Bit-Interleaved Coded Modulation

TLDR: The theoretical foundations of BICM are reviewed under the unified framework of error exponents for mismatched decoding, which allows an accurate analysis without any particular assumptions on the length of the interleaver or independence between the multiple bits in a symbol.

Abstract: The principle of coding in the signal space follows directly from Shannon's analysis of waveform Gaussian channels subject to an input constraint. The early design of communication systems focused separately on modulation, namely signal design and detection, and error correcting codes, which deal with errors introduced at the demodulator of the underlying waveform channel. The correct perspective of signal-space coding, although never out of sight of information theorists, was brought back into the focus of coding theorists and system designers by Imai's and Ungerbock's pioneering works on coded modulation. More recently, powerful families of binary codes with a good tradeoff between performance and decoding complexity have been (re-)discovered. Bit-Interleaved Coded Modulation (BICM) is a pragmatic approach combining the best out of both worlds: it takes advantage of the signal-space coding perspective, whilst allowing for the use of powerful families of binary codes with virtually any modulation format. BICM avoids the need for the complicated and somewhat less flexible design typical of coded modulation. As a matter of fact, most of today's systems that achieve high spectral efficiency such as DSL, Wireless LANs, WiMax and evolutions thereof, as well as systems based on low spectral efficiency orthogonal modulation, feature BICM, making BICM the de-facto general coding technique for waveform channels. The theoretical characterization of BICM is at the basis of efficient coding design techniques and also of improved BICM decoders, e.g., those based on the belief propagation iterative algorithm and approximations thereof. In this text, we review the theoretical foundations of BICM under the unified framework of error exponents for mismatched decoding. This framework allows an accurate analysis without any particular assumptions on the length of the interleaver or independence between the multiple bits in a symbol. We further consider the sensitivity of the BICM capacity with respect to the signal-to-noise ratio (SNR), and obtain a wideband regime (or low-SNR regime) characterization. We review efficient tools for the error probability analysis of BICM that go beyond the standard approach of considering infinite interleaving and take into consideration the dependency of the coded bit observations introduced by the modulation. We also present bounds that improve upon the union bound in the region beyond the cutoff rate, and are essential to characterize the performance of modern randomlike codes used in concatenation with BICM. Finally, we turn our attention to BICM with iterative decoding, we review extrinsic information transfer charts, the area theorem and code design via curve fitting. We conclude with an overview of some applications of BICM beyond the classical coherent Gaussian channel.

Figures

Fig. 3.8 Rate regions for MLC/MSD and BICM.

Fig. 5.2 Factor graph representation of a BICM scheme of rate R = 1. In this case, C is a binary convolutional code of rate r = 1 3 and m = 3, i. e., 8-PSK or 8-QAM.

Fig. 4.8 Bit error probability union bounds and bit-error rate simulations of 8-PSK with the 8-state rate-2/3 convolutional code in a fully-interleaved Rayleigh fading channel. Interleaver length N = 30 (circles) and N = 1000 (diamonds). In solid lines, the saddlepoint approximation union bounds for N = 30, N = 100, N = 1000 and for infinite interleaving, with PEP1(d). In dashed, dashed-dotted, and dotted lines, the heuristic approximations with weight v = 1, 2, 3 respectively.

Fig. 5.11 Area theorem for the convolutional code (5, 7)8 (left) and for a set partitioning demapper in an AWGN channel with snr = 6 dB (right).

Fig. 5.16 Regular (3, 6) LDPC BICM-ID factor graph with m = 3.

Fig. 3.4 Error exponents for coded modulation (solid), BICM with independent parallel channels (dashed), BICM using metric (2.15) (dash-dotted), and BICM using metric (2.17) (dotted) for 16-QAM with Gray labeling, Rayleigh fading and snr = −25 dB. Crosses correspond to (from right to left) coded modulation, BICM with metric (2.15), BICM with metric (2.17) and BICM with metric (2.17) and s = 1.

Full text

Foundations and Trends
R

in
sample
Vol. xx, No xx (xxxx) 1–144
c
 xxxx xxxxxxxxx
DOI: xxxxxx
Bit-Interleaved Coded Modulation
Albert Guill´en i F`abregas
1
, Alfonso
Martinez
2
and Giuseppe Caire
3
1
Department of Engineering, University of Cambridge, Trumpington Street,
Cambridge, CB2 1PZ, United Kingdom, guillen@ieee.org
2
Centrum Wiskunde & Informatica (CWI), Kruislaan 413, Amsterdam,
1098 SJ, The Netherlands, alfonso.martinez@ieee.org
3
Electrical Engineering Department, University of Southern California, 3740
McClintock Av., Los Angeles, 90080 CA, USA, caire@usc.edu
Abstract
The principle of coding in the signal space follows directly from Shan-
non’s analysis of waveform Gaussian channels subject to an input con-
straint. The early design of communication systems focused separately
on modulation, namely signal design and detection, and error correct-
ing codes, which deal with errors introduced at the demodulator of
the underlying waveform channel. The correct perspective of signal-
space coding, although never out of sight of information theorists, was
brought back into the focus of coding theorists and system design-
ers by Imai’s and Ungerb¨ock’s pioneering work on coded modulation.
More recently, powerful families of binary codes with a good tradeoff
between performance and decoding complexity have been (re-) discov-
ered. Bit-Interleaved Coded Modulation (BICM) is a pragmatic ap-
proach combining the best out of both worlds: it takes advantage of
the signal-space coding perspective, whilst allowing for the use of pow-
erful families of binary codes with virtually any modulation format.

