further study their error-rate performance in practical scenarios. We focus on a frequency non-selective, memoryless discrete-time channel with output

y = hx + w = h

K X

βk (−1)ck + w

(2.2)

k=1

where h is the channel coecient known perfectly to the receiver and w the AWGN with variance σ 2 = N0 /2 per dimension, with N0 being the noise spectral density. (More complicated channels will be studied in other chapters.) Let E[·] denote the mathematical expectation. We assume that E[|h|2 ] = 1 and so the received P 2 2 SNR can be dened as SNR = E[|x|2 ]/E[|w|2 ] = K k=1 |βk | /E[|w| ]. When the transmission rate is R bits/dim, the energy per bit to noise power spectral density ratio is given by Eb /N0 = SNR/(2R).

2.2.2 Receiver Principles The optimal receiver is the overall maximum likelihood (ML) receiver that usually has prohibitive computational complexity. We therefore consider a suboptimal receiver mainly consisting of two separated functional blocks: a channel demapper and a bank of decoder(s) which are connected by interleavers and de-interleavers. The decoding process is divided into two main phases: The demapper rst generates the bit-level log-likelihood ratios (LLRs) of {ck } from the channel outputs (2.2) (and a priori information if available). Then the decoder performs the a

posteriori probability (APP) decoding based on the demapper outputs and the constraints of the binary component codes. In the detection process, the decoder feedback can be exploited as a priori information to rene the demapping performance. This follows the turbo principle that was proposed for turbo codes [11] and later generalized to coded multi-ary systems [15, 24, 39]. The demapping/decoding operations can be repeated in an iterative manner, which can greatly improve the performance with practical component codes. Since standard APP decoding algorithms for binary codes can be applied to the decoder, we will focus only on the demapping function. For simplicity, we consider one-dimensional (1-D) SCM constellations (with real-valued {βk }) and real-valued (h, w) in this section. However, the discussions below can be extended to more complicated cases.

2.2. System Model

16

2.2.3 Demapping Algorithms Assume that an a priori LLR (possibly obtained from the decoder) for coded bit

ck is available :

µ γk = ln

Pr(ck = 0) Pr(ck = 1)

¶ ,

(2.3)

∀k.

Clearly, the a priori probability for ck can be computed as Pr(ck = 0) = 1−Pr(ck = K Q exp(γk ) Pr(ck ), 1) = 1+exp(γ and the a priori probability of symbol s ∈ X is Pr(s) = k) k=1

where Pr(ck ) takes value of Pr(ck = 0) or Pr(ck = 1), depending on the mapping rule. We consider the following two demapping methods for SCM.

2.2.3.1 APP Demapping The APP method follows those in [15, 24] to demodulate ordinary multi-ary schemes. The demapper computes the extrinsic LLR for each coded bit based on the APP principles as

µ P λAP ≡ ln k

Pr(ck = 0|y, {γk }) Pr(ck = 1|y, {γk })

¶

 P  s∈X 0 − γk = ln  Pk

p(y|s) Pr(s) p(y|s) Pr(s)

   − γk (2.4)

s∈Xk1

where Xkb denotes the subset of the signaling points s ∈ X whose k th label bit has value b ∈ {0, 1} and

µ ¶ 1 |y − hs|2 p(y|s) = √ exp − 2σ 2 2πσ

(2.5)

is the channel transition probability. The APP demapping is the optimal demapping method and can be applied to general mapping rules. The related complexity is O(2K ) since |X | = 2K .

2.2.3.2 Gaussian-Approximation-based (GA) Demapping The demapping complexity of SCM can be reduced by adopting an alternative GA method whose normalized complexity is independent of K . The key is to model (2.2) as a system with binary input (−1)ck corrupted by distortion ζk :

y = hβk (−1)ck + ζk , where

ζk = h

X m6=k

βm (−1)cm + w

(2.6)

(2.7)

2.3. Capacity Analysis

17

is the interference-plus-noise with respect to (w.r.t.) ck . Further, we approximately treat ζk as a Gaussian noise. The demapper output is then computed as

³ ´ 2 k −E[ζk ]) (2πV[ζk ])−1/2 exp − (y−hβ2V[ζ y − E[ζk ] k] ³ ´  = 2βk h = ln  2 k −E[ζk ]) V[ζk ] (2πV[ζk ])−1/2 exp − (y+hβ2V[ζ k] 

λGA k

(2.8)

where E[ζk ] and V[ζk ] denotes the mean and variance of the distortion term w.r.t.

ck , and can be found from the decoder feedback as: E[ζk ] = h

X

βm E[(−1)cm ]

(2.9)

m6=k

V[ζk ] = h2

X

|βm |2 V[(−1)cm ] + σ 2

(2.10)

m6=k

where

E[(−1)cm ] = (+1) Pr(cm = 0) + (−1) Pr(cm = 1) = tanh(γm /2) (2.11) V[(−1)cm ] = 1 − tanh2 (γm /2).

(2.12)

The complexity of the GA demapping is O(1) per bit and is independent of k and K [77]. Hence, the GA approach substantially reduces the overall receiver complexity compared to the APP approach. This enables the use of very large signal constellations. An interesting feature of the GA approach is that it is equivalent to a linear minimum mean square error (LMMSE) detector. This is illustrated in the Appendix in Section 2.9.

2.3 Capacity Analysis Conventional coded modulation schemes such as TCM and BICM are based on equispaced constellations with equiprobable signal points. They suer from the so-called shaping gap between the achievable rate and the Shannon limit. To narrow this gap, shaping techniques (that produce signals with a Gaussian-like distribution) can be applied by assigning unequal probabilities on dierent signal points [35, 36, 104]. The resultant advantage is referred to as the shaping gain [36]. In conventional shaping schemes, coding and shaping operations are separated and special shaping codes and algorithms are needed. SCM provides a more straightforward shaping method. With SCM, K BPSK

2.3. Capacity Analysis

18

Figure 2.2: Constrained capacities achieved by one-dimensional (1-D) SCM with equal-power allocations. symbols are linearly superimposed before transmission. Consequently, when K is suciently large, the transmit signal exhibits an approximate Gaussian distribution that is near-optimal for many types of channels. This results in a shaping gain" over conventional uniform signaling-based schemes. Fig. 2.2 illustrates the constrained capacity (also referred to as coded modulation (CM) capacity [19]) P   p(y|s) s∈X  (2.13) CCM = I(x; y) = K − Ey,x log p(y|x) achieved by SCM with equal power allocations1 β1 = · · · = βK and dierent K , where I denotes mutual information and Ey,x [·] denotes expectation w.r.t. the joint distribution of x and y . It is seen that the results of SCM get closer to the Shannon limit as K is increased. This is because the distribution of transmit signal, which is binomial (when {βk } are equal), approaches the Gaussian distribution when K is large. As a matter of fact, the Shannon limit can be achieved by SCM at any rate provided that K is suciently large. (See [23, 113] for related discussions.) In principle, to realize the shaping gain, coding and modulation should be jointly designed and an overall ML decoder should be used. However, in practice, 1

Equal power allocation is assumed here for simplicity. It is possible to apply optimized, unequal power allocation to increase the maximum achievable rates.

2.3. Capacity Analysis

19

suboptimal encoder/decoders are always used and so the constrained capacity may not be achieved. It is of more interest to investigate the maximum achievable rate with encoder/decoder constraints. This motivates the work in this section. We will focus on the receiver with separated demapping and decoding as outlined in Section 2.2.2. We show that ML-SCM outperforms SL-SCM from the point of view of capacity. In particular, we analyze the performance loss with the suboptimal GA demapper using the results obtained for general mismatched decoding problems in [13]. We also outline a new design rule for ML-SCM based on the capacity analysis. In this section, we assume 1-D SCM constellations again. Further, we assume AWGN channels, i.e., h = 1 in (2.2).

2.3.1 Maximum Achievable Rates 2.3.1.1 SL-SCM with APP Demapping SL-SCM is a special BICM where all information bits are protected by a single code at the transmitter and are decoded simultaneously at the receiver. We assume that an innite-length interleaver is used and the coded bits, with uniform distribution in the set {0, 1}, are independent of each other. In this case,

γk = 0,

∀k

in (2.4), which means that there is no a priori information before demapping. Following [19], each bit position k can be treated as an equivalent sub-channel with binary input ck and output λk . There are K parallel sub-channels in total. They are used to transmit an overall codeword and the overall capacity is computed as

CSL−SCM =

K X

(2.14)

I(y; ck )

k=1



P



p(y|s)   s∈X   = K− Ey,ck log P  p(y|s) k=1 K X

(2.15)

c

s∈Xk k

where the expectation Ey,ck [·] is w.r.t. the joint distribution of y and ck . According to the chain rule [28] and the fact that condition may decrease entropy,

CSL−SCM ≤

K X k=1

I(y; ck |c1 , · · · , ck−1 ) = I(y; c1 , c2 , · · · , cK ) = I(y; x).

(2.16)

2.3. Capacity Analysis

20

This means that SL-SCM may not achieve the constellation-constrained capacity.

2.3.1.2 ML-SCM with Multistage Decoding (MSD) and APP Demapping For ML-SCM, we can employ two decoding strategies. The rst is the parallel decoding on individual levels (PDL) [104] where all layers are demapped simultaneously. There is no cooperation among layers and hence the decision of one layer will not help other layers. This is clearly a suboptimal approach for ML-SCM and the achievable capacity with PDL is the same as that of SL-SCM. Another strategy is to employ multistage decoding (MSD) to decode the K layers in a layer-by-layer manner. Each layer can be treated as an independent sub-channel with binary input. Consider layer-k . Assume that layer-1 to k−1 have already been successfully decoded. Then, there is perfect a priori information for

{cm , ∀m < k} but no a priori information for {cm , ∀m > k}, i.e., γm = ±∞, ∀m < k;

and γm = 0, ∀m > k.

(2.17)

The available a priori information can be incorporated into the demapping function in (2.4) to rene the estimates of un-decoded layers. It can be easily veried that, for ML-SCM, MSD is equivalent to the SIC decoding that is widely used for MAC systems. With SIC, the decoded layers' signals can be ideally canceled from the received signal, yielding

y˜k = y −

k−1 X

cm

βm (−1)

=

m=1

K X

βm (−1)cm + w.

(2.18)

m=k

Then the constellation size of the useful signal

P m

βm (−1)cm reduces to |X˜k | =

2K−k+1 . Clearly, the APP demapping complexity decreases as k increases. The capacity for layer-k with APP demapping is now computed as

CkAP P = I(˜ yk ; ck )



 P

  z∈X˜k  = 1 − Eck ,˜yk  log  P c

p(˜ yk |z)



  . p(˜ yk |z) 

(2.19)

z∈X˜k k

The achievable total rate is

CML−SCM =

K X k=1

CkAP P .

(2.20)

2.3. Capacity Analysis

21

It is achieved if and only if the rate of the component code at each layer is chosen according to (2.19). From the chain rule of mutual information [28], CML−SCM =

I(x; y), i.e., the constrained capacity can be achieved by ML-SCM with MSD.

2.3.1.3 ML-SCM with MSD and GA Demapping The GA demapping method can also be combined with MSD. Following Section 2.2.3.2, we can rewrite (2.18) as

y˜k =

K X

βm xm + w = βk xk + ξk ,

(2.21)

m=k

where xk ≡ (−1)ck ∈ {+1, −1}, ∀k , and

ξk =

K X

β m xm + w

(2.22)

m=k+1

is the interference-plus-noise w.r.t. layer-k after perfectly canceling the decoded interferences. From (2.17), with GA demapping, ξk is approximated as zero-mean P 2 2 2 Gaussian noise with variance σξ,k = K m=k+1 βm + σ and the LLR for each coded bit is computed as (3.26)

λk =

2βk y˜k . 2 σξ,k

(2.23)

Let us assume that a binary component code with length N is employed at layer-

k and temporally introduce the time index n. After demapping, the component decoder nds the codeword x?k = [x?k [1], · · · , x?k [N ]] using the maximum likelihood rule, i.e.,

x?k

= arg max xk

= arg max xk

N X n=1 N X

xk [n]λk [n]

(2.24)

xk [n]˜ yk [n].

(2.25)

n=1

From the central limit theorem, when K − k is suciently large and {βk } are properly chosen, the interference signal ξk is indeed approximate-Gaussian distributed. In this case, GA demapping can yield performance close to that of the APP method. However, this is not true for a small or moderate K . In general, the above GA assumption leads to mismatched decoding that degrades performance. We now investigate the capacity loss due to this mismatched strategy. We rst show that the decoding of layer-k of ML-SCM with the GA metric is

2.3. Capacity Analysis

22

in essence equivalent to minimum distance decoding (MDD). The decoding metric with MDD is

d(˜ yk [n], xk [n]) = |˜ yk [n] − βk xk [n]|2

(2.26)

which is the squared Euclidean distance between y˜k [n] and xk [n]. The MDD approach nds a binary codeword that is nearest to the received signal {˜ yk [n]}, i.e.,

x?k = arg min xk

N X

d(˜ yk [n], xk [n]).

(2.27)

n=1

By inspection, it is easily understood that the decoding rule in (2.27) is equivalent to (2.24). Thus, it is sucient to investigate the maximum achievable rate with MDD. Note that the MDD approach is optimal when the noise is zero-mean Gaussian distributed. However, in ML-SCM schemes with moderate number of layers,

ξk is not Gaussian and so MDD leads to suboptimal, mismatched decoding. We follow the mismatched capacity analysis in [13] to nd the maximum achievable rate here. Before we go further, let us dene A = {+1, −1}, xK k = (xk , · · · , xK ) ∈

AK−k+1 = A × A × · · · × A. From w ∼ N (0, σ 2 ), it can be easily shown that p(˜ yk |xK yk | − xK k ) = p(−˜ k ),

K−k+1 ∀xK . k ∈ A

(2.28)

Then, for a particular xk , the channel transition probability can be written as

p(˜ yk |xk ) =

X

K p(˜ yk |xk , xK k+1 )p(xk+1 )

(2.29)

K p(−˜ yk | − xk , −xK k+1 )p(−xk+1 )

(2.30)

K p(−˜ yk | − xk , xK k+1 )p(xk+1 )

(2.31)

K−k xK k+1 ∈A

=

X

K−k xK k+1 ∈A

=

X

K−k xK k+1 ∈A

(2.32)

= p(−˜ yk | − xk ),

where we have assumed that Pr(xm = +1) = Pr(xm = −1) = 1/2, ∀m, and

{xm , ∀m} are independent. Therefore, the channel xk → y˜k is a binary-input symmetric channel (BSC). In this case, we can apply the results in [13] to nd the maximum achievable rate with MDD as the general mutual information   +∞ −αd(˜ yk ,xk ) X Z e d˜ Pr(xk )p(˜ yk |xk ) log  P yk CkGA = max −αd(˜ yk ,z) α>0 Pr(z)e x ∈A k

−∞

z∈A

(2.33)

2.3. Capacity Analysis

23

Figure 2.3: Capacities achieved by SL- and ML-SCM over AWGN channels. β1 = 3/2, β2 = 1. where Pr(xk ) = 1/2, α is an auxiliary variable to be optimized. From our earlier discussions, CkGA is also the performance limit of layer-k with the GA demapper. It can be computed numerically.

2.3.1.4 Examples Fig. 2.3 compares the capacities achieved by SL- and ML-SCM (with MSD and APP demapping) over AWGN channels. The same constellation with weighting factors {βk } = {1, 3/2} is considered for the two schemes. We assume that layer1 is decoded rst in ML-SCM, followed by layer-2. The Shannon limit and the achievable rates of each sub-channel of ML-SCM are also included for reference. From Fig. 2.3, with the same SCM signaling, ML-SCM outperforms SL-SCM. (At high SNR, the two schemes achieve very similar performance as the maximum achievable rate approaches the entropy of the transmit signal, which is the same in the two schemes.) The capacity results also suggest that with properly chosen

{βk }, we can apply the same binary component code to all the layers of ML-SCM. For example, to achieve a total rate of 1.1 bits/dim, a binary code of rate 0.55 can be used together with the weighting factors in Fig. 2.3. By contrast, in conventional MLC with QAM constellations, codes of dierent rates are required to achieve capacity. Fig. 2.4 shows the maximum achievable rate of layer-1 (the mismatch-decoded

2.3. Capacity Analysis

24

Figure 2.4: Maximum achievable rates for the mismatch-decoded layer (layer-1) in a 2-layer SCM with APP and GA demapping. The number included in the legend is the value of ρ = β1 /β2 . layer) with GA demapping in ML-SCM with K = 2 and dierent {βk }. The capacity achieved by the APP demapper is also included. (Note that when decoding layer-2, only the AWGN is present and so the GA and APP methods yield the same performance.) From Fig. 2.4, the loss in achievable rates with GA demapping depends on the values of {βk }. If ρ ≡ β1 /β2 is large, the rate loss due to GA is marginal, whereas if ρ is small, the loss is more signicant. (The results of the two methods for β1 /β2 = 3/2 are indistinguishable in Fig. 2.4.) Intuitively, this may be explained by that the interference-plus-noise ξk for layer-1 in (2.21) is more Gaussian-like when β2 is smaller and so GA becomes more accurate in this case. Note that, on the other hand, for ρ < 1, we can reverse the decoding order. Thus, for a two-layer ML-SCM scheme, the maximum penalty due to GA demapping is reached at ρ = 1.

2.3.2 Optimization of ML-SCM ML-SCM can be designed with high exibility. For example, we can set {βk } =

{1, 2, 4, · · · } to construct PAM-type constellations. The constrained capacity can then be achieved by assigning codes with proper rates to dierent layers according to the above capacity analysis, as in conventional MLC schemes [104]. However, such schemes usually require several component codes with dierent, irregular rates, which is not convenient for practical implementations.

2.3. Capacity Analysis

25

In this section, we consider another direction which assumes the same component code (of rate r) for all layers. This code can be any practical codes such as the turbo code. Our target is to nd {βk } that minimizes the total power P 2 { K k=1 |βk | } under the constraint that the bit-error-rate (BER) for each layer is no larger than a given threshold. We introduce an optimization method based on the capacity analysis above. We rst review several existing methods in the literature:

• A simulation-based layer-by-layer searching method is suggested in [66, 67]. It can provide performance close to that with (the optimal) exhaustive search method but is time-consuming.

• The linear programming method designed in [20] and [64] for code-division multiple-access (CDMA) can be applied to SCM due to the similarity between CDMA and SCM. However, this method are based on a large-system assumption and does not perform well for small number of layers K .

• Another approach is the capacity (CAP)-rule method discussed in [104] for conventional MLC. We can adapt it to design ML-SCM with a xed component code of rate-r. With this method, {βk } is selected such that the capacity CkAP P achieved by each layer (with MSD and APP demapping) is equal to r:

CkAP P = r,

k = 1, 2, · · · , K.

(2.34)

This method may be the most widely used method for MLC design. However, it assumes optimal component codes and decoders which are generally not the case. Later we show that this method is suboptimal for practical codes. In what follows, we propose a new method. The basic assumption is that, for a given binary code, the BER is determined by the mutual information contained in the decoder inputs. (This assumption has been adopted in the extrinsic information transfer (EXIT) analysis, and, has recently been applied to the bit-loading problem of coded OFDM [62].) The key to the proposed method is to treat CkAP P and CkGA computed in (2.19) and (2.33) as the measure of the information provided by the demapper to the decoder. The method works as follows:

Initialization : Find the input mutual information I0 to achieve the target BER for the component code over a binary-input (BI) AWGN channel. Assume that

2.3. Capacity Analysis

26

Table 2.1: Weighting Factors for 2- and 3-layer SCM Schemes with the Turbo Code K 2 3

MI Rule APP Demapping GA Demapping (1, 1.4717) (1, 1.4777) (1, 1.4717, 2.138) (1, 1.4777, 2.188)

CAP Rule APP & GA Demapping (1, 1.4265) (1, 1.4265, 2.021)

the noise power σ 2 = 1. Find a βK such that in y˜K = βK (−1)cK + w,

CK = I0 , where Ck , ∀k, can be either CkAP P in (2.19) or CkGA in (2.33), depending on the demapping method assumed.

Iterations : For k = K − 1, K − 2, . . ., 1, x {βK , · · · , βk+1 } and nd a βk such that in the system y˜k =

PK

m=k

βm (−1)cm + w, Ck = I0

(2.35)

when layer-k is decoded rst. We will refer to this method as the mutual information (MI)-rule method below. It has similar complexity as the CAP-rule approach but is more eective because the suboptimality of the code/decoder is taken into account. We observe that it can yield optimization results very close to the simulation-based method in [66, 67] for capacity-approaching codes but at much lower complexity. We now show examples to demonstrate the eectiveness of the proposed method. We consider K = 2 and 3 ML-SCM schemes with total rate of R = 1 and 1.5 bits/dim, respectively. The rate r = 1/2 turbo code (23, 35)8 with information block length 32768 is used as the component code. The number of iterations in the turbo decoding is 18. The target BER = 10−6 and the mutual information I0 required to achieve this BER is I0 = 0.543 for a BI-AWGN channel. We apply the CAP-rule and the proposed MI-rule to determine the weighting factors. The results are listed in Table 2.1. The simulated BER results with the APP and GA demapping methods are presented in Fig. 2.5 and 2.6, respectively. For curves marked with Iterative, the number of iterations between decoding and demapping is 2 and 4 for R = 1 and 1.5 bits/dim, respectively. It is shown that the MI-rule yields better performance. The advantage becomes more evident as the rate increases and/or when GA demapping is applied. Note that when the MSD

2.4. Error-Bound Analysis

27

Figure 2.5: Performance of turbo-coded 2- and 3-layer ML-SCM with dierent design rules and APP demapping. method is used, there is error propagation between layers. This can be alleviated by iterative decoding, as shown in Fig. 2.5. It is also worth pointing out that, with the MI-rule method, the achievable performance with GA demapping is very close to that with APP demapping, implying that the GA method can provide a good performance-complexity tradeo. This is consistent with the capacity results in Fig. 2.4 that the loss due to GA is not signicant if β1 /β2 is large.

2.4 Error-Bound Analysis The capacity analysis can provide important insights into designing SCM with capacity approaching codes such as turbo codes. However, the turbo-like codes usually lead to relatively high decoding complexity and long decoding delay. In practice, weak component codes (e.g., convolutional codes) may be used to save cost and reduce latency, especially in delay-limited applications. In this case, the capacity analysis is not very relevant and it is more useful to employ the conventional error-bound analysis to investigate performance.

2.4.1 Analysis of SL-SCM The union bound [19] and error-oor (EF) bound [24] derived for BICM can be applied to SL-SCM. With SCM, the signal constellation and mapping rule are

2.4. Error-Bound Analysis

28

Figure 2.6: Performance of turbo-coded 2- and 3-layer ML-SCM with dierent design rules and GA demapping. fully characterized by β = [β1 , β2 , · · · , βK ]> , where

00>

” denotes matrix transpose.

Following [19], we consider the case where h and w in (2.2) are complex numbers. The union bound on BER for SL-SCM writes

Pb =

∞ 1 X WI (d)f (d, β), kc d=d

(2.36)

H

where a rate kc /nc convolutional code with free Hamming distance dH is assumed;

WI (d) is the total information weight of all error events at Hamming distance d; and f (d, β) denotes the pairwise error probability (PEP) 1 f (d, β) ≤ 2πi

Z

δ+i∞

[ψub (s, β)]d

δ−i∞

ds ; s

(2.37)

δ is a positive real number in the region of convergence of ψub (s, β), K 1 1 XX X X ψub (s, β) = Φ∆(x,z) (s), K2K k=1 b=0 ¯ b b

(2.38)

x∈Xk z∈Xk

¯b = 1−b; and Φ∆(x,z) (s) is the Laplace transform of the probability density function (p.d.f.) of the metric dierence ∆(x, z) = ln(p(y|x)) − ln(p(y|z))

(2.39)

2.4. Error-Bound Analysis

29

Figure 2.7: Error bounds for an SL-SCM with β2 /β1 = 2 over AWGN channels with APP demapping. R = 1 bit/dim. between x and z computed from (2.5). For a Rician fading channel with Ricianfactor K

Φ∆(x,z) (s) =

³ ´ s(1−sN0 )K|x−z|2 /(K+1) exp − 1+s(1−sN 2 0 )|x−z| /(K+1) 1 + s(1 − sN0 )|x − z|2 /(K + 1)

.

(2.40)

The computational details for (2.37) can be found in [12]. When iterative decoding is applied to SL-SCM, the decoder feedback is used as the a priori information for the demapper, and vice versa. The BER can be signicantly reduced over iterations. From [24], the asymptotical performance is achieved when ideal a priori information is obtained from the decoder. This leads to the error oor (EF) of the system that can be lower bounded by expurgating the irrelevant error events in (2.38) as K 1 1 XX X ΦM(x,˜z) (s) ψef (s, β) = K2K k=1 b=0 b

(2.41)

x∈Xk

where z˜ is the signal point whose label bits dier from those of x only at the k th bit position. The simulation results and error bounds for an SL-SCM scheme with and without iterative decoding are demonstrated in Fig. 2.7. The 4-state convolutional code (5, 7)8 is used as the component code and the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [5] is applied to the decoder. K = 2. R = 1 bit/dim. It is shown that both the union bound and EF bound are tight for high SNRs.

2.4. Error-Bound Analysis

30

2.4.2 Analysis of ML-SCM For ML-SCM, we focus on the EF performance. We assume ideal feedback from the decoders. In this case, the inter-layer interference can be perfectly removed and each layer can be seen as a coded BPSK system. With this view, the error bound for layer-k is given by (k) Pb

∞ 1 X WI (d)f (d, βk ) = kc d=d

(2.42)

H

where f (d, βk ) is the PEP for the coded BPSK system with input alphabet {+βk , −βk }:

1 f (d, βk ) ≤ 2πj

Z

α+j∞ α−j∞

£

¤d ds Φ∆(+βk ,−βk ) (s) . s

(2.43)

The overall BER of ML-SCM is approximated as K 1 X (k) P . Pb ≈ K k=1 b

(2.44)

Note that (2.42)-(2.44) only give an approximation (and not an actual bound) of the error rate with iterative decoding. This is because, the ideal feedback assumption above underestimates the BER while the bounding in (2.43) overestimates the BER. However, for convenience, we still refer to (2.44) as the EF bound for ML-SCM as it gives very accurate prediction of BER at high SNR. Examples of ML-SCM over AWGN channels are shown in Fig. 2.8. (Examples for fading channels will be presented later in Fig. 2.12.) The rate-1/2 convolutional code (5, 7)8 is used and the transmission rate R = 1 bit/dim. Two dierent power allocation schemes {βk } are considered, among which the equal power allocation yields a lower EF. Note that GA demapping is used and the simulation results approach the EF bounds.

