Space-Time Block Codes for Wireless Systems - The ...

COMMUNICATION SCIENCES
INSTITUTE
“Space-Time Block Codes for Wireless Systems:
Construction, Performance Analysis, and Trellis
Coded Modulation”
by
Jifeng Geng
CSI-04-05-01
USC VITERBI SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
ELECTRICAL ENGINEERING — SYSTEMS
LOS ANGELES, CA 90089-2565

SPACE-TIME BLOCK CODES FOR WIRELESS SYSTEMS:
CONSTRUCTION, PERFORMANCE ANALYSIS, AND TRELLIS CODED
MODULATION
by
Jifeng Geng
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
May 2004
Copyright 2004
Jifeng Geng

Dedication
This dissertation is dedicated to my family for their everlasting support and love.
ii

Acknowledgements
I would like to thank my advisor Urbashi Mitra, who forgoes many of my shortcomings
and provides valuable support and guidance throughout this work.
iii

Contents
Dedication
Acknowledgements
List Of Tables
List Of Figures
Abstract
ii
iii
vi
vii
x
1 Introduction 1
2 Space-Time Block Codes in Multipath CDMA Systems 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The Uplink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 The Uplink System Model . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 The Uplink Chernoff Bound and Code Design Criteria . . . . . . . 15
2.2.3 The Correlated Codeword Sequence Difference Matrix Φ . . . . . . 17
2.2.4 Code Design Principles . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5 The Effect of Multipath . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The Downlink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 The Downlink Model . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Joint Maximum Likelihood Decoder . . . . . . . . . . . . . . . . . 28
2.3.3 Single user ML-based Decoder . . . . . . . . . . . . . . . . . . . . 30
2.4 Non-spread systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Optimal Minimum Metric Codes . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 OMM Codes for the Uplink . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 OMM Codes for the Downlink . . . . . . . . . . . . . . . . . . . . 45
2.6 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.2 Sensitivity to Correlation . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.3 Suboptimal Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
iv

3 On Suboptimal Linear Decoders for Space-Time Block Codes 56
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Uplink Signal Model for Spread Systems . . . . . . . . . . . . . . . . . . . 57
3.3 Linear Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 The Mapper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 The Effect of Large Number of Receive Antennae . . . . . . . . . . . . . . 65
3.6 Fisher’s Information Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.7.1 The BLUE and LS Decoders . . . . . . . . . . . . . . . . . . . . . 70
3.7.2 The LM Decoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7.3 The LMMSE Decoders . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7.4 The Effect of Number of Receive Antennae L r . . . . . . . . . . . 71
3.7.5 The Effect of Number of Active Users K . . . . . . . . . . . . . . . 72
3.7.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Nonlinear Hierarchical Codes 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Signal Model and Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Building Nonlinear Hierarchical Codes . . . . . . . . . . . . . . . . . . . . 82
4.4 The Hierarchical Structure of Optimal 2 × 2 QPSK Codes . . . . . . . . . 91
4.5 Code Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Regular Multiple Trellis Coded Modulation 98
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 RMTCM Design Parameters and Procedures . . . . . . . . . . . . . . . . 100
5.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5 RMTCM Designs Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5.1 2 × 2 BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.2 2 × 2 QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5.3 2 × 2 8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.5.4 3 × 3 BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.5.5 3 × 3 QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.6 3 × 3 8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6 Conclusions 125
Reference List 126
Appendix A
Code Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Appendix B
RMTCM Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
v

List Of Tables
2.1 The distance spectra for rate 1 BPSK 2 × 2 Class 1 OMM code. . . . . . 26
2.2 OMM codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Detailed Structure of rate 1 OMM 2×2 8PSK and 16PSK Codes. ( u n = e i 2π n ) 39
2.4 Rate 1 OMM 3 × 3 BPSK Codes. . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Three space-time code structures and their performance comparison. . . . 54
3.1 CPU time in seconds for 200 runs. . . . . . . . . . . . . . . . . . . . . . . 72
4.1 Code list of the 2 × 2 QPSK NHC. . . . . . . . . . . . . . . . . . . . . . . 92
4.2 List of Hierarchical Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 The length M of single error event for given b and g. . . . . . . . . . . . . 108
5.2 Rate 1/2 MTCM designs for 2 × 2 BPSK. . . . . . . . . . . . . . . . . . . 112
5.3 Regular MTCM design for 2 × 2 BPSK |S 3 | = 8 codewords. . . . . . . . . 113
5.4 MTCM design for 2 × 2 QPSK |S 4 | = 16 codewords. . . . . . . . . . . . . 114
5.5 Hierarchical structure in Hamming distance. . . . . . . . . . . . . . . . . . 118
A.1 2 × 2 BPSK codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2 2 × 2 QPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.3 2 × 2 8PSK code list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.4 2 × 2 8PSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.5 3 × 3 BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.6 3 × 3 QPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.7 3 × 3 QPSK code list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.8 3 × 3 8PSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.9 3 × 3 8PSK code list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.10 4 × 3 BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.11 4 × 3 QPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.12 4 × 3 QPSK code list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
vi

List Of Figures
2.1 Transmitter encoder block diagram for user k. . . . . . . . . . . . . . . . . 11
2.2 Receiver decoder block diagram. . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Union bound as a function of path delay τ. . . . . . . . . . . . . . . . . . 26
2.5 Distinction between spread and non-spread transmission. . . . . . . . . . 33
2.6 Union bound versus simulation for rate 1 OMM 2 × 2 BPSK Class 1 code. 35
2.7 Distance spectra for rate 1 OMM 2 × 2 BPSK codes. . . . . . . . . . . . . 38
2.8 Rate 1 OMM 2 × 2 BPSK block codes at SNR=6dB. . . . . . . . . . . . . 38
2.9 Rate 1 OMM 2 × 2 BPSK and 16PSK codes for ρ c = 0.3. . . . . . . . . . 39
2.10 Rate 1 OMM 3 × 3 BPSK codes at SNR=6dB. . . . . . . . . . . . . . . . 41
2.11 Rate 1 OMM 3 × 3 BPSK codes. . . . . . . . . . . . . . . . . . . . . . . . 41
2.12 Rate 2 OMM 2 × 2 QPSK Codes at ρ c = 1.0. . . . . . . . . . . . . . . . . 43
2.13 Rate 2 OMM 2 × 2 QPSK Codes at ρ c = 0.3. . . . . . . . . . . . . . . . . 43
2.14 Rate 2 OMM 2 × 2 QPSK Codes, Gold sequence, single user, flat fading. . 44
2.15 Rate 2 OMM 2 × 2 QPSK Codes, Gold sequence, multipath fading Γ =
[0, 5; 0, 5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.16 Sequence decoder M = 4, Rate 2 OMM 2×2 QPSK Codes, Gold sequence,
multipath fading Γ = [0, 5, 7, 10; 0, 5, 7, 10]. . . . . . . . . . . . . . . . . . . 45
2.17 Rate 1 OMM 4 × 2 BPSK Codes for ρ c = 0.3. . . . . . . . . . . . . . . . . 46
2.18 Multiuser rate 1 OMM 2 × 2 BPSK codes. . . . . . . . . . . . . . . . . . . 48
2.19 Decoder performance comparison with no near-far effect. . . . . . . . . . . 48
2.20 Decoder performance comparison with near-far effect. . . . . . . . . . . . 49
2.21 Equivalent ρ e as a function of SNR and ρ c . . . . . . . . . . . . . . . . . . 49
3.1 Decoder Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Decoder Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 The BLUE and LS estimators lose diversity. . . . . . . . . . . . . . . . . . 70
vii

3.4 The BLUE and LS estimators have large variance. . . . . . . . . . . . . . 71
3.5 Ignoring R incurs 0.3dB loss. . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.6 LMMSE2 decoder has slight advantage over LMMSE1 decoder at high SNR. 73
3.7 LMMSE2 decoder has less variance than LMMSE1 decoder. . . . . . . . . 73
3.8 Two receive antennae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.9 Two active users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.10 Four active users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 The relationship between {a, b} and {a ′ , b ′ }. . . . . . . . . . . . . . . . . . 88
4.2 Hierarchical structure of the 2 × 2 QPSK NHC. . . . . . . . . . . . . . . . 91
4.3 3×3 BPSK, 8PSK NHCs (8 codewords) vs best 8PSK unitary group codes
(8 codewords). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 3 × 3 8PSK NHCs (64 codewords) vs best 21PSK unitary group codes (63
codewords). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5 4×3 BPSK (8 codewords) and QPSK (64 codewords) NHCs vs orthogonal
codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1 The shift-register configuration to generate state trellis. . . . . . . . . . . 101
5.2 The shift-register configuration for 2 inputs and 3 registers and corresponding
trellis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Rate 1/2 MTCM designs for 2 × 2 BPSK. . . . . . . . . . . . . . . . . . . 112
5.4 2 × 2 BPSK, Rate 2/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5 2 × 2 QPSK, |S 4 | = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.6 2 × 2 QPSK, |S 4 | = 16, Ricean fading channels (h = 1). . . . . . . . . . . 115
5.7 2 × 2 QPSK, |S 5 | = 32, quasi-static vs fast block fading channels. . . . . . 116
5.8 Bit map of 2 × 2 QPSK, |S 4 | = 16, for serial concatenation. . . . . . . . . 118
5.9 2 × 2 QPSK, |S 4 | = 16, serial concatenation vs RMTCM. . . . . . . . . . 119
5.10 Rate 6/2, 2 × 2 8PSK, |S 7 | = 128 vs |S 8 | = 256. . . . . . . . . . . . . . . . 120
5.11 3 × 3 BPSK, |S 3 | = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.12 3 × 3 BPSK, d ∗ free
of 2 × 4 × 1. . . . . . . . . . . . . . . . . . . . . . . . . 122
5.13 Rate 3/3, 3 × 3 BPSK, S 4 , S 5 and S 6 . . . . . . . . . . . . . . . . . . . . . 122
5.14 3 × 3 QPSK, |S 5 | = 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.15 3 × 3 QPSK, |S 7 | = 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.16 3 × 3 8PSK, rate=6/3, |S 7 | = 128 vs |S 8 | = 256. . . . . . . . . . . . . . . . 124
B.1 2 × 2 BPSK, Rate 1/2, I2S2O2p1. . . . . . . . . . . . . . . . . . . . . . . 137
viii

B.2 2 × 2 BPSK, Rate 1/2, I2S2O4p1. . . . . . . . . . . . . . . . . . . . . . . 138
B.3 2 × 2 BPSK, Rate 1/2, I2S2O4P 1. . . . . . . . . . . . . . . . . . . . . . . 138
B.4 2 × 2 BPSK, Rate 2/2, 2 × 2 × 2. . . . . . . . . . . . . . . . . . . . . . . . 138
B.5 2 × 2 BPSK, Rate 2/2, 2 × 4 × 1. . . . . . . . . . . . . . . . . . . . . . . . 138
B.6 2 × 2 QPSK, Rate 1/2, 8 × 2 × 1. . . . . . . . . . . . . . . . . . . . . . . . 138
B.7 2 × 2 QPSK, Rate 2/2, 4 × 2 × 2. . . . . . . . . . . . . . . . . . . . . . . . 139
B.8 2 × 2 QPSK, Rate 2/2, 4 × 4 × 1. . . . . . . . . . . . . . . . . . . . . . . . 139
B.9 2 × 2 QPSK, Rate 3/2, 2 × 2 × 4. . . . . . . . . . . . . . . . . . . . . . . . 139
B.10 2 × 2 QPSK, Rate 3/2, 2 × 4 × 2. . . . . . . . . . . . . . . . . . . . . . . . 140
B.11 2 × 2 QPSK, Rate 3/2, 2 × 8 × 1. . . . . . . . . . . . . . . . . . . . . . . . 140
B.12 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 2 × 8. . . . . . . . . . . . . . . . . . 140
B.13 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 4 × 4. . . . . . . . . . . . . . . . . . 140
B.14 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 8 × 2. . . . . . . . . . . . . . . . . . 141
B.15 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 16 × 1. . . . . . . . . . . . . . . . . 141
B.16 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 2 × 8. . . . . . . . . . . . . . . . . . 142
B.17 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 4 × 4. . . . . . . . . . . . . . . . . . 142
B.18 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 8 × 2. . . . . . . . . . . . . . . . . . 143
B.19 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 2 × 16 × 1. . . . . . . . . . . . . . . . . 144
ix

Abstract
Space-time codes are effective means by which to combat fading by exploiting diversity
in both spatial and temporal dimensions. In this work, code performance and design
are considered in the context of Direct-Sequence Code-Division Multiple-Access (DS-
CDMA) systems. The performance criteria, diversity gain and coding gain, are redefined
as functions of the spreading code correlation. Uplink and downlink scenarios are both
analyzed in multipath fading channels. Optimal codes are searched for both spread and
non-spread systems. To reduce the decoding complexity, suboptimal linear decoders are
classified and analyzed. The asymptotic achievable diversity level is derived when the
number of receive antennae is large. Motivated by the structures found in computer
optimized codes, via distance preserving isometries, Space-Time Block Codes (STBCs)
are constructed bottom-up by optimizing the coding gain layer-by-layer and reusing the
optimized structure. STBCs constructed in this way are dubbed Nonlinear Hierarchical
Codes (NHCs). NHCs are essentially a generalization of Slepian’s group codes for
the Gaussian channel to the multiple-input multiple-output quasi-static fading channel.
NHCs consistently offer comparable or significantly better performance than constrained
designs (orthogonal or unitary). Systematic Regular MTCM (RMTCM) design procedures
are proposed to exploit the layered structure of NHCs which leads to optimized
designs for various rates/block sizes/constellation sizes. The match between the distance
spectrum of NHCs to the regular trellis structure optimizes coding gain directly, and full
diversity is naturally maintained given the full rank of the constituent NHCs. Several
factors which affect performance in both quasi-static and fast block fading channels are
analyzed and exploited to improve the overall performance. Set expansion is used to improve
high rate RMTCM designs. RMTCM offers the best tradeoff between performance
and complexity among STTC and (interleaved) serial concatenated system.
x

Chapter 1
Introduction
Methods for introducing diversity have long been used to combat fading in wireless channels,
e.g., rake receiver and Maximal Ratio Combining are classical receiver diversity
schemes [Pro95]. Recently, transmit diversity techniques have received significant attention
due to their efficiency in bandwidth and power [GFBK99, Ala98, TSC98, Hug00].
These works are, in part, motivated by information theoretic analyses [Tel95, FG98],
which show that capacity increases linearly in the minimum of the number of transmit
and receive antennae. These capacity results are shown under the assumptions of statistically
independent flat Rayleigh channels between all transmit/receive antenna pairs.
Significant work has been done for non-spread space-time systems. Space-time code
design criteria are provided in [GFBK99, TSC98]. Alamouti [Ala98] proposed a simple
orthogonal space-time block code for 2 transmit antennae, it was later generalized in
[TJC99] to orthogonal space-time block codes for more transmit antennae. Motivated by
information theoretic results, unitary space-time codes are studied in [HM00]. A systematic
way to construct full-diversity unitary group codes based on representation theory
of fixed-point-free groups is presented in [SHHS01]. Differential space-time modulation
based on unitary group codes are studied in [Hug00, HS00]. A layered space-time architecture
is suggested in [Fos96, GARH01]. A number of these non-spread block codes
are used as benchmarks for the performance evaluation of our optimized codes found by
computer search.
There is also a lot of work to generalize Space-time coding to Direct-Sequence Code-
Division Multiple-Access (DS-CDMA) systems, but very few has offered the unique view
of performance criteria and code design as a function of spreading code in both the uplink
and downlink of spread systems, and as a consequence, could not take full advantage of
1

spreading. The focus of [HMP01] is to provide full diversity gain by employing existing
schemes. In the sequel, it is shown that full diversity is trivially satisfied; thus a focus
on optimizing coding gain is more appropriate. Space-time spreading (STS) [HMP01]
relies heavily on orthogonal spreading codes, whose orthogonality cannot be maintained
in a practical asynchronous frequency-selective channel. In contrast, our optimized codes
maintain their good performance over a wide range of spreading code correlation values
and multipath profiles, and hence have greater utility in real channels. In [DSAM03],
spreading is combined with the so-called algebraic constellations, the performance is
evaluated via simulation. Although multipath is considered in the modelling, the effects of
spreading and multipath in performance analysis and code design are not analyzed. Chipinterleaving
and block-spreading are used in [ZGM02] to suppress the effect of multipath
fading in spread systems. Low-dimensional spread modulation was considered in [BV03],
the signal model was similar to the uplink in this work except that very short spreading
codes were used. The main conclusion of [BV03] is similar to Proposition 2 for the uplink
in this work: single user code design is sufficient to achieve full diversity in the uplink of
a multiuser environment. We will show that the decoupling of code designs among users
in the uplink is not a characteristic of low-dimensional spreading, but a consequence of
independent fading experienced by different users in the uplink. In [WP99], optimal
decoder and several suboptimal decoders, like linear and blind decoders, are presented
and analyzed.
In Chapter 2 of this work, we present a general model for space-time DS-CDMA signals
in a multipath fading channel, and focus on performance analysis and code design as a
function of spreading code correlation and multipath fading. Both uplink and downlink
scenarios are considered. One consequence of multipath fading is that the optimal decoder
is a block sequence decoder, however, with typical channel assumptions, we show that the
code design based on the worst case pairwise error probability analysis remains a block
optimization problem. The presence of multiple access and spreading make code design
for spread systems different from that of non-spread systems. The differences in both the
design criteria and the resultant optimal codes for spread and non-spread systems are
highlighted.
2

Based on the design criteria, which is related to spreading code correlation, this work
further attempts to optimize codes based on coding gain. The found codes, deemed Optimal
Minimum Metric (OMM) code, provide solid gains over known codes which have
more structure, like orthogonal codes [TJC99, Ala98] and unitary group codes [SHHS01],
in flat/multipath fading channels and single-user/multiuser systems. The structure of optimized
codes leads to the construction of Nonlinear Hierarchical Codes (NHC) [GM03b],
which is the focus of Chapter 4. The effect of the multipath fading channel on code performance
is evaluated. Stronger statement than those in [TNSC99] are made regarding
diversity gain and coding gain.
Chapter 2 only considers the optimal Maximum Likelihood (ML) decoder. When the
numbers of active users and space-time block codes (STBC) are large, the ML decoder is
impractical due to large decoding complexity. In Chapter 3, we consider several different
suboptimal linear decoders for STBC. The suboptimal decoders considered are structurally
similar to those of [NSTC98], which employ a two stage decoding algorithm. In
the first stage, a linear equalizers suppresses MAI and spatial inter-symbol interference.
Subsequently, a mapper is implemented to map the linear filter soft outputs to a valid
STBC(see Figure 3.2). The asymptotic achievable diversity level is evaluated for each
linear decoders when the number of receive antennae grows. The Fisher Information
Matrix for random code vector is also considered to reveal the favorable code structure
for Linear Minimum Mean Squared Error (LMMSE) estimator.
The performance criteria for a space-time system, diversity gain and coding gain,
demand a significantly different code design strategy than that for Gaussian channels,
because the code space defined by the coding gain is a non-linear, non-metric space.
The space (transmit antenna) dimension also introduces more degrees of freedom in the
code space than using the time dimension only. In Chapter 2, exhaustive computer
searches are used to seek the optimal STBC code set for certain STBCs with small size
and low rate, but the search is NP-hard and quickly becomes infeasible when the block
code size grows large and code rate is high. The computer optimized codes exhibit
very nice layered structure which motivates the construction of NHC in Chapter 4. The
construction of NHC is fundamentally different from that of constrained designs (unitary,
orthogonal) by manipulating the relationship between codewords rather than properties
of the individual codewords. Two questions are to be addressed in Chapters 4 and
3

5, respectively: first, for an arbitrary size STBC from an nPSK constellation, how do
we find the optimal (or near optimal) code set of various rates resulting in a distance
spectrum with good structure? Secondly, for such found codes, how do we introduce
memory among blocks to systematically utilize the structure of the distance spectrum?
Distance spectrum is the enumeration of the distance measure of interest between all
possible codeword pairs.
The methodology of NHC construction via isometries for MIMO quasi-static fading
channels is an extension of Slepian’s group codes [Sle68], where group structure (and
hence distance uniformity), is conserved in most cases. A sufficient condition for NHCs
to achieve geometric uniformity [GDF91] is established. The NHC construction starts
from some initial codeword matrices with no constraint on the size and structure, and
by means of isometries, systematically builds higher rate codes from lower rate codes,
while preserving the good structure of the lower rate code. It is the relationship between
codewords, rather than the structure of each individual codeword, that is exploited to
optimize coding gain. The found NHCs match those found via exhaustive computer
search, and exhibit not only optimized coding gain at each layer, but also great symmetry
and structure to be exploited in Chapter 5 of this work.
STBCs are usually insufficient to provide the needed protection from the effects of a
fading channel, therefore memory is needed. Space-time trellis codes (STTC) [TSC98,
BBH02, YB02, Gri99] usually rely on a convolutional encoder to introduce memory among
symbols, but there are no good design rules to optimize coding gain rather than exhaustive
computer search. Multiple Trellis Coded Modulation (MTCM) [BDMS91, DS87, SF02,
JS02] serves as an ideal bridge to combine the advantages of both block codes and trellis
codes, and offers simple and intuitive design rules to optimize coding gain systematically.
The MTCM designs in [SF02, ATP98] use STBCs as an inner code of a convolutional
encoder. However, such serial concatenation usually cannot take full advantage of the
structure in STBCs, and therefore introduces memory among blocks “blindly”. Chapter
5 of this work proposes systematic MTCM designs where memory is introduced between
block codes directly, instead of through an outer convolutional encoder. Because of
the symmetric structure in the trellis and constituent NHCs, such designs are dubbed
RMTCM designs.
4

In our RMTCM design, a regular trellis constructed specifically for fading channels
is generated by shift-register chains. NHCs, which admit a natural set partitioning as in
[Ung82], form the constituent codes. The parameters of the regular trellis and set partitioning
are matched to exploit the layered structure of the constituent NHCs optimally.
Several factors which affect performance in both quasi-static and fast block fading channels
are analyzed and exploited to improve the overall performance. Set expansion is used
to improve high rate MTCM designs. In each design, the performance/rate/complexity
tradeoffs are gracefully balanced and optimized. The general design procedure can be
used to serve two purposes: for a given set of NHCs, provide MTCM designs with various
rates, and, for a given rate, provide MTCM designs with various complexity and
performance.
The MTCM design in [JS02] is a special case of the RMTCM design in this work,
where the constituent codes are restricted to orthogonal codes and signal expansion is
limited to a special isometry; the set partitioning and trellis structure of [JS02] are not
systematic. RMTCM design can achieve a half dB gain over the rate 3 MTCM design in
[JS02] by doubling the number of states via adjusting trellis parameters. With even more
states, as much as 2dB gain can be achieved. The observation, that parallel transitions
should be avoided for better performance in quasi-static channels, is made in [JS02] for
orthogonal constituent codes only, we will show that it holds for more general cases.
The design principles of RMTCM do not rely on quasi-static fading channels. Thus
they can be employed to design RMTCM even for fast block fading channels, where fast
block fading is achieved due to perfect block interleaving. Both analysis and simulation
show that parallel transitions are the major detrimental factor which should be avoided
for fast block fading channels. A similar conclusion is drawn for MTCM design over
Single-Input-Single-Output fast-fading (due to perfect interleaving) channels [BDMS91].
This thesis is organized as follows: Chapter 2 discusses space-time block codes in
multipath CDMA systems. Chapter 3 discusses suboptimal linear decoders. Chapter 4
discusses the construction and properties of NHC. Chapter 5 discusses the design procedure
and performance analysis of RMTCM. Chapter 6 summarizes this work and points
out future work.
5

Chapter 2
Space-Time Block Codes in Multipath CDMA Systems
2.1 Introduction
There is a lot of work to generalize Space-time coding to Direct-Sequence Code-Division
Multiple-Access (DS-CDMA) systems, but very few has offered the unique view of performance
criteria and code design as a function of spreading code in both the uplink
and downlink of spread systems, and as a consequence, couldn’t take full advantage of
spreading. The focus of [HMP01] is to provide full diversity gain by employing existing
schemes. In the sequel, it is shown that full diversity is trivially satisfied; thus a focus
on optimizing coding gain is more appropriate. Space-time spreading (STS) [HMP01]
relies heavily on orthogonal spreading codes, whose orthogonality cannot be maintained
in a practical asynchronous frequency-selective channel. In contrast, our optimized codes
maintain their good performance over a wide range of spreading code correlation values
and multipath profiles, and hence have greater utility in real channels. In [DSAM03],
spreading is combined with the so-called algebraic constellations, the performance is evaluated
via simulation. Although multipath is considered in the modelling, the effects of
spreading and multipath in performance analysis and code design are not analyzed. Chipinterleaving
and block-spreading are used in [ZGM02] to suppress the effect of multipath
fading in spread systems. Low-dimensional spread modulation was considered in [BV03],
the signal model was similar to the uplink in this work except that very short spreading
codes were used. The main conclusion of [BV03] is similar to Proposition 2 for the uplink
in this work: single user code design is sufficient to achieve full diversity in the uplink of
a multiuser environment. In [WP99], optimal decoder and several suboptimal decoders,
like linear and blind decoders, are presented and analyzed.
6

In this work, we present a general model for space-time DS-CDMA signals in a multipath
fading channel, and focus on performance analysis and code design as a function
of spreading code correlation and multipath fading. Both uplink and downlink scenarios
are considered. Furthermore, the framework is general enough to encompass non-spread
systems with slight modification (Section 2.4). Note that multipath fading is a consequence
of signal bandwidth exceeding channel coherence bandwidth. Rich multipath fading
is usually observed in spread systems where signal usually occupy large bandwidth which
exceeds channel coherence bandwidth. Non-spread systems may also experience multipath
fading [TNSC99], although less severe than spread systems. One consequence of multipath
fading is that the optimal decoder is a block sequence decoder, however, with typical
channel assumptions, we show that the code design based on the worst case pairwise error
probability analysis remains a block optimization problem. The presence of multiple access
and spreading make code design for spread systems different from that of non-spread
systems. The differences in both the design criteria and the resultant optimal codes for
spread and non-spread systems are highlighted. Due to the assumption of independent
channels, at the uplink, the multiuser coding problem decouples to multiple single user
coding problems; in contrast, the downlink space time coding problem is truly a multiuser
problem. At the uplink, all the transmissions of different users are asynchronous, but this
does not affect the performance analysis and code design due to the decoupling among
users.
Based on the design criteria, which is related to spreading code correlation, this work
further attempts to optimize codes based on coding gain. The found codes, deemed Optimal
Minimum Metric (OMM) code, provide solid gains over known codes which have
more structure, like orthogonal codes [TJC99, Ala98] and unitary group codes [SHHS01],
in flat/multipath fading channels and single-user/multiuser systems. The search for optimized
codes is simplified through “isometries”, transformations to which the design
metrics are invariant. For more detailed discussion on isometry, please refer to Section
4.2. We also analyze some methods to construct space-time codes and show their performance
with respect to spreading code correlation. The structure of optimized codes
leads to the construction of nonlinear hierarchical codes [GM03b].
The effect of the multipath fading channel on code performance is evaluated. As expected,
with perfect CSI, multipath provides a higher diversity level. Spread systems can
7

easily achieve maximum diversity (the number of transmit antenna times the number of
multipaths for one receive antenna), while non-spread systems must satisfy more stringent
conditions to achieve full diversity. A similar, but weaker result was presented in
[TNSC99], which showed that a space-time code will achieve at least the same diversity
in a multipath fading channel as it does in a flat fading channel. Herein, the exact diversity
level is easily determined. Based on the observation on diversity level, [TNSC99]
concludes that a space-time code designed for a slow flat fading channel will continue
to perform well in a multipath fading channel, we will further show that this conclusion
also holds in terms of coding gain when the multipath delay spread is relatively small in
comparison to the symbol interval duration.
This chapter is organized as follows. Section 2.2 derives the performance criteria and
code design principles in the uplink. Section 2.3 discusses the downlink. Section 2.4
briefly describes the non-spread system. Section 2.5 tabulates some of the optimized
codes found via computer search, compares their performance with some known codes,
and analyzes properties of these codes. Section 2.6 considers some application issues
for the quasi-static channel: invariant operations, the sensitivity of code performance
to spreading code correlation, and some suboptimal methods for constructing spacetime
codes. Finally, conclusions and some suggestions for future work are discussed in
Section 2.7.
2.2 The Uplink
In this section, we provide a general space-time model for the uplink of DS-CDMA systems.
The generality of the formulation incurs some added complexity, but offers the
advantage of enabling analysis of coding gain as a function of the spreading code correlation.
If the channel coefficients of different receive antennae are uncorrelated and
the Maximum-Likelihood decoder is employed, multiple receive antennae will not affect
code design based on worst case PEP analysis (but do affect performance). Recent work
shows that as the number of receive antennae increases, the multi-input multi-output
fading channel model approaches a multiple parallel AWGN channel model; as such different
design metrics should be considered for the large number of receive antennae case
[GF01, CYV99]. Herein, we consider a small to moderate number of receive antennae,
8

hence the worst case PEP is still a valid design criterion. Furthermore, due to our consideration
of the maximum likelihood decoder, we focus on a single receive antenna. Note
that for other decoding rules like Minimum Mean Squared Error (MMSE), the number
of receive antennae does matter in code design [GM01].
Due to the large dimensionality of our problem, a variable X might have up to 4
indices. To assist the reader in deciphering the notation X l(k)
i
(t), we generally conform
to the following rules: the subscript i is the transmit antenna index, the first superscript
l is the multipath index, the second superscript (k) is the user index, and finally t is
the time index.
Sometimes a matrix or vector is independent of the time index t or
transmit antenna index i, but we still keep that index to help track the dimension of the
corresponding matrix or vector. In this case, we will point it out explicitly. Two kinds of
codeword matrices are constantly used throughout the chapter, the calligraphic D, which
represents the block code in its original block form, and the regular D, which represents
a reshaped version of D. A prefix ∆ in front of D or D denotes the difference between
two codewords.
We summarize the constants used in this chapter for quick reference:
K: the # of active users
L t : the # of transmit antennae
L r : the # of receive antennae
L c : the # of maximum resolvable multipaths for a single transmit antenna
N c : the time dimension of block code in units of the symbol period
L u : the # of samples per symbol period, or sampling rate, for non-spread systems,
or spreading gain for chip-sampled DS-CDMA systems
τ max : the largest path delay
L s : the # of samples between two consecutive transmissions
Note that L s is used for non-spread systems only, for spread systems, L s = L u .
In this chapter, a unique integer, the code index, is used to represent a codeword. If
nPSK is employed, u is the n th root of unity u = e j 2π n and d i (t) = u ci(t) , a block code
matrix of size N c × L t can be represented by the index obtained from a base n number:
(c Lt (N c )c Lt−1(N c )c Lt−2(N c ) · · · · · · + c 2 (1)c 1 (1)) n . (2.1)
9

For example, a BPSK 2 × 2 block code,
⎡
⎣ 1
−1
⎤
−1 ⎦ ,
−1
can be represented by code index,
c 2 (2) + c 1 (2)n + c 2 (1)n 2 + c 1 (1)n 3 = 1 + 1 × 2 + 1 × 4 + 0 × 8 = 7.
2.2.1 The Uplink System Model
In the uplink between K active users and one base station, we will show how optimizing
the performance of a sequence block decoder leads to the design of each block code
individually and also how this problem simplifies into multiple, single user, coding problems.
We shall consider an asynchronous multipath channel model. The transmitter is
equipped with L t transmit antennae and the receiver is equipped with 1 receive antenna.
Figure 2.1 depicts the encoder block diagram for a single block code. For each block, the
transmitter maps k c information bits of user k, [I (k) (1), I (k) (2), . . . , I (k) (k c )], to one of
K c = 2 kc space-time block codewords,
D (k) = [d (k)
i
(t)] ∈ C Nc×Lt , 1 ≤ i ≤ L t , 1 ≤ t ≤ N c (2.2)
where N c is the block length in terms of symbol duration. Each d (k)
i
(t) is spread by
a corresponding spreading code s (k)
i
(t) and transmitted via the corresponding transmit
antenna T X i . Note that each row of the codeword matrix is transmitted simultaneously.
A total of M such blocks are transmitted continuously, no memory exists among these
blocks. The code rate is
R c = k c
N c
. (2.3)
In our formulation, each d (k)
i
(t) can be taken from any point on the complex plane, but
in the following discussion of code design, d (k)
i
(t) is constrained to PSK constellations for
practical purposes. The spreading code s (k)
i
(t) is a column vector of length L u , which has
support over one symbol period. The typical entries of s (k)
i
(t) are {±1/ √ L u }, but this is
not required in the following derivation and analysis. If one uses different spreading codes
at different symbol times for the same antenna, then s (k)
i
(t 1 ) ≠ s (k)
i
(t 2 ), this model applies
10