BICM avoids the need for the complicated and somewhat less flexi-
ble design typical of coded modulation. As a matter of fact, most of
today’s systems that achieve high spectral efficiency such as DSL, Wire-
less LANs, WiMax and evolutions thereof, as well as systems based on
low spectral efficiency orthogonal modulation, feature BICM, making
BICM the de-facto general coding technique for waveform channels.
The theoretical characterization of BICM is at the basis of efficient cod-
ing design techniques and also of improved BICM decoders, e.g., those
based on the belief propagation iterative algorithm and approximations
thereof. In this monograph, we review the theoretical foundations of
BICM under the unified framework of error exponents for mismatched
decoding. This framework allows an accurate analysis without any par-
ticular assumptions on the length of the interleaver or independence
between the multiple bits in a symbol. We further consider the sensi-
tivity of the BICM capacity w ith respect to the signal-to-noise ratio
(SNR), and obtain a wideband regime (or low-SNR regime) character-
ization. We review efficient tools for the error probability analysis of
BICM that go beyond the standard approach of considering infinite in-
terleaving and take into consideration the dependency of the coded bit
observations introduced by the modulation. We also present bounds
that improve upon the union bound in the region beyond the cutoff
rate, and are essential to characterize the performance of modern ran-
domlike codes used in concatenation with BICM. Finally, we turn our
attention to BICM with iterative deco ding, we review extrinsic infor-
mation transfer charts, the area theorem and code design via curve
fitting. We conclude with an overview of some applications of BICM
beyond the classical coherent Gaussian channel.

Contents
List of Abbreviations, Acronyms and Symbols iii
1 Introduction 1
2 Channel Model and Code Ensembles 5
2.1 Channel Model: Encoding and Decoding 5
2.2 Coded Modulation 8
2.3 Bit-Interleaved Coded Modulation 9
2.A Continuous- and Discrete-Time Gaussian Channels 12
3 Information-Theoretic Foundations 16
3.1 Coded Modulation 17
3.2 Bit-Interleaved Coded Modulation 23
3.3 Comparison with Multilevel Coding 29
3.4 Mutual Information Analysis 36
3.5 Concluding Remarks and Related Work 46
i

ii Contents
4 Error Probability Analysis 49
4.1 Error Probability and the Union Bound 50
4.2 Pairwise Error Probability for Infinite Interleaving 58
4.3 Pairwise Error Probability for Finite Interleaving 71
4.4 Bounds and Approximations Above the Cutoff Rate 81
4.5 Concluding Remarks and Related Work 84
4.A Saddlepoint Location 86
4.B Asymptotic Analysis with Nakagami Fading 87
5 Iterative Decoding 89
5.1 Factor Graph Representation and Belief Propagation 91
5.2 Density Evolution 95
5.3 EXIT Charts 100
5.4 The Area Theorem 104
5.5 Improved Schemes 108
5.6 Concluding Remarks and Related Work 118
5.A Density Evolution Algorithm for BICM-ID 119
6 Applications 122
6.1 Non-Coherent Demodulation 122
6.2 Block-Fading 124
6.3 MIMO 127
6.4 Optical Communication: Discrete-Time Poisson Channel 129
6.5 Additive Exponential Noise Channel 130
7 Conclusions 133
References 136

List of Abbreviations, Acronyms and Symbols
APP A posteriori probability
AWGN Additive white Gaussian noise
BEC Binary erasure channel
BICM Bit-interleaved coded modulation
BICM-ID Bit-interleaved coded modulation with iterative decoding
BIOS Binary-input output-symmetric (channel)
BP Belief propagation
CM Coded modulation
EXIT Extrinsic information transfer
FG Factor graph
GMI Generalized mutual information
ISI Inter-symbol interference
LDPC Low-density parity-check (code)
MAP Maximum a posteriori
MIMO Multiple-input multiple-output
MLC Multi-level coding
MMSE Minimum mean-squared error
MSD Multi-stage decoding
PSK Phase-shift keying
iii