2.4.3 Impact of SCM Parameters We now consider the high-SNR regime and examine the impact of system parameters on the asymptotic EF performance.

2.4.3.1 SL-SCM versus ML-SCM With SCM, it is easy to show from (2.1) that

|x − z˜| = 4|βk |2 ,

∀x ∈ Xkb

(2.45)

2.4. Error-Bound Analysis

31

Figure 2.8: EF bounds of ML-SCM over AWGN channels with GA demapping. R = 1 bit/dim. {βk } = {1, 1} and {1, 2} are compared. where the notations x and z˜ follow from (2.41). Using Cherno bounding [19, 87], the EF of SL-SCM can be upper bounded by

³

 PK

SL−SCM Pb,Che =

where

WI (dH )  kc

k=1

Φef,k

1 2N0

´  dH

K



¡ ¢ Φef,k (s) = exp 4|βk |2 s(N0 s − 1)

(2.46)

(2.47)

for AWGN channels (K → ∞) and

Φef,k (s) =

1 1 + 4|βk

|2 s(1

− N0 s)

(2.48)

for Rayleigh fading channels (K = 0). Similarly, for the ML-SCM, the Cherno bound can be obtained from (2.42) as ML−SCM Pb,Che

¶¶dH µ K µ WI (dH ) 1 X 1 = . Φef,k kc K k=1 2N0

(2.49)

By Jensen's inequality, it can be shown that for both AWGN and Rayleigh fading channels ML−SCM SL−SCM ≤ Pb,Che Pb,Che

(2.50)

where the equality holds when {βk } are equal. This implies that SL-SCM may

2.4. Error-Bound Analysis

32

outperform ML-SCM in the EF performance at high SNR. This will be illustrated later in Fig. 2.11 where the convergence property of iterative decoding is also compared.

2.4.3.2 Component Codes The SCM performance can be improved by using component codes with large minimum Hamming distance dH . This is especially important for fading channels where dH equals to the maximum achievable diversity order. Maximization of

dH for a given coding rate can be a tough task and may lead to codes with high decoding complexity. Alternatively, we consider to construct component code by simply concatenating a high-rate binary code with a repetition code. The increase of dH is proportional to the length of repetition code. Since the rate of the component code is decreased, in order to maintain a system throughput, a larger constellation (with a larger K ) should be used. This can increase the demapping cost signicantly with conventional coded modulation schemes where non-linear mapping and APP demapping (with complexity O(2K )) are used. With SCM, however, this can be greatly alleviated by using the GA demapper whose overall complexity grows only linearly with K . Another approach to increase the diversity order in coded modulation is the signal space diversity technique that rotates the transmit signal vector using a rotation matrix. SCM with GA demapping has a complexity advantage when combined with this technique, as shown in [98]. However, the concatenating approach mentioned above is more straightforward and may achieve a better performancecomplexity tradeo. It can achieve the same diversity order as the rotation approach with only marginal complexity increase. By contrast, more sophisticated signaling processing techniques are required in the rotation approach [98].

2.4.3.3 Weighting Factors We also show that, for a given component code, the asymptotic performance can be further optimized through adjusting the weighting factors β . From (2.46), (2.47) and (2.48), the Cherno bound on BER of SL-SCM can be rewritten as SL−SCM = Pb,Che

where

WI (dH ) dH D kc

¶ µ K 1 X |βk |2 D= exp − K k=1 N0

(2.51)

(2.52)

2.5. Analysis of Iterative Decoding

33

for AWGN channels and

¶ K µ 1 X 1 D= K k=1 1 + |βk |2 N0

(2.53)

for Rayleigh fading channels. Using the Lagrange method, it is easy to verify the following result.

Proposition : For SL-SCM with power constraint

PK k=1

|βk |2 = 1, the mini-

mum value of D is achieved by |βk |2 = 1/K . This implies that to achieve the lowest error oor, weighting factors with equal amplitudes {|βk |} should be used, i.e., the coded bits should be equally protected. A similar observation is made in [87]. In Section 2.7, we will show another approach to reducing the EF by using hybrid signaling schemes. The above discussions focus on the asymptotic EF performance that is only achievable when iterative decoding converges to ideal decoder feedback. However, they cannot provide insights into the convergence property of iterative decoding. Intuitively, the convergence can be faster if the performance at the rst iteration is better since it serves as the starting point of iterative process. From the union bound analysis, the key parameter is the minimum Euclidean distance

|x − z| where the notations x and z follow from (2.38). Regarding the convergence speed, the worst case is equal {βk } where the minimum Euclidean distance can be 0 and the optimal case is the equal-distance QAM constellation with

{βk } = {1, 2, 4, · · · }. However, from the earlier discussions, the QAM scheme can lead to a higher EF. For practical applications, it is of interest to optimize {βk } for a given target BER. This can be achieved by employing the evolution method discussed in the next section and some searching methods. For high-rate cases, the optimized {βk } are usually unequal.

2.5 Analysis of Iterative Decoding This section examines a fast evolution method to track the convergence behavior of the iterative decoding in SCM schemes. We consider SCM with a large constellation (a large K ), where APP demapping is too complicated and GA demapping has to be used. We introduce a mutual information (MI)-based semi-analytical method that is in spirit similar to the EXIT chart method [18]. We assume innite interleaving lengths (so, similar to the EXIT chart method, the results can only approximately characterize systems with nite codeword lengths).

2.5. Analysis of Iterative Decoding

34

Note that for ML-SCM over AWGN channels, an SNR evolution technique can be used to analyze the convergence performance. This technique was rst proposed for interleave-division multiple-access (IDMA) systems over at-fading channels [64] and later generalized to single-carrier systems over multi-path channels with turbo equalization [111]. One common feature of those schemes is that the demapper outputs can be modeled as the outputs from an AWGN channel which can be fully characterized by a single parameter, i.e., the SNR. However, for more general cases such as single-carrier systems over fast-fading channels or OFDM systems over frequency-selective fading channels, SNR only is not sucient since the distribution of channel coecients also aects performance. In this case, it is more accurate to use the MI evolution method discussed below. Again, we consider 1-D SCM schemes in this section.

2.5.1 Analysis of SL-SCM We begin with SL-SCM. We rst consider the characterization of the input-output relationship of an APP decoder.

2.5.1.1 Decoder Following the common treatment in [18], we model the decoder outputs {γk } (or inputs {λk }) as independent, identically distributed (i.i.d.) random variables following a consistent, symmetric, Gaussian distribution. Let γ be the extrinsic LLR for a coded bit c generated by the decoder, and σγ2 its variance. Then the probability density function (pdf) of γ given c can be written as [18]

µ ¶ |γ − (−1)c σγ2 /2|2 1 p(γ|c) = √ exp − . 2σγ2 2πσγ

(2.54)

It is easily shown that

pγ (γ|c) = pγ (−γ|¯ c). This Gaussian modeling can greatly simplify performance analysis because a variety of equivalent measures (e.g., mutual information, variance, delity [7, 90]) can be used to fully characterize the information exchanged during iterative decoding/demapping. With the above assumptions, the mutual information between c and γ can be evaluated as [28]

Z

∞

Iγ = I(c; γ) = 1 +

p(γ|c = 0) log −∞

p(γ|c = 0) dγ, p(γ|c = 0) + p(γ|c = 1)

(2.55)

2.5. Analysis of Iterative Decoding

35

Figure 2.9: BER and variance σc2 v.s. input mutual information Iλ for the decoder of (5, 7)8 coded BPSK schemes over AWGN and fully-interleaved Rayleigh fading channels. and σc2 , the average of the conditional variance of (−1)c given γ , can be computed as

σc2 ≡ E[V[(−1)c ]] = E[1 − tanh2 (γ/2)] Z ∞ p(γ|c = 0) tanh2 (γ/2)dγ = 1− −∞

(2.56a) (2.56b)

where E[·] denotes expectation w.r.t. the distribution of γ . Note that, with the Gaussian modeling, either σc2 or Iγ can fully characterize the decoder outputs. Numerical or Monte-Carlo methods [45] can be used to evaluate the above integrals. Similarly, we can characterize the decoder inputs by the mutual information Iλ between {λk } and {ck }. Now, we can characterize the input-output relationship of the decoder by

σc2 = TDEC (Iλ ).

(2.57)

The BER can be treated as a function of σc2 as

BER = b(σc2 ) = b(TDEC (Iλ )).

(2.58)

The discussions above assume a consistent Gaussian distribution of the extrinsic LLRs. In practice, this assumption may not hold true. However, we observe that the function TDEC (·) is robust to the distribution of {λk }. In other words,

TDEC (·) remains almost the same for {λk } with dierent distributions. This is illustrated in Fig. 2.9 where a 4-state convolutional code (5, 7)8 is assumed and {λk }

2.5. Analysis of Iterative Decoding

36

are generated from dierent sources, i.e., the AWGN and Rayleigh fading channels. Therefore, we can obtain TDEC (·) and g(·) by simply applying Monte-Carlo simulation to a BPSK scheme over an AWGN channel.

2.5.1.2 Demapper We now consider the SL-SCM with GA demapping. From (2.8), we have

λk = where

νk =

2βk (hβk (−1)ck + hνk + w) V[ζk ] X

(2.59)

βm ((−1)cm − E[(−1)cm ])

m6=k

represents the inter-layer interference in λk . To simplify the analysis, we model νk as a zero-mean Gaussian random variable in what follows. For general channels (except the AWGN channel), νk and w have dierent impact on performance as

νk is in essence an interference. Due to the overall interleaving in SL-SCM, we approximately have σc2k = σc2 ,

∀k,

i.e., the bit variance (feedback from the decoder) is independent of the bit position. Now, νk can be characterized by a zero-mean Gaussian interference with average power 2 σζ,k =

X

2 2 βm σcm =

m6=k

X

2 2 βm σc .

(2.60)

m6=k

From (2.59), given the distribution of h, the demapping performance is jointly determined by the interference νk and channel noise w. To characterize the relative power of νk and w, we dene the signal-to-interference ratio (SIR) and signal-tonoise ratio (SNR) with respect to bit position k as

βk2 sirk = 2 , σζ,k

βk2 snrk = 2 . σ

(2.61)

Suppose that the distribution of the channel coecient h is given and xed. Then the average mutual information between λk and ck is determined by sirk and snrk :

Iλ,k = I(λk ; ck ) = T (sirk , snrk )

(2.62)

where T (·, ·) is a two-variable function to characterize the relationship between

Iλ,k and (sirk , snrk ). To valuate T (·, ·), we consider a testing channel characterized

2.5. Analysis of Iterative Decoding

37

by the transition probability

1

p(y|x, h) = q

h2 2π( sir k

Ã

|y − hx|2 exp − h2 2( sirk + snr1 k ) + snr1 k )

! .

(2.63)

Then, the average mutual information between λk and ck is determined by sirk and snrk :

Iλ,k = I(x; y|h) X Z = 1−

(2.64)

Z p(h)p(x)p(y|x, h) log

x∈{±1}y∈R h∈R

p(y| + 1, h) + p(y| − 1, h) dydh. p(y|x, h)

The above integral can be evaluated numerically or by using the Monte-Carlo method. (The detailed treatments are similar to that used in the EXIT chart technique.) Then T (·, ·) can be characterized by a look-up table. From the above discussions, we can characterize the demapper as a transfer function σc2 → {Iλ,k }. With SL-SCM, the demapper outputs are de-interleaved before input to the decoder. Suppose a random interleaver is applied. Then the decoding performance can be approximately characterized by the average mutual information contained in the decoder inputs, i.e.,

Iλ =

K 1 X Iλ,k . K k=1

(2.65)

2.5.1.3 Evolution We assume that the three functions T (·, ·), TDEC (·) and g(·) are known. (They all can be obtained by simulating binary-input systems.) Then the iterative decoding performance of SL-SCM can be tracked as follows:

Initialization : Set σc2 = 1. Recursion : Update σc2 as à σc2 = TDEC

K 1 X T K k=1

Ã

βk2 βk2 P , 2 2 2 m6=k βm σc σ

!! .

(2.66)

Termination : After a preset number of recursions, estimate the BER by substituting the nal value of σc2 into (2.58). Although the method here is semi-analytical, it has much lower complexity than Monte-Carlo simulation in evaluating the overall performance. The conversion of

2.5. Analysis of Iterative Decoding

38

SIRs and SNRs to MI in (2.62) can greatly simplify the problem. If T (·, ·) is pre-calculated and stored in a look-up table, then the evolution method can be used to quickly predict the performance for SL-SCM with arbitrary parameters

{βk }. Alternatively, we can apply the conventional EXIT chart technique but the complexity can be high. This is because, with the conventional EXIT method, we need to apply the Monte-Carlo method to generate the EXIT functions for every set of {βk }. Note that, similar to the EXIT chart technique, the MI-evolution method can only approximately predict the performance. The reasons are as follows. First, interleavers with innite length are assumed, which is optimistic for performance. Second, the Gaussian assumption of the interference may be pessimistic.

2.5.2 Analysis of ML-SCM ML-SCM diers from SL-SCM in that each bit position is independently encoded and interleaved, as shown in Fig. 2.1. Dierent layers are protected unequally if unequal power allocations {|βk |2 } are used, which means that multiple variables need to be tracked. In this case, due to the unequal protection,

σc2k 6= σc2m ,

if

|βk | 6= |βm |.

(2.67)

Given {σc2k }, the output MI of the demapper can be found using (2.61) and (2.62) in the same way as in SL-SCM. The key is to evaluate σc2k separately for the dierent layers that are independently decoded:

σc2k = TDEC (Iλ,k ).

(2.68)

The evolution process is similar to that for SL-SCM and the details are omitted here.

2.5.3 Simplications in AWGN Channel The AWGN channel is a special case of the channel model in (2.2) with h = 1. In this case, the evolution method can be simplied since the self interference and channel noise have the same impact on performance and so they can be combined into one single term. To see this, let us take SL-SCM as an example. The interference-plus-noise power (in the demapper inputs) normalized by the useful signal power is

P σζ2ˆ k

=

m6=k

|βm |2 σc2 + σ 2 . |βk |2

(2.69)

2.5. Analysis of Iterative Decoding

39

Figure 2.10: Simulation (solid lines) versus MI evolution results (dashed lines) for SL- and ML-SCM schemes with K = 8 and {βk } = {1 × 8} over AWGN channels. Now the demapper output for bit-position k can be regarded as the output for an equivalent AWGN channel

yˆk = (−1)ck + ζˆk where ζˆk ∼ N (0, σζ2ˆ ) is the equivalent channel noise. Due to the overall interleavk

ing, we can approximately model {c1 , c2 , · · · , cK } ⇒ {ˆ y1 , yˆ2 , · · · , yˆK } as parallel, binary-input, AWGN channels. The average mutual information contained in the decoder inputs can be found as

I=

K 1 X T (0, 1/σζ2ˆ ). k K k=1

Thus, the MI table in (2.62) can be reduced to one-dimensional. In this case, the SNR 1/σζ2ˆ fully characterizes the demapper outputs and the MI evolution method k

is equivalent to the conventional SNR evolution method for IDMA over AWGN channels.

2.5.4 Examples Fig. 2.10 compares the simulation and MI evolution results for SL- and MLSCM schemes with K = 8 and equal power allocation (βk = 1, ∀k ) over AWGN channels. The component code is the concatenation of a rate-1/2 convolutional code (23, 35)8 with a rate-1/4 repetition code. The system rate is R = 1 bit/dim.

2.5. Analysis of Iterative Decoding

40

Figure 2.11: Simulation results for SL- and ML-SCM with {βk } = {1 × 6, 1.4 × 2} over fully-interleaved Rayleigh fading channels. The information block length is 8192 for SL-SCM and 1024 per layer for ML-SCM such that the frame length is the same in the two schemes. We can see that the evolution results are quite accurate for both SL- and ML-SCM. As the dierent bit positions of ML-SCM correspond to independent codewords, ML-SCM can be decoded in a layer-by-layer manner and so the decoding of the bits in one layer can help others. By contrast, in SL-SCM, all bits are protected by one common codeword and so they can only be decoded in a one-shot process. We can see that ML-SCM has quicker convergence (in terms of number of iterations) than SL-SCM. Similar observations can be made in Fig. 2.11 where SL- and ML-SCM with unequal power allocations over Rayleigh fading channels are compared. The component code is the concatenation of the convolutional code (5, 7)8 and the rate-1/4 repetition code. R = 1 bit/dim. The information block length is 16384 for SLSCM and 2048 per layer for ML-SCM. We can see that ML-SCM achieves faster convergence of iterative decoding, but SL-SCM achieves a lower error oor (as explained in Section 2.4.3.1 using error bounds). Fig. 2.12 compares the EF bounds, MI evolution and simulation results for ML-SCM with dierent rates. The component code is the same as that in Fig. 2.11. We consider R = 1, 1.5 and 2 bits/dim. The number of layers is 8R. The weighting factors are listed in Table 2.2. The number of iterations It = 12 for all curves. The information block length is 1024 per layer. We can observe good agreements among the performance results obtained using dierent methods.

2.6. Design of SCM Systems

41

Figure 2.12: Simulation, MI evolution, and EF bound results for ML-SCM over fully-interleaved Rayleigh fading channels. Table 2.2: Weighing Factors for SCM Schemes With the Convolutional Code R 1 1.5 2

Weighing factors 1 × 6, 1.4 × 2 1 × 6, 1.5774 × 3, 2.0736 × 1, 1.4848 × 2 1 × 6, 1.5774 × 2, 2.0989 × 3, 1.12129 × 1, 2.4883 × 1, 2.7258 × 1, 3.5832 × 2, 3.9252 × 1

We also consider the impact of component code and interleaving length on performance. In Fig. 2.13, we show the performance of K = 2 ML-SCM schemes with dierent rate-1/2 convolutional codes (5, 7)8 , (23, 35)8 and (133, 171)8 . The number of states is 4, 16 and 64, respectively. Dierent information block lengths (i.e., half of the interleaving lengths) are compared. We can see that the scheme with the 4-state code (5, 7)8 is the least sensitive to the block length. This is due to the fact that the codes with more states lead to more correlated coded bits and hence a longer interleaver is required to reduce the correlation involved. Note also that when the interleaving length is large, the code (133, 171)8 can yield the lowest error oor.

2.6 Design of SCM Systems The various analysis tools outlined above can be used to predict the performance of SCM with and without iterative decoding. They can also be used to optimize SCM

2.6. Design of SCM Systems

42

Figure 2.13: Simulation results for ML-SCM with dierent component codes and block lengths over AWGN channels. R = 1 bit/dim. K = 2, β1 = β2 = 1.

Figure 2.14: QPSK with (a) Gray and (b) anti-Gray mapping. schemes with dierent component codes. The objective is to nd the weighting factors {βk } that can minimize the required transmitted power for given component code and target BER. In general, when capacity approaching codes (e.g., the turbo code) are used, near-optimal {βk } can be found using the mutual information rule in Section 2.3 in a layer-by-layer manner. When relatively weak codes (e.g., convolutional codes) are considered, the error bounding and evolution analysis (in Section 2.4 and 2.5) are more relevant. They can be used together with some searching techniques to optimize {βk }. In particular, exhaustive search can be used for small K and the linear programming [64] and interior point [106] methods can be employed for large K . For the two-dimensional SCM schemes characterized by (2.70), the weighting factors {αm } can be optimized similarly.

2.7. Two-Dimensional (2-D) SCM

43

Figure 2.15: SCM-1. (α1 = α2 = 1. PAPR = 3.01 dB.)

Figure 2.16: SCM-2. (α1 = 1, α2 = 2. PAPR = 2.55 dB.)

2.7 Two-Dimensional (2-D) SCM In the above, we have focused on 1-D SCM constellations with real-valued signal points. In practice, it is necessary to use 2-D schemes with complex-valued signal points to improve bandwidth eciency. One approach is to treat a 2-D SCM as two independent, parallel 1-D SCM schemes, each corresponding to the in-phase or quadrature component. The discussions in the previous sections apply to such 2-D SCM schemes. Alternatively, 2-D SCM can be constructed by superimposing QPSK signals. Every two coded bits, denoted by (c2m−1 , c2m ), are rst mapped to a QPSK symbol

xm . Then M QPSK symbols are weighted summed to form the transmit symbol x. The SCM mapping given by (2.1) is modied as x=

M X

αm xm

(2.70)

m=1

where {αm } are constant weighting factors. The resultant constellations can be seen as special cases of the generalized hierarchical (or multi-resolution) constellations [50, 80] which have been considered for multimedia applications, downlink transmissions, adaptive modulation, etc. For each QPSK symbol in (2.70), both the Gray and anti-Gray mapping [16], as illustrated in Fig. 2.14, can be used.

2.7. Two-Dimensional (2-D) SCM

44

Figure 2.17: SCM-3. (α1 = 1, α2 = 2eiπ/4 . PAPR = 1.947 dB.)

Figure 2.18: SCM-4. (α1 = 1, α2 = 3/2. PAPR = 2.87 dB.) (Note that for QPSK, the Gray mapping can also be written in the form of (2.1) √ with {βk } = {1, −1}.) Figs. 2.15-2.18 illustrates four examples with M = 2 and dierent {αm } sets, which are constructed from Gray-mapped QPSK. They have dierent properties as follows and their performance will be compared later.

• SCM-1 employs equal power allocation between the two QPSK layers. There are only 9 distinct signal points in the constellation since some bit combinations are mapped to the same symbol. From Section 2.4.3, such a scheme can minimize the error oor. However, due to the overlapped signal points, the non-iterative decoding performance is poor.

• SCM-2 is a standard 16-QAM constellation. It is suboptimal for error oor minimization. However, it can yield the best performance for non-iterative decoding.

• SCM-3 has the lowest PAPR among the four. It has poorer non-iterative decoding performance than SCM-2 since the minimum distance between signal points is smaller in SCM-3. However, when iterative decoding is used, its error oor is exactly the same as that of SCM-2, which can be veried by the EF bounding analysis.

2.7. Two-Dimensional (2-D) SCM

45

Figure 2.19: Performance of SL-SCM with dierent 2-D signaling schemes and APP demapping with (It = 10) and without (It = 1) iterative decoding over AWGN channels.

• SCM-4 can yield a tradeo between the performance with and without iterative decoding. We can also construct hybrid SCM signaling by using dierent mappings at dierent QPSK layers. An interesting application of such construction is to reduce the error oor. Consider the 2-D SCM in (2.70) where each xm is a QPSK symbol. As shown in [66], unequal power allocations {|αm |2 } are necessary (to ensure good convergence of iterative decoding) when the rate is high. In this case, the EF performance is dominated by the layer with the lowest power. On the other hand, it is shown in [16] that anti-Gray mapping is advantageous to reduce the number of nearest neighbors in a QPSK constellation if ideal decoder feedback is available. Therefore, hybrid SCM with anti-Gray mapping employed at low-power layers may potentially reduce the overall error oor. In Fig. 2.19, we show examples of SL-SCM based on the dierent 2-D constellations described above. The hybrid mapping is constructed by superimposing a QPSK with Gray mapping and a QPSK with anti-Gray mapping (α1 = α2 = 1). The rate-1/2 convolutional code (23, 35)8 is used and the system rate is R = 2. We can see that the simulation results agree with the discussions above. For the above QPSK-based 2-D SCM scheme, we can encode each QPSK layer independently. This leads to an ML-MLC scheme where each layer is a BICM scheme with QPSK signaling. Then, following [19, 104], the capacity CML−SCM

2.8. Summary

46

Figure 2.20: Capacities of ML-SCM based on QPSK with dierent mapping rules over AWGN channels. with MSD and APP demapping can be computed as:

CML−SCM =

M X

à 1 X

m=1

q=0

! I(c2m−q ; y|c1 . . . c2m−2 ) .

(2.71)

It can be easily veried that CML−SCM ≤ I(x; y). The maximum achievable rate of the above 2-D SCM scheme is aected by the mapping rule for the QPSK modulation. The achieved rates with M = 2 and 3 and dierent QPSK mapping √ rules are shown in Fig. 2.20. We set α1 = α2 = 1/ 2 for M = 2, and {αm } =

{0.3637, 0.5283eiπ/6 , 0.7672eiπ/3 } for M = 3. From Fig. 2.20, we can see that the CML−SCM achieved with Gray mapping (for QPSK) approaches the constrained capacity (the dierence is not visible), signicantly outperforming that with antiGray mapping. Therefore, Gray mapping is a better choice for constructing 2-D SCM schemes with capacity-approaching component codes. However, in cases with weak codes, anti-Gray mapping can indeed exhibit advantages in other aspects such as the error oor performance, as shown in Fig. 2.19.

2.8 Summary We have made a comprehensive study of two types of SCM schemes over memoryless channels. The basic features of SCM are reviewed. The information-theoretic analysis demonstrates that ML-SCM is advantageous when capacity approach-

2.9. Appendix: Equivalence of the LMMSE and GA Approaches

47

ing codes are employed. The error bound analysis can be used to analyze the asymptotic performance of general SCM with practical codes and receiver, based on which we can extract several useful guidelines for system designs. The convergence behavior of the iterative decoding is also investigated by a mutual information evolution technique, which can be used for quick performance prediction. The examples demonstrate that both SL- and ML-SCM are attractive for high-rate transmissions.

2.9 Appendix: Equivalence of the LMMSE and GA Approaches In this Appendix, we show that the GA demapper is equivalent to an LMMSE detector. For notational clarity, let xk = (−1)ck , x = [x1 , · · · , xK ]> and β =

[β1 , · · · , βK ]> be real numbers and h = 1. We can rewrite (2.2) as y = β > x + w.

(2.72)

Following [56], the LMMSE estimate of x is given by

ˆ = µx + Vx βR−1 (y − β > E[x]) x

(2.73)

where µx = [E[x1 ], · · · , E[xK ]]> and Vx = diag{V[xk ]}, respectively, represent the

a priori mean and covariance matrix of x that are computed from the decoder feedback {γk }, and

R = V[y] = β > Vx β + σ 2 =

K X

βq2 V[xq ] + σ 2

(2.74)

q=1

is the variance of y . Let

βk2 V[xk ]

φk ≡

K P q=1

βq2 V[xq ]

+

. σ2

ˆ can be written as Then, it can be shown that each entry x ˆk of x xˆk ≡ E[xk ] +

βk V[xk ] K P q=1

βq2 V[xq ]

+

(y − σ2

K X

βq E[xq ])

q=1

K

X φk ≡ E[xk ] + (y − βq E[xq ]). βk q=1

(2.75)

2.9. Appendix: Equivalence of the LMMSE and GA Approaches

48

To generate the LLR for bk , we write

φk xˆk = φk xk + E[xk ](1 − φk ) + βk |

à X

! (2.76)

βq (xq − E[xq ]) + w

q6=k

{z

}

ξk

where ξk is the additive noise with mean and variance

E[ξk ] = (1 − φk )E[xk ],

(2.77)

V[ξk ] = φk (1 − φk )V[xk ].