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User k’s Information bits
ST Block Coding
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Figure 2.1: Transmitter encoder block diagram for user k.
to long spreading codes which span several symbol times. For a non-spread system, which
will be discussed briefly in Section 2.4, this vector is composed of higher rate samples of
the modulation waveform and is the same for all transmit antennae; for a spread system,
s (k)
i
(t) is a well-designed spreading code, with a possibly different code for each antenna.
We note that near orthogonality between spreading codes is not a requirement in our
scheme, thus assigning unique spreading codes to different antennae of each user is not a
challenge.
The receiver block diagram is shown in Figure 2.2. In multipath fading channels, the
optimal decoder should consider the entire received sequence of M blocks due to interblock
interference from the previous and subsequent blocks. If the M blocks are among a
continuous transmission of blocks, we assume that the sequence length M is long enough
such that the edge effects can be safely neglected.
Matched
Filter Bank
Sequence
Decoder
S-T0UVVVU R SXW R
MONQP
Figure 2.2: Receiver decoder block diagram.
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Figure 2.3: Channel model.
A chip spaced tap delay line model [Pro95] is used to model the channel between
each transmit/receive antenna pair, as depicted in Figure 2.3. The number of resolvable
11

multipaths corresponding to each transmit and receive antenna pair is denoted by L c .
For simplicity, we will assume that the channels between different transmit/receive antenna
pairs have the same number of multipaths. Our model handles different number
of multipaths equally well, but that adds unnecessary complexity to the formulae. Note
that L c may be less than or equal to the largest path delay τ max . For user k, the delay
corresponding to multipath l associated with transmit antenna i is τ l(k)
i
. The channel
tap amplitude and phase associated with τ l(k)
i
at symbol time t are represented by a
complex number h l(k)
i
(t), which is called the channel coefficient and modelled as a circularly
symmetric complex Gaussian random process. We consider quasi-static fading only,
therefore h l(k)
i
(t) is constant within the block, but independent from block to block. All
unique channel coefficients are assumed to be i.i.d.. The chip spaced multipath model is
an idealized model for real channels and is invoked here to facilitate an analytical understanding.
The delay τ l(k)
i
is assumed constant within M blocks because it is determined by
the propagation delay of corresponding multipath, which is fairly constant for a moderate
mobility environment. Both transmit and receive antennae are assumed to be sufficiently
separated such that the channel coefficients for different transmit/receive antennae are
statistically independent. We use the delay profile matrix
to describe the spread of the path delays for user k and
⎡
⎤
τ 1(k)
1 · · · τ Lc(k)
1
Γ (k) =
.
⎢ . .. . ⎥
⎣
⎦ , (2.4)
τ 1(k)
L t
· · · τ Lc(k)
L t
Γ = [Γ (1) , . . . , Γ (K) ] (2.5)
to describe all users’ delay profiles. Note that
(t) ∈ R (Lu+τmax)×1 as the delayed duplicate of s (k) (t) corresponding to mul-
Denote g l(k)
i
tipath l, it has τ l(k)
i
τ max = max τ l(k)
i
, ∀i, l, k (2.6)
zeros preceding s (k)
i
(t) and possibly some trailing zeros to make its
i
12

length (L u + τ max ). For example, when using one sample per chip, the spreading code
s (k)
i
(t) and its delayed version g l(k)
i
(t) with τ l(k)
i
= 2 may look like
s (k)
i
(t) =
1 √ Lu
[ 1, −1, −1, 1, . . . , 1, −1 ] T
g l(k)
i
(t) =
1 √ Lu
[ 0, 0, 1, −1, −1, 1, . . . , 1, −1 0 . . . 0] T (2.7)
The received signal samples due to d (k)
i
(t) are
r(t) = √ σ t
∑
K ∑L t L c
∑
k=1 i=1 l=1
g l(k)
i
(t)d (k)
i
(t)h l(k)
i
(t) + n(t) ∈ C (Lu+τmax)×1 (2.8)
where n(t) is complex white Gaussian noise ∼ CN (0, I). Note that the length of r(t)
extends beyond a symbol period L u due to multipath fading.
We use CN (m, R) to
denote a complex Gaussian random vector with mean vector m and covariance matrix
R.
We will group r(t) in the order of multipath, user and block to form a final observation
vector r for the whole sequence of received blocks. We define
G (k) (t) = [g 1(k)
1 (t), . . . , g 1(k) (t), . . . . . . , g Lc(k)
1 (t), . . . , g Lc(k)
L t
(t)] ∈ R (Lu+τmax)×LcLt (2.9)
L t
group 1
group L c
{ }} { { }} {
D (k) (t) = diag[ d (k)
1 (t), . . . , d(k)
L t
(t), . . . . . . d (k)
1 (t), . . . , d(k)
L t
(t)] ∈ C LcLt×LcLt (2.10)
h (k) (t) = [h 1(k)
1 (t), . . . , h 1(k) (t), . . . . . . , h Lc(k)
1 (t), . . . , h Lc(k)
L t
(t)] T ∈ C LcLt×1 (2.11)
L t
For symbol time t, matrix G (k) (t) collects the spreading codes corresponding to all multipaths
for user k, D (k) (t) contains L c duplicates of user k’s data, and h (k) (t) collects the
channel coefficients of all the multipaths for user k.
Equation (2.8) can be written in matrix form as,
r(t) = √ σ t G(t)D(t)h(t) + n(t) ∈ C (Lu+τmax)×1 (2.12)
13

where
G(t) = [G (1) (t), . . . . . . , G (K) (t)] ∈ R (Lu+τmax)×KLcLt (2.13)
D(t) = diag[D (1) (t), . . . . . . , D (K) (t)] ∈ C KLcLt×KLcLt (2.14)
h(t) = [h (1) (t) T , . . . . . . , h (K) (t) T ] T ∈ C KLcLt×1 (2.15)
Note that inter-symbol interference (ISI) occurs on the last τ max samples of r(t − 1)
and the first τ max samples of r(t).
The received vector due to block m is obtained
by “concatenating” the r(t), t = 1, . . . , MN c into a larger vector with τ max overlapping
samples between adjacent r(t − 1) and r(t),
⎡
⎤ ⎡
⎤
G((m − 1)N
r m = √ c + 1) ∅ ∅ D((m − 1)N c + 1)
[
]
σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥ h((m − 1)N
⎣
⎦ ⎣
⎦
c + 1)
∅ ∅ G(mN c ) D(mN c )
(2.16)
⎡
⎤
n((m − 1)N c + 1)
+ ⎢ . ⎥
⎣
⎦ √ σ t G m D m h m + n m ∈ C (NcLu+τmax)×1 . (2.17)
n(mN c )
Note that
⎡
⎤
G((m − 1)N c + 1) ∅ ∅
G m =
.
⎢ ∅ .. ∅ ⎥
⎣
⎦
∅ ∅ G(mN c )
(2.18)
is not block diagonal, there are τ max overlapping rows between the adjacent blocks of G(t).
The matrix G m is described in this fashion for notational convenience. If a matrix is block
diagonal, the notation 0 is used (versus ∅. The following block matrix shows how the
last τ max rows of G(1) overlaps with the first τ max rows of G(2), the overlapping section
are designated by a prime.
⎡ [
G(1)
[
⎢ G ′ (1)
⎣ [
0
]L u×KL tL c
[
]τ max×KL tL c
[
]L u×KL tL c
[
]
0
]
G ′ (2)
]
G(2)
L u×KL tL c
τ max×KL tL c
L u×KL tL c
⎤
⎥
⎦
14

Due to quasi-static fading, h((m − 1)N c + 1) = · · · = h(mN c ), hence h m consists of only
one h((m − 1)N c + 1). The noise vector n m is complex Gaussian n m ∼ CN (0, I).
Finally, the observation vector of the whole sequence
⎡
⎤ ⎡
⎤ ⎡ ⎤ ⎡ ⎤
G
r = √ 1 ∅ ∅ D σ t ⎢ ∅ . 1 0 0 h 1 n 1
. . . ∅ ⎥ ⎢ 0 .. 0 ⎥ ⎢ . ⎥
⎣
⎦ ⎣
⎦ ⎣ ⎦ + ⎢ . ⎥ (2.19)
⎣ ⎦
∅ ∅ G M 0 0 D M h M n M
√ σ t Ḡ ¯Dh + n ∈ C (MNcLu+τmax)×1 . (2.20)
Note that ¯D represents a sequence of M block codes.
A matched filter bank, which is matched to each multipath of each user at every
symbol time t, is constructed. The matched filter outputs, which form a sufficient statistic
(ignoring any edge effect from the first and last blocks) for the decoder, can be written
as
y = ḠT r = √ σ t ¯R ¯Dh + m ∈ C
MN cL cKL t×1 , (2.21)
where
¯R ḠT Ḡ = [R ij ] ∈ R MNcLcKLt×MNcLcKLt (2.22)
is the spreading code correlation matrix, note that the
R ij G H i G j , 1 ≤ i, j ≤ M (2.23)
are sparse or zero matrices when i ≠ j, m ḠT n is distributed as CN (0, ¯R).
2.2.2 The Uplink Chernoff Bound and Code Design Criteria
We derive the Chernoff bound of pairwise error probability (PEP) when the Maximum
Likelihood (ML) decoder is employed at the receiver as in [GFBK99, TSC98] under the
assumption of perfect channel state information at the receiver. The matched filter output
vector is y ∼ CN ( √ σ t R ¯Dh, R). The averaged probability of decoding codeword sequence
¯D β while codeword sequence ¯D α is transmitted is upper bounded by
E h [P ( ¯D α → ¯D 1
β |h)] ≤ ∣
∣I + σt
4 Σ h( ¯D α − ¯D β ) H ¯R( ¯D α − ¯D β ) ∣ , (2.24)
15

where D α and D β are two codeword sequences as defined in Equation (2.20) and Σ h =
E[hh H ] ∈ R MKLtLc×MKLtLc is the correlation matrix of the channel coefficients. Due to
independent fading, Σ h is a diagonal matrix, where each diagonal entry corresponds to
the power of a particular multipath for a particular user at certain block m.
We denote ∆ ¯D = ¯D α − ¯D β . The correlated codeword sequence difference matrix, ¯Φ,
is defined as
¯Φ = ∆ ¯D H ¯R∆ ¯D ∈ C
MKL cL t×MKL cL t
(2.25)
Φ ij = ∆D H i R ij ∆D j , 1 ≤ i, j ≤ M ∈ C KLcLt×KLcLt (2.26)
Notice that Φ mm , m = 1, . . . , M are the dominant blocks in Φ.
For high SNR, the Chernoff bound can be approximated [GFBK99, TSC98] by
E h [P (D α → D β |h)] ≤
(
( σt
) −r r∏
) −1
λ i , (2.27)
4
i=1
where λ i ’s are the nonzero eigenvalues of Σ h ¯Φ and r is the number of λi ’s. Taken over all
codeword pairs, the minimum possible number of non-zero eigenvalues of Φ determines
the diversity gain,
∆ H = min rank(¯Φ). (2.28)
∆ ¯D
For those pairs of codewords which achieve ∆ H , the smallest product of non-zero eigenvalues
of Φ determines the coding gain,
∆∏
H
∆ p = min λ i . (2.29)
∆ ¯D
i=1
In the case of full diversity,
∆ p = min
∆D det(¯Φ). (2.30)
While diversity gain and coding gain are the classical worst-case space-time code design
criteria, the distance spectrum of the code set is a more accurate performance indicator for
a small number of receive antennae [AF01]. Union bound is used in [GVM02] to optimize
the whole distance spectrum, the resultant codes can achieve better performance than
the corresponding OMM code.
16

If we assume that the channel coefficients are independent for different transmit antennae,
a nontrivial Σ h will be a diagonal matrix with full rank, and the code design
will be independent of channel statistics. To simplify the analysis, in the sequel we will
assume this and focus on ¯Φ only.
2.2.3 The Correlated Codeword Sequence Difference Matrix Φ
We assume each transmit antenna uses the same fixed spreading code during the M
blocks
s i (1) = s i (2) = · · · = s i (MN c ) (2.31)
G 1 = · · · = G M and R mm exhibits a banded block Toeplitz structure. For the quasi-static
fading channels, Φ mm can be put in a more compact form.
If nL u ≤ τ max < (n + 1)L u , then R mm = G T mG m ∈ R NcKLcLt×NcKLcLt has the form
⎡
⎤
R mm (0) R mm (1) . . . R mm (n + 1) 0
R mm (1) T
.
R(0) .. . .. Rmm (n + 1)
R mm =
. .
.. . .. . .. .
, (2.32)
⎢
⎣
R mm (n + 1) T . .. . .. . .. Rmm (1) ⎥
⎦
0 R mm (n + 1) T . . . R mm (1) T R mm (0)
where R mm (0) captures the “on-time” correlation of the spreading codes,
R mm (0) = G(t) T G(t) ∈ R KLcLt×KLcLt , (2.33)
and R mm (t), 1 ≤ t ≤ n captures the “off-time” correlation,
⎡ ⎤
R mm (t) = ⎣ G(t) ⎦
0
⎤
⎣ 0 ⎦ ∈ R LcKLt×LcKLt (2.34)
G(t)
T ⎡
where, G(t) ∈ R (Lu+τmax)×KLcLt is defined in Equation (2.13), 0 ∈ R nLu×KLcLt is an
all-zero matrix whose size depends on n.
17

For the quasi-static fading channel, Φ mm can be simplified as 1
n+1
∑
Φ mm = ∆DL H c
∆D Lc ⊙ R mm (0) + ∆DL H c
Q n+1 ∆D Lc ⊙ (R mm (t) + Rmm(t)) T (2.35)
t=1
where
and
L c blocks
{ }} {
∆D Lc = [ ∆D, . . . , ∆D] (2.36)
⎡
⎤
0 1 · · · 0
. . .. . .. .
Q =
⎢
.
⎣
. .. . (2.37)
1 ⎥
⎦
0 · · · · · · 0
Regarding the rank of Φ and Φ mm , we have the following proposition that is independent
of the space-time code selected.
Proposition 1 If ∆D m ≠ 0 and at least one of ∆D n ≠ 0, n ≠ m,
min rank ¯Φ > rank Φ mm , ∀m = 1, . . . , M, (2.38)
∆ ¯D
provided one of the following conditions is satisfied:
1. The difference between no pairs of distinct path delays is an integer multiple of the
spreading code length L u
|τ lp(kp)
i p
− τ lq(kq)
i q
| ≠ nL u n = 1, 2, . . . ,
1 ≤ i p ≠ i q ≤ L t or
1 ≤ k p ≠ k q ≤ K.
(2.39)
2. If the spreading codes are not time-varying, i.e. s (k)
i
(1) = · · · = s (k)
i
(MN c ), ∀ i, k,
then
the set of spreading codes {s (k)
i
, 1 ≤ i ≤ L t , 1 ≤ k ≤ K} are linearly independent.
(2.40)
1 The operator ⊙ is the Schur product, and ⊗ is the Kronecker product.
18

3. Without loss of generality, assume τ lp(kp)
i p
= τ min and τ lq(kq)
i q
= τ max , at least one of
∆d (kp)
i p
(t m + (m − 1)N c ) ≠ 0, 1 ≤ t m ≤ N c , and (2.41)
∆d (kq)
i q
(t m + (m − 1)N c ) ≠ 0, 1 ≤ t m ≤ N c . (2.42)
Proof:
Recall from Equations (2.25) and (2.22), ¯Φ = ∆ ¯D H ¯R∆ ¯D and ¯R = Ḡ T Ḡ. Therefore
¯Φ = (Ḡ∆ ¯D) H Ḡ∆ ¯D, (2.43)
rank ¯Φ = rank Ḡ∆ ¯D (2.44)
where
Similarly,
⎡
⎤
G
Ḡ∆ ¯D 1 ∆D 1 ∅ ∅
=
.
⎢ ∅ .. ∅ ⎥
⎣
⎦ .
∅ ∅ G M ∆D M
rank Φ mm = rank G m ∆D m (2.45)
Let’s consider the two blocks from Ḡ∆ ¯D:
G m ∆D m = [g lp(kp)
i p
(t m + (m − 1)N c )∆d (kp)
i p
(t m + (m − 1)N c )], 1 ≤ t m ≤ N c , (2.46)
G n ∆D n = [g lq(kq)
i q
(t n + (n − 1)N c )∆d (kq)
i q
(t n + (n − 1)N c )], 1 ≤ t n ≤ N c . (2.47)
We want to prove that G m ∆D m and G n ∆D n span different column spaces of Ḡ∆ ¯D. It
is sufficient to show that there exists at least one vector g lp(kp)
i p
(t m +(m−1)N c )∆d (kp)
i p
(t m +
(m−1)N c ) from G m ∆D m that is not a linear combination of any components of {g lq(kq)
i q
(t n +
(n − 1)N c )∆d (kq)
i q
(t n + (n − 1)N c )}, i.e., the columns of G n ∆D n from Ḡ∆ ¯D.
1. For any non-zero vector g lp(kp)
i p
(t m + (m − 1)N c )∆d (kp)
i p
(t m + (m − 1)N c ), in Ḡ∆ ¯D,
the non-zero segment start at sample time
t m L u + (m − 1)N c L u + τ lp(kp)
i p
,
19

and the non-zero segment of any linear combination of {g lq(kq)
i q
(t n +(n−1)N c )∆d (kq)
i q
(t n +
(n − 1)N c )} can only start at sample time
t n L u + (n − 1)N c L u + τ lq(kq)
i q
.
If these times do not coincide, i.e.,
|[(t m − t n ) + (m − n)N c ]L u + τ lp(kp)
i p
⇐⇒|τ lp(kp)
i p
− τ lq(kq)
i q
| ≠ 0 (2.48)
− τ lq(kq)
i q
| ≠ [(t m − t n ) + (m − n)N c ]L u (2.49)
the vector g lp(kp)
i p
(t m + (m − 1)N c )∆d (kp)
i p
(t m + (m − 1)N c ) can never be a linear
combination of {g lq(kq)
i q
(t n + (n − 1)N c )∆d (kq)
i q
(t n + (n − 1)N c )}. Condition (2.39) is
even stronger than Equation (2.49).
2. If Condition (2.39) is violated, the non-zero segment of
g lp(kp)
i p
(t m + (m − 1)N c )∆d (kp)
i p
(t m + (m − 1)N c )
is aligned with the non-zero segments of possibly several vectors from {g lq(kq)
i q
(t n +
(n−1)N c )∆d (kq)
i q
(t n +(n−1)N c )}, Condition (2.40) guarantees that g lp(kp)
i p
(t m +(m−
1)N c )∆d (kp)
i p
(t m + (m − 1)N c ) still cannot be a linear combination of {g lq(kq)
i q
(t n +
(n − 1)N c )∆d (kq)
i q
(t n + (n − 1)N c )} due to the linear independence amongst the set
of spreading codes.
3. Condition (2.41) guarantees that the non-zero segment of
g lp(kp)
i p
(t m + (m − 1)N c )∆d (kp)
i p
(t m + (m − 1)N c )
starts earlier than the non-zero segment of any of the vectors from subsequent blocks
{g lq(kq)
i q
g lp(kp)
i p
(t n + (n − 1)N c )∆d (kq) (t n + (n − 1)N c ), n > m} in Ḡ∆ ¯D, therefore no linear
i q
combination of {g lq(kq)
i q
(t n + (n − 1)N c )∆d (kq)
i q
(t n + (n − 1)N c ), n > m} can yield
(t m + (m − 1)N c )∆d (kp) (t m + (m − 1)N c ). The effects of Condition (2.42) can
be shown in a similar fashion when n < m.
i p
✷
20

The conditions of Proposition 1 are generally met in real applications and thus are
not overly stringent. An important consequence of Proposition 1 is that, the minimum
rank of ¯Φ occurs when all but one block from the sequence is identical and is determined
by Φ mm of that single block. As such, the sequence coding problem decouples to a block
coding problem. Thus, we will focus on the properties of Φ mm from now on. The strict
inequality in Proposition 1 is critical, otherwise, the minimum rank of ¯Φ may occur when
two blocks from the sequence are different, therefore, the joint design of these two blocks
could improve coding gain of single block design and there is no decoupling of the blocks
with regards to design.
2.2.4 Code Design Principles
This section discusses code design principles for quasi-static fading channels for the uplink
(or the approximated downlink scenarios).
Proposition 2 In the uplink, if every distinct pair of users have linearly independent
spreading codes, under the assumption of independent fading for different users, the multiuser
coding problem decouples to multiple single-user coding problems.
Proof:
Recall that Φ mm = ∆D m R mm ∆D m ∈ C KLcLt×KLcLt . Whenever the codeword difference
of a certain user in ∆D m is set to zero, L c L t rows and columns of Φ mm will be zero. We
consider the smaller matrix with the zero rows and columns removed. The rank of this
matrix is never greater than the original. Continue removing rows and columns of users
until there are only two remaining users, since every two users uses linearly independent
spreading codes, removing one user always reduces rank Φ mm . So the minimum rank of
Φ mm (worst case) can only occur when one user contributes to ∆D m solely. Hence we
need only consider one user in the code design for the uplink. ✷
Note that Proposition 2 does not require linearly independent spreading codes to be
employed for different transmit antennae of the same user. From now on, only one user
will be considered in the code design task at the uplink, the superscript of the user index
will be dropped if no confusion is caused. Note that this decoupling will not exist in the
downlink (See Section 2.3).
21

Let us consider the issue of achievable diversity in multipath channels. We assume
that the sampling rate L u is high enough to resolve the available diversity (the column
rank of G m ):
N c L u + τ max > N c L c L t . (2.50)
Whether the correlation matrix R mm achieving full rank plays an important role in the
following discussion, thus we have the following proposition:
Proposition 3 If
(a) N c L u + τ max > N c L c L t (2.51)
(b) {s i (t), 1 ≤ i ≤ L t } are linearly independent ∀t ∈ [(m − 1)N c + 1, mN c ] (2.52)
then R mm achieves full rank N c L c L t .
Note that condition (a) is simply the aforementioned sampling rate constraint and
condition (b) implies that different transmit antenna of the same user use linearly independent
spreading codes at each symbol time within the block.
Proof:
Consider any two spreading codes in G m , if their nonzero segments are not aligned,
they are linearly independent; if their nonzero segments are aligned, they are linearly
independent by condition (2.52). ✷
To achieve full diversity gain, we have the following proposition.
Proposition 4 In quasi-static fading channels, if R mm is full rank and
for all i, ∆d i (t) ≠ 0, for at least one (m − 1)N c + 1 ≤ t ≤ mN c (2.53)
block m achieves full diversity gain L c L t .
Proof:
Recall from Equation (2.16) and (2.10), that for the quasi-static channel,
⎡
⎤
∆D((m − 1)N c + 1)
∆D m = ⎢ . ⎥
⎣
⎦
∆D(mN c )
22

where
group 1
group L c
{ }} { { }} {
∆D(t) = diag[ ∆d 1 (t), . . . , ∆d Lt (t), . . . . . . ∆d 1 (t), . . . , ∆d Lt (t)] ∈ C LcLt×LcLt
Condition (2.53) guarantees that the null space of ∆D m is empty, i.e.,
∀v ≠ 0, ∆D m v = u ≠ 0
when R m is full rank, it is positive definite, i.e.,
∀u ≠ 0, u H R m u > 0
hence
∀v ≠ 0, v H ∆D H mR m ∆D m v = v H Φ mm v > 0
this means Φ mm is positive definite and thus is full rank L t L c . ✷
Under the assumption of equicorrelated spreading codes (a model commonly employed
to gain intuition about spread systems), we can also make a statement about coding gain
for flat fading channels.
Proposition 5 (Coding Gain)
In the quasi-static flat fading channel (L c = 1), if R mm (0) ∈ R KLt×KLt
has equal
cross-correlation entries − 1
KL t−1 < ρ c < 1, and for all users condition (2.53) is satisfied,
full diversity gain is achieved. The coding gain is a monotonically decreasing function of
positive ρ c and is a monotonically increasing function of negative ρ c .
Proof:
Denote R mm (0) explicitly as a function of ρ c by R mm (ρ c ). For this case, Φ mm can be
rewritten as Φ mm = ∆D H ∆D⊙R mm . For − 1
KL t−1 < ρ c < 1, R mm (ρ c ) is positive definite.
Since condition (2.53) holds for all users, ∆D H ∆D has no diagonal entries equaling zero,
by [HJ91], ∆D H ∆D ⊙ R mm (ρ c ) is positive definite hence full rank.
For |ρ 1 | < |ρ 2 |, R mm (|ρ 1 |/|ρ 2 |) is easily seen to be positive definite. We next plug
∆D H ∆D⊙R mm (ρ 2 ) and R mm (|ρ 1 |/|ρ 2 |) into Oppenheim’s inequality (page 480 of [HJ91])
23

as A and B respectively, and observe that all diagonal entries of R mm (|ρ 1 |/|ρ 2 |) are equal
to unity, then we have
(
( ))
det(∆D H ∆D ⊙ R mm (ρ 2 )) ≤ det ∆D H |ρ1 |
∆D ⊙ R mm (ρ 2 ) ⊙ R mm
|ρ 2 |
= det(∆D H ∆D ⊙ R mm (ρ 1 ))
the last equation holds if ρ 1 , ρ 2 have the same sign. Due to the positive definiteness of
∆D H ∆D ⊙ R mm (ρ 1 ), we have a strict inequality if ρ 1 , ρ 2 have the same sign
det(∆D H ∆D ⊙ R mm (ρ 2 )) < det(∆D H ∆D ⊙ R mm (ρ 1 )).
✷
We comment that Proposition 5 is about performance evaluation, where all the active
users should be considered (condition (2.53) is required to hold for all users); while in
code design, the worst case analysis tells us that if each user optimizes his own code,
the overall performance improves, so code design is a single user problem. Increasing the
spreading typically reduces |ρ c | and yields a larger coding gain by Proposition 5. We can
trade off performance for spectrum.
2.2.5 The Effect of Multipath
For quasi-static channels, as was expected, multipath contributes more diversity gain
for perfect CSI. For spread systems, for each block, each path typically contributes L t
diversity levels. Now we look into the detailed structure of R mm (0) and discuss why, for
small delays and quasi-static channels, codes with good coding gain for flat channels still
have good coding gain for multipath channels.
Recall that in Equation (2.35),
n+1
∑
Φ mm = ∆DL H c
∆D Lc ⊙ R mm (0) + ∆DL H c
Q n+1 ∆D Lc ⊙ (R mm (t) + Rmm(t))
T
t=1
24

For small multipath delays, a typical spread system has R mm (t) ≃ 0, t ≠ 0, therefore
Φ mm ≃ ∆D H L c
∆D Lc ⊙ R mm (0). We partition R mm (0) as follows
⎡
⎤
R (1,1) · · · R (1,Lc)
R mm (0) =
.
⎢ . .. . ⎥
⎣
⎦ where R (i,j) = [s i 1, . . . , s i L t
] T [s j 1 , . . . , sj L t
]. (2.54)
R (Lc,1) · · · R (Lc,L c)
R (i,j) ≃ 0, i ≠ j, that is, well designed spreading codes have small cross-correlations
for multiple lags. This implies det(Φ mm ) ≈ ∏ L c
j=1 det(∆DH ∆D ⊙ R (j,j) ). Furthermore,
if R (i,i) ≃ R (j,j) , i ≠ j, then we can further approximate det(Φ mm ) ≈ [det(∆D H ∆D ⊙
R (1,1) )] Lc . Note that det(∆D H ∆D ⊙ R (1,1) ) corresponds to the coding gain for a single
path system. Hence we see that good flat fading channel codes tend to give large coding
gain even for multipath channels.
For a spread system, based on the consideration of random sequences, we can use the
following spreading code correlation model[AM00]:
(s l 1
i1
) T (s l 2
i2
) =
where ρ c is a common cross-correlation value.
⎧
1 if τ l 1
i 1
= τ l 2
i 2
and i 1 = i 2 ,
⎪⎨ ρ c if τ l 1
i 1
= τ l 2
i 2
and i 1 ≠ i 2 ,
ρ c × Lu−|τ l 1
i1 −τ l 2
i2 |
L u
if 0 < |τ l 1
i 1
− τ l 2
i 2
| < L u ,
⎪⎩ 0 else.
(2.55)
To examine the effects of multipath on well-designed single path codes, we consider
a specific code: the 2 × 2 rate 1 BPSK Class 1 code, which is discussed in depth in
Section 2.5. If ρ c = 0.3, L u = 32, for L c = 1 with delay profile [0; 0], and for L c = 2 with
delay profile [0 1; 0 1], we compare the distance spectra (the enumeration of the coding
gains of all codeword pairs) in Table 2.1. As we can see, the coding gain for L c = 2
is roughly the square of the coding gain for L c = 1, which agrees with our approximate
analysis above. For the L c = 2 case, we have searched for optimal codes with delay profile
[0, τ1 2; τ 2 1, τ 2 2], where 0 ≤ τ i l ≤ 30, the optimal codes all turn out to be the same as that
for the flat fading channel. An exhaustive search for L c = 3 and τ max ≤ 5 shows that the
flat fading channel code is still optimal. Figure 2.4 shows the union bounds for 2 × 2 rate
25

L c = 1
0 7 9 14
0 0.00 30.56 16.00 30.56
7 30.56 0.00 30.56 16.00
9 16.00 30.56 0.00 30.56
14 30.56 16.00 30.56 0.00
L c = 2
0 7 9 14
0 0 765.06 214.58 765.06
7 765.06 0 771.70 214.58
9 214.58 771.70 0 771.70
14 765.06 214.58 771.70 0
Table 2.1: The distance spectra for rate 1 BPSK 2 × 2 Class 1 OMM code.
Delay profile [0 τ; 0 τ]
2.2 x 10−3 τ
2
1.8
1.6
Union Bound
1.4
1.2
Class2 [1 7 8 14]
Class1 [0 7 9 14]
1
0.8
0.6
0 5 10 15 20 25 30 35
Figure 2.4: Union bound as a function of path delay τ.
1 BPSK Classes 1 and 2 as a function of path delay τ with delay profile [0, τ; 0, τ]. The
performance of the two codes are quite robust to the value of the path delay τ.
2.3 The Downlink
In the downlink, without loss of generality, we consider the situation where the base
station communicates with user 1. From the sequence decoder analysis for the uplink
model in Section 2.2, we observe that the performance of sequence decoder, which is
a function of Φ, is dominated by the performance of the single block code, which is
a function of Φ mm . Furthermore, sequence code design decouples to single block code
design. In the downlink, similar observations can be made and we jump directly to single
26

lock decoder analysis. We will consider two decoder structures, the jointly optimal ML
decoder for all users and a single user ML–based decoder. The key difference between the
uplink and the downlink is the fact that in the downlink, all users experience the same
channel, thus resulting in a multiuser coding problem. Note that the following downlink
model is general enough to encompass the case where different transmit antennae of the
same user employ distinct spreading codes, s (k)
i
(t) ≠ s (k)
i
(t), and the case where different
transmit antennae of the same user employ identical spreading codes, s (k)
i
(t) = s (k)
i
(t).
2.3.1 The Downlink Model
At the base station, each user’s information is encoded in the same way as shown in
Figure 2.1, but the transmitted signal for each user is combined on each transmit antenna,
and the sum signals are transmitted. We will initially assume that user 1 knows all users’
spreading codes and the common channel state information, while this assumption is
impractical, the decoder will provide a bench mark for the suboptimal decoder discussed
in the sequel.
information.
The matched filter outputs are constructed using the spreading code
At symbol time t, the received signal due to d (k)
i
(t) is,
r(t) = √ σ t
∑
∑
K ∑L t L c
k=1 i=1 l=1
g l(k)
i
(t)d (k)
i
(t)h l i(t) + n(t) ∈ C (Lu+τmax)×1 . (2.56)
A single set of channel coefficients h l i (t) appear in (2.56), this is due to the fact that each
user experiences the same channel in the downlink. Written in matrix form
r(t) = √ σ t G(t)D(t)h(t) + n(t) ∈ C (Lu+τmax)×1 , (2.57)
where
G(t) = [G (1) (t), . . . , G (K) (t)] ∈ R (Lu+τmax)×KLcLt , (2.58)
and G (k) (t) is defined in Equation (2.9). The codeword matrix D(t) is
⎡ ⎤
D (1) (t)
D(t) = ⎢ . ⎥
⎣ ⎦ ∈ CKLcLt×LcLt , (2.59)
D (K) (t)
27

where D (k) (t) is defined in Equation (2.10), finally, h(t) h (1) (t) is defined in Equation
(2.11).
In comparison with the uplink case in Equation (2.12), we see that D(t)’s have the
same nonzero entries but are arranged in a different form; h(t) in the downlink is similar
to h(t) in the uplink except that other users’ channel coefficients are not present, since
they are the same as those of user 1.
Similarly, “concatenating” the r(t), t = 1, . . . , N c into a larger vector, r, we have the
received signal for user 1,
⎡
⎤ ⎡ ⎤ ⎡ ⎤
G(1) ∅ ∅ D(1)
n(1)
r = √ σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ h(t) + ⎢ . ⎥ (2.60)
⎣ ⎦
∅ ∅ G(N c ) D(N c ) n(N c )
√ σ t GDh + n ∈ C (NcLu+τmax)×1 . (2.61)
Note h h(t) is constant within one block for the quasi-static channel, and adjacent
blocks of G(t) are offset by L u rows.
The matched filter output is
y = G T r = √ σ t RDh + m ∈ C NcKLcLt×1 , (2.62)
where R = G T G is the correlation matrix of spreading codes, m = G T n has distribution
N (0, R).
2.3.2 Joint Maximum Likelihood Decoder
The jointly optimal ML decoder for one block is employed to decode all user’s information
jointly. The matched filter output vector is y ∼ N ( √ σ t RDh, R). Similar to the uplink
case, the averaged pairwise error probability is upper bounded by
1
E h [P (D α → D β |h)] ≤ ∣
∣I + σt
4 Σ h(D α − D β ) H R(D α − D β ) ∣ , (2.63)
where
Σ h = E[hh H ]. (2.64)
28