(2.78)

Finally, by treating ξk as a Gaussian random variable, the extrinsic LLR is computed from (2.76) as

µ λk ≡ ln

Pr(ˆ xk |xk = +1) Pr(ˆ xk |xk = −1)

¶ = 2φk

xˆk − E[ξk ] V[ξk ]

(2.79)

By substituting (2.75) into (2.79), we get

λk =

2φk βk (1 − φk )V[xk ]

Ã

βk = 2P 2 βq V[xq ] + σ 2

(y − Ã

X

which is equivalent to the GA method in (2.8).

(2.80a)

βq E[xq ]

q6=k

(y −

q6=k

!

X q6=k

! βq E[xq ]

(2.80b)

Chapter 3

SCM with Clipping and Iterative Decoding 3.1 Introduction From Chapter 2, SCM with large constellations and proper power allocations can generate a Gaussian-like transmit signal. This provides a straightforward approach to achieving the shaping gain. The work in [23, 32, 66, 67] shows that such a concept is realizable with practical encoding and decoding methods. Simulation results show that SCM can operate within the shaping gap over AWGN channels [66, 67], surpassing the theoretical limit of the uniform signaling-based schemes. In Chapter 2, we have shown that SCM also provides an eective means of increasing diversity gains over fading channels by using low-rate codes and large constellations. However, there is a practical concern with SCM: the Gaussian-like transmit signal has a high PAPR, which may cause a problem for radio frequency power amplier eciency [46]. The same PAPR problem also exists in other shaped coded modulation schemes [36, 102] and OFDM systems. (The latter will be detailed in Chapter 4.) For OFDM systems, a number of PAPR reduction techniques have been studied (see [46] and references therein). They can be broadly classied into two categories. The rst includes but is not restricted to coding, partial transmit sequence, and selective mapping techniques that incur redundancy and spectral eciency loss. The second category includes pre-distortion methods that do not introduce redundancy. In particular, deliberate clipping [4, 59, 96] is a straightforward and ecient approach. It is generally more eective than other alternatives for PAPR suppression. In this chapter, we investigate the use of clipping in single-carrier SCM schemes. We show by capacity analysis that the theoretical penalty due to clipping is marginal for practical PAPR values. (In fact, with the same PAPR, the capacity of clipped SCM signaling is higher than that of uniform signaling in the lowto-medium rate region, see Fig. 3.4 below.) This is also veried by simulation results.

3.2. System Model

50

Figure 3.1: Encoder of a two-dimensional ML-SCM system. We also devise a practical soft compensation method to recover the performance loss due to clipping by exploiting the characteristics of clipping noise. This method can provide (at much lower complexity) performance close to the optimal APP method. It can be easily incorporated into the overall iterative receiver structure based on the GA demapping method outlined in Chapter 2. A variety of numerical results for AWGN and fading channels are provided. It is shown that the proposed method can eectively recover the performance loss incurred by clipping. Without loss of generality, we only consider two-dimensional multi-level SCM (ML-SCM) system in this chapter.

3.2 System Model 3.2.1 Encoding Consider an M -layer ML-SCM system as illustrated in Fig. 3.1. A data sequence

u is partitioned into M subsequences {um }. Then each um is encoded by a binary encoder (ENC-m) at the mth layer, resulting in a coded bit sequence vm = {vm [n]} of length 2J , where vm [n] ∈ {0, 1} and J is the frame length. The randomly permuted version cm of vm , from interleaver-m (Πm ), is then mapped to a QPSK √ Im sequence xm [n] = xRe −1, m [n] + ixm [n] (using the Gray mapping), where i = the superscripts  Re  and  Im  are used to denote the real and imaginary parts of cm [2n+1] cm [2n] . It is and xIm complex numbers respectively, xRe m [n] = (−1) m [n] = (−1) Im clear that xRe m [n] ∈ {+1, −1} and so does xm [n].

The output signal at time n is a linear superposition of M QPSK symbols:

x[n] =

M X m=1

αm xm [n],

n = 0, 1, · · · , J − 1,

(3.1)

3.2. System Model

51

where {αm } are constant weighting factors. The overall rate is R = 2

PM

m=1 rm

in

bits per two dimensions, where rm is the rate of the mth binary component code. The selection of {αm } will be discussed in the next subsection.

3.2.2 Peak-to-Average Power Ratio Let | · | be the amplitude. The PAPR (in decibel) of x[n] is dened as µ ¶ max{|x[n]|2 } PAPR = 10 log10 . E[|x[n]|2 ]

(3.2)

We assume that all the interleaved coded bits {cm [n]} are i.i.d. random variables with Pr(cm [n] = 0) = Pr(cm [n] = 1) = 1/2. The PAPR can be very high when

M is large. For an SCM scheme with {α1 = · · · = αM }, the PAPR is 10 log10 M dB. In order to suppress the PAPR, we can clip x[n] to x ¯[n] before transmission according to the following rule:

½ x¯[n] = g(x[n]) ≡

x[n], |x[n]| < A Ax[n]/|x[n]|, |x[n]| ≥ A

(3.3)

where A > 0 is the clipping threshold. Dene the clipping ratio (CR) in decibel as

µ CR = 10 log10

A2 E[|x[n]|2 ]

¶ .

(3.4)

The PAPR of the transmitted signal is given by PAPR = 10 log10 (A2 /E[|¯ x[n]|2 ]) . We select A according to the desired PAPR value. The performance of an ML-SCM scheme can be improved by properly choosing the power allocation factors {|αm |2 }. In this chapter, {|αm |2 } are determined using the simulation-based power allocation method [66], [67] for small M and the linear programming method [64] for large M . The phase angles ]αm can be used to shape the signal constellations and adjust the PAPR. For example, for a 4-level SCM with |αm | = 1, ∀m, the maximum PAPR = 6.02 dB is reached at ]αm = 0, ∀m. On the other hand, if ]αm = mπ/8, ∀m, then the PAPR is reduced to 5.16 dB.

3.2.3 Received Signal Model The clipped signal x ¯[n] is then transmitted over a memoryless channel. The received signal is given by

y[n] = h[n]¯ x[n] + w[n],

n = 0, 1, · · · , J − 1

(3.5)

3.3. Eect of Clipping on the Achievable Rates

52

where h[n] is the channel coecient and w[n] is an AWGN with variance σ 2 per dimension. The ratio of energy per bit (Eb ) to the noise power spectral density (N0 = 2σ 2 ) is given by Eb /N0 = E[|¯ x[n]|2 ]/(2Rσ 2 ). When M is large, x[n] can be approximated by a Gaussian random variable from the central limit theorem. Using Price's theorem for non-linear systems with Gaussian inputs [78], we can model the clipping operation in (3.3) as a linear process (3.6)

x¯[n] = κx[n] + d[n]

where κ is a constant attenuation factor, and d[n] is a Gaussian-distributed distortion term with mean zero and variance σd2 per dimension, which is statistically uncorrelated with x[n]. In general, κ and σd2 depend on CR and the statistics of

x[n], and can be calculated as [22, 74] κ=

E[x∗ [n]¯ x[n]] , E[|x[n]|2 ]

(3.7)

E[|¯ x[n]|2 ] − κ2 E[|x[n]|2 ] , 2 where ∗ denotes complex conjugate. Then (3.5) can be rewritten as σd2 =

(3.8)

(3.9)

y[n] = κh[n]x[n] + h[n]d[n] + w[n]. The above modeling will be used in Section 3.4.

3.3 Eect of Clipping on the Achievable Rates We now investigate the impact of clipping on the performance limits of SCM systems. Consider a memoryless channel characterized by Y = HX + W , where

H , X and W are, respectively, the channel coecient, transmitted signal and AWGN. Assume that the receiver has perfect knowledge of H . Then for a given distribution of X , the (constellation-) constrained capacity is quantied by the average mutual information [28, 99],

C = EH [I(X; Y |H = h)] = EH [H(Y |H = h) − H(Y |X, H = h)] = −EH [EY [log(p(Y |H = h))]] − log(2πeσ 2 ) where H(·) denotes the dierential entropy function H(X) = −

(3.10)

R

p(x) log p(x)dx

and EH [·] and EY [·] denote expectation with respect to the distribution of H and

Y , respectively. We can apply numerical methods to evaluate C . We examine the

3.3. Eect of Clipping on the Achievable Rates

53

Figure 3.2: Capacities of clipped Gaussian input signals over (a) AWGN and (b) Rayleigh fading channels. cases of continuous and discrete X separately below. Note that we focus only on the constellation-constrained capacity given by (3.10) in this chapter. For the ML-SCM schemes considered here, each layer employs a BICM scheme with Gray-mapped QPSK modulation. In theory, such ML-SCM cannot achieve the constrained capacity. However, the loss is marginal since the rate loss of BICM with Gray mapping [19] is negligible. (See also the discussion related to Fig. 2.20.)

3.3.1 Continuous Input Signal We rst consider X as a clipped version of a complex, Gaussian random variable with zero mean and variance 1/2 per dimension. The clipping rule given by (3.3)

3.3. Eect of Clipping on the Achievable Rates

54

is used and the resultant PAPR is

A2

PAPR = R |x|≤A

|x|2

exp(−|x|2 ) π

dx + A2

R |x|>A

exp(−|x|2 ) dx π

.

The probability density function of X is given by ½ exp(−|x|2 )/π, |x| < A p(x) = 2 exp(−|A| )δ(|x| − A)/(2πA), |x| = A

(3.11)

(3.12)

where δ(·) is the Dirac delta function. The maximum achievable mutual information can be computed as

Z p(y|H = h) log p(y|H = h)dy − log(2πeσ 2 )

I(X; Y |H = h) = − Y

where the distribution p(y|H = h) is evaluated using integral as µ ¶ Z −|y − hx|2 1 p(y|H = h) = exp p(x)dx, ∀y ∈ C 2 2σ 2 X 2πσ

(3.13)

(3.14)

where C denotes the complex number eld. Numerical methods can be employed to evaluate the integrals above. Specically, when A = ∞,the PAPR is innite and

µ

|h|2 I(X; Y |H = h) = log 1 + 2 2σ

¶ (3.15)

which is the channel capacity. To achieve the capacity computed above, an overall maximum likelihood (ML) decoder should be used. An alternative, low-complexity but suboptimal approach is to simply treat the clipping distortion d[n] in (3.9) as an equivalent AWGN. With this method (referred to as EAWGN below), the maximum achievable rate is restricted by

I(X; Y |H = h) = log(1 + SNDR), where

SNDR =

|h|2 κ2 E[|X|2 ] 2(σ 2 + |h|2 σd2 )

is the signal-to-noise-and-distortion-ratio conditioned on h, and √ πCR −CR2 + κ=1−e erfc(CR), 2 µ ¶ E[|X|2 ] κ2 2 σd = 1− , 2 2 1 − e−CR √ R∞ 2 for Gaussian signals, where erfc(z) = 2/ π z e−t dt.

(3.16) (3.17)

(3.18) (3.19)

3.3. Eect of Clipping on the Achievable Rates

55

Figure 3.3: SCM constellations (a) without and (b) with clipping. M = 5, {|αm |} = {1, 1.4565, 2.1218, 3.0912, 4.5031} and {]αm = mπ/10, ∀m}. The numerical results based on (3.10) for the clipped Gaussian signaling and the Shannon limit are shown in Fig. 3.2. We set CR = 3 and 2 dB, and the resultant PAPR is 3.64 and 3 dB, respectively. The performance of the clipped Gaussian signaling is close to the Shannon limit if the optimal ML decoding is used. We have also included in Fig. 3.2 the results for the EAWGN approach. It is seen from Fig. 3.2 that this suboptimal approach leads to signicant performance loss. This motivates us to develop improved techniques.

3.3.2 Discrete Input Signal Next we examine the SCM systems where X is a discrete variable. For the M -layer scheme described in Fig. 3.1, the input signal before clipping is the summation of M complex random variables. When M is large, the distribution of the unclipped signal is approximately complex Gaussian. Hence, the above analysis on the clipped Gaussian signaling can provide insights into such cases. We now focus on SCM schemes with a small-to-medium M . In this case, the transmit signal X is drawn from a discrete constellation X with the probability mass function

p(x) =

1 , |X |

∀x ∈ X

(3.20)

where |X | denotes the size of the constellation. Therefore, the distribution of the continuous received signal Y can be computed using summation as µ ¶ X 1 −|y − hx|2 exp , ∀y ∈ C. p(y|H = h) = 2πσ 2 |X | 2σ 2 x∈X

(3.21)

The capacity can be evaluated by substituting (3.21) into (3.13). Note that the constellation X has dierent shapes in the clipped and unclipped cases.

3.3. Eect of Clipping on the Achievable Rates

56

Figure 3.4: Capacities of clipped SCM schemes over (a) AWGN and (b) Rayleigh fading channels. (The optimal ML decoding is assumed.) We take a 5-layer scheme as an example. It employs a non-equispaced 45 =

1024-ary signal constellation that is fully determined by {αm }, as depicted in Fig. 3.3. The related PAPR without clipping is 5.39 dB. We set CR = 3.5 dB, and the resultant PAPR = 3.68 dB. The capacities achieved over AWGN and Rayleigh fading channels are shown in Fig. 3.4. The performance of the conventional 1024QAM1 signaling (PAPR ≈ 4.5 dB) and the Shannon limit are also included for comparison. From Fig. 3.4, we can make the following observations:

• Without clipping, the dierences between the SCM capacity, 1024-QAM 1 In this thesis, a conventional QAM constellation represents a square QAM constellation in which signal points are equispaced and utilized with equal probabilities.

3.4. Iterative Decoding of Clipped SCM

57

capacity and Shannon limit are not signicant for low-to-medium rates (e.g., up to about 9 bits per two dimensions). The SCM capacity is also higher than the 1024-QAM capacity for the most part in this region. For a target rate of 5 bits per two dimensions, the dierence between the required Eb /N0 of SCM and the Shannon limit is only 0.21 dB for AWGN channels. In this case, SCM can achieve a shaping gain of about 0.7 dB over the 1024-QAM signaling.

• Although clipping degrades the achievable rate, the eect is not serious if optimal decoding is applied. We see that in AWGN channels, at a rate of 5 bits per two dimensions, about 0.2 dB loss in Eb /N0 is introduced by clipping at PAPR = 3.68 dB. The clipped SCM, when compared with the conventional 1024-QAM, has lower PAPR but higher capacity in the vicinity of 5 bits per two dimensions. This reveals that the clipped SCM scheme can provide a good trade-o between PAPR and achievable rate. Note that here we focus on the impact of clipping on the maximum achievable rates, assuming a xed distribution of the transmit signal X . The study in [88, 91] reveals that the capacity of a peak-power-limited quadrature Gaussian channel is achieved by a distribution with discrete amplitude, uniform independent phase (DAUIP). With the SCM scheme, it is possible to optimize the distribution of X to approach the capacity predicted by [88] through adjusting the weighting factors

{αm }. This will be considered in our future work.

3.4 Iterative Decoding of Clipped SCM Now we turn our attention to practical SCM systems. For simplicity, we only discuss AWGN channels with h[n] = 1, ∀n. The discussions below can be easily extended to more general cases. In Section 3.3, we have shown that, the capacity penalty incurred by clipping is not severe for a reasonable clipping ratio. But, as seen from the EAWGN curves in Fig. 3.2, clipping can cause serious problems if treated improperly. The following observations suggest a possible approach to this issue:

• If |y[n]|, the amplitude of the received signal, is large, then the clipping probability (i.e., the probability of x[n] being clipped) is high. • If |y[n]| is small, then the clipping probability is small.

3.4. Iterative Decoding of Clipped SCM

58

Figure 3.5: Block diagram of the iterative decoding/detection algorithm. We may exploit these facts to compensate for the clipping eect. This is the underlying rationale for the soft compensation method presented below.

3.4.1 Overall Iterative Detection Principle As illustrated in Fig. 3.5, the receiver consists of one elementary signal estimator (ESE) and M soft-input soft-output (SISO) decoders (DECs), connected by interleavers and de-interleavers. The turbo-type iterative process is applied with the ESE outputs used as the DEC inputs and vice versa. We will focus on the ESE that handles the inter-layer interference and clipping distortion.

3.4.2 Optimal Realization of the ESE The function of the ESE is to generate LLRs for the coded bits {cm [n]}, ignoring the coding constraint (i.e., as if {cm [n]} is an un-coded sequence). The optimal APP demapping method can be applied here. Let us consider the bit cm [2n] that is mapped to the real part of a QPSK symbol xm [n] at layer-m. We have

µ λm [2n] = ln

Pr(xRe m [n] = +1|y[n], {γm [n]}) Re Pr(xm [n] = −1|y[n], {γm [n]})

¶ − γm [2n],

(3.22)

where γm [n] = ln (Pr(cm [n] = 0)/ Pr(cm [n] = 1)) is the a priori LLR for cm [n]. (The case for cm [2n + 1] is similar.) During iterative decoding, {γm [n]} are approximated by the feedback LLRs from the DECs. (See Fig. 3.5.) Given the signal constellation at the transmitter, (3.22) can be evaluated following the standard procedures [15, 24, 39]. The related complexity is proportional to the constellation size. The constellation size of SCM (with and without clipping) grows exponentially with the number of layers M , which can be a serious concern for large M . In the following, we present several suboptimal, low-cost alternatives.

3.4. Iterative Decoding of Clipped SCM

59

3.4.3 ESE Based on GA for Unclipped SCM In Chapter 2, we have discussed the suboptimal GA method for unclipped SCM with real weighting factors {αm }. We now consider the case with complex {αm } and extend the GA method. Consider an unclipped SCM with received signal (3.23)

y[n] = x[n] + w[n] over AWGN channels. Let us concentrate on layer-m. We rst generate

yˆm [n] =

∗ αm y[n] = |αm |xm [n] + ζˆm [n], |αm |

(3.24)

∗ where ζˆm [n] = αm ζm [n]/|αm | represents the interference-plus-noise w.r.t. xm [n]

and ζm [n] is given by:

ζm [n] =

X

(3.25)

αm0 xm0 [n] + w[n].

m0 6=m

We approximate ζˆm [n] by a complex Gaussian variable. Then we can compute the

extrinsic LLRs by (with much lower complexity than the optimal APP method) ³ ´−1/2 ³ ´ Re [n]−|α |−E[ζ ˆRe [n]])2 (ˆ ym m Re m ˆ exp − Re [n]] 2V[ζˆm  2πV[ζm [n]]  λm [2n] = ln  ³  ´−1/2 ³ ´ Re Re 2 ˆ m |−E[ζm [n]]) Re [n]] 2πV[ζˆm exp − (ˆym [n]+|α 2V[ζˆRe [n]] m

= 2|αm |

Re − E[ζˆm [n]] . V[ζˆRe [n]]

yˆRe m [n]

(3.26)

m

Re The computational details are as follows. From (3.26), the key is to nd E[ζˆm [n]] Re and V[ζˆm [n]]. First, a denition. Let x be a complex random variable and E[x]

be its mean. Dene the covariance matrix of x as · ¸ V[xRe ] C[xRe , xIm ] C[x] = , C[xRe , xIm ] V[xIm ]

(3.27)

where C[xRe , xIm ] = E[xRe xIm ]−E[xRe ]E[xIm ]. Following [77], E[xm [n]] and C[xm [n]] can be estimated as

µ

¶ µ ¶ γm [2n] γm [2n + 1] E[xm [n]] = tanh + i tanh , 2 2  ³ ´  1 − tanh2 γm2[2n] 0 ³ ´ , C[xm [n]] =  0 1 − tanh2 γm [2n+1] 2

(3.28a) (3.28b)

3.4. Iterative Decoding of Clipped SCM

60

where γm [2n] and γm [2n + 1] are the extrinsic LLRs generated by the DECs, we have assumed that the real and imaginary parts of xm [n] are uncorrelated, and thus the o-diagonal entries of C[xm [n]] are zeros. The initial values of γm [2n] and γm [2n + 1] are set to zeros, implying no a priori information. From (3.1) and (3.23), we have

E[y[n]] = C[y[n]] =

M X m=1 M X

αm E[xm [n]],

(3.29a)

> Bm C[xm [n]]Bm + σ 2 I,

(3.29b)

m=1

·

¸ Re Im αm −αm . Then, from (3.23), where I is the 2 × 2 identity matrix and Bm = Im Re αm αm we have E[ζm [n]] = E[y[n]] − αm E[xm [n]],

(3.30a)

> C[ζm [n]] = C[y[n]] − Bm C[xm [n]]Bm .

(3.30b)

Now, we can generate ∗ αm E[ζm [n]], |αm | 1 C[ζˆm [n]] = B > C[ζm [n]]Bm . |αm |2 m

E[ζˆm [n]] =

(3.31a) (3.31b)

Re Re Finally, E[ζˆm [n]] and V[ζˆm [n]] can be obtained from (3.31) and cm [2n] is estimated

using (3.26). We now consider the computational cost for the GA method as described by (3.26)-(3.31). Some simple methods, such as sharing the results of (3.29) for all layers, can be applied to reduce the computational cost. Also, some intermediate results can be reused to speed up the computations. It can be shown that the total computational cost of the ESE is about 15 real multiplications, 13 real additions and a few other operations (e.g., tanh(x) function) per coded bit (Note : Each

xm [n] carries two coded bits), which is independent of the number of layers M . For the case of real {αm }, the complexity of the GA method can be further reduced. In contrast, the complexity of the APP method is linear with 4M , i.e., the size of the signal constellation. Compared with the DEC cost, the above ESE cost is quite moderate. As an example, the APP decoding [5] for a 16-state rate-1/2 convolutional code requires about 64 real multiplications and 32 real additions per coded bit. In this case, the

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61

overall complexity of the receiver is dominated by the DECs. Considering that the APP decoding has become a standard function in modern systems employing turbo-like codes, we expect that the cost of the above GA method is acceptable for real applications.

3.4.4 Modied GA Method (MGA) for Clipped SCM We now proceed to consider clipped SCM systems. In this case, the basic GA method in Section 3.4.3 deteriorates if the clipping distortion is ignored. Based on the modeling in (3.9), an improved method is to treat the clipping distortion

d[n] as an equivalent AWGN. Again, we focus on layer-m. Rewrite (3.9) as y[n] = καm xm [n] + ζ˜m [n] with

ζ˜m [n] = κ

X

αm0 xm0 [n] + d[n] + w[n].

(3.32)

(3.33)

m0 6=m

Now the GA method can be applied. The performance of this modied GA (MGA) method is still not satisfactory, but it can be used in the initial stage of the iterative decoding, as illustrated later.

3.4.5 Soft Compensation for Clipped SCM Next, we derive a soft compensation (SC) technique that uses a joint process to treat the inter-layer interference and clipping noise. We rewrite (3.5) as

y[n] = x[n] + w[n] + z[n] = αm xm [n] + ζm [n] + z[n]

(3.34)

where ζm [n] is given by (3.25) and z[n] = x ¯[n] − x[n] represents the clipping noise. Consider the detection of xm [n] based on (3.34). We propose the following suboptimal SC method which has complexity only slightly higher than the GA method. We again approximate ζm [n] + z[n] by an additive complex Gaussian variable. It is important to note that here the statistics of ζm [n] + z[n] are dierent for dierent hypotheses on xm [n], since the clipping eect depends on the hypothesis. For simplicity, we assume real-valued {αm } and omit the time index n from now on. We rewrite (3.34) as y = αm xm + ζm + z and introduce the following notations: Re + z Re under the hypothesis • E + and E − are, respectively, the means of ζm

xRe m = +1 and −1.

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Re • V + and V − are, respectively, the variances of ζm + z Re under the hypothesis

xRe m = +1 and −1. Similarly to (3.26), we have

³ ´ Re + 2 m −E ) exp − (y −α 2V + ´ ³ λm = ln  Re − 2 m −E ) (2πV − )−1/2 exp − (y +α 2V − µ +¶ 1 V (y Re − αm − E + )2 (y Re + αm − E − )2 = − ln − + . 2 V− 2V + 2V − 

−1/2

(2πV + )

(3.35)

The conditional means E ± and variances V ± in (3.35) can be estimated using the method in the next subsection. Notice that (3.35) is only slightly more complicated than (3.26).

3.4.6 Evaluation of the Conditional Means and Variances We approximately treat ζm and z in (3.34) as independent variables. Since ζm is exactly the same as that in (3.25), the key is to nd the statistics of z . From (3.3),

z is given by ½ z=

0, |x| < A . Ax/|x| − x, |x| ≥ A

(3.36)

± If the conditional probability mass function p(x|xRe m = ±1) is available, then E

and V ± can be evaluated using numerical integration. However, this is dicult in practice due to the excessive computational cost. We propose the following suboptimal strategy employing Gaussian approximation. For clarity, we will omit the superscripts  ±  and the method below is applied to both the hypotheses

xRe m = +1 and −1. From (3.34), we can express x as x = ζm − w + αm xm .

(3.37)

Then µ ≡ E[x] and V ≡ C[x] can be obtained from (3.37), if E[xm ] and C[xm ] are given and E[ζm ] and C[ζm ] are available. We now treat x as a complex Gaussian random variable. Then E[z] and C[z] are fully determined by (µ, V ) from (3.36). We denote these relationships using the two functions below:

E[z] = φ(µ, V ),

(3.38a)

C[z] = ϕ(µ, V ).

(3.38b)

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63

Table 3.1: Parameters of the ML-SCM Schemes in Fig. 3.6

M 4 5

R 4 5

{|αm |} 1, 1.4523, 2.1086, 3.0626 1, 1.4565, 2.1218, 3.0912, 4.5031

{]αm } mπ/8, ∀m mπ/10, ∀m

PAPR(dB) 4.74 5.39

In general, the functions in (3.38) can be generated numerically using the Monte Carlo method. We can create two look-up tables to characterize them. Assuming that these two tables are available, then the SC cost is only slightly higher than the GA method in Section 3.4.3. Since (µ, V ) involves ve parameters, we need two 5-dimensional (5D) tables. We now consider an approximate technique to reduce memory cost using

V ≈ vI

(3.39)

where v = (V[xRe ] + V[xIm ])/2 and I denotes the 2 × 2 identity matrix. This is to approximately characterize x using a symmetric complex Gaussian distribution

CN (µ, vI). Now there are only three parameters involved. Furthermore, it is easily shown that

µ φ(|µ|, vI), |µ| ϕ(µ, vI) = Ψϕ(|µ|, vI)Ψ> , φ(µ, vI) =

where

· ¸ 1 µRe −µIm Ψ= . |µ| µIm µRe

(3.40a) (3.40b)

(3.41)

Therefore we only need two 2-dimensional (2-D) tables to characterize φ(|µ|, vI) and ϕ(|µ|, vI) and then use (3.40) to nd φ(µ, vI) and ϕ(µ, vI) for the SC method. The SC method is essentially a turbo-type clipping noise cancellation technique based on the extrinsic information produced by the SISO decoders. This distinguishes it from the decision-aided clipping noise cancellation techniques in [22], [71] and the signal reconstruction techniques in [57], [25].

3.4.7 Examples We now provide several examples of clipped SCM schemes. We will always assume that the interleavers are randomly generated. We observe that the MGA method based on the Price's approximation (see Section 3.4.4) provides improved performance at the starting stage of the iterative process. In this case, the feedback is not reliable and the SC method (see Section 3.4.5) is not eective. However, after

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64

Figure 3.6: Performance of the clipped ML-SCM with a doped code and dierent detection methods over AWGN channels. (a) M = 4, (b) M = 5. J = 105 . For MGA w/ SC, the MGA method is rst used for ItM = 1 iteration and then the SC method is used for ItS = 5 iterations. a few iterations, the feedback from the DECs becomes reliable and the SC method becomes more eective. In the following, we adopt a hybrid strategy in which the MGA method is executed for ItM iterations, followed by the SC procedure for ItS iterations. The values of ItM and ItS are obtained experimentally.

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65

3.4.7.1 AWGN Channels We rst consider AWGN channels. The rate-1/2 doped code introduced in [17] is chosen as the component code for each layer. We set M = 4 and 5. The parameters and the PAPR without clipping are given in Table 3.1. Clipping with

CR = 3.5 dB is applied to both systems to reduce the PAPR to about 3.68 dB. (Note that the conventional 64-QAM signaling has the same PAPR of 3.68 dB.) The maximum number of iterations in the component DECs [17]2 is set to 200, and the maximum number of iterations between the DECs and the ESE is set to 6. The entropy-based stopping criterion introduced in [66] is used to terminate the iterations. The BER performance with dierent detection methods is shown in Fig. 3.6 (a) and (b) respectively for M = 4 and 5. For comparison, the Shannon limit, the 1024-QAM capacity, the capacity limits of the related SCM signaling and the SNR evolution results for the unclipped case are also included. We can see that when clipping is not used, the GA method and the optimal APP method perform similarly, and the achieved performance is quite close to the channel capacity. At BER of 10−5 , the dierence between the required Eb /N0 and the Shannon limit is only 0.9 and 1.2 dB for M = 4 and 5, respectively3 . Also, we can see that the results predicted by SNR evolution are in good agreement with the simulation results. On the other hand, from Fig. 3.6, the SCM performance deteriorates when clipping is used. With the optimal APP detection, the loss is within 0.2 dB (at

BER = 10−5 ) for the two examples. This is roughly in line with the loss in capacity due to clipping. For the suboptimal GA and MGA methods, the performance loss increases signicantly, with the MGA method performing better. Using the SC method, however, the dierences in performance are reduced to about 0.3 and 0.5 dB for M = 4 and 5, respectively. This implies that the SC method can recover most of the performance loss due to clipping and hence provide an ecient solution. The performance loss of the SC method due to the approximation in (3.39) is also shown in Fig. 3.6. Each dimension is quantized to 20 levels. We can see that the dierence between the 5-D and 2-D methods is marginal (within 0.1 dB). For 2 The

component code employed here is a serial concatenated code with doping and its performance can be improved by iterative decoding [17], following the turbo principles. 3 For comparison, the best simulation results (to the authors' knowledge) based on trellis shaping and equispaced QAM constellations for both R = 4 and 5 bits per two dimensions are about 0.8 dB away from the channel capacity, as reported in [63], [102]. However, the associated PAPRs in [63], [102] are relatively high (7.26 and 8.93 dB for R = 4 and 5 bits per two dimensions, respectively).

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Table 3.2: Parameters of the ML-SCM Schemes at R = 2 bits per two dimensions in Fig. 3.7 S 1 4 8

M 2 8 16

{αm value × layer number} 1 × 1, 1.25 × 1 1 × 6, 1.44 × 2 1 × 12, 1.22 × 4

PAPR(dB) 2.96 8.90 11.92

Figure 3.7: Performance of SCM at R = 2 bits per two dimensions over fullyinterleaved Rayleigh fading channels. J = 2048. The number of iterations is 10. For the clipped cases with SC, we set ItM = 6, and ItS = 4. the latter, two small tables of size 20 × 20 are used for φ(·, ·) and ϕ(·, ·). Based on the above observations, we only consider the suboptimal detection methods and use 2D tables for SC in the rest of this chapter.

3.4.7.2 Fading Channels Next we consider SCM over fully-interleaved Rayleigh fading channels. The detection methods outlined above can be easily extended here. For simplicity, we only consider real {αm }, and the component code is realized by the serial concatenation of a rate-1/2 non-systematic convolutional code with generator polynomials

(23, 35)8 and a length-S repetition code. The repetition coding is introduced here to increase diversity gain. We x the total rate R = 2 bits per two dimensions (i.e., M = 2S ) and the frame length J = 2048. Three schemes with dierent S as tabulated in Table 3.2 are compared. From Fig. 3.7, we can see that the SCM performance in fading channels can be signicantly improved by introducing repetition coding. A gain of about 2.8

3.5. Comparison with BICM

67

dB at BER = 10−5 can be achieved by increasing S from 1 to 4. Note that the receiver cost increases slightly with S , which can be seen as follows.

• The normalized cost (per information bit) of a convolutional decoder is independent of S .

• The cost related to the repetition code is negligible [77]. • The ESE cost grows linearly with S . • The dominant factors of the receiver cost are usually related to the convolutional decoders. In Fig. 3.7, the results are also shown for the clipped schemes with S = 4 and 8 at a PAPR of 2.96 dB, the same as that of the unclipped scheme with S = 1. (The CRs are 2.1 and 2.02 dB respectively for S = 4 and 8.) In order to reduce the SC cost, we have adopted the following two strategies with marginal performance loss. Firstly, since the number of layers is large, we assume that dierent hypotheses of the bit to be estimated (e.g., xRe m = +1 or −1) have negligible eect on the estimation of clipping noise. Thus we use the extrinsic information only in evaluating (3.38). Secondly, we update the soft estimation of clipping noise only when all the extrinsic information from the DECs is renewed. It is seen that the SC technique can reduce the error oor. The clipped schemes with SC outperform the unclipped scheme with S = 1 by about 2 dB at BER =

10−5 . Note that with S = 8 , a large portion of the transmitted symbols are clipped and the PAPR is reduced by about 9 dB, but the performance loss is within 1 dB. This demonstrates that the SC technique can work eectively for deep clipping.

3.5 Comparison with BICM It is interesting to compare SCM with other alternative coded modulation schemes. In the following, we will focus on comparison with BICM [19, 73, 114], that has attracted tremendous attention recently for its performance advantages in fading channels. Denote c[n] = {c1 [n], c2 [n], · · · , cK [n]} where ck [n] ∈ {0, 1} is the k th coded bit carried by x[n]. With BICM, the transmitted signal x[n] is generated using, instead of (3.1), a more general mapping rule: x[n] = µ(c[n]). The image of

µ(·) is usually a 2K -ary constellation of uniformly distributed signaling points, but the principle can be generalized to non-uniform constellations [40, 61, 70]. With this view, some comments are in order.

3.5. Comparison with BICM

68

• The ML-SCM in Fig. 3.1 involves multiple encoders while a BICM scheme usually involves only one overall encoder. For turbo-like component codes, optimized ML-SCM and BICM schemes can achieve similar performance.

• With QPSK modulation at each layer, ML-SCM optimization only involves M = K/2 weighting factors {αm }. BICM optimization is a much more complicated issue involving 2K constellation points [86, 87, 93].

• SCM can be detected using the GA demapping with complexity O(M ). We observe that in many cases, the performance of the suboptimal GA method can approach that of the optimal APP method. By contrast, for general BICM schemes with non-linear mapping, the GA method can not be applied and the APP demapping with complexity O(2K ) has to be used4 . Therefore, SCM may exhibit a complexity advantage for large M .

• As demonstrated in Fig. 3.7, given a target rate, we can achieve a diversity gain in SCM by decreasing the rate of each layer (and increasing M accordingly). The design and detection complexities of SCM grow linearly with

M . For BICM, we can increase the diversity gain by using larger constellations or rotating the transmit signal vector [98], but the design and detection complexities increase very quickly.

• Both SCM and BICM suer from the high PAPR problem when OFDM is involved. Compared with BICM, SCM is more robust when clipping and soft compensation are applied, as explained in Chapter 4. In the following, we present several comparison examples based on turbo and convolutional codes.

3.5.1 Comparison in AWGN Channels We rst compare ML-SCM and BICM over AWGN channels. We consider R = 2 bits per two dimensions. The rate-1/2 turbo code [11] (23, 35)8 is employed in both schemes. For ML-SCM, we set M = 2, {αm } = {1, 1.51}; PAPR = 2.83 dB; the number of iterations is 9 in the DECs and 2 between the DECs and the ESE. For 4 In

several special cases of high-order modulations, e.g., the QAM schemes with Gray mapping, it is possible to employ some approximate, low-complexity demapping methods [97] with performance close to that of the APP method. However, in general cases, soft demapping has to be performed through evaluating the metrics associated with the 2K signal points, which involves an exponential complexity w.r.t. the number of coded bits per symbol K .

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69

Figure 3.8: Performance of turbo-coded ((23, 35)8 ) ML-SCM and BICM at R = 2 bits per two dimensions over AWGN channels. BICM, the Gray mapping is applied to the 16-QAM constellation5 ; PAPR = 2.55 dB; the number of iterations in the DEC is 18. The two schemes have nearly the same complexity. Clipping is not considered here since the PAPRs are not signicant. The performance with J = 2048 and 32768 is shown in Fig. 3.8. The BICM performance is better when J is small while the ML-SCM performance surpasses that of BICM for a large J. (Similar observations can be made for convolutional codes-based schemes over AWGN channels.) One reason for this is that the suboptimal GA detection is used in ML-SCM while the optimal APP demapping is used in BICM. Another reason is that the interleaver length for ML-SCM is only 1/M of that of BICM at xed J , which aects interleaving gain [8]. (However, when J is large, the impact of interleaver length becomes less signicant.)

3.5.2 Comparison in Fading Channels Fig. 3.9 compares ML-SCM with BICM over fading channels at rates R = 2, 3 and 5 bits per two dimensions. The same component code with S = 4 as that in Fig. 3.7 is again used here for ML-SCM. For comparison, three BICM schemes with iterative decoding (BICM-ID) reported in [24, 93, 73] are also simulated. For BICM-ID, the (23, 35)8 convolutional code is directly used for R = 2, and 5

Gray mapping yields the best known performance for BICM with turbo codes and QAM constellations [93].

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70

Table 3.3: Parameters of the ML-SCM and BICM-ID Schemes in Fig. 3.9 R

M

2 3 5

8 12 20

ML-SCM {αm value × layer number}

1 × 6, 1.44 × 2 1 × 6, 1.58 × 3, 2.07 × 2, 2.27 × 1 1 × 6, 1.58 × 3, 2.07 × 1, 2.27 × 1, 2.73 × 1, 2.99 × 1, 3.27 × 1, 3.58 × 1, 4.30 × 2, 5.65 × 1, 6.19 × 2

PAPR (dB) 9.03 10.49 11.79

BICM-ID Signaling Scheme PAPR (dB) 16-QAM MSP [24] 2.55 16-QAM MSP [24] 2.55 64-QAM SP [101] 3.68

Figure 3.9: Performance of ML-SCM and BICM-ID over fully-interleaved Rayleigh fading channels. J = 2048. For the ML-SCM without clipping, the number of iterations is 12 for R = 2 and 3, and 16 for R = 5. For the clipped ML-SCM, we set ItM = 6, ItS = 6 for R = 2 and 3, and ItM = 6, ItS = 10 for R = 5. The number of iterations is 10 for all the BICM-ID results. punctured to rate 3/4 and 5/6 (using the optimal puncture patterns in [79]) for

R = 3 and 5, respectively. The related parameters, including signaling schemes and PAPRs without clipping, are given in Table 3.3, where SP denotes the setpartitioning mapping [101] and MSP the modied SP mapping [24]. Clipping (with CR = 3 dB, PAPR ≈ 3.55 dB) is applied to ML-SCM, but not used for BICM-ID since the related PAPRs are small. From Fig. 3.9, we can see that at R = 2, BICM-ID exhibits better performance and lower PAPR (2.55 dB). For higher rates, clipped ML-SCM is inferior to BICMID at low SNRs. However, ML-SCM provides a lower error oor at high SNRs, since it can achieve higher diversity gains with low-rate component codes. Here we would like to point out that the performance of BICM-ID provided in Fig. 3.9 are based on the best schemes known to us. It may be possible to improve the BICM-ID performance (especially the error oor) through further optimization of the mapping rule and interleaver pattern.

3.6. Summary

71

With real {αm } and the GA method, the detection complexity of SCM is about 6 real multiplications, 6 real additions, and a tanh(·) operation per coded bit [77]. As a comparison, in BICM-ID schemes employing 2K -ary constellations and the log-MAP demapping [24], the demapping complexity is about 2K comparisons,

3 × 2K real additions and 2K table look-ups per coded bit, which can be very high when K is large (e.g., K = 6 for the 64-QAM signaling). We can show that, taking into account the APP decoding cost and the numbers of iterations needed, the overall complexities of the two schemes are comparable.

3.6 Summary We have investigated a peak-power-limited ML-SCM system based on clipping. By evaluating the mutual information achieved with the clipped input signal, we have shown that signicant shaping gains can be achieved with reasonable clipping thresholds. To combat the clipping eect for practically-coded systems, we have derived an ecient iterative SC method. Simulation results show that a good tradeo between PAPR and BER can be achieved with the proposed method. We have also compared ML-SCM with BICM over various types of channels. It has been shown that, compared to BICM, ML-SCM provides a simpler and more ecient means of achieving diversity gains for high-rate applications.

Chapter 4

SCM-OFDM With Clipping and Soft Compensation 4.1 Introduction OFDM is an ecient multi-carrier transmission technique for wideband, multipath channels. By dividing the channel into a number of orthogonal, parallel sub-channels, OFDM enables high-rate data transmissions free of inter-symbolinterference. Due to its high spectral eciency, robustness to multi-path eect, exible link-adaptation and low receiver complexity, OFDM has been adopted by many wireless and wireline communication standards such as the digital video broadcast (DVB), WiMAX (IEEE 802.16) and asymmetric digital subscriber line (ADSL). However, the OFDM signal exhibits a high PAPR, similar to that in SCM schemes with large constellations as considered in Chapter 3. This makes OFDM more prone to the non-linear eect of transmitter devices than conventional singlecarrier schemes. Following Chapter 3, we can apply deliberate clipping at the transmitter to reduce the PAPR of OFDM signals. The problem of clipping is BER degradation [59, 96] due to non-linear distortion. Interestingly, such performance loss should not be serious in theory. It has been shown in Chapter 3 that for Gaussian signaling over AWGN channels, the mutual information is only marginally aected by clipping. Thus, the problem boils down to how to realize the promise of information-theoretic analysis. The iterative compensation [57, 96] appears a promising direction following the success of turbo-type iterative processing in recent years. Unfortunately, the eectiveness of the conventional iterative compensation methods is far from satisfactory, especially in high-rate OFDM systems with multiary signaling. The work in [25] shows that the iterative compensation performance depends heavily on the signaling schemes (characterized by constellation and mapping rule) employed. In particular, for some popular multi-ary signaling schemes [24, 93, 86], iterative compensation can provide only limited improvement. These ndings are mainly obtained by simulation studies or using semi-analytical analy-

4.2. System Model

73

sis tools such as EXIT charts. So far and to the best of our knowledge, there has been no rigorous analysis on the impact of signaling schemes in clipped OFDM systems. In light of the above, this chapter considers the performance improvement for high-rate, coded OFDM systems with clipping. We begin with a general coded OFDM system based on BICM and study treatments for the clipping-induced distortion. The conventional treatments [22, 57, 96] utilize only partial feedback information such as symbol mean from the decoder. We propose a novel iterative soft compensation method to combat the clipping eect. We show that the performance can be noticeably improved by considering the symbol variance. We then analyze the residual clipping noise power with the proposed method. We show that this power is a monotonously increasing function of the average symbol variance in the iterative process. The latter is aected by the signaling schemes. Our main contribution is to prove that SCM can minimize the symbol variance among all possible signaling schemes when the decoder feedbacks are i.i.d.. This indicates that SCM is advantageous for clipping eect mitigation with iterative compensation. Consequently, SCM-OFDM may potentially outperform other alternatives for clipped transmission, as conrmed by examples in Section 4.6. We further show that the symbol variance can be reduced by using the multilevel SCM (ML-SCM) scheme with unequal power allocation. A mutual-information (MI) evolution technique is derived to characterize the convergence behavior of the iterative compensation. This technique can be employed to quickly predict the overall performance and optimize the system parameters. Numerical results are provided to demonstrate that SCM-OFDM oers a very attractive option for high-rate applications.

4.2 System Model Consider an OFDM system employing BICM-type coded modulation based on binary component codes. The transmitter structure is illustrated in the upper part of Fig. 4.1. The information bits are rst encoded by a binary encoder (ENC). The resultant coded bits are then randomly interleaved and packed into groups c[n] = (c1 [n], c2 [n], · · · , cK [n]) of K bits. The signal mapper maps each group c[n] to a symbol X[n] to be carried by the nth sub-carrier. The related mapping rule will be detailed later. Denote by N the number of OFDM sub-carriers and dene a column vector

4.2. System Model

74

Figure 4.1: Block diagram of coded OFDM systems with clipping, where Π denotes interleaver and Π−1 de-interleaver.

X = [X[0], X[1], · · · , X[N − 1]]> of length N . Then {X[n]} are modulated onto sub-carriers using the inverse discrete Fourier transform (IDFT). The resultant signal is over-sampled into x = [x[0], · · · , x[LN −1]]> , where L is the over-sampling factor [59],

x = F †X

(4.1)

√ 2πnm √ F is an N × LN matrix with (n, m)th entry given by ei LN / LN , i = −1 and “(·)† ” denotes conjugate transpose of matrix. We will call X a frequency-domain signal and x a time-domain signal. We will use capital letters for the signals related to X in the frequency domain and lower case letters for those related to

x in the time domain. In order to reduce the PAPR, each entry x[m] of x is clipped before transmission. Several non-linear functions g(·) can be used for this purpose. A typical choice is the amplitude clipping as considered in Chapter 3. (Another example is the Catersian clipping as discussed in the Appendix.)

½ g(x[m]) =

x[m], if Ax[m]/|x[m]|, if

|x[m]| ≤ A |x[m]| > A

(4.2)

where A is a clipping threshold. The clipping ratio is CR = 10 log10 (A2 /E[|x[m]|2 ]). With abuse of notation, we write the clipped signal vector as g(x). The PAPR of the transmit signal can be investigated using the method in [74]. For reference, the complementary cumulative distribution function (CCDF) of the PAPR is shown in Fig. 4.2 for dierent CR. It is seen that the PAPR is less than 5.9 dB for 99.9% of the OFDM blocks when amplitude clipping with CR = 0 dB

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75

Figure 4.2: CCDF of the PAPR in clipped OFDM systems. The oversampling factor L = 4, the number of sub-carriers N = 256 and the clipping ratio CR = 0, 1, · · · , 6 dB (from left to right). is applied, in contrast to 11.3 dB in the unclipped case. As discussed in [2, 74], clipping introduces both in-band non-linear distortion and out-of-band radiation. The latter can be treated by band-pass ltering the clipper outputs before transmission, which may incur PAPR re-growth. (In order to further reduce PAPR, an iterative clipping and ltering process can be applied [2].) We focus on the in-band distortion that degrades the BER performance. We assume that ideal bandpass ltering is applied to g(x) to remove the out-of-band radiation. Cyclic prex is then added to the ltered signal to prevent inter-blockinterference (IBI) and the resultant signal is transmitted. For simplicity, we also assume that

• clipping is the only source of non-linear distortion, which implies that ideal power ampliers are used; and

• the cyclic prex is suciently long so that the IBI can be perfectly removed.

4.3 Iterative Soft Compensation Consider an equivalent discrete-time multi-path channel with channel coecient vector h and AWGN vector w. We assume that h is known perfectly to the receiver and denote by y the time-domain received signal vector. The receiver is shown in the lower part of Fig. 4.1. It consists of an ESE and a DEC, connected

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76

by an interleaver (Π) and a de-interleaver (Π−1 ). Let us rst ignore the soft compensation (SC) module in Fig. 4.1. In the receiver, the cyclic prex is removed and then a discrete Fourier transform (DFT) is applied to y to produce an inband frequency-domain signal vector Y = [Y [0], Y [1], · · · , Y [N − 1]]> . Denote

H = diag(H[0], H[1], · · · , H[N − 1]) where {H[n]} form the DFT of h. The frequency-domain signal model is then written as

Y = HF g(x) + W = HF g(F † X) + W

(4.3)

where g(x) is the clipped version of x and W = F w is a vector of complex AWGN.

4.3.1 Overall Iterative Decoding Principle The ESE and DEC operate iteratively following the turbo principles. The function of the ESE is to generate the extrinsic LLR for each coded bit as µ ¶ Pr(ck [n] = 0|Y , {γk [n]}) λk [n] = ln − γk [n], ∀k, ∀n Pr(ck [n] = 1|Y , {γk [n]})

(4.4)

where {γk [n]} denote the a priori information for all coded bits from the DEC. The DEC takes the de-interleaved version of {λk [n]} in (4.4) as inputs and performs standard APP decoding. The extrinsic LLRs produced by the DEC (after interleaving) are denoted by µ ¶ Pr(ck [n] = 0|{λk [n]}, C) γk [n] = ln − λk [n], Pr(ck [n] = 1|{λk [n]}, C)

∀k, ∀n.

(4.5)

In contrast to (4.4), (4.5) is evaluated based on coding constraints C . Since the APP decoding is standard, we focus only on the ESE function below. The optimal ESE strategy follows the maximum a posteriori (MAP) principle [56] to compute bit-level LLR λk [n] from (4.3). In general, it has no closed-form solution and involves excessive complexity. So, we resort to a strategy that rst mitigate the clipping eect and then generate λk [n]. With iterative decoding/detection, the clipping eect can be mitigated by estimating x using the Bayesian estimation principles [56]. This is optimal in the sense of minimum mean square error (MMSE) but is still of high complexity. Interestingly, we observe that the performance of the Bayesian estimator may be approached by the relatively low-cost soft compensation method presented below.

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This is illustrated in the Appendix in Section 4.8 using an example with Cartesian clipping (that clips the in-phase and quadrature components separately [71]) where the closed-form expressions of both the Bayesian and soft compensation methods can be derived. Subsequently, we will focus only on amplitude clipping and the soft compensation method.

4.3.2 Soft Compensation Let us now consider the SC module in Fig. 4.1. Similarly to the treatments in Section 3.2.3, we can apply Price's theorem [78] to model the clipped signal vector

g(x) as the sum of the scaled version of x and an uncorrelated clipping noise d: g(x) = κx + d

(4.6)

where κ is a constant scalar computed as

κ=

E[x† g(x)] E[||x||2 ]

(4.7)

and || · || denotes the Frobenius norm of a vector. Then (4.3) can be rewritten as

Y = κHX + HF d + W .

(4.8)

Suppose that a soft estimate d˜ of d is obtained. We subtract HF d˜ from Y to compensate for the clipping eect1 , yielding

Z = κHX + Ξ

(4.9)

˜ + W. Ξ = HF (d − d)

(4.10)

where Note that H in (4.9) is a diagonal matrix. With interleaving, we can assume that

{X[n]} are uncorrelated. For simplicity, we can also model the residual clipping noise d − d˜ as an AWGN uncorrelated with the useful sinal X . Then the symbolwise APP algorithm [15, 24, 39] can be applied to the demapper (see Fig. 4.1) to 1 Note that the soft compensation method in this section is similar to that in Chapter 3. However, they are based on dierent modeling of the clipping noise. We observe that, for OFDM systems, the method presented here based on Price's modelling can lead to better performance.

4.3. Iterative Soft Compensation generate the extrinsic LLR for each coded bit as µ ¶ Pr(ck [n] = 0|Z[n], {γk [n]}) λk [n] = ln − γk [n], Pr(ck [n] = 1|Z[n], {γk [n]})

78

∀k, ∀n,

(4.11)

where Z[n] denotes the nth entry of Z , and γk [n] is the a priori LLR about

ck [n] which can be obtained as the extrinsic LLR from the DEC (see (4.5)). The operation in (4.11) has complexity O(2K ). At the start, without any information about d, we simply set d˜ = 0 in (4.9). This is a very rough starting point and d˜ will be rened iteratively as discussed below. Note that (4.11) and (4.4) have similar expressions but (4.11) greatly simplies the detection operations since symbol-wise demapping is used.

4.3.3 Proposed Clipping Noise Estimation Method Using the DEC feedback {γk [n]}, d˜ in (4.10) can be updated in the ESE. The following two methods are widely used by other authors for this purpose:

• Method I : the method in [74] that treats the clipping noise d in (4.8) as an AWGN independent of the wanted signal. With this treatment, we always set d˜ = 0. This method is also discussed in Section 3.4.4.

• Method II : the clipping noise cancellation method in [22, 71] that estimates d as d˜ = g(E[x]) − κE[x], where E[x] denotes the mean of x computed from {γk [n]} (see also the discussions at the end of this section). We observe that these methods work well with high clipping ratio (CR) and low data rate R but their performance degrades rapidly as CR decreases or R increases. This is because they utilize only part of the a priori information generated during iterative detection. We now present an alternative soft estimation method to improve performance. The key to the proposed method is to treat both x and d as Gaussian random variables and then estimate their distributions in the iterative decoding process. Let x be an entry in x. From (4.1), x is a weighted sum of N random variables. Following the central limit theorem, it is reasonable to model x as a complex Gaussian random variable. For simplicity, we assume that the real and imaginary parts of x have the same variance V[x]/2, where V[x] = E[|x − E[x]|2 ] is referred to as the symbol variance. The mean E[x] and the variance V[x] can be estimated from {γk [n]}, as detailed later. With the Gaussian approximation, the distribution of the clipping noise d = g(x) − κx can be estimated from E[x] and V[x].