Similarly the key matrix Φ is defined as,
Φ = (D α − D β ) H R(D α − D β ) = ∆D H R∆D ∈ C LcLt×LcLt . (2.65)
Again, if nL u ≤ τ max < (n + 1)L u , and each transmit antenna uses the same fixed
spreading code at each symbol time, the correlation matrix R has the same structure as
R defined in (2.32).
For the quasi-static fading channel
⎡ ⎤
∆D(1)
∆D(2)
Φ =
⎢ . ⎥
⎣ ⎦
∆D(N c )
H
⎡
⎤
R(0) R(1) . . . R(n + 1) 0 ⎡ ⎤
R(1) T
.
R(0) .. . .. ∆D(1)
R(n + 1)
. 0 .. . .. . .. ∆D(2)
0
.
⎢
⎣
R(n + 1) T . .. . .. . .. ⎢ . ⎥
R(1) ⎥ ⎣ ⎦
⎦ ∆D(N c )
0 R(n + 1) T 0 R(1) T R(0)
(2.66)
Note that the difference between the uplink Φ and the downlink Φ resides only in the
form of ∆D(t) (defined in Equation (2.14)) and the ∆D(t) (defined in Equation (2.59)).
Note that R(0) and R(t) can be partitioned into K × K block matrices,
⎡
⎤
R (1,1) (0) · · · R (1,K) (0)
R(0) =
.
⎢ . .. . ⎥
⎣
⎦
R (K,1) (0) · · · R (K,K) (0)
thus ¯Φ can be simplified as
⎡
R(t) = ⎢
⎣
⎤
n (t) · · · R n
(1,K) (t)
.
. .. . ⎥
⎦ , (2.67)
n (t) · · · R (K,K) (t)
R (1,1)
R (K,1)
n
Φ =
where
K∑
K∑
i=1 j=1
(∆D (i)
L c
) H ∆D (j)
L c
⊙ R (i,j) (0)+
n∑
t=1
(∆D (i)
L c
) H Q n+1 ∆D (j)
L c
∆D (k)
(
⊙
R (i,j)
n
(t) + (R (i,j)
n (t)) T ) ∈ C LcLt×LcLt , (2.68)
L c
{ }} {
L c
= [ ∆D (k) , . . . , ∆D (k) ] ∈ C Nc×LcLt , (2.69)
with D (k) defined in Equation (2.2) and Q defined in Equation (2.37).
29

For this special case of fixed spreading codes, we see clearly the summation of contributions
from different users’ codeword difference matrices and correlation matrices in the
final Φ. The matrix Φ cannot be decomposed in such a way as to decouple the contributions
of each of the active users. Again, we emphasize that this is a multiuser coding
problem. Trying to design a code which is globally optimal in such a high dimensional
space is computationally expensive. We want to point out the connection between downlink
Φ and uplink Φ: in downlink Φ, there are K terms (i = j), which are equivalent to
Φ, and these K terms have relatively large R (i,j) (0). This suggests that a good uplink
code will perform reasonably well in the downlink.
2.3.3 Single user ML-based Decoder
If user 1 is only interested in his own data and treats other user’s data as colored Gaussian
noise, then we can form a single user approximate ML decoder.
We separate the contribution of user 1’s data and the interferers’ data in Equation
(2.57)
r j (t) = √ σ t G 1 (t)D 1 (t)h(t) + √ σ t G 2 (t)D 2 (t)h(t) + n j (t) ∈ C (Lu+τmax)×1 , (2.70)
where
G 1 (t) = G (1) (t) ∈ R (Lu+τmax)×LcLt G 2 (t) = [G (2) (t), . . . , G (K) (t)] ∈ R (Lu+τmax)×(K−1)LcLt ,
and
(2.71)
D 1 (t) D (1) (t) ∈ C LcLt×LcLt (2.72)
⎡ ⎤
D (2) (t)
D 2 (t) ⎢ . ⎥
⎣ ⎦ ∈ C(K−1)LcLt×LcLt (2.73)
D (K) (t)
30

“Concatenating” the r j (t), t = 1, . . . , N c into a large vector,
⎡
⎤ ⎡ ⎤
G
r = √ 1 (1) ∅ ∅ D 1 (1)
σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ h
∅ ∅ G 1 (N c ) D 1 (N c )
⎡
⎤ ⎡ ⎤ ⎡ ⎤
G
+ √ 2 (1) ∅ ∅ D 2 (1) n j (1)
σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ h + ⎢ . ⎥
⎣ ⎦
∅ ∅ G 2 (N c ) D 2 (N c ) n j (N c )
(2.74)
= √ σ t G 1 D 1 h + √ σ t G 2 D 2 h + n ∈ C (NcLu+τmax)×1 (2.75)
The matched filter, which consists of user 1’s spreading codes only, outputs
y = √ σ t G T 1 G 1 D 1¯h +
√
σt G T 1 G 2 D 2 h + m ∈ C LcLt×1 , (2.76)
where
m = G T 1 n. (2.77)
Treating √ σ t G T 1 G 2D 2 h + m as Gaussian noise ˜m, we have
E[ ˜m] = 0, E[ ˜m ˜m H ] = σ t G T 1 G 2 G T 2 G 1 + G T 1 G 1 . (2.78)
matrix
where
Following our prior approach, we end up with the new correlated codeword difference
˜Φ = ∆D1 H (ŘT 1 Ř−1 2 Ř 1 )∆D 1 ∈ C LcLt×LcLt (2.79)
Ř 1 = G T 1 (t)G 1 (t) ∈ C LcLt×LcLt ,
Ř 2 = σ t G T 1 (t)G 2 (t)G T 2 (t)G 1 (t)+G T 1 (t)G 1 (t) ∈ C LcLt×LcLt
(2.80)
The matrix ˜Φ can be simplified and approximated to
˜Φ ≃ (∆D (1) ) H ∆D (1) ⊙ ˜R ∈ C LcLt×LcLt , (2.81)
31

where
˜R = ŘT 1 Ř−1 2 Ř 1 . (2.82)
This formulation converts this joint coding problem to a equivalent single user coding
problem with a redefined correlation matrix ˜R. We will show in Section 2.5.2 that ˜R
acts like a modified R in the uplink coding problem, thus we can apply the same code
design principles and search for codes which are equally applicable to both the uplink
and downlink scenario. We note that Ř1 and Ř2 can be estimated at the downlink in
order to construct the decoder.
2.4 Non-spread systems
The system model in Section 2.2.1 can be applied to non-spread systems with slight modification.
The system now consists of a single user and the spreading code is interpreted
as samples of the modulation waveform with length L u . All transmit antennae employ
the same modulation waveform, the entries of which can take on arbitrary real values.
This perspective offers great advantage in analyzing non-spread systems in multipath
fading channels. An important difference between spread and non-spread systems is that
in non-spread systems, to enhance spectral efficiency, pulse shapes have duration longer
than that of a symbol interval, see Figure 2.5 for the distinction between spread and
non-spread transmissions. Thus there is significant overlap between the contributions of
consecutive symbols, which causes Inter-Symbol Interference (ISI). To characterize this
situation, a new parameter L s , the number of samples between consecutive transmissions,
is introduced. We highlight the difference due to the new parameter in the system model.
In Equation (2.16), while concatenating r(t), t = 1, . . . , N c into r, due to the ISI, the
adjacent blocks of G(t) are offset by L s instead of L u . The final received vector is
r = √ σ t GDh + n ∈ C ((Nc−1)Ls+Lu+τmax)×1 . (2.83)
The dimension of G m ∆D m changes accordingly
G m ∆D m ∈ C ((Nc−1)Ls+Lu+τmax)×NcLtLc . (2.84)
32

Spread
£¥¤ ¦¥§
Non-spread
ªX«
¨¥©
Figure 2.5: Distinction between spread and non-spread transmission.
In Equation (2.50), to resolve the available diversity, the sampling rate is required to
satisfy
(N c − 1)L s + L u + τ max > N c L t L c . (2.85)
The diversity gain is still the rank of F for quasi-static fading channels. For nonspread
systems, F has dimension ((N c − 1)L s + L u + τ max ) × L t L c . Due to the same
modulation waveform, full diversity L t L c is not as easily achieved, we have the following
proposition regarding the diversity level:
Proposition 6 For non-spread systems in quasi-static multipath fading channels, the
diversity level
rank Φ mm ≥ max(the number of distinct path delays τ (l)
i
, flat fading diversity). (2.86)
Proof:
Recall Φ mm = F H m F m where F m = G m ∆D m . Columns in F m with distinct delays
have different numbers of heading zeros and hence are linearly independent, therefore,
rank Φ mm ≥ the number of distinct path delays τ (l)
i
. Aligning the nonzero segments of
columns in F m reduces rank F m , when all columns are aligned, rank F m = flat fading diversity,
therefore rank Φ mm ≥ flat fading diversity. ✷
A weaker statement is made in [TNSC99] with much more complicated analysis.
33

2.5 Optimal Minimum Metric Codes
In this section, we analyze the properties of optimized codes found via computer search.
For simplicity, the setup for search is based on flat fading channels (L c = 1), but as
predicted in Section 2.2.5, we will show the consistency of the performance of the found
codes in both multipath and flat fading channels. Recall from Equation (2.35), in the
uplink for flat quasi-static channels, Φ mm (we will drop the subscript mm for convenience)
takes the following simple form
Φ(D α , D β ) = (D α − D β ) H R mm (0)(D α − D β ) = ∆D H ∆D ⊙ R mm (0) ∈ C Lt×Lt (2.87)
and from Equation (2.68) in the downlink, Φ takes the following form
Φ(D α , D β ) =
K∑ K∑
(∆D (i) ) H ∆D (j) ⊙ R (i,j) (0) ∈ C Lt×Lt (2.88)
i=1 j=1
In the sequel, all codes are required to achieve full diversity. Optimum minimum metric
(OMM) codes are optimized by maximize the minimum coding gain. Usually more than
one set of codes can achieve the above criteria. In our study, the distance spectrum
and the union bound are used to further evaluate the performance of these codes. The
distance spectrum is the enumeration of the coding gains of all codeword pairs. At high
SNR, for a finite number of block codes, union bound is asymptotically tight, hence a
good indicator of the performance. Figure 2.6 compares the union bound with simulated
performance for OMM 2 × 2 BPSK Class 1 code (see Figure 2.7 for Class 1 code). In
spread systems, codes are optimized for ρ c = 0.3.
2.5.1 OMM Codes for the Uplink
All the found codes are summarized in Table 2.2. Note that several classes of codes may
satisfy the same OMM criteria, only the best in terms of the union bound at specified ρ c
are listed in the table. The rate 1 2 × 2 BPSK codes consist of three equivalence classes,
whose distance spectra are listed in Figure 2.7. Their union bound versus ρ c is plotted
in Figure 2.8.
We observe:
34

UB versus SIM, 2x2 BPSK Class 1, ρ c
=0.3
10 0 SNR
Union Bound
Simulation
10 −1
SER
10 −2
10 −3
10 −4
0 2 4 6 8 10 12 14 16
Figure 2.6: Union bound versus simulation for rate 1 OMM 2 × 2 BPSK Class 1 code.
35

index ρ c rate size constl OMM
1 1 1 2 × 2 BPSK [1,7,8,14]
2 1 1 2 × 2 QPSK [1,41,156,247]
3 1 1 2 × 2 8PSK [3,1314,2167,3414]
4 1 1 2 × 2 16PSK
[
[8,14469,33744,47965]
]
1 84 166 248
5 1 1 3 × 3 QPSK
[
282 335 429 499
]
2 42 93 117
6 1 1.5 2 × 2 QPSK
128 168 223 247
⎡
⎤
0 20 42 62
7 1 2 2 × 2 QPSK
⎢ 70 82 107 127
⎥
⎣133 145 173 185⎦
195 215 236 248
8 0.3 1 2 × 2 BPSK [0,6,11,13]
9 0.3 1 2 × 2 QPSK [1,87,174,248]
10 0.3 1 2 × 2 8PSK [3,1314,2447,3246]
11 1 1 2 × 2 16PSK
[
[8,14469,33744,47965]
]
0 31 99 124
12 0.3 1 3 × 3 QPSK
[
421 442 454 473
]
2 42 93 117
13 0.3 1.5 2 × 2 QPSK
128 168 223 247
⎡
⎤
0 10 37 47
14 0.3 2 2 × 2 QPSK
⎢ 82 88 119 125
⎥
⎣133 143 160 170⎦
15 0.3 1 4 × 2 BPSK
215 221 242 248
⎡
⎤
0 15 51 60
⎢ 86 89 101 106
⎥
⎣149 154 166 169⎦
195 204 240 255
Table 2.2: OMM codes.
36

1. As observed by others, the space defined by coding gain is not a metric space, since
the triangle inequality does not hold. Figure 2.7 shows that the Class 2 code has
four “edges” with coding gain 16 and two “diagonals” with coding gain 64.
2. These codes are metric uniform, i.e. the distance profiles are the same for all
codewords. We avoid the term geometrically uniform, as Forney’s definition of a
geometrically uniform signal set [GDF91] is predicated upon the presence of a metric
space.
3. The coding gain corresponds to the “length” of the shortest branch in the distance
spectrum. Two codes with the same coding gain do not necessarily have the same
performance, as they might have different distance spectra. This fact can be easily
observed from the union bound plots in Figure 2.8.
4. Performance is generally a function of ρ c . From the distance spectra figures, it is
easy to note that Class 1 and Class 3 have performance dependent on ρ c , while
Class 2 does not. This is due to the fact that Class 2 is an orthogonal code set.
Class 2 is Alamouti’s orthogonal code set, which is optimal by either OMM or OUB
standards. Actually all orthogonal codes [Ala98, TJC99] have coding gain independent
of spreading coding correlation, because the codeword difference matrices of
these orthogonal codes are still orthogonal.
5. The sensitivity of code performance to ρ c is a function of the coefficient of ρ 2 c. For
Class 1, the coefficient of ρ 2 c is 16; for Class 3, it is 64. Thus we expect Class 3 to be
more sensitive to ρ c than Class 1. This conjecture is verified from the union bound
plot in Figure 2.8.
6. Class 3 is not an OMM code for all ρ c , because when ρ c exceeds √ 3/2, 64 − 64ρ 2 c
will be less than 16.
7. Class 1 outperforms Class 2 for ρ c < 0.6.
The same set of optimal codes for rate one 8PSK and 16PSK are optimal for the
spread and non-spread systems. The 16PSK OMM codes have several distinguishing
properties:
1. It is 0.3dB better than the BPSK OMM codes (orthogonal)(see Figure 2.9).
37

16
0 9
32-16ρ 2 32-16ρ 2
16 16
16
1 7
64 64
16 16
16
0 6
64-64ρ 2 64-64ρ 2
16 16
16
14 7
16
8 14
16
9 15
Class 1 Class 2 Class 3
Figure 2.7: Distance spectra for rate 1 OMM 2 × 2 BPSK codes.
0.08
0.075
BPSK C1
BPSK C2
BPSK C3
SNR=6dB
0.07
Union Bound
0.065
0.06
0.055
0.05
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ρ c
Figure 2.8: Rate 1 OMM 2 × 2 BPSK block codes at SNR=6dB.
38

[
3 1314 2447 3246
8PSK u
0
8 u 0 ] [
8 u
2
8 u 4 ] [
u 0 8 u 3 8 u
4
8 u 6 ] [
8 u 4 8 u 2 8 u
6
8 u 2 ]
8 u 1 8 u 7 8
8 u 5 8 u 6 8
8 14469 33744 47965
16PSK [ u 0 16 u 0 ] [
16 u
3
16 u 8 ] [
u 0 16 u 8 16 u
8
16 u 3 ] [
16 u 8 16 u 5 16 u
11
16 u 13
16 u 0 16 u 11 ]
16
16 u 5 16 u 13
16
Table 2.3: Detailed Structure of rate 1 OMM 2×2 8PSK and 16PSK Codes. ( u n = e i 2π n )
ρ c
=0.3
2X2 BPSK
2X2 16PSK
10 −2
Union Bound
10 −1 SNR
10 −3
10 −4
6 7 8 9 10 11 12 13 14 15 16
Figure 2.9: Rate 1 OMM 2 × 2 BPSK and 16PSK codes for ρ c = 0.3.
2. It is unitary, but, it does not form a group under matrix multiplication, neither is it a
subset of group unitary codes [SHHS01]. The detailed structure of the OMM 8PSK
and 16PSK (Table 2.3) suggests that as more degrees of freedom are introduced,
code tends to be more unitary. This agrees with the general structure suggested in
[MH99].
3. The 16PSK optimal code is actually “correlation free”. It is OMM for all ρ!
The rate 1 OMM 3 × 3 BPSK codes are listed in Table 2.4. We make the following
observations:
1. There are 3 equivalence classes which are OMM codes for ρ c = 0.3, but their multiplicity
of the minimum metric in the distance spectra, N p , are different. The
39

Class ρ ∆ p (ζ v ) N p code index
C1 ρ = 0.3 122.24(0.6431) 8 [0,31,99,124,421,442,454,473]
C2 ρ = 0.3 122.24(0.6431) 12 [0,31,99,124,394,433,470,493]
C3 ρ = 0.3 122.24(0.6431) 16 [0,31,99,124,394,405,489,502]
C4 ρ = 1.0 64(0.5774) 22 [1,84,166,248,282,335,429,499]
Table 2.4: Rate 1 OMM 3 × 3 BPSK Codes.
multiplicity of a certain coding gain is the number of codeword pairs which achieve
that coding gain. Larger multiplicity for small coding gains results in higher probability
of decoding error, thus worse performance. For minimum metric 122.24,
Class 1 has multiplicity 8, Class 2 has multiplicity 12 and Class 3 has multiplicity
16. The union bound plot confirms that Class 1 is slightly better than Class 2 and
Class 2 is slightly better than Class 3.
2. Class 4 is OMM code for ρ c = 1.
3. The performances of all codes are correlation dependent, as shown in Figure 2.10.
Class 1, 2, and 3 are more sensitive to correlation than Class 4.
4. The union bound versus SNR plot of Class 1 and Class 4 is in Figure 2.11. Class 1
is more sensitive to correlation than Class 4. At ρ c = 1, Class 1 is 1dB worse than
Class 4, but at ρ c = 0.3, it is 1dB better than Class 4.
5. In Figure 2.11, the best rate 1, 3 × 3 unitary group code listed in [SHHS01], which
is a cyclic group achieving diversity product ζ v = 0.5134 and using the 8PSK constellation,
is compared with Class 4, which is not unitary but constant envelop
modulation, has diversity product ζ v = 0.5774 and uses only the BPSK constellation.
We compare the rate 2 2 × 2 QPSK OMM codes with the orthogonal code.
1. In a non-spread system (ρ c = 1), the performances of OMM spread, OMM nonspread
and orthogonal codes are compared in Figure 2.12. The orthogonal code is
about 0.3dB better than the OMM non-spread code, both are better than OMM
spread code, which is optimized for spread systems and loses full diversity in a nonspread
system. The orthogonal code turns out to have better distance spectrum
than the OMM non-spread code [GVM02], therefore has better performance.
40

SNR=6dB
10 −1 ρ c
Union Bound
3X3 BPSK C1
3X3 BPSK C2
3X3 BPSK C3
3X3 BPSK C4
10 −2
−0.5 0 0.5 1
Figure 2.10: Rate 1 OMM 3 × 3 BPSK codes at SNR=6dB.
10 0
10 −1
SNR
3X3 Unitary Group ρ c
=1
3X3 BPSK C1 ρ c
=1
3X3 BPSK C4 ρ c
=1
3X3 BPSK C1 ρ c
=0.3
Union Bound
10 −2
10 −3
10 −4
0 2 4 6 8 10 12 14
Figure 2.11: Rate 1 OMM 3 × 3 BPSK codes.
41

2. In a spread system (ρ = 0.3), the performances of OMM spread, OMM non-spread
and orthogonal codes are compared in Figure 2.13. OMM spread code is about
0.3dB better than that of OMM non-spread code, and 0.6dB better than that of
orthogonal code.
3. The performance of OMM spread and orthogonal codes are compared in the following
setup with Gold sequence of length L u = 31:
(a) (Figure 2.14) Single user K = 1, flat fading L c = 1. Because Gold sequence
has cross-correlation even smaller than 0.3, the OMM spread code is 0.7dB
better than the orthogonal code.
(b) (Figure 2.15) Single user K = 1, multipath fading channel L c = 2 with delay
profile (defined in Equation (2.4) and (2.5))
⎡ ⎤
Γ = ⎣ 0 5 ⎦ .
0 5
The OMM spread code is about 1.2dB better than the orthogonal code, the
gain almost doubles than that in flat fading channels.
(c) (Figure 2.15) Two user K = 2, multipath fading channel L c = 2 with delay
profile
⎡
⎤
Γ = ⎣ 0 5 7 10 ⎦ .
0 5 7 10
The OMM spread code is still about 1.2dB better than the orthogonal code.
The two user system is about 1.2dB worse than the single user system.
(d) (Figure 2.16) Sequence decoder M = 4, two user K = 2, multipath fading
channel L c = 2 with delay profile
⎡
⎤
Γ = ⎣ 0 5 7 10 ⎦ .
0 5 7 10
The OMM spread code is about 0.7dB better than the orthogonal code.
42

10 0 SNR
OMM spread
OMM non−spread
Orthogonal
Union Bound
10 −1
10 −2
6 7 8 9 10 11 12 13 14 15 16
Figure 2.12: Rate 2 OMM 2 × 2 QPSK Codes at ρ c = 1.0.
OMM spread
OMM non−spread
Orthogonal
10 −1
Union Bound
10 0 SNR
10 −2
10 −3
6 7 8 9 10 11 12 13 14 15 16
Figure 2.13: Rate 2 OMM 2 × 2 QPSK Codes at ρ c = 0.3.
43

10 0 SNR
OMM spread
orthogonal
10 −1
Union Bound
10 −2
10 −3
6 7 8 9 10 11 12 13 14 15 16
Figure 2.14: Rate 2 OMM 2 × 2 QPSK Codes, Gold sequence, single user, flat fading.
10 −1
K=1, OMM spread
K=1, orthogonal
K=2, OMM spread
K=2, orthogonal
Union Bound
10 0
10 −2
10 −3
SNR
10 −4
10 −5
6 7 8 9 10 11 12 13 14 15 16
Figure 2.15: Rate 2 OMM 2 × 2 QPSK Codes, Gold sequence, multipath fading Γ =
[0, 5; 0, 5].
44

10 0 SNR
OMM spread
orthogonal
10 −1
SER
10 −2
10 −3
0 2 4 6 8 10 12
Figure 2.16: Sequence decoder M = 4, Rate 2 OMM 2 × 2 QPSK Codes, Gold sequence,
multipath fading Γ = [0, 5, 7, 10; 0, 5, 7, 10].
For ρ c = 0.3, the rate 1, OMM 4 × 2 BPSK code is also found. In Figure 2.17, its
union bound shows about a 0.4dB gain over that of two continuous rate 1 OMM 2 × 2
BPSK Class 1 codes.
2.5.2 OMM Codes for the Downlink
As pointed out in Section 2.3.1, the optimal approach for the downlink is a joint multiuser
code design. Recall that we considered two cases in Section 2.3.2 and 2.3.3. First,
if the MAI in the correlated codeword difference matrix in (2.68) is ignored, the resultant
expression is identical to the uplink single user code and thus the code design problem
is also equivalent to the single user problem. Secondly, approximating the MAI as Gaussian
noise, we also get an equivalent single user code design problem with the modified
45

ρ c
=0.3
10 0 SNR
Rate 1 4X2
2 Rate 1 2X2
10 −1
Union Bound
10 −2
10 −3
0 2 4 6 8 10 12 14
Figure 2.17: Rate 1 OMM 4 × 2 BPSK Codes for ρ c = 0.3.
correlation matrix in (2.82). To understand this equivalence, we present an example. If
equicorrelated spreading codes are used for K users, the equivalent Ř is
Ř =
⎡
1
⎣ a
2σ t (K − 1)ρ 2 (1 − ρ) + 1 − ρ 2
b
⎤
b ⎦
a
where (2.89)
a = σ t (K − 1)ρ 4 − 2σ t (K − 1)ρ 3 + σ t (K − 1)ρ 2 − ρ 2 + 1 (2.90)
b = −σ t (K − 1)ρ 4 + 2σ t (K − 1)ρ 3 − ρ 3 − σ t (K − 1)ρ 2 + ρ (2.91)
This is equivalent to the single user case with correlation ρ e = b/a. In Figure 2.21, we plot
ρ e as a function of σ t and ρ. As seen from the plot, for a large range of σ t ∈ (0dB, 15dB)
and ρ ∈ (−0.2, 1), ρ e roughly lies within (−0.6, 0.6), for which case, the Class 1 codes in
Figure 2.7 are the best OMM rate 1 BPSK codes. Observe that ρ e is not a monotonic
function of either ρ or the SNR. As either K or σ t goes to ∞, Ř approaches,
⎡ ⎤
lim Ř = lim Ř = 1 − ρ ⎣ 1 −1 ⎦ (2.92)
σ t→∞ K→∞ 2 −1 1
46

where Class 2 in Figure 2.7 becomes the best OMM code. We note that the MAI approximation
is experienced in two ways: Ř has a modified correlation ρ c and there is a
reduction in effective SNR which is captured by the constant γ where Ř = γŘ, Ř is such
that Ř ii = 1.
For ρ c = 0.3, if distinct spreading codes are employed for different transmit antennae
of the same user, the multiuser rate 1 OMM 2 × 2 BPSK codes for two users are
C 1 = [3, 20, 40, 63, 78, 89, 101, 114, 141, 154, 166, 177, 192, 215, 235, 252],
its distance spectrum shows it is metric uniform. If identical spreading codes are employed
for different transmit antennae of the same user, the OMM codes for two users are
C 2 = [5, 18, 46, 57, 75, 92, 96, 119, 136, 159, 163, 180, 198, 209, 237, 250].
Figure 2.18 compares the performance of 4 sets of codes in the downlink with optimal
joint ML decoding: non-spread (single user) OMM code (Alamouti’s code, Class 2 in
Figure 2.7), spread single user OMM code (Class 1 in Figure 2.7), spread downlink two
user OMM code with same spreading (C 2 ) and spread downlink two user OMM code with
distinct spreading (C 1 ). The performance is improved by about 0.2dB when the next set
of codewords are used. We also consider the use of single user optimal codes (Class 1 in
Figure 2.7) with various ML-based decoders in Figure 2.19 and 2.20. The performance of
the following three decoders: joint optimal decoder, single user decoder ignoring MAI and
single user decoder approximating MAI as Gaussian noise, is investigated. In Figure 2.19,
we assume the two users have equal power, i.e., the Near-Far-Ratio (NFR) equals 0dB.
The joint ML decoder gives the best performance (about 1dB gain over that of single user
decoder ignoring MAI), the single user decoder approximating MAI gives slightly better
performance than the single user decoder ignoring MAI. If we increase user 2’s signal
power to be 4dB larger than user 1’s, i.e., NFR= 4dB, we have resulting performances as
seen in Figure 2.20. The joint ML decoder and the single user decoder approximating MAI
provide performance about 4dB and 2dB better than that of single user decoder ignoring
MAI, respectively. Thus we have the classic complexity and performance tradeoff. The
joint designs offer superior performance at the expense of a more complex decoder.
47

Code Comparison, Joint ML Decoder, ρ=0.3
10 0 SNR(dB)
SER
10 −1
Joint OMM, distinct spread
Single OMM, distinct spread
Alamouti Code, distinct spread
Joint OMM, same spread
UB on Joint OMM, distinct spread
UB on Single OMM, distinct spread
UB on Alamouti code, distinct spread
UB on Joint OMM, same spread
10 −2
0 1 2 3 4 5 6 7 8 9 10
Figure 2.18: Multiuser rate 1 OMM 2 × 2 BPSK codes.
Decoder Comparison, ρ=0.3
10 0 SNR(dB)
Joint ML Decoder
Single ML, App MAI
Single ML, Ignore MAI
SER
10 −1
10 −2
0 1 2 3 4 5 6 7 8 9 10
Figure 2.19: Decoder performance comparison with no near-far effect.
48

10 0 Decoder performance comparison, Single user OMM code, ρ c
=0.3, NFR=4dB
Joint ML Decoder
Single User ML, Aprx MAI
Single User ML, Ignore MAI
10 −1
SER
10 −2
10 −3
0 1 2 3 4 5 6 7 8 9 10
SNR(db)
Figure 2.20: Decoder performance comparison with near-far effect.
Equivalent ρ e
as a function of SNR and ρ c
1
0.8
0.6
0.4
0.2
ρ e
0
−0.2
−0.4
−0.6
−0.8
−1
15
10
SNR(dB)
5
0
−0.5
0
0.5
1
ρ c
Figure 2.21: Equivalent ρ e as a function of SNR and ρ c .
49

2.6 Practical Considerations
In this section, we consider practical issues that can affect code design. We present
some invariant operations which yield equivalent codes and their application to improve
code searches. We observe that some codes have performance independent of spreading
code correlation, this phenomenon is analyzed in some detail. Some suboptimal ways to
construct space-time block codes are also discussed.
2.6.1 Isometries
Isometries are defined as “distance preserving” transformations, where the “distance” can
be diversity gain ∆ H in (2.28) or coding gain in (2.29). For any function γ(.) on codeword
D, if
∆ p (γ(D i ) − γ(D j )) = ∆ p (D i − D j ) (2.93)
γ(.) is an isometry of coding gain ∆ p . Likewise isometry of diversity gain ∆ H can be
defined.
If we restrict to the flat fading quasi-static channel, we have Φ mm = ∆D H ∆D ⊙
R mm (0), the following isometries of ∆ H and ∆ p for spread systems can be identified (for
a discussion on isometries for non-spread systems, please refer to Section 4.2):
1. φ(D) = UD, where U is arbitrary unitary matrix U H U = I.
2. φ(D) = DP ,
• P is a unitary permutation matrix: P H P = I, each row and column of P has
only one non-zero element.
• Equi-correlated spreading codes.
3. φ(D) = [d ∗ ti ], componentwise complex conjugate
• Real spreading codes.
The right isometry P generally does not apply to spread system[Gen00], where spreading
codes have non-equal correlations, but does hold true for the academic equi-correlated
spreading code case.
Detailed proofs of the above isometries for spread systems are
omitted. We call two codes equivalent codes if they can be related by the above isometries.
50

The set of equivalent codes forms an equivalence class. Since all codes in an equivalence
class achieve the same performance, in terms of coding gain and diversity gain, we need
only investigate only one member of the equivalence class.
In the search process, once one member of equivalence class is found, all other members
can be ignored. This fact can greatly reduce the number of candidate codes. A
fundamental set is the smallest set of codes such that all possible codes are equivalent to
one code of this set. It is usually very hard to identify the fundamental set, but relatively
easy to identify a larger set which contains the fundamental set. In our case, we can
always use these isometries to force one codeword to have entries of value 1 on the top
row and left column, i.e.,
⎡
⎤
1 1 . . . 1
1 ∗ . . . ∗
.
⎢.
. .. . ⎥
⎣
⎦
1 ∗ . . . ∗
For example, if we want to search for 2 × 2 rate 1 BPSK codes, we need to search for four
2 × 2 codewords, each entry of which has two possible values, thus the total number of
possible combinations are 2 16 . By forcing one codeword to have the above structure, the
number of cases to be searched is reduced to 2 13 . For constellation size N cstl and block
size N c × L t , this can reduce the candidate set by a factor of N Nc+Lt−1
cstl
.
Another use of these isometries is to find all the equivalent codes. Due to the symmetry
of the codes, a subset of these invariant operations is sufficient to generate all the
equivalent codes. For example, the equivalence class of optimal minimal metric rate 1
BPSK 2 × 2 Class 1 code contains 8 equivalent codes, which can be generated from any
one of these equivalent codes plus a left multiplier which comes from the Dihedral Group
D 8 ,
⎡
⎣ 1 ⎤
0 ⎦
⎡
⎣ −1 ⎤
0 ⎦
⎡
⎣ 1 ⎤
0 ⎦
⎡
⎣ −1 ⎤
0 ⎦
0 1 0 1 0 −1 0 −1
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎣ 0 1 ⎦ ⎣ 0 −1 ⎦ ⎣ 0 1 ⎦ ⎣ 0 −1 ⎦ .
1 0 1 0 −1 0 −1 0
51