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According to [56], given a priori information, the optimal estimate of d (in terms of minimization of the mean square error (MSE)) is the conditional mean of d. Therefore, we propose to set

Z d˜ = E[g(x) − κx] = |x|≥0

µ ¶ |x − E[x]|2 (g(x) − κx) exp − dx. πV[x] V[x]

(4.12)

The corresponding residual clipping noise power can be estimated by

Z ˜ 2] = E[|d − d| |x|≥0

¶ µ ˜2 |x − E[x]|2 |g(x) − κx − d| dx, exp − πV[x] V[x]

(4.13)

which can be used to generate the variance of the entries of Ξ in (4.10). Note that d˜ in (4.12) can be tabulated as a function of (E[x], V[x]) for on-line evaluation, similarly as in Chapter 3. In general, this requires a two-dimensional table and increases the storage cost. Based on the above discussions, we list the clipping noise estimation procedure as follows. (i ) Estimate the mean E[X[n]] and variance V[X[n]] of X[n] (the nth entry of

X ) from {γk [n]} as detailed in the next section. (ii ) Generate the means and variances of the entries of x based on the relationship x = F † X . More specically, E[x] = F † E[X] where E[x] and E[X] are respectively the means of x and X . The variance of x[m], ∀m, is computed P −1 as V[x[m]] = 1/(LN ) N n=0 V[X[n]].

˜ (iii ) Generate the clipping noise estimate d[m] using (4.12) Steps (i)-(iii) are performed respectively by the soft mapper, IDFT, and SC modules in the ESE. The ESE/DEC operations outlined above can be repeated for a number of iterations. Interestingly, Method II mentioned earlier can be seen as an approximation of the proposed method with the assumption V[x] = 0. Clearly, it utilizes the symbol mean E[x] only while ignores the variance V[x]. We will show in Section 4.6 that the proposed method can signicantly outperform it.

4.4 Optimal Signaling for Soft Compensation 4.4.1 Residual Clipping Noise We now investigate the impact of signaling schemes on the performance of the iterative soft compensation method. From Section 4.3.3, the soft estimate d˜ is

4.4. Optimal Signaling for Soft Compensation

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aected by the DEC feedback {γk [n]}. Following the EXIT charts methods for analyzing iterative decoding systems [18], we model {γk [n]} as i.i.d. random variables. With interleaving, we also assume that {X[n]} are i.i.d. and ergodic. Let

Eγ [·] denote the expectation w.r.t. the distribution of {γk [n]}. If the average residual clipping noise power2 ¯ 2 ]/N = Eγ [||d − d|| ¯ 2 ]/(LN ) σd2 ≡ Eγ [||F (d − d)||

(4.14)

is reduced for each iteration, then the clipping eect can be alleviated. We can use σd2 as a measure for the residual clipping eect in (4.9). Clearly, we want to minimize σd2 . Similarly, dene the residual noise power in estimating X by E[X] as

σx2 ≡ Eγ [||X − E[X]||2 ]/N = Eγ [||x − E[x]||2 ]/N.

(4.15)

The second equation in (4.15) holds since x = F H X , E[x] = F H E[X] and F F H =

I . From the estimation theory [56], σx2 equals to the expectation of the symbol variance V[X[n]]:

σx2 = Eγ [V[X[n]]] ,

∀n.

(4.16)

Clearly, σx2 and σd2 can be used to measure the accuracy of the estimation of x and

d, respectively. From Section 4.3.3, the estimation of d is obtained based on that for x. Therefore, we can write σd2 as a function of σx2 :

σd2 = φ(σx2 ).

(4.17)

Fig. 4.3 shows the σd2 versus σx2 curves (obtained by the Monte-Carlo method) for dierent clipping ratios. We can see that σd2 is a monotonically increasing function of σx2 . This is well within expectation: a more accurate estimate of x would lead to a more accurate estimate of d. We also observed that the relationship between σx2 and σd2 is nearly independent of the signaling schemes for generating X . (This is because when N is suciently large, from the central limit theorem, the IDFT output x is always approximate Gaussian-distributed for arbitrary signaling schemes.) As a consequence, minimizing σd2 is equivalent to minimizing σx2 . On the other hand, the signaling scheme used to generate X can aect σx2 , as discussed below. Our goal is to nd the 2 Note

that σd2 here is dened as the power of a two-dimensional complex noise, which is dierent from that in Chapter 3. The case is similar for σ 2 in (4.65).

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Figure 4.3: σd2 versus σx2 . L = 4. N = 256. The average signal power E[|X[n]|2 ] is normalized to 1. signaling scheme that can minimize σx2 (and so σd2 ).

4.4.2 Signaling Schemes We now discuss the statistical measures related to a signaling scheme. For notational simplicity, we ignore the symbol index n temporarily. Let c = (c1 , c2 , · · · , cK ) be a binary K -tuple with ck ∈ {0, 1} and B the set of 2K such K -tuples. Let s be the image of c (in the complex plane) and X be a constellation of 2K such points3 . We will call c the label of s. Denote by µ a mapping from B to X . Some examples of (X , µ) for BICM-ID can be found in [24, 86, 93] and are illustrated in Fig. 4.4. With SCM, the mapping rule is given by

s=

K X

βk (−1)ck ,

(4.18)

k=1

where the weighting factors {βk } are (complex) constants. We can apply the SCM signaling to the transmitter in Fig. 4.1, which will result in a special case of BICM-ID [98], i.e., the SL-SCM scheme discussed in Chapter 2. We will derive the advantages of such a BICM-ID scheme for clipping eect compensation. Following the turbo principle, {ck } can be treated as binary random variables 3 In

this thesis, we allow overlapping of the signaling points in X . (This is a special case of the multiple labeling scheme in [72].) For example, for the SCM mapping dened in (4.18), let K = 2 and β1 = β2 = 1. Then c = (0, 1) → s = 0 and c0 = (1, 0) → s0 = 0. In this case, we still regard s and s0 as two dierent points in X , even if they are identical. In other words, each point in X is distinguished by its label c rather than by its position in the complex plane. This treatment is useful when we introduce symmetric conditions.

4.4. Optimal Signaling for Soft Compensation

82

Figure 4.4: Examples of 16-ary signaling schemes. and so is the mapped symbol s. Suppose a set of a priori LLRs {γk } about {ck } are available.

µ γk = ln

Pr(ck = 0) Pr(ck = 1)

¶ ,

k = 1, 2, · · · , K,

(4.19)

eγk , k = 1, 2, · · · , K. (4.20) 1 + eγk (In the soft mapper, {γk } can be taken as the feedback LLR values from the DEC. Pr(ck = 0) = 1 − Pr(ck = 1) =

At the rst iteration, we simply set γk = 0, ∀k , as there is no DEC feedback.) The mean and variance of s can be found as follows: Let s = [s0 , s1 , · · · , s2K −1 ]> be the vector of the signaling points in X and p = [p0 , p1 , · · · , p2K −1 ]> the vector of the probabilities associated with the entries in s. We express the mapping in a functional form as s = µ(c) and order the elements in s and p using the natural binary expression of their  s0  s1  s ≡  ..  .

labels c = (c1 , c2 , · · · , cK ) as below.    µ(c1 = 0, c2 = 0, · · · , cK = 0)   µ(c1 = 0, c2 = 0, · · · , cK = 1)     , = ..    .

µ(c1 = 1, c2 = 1, · · · , cK = 1)

s2K −1 





Pr(c1 = 0) Pr(c2   Pr(c1 = 0) Pr(c2     p≡ =    Pr(c1 = 1) Pr(c2 p2K −1 p0 p1 .. .

(4.21)

 = 0) · · · Pr(cK = 0) = 0) · · · Pr(cK = 1)   , ..  . = 1) · · · Pr(cK = 1)

(4.22)

where we have assumed that {c1 , c2 , · · · , cK } are independent. Then, the mean

4.4. Optimal Signaling for Soft Compensation

83

and variance of s can be expressed in vector forms as,

E[s] =

K −1 2X

pm sm = p† s,

(4.23a)

m=0 2

V[s] = E[|s − E[s]| ] =

K −1 2X

†

2

pm |sm − p s| =

m=0

K −1 2X

pm |sm |2 − s† pp† s.(4.23b)

m=0

4.4.3 Local and Global Statistics Note that E[s] and V[s] in (4.23) are computed for xed {γk }. Similar to the treatment in Section 4.4.1, assume that {γk } are i.i.d. random variables and denote by Eγ [·] the mathematical expectation w.r.t. the joint distribution of {γk }. In particular, we are interested in Eγ [V[s]],   K −1 K −1 2X 2X £ ¤ 2 † †   Eγ[pm ] |sm |2 − s† Eγ pp† s.(4.24) Eγ [V[s]]=Eγ pm |sm | − s pp s = m=0

m=0

We will call E[s] and V[s] as local mean and variance, respectively, and call Eγ [V[s]] as global variance.

4.4.4 Minimum Global Variance We now consider the minimization of the global variance by properly choosing signaling schemes.

Assumption 1 : The signaling (X , µ) is unbiased and with unit average power: −K

2

K −1 2X

sm = 0,

(4.25a)

|sm |2 = 1.

(4.25b)

m=0 −K

2

K −1 2X

m=0

Assumption 2 : The elements of {γk } are i.i.d. and satisfy the symmetric condition:

pγ (γ) = pγ (−γ),

(4.26)

∀γ ∈ {γk }.

Note that Assumption 2 implies that

Eγ [Pr(ck = 0)] = Eγ [Pr(ck = 1)] = 1/2,

∀k,

(4.27)

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84

there is a constant η such that

η = Eγ [Pr2 (ck = 0)] = Eγ [Pr2 (ck = 1)],

∀k,

(4.28)

and the elements in X have equal occurrence probabilities4 :

Eγ [pm ] = 2−K ,

m = 0, 1, · · · , 2K − 1.

(4.29)

Based on the above two assumptions, the global variance in (4.24)

£ ¤ Eγ [V[s]] = 1 − s† Eγ pp† s.

(4.30)

Theorem 1: Under Assumptions 1 and 2, the minimum global variance min Eγ [V[s]] = 2 − 4η s

(4.31)

where the minimization is over all possible selections of s (i.e., all possible signaling schemes (X , µ)).

Proof : See the next subsection. Theorem 2: Under Assumptions 1 and 2, for arbitrary K and arbitrary {βk }, the SCM signaling dened by (4.18) achieves the minimum global variance.

Proof: Given the a priori probability of ck , the local mean and variance of the variable (−1)ck can be written as

E[(−1)ck ] = (−1)0 Pr(ck = 0) + (−1)1 Pr(ck = 1) = 2 Pr(ck = 0) − 1,(4.32a) V[(−1)ck ] = 1 − E2 [(−1)ck ].

(4.32b)

From (4.18) and Assumptions 1 and 2, we have

Eγ [V[s]] =

K X

|βk |2 Eγ [1 − (2 Pr(ck = 0) − 1)2 ]

k=1 K X

=4

¡ ¢ |βk |2 Eγ [Pr(ck = 0)] − Eγ [Pr2 (ck = 0)]

(4.33a) (4.33b)

k=1

= 2 − 4η 4 According

(4.33c)

to footnote 3, two points sm and sm0 , m 6= m0 , may be identical but their occurrence probabilities are counted separately.

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85

where (4.33c) follows the average power constraint K

2 −1 K X 1 X 2 |sm | = |βk |2 = 1. 2K m=0 k=1

Some intuitive explanations of the above theorems are included in Section 4.4.6. From Theorem 2, the values of {βk } have no impact on the global variance achieved by the SCM signaling. However, they may still aect system performance from the point of view of capacity and error rates.

4.4.5 Proof of Theorem 1 In this subsection, we present the proof of Theorem 1. Dene P = Eγ [pp† ]. Then (4.30) can be rewritten as

Eγ [V[s]] = 1 − s† P s.

(4.34)

Clearly, Eγ [V[s]] is determined by two factors: s and P . We rst derive a simple expression for P . Denote by ⊗ the Kronecker product. Based on Assumption 2, we can decompose p as

·

¸ · ¸ · ¸ Pr(c1 = 0) Pr(c2 = 0) Pr(cK = 0) p= ⊗ ⊗ ··· ⊗ . Pr(c1 = 1) Pr(c2 = 1) Pr(cK = 1)

(4.35)

When K = 1,

··

¸ ¸ ¤ Pr(c1 = 0) £ Pr(c1 = 0) Pr(c1 = 1) Pr(c1 = 1)

P = Eγ · ¸ Eγ [Pr2 [c1 = 0]] Eγ [Pr[c1 = 0]Pr[c1 = 1]] = . Eγ [Pr[c1 = 0]Pr[c1 = 1]] Eγ [Pr2 [c1 = 1]]

(4.36)

From Assumption 2

Eγ [Pr(ck = 0)Pr(ck = 1)] = Eγ [Pr(ck = 0)(1 − Pr(ck = 0))] = 1/2 − η, ∀k.

(4.37)

Therefore, for K = 1,

·

¸ η 1/2 − η P = . 1/2 − η η For K = 2, from the mixed product property of matrixes [84] [i.e., (A ⊗ B)(C ⊗

4.4. Optimal Signaling for Soft Compensation

86

D) = AC ⊗ BD ], we have P = Eγ[pp† ] (4.38) ·µ· ¸ · ¸ ¸¶ ¤ £ ¤¢ Pr(c1 = 0) Pr(c2 = 0) ¡£ =Eγ ⊗ Pr(c1 = 0) Pr(c1 = 1) ⊗ Pr(c2 = 0) Pr(c2 = 1) Pr(c1 = 1) Pr(c2 = 1) ·· ¸ ¸ ·· ¸ ¸ ¤ ¤ Pr(c1 = 0) £ Pr(c2 = 0) £ =Eγ Pr(c1 = 0) Pr(c1 = 1) ⊗Eγ Pr(c2 = 0) Pr(c2 = 1) Pr(c1 = 1) Pr(c2 = 1) · ¸ · ¸ η 1/2 − η η 1/2 − η = ⊗ 1/2 − η η 1/2 − η η where we used Assumption 2. The result can be easily generalized for K > 2,

·

η 1/2 − η P = Eγ [pp ] = 1/2 − η η

¸⊗K

†

where we used the notation

⊗K

(4.39)

for the power of Kronecker product: A⊗1 =

A, A⊗2 = A ⊗ A and A⊗3 = A ⊗ A ⊗ A, etc. Incidentally, 1/4 ≤ η ≤ 1/2,

(4.40)

where 1/4 ≤ η follows Eγ [Pr(ck = 0)] = 1/2 and Eγ [(Pr(ck = 0) − Eγ [Pr(ck = 0)])2 ] ≥

0, and η ≤ 1/2 follows Pr(ck = 0) ≥ 0, Pr(ck = 1) ≥ 0 and (4.37).

Proof : From (4.34), i.e., Eγ [V[s]] = 1 − s† P s, minimizing Eγ [V[s]] is equivalent to maximizing s† P s. We may consider using s = 2K/2 umax for this purpose, where umax is the eigenvector for the maximum eigenvalue λmax of P . [umax is scaled by 2K/2 to satisfy the power constraint in (4.25b).] However, as shown below, umax does not meet the unbiased condition in (4.25a); hence, we have to resort to the eigenvector of the second largest eigenvalue. For K = 1, P in (4.39) can be decomposed as √ ¸· √ ¸ · ¸ · √ ¸· √ η 1/2 − η 1/√2 1/ √2 1/2 1/√2 1/ √2 P= = (4.41) . 1/2 − η η (4η − 1)/2 1/ 2 −1/ 2 1/ 2 −1/ 2 For K > 1, from (4.41) and the mixed product property of matrixes, the eigendecomposition of P is given by

P = Eγ [pp† ] = U ΛU >

(4.42)

√ ¸⊗K √ 1/√2 1/ √2 U= 1/ 2 −1/ 2

(4.43)

where

·

4.4. Optimal Signaling for Soft Compensation

87

and

· Λ=

¸⊗K

1/2 (4η − 1)/2

(4.44)

.

Note that U is an orthogonal matrix with U U > = I , where I is the identity matrix. The eigenvalues of P are given by the diagonal entries of Λ: 



  Λ = 2−K diag 1, 4η − 1, · · · , 4η − 1, (4η − 1)2 , · · · , (4η − 1)2 , · · · , (4η − 1)K  . | {z } | {z } K

times

K(K−1)/2

times

(4.45) Since 1 > 4η − 1 > 0 from (4.40), the maximum eigenvalue λmax = 1/2

K

corresponding eigenvector umax = 2−K/2 [1 1

and the

· · · 1]> .

Now, return to the original problem of minimizing Eγ [V[s]]. At rst glance, from the matrix theory, we may select s = 2K/2 umax for this purpose, but this violates the unbiased constraint in (4.25a). It can be veried that 1> s 6= 0 when

s = 2K/2 umax and 1> s = 0 when s is selected from (scaled versions of) other columns of U , where 1 is the all-one vector. (Note: The entries of every column of U in (4.43), except umax , consists of equal numbers of −2−K/2 and 2−K/2 .) Therefore, condition (4.25) indicates that s cannot be 2K/2 umax and must be orthogonal to umax . We therefore select s using the eigenvectors corresponding to the second largest eigenvalue 2−K (4η−1). After proper scaling of s to satisfy (4.25b) and substituting it into Eγ [V[s]] = 1 − s† P s, we obtain the minimum global variance

min Eγ [V[s]] = 1 − 2K · 2−K (4η − 1) = 2 − 4η. s

4.4.6 Discussions 4.4.6.1 Examples Here are some intuitive explanations on the above proof of Theorem 1. Let us return to (4.42): P = U ΛU > . Suppose that the 2K diagonal entries in Λ are decreasingly ordered. Denote by V the submatrix of U formed by columns 2 to

K + 1 of U . We take K = 2 as an example. From (4.43), we have Λ = diag(λ0 , λ1 , λ2 , λ3 ) µ ¶ 1 4η − 1 4η − 1 (4η − 1)2 = diag , , , , 4 4 4 4

(4.46) (4.47)

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88

Figure 4.5: Examples of signaling schemes.



 +1 +1 +1 +1 1  +1 −1 +1 −1  , U = [u0 , u1 , u2 , u3 ] =  2  +1 +1 −1 −1  +1 −1 −1 +1   +1 +1 1  −1 +1  . V = [u1 , u2 ] =  2  +1 −1  −1 −1

(4.48)

(4.49)

In this case, umax = u0 is the eigenvector that corresponds to the largest eigenvalue (λ0 = 1/4) of P . Recall that s contains the collection of points in X . Some special choices of s are discussed below. (i) s = 2K/2 u0 = 2K/2 umax = [1 1

1 1]> . This means that all the signal

points in X overlap each other (i.e., s0 = s1 = s2 = s3 = 1). In this case, the four input bit combinations {(c1 , c2 ) = (00), (01), (10), (11)} are all mapped to the signal point s = 1, as shown in Fig. 4.5(a). The global variance is indeed minimized, as Eγ [V[s]] = 1 − s† P s = 0. Notice that there is only one distinct point, which is obvious. The best estimate is simply this point. However, this choice violates the unbiased constraint in (4.25a). Furthermore, it is obviously a bad choice when viewed from the information theory as it cannot deliver any information. If we consider a memoryless

4.4. Optimal Signaling for Soft Compensation

89

AWGN channel with inputs drawn from s, then the mutual information between the channel input and output is zero when s = 2K/2 u0 . This agrees with the well-known observation that a scheme which achieves lower MMSE may lead to lower mutual information and vice versa [44]. (ii) s = 2K/2 u1 = [1

−1 1

− 1]> . In this case, Eγ [V[s]] is increased (com-

pared with (i)) to 2 − 4η . Now, the four points in X have two distinct values, i.e., s0 = s2 = 1 and s1 = s3 = −1 (see Fig. 4.5(b)), and mutual information can achieve at most 1 bit/symbol. In general, this is not a good choice for high-rate transmissions, since it is in essence the BPSK signaling. (iii) s = β1 u1 + β2 u2 where β1 and β2 are non-zero constants. An example of √ √ β1 = 2, β2 = i 2 is shown in Fig. 4.5(c)), which results in QPSK signaling. In this case, Eγ [V[s]] is still 2 − 4η since both u1 and u2 correspond to the second largest eigenvalue (λ1 = λ2 = (4η − 1)/4) of P . This choice of s can achieve higher mutual information than the BPSK signaling in case (ii). (iv) s = β1 u1 + β2 u2 + β3 u3 where β3 is non-zero. Based on the discussions above, Eγ [V[s]] is larger than 2 − 4η but the achievable mutual information may potentially be increased compared with (i)-(iii). Clearly, (i) is an extreme case that cannot deliver any information at all. For (ii) and (iii), s can be written in the form of

s = V β> where β = [β1

(4.50)

β2 ]> . From (4.49), each row of V can be seen as the binary

expression (over {+1, −1}) of a signaling point label, and the 2K rows of V list the labels of all the 2K points in s. Therefore, (ii) and (iii) are instances of the SCM signaling dened in (4.18). Both of them can achieve the minimum global variance under the unbiased constraint in (4.25a), but (iii) can achieve higher mutual information. Choice (iv) results in a larger Eγ [V[s]] so it is a worse choice under the criterion of minimum global variance. However, we should be cautious when considering the overall system performance as (iv) may also potentially lead to higher mutual information. Finding the optimal solution is a very complicated issue, which is beyond the scope here. However, we can justify the asymptotic optimality of the SCM signaling using the following arguments: (a) For the SCM signaling, s can be selected as a linear combination of the columns of V . (For general K , V consists of K linearly independent column

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vectors.) The SCM signaling always achieves the minimum global variance condition under the unbiased constraint in (4.25a). (b) When K is large, from the central limit theorem, the transmit signal is close to the Gaussian signal that maximizes the mutual information for Gaussian channels [28].

4.4.6.2 Consequences of the Theorems Consider the time index n. Let {γk [n], 1 ≤ k ≤ K, 0 ≤ n ≤ N − 1} be K × N independent realizations of {γk } and {X[n], 0 ≤ n ≤ N − 1} be N independent realizations of s. The local statistics, E[X[n]] and V[X[n]], are computed using (4.23) from {γk [n]} for a xed n. From (4.15) and (4.24), we have

σx2 = Eγ [V[s]].

(4.51)

Thus, Theorems 1 and 2 indicate that the SCM signaling is an optimal solution to minimize σx2 (and so σd2 ) among all possible signaling schemes. It is important to note that the optimality here is with respect to the clipping eect compensation only. The conclusion can be dierent for a dierent objective, e.g., the BER performance optimization. To elaborate on this point, we return to Fig. 4.1. The SCM signaling is optimal for the link from point A to point B in the iterative receiver. However, other parts of the receiver are also aected by the choice of signaling schemes for which the SCM signaling may not be optimal. For example, the SCM signaling may not be the optimal choice when the demapper is considered. Nevertheless, in cases where clipping is deep and/or transmission rate is high, the clipping eect may dominate the BER performance and the SCM does yield lower BER than other alternatives. We will verify this by numerical results in Section 4.6.

4.5 Analysis and Design of SCM-Based OFDM We have shown that the SCM signaling minimizes the average symbol variance when the DEC feedback {γk [n]} are i.i.d., which is advantageous for clipping eect mitigation. The i.i.d. assumption of {γk [n]} implies that a BICM-type scheme is considered. In this section, we further consider the case with multi-level encoder and non-i.i.d. {γk [n]}. We rst show that the symbol variance can be reduced by using multi-level SCM (ML-SCM) with unequal {βk }. We then introduce a fast evolution technique to track the convergence property of the symbol variance.

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Based on this technique, we can optimize the values of {βk } to improve the overall performance.

4.5.1 Multi-level SCM (ML-SCM) Revisited We now focus on the ML-SCM scheme which has been discussed in Chapter 2 and 3. For a xed k , we call the set {(−1)ck [n] } as layer k . All the bits in a layer are encoded by a single code and separate encoders are used for dierent layers. The transmit signal is generated using (4.18). From Chapter 2, such ML-SCM schemes have several advantages as follows:

• It can achieve the constellation-constrained capacity [104]. By contrast, BICM suers from a capacity loss [19]. This implies that, when capacityapproaching component codes are used, ML-SCM may outperform BICM with the same signaling scheme (i.e., the SL-SCM; c.f. Chapter 2).

• It can be decoded in a layer-by-layer manner. The information produced by the decoded layers can facilitate decoding of remaining layers. This is computationally more ecient compared with BICM where all coded bits are decoded simultaneously. In the next subsection, we show that ML-SCM can also lead to a lower symbol variance.

4.5.2 Global Variance in ML-SCM For ML-SCM, the following assumptions are always approximately true: (i) the DEC feedback {γk [n]} for the bits in each layer are i.i.d., and (ii) {γk [n]} have dierent distributions for dierent layers. We denote the global variance for the signals in the k th layer as

ρk = Eγ [V[(−1)ck [n] ]] = Eγ [1 − tanh2 (γk [n]/2)].

(4.52)

We also dene the average of {ρk } as K 1 X ρ≡ ρk . K k=1

(4.53)

It is easy to verify that ρk = 2 − 4ηk , ρ = 2 − 4η where ηk ≡ Eγ [Pr2 (ck = 0)] and P η ≡ K1 K k=1 ηk .

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From (4.18) and (4.52), the global symbol variance for an ML-SCM scheme is given by

Eγ [V[s]] =

K X

|βk |2 ρk .

(4.54)

k=1

When {|βk |} are dierent, without loss of generality, we order {|βk |} as (4.55)

1 ≥ |β1 | ≥ |β2 | ≥ · · · ≥ |βK | ≥ 0.

The decoder for a layer with a larger |βk | should naturally lead to a smaller variance

ρk as the signals from this layer are better protected. Therefore, {ρk } will follow the order below.

0 ≤ ρ1 ≤ ρ2 ≤ · · · ≤ ρK ≤ 1. (4.56) PK 2 Recall the average power constraint in (4.25): k=1 |βk | = 1. Using the Chebyshev's inequality [112] and from (4.53), (4.55) and (4.56), we can show that, for ML-SCM, (4.57)

Eγ [V[s]] ≤ ρ = 2 − 4η.

It is interesting to compare (4.57) with Eγ [V[s]] = 2−4η in Theorem 2. Theorem 2 is based on the assumption that {γk [n]} are all i.i.d., and so {ρk } are all the same, i.e., ρk = ρ, ∀k . Here, in (4.57), we have relaxed this restriction and allow {ρk } to be dierent. From (4.57), we can see that when the overall quality (characterized by ρ) of the DEC feedback is the same, ML-SCM may lead to a lower global variance than BICM with the same signaling scheme. In addition, this global variance can be minimized through adjusting {βk }. This is the main motivation for the discussion in the subsequent sub-sections.

4.5.3 Performance Prediction The discussions above assume a given distribution of the DEC feedback. With iterative decoding, this distribution, characterized by {ρk }, changes as the iteration (it)

proceeds. Let It be the number of iterations and ρk

the value of ρk after it

iterations. Then the global variance after It iterations is given by

E(It) γ [V[s]]

=

K X

(It)

|βk2 |ρk .

(4.58)

k=1 (It)

Clearly, it is of interest to minimize Eγ [V[s]] since it determines the residual clipping noise after the nal iteration. (It)

In SCM schemes, the values of {ρk } are aected by {βk }. In general, their

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relationship cannot be expressed in a closed form due to the non-linear nature of (It)

iterative decoding. However, for given {βk }, {ρk } can be obtained recursively by using a fast evolution process, as discussed below. Then one can optimize {βk } to minimize (4.58) based on the evolution analysis. (it)

We now discuss how to eciently track {ρk }. In SL-SCM, the standard EXIT chart technique [18] can be adapted for this purpose and so we omit the details. In what follows, we focus on ML-SCM schemes where the standard EXIT chart technique may meet diculties as multiple decoders are involved. We will (it)

outline an evolution technique to track {ρk } during the iterative process. This technique follows the MI evolution method in Chapter 2 where unclipped singlecarrier schemes over at-fading channels are considered. Here, we extend it to clipped SCM-OFDM over frequency-selective fading channels with iterative soft compensation.

4.5.3.1 Demapping Based on GA Before we go further, we rst review the GA demapping method for SCM. This technique can reduce the receiver cost compared with the APP approach. It is also easier to analyze. From (4.18), an SCM signal can be expressed as

X[n] =

K X

βk Xk [n]

k=1

where each Xk [n] = (−1)ck [n] ∈ {+1, −1} is a BPSK signal. We refer to {Xk [n]} as the k th layer. We focus on a particular Xk [n] and rewrite the nth entry of Z in (4.9) as (4.59)

Z[n] = κH[n]βk Xk [n] + ζk [n] where

ζk [n] = κH[n]

X

(4.60)

βm Xm [n] + H[n]D[n] + W [n]

m6=k

˜ . For − d) simplicity, we assume that κ, {βk } and H[n] in (4.59) are real numbers. Now, is the distortion component and D[n] denotes the nth entry of

1 F (d L

we treat (4.59) as a BPSK-input system and model the distortion ζk [n] as an independent Gaussian variable. The statistics of {ζk [n]} in (4.59) can be found using the DEC feedback {γm [n]}. Then the demapper output is calculated as

λk [n] =

2κβk H[n] (Re(Z[n]) − E[Re(ζk [n])]). V[Re(ζk [n])]

(4.61)

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The GA treatment can signicantly reduce the demapping complexity, as discussed in Chapter 2.

4.5.3.2 MI Evolution Now we are ready to introduce the MI evolution analysis. Substituting (4.59) into (4.61), we can write the ESE output (i.e., the DEC input) as

λk [n] =

2κβk H[n] (κH[n]βk Xk [n] + Re(ζk [n]) − E[Re(ζk [n])]). V[Re(ζk [n])]

(4.62)

In (4.62), Re(ζk [n]) − E[Re(ζk [n])] represents the residual interference-noise component. Its average power can be measured using the global variance as follows

Eγ [V[Re(ζk [n])]] = Eγ [|Re(ζk [n]) − E[Re(ζk [n])]|2 ] = |κβk |2 (|H[n]|2 PI,k + PW,k )

(4.63a) (4.63b)

where PI,k and PW,k are the relative power (normalized by |κβk |2 ) of the interference and noise components, respectively, and can be computed from (4.60) as

PI,k =

X |βm |2 σd2 E [V[X [n]]] + , γ m |βk |2 2|κβk |2 m6=k

(4.64)

σ2 2|κβk |2

(4.65)

PW,k =

where we have approximated the average power of D[n] as the residual clipping noise power σd2 and σ 2 denotes the average power of the (complex) channel noise. Then, the SNR for (4.62) w.r.t. the wanted signal Xk [n] is

snrk [n] =

|κβk H[n]|2 |H[n]|2 = . Eγ [V[Re(ζk [n])]] |H[n]|2 PI,k + PW,k

(4.66)

We make three assumptions:

• The DEC inputs at dierent time indexes are uncorrelated. Similarly, the feedbacks from the DEC at dierent time indexes are also uncorrelated.

• The input sequence of the DEC is characterized by {snrk [n]} in (4.66). (This holds true when Gaussian assumption is applied to {ζk [n]}.) • The distribution of {H[n]} is given. (A typical case is that {H[n]} follows Rayleigh fading.)

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Figure 4.6: Block diagram of the Monte Carlo simulation for generating TMI (·, ·). Note that the rst assumption above can be ensured through the use of random interleavers. It also implies innitely long codeword length. Based on the above three assumptions and from (4.66), the pair (PI,k , PW,k ) fully determines the quality of the ESE outputs. In the iterative decoding process, PW,k is a constant number but PI,k decreases as the iteration proceeds. The key to the MI evolution analysis is to characterize the mutual information contained in the ESE outputs as a function of PI,k , PW,k .

Iλ,k ≡ I(Xk [n], λk [n]) = TMI (PI,k , PW,k ).

(4.67)

In general, TMI (·) cannot be expressed in a closed form, but can be characterized by a look-up table created by the Monte Carlo simulation. The block diagram of the simulation is depicted in Fig. 4.6, where we have used an equivalent channel model as

Zk [n] = H[n](Xk [n] + Nk [n]) + Wk [n]

(4.68)

where Xk [n] ∈ {+1, −1} is the BPSK signal, Nk [n] ∼ N (0, PI,k ) and Wk [n] ∼

N (0, PW,k ), respectively, represent the (normalized) interference and channel noise. Then the mutual information dened in (4.67) can be estimated by [45]

Iλ,k

µ µ ¶¶ N −1 1 X 2H[n]Zk [n] =1− log2 1 + exp −Xk [n] × , N → +∞. N n=0 |H[n]|2 PI,k + PW,k (4.69)

Note that by applying the Monte-Carlo method to dierent (PI,k , PW,k ), we can create a two-dimensional look-up table to characterize the conversion from (PI,k , PW,k ) to Iλ,k . Now the remaining problem is how to track PI,k . From our earlier denitions, PK 2 ρk = Eγ [V[Xk [n]]], k = 1, 2, · · · , K . Now σx2 = k=1 |βk | ρk . Therefore, from (4.17), σd2 can be found from {ρk } as

σd2 = φ

ÃK X k=1

! |βk |2 ρk

.

(4.70)

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From (4.64) and (4.70), PI,k is determined by {ρk }. On the other hand, ρk is the variance of the DEC feedback of the k th layer. Following the treatment in Chapter 2, we characterize it as a function of the mutual information contained in the DEC inputs (i.e. ESE outputs): (4.71)

ρk = TDEC (Iλ,k ).

Similarly, the BER performance of the DEC can be characterized by a function as (4.72)

BERk = b(ρk ).

To summarize, we can characterize the iterative decoding process using the following procedure. (i) Initialization : Set it = 1 and ρk

(it)

= 1, PW,k =

σ2 , 2|κβk |2

k = 1, 2, · · · , K .

(ii) Recursion: For the itth iteration: Find the normalized interference power for the ESE:

(it)

PI,k

φ X |βm |2 (it) = ρ + |βk |2 m m6=k

³P

K m=1

(it)

|βm |2 ρm

´

2κ|βk |2

,

k = 1, 2, · · · , K.

Find the mutual information in the DEC inputs: (it)

(it)

Iλ,k = TMI (PI,k , PW,k ),

k = 1, 2, · · · , K.

Find the output variance of the DEC: (it+1)

ρk

(it)

= TDEC (Iλ,k ),

k = 1, 2, · · · , K.

If it < It, set it ← it + 1 and go to (ii); otherwise, go to (iii). (iii) Termination : Substitute ρk

(It)

into (4.58) to nd the nal symbol variance;

and output the BER for each layer: (It)

BERk = b(ρk ),

k = 1, 2, · · · , K.

The above MI evolution technique can be extended to general cases such as those with complex {βk } and κ. It leads to more accurate results than the EXIT

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97

chart technique with ML-SCM. (Note: We may use multi-dimensional EXIT functions [94] to improve the EXIT chart technique, but this can be very complicated for large K . The MI evolution technique simplies the problem due to the signalplus-noise modelling in (4.62).) It is worth mentioning that in the MI evolution method, only the binary-input channel model in Fig. 4.6 needs to be simulated. Then using the stored look-up tables to characterize (4.71), one can predict the performance of ML-SCM with arbitrary weighting factors {βk } and channel SNR. This is more convenient than the EXIT chart method in BICM where Monte-Carlo simulations of a multi-ary system have to be performed whenever the signaling scheme or channel SNR is changed.

4.5.4 Symbol Variance Minimization (It)

We are interested in minimization of the symbol variance Eγ [V[s]] after It iterations so as to minimize the residual clipping eect after iteration terminates. (It)

As discussed in Section 4.5.3, Eγ [V[s]] is determined by {βk } and can be easily evaluated using the MI evolution method. Exploiting this fact, we can search the (It)

values of {βk } that can minimize Eγ [V[s]]. The optimization methods studied in [66, 64, 105] can be used for this purpose. In particular, we can resort to exhaustive search for a small K and the interior point method [105] for a large K .

4.6 Numerical Results In this section, we present numerical results to verify the above analysis. We take OFDM systems based on BICM [24, 93] and ML-SCM for examples. For the SCM schemes, we assume that K is an even number and βk−1 = iβk with

βk−1 being a real number for k = 2, 4, · · · , K . This means that the in-phase and quadrature components of the transmit signal employ the same signaling scheme. The APP and GA demapping methods are applied to the BICM and SCM schemes, respectively.

4.6.1 Residual Clipping Noise and Global Variance We rst show the eectiveness of the proposed soft compensation method by comparing it with the two alternative methods mentioned in Section 4.3.3. For illustration, we dene the signal to residual clipping noise ratio (SRCNR) from (4.9) and (4.10) as SRCNR = |κ|2 /σd2 , where we have assumed E[|X[n]|2 ] = 1.

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98

Figure 4.7: SRCNR achieved by dierent detection methods for a BICM-IDOFDM scheme. Clipping with L = 4, N = 256, and CR = 0 dB is assumed. We consider an SL-SCM-based OFDM system. From Section 4.4.1, the SRCNR is a function of the global variance σx2 . The σx2 versus SRCNR curves are plotted in Fig. 4.7. It is shown that the proposed method consistently outperforms the other two methods. This is because it estimates d as the conditional mean d˜ =

E[g(x) − κx], which is optimal when the a priori distribution of x is available. We also observed that the curves in Fig. 4.7 are nearly independent of the signaling schemes employed. We then examine the global variance σx2 achieved by dierent signaling schemes. We adopt the i.i.d., consistent Gaussian assumption in [18] to model the extrinsic LLRs {γk [n]} (from the DEC) in the iterative process. Then, the mutual information between {γk [n]} and the coded bits {ck [n]}, denoted by Iγ , can be used to fully characterize the distribution of {γk [n]}. In general, a larger Iγ implies that the extrinsic information produced by the DEC are more reliable. The σx2 achieved by ve 16-ary signaling schemes, namely, the 16-QAM with Gray, Mixed, MSP, M16a mappings and the SCM scheme illustrated in Fig. 4.45 , are compared in Fig. 4.8. From the discussions above, to minimize the clipping eect with soft compensation, σx2 should be minimized. Fig. 4.8 shows that the SCM signaling is clearly a better choice than its alternatives for clipping noise mitigation. 5

The reasons for considering these mappings are as follows. First, the M16a and MSP mapping (and the related set-partitioning (SP) mapping) can yield good performance for BICMID over AWGN channels [87, 24]. Second, it is shown in [25] that the Mixed mapping can potentially outperform other mappings for clipped transmission.

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99

Figure 4.8: Impact of signaling schemes on σx2 .

4.6.2 Simulation Results We now present BER results of clipped OFDM systems with dierent detection methods and signaling schemes.

4.6.2.1 Impact of Detection Methods Fig. 4.9 compares the proposed soft compensation method with the two alternative methods considered in Fig. 4.7. A BICM-ID-OFDM system employing the SCM signaling with K = 4, {|βk |} = {1 × 4} is considered. The 16-state, rate-1/2 convolutional code (23, 35)8 is used. The system throughput is R = 2 bits per two dimensions. The frame length is 4096. (Each frame consists of 16 OFDM blocks.) AWGN channels are assumed. The number of iterations It = 10. It is shown that the proposed method can signicantly outperform the other two approaches. This is consistent with Fig. 4.7 and also demonstrates the effectiveness of the proposed method. The proposed method based on (4.12) has roughly the same computational complexity as the two alternatives except that it requires more memory to store a look-up table. However, this extra cost can be justied by considerable performance gain.

4.6.2.2 Impact of Signaling Schemes Fig. 4.10 presents the BER of BICM-ID-OFDM schemes employing the code

(23, 35)8 and dierent signaling schemes in Fig. 4.8. The SCM signaling with K = 4, {|βk |} = {1 × 4} is compared with the 16-QAM with MSP, Mixed and

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Figure 4.9: Comparison of dierent detection methods for a clipped BICM-IDOFDM scheme with SCM signaling (K = 4, {|βk |} = {1 × 4}). L = 4, N = 256, CR = 0 dB. Gray mapping. It can be seen that the SCM signaling is more robust against the clipping eect than other choices. The performance degradations at BER = 10−5 due to clipping are about 0.18, 0.2, 0.45 and 0.9 dB for the SCM, Gray, Mixed and MSP schemes, respectively. This is roughly in line with the global variance results in Fig. 4.8. Note that in the clipped case, SCM requires slightly higher SNR to achieve BER = 10−5 than the Mixed signaling, which implies that SCM is not necessarily BER-optimal. However, we can show that SCM can outperform the latter at BER = 10−5 if more severe clipping eect is present, e.g., when repeated clipping is used to achieve a low PAPR [2]. We can also visualize the impact of signaling schemes using the EXIT charts in Fig. 4.11. Let Iλ be the mutual information between {λk [n]} and {ck [n]}. Then the EXIT curves Iγ → Iλ and Iλ → Iγ can be used to characterize the ESE and DEC, respectively. It is seen that the SCM scheme leads to a wider decoding tunnel between the ESE and DEC curves than its alternatives. This suggests that iterative decoding converges earlier in SCM. Note that the SCM scheme achieves a lower Iλ than the Mixed and MSP schemes at the high Iγ range, implying a possible error oor problem. This observation can be explained as follows: Recall that soft compensation is used in all methods. Suppose that feedback information is perfect. The clipping eect can then be fully mitigated. In this case, the performance comparison follows the unclipped scenario that the Mixed and the MSP mappings are optimized. The SCM signaling is only a special choice among

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101

Figure 4.10: Comparison of BICM-ID-OFDM schemes with dierent signaling schemes when clipping and the soft compensation are used. L = 4, N = 256 and CR = 0 dB. AWGN channels are assumed. all possible signaling schemes for BICM-ID, and so, it may not be the best one. However, when clipping is present, the SCM signaling can yield better overall performance, as demonstrated by the EXIT curves in Fig. 4.11 as well as by the simulation results in Fig. 4.10. We also show examples with a rate-1/2 turbo code (23, 35)8 in Fig. 4.12. An ML-SCM scheme with K = 4 is compared with a BICM scheme employing the 16-QAM with Gray mapping. (Note : As mentioned in [93], the Gray mapping can provide the best BER performance in turbo-coded BICM systems.) We can see that without clipping, the SCM and BICM-based schemes have comparable performance. However, with clipping, the advantage of SCM is quite evident. This is because the SCM signaling can lead to better clipping eect mitigation, as discussed in Section 4.4.

4.6.2.3 High-Rate Transmissions We also consider OFDM examples with high-rate BICM-ID and ML-SCM schemes in fading channels. We assume a slow time-varying frequency selective (STVFS) channel with 12 taps as considered in [25, 26]. The ML-SCM and BICM-ID schemes at rate R = 4 bits per two dimensions are compared. For the MLSCM scheme, the component code is constructed by concatenating the rate-1/2 convolutional code (23, 35)8 with a length-4 repetition code; K = 32. An unequal

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102

Figure 4.11: EXIT charts for the clipped BICM-ID-OFDM schemes in Fig. 4.10 at Eb /N0 = 4 dB. power allocation among the layers is applied and the values of {βk } are listed in the caption of Fig. 4.13. For the BICM-ID scheme, the 64-QAM with setpartitioning (SP) and Gray mapping is used [75]6 and the above convolutional code is punctured to rate 2/3 to achieve R = 4. Fig. 4.13 shows the simulation results for L = 4, N = 256 and CR = 2 dB. Notice that without clipping, BICM-ID with SP signaling and ML-SCM have similar performance. However, when clipping is used, ML-SCM can signicantly outperform BICM-ID. In this case, the iterative compensation is not very eective for the BICM-ID scheme with the SP signaling; hence, its performance is dramatically degraded by the clipping eect. Soft compensation still works well with the SCM signaling in this case. The Gray signaling shows good robustness against clipping eect, but it suers from a high error oor. Note that the APP and GA demapping methods are applied to the BICM-ID and ML-SCM schemes, respectively. For a 2K -ary constellation, the complexities of the APP and GA methods grow linearly with 2K and K , respectively. Hence, the schemes compared in Fig. 4.13 have comparable complexity, despite the fact that the ML-SCM scheme uses much larger constellations. From Fig. 4.10 and 4.13, we can see that the advantage of SCM becomes more evident as the rate increases. This is because, for a higher rate, the SNR value at the working point becomes higher. In other words, the channel noise 6

The SP mapping is adopted because we observed that it can yield better performance for the BICM-ID with 64-QAM over fading channels than other options available in the literature.

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103

Figure 4.12: Comparisons of turbo-coded ML-SCM-OFDM and BICM-ID-OFDM with clipping and soft compensation over AWGN channels. L = 4, N = 256. The frame length is 32768. R = 2 bits per two dimensions. The number of iterations between the ESE and DEC is 6. The number of the iterations in the turbo decoder is 6. For the ML-SCM, {|βk |} = {1 × 2, 1.5 × 2}. power level becomes relatively lower. Thus, the clipping-induced distortion, which increases with signal power, becomes a dominant factor. In this case, SCM is more advantageous as it can minimize the residual clipping distortion. We expect that SCM with iterative soft compensation can also work well in OFDM systems with more general non-linear distortions as considered in [2, 52].

4.6.2.4 Evolution versus Simulation Results Fig. 4.14 compares the simulation results and the BER predicted by the MI evolution technique outlined in Section 4.5.3. The ML-SCM-OFDM scheme with

R = 4 used in Fig. 4.13 is considered. A relatively large frame length of 16384 is employed here since we have assumed innite frame length in MI evolution. From Fig. 4.14, the evolution and simulation results agree well in both the clipped and unclipped cases. This clearly demonstrates the eectiveness of the MI evolution technique.

4.7 Summary Clipping can alleviate the high PAPR problem in OFDM systems without incurring rate loss. However, the clipping-induced distortion can cause serious performance degradation when not treated properly. In this chapter, we have considered

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 104

Figure 4.13: Performance of convolutional-coded ML-SCM-OFDM and BICMID-OFDM with clipping and soft compensation over STVFS channels. L = 4, N = 256, and CR = 2 dB. R = 4. The frame length is 4096. It = 10. For the SCM scheme, K = 32, {|βk |} = {1 × 12, 1.58 × 4, 2.10 × 6, 2.49 × 2, 2.73 × 2, 3.58 × 4, 3.93 × 2}. an iterative soft compensation method to mitigate the clipping eect. We have shown that this method can signicantly outperform conventional approaches. We have also made a rigorous analysis of the impact of signaling schemes on performance. Both the analytical and numerical results show that the SCM scheme can outperform other alternatives, especially when clipping is severe and/or transmission rate is high.

4.8 Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping In this Appendix, we compare the detection methods for OFDM systems with Nyquist-rate sampled Cartesian clipping [71] over AWGN channels. The timedomain received signal is written as

y = g(x) + w

(4.73)

where x and y represent the in-phase (or quadrature) part of the time-domain transmit and received signal, respectively, w ∼ N (0, σ 2 ). In (4.73), we have

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 105

Figure 4.14: Simulation versus evolution results for the ML-SCM-OFDM with R = 4 bits per two dimensions over fully-interleaved Rayleigh fading channels. The frame length is 16384. In the clipped case, L = 4, N = 256, and CR = 2 dB. It = 12. omitted the symbol index and g(x) is the Catersian clipping function:

 x > A,  A, g(x) = x, −A 6 x 6 A,  −A, x < −A. In OFDM, the time-domain signal {x} are obtained from the IDFT of the frequencydomain symbols {X} which are mapped from coded bits according to certain signaling schemes. By invoking the central limit theorem, x can be modeled as a −2 →, → real-valued Gaussian R.V.. Let (− µ x σx ) be the a priori mean and variance of x that are computed from the DEC feedback. Then the a priori probability density function (PDF) of x can be written as

à ! →)2 1 (x − − µ x p(x) = √ − exp − . − →2 → 2π σx 2σx

(4.74)

4.8.1 Iterative Bayesian Method Several authors [6, 29] have applied the Bayesian estimation principles to clipped → = 0 (which is true without iterative decoding) OFDM, but only the case with − µ x

→ = was considered. We now consider the Bayesian estimation with − µ 0, which x 6 enables the usage of the Bayesian method in iterative mitigation of clipping eect.

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 106 From [56], the Bayesian estimate of x (represented by the APP mean and variance) can be computed from the observation y and the a priori information as

Z 1 xp(y|x, g(·))p(x)dx p(y|g(·)) Z h → −2 i 1 − → 2 σx = V x|y, µx , σx = x2 p(y|x, g(·))p(x)dx − µx 2 , p(y|g(·)) h

− →2 i − → µx = E x|y, µx , σx =

(4.75a) (4.75b)

where p(x) is given by (4.74) and

µ ¶ 1 (y − g(x))2 p(y|x, g(·)) = √ exp − 2σ 2 2πσ is the channel transition probability. To evaluate the above integrals, dene

− →2 → σ y + σ2− µ x µ∗ = x− , →2 2 σx + σ

σ∗2

− →2 2 σx σ =− , →2 σx + σ 2



C∗ =

− → 2 σx2 y 2 +σ 2 − µ→ x − → 2 σ σx +σ 2  √ ∗− exp − → 2σ∗2 2πσ σx

 − µ2∗

 .

Let σy2 = σ 2 , and µy = y . We dene

αy =

A − µy −A − µy √ , βy = √ . 2σy 2σy

→2 →, − Similarly, we dene α∗ , β∗ from (µ∗ , σ∗2 ), αx and βx from (− µ x σx ). Then it can be veried that

Z

+∞

p(y|g(·)) = = Z

(4.76)

p(y|x, g(·))p(x)dx −∞ −α2y

e

2

[2 − erfc(αx )] + e−βy erfc(βx ) C∗ [erfc(α∗ ) − erfc(β∗ )] √ + 2 2 2πσ

+∞

xp(y|x, g(·))p(x)dx r r ¸ ¸ · 2 · e π− e−βy − π− → → → → −α2x − −βx2 = µx (2 − erfc(αx )) + µx erfc(βx ) −e σx + σx e + 2πσ 2 2πσ 2 r ¸ · C∗ π 2 2 −α∗ −β∗ µ∗ [erfc(α∗ ) − erfc(β∗ )] +√ σ∗ (e −e )+ (4.77) 2 2π −∞ −α2y

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 107 Z

+∞ −∞ −α2y

x2 p(y|x, g(·))p(x)dx

¸ − →2 π − → 2 (µx + σx ) [2 − erfc(αx )] 2 2πσ C∗ 2 C∗ 2 2 + √ σ∗2 (α∗ e−β∗ − β∗ e−α∗ ) + (µ∗ + σ∗2 ) [erfc(α∗ ) − erfc(β∗ )] 2 π · ¸ √ −βy2 − →2 − →2 e π − → −βx2 2 +√ −σx αx e + (µx + σx )erfc(βx ) 2 2πσ

e = √

·

→ −2 −α2 σ x βx e x +

√2 π

√

(4.78)

R +∞

exp(−t2 )dt. From (4.75)-(4.78), the APP means and ← − variances can be evaluated. Following [65], the extrinsic mean ← µ−x and variance σx2 where erfc(z) =

z

of x are obtained as

− →2 → σx µx − σx2 − µ x ← − µx = − , →2 σx − σx2 ¯ → ¯ − ← −2 ¯¯ σx2 σx2 ¯¯ σx = ¯ − ¯. ¯→ σx2 − σx2 ¯

(4.79a) (4.79b)

The results of (4.79) will be used for de-mapping/decoding in the next iteration by modeling

← − µ−x = x + ← w

(4.80) ← − − is an equivalent AWGN with variance σ 2 . where x is the useful signal and ← w x

4.8.2 Iterative Soft Compensation Method An alternative to the above Bayesian approach is the soft compensation method outlined in Section 4.3. For the Cartesian clipping, closed-form expressions can be derived. We rewrite (4.73) as

y = κx + d + w

(4.81)

where d is the clipping distortion, κ is a constant scaling factor follows the Price's theorem. For the Cartesian clipper, the clipping distortion is given by  x>A  A − κx, d = g(x) − κx = x(1 − κ), −A 6 x 6 A  −A − κx, x < −A.

(4.82)

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 108

Figure 4.15: Comparison of dierent clipping eect mitigation methods for Nyquist-rate sampled Cartesian clipping at CR = 4 dB. R = 2; {βk } = {1, 1.05, i, 1.05i}. The number of iterations is 10 for all curves. Other parameters are the same as those in Fig. 4.9.

→2 →, − In the soft compensation method, only the a priori information (− µ x σx ) is exploited to estimate and cancel the clipping distortion d. Introduce the notations − →

µx √ − , βx = αx = −A− 2σ→ x → − →, σ 2 ) as: (− µ x

− A− µ→ x √ − . 2σ→ x

Then d can be estimated as its mean conditioned on

x

h →2 i →, − d˜ = E d|− µ (4.83) x σx → → − → σx A+− µ A−− µ 2 2 x x →. = [erfc(αx ) − 2] + erfc(βx ) + √ (e−αx − e−βx ) + (1 − κ)− µ x 2 2 2π The residual noise is given by the variance of d that can be computed h clipping − →2 i → 2 2 − from σd = d |µx , σx − d˜2 where

→2 → →2 + − σx ) + A2 + 2Aκ− µ E[d2 ] = κ2 (− µ x x i − → −α2x h √ → − √ σx e + √ (1 − 2κ)( 2− µx + → σx αx ) − 2Aκ π i − → −βx2 h √ → − √ σx e (1 − 2κ)( 2− − √ µx + → σx βx ) + 2Aκ π i →2 erfc(αx ) h → − A2 →2 + − µ (1 − 2κ)(− µ σx ) − 2Aκ− + x x 2 i →2 erfc(βx ) h − → − A2 . →2 + − − ) + 2Aκ µ (1 − 2κ)(− µ σ x x x 2

(4.84)

4.8. Appendix: Iterative Detection of OFDM With Nyquist-Rate Sampled Cartesian Clipping 109 Then the detection can be performed based on

z ≡ y − d˜ = κx + w ˜

(4.85)

where w ˜ = d − d˜+ w is assumed as an AWGN with variance σd2 + σ 2 . The detailed treatments are similar to those in Section 4.3 and are omitted here.