2.6.2 Sensitivity to Correlation
Usually, the performance of the resultant codes is a function of the spreading code correlation
function, but this is not true for all code sets. It can be desirable for a code to be
insensitive to correlation, i.e., to maintain its optimality or relatively good performance
for all correlation values, especially in multipath fading channels where the low valued
correlation functions for spreading codes do not always exist. We observe that in our
resulting optimal codes, the 2 × 2 BPSK Class 2 code (Alamouti’s code [Ala98]), which
is an orthogonal code, and the 2 × 2 16PSK code, which is an unitary code, have performance
independent of the spreading code correlation; thus we shall call these codes
“correlation free”. Unitary codes [HM00, SHHS01, Hug00, HS00] U satisfy UU H = I,
orthogonal codes [Ala98, TJC99] are unitary codes which have more structure. The 2 × 2
orthogonal codes have following structure:
⎡
⎣ x
y ∗
⎤
y ⎦
−x ∗
where x and y are from some constellation points. Indeed all orthogonal codes are correlation
free, because the codeword difference matrix is still orthogonal. A natural conjecture
is that unitary codes are correlation free too. But this is not true! For the 2-transmitantenna
case, we investigate what are the sufficient conditions to be correlation free.
Rewriting the two codewords D α and D β in terms of their column vectors
D α = [α 1 α 2 ] D β = [β 1 β 2 ], (2.94)
the condition of being correlation free is met when the off-diagonal entries of (D α −
D β ) H (D α − D β ) are zero, which leads to
α H 1 α 2 + β H 1 β 2 − α H 1 β 2 − β H 1 α 2 = 0. (2.95)
Unitary codes force the first two terms in Equation (2.95) to be zero by definition, but
do not necessarily cancel the last two terms. An example is
⎡ ⎤ ⎡ ⎤
D α = ⎣ 1 1 ⎦ D β = ⎣ i 1 ⎦ ,
1 −1
−1 −i
52

which does not satisfy Equation (2.95) and has
⎡
⎤
6 (−2 + 2i)ρ
Φ = ⎣
c
⎦ ,
(−2 − 2i)ρ c 2
and
det(Φ) = 12 − 8ρ 2 c.
Furthermore, unitary group codes are not always correlation free. An example which
has performance dependent on correlation is G 21,4 , which is given in [SHHS01]. Two
elements of this fixed-point-free group are
⎡ ⎤
η 0 0
A = ⎢
⎣0 η 4 0 ⎥
⎦
0 0 η 16
⎡ ⎤
0 1 0
B = ⎢
⎣ 0 0 1⎥
⎦ (2.96)
η 7 0 0
where η = e 2πj/21 . These two codewords have
Φ =
⎡
⎢
⎣
⎤
2 −η −1 ρ c −η 9 ρ c
−ηρ c 2 −η −4 ρ c
⎥
⎦ ,
−η −9 ρ c −η 4 2
and
det(Φ) = 8 − 6ρ 2 c + ρ 3 c.
Interestingly, we have checked all the smallest examples for 2 × 2 irreducible fixedpoint-free
unitary group codes listed in [SHHS01], i.e., G 6,−1 , D 4,1,−1 , E 3,1 , F 3,1,−1 and
J 1,1 , and found that all of these are correlation free. It seems that the group structure
implicitly forces these unitary codes to satisfy the condition of being correlation free in
the 2 × 2 case, but does not do so for the 3 × 3 case.
2.6.3 Suboptimal Codes
Most space-time block codes used in DS-CDMA systems are adopted from non-spread
codes. By forcing some structure on space-time code sets (like BLAST [Fos96] and layered
space-time codes [GARH01]), the code design or the received signal processing can
53

structure
]
1 structure
]
2 structure
]
3
[˜c1 ˜c
space-time code
2
[˜c1 ˜c 1
[˜c1 ˜c 2
˜c 2
[
˜c 1
]
˜c 2
[
˜c 2
]
˜c 1
[
˜c 2
]
1 0 1
Φ δmin
2 δ
0 1
min
2 ρc 2 0
δ
ρ c 1
min
2 0 0
diversity gain 2 2 1
coding gain δmin 4 δmin 4 (1 − ρ2 c) 2δmin
2
Table 2.5: Three space-time code structures and their performance comparison.
be facilitated at the expense of performance loss. Based on our previous analysis, the
performance of these codes are easily analyzed as a function of spreading code correlation.
We consider the simplest 2 × 2 case. Two symbols from the nPSK constellation, c 1
and c 2 , are mapped onto three space-time structures; their key matrices Φ, diversity gain
and coding gain are delineated in Table 2.5. The smallest Euclidean distance of the nPSK
constellation is denoted by δ min . In the mapping, ˜c i , i = 1, 2 can be anyone from the set
{±c i , ±c ∗ i }, and the choice of ˜c i will not affect the following discussion.
In Table 2.5, Structure 1 is diagonal mapping, Structure 2 is horizontal mapping
and Structure 3 is vertical mapping. Structure 1 and 2 achieve full diversity of 2 and
Structure 3 achieves only 1 level of diversity due to the fact that the same symbol is
transmitted via the same antenna and experiences the same fading. Structure 1 is better
than Structure 2 because its coding gain is independent of spreading code correlation.
Alamouti’s code belongs to Structure 1. The above discussion can be easily generalized
to space-time block codes of arbitrary size. The conclusions are: we want to repeat the
same symbol at different transmit antenna and different time slots, because repeating at
the same antenna incurs a diversity level loss and repeating at the same time slot suffers
correlation loss. This explains why diagonal BLAST outperforms vertical BLAST [Fos96]
and horizontal BLAST.
2.7 Conclusions
In this chapter, we develop a general framework for space-time block code design in
multipath fading channels for both non-spread and spread systems. This framework is
facilitated by a general interpretation of the notion of a “spreading code”. From the
Chernoff bound on the pairwise error probability for a maximum likelihood sequence
54

decoder, the classical space-time coding design criteria–diversity gain and coding gain,
are determined for quasi-static fading channels. Optimizing the sequence decoder turns
out to be code design of single block, which is complicated for spread system due to
the interaction of the space-time code and the spreading code. We investigated the
effects of multipath and spreading on diversity level and code designs. We find the exact
diversity level in multipath fading channel for both spread and non-spread systems and
also demonstrate that under small delay assumptions, good space-time codes for flat
fading channels tend to perform well even in a multipath fading environment. In the
uplink, the multiuser code design decouples to multiple single user code design problems;
in the downlink a true multiuser coding problem exists. We provide various approaches to
approximate the downlink case as single user problem. Due to the high dimensionality of
computer search, only some small dimensional codes are presented here, but they exhibit
solid to substantial performance gains over known space-time codes. The OMM codes
found for two transmit antennae yield 0.5dB gain over known comparable unitary group
codes when 16PSK signals are used; the non-spread optimal codes for three transmit
antenna yield 4dB gain over known unitary group codes and the spread optimal codes
exhibit another 1dB gain over the optimal non-spread optimal code. In the downlink, the
multiuser OMM code for two users shows 0.5dB gain over the single user OMM code. Our
results suggest that significant performance gains over some existing systematic designs
are possible thus prompting the search for alternative systematic designs [GM03a].
55

Chapter 3
On Suboptimal Linear Decoders for Space-Time Block
Codes
3.1 Introduction
Throughout Chapter 2, only the optimal Maximum Likelihood (ML) decoder is considered.
When the numbers of active users and space-time block codes (STBC) are large,
ML decoder is impractical due to large decoding complexity. In this chapter we consider
several different suboptimal linear decoders for STBC.
The suboptimal decoders considered in this paper are structurally similar to those of
[NSTC98]. That is, equalization and decoding are separated (see Figure 3.1), thus a two
stage decoding algorithm is proposed. In the first stage, a linear equalizers suppresses
MAI and spatial inter-symbol interference. Subsequently, a mapper is implemented to
map the linear filter soft outputs to a valid STBC(see Figure 3.2).
Combined
Equalization
and Decoding
All Decoders
Optimal
Mapper
Seperate
Equalizatin
and Decoding
Linear
Equalization
Suboptimal
Mapper
Figure 3.1: Decoder Classification.
56

®°¯²±
³°´µ
Î
À
»
Á
Æ
Ç
¼>½ ¾G¿
Matched
Filter Bank
Linear
Filter
Mapper
to STBC
Â>Ã ÄGÅ
­¬
·°¸¹Jº
È>É ÊGË
Figure 3.2: Decoder Structure.
ÊÍÌ
3.2 Uplink Signal Model for Spread Systems
To make the equalizer nontrivial, the number of observations must be greater than or
equal to the number of estimated signals. We setup our system under this constraint.
For spread system, due to our assumption that distinct spreading codes are employed at
different transmit antennae, one receive antenna is sufficient; for non-spread system, the
number of receive antennae must not be less than the number of transmit antennae.
We consider a general framework for space-time block codes in the uplink of spread
systems. The transmitter has the same structure as Figure 2.1 in Section 2.2.1. The
receiver block diagram is shown in Figure 3.2. Synchronous transmission is assumed
and perfect channel state information is assumed available at the receiver but not at the
transmitter. A matched filter bank, which is a set of filters matched to the spreading
waveform at each transmit antenna is constructed. The output of the matched filter bank
is a set of sufficient statistics for the decoding problem. A two stage decoding scheme is
employed. The first stage corresponds to a linear equalizer which forms a soft estimate
˜d of the transmitted codeword d. The second stage maps the soft estimate to a valid
codeword which is then employed to determine an estimate of the information sequence,
[Îk (1), Îk (2), . . . , Îk (k c )].
The signal received by receive antenna r at time n can be written as
y r (n) = √ σ t R r (n)H r (n)d(n) + m r (n) ∈ C LtK×1 , (3.1)
57

where σ t is signal-to-noise ratio (SNR) normalized by L t to keep the total transmit power
constant and
S r (n) = [s 1 1(n), . . . , s 1 L t
(n), . . . . . . , s K 1 (n), . . . , s K L t
(n)] ∈ R Lu×LtK (3.2)
R r (n) = S r (n) T S r (n) ∈ R LtK×LtK (3.3)
H r (n) = diag(h 1 1r(n), . . . , h 1 L tr(n), . . . . . . , h K 1r(n), . . . , h K L tr(n)) ∈ C LtK×LtK (3.4)
d(n) = [d 1 1(n), . . . , d 1 L t
(n), . . . , d K 1 (n), . . . , d K L t
(n)] T ∈ C LtK×1 (3.5)
m r (n) = [m 1 1r(n), . . . , m k L tr(n), . . . , m K 1r(n), . . . , m K L tr(n)] T , ∈ C LtK×1 (3.6)
where s k t (n) is the spreading code used at transmit antenna t at time n for user k; R r (n)
is the spreading code correlation matrix for receive antenna r at time n 1 ; h k tr(n) ∼
CN (0, 1) 2 is the channel coefficient between transmit antenna t and receive antenna r for
user k at time n; d k t (n) is the symbol transmitted from antenna t for user k at time n;
m tr (n) is complex Gaussian noise for receive antenna t at time n, m r (n) ∼ CN (0, R r (n)).
Concatenating y r (n), n = 1, . . . , N c into a larger vector y r we get
y r = [y r (1) T , . . . , y r (N c ) T ] T (3.7)
⎡
⎤ ⎡
⎤ ⎡ ⎤ ⎡ ⎤
R
= √ r (1) 0 0 H r (1) 0 0 d(1) m r (1)
σ
.
t ⎢ 0 .. . 0 ⎥ ⎢ 0 .. 0 ⎥ ⎢ . ⎥
⎣
⎦ ⎣
⎦ ⎣ ⎦ + ⎢ . ⎥
⎣ ⎦ (3.8)
0 0 R r (N c ) 0 0 H r (N c ) d(N c ) m r (N c )
√ σ t R r H r d + m r ∈ C LtKNc×1 . (3.9)
1 Note that in a synchronous flat fading channel, R 1(n) = · · · = R Lr (n), we keep the subscript r simply
to help keep track of the matrix size. Asynchronism and multipath could result in different effective code
correlation matrices between each transmit and receive antenna pair.
2 We use CN (m, K) to denote a circularly symmetric complex Gaussian random vector with mean m
and variance matrix K.
58

Further concatenating y r , r = 1, . . . , L r over each receive antenna into a super vector y,
we have the overall received signal,
y = [y1 T , . . . , yL T r
] T (3.10)
⎡
⎤ ⎡ ⎤ ⎡ ⎤
R
= √ 1 0 0 H 1 m 1
σ
.
t ⎢ 0 .. 0 ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ d + ⎢ . ⎥
(3.11)
⎣ ⎦
0 0 R Lr H Lr m Lr
√ σ t RHd + m ∈ C LtKNcLr×1 . (3.12)
where R ∈ R LtKNcLr×LtKNcLr is the spreading code correlation matrix, H ∈ C LtKNcLr×LtKNc
is channel coefficient matrix and d ∈ C LtKNc×1 is codeword vector.
3.3 Linear Equalizer
By Equation (3.12), d is the codeword vector to be estimated and y is the observed signal
vector. The linear equalizer tries to find a linear equalizer M to approximate the true
data vector d by
˜d = My Ad + ˜m (3.13)
based on various criteria. The equalizer suppresses MAI and spatial intersymbol interference.
We focus on linear equalizers and evaluate their performance (in terms of achievable
diversity levels).
The Best Linear Unbiased Estimator (BLUE), the Minumum Variance UnBiased
(MVUB) estimator and the Maximum Likelihood (ML) estimator have the same form
(due to Gaussian noise vector m)
˜d BLUE = 1 √
σt
[
H H RH ] −1
H H y. (3.14)
Note that for one receive antenna L r = 1, random channel matrix H is diagonal, if we
assume H is invertible for most channel realizations, BLUE estimator is essentially a
59

decorrelator. Denote ‖x‖ 2 R xH R −1 x and ‖x‖ 2 x H x, the Weighted Least Square
(WLS) estimator finds the linear equalizer M to minimize weighted squared error
J W LS = ‖My − d‖ 2 R (3.15)
The WLS estimator turns out to be the same as the BLUE estimator. It is easy to show
that E[˜d BLUE ] = d, hence the BLUE estimator is unbiased. The residual error ˜m has
covariance matrix
˜R BLUE = [ σ t H H RH ] −1
(3.16)
error
The Least Square (LS) estimator finds the linear equalizer M to minimize squared
J LS = ‖My − d‖ 2 (3.17)
and turns out to be
˜d LS = 1 √
σt
[
H H R 2 H ] −1
H H Ry. (3.18)
Again it is easy to show that the LS estimator is also unbiased and the residual error has
covariance matrix
˜R LS = 1 σ t
[
H H R 2 H ] −1 [
H H R 3 H ] [ H H R 2 H ] −1
. (3.19)
Assuming the codeword vector d has zero-mean and correlation E[dd H ] = R d , and
further that the transmitted code vector d and the noise m are also independent, we can
determine the following statistics:
E[dy H ] = √ σ t H H R (3.20)
E[yy H ] = √ σ t RHR d ( √ σ t RH) H + R. (3.21)
The Linear Minimum Mean Squared Error (LMMSE) estimator finds a linear equalizer M
to minimize the Mean Squared Error (MSE) between My and the transmitted codeword
vector d
J LMMSE = E[‖My − d‖ 2 ]. (3.22)
60

The resultant LMMSE estimator has the following form,
˜d LMMSE = E[dy H ]E[yy H ] −1 y (3.23)
= √ σ t R d H H R [ σ t RHR d H H R + R ] −1
y (3.24)
= √ σ t R d
[
I − σt H H RHR d (I + σ t H H RHR d ) −1] H H y. (3.25)
Matrix inversion lemma 3 is invoked in Equation (3.25) to reduce the dimension of matrix
inversion from KL r L t N c in Equation (3.24) to KL t N c in Equation (3.25). For more than
two receive antennae, Equation (3.25) has less complexity.
The LMMSE estimator is
biased. If R d is nonsingular, another form of the LMMSE estimator is
˜d LMMSE = √ [
σ t σt H H RH + R −1 ] −1
d H H y, for nonsingular R d . (3.26)
in this case, it is easy to see that LMMSE estimator converges to BLUE estimator and
is efficient,
lim ˜d LMMSE = ˜d BLUE . (3.27)
σ t→∞
This is not true for singular R d . The residual error has covariance matrix
˜R LMMSE = σ t R d H H [ I + σ t RHR d H H] −1
R
[
I + σt HR d H H R ] −1
HRd (3.28)
when R d in nonsingular
˜R LMMSE = [ R −1
d
thus it is easy to see that
+ σ t HRH H] −1
σt H H RH [ R −1
d
+ σ t HRH H] −1
, for nonsingular Rd
(3.29)
lim ˜R LMMSE = ˜R BLUE (3.30)
σ t→∞
3 (A − CB −1 D) −1 = A −1 + A −1 C(B − DA −1 C) −1 DA −1 .
61

3.4 The Mapper
Recall that the output of linear equalizer ˜d = Ad + ˜m is the soft estimate of d. We have
A BLUE = I KLtN c
(3.31)
A LS = I KLtN c
(3.32)
[
A LMMSE = R d IKLtNc − σ t H H RHR d (I + σ t H H RHR d ) −1] σ t H H RH (3.33)
= [ σ t H H RH + R −1 ] −1 √
d σt H H RH for nonsingular R d (3.34)
The optimal mapper
is equivalent to ML decoder.
ˆd = arg min
d k
‖˜d − Ad k ‖ 2˜R
(3.35)
= arg min
d k
‖y − √ σ t RHd k ‖ 2 R (3.36)
Now we consider suboptimal mapper. The simplest mapper (MAP1) simply applies
a hard decision rule to each component of ˜d, that is we decode ˆd bit by bit,
ˆd t (n) = arg min
c k
| ˜d t (n) − c k | 2 , (3.37)
where c k is a valid constellation point. The decoded ˆd might not be a valid codeword,
in this case, the closest codeword is the one with the largest number of matches with ˆd.
However, we find that the performance of the decoder in Equation (3.37) is significantly
worse than that of the mappers to be considered and thus this mapper is not considered
for further investigation.
The second mapper (MAP2) is the minimum Euclidean distance mapper,
ˆd = arg min
d k
‖˜d − d k ‖ 2 , (3.38)
where d k is a valid codeword vector. If the noise ˜m in Equation (3.13) is a white Gaussian
noise vector, this mapper corresponds to the maximum likelihood decoder (this is a M-ary
hypothesis testing problem).
62

The third mapper (MAP3) is
this mapper is also considered in [BTT02].
ˆd = arg min
d k
‖˜d − Ad k ‖ 2 , (3.39)
MAP3 is equivalent to MAP2 for BLUE and LS estimators, but different for LMMSE
estimators.
For the ML decoder, two scenarios are considered: the optimal ML decoder (ML1)
and the ML decoder ignoring the noise color (ML2)
ˆd = arg min
d k
‖y − √ σ t RHd k ‖ 2 R, (3.40)
ˆd = arg min
d k
‖y − √ σ t RHd k ‖ 2 . (3.41)
The BLUE decoder
The LS decoder
ˆd = arg min
d k
‖˜d BLUE − d k ‖ 2 . (3.42)
ˆd = arg min
d k
‖˜d LS − d k ‖ 2 . (3.43)
For the LMMSE decoder, three possible scenarios are considered: the LMMSE estimator
plus MAP2 (LMMSE1)
the LMMSE estimator plus MAP3 (LMMSE2)
ˆd = arg min
d k
‖˜d LMMSE − d k ‖ 2 , (3.44)
ˆd = arg min
d k
‖˜d LMMSE − Ad k ‖ 2 , (3.45)
and the LMMSE estimator assuming R d = I plus MAP2 (LMMSE3)
ˆd = arg min
d k
‖˜d LMMSE (R d = I) − d k ‖ 2 . (3.46)
63

Now we evaluate the overall decoder performance via Chernoff bound on pairwise
codeword error probability conditioned on channel realization.
For the ML1 decoder, when d 1 is transmitted, the probability of decoding d 2 erroneously
is upperbounded by
(
P ML (d 1 → d 2 |H) ≤ exp − σ )
t
4 (d 1 − d 2 ) H H H RH(d 1 − d 2 ) CB ML1 (3.47)
For the ML2 decoder
P ML (d 1 → d 2 |H) ≤ exp
(
− σ t
4
[
(d1 − d 2 ) H H H R 2 H(d 1 − d 2 ) ] )
2
(d 1 − d 2 ) H H H R 3 CB ML2 (3.48)
H(d 1 − d 2 )
For the BLUE decoder
P BLUE (d 1 → d 2 |H) ≤ exp
(
− σ t
4
[
(d1 − d 2 ) H (d 1 − d 2 ) ] 2
(d 1 − d 2 ) H (H H RH) −1 (d 1 − d 2 )
)
CB BLUE (3.49)
For the LS decoder
P LS (d 1 → d 2 |H) ≤ exp
(
− σ t
4
[
(d1 − d 2 ) H (d 1 − d 2 ) ] 2
(d 1 − d 2 ) H (H H R 2 H) −1 (H H R 3 H)(H H R 2 H) −1 (d 1 − d 2 )
CB LS (3.50)
)
For the LMMSE1 decoder
(
P LMMSE2 (d 1 → d 2 |H) ≤ exp − 1 [
(ALMMSE d 1 − d 2 ) H (A LMMSE d 1 − d 2 ) ] )
2
4 (A LMMSE d 1 − d 2 ) H ˜R LMMSE (A LMMSE d 1 − d 2 )
CB LMMSE1 (3.51)
For the LMMSE2 decoder
(
P LMMSE3 (d 1 → d 2 |H) ≤ exp − 1 [
(d1 − d 2 ) H A H LMMSE A LMMSE(d 1 − d 2 ) ] )
2
4 (d 1 − d 2 ) H A H ˜R LMMSE LMMSE A LMMSE (d 1 − d 2 )
CB LMMSE2 (3.52)
64

For the LMMSE3 decoder, the Chernoff bound CB LMMSE3
is a special case of
CB LMMSE1 when R d = I.
3.5 The Effect of Large Number of Receive Antennae
Define
∑L r
X H H RH = Hr H R r H r . (3.53)
r=1
where H r and R r defined in Equation (3.9). By noting that H r is a diagonal matrix, we
have
∑L r
X ii = |(H r ) ii | 2 diagonal elements, (3.54)
r=1
L r
∑
X ij = (R r ) ij (H r ) ∗ ii(H r )jj off-diagonal elements. (3.55)
r=1
and as L r → ∞, by the assumption of i.i.d. h k tr and strong Law of Large Numbers,
lim X = L rI LtKN c
almost surely. (3.56)
L r→∞
Clearly the effect of more receive antennae is to strengthen the diagonal elements and
weaken the off-diagonal elements of X.
Similarly we can show that
∑L r
∑L r
H H R 2 H = Hr H RrH 2 r H H R 3 H = Hr H RrH 3 r (3.57)
r=1
r=1
If we assume spreading codes are equi-correlated,
lim
L HH R 2 H = L r [1 + (KL t − 1)ρ 2 ]I LtKN c
(3.58)
r→∞
65

and
lim
L HH R 3 H =
r→∞
L r [1 + 6(KL t − 1)ρ 2 + 4(KL t − 1)(KL t − 2)ρ 3 + (KL t − 1)((KL t − 2) 2 + 1)ρ 4 ]I LtKN c
.
(3.59)
Now let us explicitly consider the effect of a large number of receive antennae on the
equalizer. Since BLUE and LS estimators are unbiased, we only need to evaluate the
residual noise covariance. For BLUE estimator
for LS estimator
lim ˜R LS
L r→∞
lim ˜R BLUE = 1 I LtKN
L r→∞ σ t L
c
(3.60)
r
α
σ t L r
I LtKN c
(3.61)
= 1 + 6(KL t − 1)ρ 2 + 4(KL t − 1)(KL t − 2)ρ 3 + (KL t − 1)((KL t − 2) 2 + 1)ρ 4
(σ t L r )(1 + 2(KL t − 1)ρ 2 + (KL t − 1) 2 ρ 4 I LtKN
)
c
.
(3.62)
because the LS estimator has larger variance than the BLUE estimator in the limit
(α > 1), we expect the LS estimator has worse performance than the BLUE estimator.
The LMMSE estimator, when R d in nonsingular
lim A [
LMMSE = lim ILtKN c
+ R −1
L r→∞ L
d /(σ tL r ) ] −1
= ILtKN c
(3.63)
r→∞
is unbiased, the residual noise have covariance matrix
lim ˜R LMMSE =
L r→∞
lim
L r→∞
1 [
ILtKN
σ t L
c
+ R −1
d /(σ tL r ) ] −2 1 = I LtKN
r σ t L
c
. (3.64)
r
66

Now we consider the effect of large number of receive antennae on overall decoder
performance. Plug the above asymptotics into Equations (3.47), (3.48), (3.49), (3.50),
(3.51) and (3.52).
lim CB ML1 =
L r→∞
lim CB BLUE =
L r→∞
= lim
L r→∞ CB LMMSE2 =
lim
L r→∞ CB LMMSE1
lim CB LMMSE3 = exp
L r→∞
(
− σ tL r
4 ‖d 1 − d 2 )‖ 2 )
(
lim CB LS = lim CB ML1 = exp − σ )
tL r
L r→∞ L r→∞ 4α ‖d 1 − d 2 )‖ 2
(3.65)
(3.66)
Hence for large number of receive antennae, all the BLUE decoder and LMMSE decoders
achieve the same performance as that of optimal ML1 decoder. The LM2 and LS decoder
is worse by a factor α.
3.6 Fisher’s Information Matrix
To seek the favorable structure of codeword for the LMMSE estimator, we evaluate
Fisher’s Information Matrix (FIM) conditioned on channel realization H for random
codeword vector d. Recall that
y = √ σ t RHd + m
where m ∼ CN (0, R). We have
P (y|d, H) =
1
π LtKNc det(R) exp ( −(y − √ σ t RHd) H R −1 (y − √ σ t RHd) ) (3.67)
If we assume d ∼ CN (0, R d ),
P (d) =
1
( )
π NcLt det(R d ) exp d H R
d −1 d
(3.68)
the joint pdf of y and d is
P (y, d|H) = P (y|d, H)P (d) (3.69)
67

the FIM for random d is
[ ∂ ln P (y, d|H)
J = E
∂d ∗
= E
]
∂ ln P (y, d|H)
∂d T
[ ∂ ln P (y|d, H) ∂ ln P (y|d, H)
∂d ∗ ∂d T +
= σ 2 t H H RH + R −1
d
]
∂ ln P (d) ∂ ln P (d)
∂d ∗ ∂d T
(3.70)
(3.71)
(3.72)
Let I denote the inverse of J, the Cramer-Rao lower bound (CRLB) says
var( ˆd i ) ≥ I ii (3.73)
For large number of receive antennae, plug Equation (3.56) into J
lim J =
L σ2 t L r I LtKN c
+ R −1
r→∞ d
(3.74)
To seek the optimal code structure for the estimator, we try to minimize the overall
lower bound of estimation variance subject to constant block energy
(
) −1
min
d trace σ t H H RH + R
d
−1 subject to E[d H d] = c (3.75)
Recall R d = E[dd H ], if the eigenvalues of R d are {λ i } n 1 , the optimization problem is
equivalent to
min
d
n∑
i=1
1
σ 2 t L r + 1 λ i
subject to
By Kuhn-Tucker conditions, the minimum is achieved when
n∑
λ i = c (3.76)
i=1
λ 1 = c, λ 1 = · · · λ n = 0 (3.77)
This suggests strong correlation among d is preferred by the estimator, which agrees with
common sense: the estimation of one element helps the estimation of the others. Since
usually
σt 2 L r ≫ 1 , (3.78)
λ i
68

hence
1
σt 2L r + 1 ≃ 1
σ 2 λ i t L if λ i ≠ 0 (3.79)
r
1
σt 2L r + 1 = 0 if λ i = 0. (3.80)
λ i
This suggests the minimum rank of R d will decide the minimum of Equation (3.76),
rather than the relative distribution of non-zero eigenvalues.
To search for good codes for LMMSE decoder, the difficulty resides in the absence of
closed form expression for PEP. The above Fisher’s Information Matrix analysis suggests
that small number of non-zero eigenvalues of R d reduces the variance of linear estimator,
but another measure which quantize the effect of mapper is still missing.
Numerical
results show that the mappers are very insensitive to the variance of linear estimator.
Two unsatisfactory approaches have been tested to search for good codes suitable for
LMMSE decoder.
The first approach is based on the fact that OMM codes are not
unique, within the set of OMM codes, we try to find the code with the minimum rank
of R d . But for all known OMM codes within the same set, R d ’s turn out to have the
same rank. The second approach tries to find the code sets with minimum rank of R d , for
codes with the same rank of R d , worst case coding gain is minimized. This yields large
number of code sets, primitive analysis shows no noticeable advantage of found codes.
3.7 Simulation
In the simulation, a rate 1, 2 × 2 BPSK code set optimized for spread system [GMF03]
is used, it consists of four codewords:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎣ 1 1 ⎦ ⎣ 1 −1 ⎦ ⎣ −1 1 ⎦ ⎣ −1 −1 ⎦ . (3.81)
1 1 −1 −1 1 −1 −1 1
The codeword correlation is
⎡ ⎤
1 0 0 0
0 1 1 0
R d =
⎢
⎣0 1 1 0⎥
⎦
0 0 0 1
(3.82)
69

Apparently R d is singular, which is typical for small number of codewords and small
constallation size.
3.7.1 The BLUE and LS Decoders
Although the BLUE and LS decoders are unbiased, they cannot achieve the same diversity
level as ML decoder (Figure 3.3). An inspection of the average variance (Figure 3.4)
L t
=2, L r
=1, K=1, ρ=0.3, BLUE and LS decoder lose diversity
10 0 SNR
LS
BLUE
LMMSE1
ML1
10 −1
SER
10 −2
10 −3
0 1 2 3 4 5 6 7 8 9 10
Figure 3.3: The BLUE and LS estimators lose diversity.
var = ‖˜d − d‖ 2 (3.83)
shows the loss of diversity is due to the large variance of estimated codeword vector. On
the contrary, LMMSE decoder is biased, but has very small variance, hence achieves the
same diversity level as ML decoder. The LS decoder is slightly worse than the BLUE
decoder, and has slightly larger estimator variance.
70

250
L t
=2, L r
=1, K=1, ρ=0.3, BLUE and LS estimator has large variance
LS
BLUE
LMMSE1
200
150
VAR
100
50
0
0 1 2 3 4 5 6 7 8 9 10
SNR
Figure 3.4: The BLUE and LS estimators have large variance.
3.7.2 The LM Decoders
LM2 has about 0.3dB loss than LM1 by ignoring the color of the noise (Figure 3.5). By
considering the noise color, LM1 calculates a weighted Euclidean distance, therefore the
performance improvement is obtained by more decoding complexity.
3.7.3 The LMMSE Decoders
LMMSE2 decoder has slight advantage over LMMSE1 at high SNR (Figure 3.6), but it has
complexity comparable to ML decoder, it is not a good candidate for multiuser detection,
where the number of hypotheses grows exponentially with the number of active users K.
LMMSE2 decoder has less variance than LMMSE1 decoder (Figure 3.7). Ignoring the
codeword correlation incurs about 0.2dB loss in performance and larger variance.
3.7.4 The Effect of Number of Receive Antennae L r
Two receive antennae improve the performance of all decoders, but the BLUE and LS
decoders still cannot achieve the same level of diversity as ML and LMMSE1 decoders.
71

L t
=2, L r
=1, K=1, ρ=0.3, ignoring R incurrs 0.3dB loss
10 0 SNR
ML2
ML1
10 −1
SER
10 −2
10 −3
0 1 2 3 4 5 6 7 8 9 10
Figure 3.5: Ignoring R incurs 0.3dB loss.
# of active users ML1 LMMSE3
K=2 0.7500 0.1100
K=4 13.7190 0.2810
Table 3.1: CPU time in seconds for 200 runs.
3.7.5 The Effect of Number of Active Users K
For 2 (Figure 3.9) and 4 (Figure 3.10) active users, the single user LMMSE3 decoder
shows performance close to joint LMMSE3 decoder and has much less complexity. Note
the equalizer is still joint equalizer.
3.7.6 Complexity
Simulation results show that LMMSE3 has complexity which grows linearly with the
number of active users, while ML1 has complexity which grows exponentially with the
number of active users (See Table 3.7.6).
72

L t
=2, L r
=1, K=1, ρ=0.3
10 0 SNR
LMMSE1
LMMSE2
LMMSE3
10 −1
SER
10 −2
10 −3
10 −4
0 2 4 6 8 10 12 14 16
Figure 3.6: LMMSE2 decoder has slight advantage over LMMSE1 decoder at high SNR.
3
2.5
L t
=2, L r
=1, K=1, ρ=0.3, LMMSE2 has less variance than LMMSE1
LMMSE1
LMMSE2
LMMSE3
2
VAR
1.5
1
0.5
0
0 2 4 6 8 10 12 14 16
SNR
Figure 3.7: LMMSE2 decoder has less variance than LMMSE1 decoder.
73

L t
=2, L r
=2, K=1, ρ=0.3, two receive antennae
10 0 SNR
LS
BLUE
LMMSE1
ML1
10 −1
SER
10 −2
10 −3
10 −4
0 1 2 3 4 5 6 7 8 9 10
Figure 3.8: Two receive antennae.
K=2, [0 7 9 14]
10 0 SNR
Single LMMSE3
Joint LMMSE3
ML
SER
10 −1
10 −2
0 1 2 3 4 5 6 7 8 9 10
Figure 3.9: Two active users.
74

K=4, [0 7 9 14]
10 0 SNR
Single LMMSE3
Joint LMMSE3
ML
10 −1
SER
10 −2
10 −3
0 2 4 6 8 10 12 14
Figure 3.10: Four active users.
75

Chapter 4
Nonlinear Hierarchical Codes
4.1 Introduction
The computer optimized codes in Chapter 2 exhibit layered structure which can be characterized
by isometry (see Section 4.2 for detailed discussion on isometry). This motivates
the construction of Nonlinear Hierarchical Codes (NHC) based on isometries. The construction
of NHC is a generalization of group codes [Sle68] to the space-time scenario.
STBCs have been designed using constraints on the structure of the codewords. Such
constraints often yield systematic designs making code set generation and often analysis
quite straightforward. In particular, orthogonal codes [TJC99] and unitary group codes
[HM00, SHHS01] enforce orthogonal or unitary structure on the code blocks and are easily
generated. However, computer optimized codes, optimal minimum metric codes (OMM)
[Gen00] and optimal union bound codes (OUB) [GVM02], can offer up to several dB gains
over the aforementioned code sets, but are computationally expensive to determine. For
STBCs to be used in a serially concatenated system, this several dB gain [Gen00] is seen
directly in overall performance. For both practical and theoretical reasons, there is strong
interest in developing high performance code sets that have systematic constructions.
As noted above, the coding gain measure does not define a metric space, which complicates
systematic constructions of code sets. Operations that preserve “distance” measures
like the coding gain are deemed isometries. Isometric transformations have been
previously investigated in the context of space-time systems: to shorten search times
for codes sets [Gen00, GVM02] and to perform code set expansion for use in space-time
trellis-coded modulation design [SF02]. The computer optimized codes [Gen00, GVM02]
exhibit strong structure and symmetry which can be explained via isometries. This in
76

turn, suggests the systematic construction of nonlinear hierarchical codes (NHC) for large
block and/or constellation sizes, which we present herein.
Slepian [Sle68] studied group codes for the Gaussian channel in 1967. 1 The group
codes defined by Slepian possessed geometric uniformity and can be decomposed into a
direct sum of certain basic group codes. Starting from an initial vector, the basic group
codes can be generated by irreducible representations of a finite group. However, finding
group codes with best worst case performance is challenging as noted by Slepian. However,
we are able to generalize and extend Slepian’s design strategy for the construction of
NHCs. The methodology of NHC construction via isometries for MIMO quasi-static
fading channels is an extension of Slepian’s group codes, where group structure (and
hence distance uniformity), is conserved in most cases. A sufficient condition for NHCs
to achieve geometric uniformity [GDF91] is established. The NHC construction starts
from some initial codeword matrices with no constraint on the size and structure, and
by means of isometries, systematically builds higher rate codes from lower rate codes,
while preserving the good structure of the lower rate code. It is the relationship between
codewords, rather than the structure of each individual codeword, that is exploited to
optimize coding gain. The found NHCs match those found via exhaustive computer
search, and exhibit not only optimized coding gain at each layer, but also great symmetry
and structure to be exploited in Chapter 5 of this work.
This chapter is organized as follows: Section 4.2 presents the signal model; Section 4.3
discusses the construction and properties of NHCs. As an example, Section 4.4 describes
in detail the generation of the 2 × 2 QPSK NHC. Section 4.5 discusses the constructed
NHCs and compares their performance with corresponding orthogonal or unitary group
codes.
4.2 Signal Model and Isometries
We consider a single user system where the transmitter and receiver are equipped with
L t and L r antennae, respectively. We assume perfect synchronization. The received
1 Distinction should be made from the unitary groups codes in [HM00], where unitary group codes are
indeed matrix representations of Slepian’s groups codes.
77

signal corresponding to N c symbol intervals after appropriate front-end processing can
be formulated as
Y = √ σ t DH + N ∈ C Nc×Lr (4.1)
where
σ t = signal-to-noise ration (SNR) normalized by L t (4.2)
D = [d ti ] ∈ C Nc×Lt codeword matrix (4.3)
H = [h ij ] ∈ C Lt×Lr fading coefficient matrix (4.4)
N = [n tj ] ∈ C Nc×Lr additive white Gaussian noise matrix (4.5)
Y = [y tj ] ∈ C Nc×Lr received signal matrix (4.6)
For the current work, d ti is constrained to a finite nP SK set, that is, d ti = u c ti
where
u = e j 2π n , c ti ∈ {0, . . . , n−1}. For ease of representation, each codeword matrix is assigned
a unique index, by concatenating c ti ’s into a single base-n number:
(c 1,1 . . . c 1,Lt . . . . . . c Nc,1 . . . c Nc,L t
) n . (4.7)
The fading coefficients are complex Gaussian, and modeled as independent for different
transmit antenna i or receive antenna j. A quasi-static fading channel is considered, i.e.,
H remains constant within each block of N c symbols, but independent from block to
block. Perfect channel state information is assumed known at the receiver. The noise,
n tj ∼ CN (0, 1) is additive complex Gaussian noise, independent for different symbol times
t and different receive antennae j.
We define the codeword difference correlation matrix
Φ(D i − D j ) (D i − D j ) H (D i − D j ), (4.8)
and denote the non-zero eigenvalues of Φ(D i − D j ) by λ k . The channel vector h, which
is a concatenation of the columns of H, has mean and covariance
¯h E[h] (4.9)
Σ h E[(h − ¯h)(h − ¯h) H ] (4.10)
78

We will assume Σ h always has full rank.
The averaged (over channel) pairwise error
probability of decoding D j when D i is transmitted can be upper bounded by
E h [P (D i → D j )|h)] ≤
(
exp
−h H Σ −1
h
[
I − (I +
σ t
4 Σ hΦ) −1] )
h
|I + σt
4 Σ hΦ|
(4.11)
at high SNR, assuming Φ is full rank, which is the focus of this work, the bound can be
simplified to
E h [P (D i → D j )|h)] ≤
( )
exp −h H Σ −1
h h
| σt
4 Σ h|
1
|Φ|
(4.12)
Because the factor in front of 1/|Φ| is a function of channel mean and covariance only,
the worst case code design criteria [TSC98, GFBK99] determined by Φ are independent
of channel conditions: Rayleigh (¯h = 0) or Ricean (¯h ≠ 0), uncorrelated (Σ h = I) or
correlated (Σ h ≠ I). The special case when Σ h is rank deficient is not discussed in this
work. The design criteria are diversity gain
∆ H (D i − D j ) = rank Φ(D i − D j ), (4.13)
and coding gain 2
∆ p (D i − D j ) =
∆ H ∏
k=1
λ k (4.14)
= det Φ(D i − D j ) assuming full diversity gain. (4.15)
Another interesting distance measure is
∆ H
∑
∆ s (D i − D j ) = λ k = trace(Φ(D i − D j )) (4.16)
k=1
which characterizes the performance of a space-time system in Gaussian channel or when
the number of receive antennae is large [SF02] and will be used in the sequel for signal
set expansion (see Equation (5.8)). For a set of codewords, the worst case design goals
are
2 There are other equivalent definitions of coding gain, we adopt this form for ease of presentation.
79

1. full diversity
∆ H (D i − D j ) = L t , ∀i ≠ j (4.17)
2. maximize minimum coding gain
max min ∆ p (D i − D j ), ∀i ≠ j (4.18)
To define our code construction method, we require the notion of an isometry. Isometries
are defined as “distance preserving” transformations, where the “distance” ∆ x can
be ∆ H , ∆ p or ∆ s .
Definition 1 A function φ of codeword D is an isometry for ∆ x if
∆ x (γ(D i ) − γ(D j )) = ∆ x (D i − D j ). (4.19)
We assume that all isometries map the all zero matrix to the all zero matrix, i.e., γ(0) = 0.
Lemma 1 For ∆ x = ∆ H , ∆ p , or ∆ s , the followings are isometries: 3
1. γ(D) = UD, where U is arbitrary unitary matrix U H U = I.
2. γ(D) = DP , where P is arbitrary unitary matrix: P H P = I.
3. γ(D) = [d ∗ ti ], componentwise complex conjugation.
Proof:
If we can show that the three isometries do not change the eigenvalues of Φ(D i − D j ),
then the “distance” ∆ x (x = H, p or s) is preserved. Let λ and v be an eigenvalue and
corresponding eigenvector of Φ,
Φv = λv. (4.20)
3 This work focuses on non-spread systems, which applies equally well to the uplink of a multiuser
Direct-Sequence Code-Division Multiple-Access (DS-CDMA) system when different transmit antennae of
one user employ the same spreading code [Gen00]. The system with different spreading codes employed
for different transmit antennae is called a spread system. Not all isometries for non-spread systems apply
to spread systems, some limitations have to be applied. Isometry 1 holds as before. Isometry 2 requires:
P is a unitary permutation matrix (P H P = I, each row and column of P has only one non-zero element)
and equi-correlated spreading codes. Isometry 3 requires real valued spreading codes. Hence for spread
systems, pragmatic isometries are essentially limited to U only.
80

1. Isometry U does not change Φ at all: Φ(UD i −UD j ) = (D i −D j ) H U H U(D i −D j ) =
Φ(D i − D j ).
2. Let v = P v ′ , then ||v ′ || = ||v|| = 1 and
ΦP v ′ = λP v ′ =⇒ P H ΦP v ′ = λv ′ ,
therefore λ and v ′ are an eigenvalue and corresponding eigenvector of Φ(D i P −
D j P ) = P H ΦP . Thus the eigenvalues are preserved.
3. The eigenvalues of a Hermitian matrix, Φ, are real. Taking complex conjugation on
both sides of Equation (4.20), Φ ∗ v ∗ = λv ∗ , therefore, λ and v ∗ are an eigenvalue
and corresponding eigenvector of Φ ∗ .
In summary, the eigenvalues of Φ are not changed by the these isometries and, thus the
“distance”, ∆ x is preserved. ✷
An interesting observation is that for 2 × 2 block codes, the effect of complex conjugation
can be “emulated” by combinations of U and P . Here is an example:
⎡ U ⎤ ⎡ 1 ⎤ ⎡ P ⎤ ⎡ 7 ⎤
⎣ 1 0 ⎦ × ⎣ 1 1 ⎦ × ⎣ 0 1 ⎦ = ⎣ 1 1 ⎦ (4.21)
0 e −j π 4 1 e j π 4 1 0 1 e −j π 4
For larger block sizes, it may be challenging to find these equivalent combinations of U
and P to emulate the effect of complex conjugation.
To ensure closure for the isometries, we further require that U and P have only
one non-zero element on each row and column, which is drawn from the same nPSK
constellation as the original codeword D. With this constraint, there are n Lt L t ! possible
matrices of size L t ×L t for the left isometries, and n Nc N c ! possible matrices of size N c ×N c
for the right isometries. Either left or right isometries form a non-abelian group; the fixedpoint-free
groups of these isometries are well-studied[SHHS01]. Isometries consisting of all
the combinations of U and P also form a non-abelian group with cardinality n Lt L t !n Nc N c !,
which includes left (or right) only isometries as a subgroup.
Codewords which can be transformed from one to another via isometries form an
equivalence class, which shall be called “double cosets” if both U and P are used, and
“cosets” if only U is used. When no confusion is caused, we will simply use “coset” to
81

epresent both for convenience. All codewords within a single equivalence class have the
same ∆ p (D), which will be called the coding gain of the coset; but the converse is generally
not true. A distinction between our defined cosets and those classical cosets defined in
group theory [Arm87] is that in group theory all cosets have a common cardinality, while
here, different cosets may have different cardinalities. The codeword space is naturally
decomposed into cosets.
The coset leader, l, has the smallest codeword index within the coset. A double coset
with cardinality m will be designated by its coset leader as DC l (m), similarly a coset
will be designated as C l (m). A double coset usually splits into several cosets with equal
cardinality. An example for 2 × 2 QPSK codes is given below, where the optimal rate 2
code (16 codewords) is found within DC 2 (64):
DC 0 (64) = C 0 (16) ∪ C 17 (16) ∪ C 34 (16) ∪ C 51 (16)
DC 1 (128) = C 1 (32) ∪ C 3 (32) ∪ C 18 (32) ∪ C 35 (32)
DC 2 (64) = C 2 (32) ∪ C 19 (32)
A coset is actually a trivial set of group codes with the largest possible cardinality.
4.3 Building Nonlinear Hierarchical Codes
Given the nonlinear, non-metric, codeword space, finding a good set of codewords is not
a simple task. With the aid of isometries, we propose the following greedy algorithm to
build nonlinear hierarchical codes (NHCs):
1. Pick a good initial codeword set S k with 2 k codewords.
2. Generate S
k ′ from S k by the following criterion
S ′ k = {U mS k P n |max
m,n min
i,j ∆ p(D ′ i − D j ), ∆ H = L t , D ′ i ∈ S ′ k , D j ∈ S k } (4.22)
3. Let S k+1 = S k ∪ S
k ′ , k = k + 1 and go to step 2.
Step 2 generates a new set S ′ which is isomorphic to the base set S and is as “far
away” from the base set as possible. The procedure stops when the desired rate is reached
or S ′ is empty. The following properties can be shown:
82

1. Cardinality: |S k | = 2 k
2. S
k ′ and S k have the identical distance spectrum with respect to ∆ H , ∆ p and ∆ s .
3. Coding gain: ∆ p (S k+1 ) ≤ ∆ p (S k )
Step 2 and 3 extend the size of any code set while maintaining full diversity and good
coding gain. The initial codeword set may be a single codeword matrix, in this case,
all the codewords generated by the above procedure belong to the same double coset.
Since each double coset is of the largest possible cardinality for group codes, the above
procedure is a systematic way of carving out a good subset (or subgroup) from this double
coset. We could start from a codeword set with members from different double cosets,
but this set is generally difficult to expand in Step 2, because members from the same
double coset lead to desirable symmetry properties 4 . To simplify code construction and
analysis, we focus on codewords from the same double coset. Step 2 can be relaxed by
accepting S ′ whose “distance” from S is larger than a certain threshold τ:
S ′ k = {U mS k P n |min
i,j ∆ p(D ′ i − D j ) >
m,n
τ, ∆ H = L t , D ′ i ∈ S ′ k , D j ∈ S k }. (4.23)
Of course, multiple candidate sets would be found, but this makes it easier to find code
sets of larger cardinality. The above procedure enables us to start from a single codeword
and build a good set of codes for different rates with limited complexity by reusing the
good sub-structure again and again. At each layer of the construction/expansion, the
associated distance spectra have good properties and coding gains are nearly-optimized
and we conjecture could in fact be optimal for several cases.
are summarized in Table 4.2 of Section 4.5.
The constructed NHCs
The beauty of the NHC is the graceful
tradeoff between performance and rate while maintaining good structure. Properties 2
and 3 induce a natural set partitioning [Ung82] on NHCs which lays the foundation for
space-time trellis-coded modulation design. This is the topic of Chapter 5.
To obtain a good hierarchical code, two important issues need to be addressed:
1. Choice of initial codeword or lower rate code
2. Choice of isometries
4 There are exceptions where codewords from different double cosets give good performance.
83

We may construct the whole set from a single codeword, in this case, all codewords belong
to the same double coset, the S k of the current step always serves as the lower rate code
for next step; we may also start from a well constructed set of codewords, which may be
a NHC of lower constellation, an orthogonal code or a unitary group code, in this case
the constructed NHC may not belong to the same double coset.
The interaction between an initial codeword and isometries is complicated when double
cosets instead of cosets are considered. If we require that all codewords are drawn from
the same double coset, choosing a good initial codeword is equivalent to selecting a good
double coset. This significantly simplifies the task of choose a good initial codeword, but
still does not solve the problem completely. Our strategy is to seek the double cosets with
the maximum coding gain as motivated by the following two arguments. Consider a pair
of codewords, D and D ′ , from the same double coset such that they maximize the coding
gain ∆ p (D − D ′ ), the maximizing pair is simply D and −D with ∆ p (2D) = det(4D H D).
Maximizing ∆ p (2D) leads to the coset with the largest coding gain,
D ∗ = arg max
D
det(Φ(D − (−D)) = arg max
D det(DH D) (4.24)
Our second argument behind using the maximal determinant coset is as follows. Consider
cosets formed by the left isometry U, then our goal for the initial codeword D is:
D ∗ = arg max det(Φ(UD − D)) (4.25)
D,U
= arg max
D
det(DH D)max det((U −
U I)H (U − I)) (4.26)
The optimization can be decoupled to choosing the correct coset max
D
det(DH D) and
the correct isometry max det((U −
U
I)H (U − I)). Note that the condition max det((U −
U
I) H (U − I)) implies det(U − I) ≠ 0, which is the condition of U being fixed point free
[HM00]. Our simple argument suggests that fixed point free matrices/codes may not be
the best codewords, but they are desirable isometries. Thus, fixed point free isometries
should be coupled with maximal determinant code words to transform them into constant
envelope modulation codewords. A similar argument cannot be extended to double cosets.
Empirically, we have found that our strategy of starting with the coset with the largest
associated determinant does indeed yield the best codes.
84

We have observed that good codes from smaller constellations are embedded in codes
from larger constellations and usually serve as better lower rate codes for the basis of
expansion. Relaxing the maximization in Equation (4.22) to the threshold in Equation
(4.23) also makes further expansion easier.
Left or right isometries alone are usually insufficient to generate good higher rate
codes. The theory of fixed-point free group codes enables one to eliminate inappropriate
isometries and reduce the search space, thus enabling the practical search/design of block
codes with very large cardinality. Recall that complex conjugation can be emulated in
2 × 2 block codes by a suitable combination of left and right isometries, but this does not
hold for general block sizes. We endeavor to avoid complex conjugation, or cross-coset
codewords for that matter, because the fine structure maintained by focusing on left and
right isometries is broken and thus makes generating a good S
k ′ from S k challenging.
In general, we propose cross-coset codewords and complex conjugation as methods for
expansion only in the last step of the design process.
Given the nature of the problem (nonlinear, non-metric space, non-unique transformation),
carrying out the procedure requires some trial and error; however very good codes
can be found via a few steps and with fairly small complexity. A brute force search for
2 K codewords of size L t ×N c from nPSK constellations requires testing O(n LtNc2K) cases.
In contrast, the hierarchical design requires testing only O(Kn Lt+Nc L t !N c !) cases. There
is inherent redundancy in the isometries. That is, the same code set S ′ can be generated
by different isometries U m and P n from the same base set S; similarly a single initial code
set S can yield multiple code sets S ′ , each with the maximal coding gain. Because we
are interested in finding only one optimized code set, any other code sets with equivalent
performance (distance spectrum) are deemed redundant. By exploiting the properties of
isometries, many redundancies can be removed from the search as follows:
1. About half of the left isometries U are redundant. Because U H U = I and U r = I
for some r, the following pair of sets share the identical distance spectrum,
S ∪ US ↔ S ∪ U H S (4.27)
S ∪ U t S ↔ S ∪ U r−t S (4.28)
If U is included in the search, U H becomes redundant.
85

2. Most right isometries P are redundant. For example, the 2×2 QPSK NHC is found
in the double coset DC 2 (64), which can be decomposed into two cosets
DC 2 (64) = C 2 (32) ∪ C 19 (32) (4.29)
⎡ ⎤ ⎡ ⎤
= C 2 (32) ⎣ 1 ⎦ ∪ C 2 (32) ⎣ 1 ⎦ . (4.30)
1
i
In other words, applying all other right isometries on C 2 (32) yields no new cosets,
therefore, only two right isometries need to be included in the search.
For 2 × 2 QPSK, the number of left/right isometry pair is reduced from 32 × 32 to 18 × 2,
for 3 × 3 8PSK, the reduction is significant: from 3072 × 3072 to 1564 × 64.
As noted earlier, the NHCs are strongly analogous to Slepian’s group codes for Gaussian
channels [Sle68] which are geometrically uniform signal sets [GDF91]. Our extension
of Slepian’s group codes to MIMO quasi-static fading channels considers designs which
endeavor to maintain geometric uniformity via isometries. The strong analogy between
NHC and Slepian’s group code is evident:
NHC
Coding gain ∆ p
Initial block code
Isometry U and P
Double coset
Slepian’s group code
Euclidean distance
Initial vector
Real orthogonal matrices O
Fundamental region
NHCs attempt to generalize Slepian’s group codes to MIMO fading channel in a
practical way. The group structure between the group of real orthogonal matrices {O i } is
replaced by hierarchical structure for ease of construction. As remarked by Slepian, “the
initial vector problem and the fundamental region” are complicated, general solution of
these problems are not available yet.
We refashion Forney’s definition [GDF91] for geometrically uniform (GU) group codes
for MIMO quasi-static fading channels:
Definition 2 The codeword set S is geometrically uniform if, given any two codeword
matrices, D and D ′ in S, there exists an isometry γ that transforms D to D ′ while leaving
S invariant; i.e. γ(D) = D ′ and γ(S) = S.
86

The isometry γ could be a combination of the afore-mentioned three isometries.
Specifically, for NHC S k to be GU, we need ∀ a, b ∈ S k , ∃ U j , P j such that
b = U j aP j (4.31)
S k = U j S k P j . (4.32)
GU is a strong kind of symmetry, a less strict symmetry is distance uniform (DU).
Definition 3 The codeword set S is distance uniform if the distance measures (e.g. the
coding gain) of any codeword to others do not depend on the choice of the codeword, in
other words, the distance profile from one codeword to all others are all the same.
Geometrical uniformity implies distance uniformity, but the reverse may not be true.
An counter example is the 3 × 3 BPSK NHC (see Table A.5), the following subset is DU,
{1, 172, 304, 413} (4.33)
but not GU, because {1, 172} and {304, 413} belong to double coset 1 and 10, respectively,
the isometries to transform codewords from double coset 1 to 10 do not exist. We observe
that most, but not all, of our found NHCs are in fact DU. We ask what are the conditions
for a NHC to be GU? Lemma 2 and Theorem 1 answer this question. In the sequel, we
use the notation
U{code set S}P = {code set S ′ } (4.34)
to represent the 1-1 isometry transformation that maps each codeword a from S to a
codeword a ′ in S ′ such that
UaP = a ′ , ∀ a ∈ S, a ′ ∈ S ′ . (4.35)
Without loss of generality, we assume a and b are two arbitrary elements from S k and
the following transformation holds (see Figure 4.1)
{a ′ , b ′ } = U k {a, b}P k . (4.36)
87

★
✥
S k
✧
a
U j , P j ✲
b
✦
U k , P k
U k , P k
★
S
k
′ ❄
a ′ ❄
b ′
✧
✥
✦
Figure 4.1: The relationship between {a, b} and {a ′ , b ′ }.
Lemma 2 If S k is GU, and I U and I P are the isometry sets which satisfy Definition
2 for S k , then for arbitrary left and right isometries U k and P k , S
k ′ = U kS k P k is also
GU. A set of isometries which satisfies Definition 2 for S
k ′ is {U kU j Uk H, U j ∈ I U } and
{Pk HP jP k , P j ∈ I P }.
Proof:
First, let us find the isometries to transform a ′ to b ′ ,
b ′ = U k bP k
= U k U j aP j P k
= U k U j U H k a′ P H k P jP k .
If we can show S ′ k is invariant under isometries U kU j U H k
holds because
and P H k P jP k , S ′ k
is GU. This
S ′ k = U kS k P k
= U k U j S k P j P k
= U k U j U H k S′ k P H k P jP k .
✷
88

Theorem 1 If S k is GU, and I U and I P are the isometry sets which satisfy Definition
2, then a set of sufficient conditions for S k+1 = S k ∪ S ′ k , where S′ k = U kS k P k , to be GU
is
S k = (U k U j U k )S k (P k P j P k ) and (4.37)
= (U H k U jU k )S k (P k P j P H k
) and (4.38)
= (U k U j U H k )S k(P H k P jP k ), ∀ U j ∈ I U , ∀ P j ∈ I P . (4.39)
Proof:
Given S ′ k = U kS k P k and b = U j aP j , we first identify the isometries to transform one
codeword to another and the effects of these isometries on the corresponding code set.
b = U j aP j S k = U j S k P j (4.40)
b ′ = U k U j Uk H a′ Pk H P jP k S k ′ = U kU j Uk H S′ k P k H P jP k (4.41)
b ′ = U k U j aP j P k S k ′ = U kU j S k P j P k (4.42)
a = Uj H Uk H b′ Pk H P j H S k = Uj H Uk H S′ k P k H P j H , (4.43)
For S k+1 = S k ∪ S ′ k
to be GU, it is sufficient to establish the following three points
1. Given isometries U j and P j preserve S k (Equation (4.40)), they should also preserve
S ′ k . This requires S ′ k
= U j S ′ k P j
which is Equation (4.38).
⇔ U k S k P k = U j U k S k P k P j
(4.44)
⇔ S k = Uk HU jU k S k P k P j Pk H, 2. Given isometries U k U j U H k and P H k P jP k preserve S ′ k
also preserve S k . This requires
(Equation (4.41)), they should
S k = U k U j U H k S kP H k P jP k , (4.45)
which is Equation (4.39).
89

3. Given Isometries U k U j and P j P k transform S k to S
k ′ (Equation (4.42)), they should
transform S ′ k to S k at the same time. This requires
S k = U k U j S ′ k P jP k = U k U j U k S k P k P j P k , (4.46)
which is Equation (4.37).
Similarly, given Isometries Uj HU k
H and P k HP j transform S
k ′ to S k, they should transform
S k to S
k ′ at the same time. This yields the same condition as Equation (4.37).
✷
We observe that the definition of GU is not constructive. Given a set of isometries
which satisfies definition GU for S k , Lemma 2 and Theorem 1 are able to construct a set
of isometries which satisfies the definition of GU for S
k ′ or S k+1 = S k ∪ S
k ′ . Conditions
specified in Theorem 1 may be stronger than necessary, namely, if the set of isometries
specified in Theorem 1 fails the test of GU, other set of isometries may still pass the
test. Nevertheless, Theorem 1 provides a sufficient condition which is easy to test. For
example, to test if the 2 × 2 BPSK NHC (Table A.1) is GU, we can start from S 1 :
1. S 1 = {1, 14} is obviously GU where I U = {−I}, I P = {I}.
2. Is S 2 = {1, 14, 7, 8} GU? From the structure of S 1 ,
⎡ ⎤ ⎡ ⎤
U j = ⎣ −1 0 ⎦ , P j = ⎣ 1 0 ⎦
0 −1 0 1
and from Table A.1
⎡ ⎤ ⎡ ⎤
U k = ⎣ 0 1 ⎦ , P k = ⎣ 1 0 ⎦ .
−1 0 0 1
Since U k U j U k = P k P j P k = P
k ′ P jP k = P k P j P
k ′ = I and U k ′ U jU k = U k U j U
k ′ = −I, it
is easy to verify that S 1 is invariant in Theorem 1, therefore S 2 is GU. Now we have
⎧⎡
⎤ ⎡ ⎤ ⎡ ⎤⎫
⎨
I U = ⎣ −1 0 ⎦ , ⎣ 0 1 ⎦ , ⎣ 0 −1 ⎬
⎦
⎩ 0 −1 −1 0 1 0 ⎭
⎧⎡
⎤⎫
⎨
I P = ⎣ 1 0 ⎬
⎦
⎩ 0 1 ⎭ . 90

2
ÕOØÙ
168 30 180
×
42
ÔÖÕ
ìíïîIð0ñ
òóïôöõL÷
128
å æçè é¦êëêå æ¦è é
54
àãâCä
ßáà
156
ÚÛ@Ü
93
247
105
195
ÛÝÞ
ÏÐ@Ñ
ÐÒÓ
117
223
65
235
Figure 4.2: Hierarchical structure of the 2 × 2 QPSK NHC.
3. Is S 3 = {1, 14, 7, 8, 2, 13, 4, 11} GU? For each U j ∈ I U , we have
⎡ ⎤
U j = ⎣ −1 0 ⎦ ,
0 −1
⎡ ⎤
U j = ⎣ 0 1 ⎦ ,
−1 0
⎡ ⎤
U j = ⎣ 0 1 ⎦ ,
−1 0
⎡ ⎤
U k U j U k = U k ′ U jU k = U k U j U k ′ = ⎣ −1 0 ⎦ .
0 −1
⎡ ⎤
U k U j U k = U k ′ U jU k = U k U j U k ′ = ⎣ 0 −1 ⎦ .
1 0
⎡ ⎤
U k U j U k = U k ′ U jU k = U k U j U k ′ = ⎣ 0 1 ⎦ .
−1 0
Since P k = P j = I, P k P j P k = P ′ k P jP k = P k P j P ′ k = I. It is easy to verify that S 2 in
invariant in Theorem 1, therefore S 3 is GU.
4.4 The Hierarchical Structure of Optimal 2×2 QPSK Codes
As a case-study, we consider the construction of the 2 × 2 QPSK NHC. We shall see that
the NHC is in fact the optimal union bound code 5 found by brute force search [GVM02].
The hierarchical structure of the 2×2 QPSK NHC is depicted in Figure 4.2, the codewords
are delineated in Table 4.1.
The steps to generate 2 × 2 QPSK 16 codeword set are described below:
5 The union bound of its block error rate is lowest among all possible STBCs with the same block size
and constellation size.
91