4.8.3 Examples In Fig. 4.15, we show examples of SL-SCM-based OFDM systems with the rate-

1/2 convolutional code. From Fig. 4.15, Method I and II (c.f., Section 4.3.3) can compensate for the clipping eect to some extent but the performance can still be improved. For Eb /N0 > 6 dB, the Bayesian method yields the best performance but has the highest computational complexity. The soft compensation method provides slightly degraded performance at lower complexity.

Chapter 5

SCM over Linear Vector Channels

5.1 Introduction Modern communication systems should be able to provide high-rate, reliable data transmissions with limited spectrum resources. Information-theoretic analysis [28, 37, 95] suggests that allowing multiple users and/or data streams to share the same frequency band can signicantly improve the spectrum eciency. The nonorthogonal multiple-access system [106] and multi-input multi-output (MIMO) system [48, 110, 113] are ecient approaches to realizing this principle. One challenging issue in such systems is the interference among the concurrently transmitted signals. For example, in multi-user MIMO systems operating over multi-path channels, the multiple-access interference (MAI), cross-antenna interference (CAI) and intersymbol interference (ISI) constitute major sources of impairments to performance. The MAP detection is the optimal approach to recover information from the interfered received signals. However, it has an exponential complexity and is impractical for most applications. For example, consider a Q−user scheme with nt transmit antennas per user over a channel with L taps. Let the signal at each transmit antenna be drawn from a constellation of size 2K . At the receiver, the signals from dierent users/antennas/paths are superimposed to form a super constellation of size 2KQLnt . The optimal MAP approach needs to compute the metric between the received symbol and every point in the constellation. The related complexity O(2KQLnt ) can be prohibitively high even for modest

K , Q, L and nt values. To handle this complexity issue, solutions with dierent complexity-performance tradeo have been studied, among which the iterative linear minimum mean-square error (LMMSE) method [30, 31, 47, 100, 107, 108] has received much attention. On the other hand, coded modulation has been widely adopted to improve the power- and spectrum-eciency of high-rate systems. BICM [114], MLC [53] as well as SCM can be applied. Signaling schemes are the crucial design parameters for such schemes. Much progress has been made in the design and optimization of signaling schemes for memoryless channels where only the additive channel noise

5.2. Iterative LMMSE Detection

111

is present and the optimal MAP detection is used. However, for more complicated channels where the MAP method becomes impractical, there have been limited discussions on the selection of signaling schemes. In light of the above, this chapter focuses on high-rate coded modulation systems over multi-user/multi-antenna/multi-path channels. We apply a generic linear vector model to characterize such systems and adopt the iterative LMMSE receiver. We introduce an evolution method to quickly predict the performance and consider the impact of signaling schemes. Maximizing the output signalto-interference-and-noise ratio (SINR) of the LMMSE estimator is used as the criterion for choosing signaling schemes. Under this criterion, we reveal the advantages of SCM. We also show that QAM with Gray mapping (that was regarded as a poor option for AWGN channels when iterative decoding is used) can achieve performance close to that of SCM.

5.2 Iterative LMMSE Detection 5.2.1 System Model The multiple-user/MIMO systems over multi-path channels can be characterized using a linear vector model as illustrated in Fig. 5.1. (The detailed matrix manipulations will be discussed later.) The source data is rst encoded by the ENC using a binary FEC code, and permuted by a random interleaver (Π) to produce a bit sequence c. Then c is segmented into N sub-blocks c = {c[0], c[1], · · · , c[N − 1]} where each c[n] is a sub-block of K bits c[n] = {c1 [n], c2 [n], · · · , cK [n]}. The mapper then maps each c[n] onto a signaling point x[n] in a constellation X of size

2K using a mapping rule µ. The above transmitter scheme follows the principles P ck [n] . of BICM-ID [24]. SL-SCM is a special case with x[n] = K k=1 βk (−1) Let matrix H represent the multiplicative eect of the linear channel. The received signal is given by

y = Hx + η,

(5.1)

where x = [x[0], x[1], · · · , x[N − 1]]> is the transmitted signal vector and η is a vector of AWGN with covariance matrix N0 I .

5.2.2 Overall Iterative Detection Principles The iterative receiver structure is shown in the lower part of Fig. 5.1. Similar to that in Chapter 4, the ESE computes the extrinsic LLR λk [n] for each ck [n] from

5.2. Iterative LMMSE Detection

112

Figure 5.1: Transmitter and receiver structure of a coded multi-ary system over linear channels. Π denotes interleaver and Π−1 de-interleaver.

Figure 5.2: LMMSE approach to the ESE. the channel observation and DEC feedback µ ¶ Pr(ck [n] = 0|y, {γk [n]}) λk [n] ≡ ln − γk [n]. Pr(ck [n] = 1|y, {γk [n]})

(5.2)

The DEC performs the APP decoding using {λk [n]} as inputs, producing the

extrinsic LLRs {γk [n]} as dened in (4.5). After decoding, the ESE operations can be executed again to rene the estimates {λk [n]} using the feedbacks {γk [n]}. This process continues iteratively. Hard decisions are made after the nal iteration to produce the data estimates. Related discussions on such iterative detection process can be found in [1, 30, 107, 81, 108, 111]. In what follows, we focus on the ESE function as the DEC function is standard.

5.2.3 ESE Function The optimal MAP solution to the ESE function in (5.2) is usually highly complicated, since, after the linear transform H , the constellation of Hx can be signicantly expanded. The iterative LMMSE detector is a low-cost, suboptimal alternative. As shown in Fig. 5.2, the detection process in the ESE can be divided into the following three steps.

5.2. Iterative LMMSE Detection

113

Step 1. Soft Mapping : In this step, the means {E[x[n]]} and variances {V[x[n]]} of the entries of x are generated using the DEC feedback LLRs {γk [n]}. This is a preparation stage for the LMMSE estimation and the details are the same as those in Section 4.4.2.

Step 2. LMMSE Estimation : Dene the mean of x as

E[x] = [E[x[0]], E[x[1]], · · · , E[x[N − 1]]]> .

(5.3)

Following [111], we assume that the entries of x are independent, and the covariance matrix of x is

V = vI,

(5.4)

where I is the identity matrix and v is the average of the symbol variance computed by Step 1:

N −1 1 X v= V[x[n]]. N n=0

(5.5)

The LMMSE estimate of x is [56]

ˆ ≡ E[x|y] = E[x] + V H † R−1 (y − E[y]) x

(5.6)

where E[y] ≡ HE[x], and R is the covariance matrix of y :

R = E[(y − E[y])(y − E[y])† ] = HV H † + N0 I.

(5.7)

Step 3. Soft De-Mapping : Finally, we compute the LLR {λk [n], ∀k} dened in ˆ , as (5.2). Following [100, 107, 111], we can write x ˆ[n], the nth entry of x

xˆ[n] = φ[n]x[n] + ξ[n]

(5.8)

where ξ[n] is modeled as a Gaussian noise and φ[n] is selected such that x[n] and ξ[n] are uncorrelated. Let h[n] be the nth column of H . Then

φ[n] ≡ vh[n]† R−1 h[n]

(5.9)

ξ[n] ≡ E[x[n]] + vh† [n]R−1 (y − E[y] − h[n]x[n]).

(5.10)

and It can be shown that

E[ξ[n]] = (1 − φ[n])E[x[n]]

(5.11)

5.2. Iterative LMMSE Detection V[ξ[n]] = vφ[n](1 − φ[n]).

114 (5.12)

Treating (5.8) as a memoryless system with channel gain φ[n] and noise ξ[n], we can evaluate (5.2) based on the APP demapping principles as [39]:  P  Pr(ˆ x[n]|x[n] = s) Pr(x[n] = s)   s∈Xk(0)  − γk [n] P λk [n] = ln  (5.13)  Pr(ˆ x[n]|x[n] = s) Pr(x[n] = s)  (1)

s∈Xk

´ ³  P  |ˆ x[n]−φ[n]s−E[ξ[n]]|2 Pr(x[n] = s) exp − V[ξ[n]]  s∈X (0)   k  ³ ´ = ln  P  − γk [n](5.14) |ˆ x[n]−φ[n]s−E[ξ[n]]|2   exp − Pr(x[n] = s) V[ξ[n]] (1)

s∈Xk (0)

(1)

where Xk and Xk are the sets of constellation points whose k th bit position carries 0 and 1, respectively, and {Pr(x[n] = s)} are computed using the a

priori LLRs {γk [n]}. When SCM is used, alternatively, we can apply the GA demapping method discussed in Chapter 2 to evaluate λk [n].

5.2.4 Complexity Analysis The above LMMSE procedure is a low-cost alternative to the MAP approach. The main problem for the MAP method is that the signal constellation expands after transmission over a linear channel characterized by (5.1). To see this, let us consider a single-path MIMO channel with nt transmit antennas and nr receiver antennas. Suppose a 2K -ary constellation is used for each entry of the transmit signal vector x. Then the constellation size of the useful signal Hx in (5.1) is

2Knt . The complexity of the MAP method is O(nr 2Knt /nt ) per symbol, which can be extremely high. The complexity can be greatly reduced by using the LMMSE approach. For the above MIMO channel, H is an nr × nt matrix and R is an nr × nr matrix. The complexity involved in the matrix inversion R−1 is O(n3r ). The overall complexity per symbol can be roughly estimated as O(2K + n3r /nt ) (with O(n3r /nt ) for evaluating (5.6) and O(2K ) for the soft mapping and demapping). This can be much lower than the complexity O(nr 2nt K /nt ) of the optimal MAP detector. We can also use (5.1) to represent a multi-path MIMO channel. In this case, H is an nr N × nt N matrix with a Toeplitz-type structure. The detection complexity becomes O(2K + (nr N )3 /(nt N )) per symbol that can be very high. However, using a cyclic prexing technique, H can be converted into a circulant matrix. Then R−1 can be computed using the FFT technique, with complexity reduced to

5.3. Evolution Analysis

115

Figure 5.3: Evolution model of the iterative LMMSE receiver.

O(2K + (1 + nr /nt ) log N + n2r ) per symbol. Interested readers are referred to [111] (where the treatments for general multi-user/multi-antenna/multi-path systems are discussed) for details.

5.3 Evolution Analysis We proceed to the analysis problem for the LMMSE receiver outlined above. We consider an MI evolution method similar to those discussed for SCM in memoryless channels (c.f. Chapter 2) and clipped OFDM systems (c.f. Chapter 4). Here, we extend the discussions to general multi-ary schemes with iterative LMMSE detection. We also show that the impact of signaling schemes on performance can be characterized by the symbol variance discussed in Chapter 4 and SCM is advantageous in this respect. For simplicity, we rst assume a single-user, single-antenna, multi-path channel and then extend the discussions to more general linear channels.

5.3.1 Transfer Functions We employ the evolution model in Fig. 5.3 to characterize the receiver in Fig. 5.1. We assume xed H , N0 and innite interleaving lengths. For each of the four modules in Fig. 5.3, we dene a transfer function to characterize its performance.

5.3.1.1 DEC Similarly to [18], we can use the EXIT function

Iγ = TDEC (Iλ )

(5.15)

to fully characterize the DEC. Here, Iλ is dened as the mutual information between the DEC inputs {λk [n]} and the coded bits {ck [n]}, and Iγ is dened sim-

5.3. Evolution Analysis

116

ilarly for the DEC outputs {γk [n]}. The BER performance of the DEC can also be characterized by a monotone decreasing function g(·) as

BER = b(Iγ ).

(5.16)

5.3.1.2 Soft Mapper Assume innite interleaving length. Then the average variance v dened in (5.5) equals to the expectation of symbol variance, i.e., v = Eγ [V[x[n]]], where the expectation Eγ [·] is w.r.t. the distribution of {γk [n]} and V[x[n]] is computed by (4.23). The soft mapper takes the extrinsic LLRs {γk [n]} as inputs and produces the soft estimates of x[n]. Thus, it is reasonable to characterize the soft mapper by the average variance v of its outputs, i.e.,

v = Eγ [V[x[n]]] = TSM (Iγ ).

(5.17)

Note that from [56], v is also equal to the mean square error (MSE) in estimating

{x[n]} as E[x[n]] computed using the DEC feedback {γk [n]}. At the beginning of the iterations, since there is no a priori information from the DEC (i.e., Iγ = 0),

v = TSM (0) = 1, where we have assumed that the average transmitted power E[|x[n]|2 ] = 1. When perfect a priori information is available, we have Iγ = 1 and v = TSM (1) = 0.

5.3.1.3 LMMSE Estimator The LMMSE estimate x ˆ[n] in (5.8) can be viewed as the output signal of an equivalent channel with multiplicative coecient φ[n] and additive noise ξ[n]. The SINR based on the modeling in (5.8) can be computed as

|φ[n]|2 φ[n] = V[ξ[n]] (1 − φ[n])V[x[n]] † −1 h[n] R h[n] = . 1 − vh[n]† R−1 h[n]

Γ[n] ≡

(5.18)

Let

R[n] = v

X

h[n0 ]h[n0 ]† + N0 I = R − vh[n]h[n]† .

(5.19)

n0 6=n

From the matrix inversion lemma,

R−1 = R[n]−1 −

vR[n]−1 h[n]h[n]† R[n]−1 . 1 + vh[n]† R[n]−1 h[n]

(5.20)

5.3. Evolution Analysis

117

Then from (5.18), it can be shown that

Γ[n] = h[n]† R[n]−1 h[n].

(5.21)

For cases where the size of the covariance matrix R[n] is small, (5.21) can be evaluated directly. Otherwise, the transform-domain method outlined below can be used. For illustration, consider an L-tap multi-path channel with coecients

{h0 , h1 , · · · , hL−1 }. It is shown in [111] that Γ[n] can be approximated as follows: Γ[n] ≈ Γ =

u , 1 − vu

∀n

(5.22)

where

N −1 1 X |g[n]|2 u= (5.23) N n=0 v|g[n]|2 + N0 ¡ 2πnl ¢ P being the frequency-domain channel gain and with g[n] = L−1 l=0 hl exp −i N √ i = −1. From the above discussions, Γ is a deterministic function of v , H and N0 ,

denoted by

Γ = TMMSE (v, H, N0 ).

(5.24)

5.3.1.4 Soft De-Mapper As illustrated in Fig. 5.3, the soft demapper takes the LMMSE estimates and DEC feedback (characterized by Γ and Iγ , respectively) as inputs and produces the extrinsic LLRs {λk [n]}. Thus, its performance can be characterized by the following transfer function.

Iλ = TDEM (Γ, Iγ ).

(5.25)

5.3.2 Evolution Analysis Among the four transfer functions above, only TMMSE (v, H, N0 ) is a function of channel matrix. Fortunately, TMMSE (v, H, N0 ) has a closed-form expression in (5.22) and thus it can be quickly evaluated on the y (rather than pre-simulated). The other three transfer functions TDEC (Iλ ), TSM (Iγ ) and TDEM (Γ, Iγ ) are not functions of H . They can be obtained by applying the Monte-Carlo method to an AWGN channel and characterized by look-up tables. Therefore, given {H, N0 }, the overall receiver performance can be predicted by the following evolution procedures, where TDEC (Iλ ), TSM (Iγ ) and TDEM (Γ, Iγ ) are assumed to be known and

TMMSE (v, H, N0 ) is computed on line.

5.3. Evolution Analysis

118

Initialization : Set Iγ = 0. Recursion : Update Iγ as Iγ = TDEC (TDEM (TMMSE (TSM (Iγ ), H, N0 ) , Iγ )) .

Termination : After a preset number of recursions, estimate the BER by substituting the nal value of Iγ into (5.16). Later we will see that the above evolution method can provide quick and accurate performance prediction. We can also nd the average performance over block-fading channels (averaged over the distribution of H ) by applying the above method to randomly generated samples of H . Note that an alternative to the above four-module approach is to characterize the whole ESE block using a single pre-simulated EXIT function. This is feasible for a xed channel matrix H , but becomes intractable for random H . When perfect a priori information is available, we have Iγ = 1 and v =

TSM (1) = 0, and Γ converges to the upper limit PL−1 N −1 |hl |2 1 X |g[n]|2 Γ=u= = l=0 N n=0 N0 N0

(5.26)

where the last equality follows Parseval's theorem. In this case, the asymptotic performance can be predicted by the error-oor bounding analysis in Chapter 2.

5.3.3 Impact of Signaling Schemes We now consider the impact of signaling schemes on system performance for a given component code.

Proposition: The output SINR of the LMMSE estimator is a monotonously decreasing function of the average variance v at the output of the soft mapper.

Proof : From the matrix derivation property dA−1 = −A−1 dAA−1 , we have ¡ ¢ dΓ[n] d (h[n])† (R[n])−1 h[n] = dv dv dR[n] = −(h[n])† (R[n])−1 (R[n])−1 h[n] dv X † =− (h[n]) (R[n])−1 (h[n0 ])† h[n0 ](R[n])−1 h[n]

(5.27a) (5.27b) (5.27c)

n0 6=n

≤ 0.

(5.27d)

Therefore, the SINR is a monotonously decreasing function of the average symbol

5.3. Evolution Analysis

119

Figure 5.4: Comparison of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICM-ID with 16-QAM MSP signaling over Proakis B channel. The dashed and solid curves represent the evolution and simulation results, respectively. The number of iterations It = 10. variance v . Thus, v should be minimized when the LMMSE estimation performance is concerned. On the other hand, v is determined jointly by the DEC feedback and the signaling scheme. As proven in Chapter 4, with i.i.d. DEC feedback LLRs,

SCM achieves the minimum v among all possible signaling schemes. This means that SCM maximizes the output SINR for the LMMSE estimator. In this respect, many commonly known signaling schemes (such as the QAM with MSP mapping [24]) are suboptimal. It will be shown later that the QAM with Gray mapping yields performance close to that of SCM. In multi-path channels, the ISI is the dominant factor that aects performance. The LMMSE estimation is used to mitigate this interference. We expect that SCM can lead to improved overall performance in such channels.

5.3.4 Extension to MIMO Channels We now consider an MIMO multi-path channel with nt transmit antennas and nr receiver antennas. In this case,

 H0 · · · H2 H1  H1 H0 · · · H2    H =  .. .. . . ..   . . .  . HN −1 · · · H1 H0 

5.3. Evolution Analysis

120

Figure 5.5: Performance of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICM-ID with 16-QAM signaling schemes (Gray, Mixed and MSP) over random multi-path channels. It = 10. Information block length is 8192. where Hn , ∀n, is a nr × nt matrix which characterizes the nth tap of the MIMO multipath channel. The dierence between the MIMO and single-antenna cases is that, due to the random fading eect, the signal from dierent transmitter antennas may undergo dierent fading states. Then the LMMSE estimator outputs have dierent SINRs for the nt transmitter antennas. Thus, to characterize the performance, we need to track nt SINR values {Γn }, where Γn denotes the SINR of the LMMSE estimates for transmit antenna n. Similarly to (5.22), we can compute {Γn } as

Γ ≡ diag(Γ1 , Γ2 , · · · , Γnt ) = U (I − vU )−1 where

(5.28)

N −1 1 X U= G[n]† (vG[n]G[n]† + N0 I)−1 G[n], N n=0

with

G[n] =

N −1 X l=0

Hl exp(−

i2πnl ) N

being an nr × nt matrix that characterizes the frequency response of the MIMO channel. For the evolution procedure, the soft mapper and decoder parts are the same as those discussed earlier. The output SINR of the LMMSE estimator is now characterized by (5.28) instead of (5.22). Consider the case where the symbols from dierent transmit antennas are mapped from randomly interleaved

5.4. Numerical Results

121

Figure 5.6: FER of single-user MIMO systems with SL-SCM and BICM-MSP over single-path channels. The receiver has nr = 4 antennas. 16-QAM is always assumed. (For SL-SCM, {βk } = {1, 2, i, 2i}.) Information block length = 2048. It= 10. System throughputs R = 2 for nt = 1 and R = 8 for nt = 4. coded bits from one common codeword. At the receiver, the demapper outputs for the nt transmit antennas should be de-interleaved before entering the DEC. Thus, the decoder inputs are characterized by the following average mutual information nt 1 X Iλ = TDEM (Γm , Iγ ). nt m=1

(5.29)

The evolution analysis and the impact of signaling schemes on performance are similar to those in the single-antenna case. We omit the details.

5.4 Numerical Results In this section, we present simulation results to verify the above analysis. We always assume a rate-1/2, 4-state convolutional code (5, 7)8 and SL-SCM schemes. Here, the weighting factors {βk } of SCM are chosen in an ad hoc manner and they may be optimized to further improve the performance.

5.4.1 Single-User Multi-path Channels We rst consider single-user single-antenna schemes over multi-path channels. Fig. 5.4 presents the BER of SL-SCM and BICM-ID systems over Proakis B channel [a 3-tap multi-path channel with tap coecients (0.407, 0.815, 0.407)]. The

5.4. Numerical Results

122

Figure 5.7: BER of SL-SCM with {βk } = {1, 3/2, i, 3/2i} and BICM-ID with 16-QAM MSP mapping over quasi-static MIMO multi-path channels. R = 4. It = 20. Information block length is 8192. information block length is 65536. The system throughput R = 2 bits/channel use. For the SCM, K = 4, {βk } = {1, i, 3/2, 3/2i}. The simulation and evolution results are compared. It is seen that SCM signicantly outperforms the MSP signaling in Proakis B channel with iterative LMMSE detection. This demonstrates that the advantage of SCM in maximizing the output SINR of the LMMSE estimator can indeed lead to signicant performance improvement. It is also shown that the evolution results agree well with the simulation results. Note that for an AWGN channel, the ESE reduces to a symbol-by-symbol APP demapper, since there is no ISI. Then the maximum SINR property has no eect on performance in this case and the MSP signaling can outperform SCM, as shown in Fig. 5.4. In the above, we have assumed a xed multi-path channel. We now consider a quasi-static multi-path channel where the channel matrix H remains constant in a frame period but changes randomly from one frame to another. Fig. 5.5 presents the average performance over a 16-tap random multi-path channel where each tap coecient is a zero-mean complex Gaussian random variable with variance 1/16. The information block length is 8192. We observe that SCM is advantageous in this case. The 16-QAM Gray signaling leads to performance close to that of SCM, agreeing well with the variance results in Fig. 4.8. The prediction results based on the evolution analysis are also quite close to the simulation results.

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123

Figure 5.8: FER for Q = 2-user MIMO systems with SL-SCM and BICM-ID over multi-path channels. For SL-SCM, {βk } = {1, 2, i, 2i}. For BICM-ID, 16-QAM is used. Information block length = 1024 for each user. It = 20.

5.4.2 MIMO Channels We also consider MIMO systems where each user employs a BLAST-type scheme [1]. Each coded bit sequence is randomly interleaved and mapped onto a constellation X . The resultant symbol sequence is serial-to-parallel converted and then transmitted in parallel from dierent antennas. Dierent users are distinguished by their interleavers, following the principles of IDMA [77]. Let r be the rate of the binary encoder, Q the number of users and nt the number of transmit antennas per user. We dene the system throughput as

R = rQKnt .

(5.30)

For MIMO channels, we assume that the channel coecients are i.i.d. and quasistatic. Fig. 5.6 compares the frame-error-rate (FER) for single-user MIMO systems with SCM and BICM-MSP over single-path channels. The number of receiver antennas is xed to nr = 4. We consider nt = 1 and 4 transmit antennas. With nt = 1, the LMMSE receiver reduces to an APP demapper. The 16QAM MSP signaling used in Fig. 5.6 is optimized for the MAP detection [24] and, indeed, we can see from Fig. 5.6 that BICM-MSP outperforms SCM in this case. Note that the advantage of SCM w.r.t. LMMSE detection is not relevant in this case. When nt = 4, SCM can signicantly outperform BICM-MSP, which is mainly

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124

Figure 5.9: FER of Q = 2-user MIMO systems employing dierent SL-SCM schemes for R = 8 and R = 12. For curves marked by (1, 2) , {βk } = {1, i, 2, 2i}. The cases of other curves are similar. It = 10. Information block length is 8192 and 12288 respectively for R = 8 and R = 12. attributed to the maximum SINR property of SCM discussed in Section 5.3.3. Interestingly, when nt increases from 1 to 4, the SCM performance improves, thanks to the diversity gain oered by a larger nt . In the meanwhile, the BICMMSP performance degrades, indicating that iterative LMMSE detection does not perform well with the MSP signaling. Fig. 5.7 shows the BER of SCM and BICM-ID with 16 QAM MSP mapping over 2×2 multi-path channels with L = 4 taps. It is seen that SCM performs better again. The simulation and evolution results agree well, verifying the eectiveness of the evolution technique for MIMO channels. Fig. 5.8 presents the FER for a Q = 2-user MIMO system over multi-path channels with L = 4 taps. Each user has nt = 2 transmit antennas. The receiver has nr = 4 antennas. 16-QAM is assumed. The performance with the ve dierent signaling schemes in Fig. 4.4 is compared. It is seen that SCM yields the best performance in this case. The performance gap between SCM and BICM-MSP is more than 3 dB at FER = 10−4 . Fig. 5.9 shows the FER for Q = 2-user MIMO system with dierent SCM schemes. We set nt = 2, nr = 4, and assume L = 4-path channels. Two system throughputs R = 8 and 12 are considered. For R = 12, the rate-1/2 convolutional code is punctured to achieve a rate of r = 3/4. It is seen that for R = 8, the SCM scheme with equal power allocations {|βk |} = {1} performs better. However,

5.5. Summary

125

when the rate is increased to R = 12, the QAM-type scheme with {|βk |} = {1, 2} provides a better choice. This is because the scheme with equal {|βk |} leads to poor convergence of the iterative decoding when rate is high.

5.5 Summary We have shown that the iterative LMMSE detection can provide a solution with good tradeo between performance and complexity for high-rate coded modulation schemes over linear vector channels. Its performance can be predicted by a fast evolution method. We show that SCM can maximize the output SINR of the LMMSE estimator. Consequently, SCM can outperform other conventional signaling schemes over multi-user/multi-antenna/multi-path channels where the interference dominates performance. The simulation results agree well with the analysis.