[
2
] [
168
] [
42
] [
128
]
1 1 −1 −1 1 −1 −1 1
1 −1 −1 1 −1 −1 1 1
[
93
] [
247
] [
117
] [
223
]
i i −i −i i −i −i i
−i i i −i i i −i −i
[
30
] [
180
] [
54
] [
156
]
1 i −1 −i 1 −i −1 i
−i −1 i 1 i −1 −i 1
[
105
] [
195
] [
65
] [
235
]
i −1 −i 1 i 1 −i −1
−1 i 1 −i 1 i −1 −i
Table 4.1: Code list of the 2 × 2 QPSK NHC.
1. Pick an initial codeword, say 2 from DC 2 .
2. Generate another codeword (rate r = 0.5, coding gain ∆ p = 64.00)
⎡
⎣ −1
⎤ ⎡ ⎤
⎦ {2} ⎣ 1 ⎦ = {168}
−1 1
3. Generate 4 codewords (r = 1.0, ∆ p = 16.00)
⎡
⎣
−1
⎤ ⎡ ⎤
1
⎦ {2, 168} ⎣ 1 ⎦ = {42, 128}
1
4. Generate 8 codewords (r = 1.5, ∆ p = 16.00)
[
−1
]
[
1
1
{2, 168, 42, 128}
1
]
= {93, 247, 117, 223}
5. Generate 16 codewords, (r = 2.0, ∆ p = 4.00)
⎡
⎣ 1
⎤
⎡
⎦ {2, 168, 42, 128, 93, 247, 117, 223} ⎣ 1
−i
⎤
⎦
i
= {30, 180, 54, 156, 105, 195, 65, 235}
92

size const coset S 1 S 2 S 3 S 4 S 5 S 6 S 7 Table
2 × 2 BPSK 1 64.00 16.00† A.1
2 × 2 QPSK 2 64.00 16.00 16.00 4.00† A.2
2 × 2 8PSK 4 64.00 16.00 16.00 4.00 1.37 0.34† A.4,A.3
2 × 2 16PSK 8 64.00 16.00 16.00 4.00 1.37 0.34 0.09
3 × 3 BPSK 1, 10 256.00 64.00 64.00‡† A.5
3 × 3 QPSK 397 1280.00 160.00 64.00 32.00‡ 8.00‡ A.6,A.7
3 × 3 8PSK 10794 1620.08 202.51 104.97 32.00‡ 7.03‡ 2.34‡ 0.69‡ A.8,A.9
4 × 3 BPSK 83 4096.00 512.00 512.00 64.00‡† A.10
4 × 3 QPSK 8714 4096.00 512.00 512.00 384.00 128.00 32.00‡ A.11,A.12
‡ Code set which is not distance uniform. † Code set which achieves Singleton bound.
Table 4.2: List of Hierarchical Codes
The initial codeword 2 is not picked by chance, it belongs to the double coset DC 2
which has codewords of the largest determinant. Surprisingly, all of the optimal union
bound codes found by exhaustive search [GVM02] exhibit this hierarchical structure. We
note that our construction does not enforce group structure, however, the 16 generated
codewords fulfill the requirement of geometric uniformity as is clearly seen in Figure 4.2.
4.5 Code Lists
This section lists the hierarchical codes for various block sizes, constellation sizes and the
corresponding MTCM designs. Table 4.2 summarizes the largest full diversity NHC sets
for various block and constellation sizes. The double coset and minimum coding gain of
each layer is also listed. If the target is not the largest full diversity set, better NHCs
could be constructed for smaller sets. The detailed structure of each code set is tabulated
in the tables in the Appendix.
For 2 × 2 NHCs, the lower constellation code sets are “embedded” in the higher
constellation code sets. For example, the 16PSK NHC is identical to 8PSK NHC up to
S 6 , the 8PSK NHC is identical to QPSK NHC up to S 4 . However this embedded structure
does not always exist in NHCs of larger size. Note that for 3 × 3 BPSK NHC, S 3 is a
cross-coset design: the union of the two S 2 ’s from coset 1 and 10, respectively.
Several of the NHCs also achieve the Singleton bound
|S k | ≤ n Nc(Lt−∆ H+1)
(4.47)
93

which are very likely to be optimal code sets. The Singleton bound is an upper bound
on the cardinality of the N c × L t STBC set with diversity gain L t and constructed from
a constellation with size n. We compare the performance of our nonlinear hierarchical
space-time block codes with the best known unitary or orthogonal codes via union bounds.
For 3 × 3 STBCs, the best unitary group codes (UGC) [HM00] with cardinality 8 has the
following structure: let u = e j π 4 , the UGC is a cyclic group generated by
⎡ ⎤
u 0 0
⎢
⎣0 u 0 ⎥
⎦ . (4.48)
0 0 u 3
In Figure 4.3, Our BPSK NHC achieves the same performance as the 8PSK UGC; our
8PSK NHC yields about a 0.9dB gain over the UGC. The best 3 × 3 UGC with 63
codewords is G 21,4 [SHHS01], which is constructed from 21PSK in the form: A l B k , l =
0, . . . , 20, k = 0, 1, 2, where η = e 2π
21 i ,
⎡
η
A = ⎢
⎣
η 4
⎤
⎥
⎦
η 16
⎡ ⎤
0 1 0
B = ⎢
⎣ 0 0 1⎥
⎦ . (4.49)
η 7 0 0
Figure 4.4 shows that the 64 code set NHC S 6 constructed from 8PSK is less than half
dB away from the 63 code set G 21,4 in performance. Considering that S 6 uses only
8PSK while G 21,4 uses 21PSK, S 6 offers very good performance. In Figure 4.5, 4 × 3
BPSK/QPSK NHCs (8/64 codewords) achieve about a 2dB gain over the corresponding
orthogonal designs[TH01], which have the following structure
⎡
s 1 s 2 s 3
−s ∗ 2 s ∗ 1 0
⎢
⎣−s ∗ 3 0 s ∗ 1
⎤
. (4.50)
⎥
⎦
0 s ∗ 3 −s ∗ 2
94

3X3 NHC vs UGC, 8 Codewords
10 0 SNR
NHC(BPSK)
NHC(8PSK)
UGC(8PSK)
10 −1
Union Bound
10 −2
10 −3
10 −4
0 2 4 6 8 10 12 14
Figure 4.3: 3 × 3 BPSK, 8PSK NHCs (8 codewords) vs best 8PSK unitary group codes
(8 codewords).
3X3 NHC vs UGC
UGC |G 21,4
|=63 (21PSK)
NHC |S 6
|=64, (8PSK)
10 0
Union Bound
10 1 SNR
10 −1
10 −2
0 2 4 6 8 10 12 14
Figure 4.4: 3 × 3 8PSK NHCs (64 codewords) vs best 21PSK unitary group codes (63
codewords).
95

4X3 BPSK(8 Codewords) and QPSK(64 Codewords)
10 0
Orthogonal Codes, QPSK
Hierarchical Codes, QPSK
Orthogonal Codes, BPSK
Hierarchical Codes, BPSK
Union Bound
10 1
10 −1
10 −2
SNR
10 −3
10 −4
0 2 4 6 8 10 12 14
Figure 4.5: 4 × 3 BPSK (8 codewords) and QPSK (64 codewords) NHCs vs orthogonal
codes.
All the 2 × 2 NHCs happen to be orthogonal, or should be called super-orthogonal
[JS02]. The super-orthogonal STBC is like a simplified NHC. For example, the 2 × 2 and
4 × 4 super-orthogonal STBCs have the following structures respectively
⎡ ⎤ ⎡ ⎤
C(x 1 , x 2 , θ) = ⎣ x 1 x 2
⎦ ⎣ ejθ 0
⎦ (4.51)
−x ∗ 2 x ∗ 1 0 1
⎡
⎤ ⎡
⎤
x 1 x 2 x 3 0 e jθ 1
0 0 0
−x ∗ 2 x ∗ 1 0 −x 3
0 e jθ 2
0 0
C(x 1 , x 2 , x 3 , θ 1 , θ 2 , θ 3 ) =
(4.52)
⎢
⎣ x ∗ 3 0 −x ∗ 1 −x 2
⎥ ⎢
⎦ ⎣ 0 0 e jθ 3
0⎥
⎦
0 −x ∗ 3 x ∗ 2 −x 1 0 0 0 1
In comparison with NHCs, the super-orthogonal STBCs have the following limits:
1. Based on orthogonal code, the super-orthogonal STBCs do not exist for all possible
block size. As shown by NHCs, the orthogonal code are not always the best starting
base set.
2. Using only a limited form of right isometry P , the choice of parameter θ in P is not
systematic.
96

3. Lack of a hierarchical structure identifiable by isometry, the set partitioning of
super-orthogonal STBCs is not systematic.
97

Chapter 5
Regular Multiple Trellis Coded Modulation
5.1 Introduction
In the design of STBCs, memory or redundancy is inherent within each block. But this
is usually not strong enough in a fading channel, therefore, more memory needs to be
introduced between STBCs.
Space-time trellis codes (STTC) [TSC98, BBH02, YB02, Gri99] usually rely on a
convolutional encoder to introduce memory among symbols, but there are no good design
rules to optimize coding gain rather than exhaustive computer search. Multiple Trellis
Coded Modulation (MTCM) [BDMS91, DS87, SF02, JS02] serves as an ideal bridge to
combine the advantages of both block codes and trellis codes, and offers simple and
intuitive design rules to optimize coding gain systematically. Both STBC and STTC
can be viewed as certain kind of degenerated MTCM: STBC is equivalent to a single
state MTCM, and STTC is MTCM with a vector signal instead of a matrix block on
each transition. The MTCM designs in [SF02, ATP98] use STBCs as an inner code of a
convolutional encoder. Two important features of MTCM design over quasi-static MIMO
channels are pointed out in [SF02]: set expansion via isometries to generate large enough
“piece-wise full-rank” signal sets and simple labeling rules to guarantee full diversity.
However, such serial concatenation usually cannot take full advantage of the structure in
STBCs, and therefore introduces memory among blocks “blindly”. Chapter 5 of this work
proposes systematic MTCM designs where memory is introduced between block codes
directly, instead of through an outer convolutional encoder. Because of the symmetric
structure in the trellis and constituent NHCs, such designs are dubbed RMTCM designs.
98

In our RMTCM design, a regular trellis constructed specifically for fading channels
is generated by shift-register chains. NHCs, which admit a natural set partitioning as in
[Ung82], form the constituent codes. The parameters of the regular trellis and set partitioning
are matched to exploit the layered structure of the constituent NHCs optimally.
Several factors which affect performance in both quasi-static and fast block fading channels
are analyzed and exploited to improve the overall performance. Set expansion is used
to improve high rate MTCM designs. In each design, the performance/rate/complexity
tradeoffs are gracefully balanced and optimized. The general design procedure can be
used to serve two purposes: for a given set of NHCs, provide MTCM designs with various
rates, and, for a given rate, provide MTCM designs with various complexity and
performance.
The MTCM design in [JS02] is a special case of the RMTCM design in this work,
where the constituent codes are restricted to orthogonal codes and signal expansion is
limited to a special isometry; the set partitioning and trellis structure of [JS02] are not
systematic. RMTCM design can achieve a half dB gain over the rate 3 MTCM design in
[JS02] by doubling the number of states via adjusting trellis parameters. With even more
states, as much as 2dB gain can be achieved. The observation, that parallel transitions
should be avoided for better performance in quasi-static channels, is made in [JS02] for
orthogonal constituent codes only, we will show that it holds for more general cases.
The design principles of RMTCM do not rely on quasi-static fading channels. Thus
they can be employed to design RMTCM even for fast block fading channels, where fast
block fading is achieved due to perfect block interleaving. Both analysis and simulation
show that parallel transitions are the major detrimental factor which should be avoided
for fast block fading channels. A similar conclusion is drawn for MTCM design over
Single-Input-Single-Output fast-fading (due to perfect interleaving) channels [BDMS91].
In regular MTCM design, a sequence of STBCs are organized into frames and memory
is added between STBCs in a smart way to exploit the distance spectrum of constituent
code. This chapter is organized as follows. Section 5.3 provides the general regular
MTCM design procedure for a given hierarchical block code and rate. The performance
of such a design is analyzed in quasi-static and fast block fading channels (Section 5.4),
the connections between performance and design parameters are revealed and general
guidelines to improve regular MTCM designs in two fading channels are summarized.
99

The high rate performance is poor, Section 5.3 also proposes signal set expansion to
improve high rate performance.
5.2 Signal Model
In RMTCM design, a sequence of STBCs are organized into frames. We consider two
extreme channel conditions: if channel is constant for all blocks, it is deemed quasi-static
fading; if the channel is constant within a block, but independent from block to block
(due to perfect block interleaving), it is deemed fast block fading. In both fading channels,
the received signal for one block is still characterized by Equation (4.1). A single error
event (SEE) consists of two paths starting from the same state and merging only once
after M blocks.
Quasi-static Fading Channels In quasi-static fading channels, the received signal model
for the M-stage single error event is
Ȳ = √ σ t ¯DH + ¯N ∈ C
MN c×L r
(5.1)
where Ȳ = [Y T
1 , Y T
2 , . . . , Y T M ]T , ¯D = [D T 1 , DT 2 , . . . , DT M ]T and ¯N = [N T 1 , N T 2 , . . . , N T M ]T ,
Y m , D m , N m and H are defined in Equations (4.1)-(4.5).
Fast Fading Channels The received signal model for the M-stage single error event is
similar to that in quasi-static fading channel
Ȳ = √ σ t ¯D ¯H + ¯N ∈ C
MN c×L r
(5.2)
except that ¯D = diag(D 1 , D 2 , . . . , D M ) and ¯H = [H T 1 , HT 2 , . . . , HT M ]T . Note that ¯D
is block diagonal matrix in fast fading channels.
5.3 RMTCM Design Parameters and Procedures
NHCs admit a natural set partitioning as in [Ung82], and hence are a perfect candidate
for MTCM design. The set partitioning is simply the reverse procedure of construction
which will not be repeated here. Another design factor is the state trellis. Because the
only purpose of the state trellis is to introduce memory among the block codes, we will
100

S3 S2 S1
0 0 0
log 2
g
registers
0 0 1
0 1 0
S1
S3
0 1 1
log 2
b
register
chains
S1
S2
Sa+1
Sa+2
S2a+1
S2a+2
S2
1 0 0
1 0 1
1 1 0
Sa
S2a
1 1 1
Figure 5.1: The shift-register configuration
to generate state trellis.
Figure 5.2: The shift-register configuration
for 2 inputs and 3 registers and corresponding
trellis.
keep the state trellis as simple as possible. If the first a = log 2 b out of log 2 I input bits
are used to generate state trellis, the rest are used for parallel transitions, for N = log 2 S
registers, the simplest configuration of shift-registers to generate a state trellis is (see
Figure 5.1):
1. There are a = log 2 b feed-forward shift-register chains
2. Each of the first log 2 b bits is feed into the beginning of only one shift-register chain
3. No cross-links between two shift-register chains exist
4. Registers s k , k = 1, . . . , N, are assigned to each chain in a round-robin fashion,
states are defined as (s N , . . . , s 1 ).
Figure 5.2 shows the trellis generated by a shift-register configuration of two inputs
and three registers. For a regular trellis, there are b branches emerging from any given
state, and each branch contains p parallel transitions. Every b starting states are fully
connected to b ending states, all the branches connecting these states form a trellis group.
The number of trellis groups is denoted by g. To prevent catastrophic RMTCM designs, g
must be greater than 1, this means that the first shift-register chain must contain at least
2 registers (Figure 5.1). The set of parameters {g, b, p} describes the internal structure
of a regular trellis. Another set of parameters {I, S, O} which characterize the external
101

ehavior of a regular trellis are: the number of inputs I, the number of outputs O and
the number of states S. These two sets of parameters are related by:
O = gbp (5.3)
I = bp (5.4)
S = gb (5.5)
The overall code rate is
r = 1 N c
log 2 I. (5.6)
The E b /N 0 is related to SNR σ t by
E b
N 0
= L tσ t
r . (5.7)
When the frame error rates (FER) of designs with various rates are compared, if code
efficiency is of more importance, E b /N 0 should be used; if signal power constraint L t σ t
and achievable diversity level are of more importance, SNR L t σ t should be used. In this
work, we find plotting the FER curves with respect to SNR L t σ t is more instructive.
Using Equation (5.7), readers could easily obtain the corresponding curves with respect
to E b /N 0 .
If we want to assign the NHC set S K to this trellis, the number of trellis outputs
should match the cardinality of S K . Given O = |S K |, the procedure for RMTCM design
is:
1. Factor O = gbp
2. Generate the regular trellis by parameters {g, b}
3. Partition S K to g subsets, assign each subset S k to each trellis group
4. Perform Catastrophe test
The particular factorization of O determines the code rate r by Equations (5.3) and
(5.6). Once the code rate r is determined, the only free design parameter is b, which will
in turn determine p by Equation (5.4) and S by Equation (5.5). The labeling rule of
Step 4 guarantees full diversity [SF02]. The coding gain is optimized through the NHC
102

construction, the set partitioning principle [Ung82] and the matching between the layered
distance spectrum of NHC and the structure of the regular trellis. Such a RMTCM
design with g = xx, b = yy and p = zz will be labeled as xx × yy × zz. There is
still some freedom in assigning STBCs from S k to the transitions in a trellis group, we
find the “cyclic-shift rule” works very well: in every trellis group, there are b starting
states, if the transitions from one starting state are labeled by block codes in the order
D k1 , . . . , D kI , the transitions from the next starting state are labeled by a p-cyclic shift
of D k1 , . . . , D kI , namely, D kp+1 , . . . , D kI , D k1 , . . . , D kp . Special caution should be taken
to avoid catastrophic codes caused by bad labeling.
Due to the labeling rule that transitions diverging from and converging to the same
state are uniquely labeled by codewords from the same set S k , full diversity over S k instead
of S K is sufficient to guarantee full diversity of the MTCM design. This motivates
the expansion of the signal set further without the requirement of full diversity. A natural
criterion used in the expansion is ∆ s and the procedures to expand the signal set are
similar to those based on ∆ p :
1. Expand S k to S ′ k
by the following criterion
S ′ k = {U mS k P n |max
m,n min
i,j ∆ s(D ′ i − D j ), ∆ H < L t , D ′ i ∈ S ′ k , D j ∈ S k } (5.8)
2. Let S k+1 = S k ∪ S
k ′ , k = k + 1 and go to step 1.
The procedure stops when the desired rate is reached or S ′ is empty.
The expansion
can also be based on ∆ p . For two transmit antennae, the two approaches are essentially
identical. For more than two transmit antennae, the expanded NHCs based on two criteria
have similar performance.
Notice that full diversity ∆ H = L t is no longer required in signal set expansion. If
the largest full diversity subset is S m , the highest achievable rate of RMTCM design with
full diversity is r m = m N c
. Similar properties hold for expanded sets m ≤ k ≤ K:
1. Cardinality: |S k | = 2 k .
2. S ′ k and S k have identical distance spectrum with respect to ∆ H , ∆ p and ∆ s .
3. ∆ s : ∆ s (S k+1 ) ≤ ∆ s (S k ).
103

All the NHCs found in this work satisfying the following property (See code lists in
Table 4.2)
∆ x (S k+1 ) ≤ ∆ x (S k ), x = H, p or s. (5.9)
Because ∆ H and ∆ p characterize the performance of NHCs in Ricean channels 1 and
∆ s characterize the performance in Gaussian channels, this property suggests that the
set partitioning of NHCs is optimal in both Ricean and Gaussian channels. Although
designed for Ricean channel, RMTCM designs should yield reasonably good performance
even in a Gaussian channel.
We briefly discuss two problems regarding trellis design before concluding this section.
First, can feedback be added to the shift-register chains? The answer is yes, but proceed
with caution. Section 5.4 presents an important design parameter M m : the minimum
length M m of a single error event. If feedback is added at the middle of the register chain
rather than the beginning, M m is reduced and performance will be degraded. Feedback to
the beginning will not affect the frame error rate performance, but shuffles the mapping
between input bit sequences and output block codes and makes the overall structure
recursive, and could therefore potentially benefit a serial concatenated system [BDMP98].
The second question is, why not serially concatenate an optimal binary convolutional
encoder [LC83] with a “good” mapper, which maps the convolutional encoder’s output
bit sequence to a block code? The simple answer is because serial concatenation usually
cannot take advantage of the distance spectrum and the underlying finite state machine
directly and hence yields inferior performance with respect to RMTCM (see Figure 5.9 in
Section 5.5.2 for a direct comparison). Consider the following, let u denote the uncoded
input bits to the convolutional encoder, S denote the states, and v denote the output
bit sequence, f(.) denote the mapper from the output bit sequence to STBCs. For serial
concatenation, the overall function to map input bits to a STBC is
f(v(u, S)), (5.10)
while for RMTCM, the overall function is
f ′ (u, S). (5.11)
1 Note that we consider Rayleigh channels to be a special case of Ricean channels.
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Since f ′ is a direct function of inputs u and states S, rather than through an intermediate
(linear) function v, RMTCM is better adapted to match the underlying states to the
output code space, namely, the set of NHCs. Secondly, the code space of NHCs is rather
large due to the high dimensionality of the block code. For example, the full diversity 3×3
8PSK NHC has 128 block codes, which is much larger than the number of constellation
points for PSK. RMTCM can naturally exploit this large code space, while a mapper f(.)
in serial concatenation appears to be inefficient and sometimes awkward. There seems
to exist the possibility to take advantage of both the well-designed binary convolutional
codes and the large code space of NHCs via a smart mapper as in pragmatic TCM
[VWZP89], but this topic is beyond the scope of this work.
5.4 Performance Analysis
We next analyze the performance of our space-time MTCM systems. We consider a single
error event (SEE) with M blocks. Recall that ¯D is the sequence of M transmitted STBCs
where M is the length of a SEE.
Quasi-static Fading Channels In quasi-static fading channels, the pairwise error probability
of decoding ¯D ′ when ¯D (see Equation (5.1)) is transmitted, after averaging
over the channel, can be upper bounded by
P ( ¯D → ¯D ′ ) ≤
( σt
) −Lt 1
∣
4 ∣( ¯D − ¯D ′ ) H ( ¯D − ¯D ′ ) ∣ . (5.12)
Fast Fading Channels In fast fading channels, the channel-averaged pairwise error
probability of decoding ¯D ′ when ¯D (see Equation (5.2)) is transmitted can be upper
bounded by
P ( ¯D → ¯D ′ ) ≤
( σt
) −MLt 1
∣
4 ∣( ¯D − ¯D ′ ) H ( ¯D − ¯D ′ ) ∣ . (5.13)
Comparing Equations (5.12) and (5.13), the diversity gain is
⎧
⎪⎨ L t quasi-static fading
∆ H =
⎪⎩ ML t fast block fading
(5.14)
105

Whenever there exist parallel transitions, M = 1, and the achievable diversity level in
fast block fading channels is limited to that of quasi-static fading channels. Therefore,
in fast block fading channels parallel transitions should be avoided (this is verified by
simulation in Figure 5.7). The coding gain is
⎧
⎪⎨
∆ p = det[( ¯D − ¯D ′ ) H ( ¯D − ¯D
det[ ∑ M
′ i=1
)] =
(D i − D i ′)H (D i − D i ′ )] quasi-static fading
⎪⎩
∏ M
i=1 det[(D i − D i ′)H (D i − D i ′)]
fast block fading
In both cases, increasing the length, M, of a SEE is preferred.
(5.15)
The following two theorems justify our approach to MTCM design with STBCs in
quasi-static fading channels: cascading well-designed block codes. First, by Minkowski’s
Inequality [HJ91], for positive definite matrices (D i − D ′ i )H (D i − D ′ i ),
[ M
]
∑
det (D i − D i) ′ H (D i − D i)
′ ≥
i=1
M∑
det [ (D i − D i) ′ H (D i − D i) ′ ] (5.16)
i=1
This suggests that the coding gain of an error path grows at least with the sum of coding
gains of individual stages. Secondly, the monotonicity theorem [HJ91] yields
µ k ((D i − D ′ i) H (D i − D ′ i) + (D j − D ′ j) H (D j − D ′ j)) ≥ µ k ((D i − D ′ i) H (D i − D ′ i)) (5.17)
for positive semi-definite (D j − D j ′ )H (D j − D j ′ ), where µ k’s are eigenvalues arranged in
increasing order. Similar observations for specifically orthogonal STBCs are made in
[JS02], here we have shown this property holds for more general STBCs. For an example
of Minkowski’s inequality, see Figure 5.12 and Equation (5.33).
The minimum coding gain of all possible SEEs is defined as
d free = min
j
det( ¯D j − ¯D ′ j) H ( ¯D j − ¯D ′ j) (5.18)
where ¯D j and ¯D j ′ form a single error event. The coding gain associated with the shortest
path diverging from the zero state and converging at the zero state will be called d ∗ free . We
notice that the code is nonlinear, hence d ∗ free is not necessarily equal to d free. However,
d ∗ free agrees with the simulated performance fairly well and inspection of d∗ free
reveals very
106

helpful insight for code design (see Section 5.5.4). A SEE may be caused by parallel transitions
(M = 1), in this case, Equation (5.18) takes a simpler form d free = ∆ p (S k ), where
S k is the subset used to label a trellis group; otherwise (M > 1), the minimum length of
such a SEE not caused by parallel transitions, M m , is related to design parameters by
the following theorem:
Theorem 2 The minimum length M m of a single error event not caused by parallel
transitions for a regular trellis can be determined by
M m = ⌊log b g⌋ + 2. (5.19)
Proof:
In the finite state machine, the number of shift-register chains is log 2 b, the number of
shift-registers is log 2 S. By the construction rules of a regular trellis, the minimum length
of registers per chain is ⌊log 2 S/ log 2 b⌋ = 1 + ⌊log b g⌋. Starting from the same state, the
minimum number of stages for the minimum-length shift-register chain to converge to
another identical state due to different inputs is obviously 2 + ⌊log b g⌋. An easier way to
envision this is to consider the smallest number of stages of a “1” propagate through the
all-zero shift register chains. This happens when the “1” is feed into the shortest chain
with length 1 + ⌊log b g⌋. It takes 1 + ⌊log b g⌋ stages for the “1” to be shifted to the end
of the chain, and one more stage to be shifted out of the chain. ✷
Intuitively, increasing the number of trellis groups, g, forces two diverging paths to
traverse more branches before converging, and hence enlarges M m . Larger M m usually
results in better performance. The effects of g on M m is listed in Table 5.1. The point
where M m jumps is where we expect performance improvement. The sweet point for b =
2, 4, 8 seems to be g = 8, 4, 8, respectively, where the jumps occur still within a reasonable
number of states. Quadrupling the block set, coupled with RMTCM modulation, can
typically achieve 2dB gain over the original block code without memory, which seems to
be a good tradeoff between performance and complexity. Therefore, this work considers
mostly g = 4. For high rate codes, signal set expansion (Equation (5.8)) is used to
increase g.
Theorem 3 If we assume that g and b are powers of 2 (as in the case of RMTCM) and
there are no parallel transitions (p = 1), then the number of distinct paths T (n) starting
107

M m g=1 g=2 g=4 g=8 next jump
b=2 2 3 4 5 g=16
b=4 2 2 3 3 g=16
b=8 2 2 2 3 g=64
b=16 2 2 2 2 g=16
Table 5.1: The length M of single error event for given b and g.
from zero state and ending at a reachable state i after n stages is only a function of n
(independent of i):
⎧
⎪⎨ 1 n = 1
(b ≥ g) T (n) =
⎪⎩ b n−1
g
n > 1
⎧
⎪⎨ 1 n ≤ log b g + 1
(b < g) T (n) =
⎪⎩ b n−1
g
n > log b g + 1
(5.20)
(5.21)
Proof:
The shift register chains employ only the linear operation: addition modulo-2, ⊕. Let the
set of possible state sequences starting from the zero state be W = { ¯w| ¯w = (w 0 , . . . , w n ), w 0 =
0}, and define the following binary operation of two state sequences as element-wise addition
modulo-2,
ā ⊕ ¯b (a 1 ⊕ b 1 , . . . , a n ⊕ b n ). (5.22)
It is easy to verify the following properties
1. Closure: if ¯w 1 , ¯w 2 ∈ W, ¯w 1 ⊕ ¯w 2 ∈ W, because of the linear shift register chain
structure. The state a k is a linear function of previous state a k−1 and input u k−1 ,
similarly, b k is a linear function of b k−1 and input v k−1
a k = f(a k−1 , u k−1 ) by trellis structure (5.23)
b k = f(b k−1 , v k−1 ) by trellis structure (5.24)
a k ⊕ b k = f(a k−1 ⊕ b k−1 , u k−1 ⊕ v k−1 ) by linearity (5.25)
108

this suggests that for any previous state a k−1 ⊕ b k−1 , there exists a valid input
u k−1 ⊕v k−1 to bring the system to next state a k ⊕b k , therefore (a k−1 ⊕b k−1 , a k ⊕b k )
is a valid state sequence. By induction, ¯w 1 ⊕ ¯w 2 is a valid sequence.
2. Associativity: ( ¯w 1 ⊕ ¯w 2 ) ⊕ ¯w 3 = ¯w 1 (⊕ ¯w 2 ⊕ ¯w 3 ).
3. Identity: the all zero state sequence ¯0, ¯w + ¯0 = ¯w.
4. Inverse: every state sequence ¯w is its own inverse ¯w ⊕ ¯w = ¯0, ¯w −1 = ¯w.
5. Commutativity: ¯w 1 ⊕ ¯w 2 = ¯w 2 ⊕ ¯w 1 .
therefore, W forms an abelian group under the binary operation ⊕ defined in Equation
(5.22). Denote the set of all the state sequences starting from the zero state and ending
in state i by W i = { ¯w|w 0 = 0, w n = i}, then we want to show that all W i ’s have equal
cardinality T (n). The subset W 0 actually forms a subgroup. This can be verified by the
subgroup theorem [Arm87]: if ā, ¯b ∈ W 0 , then
ā ⊕ (¯b) −1 =ā ⊕ ¯b (by inverse property 4, (¯b) −1 = ¯b) (5.26)
∈W (by closure property 1) (5.27)
∈W 0 (by a n ⊕ b n = 0). (5.28)
By the subgroup theorem, W 0 is a subgroup, and W i ’s, i ≠ 0, are cosets. All cosets have
equal cardinality, therefore T (n) is independent of ending state i.
If b ≥ g, at stage n = 1, all reachable states {0, . . . , b−1} have T (1) = 1. At stage n =
2, all states become reachable, and T (2) = b/g. For any stage n > 2, T (n) = bT (n − 1).
This proves Equation (5.20).
If b < g, at stages n = 1, . . . , log b g+1, all reachable states {0, . . . , b−1} have T (1) = 1.
At stage n = log b g + 1, all states become reachable. For any stage n > log b g + 1,
T (n) = bT (n − 1). This proves Equation (5.21). ✷
Corollary 1 For any state with T (n) ≥ 2, the number of error events not caused by
parallel transitions is ( )
T (n)
2 = (b 2n−2 − b n−1 )/(2g 2 ).
The proof is straightforward because any two distinct paths ending at the same state
form an error event. Because an error event may contain several SEEs, Corollary 1
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provides a lower bound on the number of SEEs. In Theorem 3, the existence of error
events is equivalent to T (n) ≥ 2, which in turn is equivalent to n ≥ log b g + 2. This
provides another proof of Theorem 2 when b and g are powers of 2. Theorem 3 and
Corollary 1 characterizes the number of distinct paths, T (n), and the number of error
events as a function of b, g and n. A larger g always reduces the number of SEEs,
and therefore improves system performance. Because SEEs with short length tend to
dominate performance, a more direct improvement of performance due to increasing g is
via reducing T (n) for small n.
We want to point out that the RMTCM design is a nonlinear code once the effects
of labeling is taken into account. Theorems 2, 3 and Corollary 1 characterize the general
structure of regular trellis without considering the effects of labeling. Therefore they
apply to any code with a regular trellis structure.
For a RMTCM design, the ultimate design goal is to increase d free . There are two
means by which to increase d free : if d free is caused by parallel transitions (M = 1),
reducing the number of parallel transitions p can increase d free due to improving the
performance of the constituent NHC ∆ p (S k ) ≥ ∆ p (S k+1 ); otherwise, increasing the minimum
length M m of SEE not caused by parallel transitions can increase d free due to
the observations on coding gain in Equation (5.15). From a design point of view, for a
given rate, increasing g may increase M m ; increasing b decreases p. A negative point of
increasing b, by Equation (5.19), is that M m might be decreased (See Table 5.1). Empirical
results suggest that the advantage of reducing p usually outweighs the disadvantage
of reduced M m . For RMTCM in fast block fading channels, p should be reduced until
p = 1; while in quasi-static fading channels, p should be reduced to a sweet point where
large performance gains can be achieved by relative small decrease of p. The sweet point
is closely related to the distance spectrum of the hierarchical codes.
The performance of RMTCM be can improved by increasing either b or g, as a consequence,
the number of states, S = gb, grows linearly with g and b. The decoder employs
the Viterbi algorithm, whose decoding complexity is linear in the number of states S.
To calculate the branch metric for Viterbi decoder, the received signal is correlated with
O = gbp block codes. Therefore both the calculation of branching metric and decoding
are linear in g and b, furthermore, the calculation of branching metric is also linear in p.
110

5.5 RMTCM Designs Examples
This section summarizes various RMTCM designs with 2×2 and 3×3 BPSK, QPSK and
8PSK NHCs. For the detailed list of NHCs, please refer to Table 4.2 and the references
therein.
MTCM designs following the rules outlined in Section 5.3 will be denoted Regular
MTCM designs, otherwise irregular MTCM designs. The performance of various MTCM
designs and comparable space-time trellis codes are compared. We use two sets of parameters
to distinguish different designs:
1. Regular MTCM designs with the three parameters g = x, b = y and p = z are
denoted by x × y × z.
2. Irregular MTCM designs with the four parameters I = w, S = x, O = y and p = z
are denoted by IwSxOyP z.
The set partitioning and labelling for 2 × 2 QPSK RMTCM designs are listed in Chapter
B.
The performance of various RMTCM designs and comparable space-time trellis codes
are compared. A block code can be viewed as a trivial trellis code with a single state and
every block on parallel transitions. For a RMTCM design x × y × z, the same rate trivial
trellis code (block code) 1 × 1 × yz which uses only one out of the g groups of STBCs is
often included for performance comparison.
If the largest full diversity subset is S m , the highest achievable rate RMTCM design
with full diversity is determined by
r m = m N c
. (5.29)
Via simulation we observe that expanding S m 4 times is sufficient to achieve very good
performance with a relative small increase in complexity.
The super-orthogonal space-time trellis codes [JS02] coincide with a subset of nonlinear
hierarchical codes and their MTCM designs. The RMTCM designs in this work
are more systematic and complete. We show several ways to improve the performance
of the super-orthogonal space-time trellis codes. With added complexity, more than 2dB
111

Label S k S Comment Figure
1/2 I2S2O2p1 S 1 2 regular, catastrophic B.1
1/2 I2S2O4p1 S 2 2 irregular, non-catastrophic B.2
1/2 2 × 2 × 1 S 2 4 regular, non-catastrophic B.3
Table 5.2: Rate 1/2 MTCM designs for 2 × 2 BPSK.
2X2 BPSK, Rate 1/2
10 0 SNR
10 −1
FER
10 −2
10 −3
r=0.5, 1x1x2, block
r=0.5, I2S2O2p1, cata
r=0.5, I2S2O4p1
r=0.5, 2x2x1
10 −4
0 2 4 6 8 10 12 14 16
Figure 5.3: Rate 1/2 MTCM designs for 2 × 2 BPSK.
improvement can be made. The RMTCM designs based on 3 × 3 NHCs have no superorthogonal
counterparts.
5.5.1 2 × 2 BPSK
Table 5.2 summarized 3 possible rate half designs based on S 1 and S 2 , the detailed
labelling is depicted by the corresponding figures in Chapter B. Figure 5.3 compares
their performance. Even the 2-state catastrophic design I2S2O2p1 is more than half dB
better than the block code 1 × 1 × 2; the 2-state non-catastrophic design I2S2O4p1 is
about 1dB better than the catastrophic design I2S2O2p1 by using S 2 instead of S 1 ; the
4-state regular MTCM design 2 × 2 × 1 is 2dB better than the 2-state non-catastrophic
design I2S2O4p1. We note that the 4-state regular MTCM design 2 × 2 × 1 achieves
more than 3.5dB gain over the block code 1 × 1 × 2 by simply introducing 2-bit memory.
112

Label S k O(= gbp) S(= gb) Figure
2/2 1 × 1 × 4 S 2 4 1
2/2 2 × 2 × 2∗ S 3 8 4 B.4
2/2 2 × 4 × 1 S 3 8 8 B.5
Table 5.3: Regular MTCM design for 2 × 2 BPSK |S 3 | = 8 codewords.
2X2 BPSK, Rate 2/2
10 0 SNR
10 −1
FER
10 −2
r=1.0, 1x1x4, block
r=1.0, 2x2x2
r=1.0, 2x4x1
10 −3
0 2 4 6 8 10 12 14 16
Figure 5.4: 2 × 2 BPSK, Rate 2/2.
The highest rate RMTCM design based on 2 × 2 BPSK NHC is determined by S 2 :
r m = 2/2 = 1. For rate 2/2, the two possible expanded MTCM designs with S 3 are
listed in Table 5.3; their performance is compared in Figure 5.4. Notice that Coset C 1 (8)
contains 8 codewords only, which is partitioned as S 2 and S 2 ′ and assigned to the two
trellis groups in designs 2 × 2 × 2 and 2 × 4 × 1. The RMTCM designs show 2dB gain over
the block codes 1 × 1 × 4, the 8-state design 2 × 4 × 1 is slightly better than the 4-state
design 2 × 2 × 2, which is reported in [JS02].
5.5.2 2 × 2 QPSK
The full diversity code sets are S 1 –S 4 , S 5 and S 6 are rank deficient expanded code sets.
The 2 × 2 QPSK code is a very good example to show the general design procedures
of RMTCM outlined in Section 5.3, the detailed labeling of each RMTCM design are
113

Label S k O(= gbp) S(= gb) Figure
1/2 8 × 2 × 1 S 4 16 16 B.6
2/2 4 × 2 × 2∗ S 4 16 8 B.7
2/2 4 × 4 × 1 S 4 16 16 B.8
3/2 2 × 2 × 4 S 4 16 4 B.9
3/2 2 × 4 × 2∗ S 4 16 8 B.10
3/2 2 × 8 × 1 S 4 16 16 B.11
4/2 1 × 1 × 16 S 4 16 1
Table 5.4: MTCM design for 2 × 2 QPSK |S 4 | = 16 codewords.
omitted, only their performance is compared.
Readers can apply the general design
procedure and the “cyclic shift rule” to reproduce all the RMTCM designs in the sequel.
There are seven possible ways to factor |S 4 | = 16, this yields seven possible RMTCM
designs with S 4 (Table 5.4). Figure 5.5 compares their performance. For a given rate
(I = bp), there might be several designs with different numbers of parallel transitions
and states. As can be seen, the rate/performance/complexity tradeoffs are gracefully
balanced. For rate r = 1.0, the two designs with states 8 and 16 have roughly the same
performance, therefore the 8-state design 4 × 2 × 2 is preferred. The reason is that the
two parallel transitions have coding gain ∆ p (S 1 ) = 64, which is large enough not to affect
the d free of the overall RMTCM design. Similarly, for rate r = 1.5, the 8-state and
16-state designs have about the same performance, which is about 2dB better than that
of 4-state design, therefore the 8-state design 2 × 4 × 2 is preferred. The performance of
these seven designs in Ricean channels (h = 1) is shown in Figure 5.6. Comparing Figure
5.6 with Figure 5.5, the performance of all codes are improved by about 4dB in Ricean
channel than the performance in Rayleigh channel. If we plug h = [1, 1] T and Σ h = I
into Equation (4.12),
E h [P (D i → D j )|h)] ≤
1 1
4 Σ h| |Φ|
| eσt
(5.30)
the SNR gain is 10 log e = 10 log 2.718 = 4.3dB, which agrees with the simulations. Note
that the purpose of this comparison is to demonstrate that the code design is independent
of Rayleigh or Ricean channel, therefore we do not normalize the power of the fading to
114

2X2 QPSK, |S 4
|=16 codewords
10 0 SNR
10 −1
FER
10 −2
10 −3
r=2.0, 1x1x16
r=1.5, 2x2x4
r=1.5, 2x4x2
r=1.5, 2x8x1
r=1.0, 4x2x2
r=1.0, 4x4x1
r=0.5, 8x2x1
10 −4
0 2 4 6 8 10 12 14 16
Figure 5.5: 2 × 2 QPSK, |S 4 | = 16.
2X2 QPSK, |S 4
|=16 codewords, Ricean h=1
10 0 SNR
10 −1
10 −2
FER
10 −3
10 −4
r=2.0, 1x1x16
r=1.5, 2x2x4
r=1.5, 2x4x2
r=1.5, 2x8x1
r=1.0, 4x2x2
r=1.0, 4x4x1
r=0.5, 8x2x1
10 −5
0 2 4 6 8 10 12 14 16
Figure 5.6: 2 × 2 QPSK, |S 4 | = 16, Ricean fading channels (h = 1).
115

2X2 QPSK, |S 5
|=32 codewords
10 0 SNR
10 −1
FER
10 −2
10 −3
10 −4
r=4/2, 1x1x16, quasi−static
r=4/2, 1x1x16, fast block,
r=4/2, tarokh 4 state, quasi−static
r=4/2, 2x2x8, quasi−static
r=4/2, 2x2x8, fast block
r=4/2, 2x4x4, quasi−static
r=4/2, 2x4x4, fast block
r=4/2, 2x8x2, quasi−static
r=4/2, 2x8x2, fast block
r=4/2, 2x16x1, quasi−static
r=4/2, 2x16x1, fast block
10 −5
0 2 4 6 8 10 12 14 16 18 20
Figure 5.7: 2 × 2 QPSK, |S 5 | = 32, quasi-static vs fast block fading channels.
unity. If the power is normalized, then plugging h ′ = [0.5, 0.5] T
Equation (4.12) yields
and Σ ′ h = 0.5Σ h into
E h [P (D i → D j )|h)] ≤
the SNR loss would be 10 log(2/ √ e) = 0.8dB.
|
√ e σ t
2
1 1
4 Σ h| |Φ|
(5.31)
The rate r = 2 single-state MTCM design of S 4 is just the block code without memory,
it has very poor performance. Expanded RMTCM designs based on S 5 and S 6 can
improve the performance at this rate. Figure 5.7 compares the performance of designs
with S 5 in both quasi-static and fast fading channels.
• In quasi-static fading channels, the 4-state design is reported in [JS02], it yields
about 1.7dB gains over corresponding 4-state QPSK design by Tarokh [TSC98].
As p decreases from 8 to 1, (and the number of states increases from 8 to 64),
performance is improved consistently by an extra 0.8dB. Similar observations can
be made for designs with S 6 . Designs based on S 6 have about 0.4dB gain over
corresponding (same b) designs based on S 5 by doubling g. The 4-state design
116

2 × 2 × 8 is reported in [JS02], the 32-state design 4 × 8 × 2 and the 8-state design
4 × 2 × 8 are 1dB and 0.5dB better than 2 × 2 × 8, respectively.
• In fast fading channels, all designs with parallel transitions achieve the same level of
diversity (∆ H = 2) as in quasi-static fading channels eventually, the design 2×8×2
shows this tendency only at SNR>16dB. The design with no parallel transitions
2 × 16 × 1 is capable of exploiting more diversity (∆ H = 12) in the channel as
predicted by Equation (5.14).
Figure 5.9 compares the serially concatenated system with RMTCM. The concatenated
system uses optimal rate 1/2 convolutional encoder from [LC83]. The three generators
are: {64, 74}, {46, 72} and {554, 744}, which generate state trellises with 8, 16
and 64 states respectively. The bit mapper maps every four output bits to one STBC
from S 4 . Overall, two information bits are transmitted over every STBC which lasts
two symbols, therefore the rate is r = 1. The mapper is designed to imitate the hierarchical
structure of coding gain ∆ p in S 4 in terms of Hamming distance d H , in the
hope that larger Hamming distance, which is the optimization goal of convolutional encoder,
will result in larger coding gain, which is a performance measure of interest in a
MIMO fading channel. Figure 5.8 shows the correspondence between bit sequence and
block codes in S 4 . Let c 1 and c 2 be two binary codewords, d be any binary vector, then
d H (c 1 − c 2 ) = d H ((c 1 + d) − (c 2 + d)). Hence the isometry for Hamming distance is
simply addition modulo-2 by any binary vector, via which, the hierarchical structure of
the Hamming distance of the bit sequence can be described in Table 5.5 in a similar manner
as the hierarchical structure of the coding gain in Table 4 of [GM04]. The mismatch
between the optimization goal of convolutional encoding and performance measure for
MIMO fading channels yields the relative performance loss of such a concatenated system:
the 8, 16 and 64 state concatenated systems are respectively, 2.5dB, 1dB and 0.5dB,
worse than the corresponding RMTCM designs.
5.5.3 2 × 2 8PSK
For 2 × 2 8PSK NHCs, up to S 6 are full diversity code sets; S 7 and S 8 are expanded code
sets. The highest achievable rate with full diversity is r m = 6/2 = 3. Various designs
with rates from 1/2 to 6/2 = 3 can be realized. We note that for a given rate, p = 2 and
117

S k d S k S
k ′ d H
S 1 (1, 1, 1, 1)
{
{(0, 0, 0, 0)}
} {
{(1, 1, 1, 1)}
}
4
(0, 0, 0, 0) (1, 1, 0, 0)
S 2 (1, 1, 0, 0)
2
⎧
(1, 1, 1, 1)
⎫ ⎧
(0, 0, 1, 1)
⎫
(0, 0, 0, 0) (1, 0, 1, 0)
⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬
(1, 1, 1, 1) (0, 1, 0, 1)
S 3 (1, 0, 1, 0)
2
(1, 1, 0, 0) (0, 1, 1, 0)
⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭
(0, 0, 1, 1) (1, 0, 0, 1)
⎧ ⎫ ⎧ ⎫
(0, 0, 0, 0) (0, 0, 0, 1)
(1, 1, 1, 1) (1, 1, 1, 0)
(1, 1, 0, 0) (1, 1, 0, 1)
⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬
S 4 (0, 0, 0, 1)
(0, 0, 1, 1)
(1, 0, 1, 0)
(0, 1, 0, 1)
(0, 1, 1, 0) ⎪⎩ ⎪⎭
(1, 0, 0, 1)
(0, 0, 1, 0)
(1, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 1, 1) ⎪⎩ ⎪⎭
(1, 0, 0, 0)
Table 5.5: Hierarchical structure in Hamming distance.
1
(0000) 2 (1111) 168
(0001) 30 (1110) 180
(1100) 42 (0011) 128
(1101) 54 (0010) 156
(1010) 93 (0101) 247
(1011) 105 (0100) 195
(0110) 117 (1001) 223
(0111) 65 (1000) 235
Figure 5.8: Bit map of 2 × 2 QPSK, |S 4 | = 16, for serial concatenation.
118

2X2 QPSK, |S 4
|=16, concat vs MTCM
10 0 SNR
10 −1
FER
10 −2
r=1.0, S=8, concat
r=1.0, S=16, concat
r=1.0, S=64, concat
r=1.0, 4x2x4, MTCM
r=1.0, 4x4x2, MTCM
r=1.0, 16x4x1, MTCM
10 −3
0 2 4 6 8 10 12 14 16
Figure 5.9: 2 × 2 QPSK, |S 4 | = 16, serial concatenation vs RMTCM.
p = 1 have about the same performance, because ∆ p (S 1 ) = 64, which is large enough
not to affect d free . Figure 5.10 compares rate 6/2 design with expanded S 7 and S 8 . By
doubling the number of states, both the 8-state design 2 × 4 × 16 and 4 × 2 × 32 are about
0.5 dB better than the 4-state design 2 × 2 × 32, which is reported in [JS02]. For b = 16
and 32, the designs based on S 8 is about 0.3dB and 0.5dB better than those based on S 7 .
The best design 4 × 32 × 2 is more than 2dB better than the design 2 × 2 × 32 reported
in [JS02].
5.5.4 3 × 3 BPSK
To the best of our knowledge, these 3 × 3 codes are new to the literature. Figure 5.11
compares the performance of designs with S 3 . Next, we analyze the d ∗ free
of two rate
r = 2/3 designs: the 8-state design 2 × 4 × 1 and the 4-state design 2 × 2 × 2. Figure 5.12
shows the shortest diverging path for design 2×4×1. We denote the coding gain between
119

Rate 6/2 2X2 8PSK, |S 7
|=128 vs |S 8
|=256 Codewords
10 0 SNR
10 −1
FER
10 −2
r=6/2, 1x1x64
r=6/2, 2x2x32
r=6/2, 2x4x16
r=6/2, 2x16x4
r=6/2, 2x32x2
r=6/2, 4x2x32
r=6/2, 4x16x4
r=6/2, 4x32x2
10 −3
10 12 14 16 18 20 22 24 26
Figure 5.10: Rate 6/2, 2 × 2 8PSK, |S 7 | = 128 vs |S 8 | = 256.
two paths with the label (C 1 , C 2 , . . . , C M ) and (C ′ 1 , C′ 2 , . . . , C′ M ) by ∆ p((C 1 , C 2 , . . . , C M )−
(C 1 ′ , C′ 2 , . . . , C′ M
)). According to Equation (5.16),
∆ p ((C 1 , C 2 , . . . , C M ) − (C ′ 1, C ′ 2, . . . , C ′ M)) >
M∑
∆ p ((C 1 ) − (C 1)) ′ (5.32)
i=1
The 8-state design has no parallel transitions, the shortest path starting from the zero
state and merging back to the zero state takes 3 stages:
d ∗ free = ∆ p((330, 413, 251) − (1, 1, 1)) = 2048 (5.33)
>> ∆ p ((330) − (1)) + ∆ p ((413) − (1)) + ∆ p ((251) − (1)) (5.34)
= 64 + 64 + 64 = 192 (5.35)
The gain achieved by elongating the number of stages in the shortest path is usually much
larger than that predicted by Minkowski’s Inequality. The 4-state design has parallel
120

3X3 BPSK, |S 3
|=8 Codewords
10 0 SNR
10 −1
10 −2
FER
10 −3
10 −4
r=3/3, 1x1x8
r=2/3, 2x2x2
r=2/3, 2x4x1
r=1/3, 4x2x1
10 −5
0 2 4 6 8 10 12 14 16
Figure 5.11: 3 × 3 BPSK, |S 3 | = 8.
transitions, the shortest path takes just one stage, hence the free distance is just the
coding gain of block code set S 1 .
d ∗ free = ∆ p(S 1 ) = 256.
Due to the large difference in d ∗ free
, the 8-state design yields 1.5dB gain over the 4-state
design.
For rate r = 3/3, Figure 5.13 compares the performance of 2×8 designs with S 4 , 4×8
designs with S 5 and 8 × 8 designs with S 6 . For the same g, the performance is always
improved by reducing the number of parallel transitions. For b = 2, increasing g does not
improve performance, because the limiting factor is parallel transitions p = 8. For b = 4,
increasing g from 2 to 4 yields about 0.5dB gains, but increasing g from 4 to 8 yields no
noticeable gains. For b = 8, more than 0.5 dB gains is observed when g increases from
2 to 4, but about 0.3 dB gains is observed when g increases from 4 to 8. We conjecture
that for most RMTCM designs with b = 2 and b = 4, g = 4 is sufficient.
121

1 172 251 330
1
1 1
304 413 486 87
172 251 330 1
330
251
413 486 87 304
251 330 1 172
413
486 87 304 413
330 1 172 251
87 304 413 486
Figure 5.12: 3 × 3 BPSK, d ∗ free
of 2 × 4 × 1.
3X3 BPSK, Rate 3/3, |S 4
| vs |S 5
| vs |S 6
|
10 0 SNR
10 −1
FER
10 −2
10 −3
r=3/3, 2x2x4
r=3/3, 2x4x2
r=3/3, 2x8x1
r=3/3, 4x2x4
r=3/3, 4x4x2
r=3/3, 4x8x1
r=3/3, 8x2x4
r=3/3, 8x4x2
r=3/3, 8x8x1
10 −4
0 2 4 6 8 10 12 14 16
Figure 5.13: Rate 3/3, 3 × 3 BPSK, S 4 , S 5 and S 6 .
122

3X3 QPSK, |S 5
|=32 codewords
10 0 SNR
10 −1
10 −2
FER
10 −3
10 −4
r=5/3, 1x1x32
r=4/3, 2x2x8
r=4/3, 2x4x4
r=4/3, 2x8x2
r=4/3, 2x16x1
r=3/3, 4x2x4
r=3/3, 4x4x2
r=3/3, 4x8x1
r=2/3, 8x2x2
r=2/3, 8x4x1
r=1/3, 16x2x1
10 −5
10 −6
0 2 4 6 8 10 12 14 16
Figure 5.14: 3 × 3 QPSK, |S 5 | = 32.
5.5.5 3 × 3 QPSK
S 1 to S 5 are full diversity code sets, S 6 , S 7 and S 8 are expanded code sets. The highest
achievable rate of RMTCM design is r m = 5/3. Figure 5.14 shows RMTCM designs
with S 5 for various rates. Figure 5.15 shows rate 5/3 RMTCM designs with S 7 . By
increasing memory and decreasing the number of parallel transitions, up to 5dB gain can
be obtained over the 1-state block code.
5.5.6 3 × 3 8PSK
S 1 to S 7 are full diversity code sets, S 8 is expanded code set. The highest achievable rate
of RMTCM design is r m = 7/3. Based on S 7 , there are four rate 4/3, five rate 5/3 and six
rate 6/3 RMTCM designs. For any rate, consistent performance improvement is observed
with increased b and the best design offers more than 5dB gains over corresponding block
code. Figure 5.16 compares the expanded high rate designs based on S 8 with designs
based on S 7 , performance improvement is large for b ≥ 8.
123

Rate 5/3, 3X3 QPSK, |S 7
|=128 Codewords
10 0 SNR
10 −1
FER
10 −2
r=5/3, 1x1x32
r=5/3, 4x2x16
r=5/3, 4x4x8
r=5/3, 4x8x4
r=5/3, 4x16x2
10 −3
0 2 4 6 8 10 12 14 16
Figure 5.15: 3 × 3 QPSK, |S 7 | = 32.
3X3 8PSK, rate=6/3, |S 7
|=128 codwords vs |S 8
|=256 codewords
10 0 SNR
10 −1
FER
10 −2
r=6/3, 1x1x64
r=6/3, 2x2x32
r=6/3, 2x4x16
r=6/3, 2x8x8
r=6/3, 2x64x1
r=6/3, 4x2x32
r=6/3, 4x4x16
r=6/3, 4x8x8
r=6/3, 4x64x1
10 −3
0 2 4 6 8 10 12 14 16
Figure 5.16: 3 × 3 8PSK, rate=6/3, |S 7 | = 128 vs |S 8 | = 256.
124

Chapter 6
Conclusions
This dissertation covers four topics of space-time codes which are involved with my Ph.D.
research: Chapter 2 discusses the code performance and design in spread systems, Chapter
3 considers suboptimal linear decoders, Chapter 4 discusses the construction and
properties of NHC, finally, Chapter 5 discusses the design procedure and performance
analysis of RMTCM. The framework of Chapter 2 was finished during my master’s research
at Ohio State University, but most analyses and proofs are tightened as part of
my Ph.D. work. Therefore it is included this dissertation.
The original intention of studying suboptimal decoders in Chapter 3 was to find code
structures more suitable for these suboptimal linear decoders. But it was realized to be
a very challenging task because no closed form upper bounds could be obtained for these
decoders.
Motivated by the hierarchical structure of optimal codes found by computer search in
Chapter 2, a methodology to build optimized codes hierarchically via distance preserving
isometries is proposed in Chapter 4. The procedure can yield arbitrary STBCs with
no constraints on the block size and code structure (unitary or orthogonal). The found
hierarchical codes yield not only optimized coding gain at each layer of the hierarchy,
but also symmetry and structure which can be exploited in system implementation and
integration. The interaction between initial codeword and isometries is still not completely
clear. Further investigation could eliminate the computer search aspect of our
construction method altogether and result in a fully systematic code design.
General RMTCM design techniques with NHCs and their expansions are presented
in Chapter 5. For a given set of nonlinear hierarchical code, various MTCM designs are
provided to show how to balance rate/performance/complexity tradeoffs. Two design
125

parameters: the number of trellis groups g and the number of emerging branches from
any state b, are used to improve the performance of MTCM design, this work endeavors
to find the sweet point where g and b are mostly effectively used to boost performance.
As much as 2dB gain is observed over other MTCM design via our systematic approach.
There is still some freedom in the trellis labeling, our approach, the cyclic-shift rule is
perhaps not the best. Proper labeling may yield a recursive structure to facilitate serial
concatenation of interleaved codes [BDMP98].
126

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130

Appendix A
Code Lists
For different block sizes and constellations, tables of nonlinear hierarchical codes are listed
in this section. There are two kinds of tables: one lists the isometries used to generate
each layer of the code, together with the minimum diversity gain ∆ H , coding gain ∆ p
and ∆ s ; the other lists the full code set. The second table is used only when the code
lists are too large to fit in the first table. The first table is divided into two sections
marked by double lines, the first section lists full diversity code, the second section lists
the expanded code (see Equation (5.8) for an explanation of expanded code).
S k
[
U
]
S k
[
P
]
S
k ′ ∆ H ∆ p ∆ s
−1 0
1 0
S 1 {1}
{14} 2 64.00 16.00
[
0 −1
]
[
0 1
]
0 1
1 0
S 2 {1, 14}
{7, 8} 2 16.00 8.00
−1 0
0 1
[ ]
[ ]
1 0
1 0
S 3 {1, 14, 7, 8}
{2, 13, 4, 11} 1 8.00 8.00
0 −1
0 1
Table A.1: 2 × 2 BPSK codes.
131

ù"ú"ú"û ù"ü"ù ù"ý0øJþ ÿLù"ý"ù ú ø¢¡;þ ÿ+øJù"û ú"ù¤£ ø ÿLþ"ý ù ø"øJþ øJý ø ù"ù"ù ø ÿLúÿ+ø ú"ý¤¡"þ ÿLý"ù"û ú"ú"û;û
ø
ù¤¡"ü"ü ¡ ø¢¡ ú"ý;ù"ú ÿLû¤£¤¡ ú"ü"ú;ú ÿLþ"þÿ ú¤¡"ý"ü û"ú¤£ ù"üÿ"ÿ þ¤£"ü ù"û"þ¤¡ ÿ¥¡"û"ü øJý ø¢£ ÿLü"ü"ú ú"þ"ùÿ
ü¤¡ÿ
ùÿ¥£"þ ÿLý"û ù"ú;þ"ù ÿ¥£"ù"þ ú"ù"ü;ù ÿLù øJý ú¤£"þ"ý £"ý"û ù"ù¤¡"ý ùÿLþ ù¤£¤£"þ ÿ"ÿLù"þ ú øJý ø ÿLú¤£"ù úÿLþ;ý
ú"ú"ý
ù"û¤¡"þ û"ù"û ù"ü;ý"ù ÿLü"þ ø ú"þÿ ù ÿ¥¡"û"ý øJý"ú"û ü"û"ù ù¤¡"ü"ý ¡"ú"þ ú"ýÿ+ø ÿLû øJþ ú"ü"ù ø ÿLþ¤¡"ù ú¤¡"ý;ý
þ¤£"ý
ú"ùÿ ùÿ+øJü ü¤¡ ù"ú§¡"ú ÿ¥£"ù¤¡ ú"ù"þ;ú ÿLù"ú"ü ú¤£¤¡ÿ øJü¤¡ ù"ù"ûÿ ù"ý"ü ù¤£ øJü ÿ"ÿLù¤¡ ú"ú"ü¤£ ÿLú¤£ÿ úÿ¥¡ÿ
¨¤¨¤¨¤¨§¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨§¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨§¨¤¨¤¨¤¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨¤¨¤¨§¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨§¨¤¨¤¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨©¨¤¨¤¨§¨©¨¤¨¤¨¤¨
S k
[
U
]
S k
[
P
]
S
k ′ ∆ H ∆ p ∆ s
−1 0
1 0
S 1 {2}
{168} 2 64.00 16.00
[
0 −1
]
[
0 1
]
0 1
1 0
S 2 {2, 168}
{42, 128} 2 16.00 8.00
[
−1 0
]
[
0 1
]
i 0
1 0
S 3 {2, 168, 42, 128}
{93, 247, 117, 223} 2 16.00 8.00
[
0 −i
] { } [
0 1
] { }
1 0 2, 168, 42, 128 1 0 30, 180, 54, 156
S 4 2 4.00 4.00
0 −i 93, 247, 117, 223 0 i 105, 195, 65, 235
⎧
⎫
⎧
⎫
[ ]
2, 168, 42, 128
⎪⎨
⎪⎬ [ ]
7, 173, 47, 133
⎪⎨
⎪⎬
1 0 93, 247, 117, 223 1 0 82, 248, 122, 208
S 5 1 4.00 4.00
0 i 30, 180, 54, 156 0 1 19, 185, 59, 145
⎪⎩
⎪⎭
⎪⎩
⎪⎭
⎧
105, 195, 65, 235
⎫
⎧
110, 196, 70, 236
⎫
2, 168, 42, 128
8, 162, 138, 32
93, 247, 117, 223
87, 253, 213, 127
30, 180, 54, 156
75, 225, 201, 99
⎪⎨
⎪⎬
⎪⎨
⎪⎬
[ ]
1 0
S 6
0 1
105, 195, 65, 235
7, 173, 47, 133
82, 248, 122, 208
19, 185, 59, 145
⎪⎩
⎪⎭
110, 196, 70, 236
[ ]
0 1
1 0
Table A.2: 2 × 2 QPSK.
150, 60, 20, 190
13, 167, 143, 37
88, 242, 218, 112
76, 230, 206, 100
⎪⎩
⎪⎭
155, 49, 25, 179
1 4.00 4.00
ù"ûÿLú £"ûÿ ù"þ;ú¤¡ ÿLü"ù¤¡ ú¤¡ ø¦¡ ÿ¥¡"ý"ú ú"ü¤¡ÿ þ"ü¤¡ ù¤¡"ù¤£ û¤¡"ú ù"ü øJü ÿ¥£"üÿ ú"þ¤£"ü ÿLþÿ¥£ ú"û"ú§£
¡"þ¤£
ù¤£"û ù"ý"þ ø ú"ù ù"ú;ý"þ ÿ+øJû"ù ú"ùÿ þ ÿ"ÿ¥¡ ø ú¤£"ý"û øJú"ù ùÿLü"û ÿ+ø"ø ù øJþ ø ÿLý"û"ù ú"ú"ú"ý ÿLù"þ"û úÿLý;û
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¡ÿLü ú"ý¤£ÿ ÿLý"ý¤¡ ù¤¡;û"ú ÿLüÿ¥¡ ú"û¤¡;ú ÿLû"ù"ü ú"ü"ûÿ þ"ü¤£ ù"û¤£ÿ û"ý¤¡ ù"ü"ú"ü ù"ý"ù"ü ú¤¡"þ¤£ ÿ¥¡ ø@ÿ øJý¤¡;ú
¡"þ ù ø@ÿLý ú"û"û ùÿ ù"ù ÿLù¤¡"û ú¤£ ø;ø ÿ¥£"ý"ý ú"ú"ù"ý ù¤£ ø ù¤£"ù"ù ø¢¡"þ ù"ù"ü"þ ÿLú"þ"þ úÿ+ø"ø ÿ"ÿLý"ý ú øJú;ù
£"ú"ú ù"þ"û¤£ þ"ùÿ ù¤£§¡¤¡ ÿLû"û¤¡ øJý"ý§¡ ÿLü¤£¤£ ú¤¡ÿLü û ø¢£ ù"ü¤¡¤¡ ü"ú"ú ù"û"þ"ü ÿLþ øJú ú"û"ý¤¡ ÿ¥£¤£¤£ ú"þ"ü§£
û"ü ù øJýÿ
¤¤¤§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤§¤¤¤©¤¤¤¤¤¤¤©¤¤§©¤¤¤¤¤¤¤©¤¤¤¤¤¤¤©§¤¤©¤¤¤¤¤¤¤©¤¤¤¤¤¤¤¤¤¤¤©¤¤¤¤¤¤¤©¤¤§©¤¤¤
ùÿ;ÿLú ÿLù"û¤¡ ú¤£ øú ÿ+øJüÿ ú"úÿLü ù ø¢£ ù¤£ÿLú øJû"ü ù"ù"þ"ü ÿLú¤¡"ü úÿ+øJú ÿLý"üÿ ú øJúÿ
ú¤£¤¡
£"ù ø ù"þ¤£"û þÿLù ù¤£;û"þ ÿ¥¡"ù"ù ú"ü"ü;þ ÿLü øJû ú¤¡¤¡ ø ¡"ý"ý ù"ü"û"þ ü"ù ø ù¤¡ ø"ø ÿLþ"ú ø ú¤£"ü"þ ÿ¥£ øJû ú"þ"þ;û
ú"ü¤£ ù"ù"ù"ú ÿ¥¡ÿ ù ø;ø¢¡ ÿLý"þÿ ú"ú¤£§¡ ÿLú"ý¤£ úÿLú"ú £"ü ù"ú"ú¤£ ù"þ"ú ùÿ"ÿ"ÿ ÿ"ÿLü"ú ú øJû"ü ÿ+ø@ÿ¥¡ ú"ù ø¦£
¡¤¡"û ù"û"ý ø £¤£"ù ù"þ;ù"þ ÿLü"þ"ù ú¤¡"ú;þ ÿLû"ü ø øJý"ù"û ü¤£"ù ù¤¡ÿLû û"û ø ú"ý"ý ø ÿ¥£"þ"ù ú"þ¤£"ý ÿLþ"ý"û ú"û"ù;û
ÿ+øJú ù ø¢¡¤£ øJúÿ ùÿ þ¤¡ ÿLú ø@ÿ ú"ý"ü§¡ ÿLý¤£"ú ú"ú"þ¤£ úÿLü ù"ý¤¡¤£ úÿ ù"ú"û"ú ÿ+ø¢£"ú ú"ù"ý"ü ÿ"ÿLû¤£ ú øJü§¡
ù"ü"ú"ý þ"þ"û ù"û0øJù ÿ¥¡"ú"ù øJý"û0ø ù"ý"ù"ý ú¤¡¤¡"û ¡ÿLý ú"ý øJù ü"ü"þ ù¤¡¤£ ø ÿLü"ý"þ ú"û"û ø ÿLû"ù"ý ú"ü¤£;ù
£"ü"þ
Table A.3: 2 × 2 8PSK code list.
132