Chapter 6

OFDM-IDMA 6.1 Introduction Consider a multi-user communication system over multi-path channels where the MAI and ISI are major sources of impairments to performance. As discussed in Chapter 5, in the single-carrier case, the iterative time-domain LMMSE detection can be applied to mitigate the MAI and ISI. However, when the number of paths is very large, the time-domain technique can be costly. Recently, it has been shown that this issue can be eciently resolved by using OFDM-IDMA [68]. In OFDMIDMA, ISI is treated by the cyclic prexing technique in OFDM [55], and MAI is handled by iterative multiuser detection (MUD) with IDMA [64]. Compared with conventional multi-carrier schemes, OFDM-IDMA has several noticeable advantages such as low-cost receiver, robustness against fading and exible rate adaptation. This chapter is concerned with the analysis and design of OFDM-IDMA. We rst investigate the information-theoretic advantages of non-orthogonal transmissions in fading multiple-access channels. We show that non-orthogonal schemes can achieve signicant performance improvement over orthogonal ones. Such improvement is referred to as multi-user gain (MUG) because it is only achievable through MUD. This provides a motivation for optimizing the performance of OFDM-IDMA systems according to theoretical prediction. We then turn attention to some practical issues. We will outline an SNR evolution technique to analyze the performance of OFDM-IDMA, with which the BER can be quickly predicted. We will apply this technique to system design and optimization. We will demonstrate the following noticeable advantages for OFDM-IDMA: First, the clipping technique can be used in OFDM-IDMA to reduce the PAPR and the performance loss can be eciently recovered. Second, power allocation can be applied to deliver the MUG promised by theoretical analysis. Third, OFDM-IDMA is robust against frequency-selective fading when low-rate coding is used. Finally, SCM can be integrated into OFDM-IDMA to maintain high throughput. Numerical examples are provided to conrm these properties.

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127

6.2 Motivation Before going into the details, we rst provide in this section a compelling motivation for the introduction of non-orthogonal transmission in fading environments. We focus on the minimum transmitted-sum-power (MTSP) to support reliable communications. We will show that non-orthogonal options can signicantly outperform orthogonal ones regarding MTSP. This advantage is referred to as MUG because it is achievable only through MUD. Consider a Q-user multiple-access system over quasi-static at-fading channels. We assume that the channel gains {hq } are i.i.d. and perfectly known at the transmitters and receiver. For simplicity, suppose that the system sum rate is R bits/s/Hz and the rate of each user is R/Q. Let the received signal in a multiple-access channel be written as

y=

Q X

(6.1)

hq xq + n

q=1

where xq is user-q 's transmit signal and n is an AWGN with variance N0 . We assume that {xq } are independent, Gaussian-distributed. We st consider the AWGN channel (i.e., hq = 1, ∀q ). Suppose that the successive interference cancelation (SIC) strategy [99] is applied at the receiver. We will use the descending decoding order indexed on q with xQ decoded rst and x1 last. Denote by ϑq the received power for user q . From information theory [99], the minimum received power for user-q with SIC is given by R Q

ϑq = (2 −1)(N0 +

q−1 X

R

ϑi ) = N0 (2 Q −1)2

R(q−1) Q

.

(6.2)

i=1

Now consider a at-fading channel with xed {hq }. We control the transmit power of user-q to R

poptimal q

ϑq N0 (2 Q − 1)2 = = |hq |2 |hq |2

R(q−1) Q

(6.3)

so as to maintain the same received power prole {ϑq } as that in (6.2) for the AWGN channel. Given {|hq |2 }, a permutation of {|hq |2 } can be obtained by reindexing its elements. For example, {|h1 |2 = 1, |h2 |2 = 2} is a permutation of

{|h1 |2 = 2, |h2 |2 = 1}. According to [99], among all possible permutations of

6.2. Motivation

128

{|hq |2 }, the minimum of the following sum of transmit powers Q X

R Q

poptimal = N0 (2 − 1) q

q=1

Q X 2 q=1

R(q−1) Q

|hq |2

(6.4)

is achieved when

|h1 |2 ≤ |h2 |2 ≤ · · · ≤ |hQ |2 ,

(6.5)

where (6.5) is referred to as the optimal decoding order. It can be shown that in this case, (6.4) gives the theoretical limit of MTSP. It is interesting to compare (6.4) with the result of an orthogonal strategy in which the signals from the Q users are orthogonal to each other and they equally share the channel resources. Let the (long-term average) rate per user be R/Q (so sum rate = R) and, according to channel capacity formula, the minimum transmit power is ¡ ¢ N0 2R − 1 orthogonal pq = . Q|hq |2

(6.6)

For example, consider time-division multiple-access (TDMA). Let the instantaneous rate be R in the active time slot for a user and so the long-term average rate is R/Q per user. In this case, the minimum instantaneous transmit power is N0 (2R − 1)/|hq |2 in the active time slot for a user and its average over Q slots is given in (6.6). It can be shown that (6.6) holds for other orthogonal schemes. Based on (6.6), the MTSP for an orthogonal scheme is Q X q=1

Q

porthogonal q

N0 (2R − 1) X 1 = . Q |hq |2 q=1

(6.7)

MUG refers to the dierence between (6.4) (with optimal decoding order) and (6.7). Intuitively, such an advantage is achieved by matching the elements of two sets {|hq |2 } and {ϑq } in a large-to-large/small-to-small manner. Such ordered matching is possible only in a non-orthogonal environment. There is no MUG in an AWGN channel where |hq |2 = 1, ∀q . Hence, MUG comes from the nearfar eect in fading channels together with the matching strategy. Due to the non-orthogonal nature, MUD is necessary at the receiver for optimal detection. The above discussions assume xed channel gains {|hq |2 }. Practical {|hq |2 } are random variables. Therefore, we are interested in the MTSP averaged over the distribution of {|hq |2 }. Fig. 6.1 shows average MTSP versus R for dierent multiple access schemes over at-fading channels, where OFDMA (orthogonal frequency-division multiple-access) is used as a representative orthogonal scheme.

6.3. System Model

129

Figure 6.1: Average MTSP versus system sum rate R for dierent multiple access schemes.

Q = 4 users are uniformly distributed in a hexagon cell. The channel eect includes path loss (the fourth law), lognormal fading (standard deviation = 8 dB) and Rayleigh fading. The distance from the farthest corner of the cell to the base station is 1. The lognormal fading is scaled such that its mean equals 1. The outage probability is 1%. We can see that there is a signicant gap (i.e., MUG) between the theoretical limit and the OFDMA performance. This gap increases with Q. Note that the theoretical limit can only be achieved by non-orthogonal transmission schemes with MUD [99]. Without MUD, performance degrades signicantly, as seen from the curves in Fig. 6.1 labeled as CDMA with single-user detection. Only at fading is considered in Fig. 6.1. In the case of frequencyselective fading, OFDMA performance can be improved by water-lling while theoretical limit can be achieved by multi-user water-lling over a non-orthogonal scheme. The trend is similar to that shown in Fig. 6.1, so the details are omitted. From Fig. 6.1, MUG becomes signicant when rate increases. This provides the motivation for the OFDM-IDMA scheme presented below, in which the orthogonality of OFDM is employed in suppressing ISI while the non-orthogonality of IDMA is explored in realizing MUG. The overall system is non-orthogonal.

6.3 System Model 6.3.1 Transmitter Principles A Q-user OFDM-IDMA system is illustrated in Fig. 6.2. The system structure follows the principles proposed in [68]. For user-q , the information data are

6.3. System Model

130

FEC-encoded into cq = {cq [n]}. The sequence cq is interleaved by a user-specic interleaver πq and then mapped to a complex vector Xq = [Xq [0], · · · , Xq [N − 1]]> using QPSK, where N is the number of sub-carriers. Each dimension of Xq [n] (denoted by XqRe [n] or XqIm [n]) represents a bit in cq . Then {Xq [n]} are modulated onto sub-carriers using the IDFT. The resultant signal is over-sampled into

xq = [xq [0], · · · , xq [LN − 1]]> , where L is the over-sampling factor and N −1 2πmn 1 X xq [m] = √ Xq [n]ei LN . LN n=0

As discussed in Chapter 4, the time-domain signal in OFDM has a high PAPR. We adopt clipping to suppress the PAPR. The clipping rule given by (4.2) is applied to each {xq [m]}. The clipped symbols {g(xq [m])} are then band-pass ltered and transmitted. Following Chapter 4, we can model clipping by a linear system as (6.8)

g(xq [m]) = κxq [m] + dq [m]. Here,

κ ≡ E[x∗q [m]g(xq [m])]/E[|xq [m]|2 ] is a constant. (6.9)

dq [m] ≡ g(xq [m]) − κxq [m]

is the clipping noise which is statistically uncorrelated with xq [m]. For convenience, we will refer to Xq and xq as the frequency- and time-domain transmit signal vectors, respectively. From the above discussions, we have

xq = F † Xq ,

(6.10)

Xq = F x q

√ / LN . (Note: Eqn. (6.10) can be computed using the fast Fourier transform technique.) The where F is an N × LN matrix with (n, m)th entry given by ei

2πmn LN

time-domain clipping noise vector is dened as dq ≡ [dq [0], · · · , dq [LN − 1]]> and its frequency-domain counterpart is given by Dq ≡ [Dq [0],· · · ,Dq [N − 1]]> = F dq .

6.3.2 Receiver Principles Assume perfect synchronization. The core of the OFDM-IDMA receiver consists of an ESE and Q APP DECs, as shown in Fig. 6.2. After the OFDM demodulation, the frequency-domain received signal can be

6.3. System Model

131

Figure 6.2: Transmitter/receiver structure for OFDM-IDMA. The QPSK modulation, cyclic prex insertion and removal for OFDM are not shown for simplicity. ENC and DEC denote encoder and decoder, respectively. represented as

r[n] = κ

Q X

Hq [n]Xq [n] +

q=1

Q X

Hq [n]Dq [n] + Z[n],

(6.11)

q=1

where Z[n] is a complex AWGN with variance N0 , Dq [n] represents the clipping noise from user q , and Hq [n] is the channel coecient related to the nth sub-carrier for user q . Note that r[n] in (6.11) is the signal after the OFDM demodulation. From the IDMA layer, the combination of the OFDM layer and physical channel can be viewed as a bank of N parallel sub-channels, each corresponding to an OFDM sub-carrier. With this view, ISI has already been resolved by the OFDM layer and we will focus on the MAI treatment in the IDMA layer. Focusing on Xq [n], we can rewrite (6.11) as

r[n] = κHq [n]Xq [n] + ξq [n] where

ξq [n] = κ

Q X k=1,k6=q

Hk [n]Xk [n] +

Q X

Hk [n]Dk [n] + Z[n]

(6.12)

(6.13)

k=1

is the distortion component in r[n] with respect to Xq [n]. From the central limit theorem, ξq [n] can be approximated by a complex Gaussian random variable when

Q is large. (For simplicity, we assume that its real and imaginary parts have the same variance.) The statistics of Xk [n] can be estimated from the DEC feedback. Assume that E[Dk [n]] and V[Dk [n]] are available, where the variance is dened as V[x] = E[|x|2 ] − |E[x]|2 . Then the iterative detection procedure in [64] can be modied as follows:

6.4. Performance Analysis and Optimization

132

(i) The ESE computes {E[ξq [n]], V[ξq [n]]} based on (6.13) as Q X

E[ξq [n]] = κ

Q X Hk [n]E[Xk [n]] + Hk [n]E[Dk [n]]

k=1,k6=q

V[ξq [n]] = |κ|

2

Q X

(6.14)

k=1

|Hk [n]|2 V[Xk [n]]

k=1,k6=q

+

Q X

|Hk [n]|2 V[Dk [n]] + N0 .

(6.15)

k=1

The extrinsic LLR about XqRe [n] is given by

à λRe q [n] ≡ ln =

Pr(r[n]|XqRe [n] = +1) Pr(r[n]|XqRe [n] = −1)

!

4 |κHq [n]| Re(e−iθq [n] (r[n] − E[ξq [n]])) V [ξq [n]]

(6.16)

where we have assumed κHq [n] = |κHq [n]|eiθq [n] , and “Re(·) denotes the real part of a number. Similarly, we can compute λIm q [n]. (ii) Taking the ESE outputs as inputs, the DECs perform APP decoding. The DEC feedbacks are then used to rene the means and variances of Xq [n] and

Dq [n]. Return to Step (i) for the next iteration. The remaining problem now is to nd E[Dq [n]] and V[Dq [n]]. One way is simply ignoring them [i.e., setting them to zeros] but this may incur considerable performance loss. A better way is to estimate them iteratively. The treatment is the same as that in Chapter 4 for single-user OFDM systems, and so we omit the details.

6.4 Performance Analysis and Optimization We now outline an analysis technique for the above scheme. It is an extension of the evolution techniques for the single-user SCM schemes we discussed in Chapter 2-5. For a particular user in OFDM-IDMA, the desired signal suers from a number of interference signals from independent frequency selective fading channels. We will develop analysis techniques to treat this issue. We will also demonstrate the application of the fast analysis technique in system optimization so as to achieve the MUG discussed earlier.

6.4. Performance Analysis and Optimization

133

6.4.1 Performance Prediction Assume that {Hq [n], ∀q, ∀n} are mutually independent. (This assumption follows the fully interleaved fading modeling in [55], which is approximately true if interleaver length is suciently long.) From (6.12) and (6.16), we have Im λRe q [n] + iλq [n] ≈

4 |κHq [n]| (|κHq [n]|Xq [n]+ e−iθq [n] (ξq [n]−E[ξq [n]])). (6.17) | {z } V[ξq [n]] δq [n]

According to the discussions in Section 6.3.2, the interference signal δq [n] in (6.17) can be seen as a zero-mean Gaussian noise with variance V[ξq [n]]. Note that due to the uctuations of channels, V[ξq [n]] is dierent for dierent n and is related to the values of {Hq [n]}. Thus, it is dicult to accurately characterize the detection process. We consider the following approximation. The key is to model δq [n] as an independent sample of an AWGN with power

Iq ≡ E[V[ξq [n]]] =

Q X

ηk (|κ|2 vx,k + vd,k ) + ηq vd,q + N0 ,

(6.18)

k=1,k6=q

where ηk ≡ E[|Hk [n]|2 ] is user-k 's average channel gain, while vx,k ≡ E[V[Xk [n]]] and vd,k ≡ E[V[Dk [n]]] are the average variances of the signal and clipping noise from user k , respectively. Then, the performance of the ESE can be approximately characterized by the SNR with respect to xq [n] in (6.17), i.e.,

Γq ≡

E[|κHq [n]Xq [n]|2 ] 2|κ|2 ηq = E[|ξq [n] − E[ξq [n]]|2 ] Iq

(6.19)

where we have assumed that all users have the same transmit power E[|Xq [n]|2 ] =

2. The average variance vx,k in (6.18) characterizes the uncertainty in user-k 's signal after the APP decoding. Thus, its value depends fully on the DEC inputs and the FEC code employed. The case for the average variance vd,k of the clipping noise is similar, but it is also related to the clipping parameters. Given the coding/clipping schemes, vx,k and vd,k can be characterized by the following two functions:

vx,k = f (Γk ),

vd,k = c(Γk ).

(6.20)

Similarly, we can characterize the relationship between BER and Γk by a function as

BERk = b(Γk ).

(6.21)

6.4. Performance Analysis and Optimization

134

The three functions in (6.20) and (6.21) can be pre-simulated by applying the Monte-Carlo method to a single-user system. Notice that f (·) and g(·) in (6.20) and (6.21) are similar to the two tables dened by (16) and (17) in [64]. However, there is a subtle dierence. In [64], the two tables can be generated by presimulation in an AWGN channel. Here the underlying OFDM layer is modeled by a bank of parallel channels with individual fading coecients. [See (6.11).] The channel for pre-simulation should be modeled accordingly. Assume that these (it)

tables are available. Let It be the total number of iterations. Denote by Γq

the

value of Γq after it iterations. The SNR evolution procedure is listed below: (i) Initialization : Set Γq = 0, q = 1, 2, · · · , Q. (0)

(ii) Recursions : For it = 0, 1, · · · , It − 1, update the SNR as

Γ(it+1) = q

2|κ|2 ηq Q P

,

(6.22)

(it) (it) (it) ηk (|κ|2 f (Γk )+c(Γk ))+ηq c(Γq )+N0 k=1,k6=q

q = 1, 2, · · · , Q. (iii) Termination : Find the BER by substituting the nal values of {Γq } into (6.21). The eectiveness of this technique will be demonstrated by the numerical examples in Section 6.5.

6.4.1.1 MUG and Power Allocation The information-theoretic analysis of MUG in Section 6.2 is for capacity-achieving codes and optimal detection. For practical OFDM-IDMA schemes, rigorous theoretical analysis of MUG is dicult due to the suboptimality of the FEC, the frequency selectiveness of the channel and the non-linear nature of the iterative receiver. However, the underlying rationale in achieving MUG suggests that it can be realized by practical OFDM-IDMA with proper design. As shown in Section 6.2, the MUG is achieved by ordered matching the sets of received signal power and channel gain. We therefore consider a two-step strategy which rst optimizes the received power prole and then applies ordered matching to allocate the transmit power. The SNR evolution technique can be used as the underlying tool to accelerate the search of received power prole. We will omit the details because the principle is very similar to that discussed in [64]. The results presented in Section 6.5 are all based on this strategy.

6.4. Performance Analysis and Optimization

135

Figure 6.3: Transmitter structure of OFDM-IDMA with SCM, where SP represents spreader.

6.4.2 Spreading and SCM OFDM-IDMA can work with any FEC coding. However, low-rate coding is particularly attractive because it can provide more robust performance in frequencyselective channels. An even simpler approach is to introduce spreading after FEC coding, which is similar to OFDM-CDMA [55]1 . With spreading, each bit is spread as chips over dierent OFDM sub-carriers. At the receiver, the information is collected from these sub-carriers. This results in diversity that averages out the fading eect. Spreading reduces throughput but this can be compensated by SCM. With SCM, each user can transmit more than one coded sequence using a multi-layer encoder, as in Fig. 6.3. In this way, high-rate transmission and very exible rate adjustment can be achieved. For example, dierent number of layers can be assigned to dierent users according to their channel conditions. The receiver principle is similar to that in Section 6.3.2 (since a layer in Fig. 6.3 is equivalent to a user in Fig. 6.2) and the details are omitted here. Later we will show that, given the transmission rates of the users, the introduction of spreading and SCM can yield noticeable performance improvement. Receiver complexity increases when more layers are involved. This is mainly due to the ESE cost that increases linearly with the spreading length. The cost related to de-spreading is marginal and the cost related to DEC, which usually dominates the receiver complexity, is independent of spreading. Overall, the cost increase is quite moderate. The above SCM-based scheme has an additional advantage regarding the clip1 The

purpose of spreading in OFDM-IDMA is purely for diversity gain rather than user separation as in OFDM-CDMA [55]. OFDM-IDMA relies on chip-level, user-specic interleaving to distinguish users.

6.5. Numerical Results and Discussions

136

Figure 6.4: Performance of OFDM-IDMA with and without clipping. The spreading length is S = 8. The rate of each user is 1/8. Q = 16. R = 2. The information block length for each user is 512. All users have the same transmit power. ping issue that it is more robust to clipping eect than other alternatives, such as OFDMA based on BICM-ID. The reason has been discussed in Chapter 4.

6.5 Numerical Results and Discussions We set the number of sub-carriers to N = 256, the over-sampling factor L = 4 and the clipping ratio CR = 0 dB. With the clipping and ltering technique, the PAPR of the transmit signal can be reduced to less than 5.9 dB for 99.9% of the OFDM blocks, in contrast to 11.3 dB in the unclipped case. For simplicity, the overhead due to cyclic prexing is ignored below. Unless otherwise stated, we assume a fully interleaved Rayleigh fading channel [55], where {Hq [n]} are i.i.d. complex-Gaussian with variance 1. The rate-1/2 convolutional code (23, 35)8 is always used for the ENC. The number of iterations is 10. We rst examine the eciency of the SNR evolution technique. Fig. 6.4 shows the performance of a Q = 16-user OFDM-IDMA system. (The parameters are listed in the caption.) It is seen that the evolution and simulation results agree with each other. It is also seen that the iterative detector outlined in Section 6.3.2 can eciently recover the performance loss due to clipping. We next show the performance of OFDM-IDMA schemes with SCM. Let Q = 2 and rate per user = 1 bit/s/Hz. Two dierent spreading lengths, that is, S = 1 and S = 8, are considered. In order to support the required transmission rate, the layer numbers for each user are set to 1 and 8 for S = 1 and 8, respectively. The

6.5. Numerical Results and Discussions

137

Figure 6.5: Performance of two-user OFDM-IDMA schemes with dierent spreading factors S . Table 6.1: Received Power Prole of the OFDM-IDMA Systems in Fig. 6.6. Layer Number Power Level (dB)

12 0

5 3.1673

5 5.5427

2 6.3345

information block lengths for each layer are 4096 and 512, respectively for S = 1 and 8 such that the interleaver lengths are the same. All layers have the same transmit power. Fig. 6.5 demonstrates that the scheme with S = 8 signicantly outperforms that with S = 1 in achieving a low BER (e.g., 10−5 ). This conrms the advantage of spreading and SCM in OFDM-IDMA systems. Fig. 6.6 demonstrates the impact of MUG as discussed in Section 6.2. We assume that the transmitters have knowledge of the path loss and log-normal fading factors (the same as those in Fig. 6.1) but have no knowledge of the Rayleigh fading coecients. We set S = 8 and consider a sum rate R = 3 so that a total of 24 layers are involved. We consider dierent numbers of users. For simplicity, we assume that all users employ the same number of layers. Thus, each user has 3 layers when Q = 8, and 6 layers when Q = 4, and so on. The linearprogramming method [64] is used for optimizing the received power prole and the results are listed in Table 6.1. The ordered matching in (6.3) and (6.5) is applied to the received power prole in Table 6.1 to determine the transmit power prole. For comparison, we consider OFDMA using a rate-R = 3 BICM-ID scheme with the 16-QAM MSP signaling [93]. Interleaved sub-carrier allocation [51] is assumed for OFDMA. Clipping is also applied to OFDMA to reduce the PAPR and an iterative detector similar to that discussed in Chapter 4 is applied to combat the clipping eect. The oversampling factor L = 4 and clipping ratio CR = 0 dB. The

6.6. Summary

138

Figure 6.6: Performance of OFDM-IDMA and OFDMA with clipping in an uplink scenario. power of the channel noise is set to 1 (i.e., 0 dB). The same maximum latency is assumed for the OFDM-IDMA and OFDMA systems with dierent number of users Q. Thus, the information block length per user in OFDMA decreases as Q increases. We can show that the two schemes have comparable complexities. From Fig. 6.6, OFDM-IDMA outperforms OFDMA, especially for a large Q. When Q = 4, OFDM-IDMA can achieve about 8 dB gain compared with OFDMA. There are several reasons for this. First, the non-orthogonal OFDM-IDMA scheme can realize the MUG. Second, OFDM-IDMA with spreading can achieve signicant diversity gains. Third, BICM-ID-based OFDMA is more sensitive to the clipping eect than OFDM-IDMA. Finally, BICM-ID performance degrades when block length reduces.

6.6 Summary In this chapter, we consider OFDM-IDMA with iterative decoding for uplink transmissions. We have shown that OFDM-IDMA can (i) alleviate the PAPR problem commonly suered by OFDM-based schemes; (ii) deliver signicant MUG compared with other orthogonal alternatives; (iii) provide robust communication in frequency-selective channels; and (iv) support high single-user throughput.

Chapter 7

Conclusions and Future Work 7.1 Conclusions In this thesis, we have made a comprehensive study on the theoretical and practical aspects of SCM over various channels. We summarize our major ndings as follows.

• We begin with the relatively simple memoryless channels. We analyze the maximum achievable rates, error bounds and convergence of iterative decoding for both SL- and ML-SCM. The results can provide fast performance evaluation and useful design guidelines for SCM with dierent types of component codes.

We show that ML-SCM is advantageous when capacity-

approaching codes are used or faster convergence of iterative decoding is required. When more practical component codes are considered, SL-SCM can achieve a lower error oor. The impact of system parameters including weighting factors, component codes, interleaving length, is also studied.

• We consider the PAPR issue of high-rate single-carrier SCM schemes with a large number of layers. We investigate the clipping method for PAPR reduction and devise several low-cost iterative detection algorithms. We show that such algorithms can eciently recover the performance loss due to clipping. We also show that SCM can achieve signicant shaping gain for AWGN channels and diversity gain for fading channels that can greatly improve performance.

• We apply SCM in both single- and multi-user OFDM systems with clipping. A novel soft compensation technique is proposed, which can outperform conventional clipping noise cancellation techniques for OFDM. Furthermore, by rigorous proof, we demonstrate that SCM-based OFDM schemes have significant advantages over schemes based on other alternative coded modulation schemes such as BICM-ID. The advantages are very evident when the transmission rate is high or a low PAPR is required. Ecient analytical methods are derived for SCM-based OFDM schemes with clipping and soft compensation.

7.2. Future work

140

• We also consider SCM over more complicated multi-user/multi-antenna /multipath channels which may cause serious interferences. An iterative LMMSE receiver is applied. The focus is on the performance analysis and the comparison with other alternative coded modulation schemes. We show that a semi-analytical evolution method can characterize the iterative detection process accurately. We demonstrate by numerical results that SCM can signicantly improve performance in hazardous communication environments. In summary, by both analytical and simulation studies, we have shown that SCM is a very attractive option for single- and multi-user, single- and multicarrier communication systems with low complexity, robust performance and high exibility.

7.2 Future work In the following, we list several possible topics for future work.

• Coded Modulation for More General Channels. In this thesis, we mainly focus on the AWGN and fully-interleaved channels and allow large codeword length. In such cases, the uctuations of channel states can be averaged out and analysis and design can be simplied. In practical delaylimited applications, the channel can be modeled as block-fading where the channel state changes independently in a nite number of blocks. It is shown in [34] that the performance (measured by the outage capacity) in such channels can be improved by using large constellations. The work in this thesis shows that SCM oers a low-complexity approach to realizing large constellations. Therefore, SCM can be an attractive option for block-fading channels.

• Adaptive Coding and Modulation (ACM). We have always assumed that the channel state information (CSI) is unknown to the transmitter in this thesis. When CSI is available at the transmitter, we may apply ACM to increase the throughput. For example, for a multi-path channel, the channel capacity is achieved by waterlling when CSI is available [21]. ACM is necessary to realize this principle. Current treatments to this issue are based on conventional QAM-type constellations. SCM may provide a more convenient approach since its encoder/decoder structures are the same for dierent rate and constellations.

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141

• SCM in Cognitive Radio. Another possible topic is in cognitive radio [43]. It is shown in [54] that dirty paper coding (DPC) and interference cancelation play important roles in achieving the capacity of cognitive radio channels. Interestingly, SCM provides an eective means of realizing DPC [10], and, furthermore, it provides a good option for iterative interference cancelation. We expect that SCM may serve as a useful building block for cognitive radio.

• Channel Estimation. We have assumed perfect channel state information (CSI) at the receiver in this thesis. In practical time-varying environments, the CSI should be estimated and tracked dynamically. It is of interest to incorporate the channel estimation function into the SCM receiver and study the impact of estimation errors on performance. The information data and CSI may be jointly estimated in an iterative manner and the performance may be improved as iteration proceeds. In order to achieve high accuracy in channel estimation, proper training schemes and channel tracking algorithms should be designed.

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