S k [
U
]
S k [
P
]
S k ′ ∆ H ∆ p ∆ s
−1 0
1 0
S 1 {4}
{2336} 2 64 16
[
0 −1
]
[
0 1
]
0 1
1 0
S 2 {4, 2336}
{292, 2048} 2 16 8
[
−1 0
]
[
0 1
]
i 0
1 0
S 3 {4, 2336, 292, 2048}
{1202, 3478, 1426, 3254} 2 16 8
[
0 −i
] { } [
0 1
] { }
1 0
4, 2336, 292, 2048
1 0
180, 2448, 404, 2224
S 4 2 4 4
0 −i
⎧
1202, 3478, 1426, 3254
⎫
0 i
⎧
1314, 3078, 1026, 3366
⎫
[ ]
4, 2336, 292, 2048 [
] 971, 2799, 1657, 3933
⎪⎨
⎪⎬
1 0 1202, 3478, 1426, 3254 e jπ/4 ⎪⎨
⎪⎬
0
747, 3023, 1881, 3709
S 5 0 1
180, 2448, 404, 2224
⎪⎩
1314, 3078, 1026, 3366
⎪⎭
0 e −jπ/4 2 1.37 2.34
635, 2911, 1769, 4045
⎪⎩
859, 2687, 1993, 3821
⎪⎭
⎧
⎫
⎧
⎫
4, 2336, 292, 2048
330, 2158, 1528, 3292
1202, 3478, 1426, 3254
106, 2382, 1240, 3580
[ ] 180, 2448, 404, 2224
1 0
⎪⎨
⎪⎬ [ ] 506, 2270, 1128, 3404
1314, 3078, 1026, 3366
1 0
⎪⎨
⎪⎬
218, 2558, 1352, 3180
S 6
0 e jπ/4 971, 2799, 1657, 3933
0 e j5π/4 2 0.34 2.34
785, 2613, 1927, 3747
747, 3023, 1881, 3709
561, 2837, 1703, 3971
⎪⎩
635, 2911, 1769, 4045 ⎪⎭
⎪⎩
897, 2725, 1591, 3859⎪⎭
859, 2687, 1993, 3821
673, 2949, 1815, 3635
[ ]
1 0
S 7 0 1
[ ]
1 0
S 8 0 1
[ ]
1 0
S 6 0 −1
[ ]
1 0
S 7
0 e jπ/4
S ′ 6 1 0.34 2.34
S ′ 7 1 0.20 1.17
Table A.4: 2 × 2 8PSK.
coset S k ⎡
U S k P S k ′ ∆ H ∆ p ∆ s
1 S 1 ⎣ 1 0 0
⎤
⎡
0 −1 0 ⎦ {1}
⎣ 1 0 0
⎤
0 −1 0⎦ {172} 3 256 20
0 0 −1
0 0 1
⎡
1 S 2 ⎣ 0 1 0
⎤
⎡
0 0 1⎦ {1, 172}
⎣ 0 0 1
⎤
−1 0 0⎦ {251, 330} 3 64 20
1 0 0
0 −1 0
⎡
10 S 1 ⎣ 1 0 0
⎤
⎡
0 −1 0 ⎦ {304}
⎣ 1 0 0
⎤
0 −1 0⎦ {413} 3 256 20
0 0 −1
0 0 1
⎡
10 S 2 ⎣ 0 1 0
⎤
⎡
0 0 1⎦ {304, 413}
⎣ 0 0 −1
⎤
1 0 0 ⎦ {486, 87} 3 64 20
1 0 0
0 1 0
1,10 S 3 N/A {1, 172, 251, 330} N/A {304, 413, 486, 87} 3 64 16
⎡
1,10 S 4 ⎣ 0 0 1
⎤
⎡
{ }
0 1 0⎦
1, 172, 251, 330 ⎣ 1 0 0
⎤
{ }
0 0 1⎦
206, 383, 181, 24
2 16 12
304, 413, 486, 87
98, 467, 297, 388
−1 0 0
⎧
⎫
0 −1 0
⎧
⎫
⎡
1,10 S 5 ⎣ 0 0 1
⎤ 1, 172, 251, 330 ⎡
⎪⎨
⎪⎬
0 1 0⎦
304, 413, 486, 87 ⎣ −1 0 0
⎤ 419, 18, 472, 373
⎪⎨
⎪⎬
0 0 1⎦
271, 190, 68, 233
1 8 8
206, 383, 181, 24
112, 221, 138, 315
−1 0 0 ⎪⎩
98, 467, 297, 388
⎪⎭ 0 1 0 ⎪⎩
321, 492, 407, 38
⎪⎭
⎧
⎫
⎧
⎫
1, 172, 251, 330
72, 229, 178, 259
304, 413, 486, 87
377, 468, 431, 30
⎡
1,10 S 6 ⎣ 1 0 0
⎤ 206, 383, 181, 24 ⎡
⎪⎨
⎪⎬
0 1 0⎦
98, 467, 297, 388 ⎣ 1 0 0
⎤ 135, 310, 252, 81
⎪⎨
⎪⎬
0 1 0 ⎦ 43, 410, 352, 461
1 8 8
419, 18, 472, 373
490, 91, 401, 316
0 0 1
0 0 −1
271, 190, 68, 233
326, 247, 13, 160
⎪⎩
112, 221, 138, 315⎪⎭
⎪⎩
57, 148, 195, 370 ⎪⎭
321, 492, 407, 38
264, 421, 478, 111
Table A.5: 3 × 3 BPSK.
133

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S k
⎡
U
⎤
S k
⎡
P
⎤
S
k ′ ∆ H ∆ p ∆ s
−1 0 0
1 0 0
S 1
⎣ 0 −1 0 ⎦ {397} ⎣0 1 0⎦ {174887} 3 1280 36
⎡
0 0 −1
⎤
⎡
0 0 1
⎤
1 0 0
0 i 0
S 2
⎣0 −1 1 ⎦ {397, 174887} ⎣i 0 0⎦ {86892, 260550} 3 160 18
0
⎡
0 −1
⎤
⎡
0 0 i
⎤
1 0 0
0 i 0
S 3
⎣0 1 0⎦ S 2
⎣1 0 0 ⎦ S 2 ′ 3 64 16
⎡
0 0 1
⎤
⎡
0 0 −i
⎤
1 0 0
i 0 0
S 4
⎣0 −1 0⎦ S 3
⎣0 1 0 ⎦ S 3 ′ 3 32 12
⎡
0 0 1
⎤
⎡
0 0 −1
⎤
1 0 0
1 0 0
S 5
⎣0 i 0 ⎦ S 4
⎣0 0 1⎦ S 4 ′ 3 8 10
0 0 −i
0 −1 0
⎡ ⎤
⎡ ⎤
1 0 0
1 0 0
S 6
⎣0 0 i⎦ S 5
⎣0 −i 0⎦ S 5 ′ 1 8 10
⎡
0 −i 0
⎤
⎡
0 0 1
⎤
1 0 0
1 0 0
S 7
⎣0 −1 0⎦ S 6
⎣0 1 0⎦ S 6 ′ 1 8 10
⎡
0 0 1
⎤
⎡
0 0 1
⎤
1 0 0
1 0 0
S 8
⎣0 1 0⎦ S 7
⎣0 0 −1⎦ S 7 ′ 1 8 8
0 0 1
0 1 0
Table A.6: 3 × 3 QPSK.
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Table A.7: 3 × 3 QPSK code list.
134

S k ⎡
U S k P S k ′ ∆ H ∆ p ∆ s
S 1 ⎣ 1 0 0
⎤
⎡
0 1 0⎦ {10794} ⎣ −1 0 0
⎤
0 −1 0 ⎦ {76702478} 3 1620.08 36.00
0 0 1
0 0 −1
⎡
S 2 ⎣ 1 0 0
⎤
⎡
0 1 1⎦ S 1 ⎣ i 0 0
⎤
0 i 0 ⎦ S 1 ′ 3 202.51 18.00
0 0 1
0 0 −i
⎡
⎡
S 3 ⎣ 1 0 0
⎤
e j π ⎤
4 0 0
0 1 0 ⎦ S 2
⎢
⎣ 0 e j π 4 0 ⎥
0 0 −1
0 0 e j 3π ⎦ S 2 ′ 3 104.97 16.59
4
⎡
⎤
⎡
S 4 ⎣ 1 0 0
⎤
1 0 0
0 −1 0⎦ S 3
⎢
⎣
0 e j 3π 4 0 ⎥
0 0 i
0 0 e j 5π ⎦ S 3 ′ 3 32.00 15.41
4
⎡
⎤
1 0 0
⎡
⎢
S 5 ⎣0 e j π ⎥
4 0
0 0 e j π ⎦ S 4 ⎣ 1 0 0
⎤
0 −i 0 ⎦ S 4 ′ 3 7.03 13.76
4
0 0 −1
⎡
⎤
⎡
1 0 0
⎢
S 6 ⎣
0 1 0 ⎥
0 0 e j 3π ⎦ S 5 ⎣ 1 0 0
⎤
0 −i 0 ⎦ S 5 ′ 3 2.34 5.51
4
0 0 −i
⎡
⎤
⎡
1 0 0
e j π ⎤
4 0 0
S 7 ⎣0 i 0 ⎦ ⎢
0 0 e j π S 6 ⎣ 0 −1 0
⎥
⎦ S
4
0 0 e j 3π 6 ′ 3 0.69 4.93
4
⎡
⎤
⎡
S 8 ⎣ 0 1 0
⎤
1 0 0
1 0 0⎦ S 7
⎢
⎣
0 e j 3π 4 0 ⎥
0 0 1
0 0 e j 7π ⎦ S 7 ′ 2 0.12 4.93
4
Table A.8: 3 × 3 8PSK.
,.-0/01#-023-4,.5413-36$70/010,.805010,9+0+470/3-.+3+ ,0,:+4/0-0,32;+

J;KL4MNL=L;L0K=O;P=Q;ORO=Q;O=Q;Q;MNL3P.M=J=K;JL0OTS;S=U;P=QVL0MTJ=Q;W=U;J=Q;OTS;O=W;O=JVL0PTJ=W;S=J;P=P;PTW;O=S;W=J;W;UNL0O=K.MXU;K;O=KTO;S;M=J=S;K;KNL0OL0SL0U.MXUTK;J=O.ML3JVLYL4MXW;P;W;M=U=QTJ;S=S.M=K=U;UNL4M(L4M;MXJ;J=K
U;M=O;S=S;JZL=L0W;S=O;J=K;J[L0P=U.M=M=O=PZL3P;J;U=J;O;M=OTS.MXK.M(L0Q=J\Q=U;Q=J;Q=P;JTU;P=Q;O=K;P;STJ=K;K=O.MXP.M]W;U=J;Q=K;U;QNL0O=S;QL0W;J=WTW;JL0W=W;W;WNL0W=J;O=W;K;O=QTK;WL0K.M=M=QNL4M(L0O;U=J;K=WTK;J=K;J;P;M=QNL0U=J;P;U=S;K=W
L=L0Q.MXS;OZL=L0S;U=K.M(L0ORK;ML3Q;U=PZL3P;OVL0M=QL0STS;U=W;K=K;K=P\J=Q;Q=Q;P=K;STS;KL0J=U;K;PTJ=O;U=J.MXK;STW;O=K;P=K;K;UNL0O=W;S=O;JVL3WTO;P=UVL3U;S;WNL0OL0O=O;S;O=UTK;O;M=K;S=S;QNL4MXW;W;P=U;W=QTJ;S=O;Q;O=W;UNL0U=Q;S;K=Q;U=W
S k ⎡
U
⎤
S k P S k ′ ∆ H ∆ p ∆ s
1 0 0 0
⎡
S 1
⎢0 −1 0 0
⎥
⎣0 0 1 0 ⎦ {83} ⎣ 1 0 0
⎤
0 1 0 ⎦ {4012} 3 4096.00 48.00
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 2
⎢0 −1 0 0
⎥
⎣0 0 1 0 ⎦ {83, 4012} ⎣ 1 0 0
⎤
0 1 0 ⎦ {989, 3106} 3 512.00 24.00
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 3
⎢0 1 0 0
⎥
⎣0 0 −1 0 ⎦ S 2 ⎣ 1 0 0
⎤
0 −1 0 ⎦ S 2 ′ 3 512.00 24.00
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 4
⎢0 1 0 0
⎥
⎣0 0 1 0 ⎦ S 3 ⎣ 0 1 0
⎤
0 0 1⎦ S 3 ′ 3 64.00 12.00
−1 0 0
0 0 0 −1
Table A.10: 4 × 3 BPSK.
S k ⎡
U
⎤
S k P S k ′ ∆ H ∆ p ∆ s
1 0 0 0
⎡
S 1
⎢0 −1 0 0
⎥
⎣0 0 1 0 ⎦ {8714}
⎣ 1 0 0
⎤
0 1 0 ⎦ {11176096} 3 4096 48
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 2
⎢0 −1 0 0
⎥
⎣0 0 1 0 ⎦ {8714, 11176096} ⎣ 1 0 0
⎤
0 1 0 ⎦ {696994, 10487816} 3 512 24
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 3
⎢0 1 0 0
⎥
⎣0 0 −1 0 ⎦ S 2 ⎣ 1 0 0
⎤
0 −1 0 ⎦ S 2 ′ 3 512 24
0 0 −1
0 0 0 −1
⎡
⎤
1 0 0 0
⎡
S 4
⎢0 1 0 0
⎥
⎣0 0 0 1⎦ S 3 ⎣ i 0 0
⎤
0 0 i⎦ S 3 ′ 3 384 24
0 i 0
0 0 −1 0
⎡
⎤
1 0 0 0
⎡
S 5
⎢0 i 0 0
⎥
⎣0 0 1 0⎦ S 4 ⎣ 0 1 0
⎤
1 0 0⎦ S 4 ′ 3 128 20
0 0 i
0 0 0 i
⎡
⎤
1 0 0 0
⎡
S 6
⎢0 1 0 0
⎥
⎣0 0 1 0⎦ S 5 ⎣ 0 i 0
⎤
i 0 0⎦ S 5 ′ 3 32 14
0 0 1
0 0 0 i
⎡
⎤
1 0 0 0
⎡
S 7
⎢0 1 0 0
⎥
⎣0 0 1 0⎦ S 6 ⎣ 1 0 0
⎤
0 1 0 ⎦ S 6 ′ 1 16 14
0 0 −1
0 0 0 1
⎡
⎤
1 0 0 0
⎡
S 8
⎢0 −1 0 0
⎥
⎣0 0 1 0 ⎦ S 7 ⎣ 0 1 0
⎤
−1 0 0 ⎦ S 7 ′ 1 16 14
0 0 −i
0 0 0 −1
Table A.11: 4 × 3 QPSK.
U=P;Q.MXO;PZL=L4M=P=K;Q=Q;JRQ=UVL3Q;O;MNL3P;K;J=W.MXQ.M]S;U=KVL3K;P=S\Q=U.MXW;K=W;OTU;P=W;Q=Q;U.M]J=O;W=K;W=S.M]W;U=U;O=U;UVLYL0O=W;P=P;O.M(L^O;P=S.MXOVL;LYL0W=JVL3S;U;OL^K.MXO.M=U=OVLYL4MXU;K;S=OVL=L^J;P=J;O;K=J;WNL0U=K;W;PL0J=K
W=S.M=Q=U;OZL3P;O;W=Q;J=K.M`L3O.MXU;J;MNL=L0P;S=P.MXS;OTS;K=O;U=S;O;MTJ;M=SL0W;M=OTSVL0M=P=J.M=PTQ=P.MXU;Q=K;PTOVL3W;J=S;P;KNL0O=SVL=L4ML3UTW;K=U;S=UVL;LYL0O=O;U=K;U;P=QTJ;U=J;P.MXP;KNL0U=Q;J;Q=SVL3UTK;O=Q;SVL3S;KNL4MXO;K;K;M=Q=U
_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_=_=_;_;_=_;_=_;_;_X_;_;_=_;_=_;_=_;_;_=_=_;_=_;_=_;_;_=_=_;_=_;_=_;_;_=_;_=_=_;_=_;_=_;_;_=_;_=_=_;_=_;_;_=_;_=_
J=K;J;J=U;JZL3P;Q;Q;M=S=O;JRWL0J=S;U=JZL=L0U;W;M=J=O;JTU;P=P;O=W.MXJ\J=J;O=O;W=W;JTS;W=J;P=K;U;STQ=S;Q=S;U=K.M]W;Q=P;W=K;P;WNL0W=K;K=W;OVL3QTW;S=J;U=S;PVLYL0O=U;Q=JVL0S=UTJ;P=U;U;O=K;WNL0U=O.M=K=O.MXQTK;U;M=W.M=M=UNL4MXU;U;W=J;JL
O=WVL0K=K;OZL3P;K;P=W;P=O;ORS=S;W=Q;O=PZL=L;L0U=P;J=J;STS;J=K;U=Q;Q=S\J;M=J=S;J=W;PTSVL3J;W=K;O;PTQL0KL0P=J;STO;S=P;U=U;J;UNL0W=Q;Q;M=S;P=WTW;WL0WL0P;UNL0O=O;J=S.M=J=WTJVL3O;UVL3Q;QNL4MXP;U.MXU;J=QTK;K=U;K;U=P;UNL4M=M=O;P=S;J=W
J=S;W.MXU;PZL3P;J;Q=S;P=S;JRU=Q;Q=OVL0MNL=L0UVL3K;J;M;M]S;J=J;K=Q;S;MTJ=J;S=Q;W=U.M]S;W=S;K=U;S.M]QL0Q=PVL3U.M]W;J=W;S=W;W;UNL0W=Q;J;M;ML3QTW.MXQ;S=P;PVLYL0O=U.M=M=Q;KL^K;Q=J;P;U=UVLYL0U=J;W;O=O.M(L^K;W=W.M;MXU;WNL4MXS;J;S=W;U=K
a=a;a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a=a;a;a=a;a=a;a;a=a;a=a;a=a;a=a=a;a;a=a;a=a;a;aXa;a;a=a;a=a;a=a;a;a=a=a;a=a;a=a;a;a=a=a;a=a;a=a;a;a=a;a=a=a;a=a;a=a;a;a=a;a=a=a;a=a;a;a=a;a=a
U=P;P;Q=S;K=JTJ;J;O=U;J=S;JTS;W=J;UL0S=OTQ;S;J=Q;Q=J;PNL;L0U=W;O=S;W;MbWVL3O;J=W;SZL3P;Q=Q;W=Q;W;JbJ;K=KVL0M=JTK;U;M=JL0P;WNL4MXU;U=U;SVL3QTJ;P=U;OL0S;QNL0U=O.MXWVL0Q=WZL3W;K=K;K;P=O;WTW=Q;P.MXS;W=QZL3O;U=Q;Q;K;M=WTW=S;JVL3W;K=Q
S=S;S;J=S;O;M]J;Q;W=O;W;M=OTS;O=W.MXU;U=OTJ;W;U=P.MXK.McL0P.MXJ;O;M=U;MbO;Q=J;U=K;OZL=L;L3K.M=ML0P
L3P.MXP;PTJ;S=S;S=P;KVLYL4M(L4MXK;W.MXQTK;J=OVL3K;S;KNL4MXW;P=K;J;Q=UZL3O;K;M=S;U=Q;KTW=O;S;K=S;S=UZL3OVL=L0Q;O=U;KTO=S.M=Q=Q;J=U
S=U;W;W=U;O=PTQ;P;PL4MXJ;STS;KL0W=O;W=OTJ;O.M(L;L3J;ONL0P;OL0U=S;Q=PRK.MXU;W=W;SZL=L0S=U;W=Q;K;P`L0S=P;J=K;STJ;S=O;K=S;O;UNL0U=Q;U=P;U;S=WTK;O;M;MXW;P;UNL4MXW;W=U;P;J=WZL3O;W=S;WVL3K;UTW=O;K;S;ML3WZL3OVL3O.M=J=S;QTO=P;U;S=K;W=Q
S=J;Q;P=U;U;M]J;J;S=KVL3S.M]S;W=U;P=P;U=JTQVL0J=K.MXS;PNL;L0UL0Q=W;W=PRU;Q=K;Q=P;JZL3P;J=Q;U=U;Q;JbJ;S;M=P=O;PTK;W=W;O=Q;P;WNL4MXS;J=P;P;O=KTK;Q=J;S=Q;K;KNL0U=J;W=U;Q;Q=WZL3W;Q=J;O;P=W;KTW=J;W;P=QVL3WZL3O;U;M=O.MXPVL^W;M=Q;P=W;KL
S;M=K;O=O;O=JTQ;U;Q=O.MXU;JTU;P=Q;Q=U.MXJTJ;K;K=U;K=W;JNL0P;J;M=P=S;J;M[L3P;U=S;J=S;SZL=L0W=S;J=U;P;JbU.M=M=K=Q;JTK;J=J;P;M=W;QNL0U=J;P=P;J;O=WTK;W=S;PL0O;UNL4M(L0OL;L0OLdL3O;S=Q;U;S=QVL^W=U;J;J=P;U=UZL3W;J=O;KVL3U;QTW=JVL4M(L0J=W
_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_=_;_;_=_;_=_;_;_=_;_=_;_=_;_=_=_;_;_=_;_=_;_;_X_;_;_=_;_=_;_=_;_;_=_=_;_=_;_=_;_;_=_=_;_=_;_=_;_;_=_;_=_=_;_=_;_=_;_;_=_;_=_=_;_=_;_;_=_;_=_
S=K;O;P=J;K;M]J.M=S=U;Q=U;OTSVL3U;JL;L0M]Q;P.MXO;O=Q;ONL;L0PL0J=K;J;M`L0O=O;P=S;OZL3P;O=W;J;M;M=PbW;S=O;U=K;PTK;O=J;Q=KVL0KNL4MXO;K=Q;Q;P=UTJ;U=K;K=O;Q;UNL0U=Q;QL0Q;S=KZL3O;S=P;Q;K=QVL^OL0W;Q=J;S=QZL3O;O=U;W;J=O;UTW=K;U;U=K;W=K
S=U;K.MXU;O;M]Q;U.MXU;P=Q.M]U;P=O;S=U;J=JTJ;O;W=W;P=K;PNL0P;K=J;O=Q.MXPRQ;U=P;WL0JZL=L4MXP;Q=O;S;PbU;P=K;J=U;JTJ;P=J;Q=WVL0WNL0U=K.MXK.M=W=KTK.MXO;O=K;W;WNL4MXU;K=P;SVL3KZL3O;W=P;S;P=S;KTW=U;U.MXQ.MXWZL3W;JL4M=P=WVL^O=P;S;S=Q;SL
S=J;KVL3U;U=PTJ.M=J=W;WL0STSVL3J;U=U;P=OTQVL0K=U;W=U;ONL;L;L3S;Q=WVL3SRS;S=K;U=U;PZL3P;K=P;U=U;O;PbO;W=U.MXJ;STK;K=U.MXO;W;UNL4M=M=O=S;Q;U=WTJVL3O;P=K;S;WNL4MXP;U=O;J;O=UZL3W;Q=Q;S;J=S;UTO=S;P.MXK;O=WZL3O;O=J;P;K=Q;QTW=WVL0O=K;J=Q
Table A.12: 4 × 3 QPSK code list.
136

Appendix B
RMTCM Design Examples
If the regular trellis is too complicated, only the partition and labelling will be shown,
readers can follow the procedures in Section 5.3 to generate the corresponding regular
trellis.
0 1 14
1 14 1
Figure B.1: 2 × 2 BPSK, Rate 1/2, I2S2O2p1.
137

0 1 14
1 7 8
Figure B.2: 2 × 2 BPSK, Rate 1/2, I2S2O4p1.
0 1 14
1 7 8
2 14 1
3 8 7
Figure B.3: 2 × 2 BPSK, Rate 1/2, I2S2O4P 1.
0 1 14 7 8
1 11 4 13 2
2 7 8 1 14
3 13 2 11 4
Figure B.4: 2 × 2 BPSK, Rate 2/2, 2 × 2 × 2.
0 1 14 7 8
1 11 4 13 2
2 14 7 8 1
3 4 13 2 11
4 7 8 1 14
5 13 2 11 4
6 8 1 14 7
7 2 11 4 13
Figure B.5: 2 × 2 BPSK, Rate 2/2, 2 × 4 × 1.
0 2 168
1 42 128
2 93 247
3 117 223
4 30 180
5 54 156
6 105 195
7 65 235
8 168 2
9 128 42
10 247 93
11 223 117
12 180 30
13 156 54
14 195 105
15 235 65
Figure B.6: 2 × 2 QPSK, Rate 1/2, 8 × 2 × 1.
138

0 2 168 42 128
1 93 247 117 223
2 30 180 54 156
3 105 195 65 235
4 42 128 2 168
5 117 223 93 247
6 54 156 30 180
7 65 235 105 195
Figure B.7: 2 × 2 QPSK, Rate 2/2, 4 × 2 × 2.
0 2 168 42 128
1 93 247 117 223
2 30 180 54 156
3 105 195 65 235
4 168 42 128 2
5 247 117 223 93
6 180 54 156 30
7 65 235 105 195
8 42 128 2 168
9 117 223 93 247
10 54 156 30 180
11 65 235 105 195
12 128 2 168 42
13 223 93 247 117
14 156 30 180 54
15 235 105 195 65
Figure B.8: 2 × 2 QPSK, Rate 2/2, 4 × 4 × 1.
0 2 168 42 128 93 247 117 223
1 30 180 54 156 105 195 65 235
2 93 247 117 223 2 168 42 128
3 105 195 65 235 30 180 54 156
Figure B.9: 2 × 2 QPSK, Rate 3/2, 2 × 2 × 4.
139

0 2 168 42 128 93 247 117 223
1 30 180 54 156 105 195 65 235
2 42 128 93 247 117 223 2 168
3 54 156 105 195 65 235 30 180
4 93 247 117 223 2 168 42 128
5 105 195 65 235 30 180 54 156
6 117 223 2 168 42 128 93 247
7 65 235 30 180 54 156 105 195
Figure B.10: 2 × 2 QPSK, Rate 3/2, 2 × 4 × 2.
0 2 168 42 128 93 247 117 223
1 30 180 54 156 105 195 65 235
2 168 42 128 93 247 117 223 2
3 180 54 156 105 195 65 235 30
4 42 128 93 247 117 223 2 168
5 54 156 105 195 65 235 30 180
6 128 93 247 117 223 2 168 42
7 156 105 195 65 235 30 180 54
8 93 247 117 223 2 168 42 128
9 105 195 65 235 30 180 54 156
10 247 117 223 2 168 42 128 93
11 195 65 235 30 180 54 156 105
12 117 223 2 168 42 128 93 247
13 65 235 30 180 54 156 105 195
14 223 2 168 42 128 93 247 117
15 235 30 180 54 156 105 195 65
Figure B.11: 2 × 2 QPSK, Rate 3/2, 2 × 8 × 1.
0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
3 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
Figure B.12: 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 2 × 8.
0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
3 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
4 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
5 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
6 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
7 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
Figure B.13: 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 4 × 4.
140

0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168
3 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173
4 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
5 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
6 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247
7 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248
8 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
9 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
10 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180
11 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185
12 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
13 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
14 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195
15 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196
Figure B.14: 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 8 × 2.
0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2
3 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7
4 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168
5 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173
6 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42
7 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47
8 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
9 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
10 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93
11 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82
12 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247
13 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248
14 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117
15 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122
16 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
17 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
18 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30
19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19
20 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180
21 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185
22 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54
23 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59
24 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
25 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
26 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105
27 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110
28 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195
29 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196
30 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65
31 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70
Figure B.15: 2 × 2 QPSK, Rate 4/2, |S 5 | = 32, 2 × 16 × 1.
141

0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 8 162 138 32 87 253 213 127 75 225 201 99 150 60 20 190
3 13 167 143 37 88 242 218 112 76 230 206 100 155 49 25 179
4 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
5 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
6 75 225 201 99 150 60 20 190 8 162 138 32 87 253 213 127
7 76 230 206 100 155 49 25 179 13 167 143 37 88 242 218 112
Figure B.16: 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 2 × 8.
0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 8 162 138 32 87 253 213 127 75 225 201 99 150 60 20 190
3 13 167 143 37 88 242 218 112 76 230 206 100 155 49 25 179
4 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
5 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
6 87 253 213 127 75 225 201 99 150 60 20 190 8 162 138 32
7 88 242 218 112 76 230 206 100 155 49 25 179 13 167 143 37
8 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
9 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
10 75 225 201 99 150 60 20 190 8 162 138 32 87 253 213 127
11 76 230 206 100 155 49 25 179 13 167 143 37 88 242 218 112
12 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
13 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
14 150 60 20 190 8 162 138 32 87 253 213 127 75 225 201 99
15 155 49 25 179 13 167 143 37 88 242 218 112 76 230 206 100
Figure B.17: 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 4 × 4.
142

0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 8 162 138 32 87 253 213 127 75 225 201 99 150 60 20 190
3 13 167 143 37 88 242 218 112 76 230 206 100 155 49 25 179
4 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168
5 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173
6 138 32 87 253 213 127 75 225 201 99 150 60 20 190 8 162
7 143 37 88 242 218 112 76 230 206 100 155 49 25 179 13 167
8 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
9 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
10 87 253 213 127 75 225 201 99 150 60 20 190 8 162 138 32
11 88 242 218 112 76 230 206 100 155 49 25 179 13 167 143 37
12 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247
13 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248
14 213 127 75 225 201 99 150 60 20 190 8 162 138 32 87 253
15 218 112 76 230 206 100 155 49 25 179 13 167 143 37 88 242
16 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
17 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
18 75 225 201 99 150 60 20 190 8 162 138 32 87 253 213 127
19 76 230 206 100 155 49 25 179 13 167 143 37 88 242 218 112
20 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180
21 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185
22 201 99 150 60 20 190 8 162 138 32 87 253 213 127 75 225
23 206 100 155 49 25 179 13 167 143 37 88 242 218 112 76 230
24 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
25 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
26 150 60 20 190 8 162 138 32 87 253 213 127 75 225 201 99
27 155 49 25 179 13 167 143 37 88 242 218 112 76 230 206 100
28 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195
29 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196
30 20 190 8 162 138 32 87 253 213 127 75 225 201 99 150 60
31 25 179 13 167 143 37 88 242 218 112 76 230 206 100 155 49
Figure B.18: 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 4 × 8 × 2.
143

0 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235
1 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236
2 8 162 138 32 87 253 213 127 75 225 201 99 150 60 20 190
3 13 167 143 37 88 242 218 112 76 230 206 100 155 49 25 179
4 168 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2
5 173 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7
6 162 138 32 87 253 213 127 75 225 201 99 150 60 20 190 8
7 167 143 37 88 242 218 112 76 230 206 100 155 49 25 179 13
8 42 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168
9 47 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173
10 138 32 87 253 213 127 75 225 201 99 150 60 20 190 8 162
11 143 37 88 242 218 112 76 230 206 100 155 49 25 179 13 167
12 128 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42
13 133 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47
14 32 87 253 213 127 75 225 201 99 150 60 20 190 8 162 138
15 37 88 242 218 112 76 230 206 100 155 49 25 179 13 167 143
16 93 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128
17 82 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133
18 87 253 213 127 75 225 201 99 150 60 20 190 8 162 138 32
19 88 242 218 112 76 230 206 100 155 49 25 179 13 167 143 37
20 247 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93
21 248 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82
22 253 213 127 75 225 201 99 150 60 20 190 8 162 138 32 87
23 242 218 112 76 230 206 100 155 49 25 179 13 167 143 37 88
24 117 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247
25 122 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248
26 213 127 75 225 201 99 150 60 20 190 8 162 138 32 87 253
27 218 112 76 230 206 100 155 49 25 179 13 167 143 37 88 242
28 223 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117
29 208 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122
30 127 75 225 201 99 150 60 20 190 8 162 138 32 87 253 213
31 112 76 230 206 100 155 49 25 179 13 167 143 37 88 242 218
32 30 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223
33 19 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208
34 75 225 201 99 150 60 20 190 8 162 138 32 87 253 213 127
35 76 230 206 100 155 49 25 179 13 167 143 37 88 242 218 112
36 180 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30
37 185 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19
38 225 201 99 150 60 20 190 8 162 138 32 87 253 213 127 75
39 230 206 100 155 49 25 179 13 167 143 37 88 242 218 112 76
40 54 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180
41 59 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185
42 201 99 150 60 20 190 8 162 138 32 87 253 213 127 75 225
43 206 100 155 49 25 179 13 167 143 37 88 242 218 112 76 230
44 156 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54
45 145 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59
46 99 150 60 20 190 8 162 138 32 87 253 213 127 75 225 201
47 100 155 49 25 179 13 167 143 37 88 242 218 112 76 230 206
48 105 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156
49 110 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145
50 150 60 20 190 8 162 138 32 87 253 213 127 75 225 201 99
51 155 49 25 179 13 167 143 37 88 242 218 112 76 230 206 100
52 195 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105
53 196 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110
54 60 20 190 8 162 138 32 87 253 213 127 75 225 201 99 150
55 49 25 179 13 167 143 37 88 242 218 112 76 230 206 100 155
56 65 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195
57 70 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196
58 20 190 8 162 138 32 87 253 213 127 75 225 201 99 150 60
59 25 179 13 167 143 37 88 242 218 112 76 230 206 100 155 49
60 235 2 168 42 128 93 247 117 223 30 180 54 156 105 195 65
61 236 7 173 47 133 82 248 122 208 19 185 59 145 110 196 70
62 190 8 162 138 32 87 253 213 127 75 225 201 99 150 60 20
63 179 13 167 143 37 88 242 218 112 76 230 206 100 155 49 25
Figure B.19: 2 × 2 QPSK, Rate 4/2, |S 6 | = 64, 2 × 16 × 1.
144