Bit-Interleaved Coded Modulation (eBook)

Fundamentals, Analysis and Design

Leszek Szczecinski, Alex Alvarado (Autoren)

eBook Download: EPUB

2014
John Wiley & Sons (Verlag)
978-1-118-69402-2 (ISBN)

Infos

Presenting a thorough overview of bit-interleaved coded modulation (BICM), this book introduces the tools for the analysis and design of BICM transceivers. It explains in details the functioning principles of BICM and proposes a refined probabilistic modeling of the reliability metrics-the so-called L-values-which are at the core of the BICM receivers.  Alternatives for transceiver design based on these models are then studied.

Providing new insights into the analysis of BICM, this book is unique in its approach, providing a general framework for analysis and design, focusing on communication theoretic aspects of BICM transceivers. It adopts a tutorial approach, explains the problems in simple terms with the aid of multiple examples and case studies, and provides solutions using accessible mathematical tools.

The book will be an excellent resource for researchers in academia and industry: graduate students, academics, development engineers, and R & D managers.

Key Features:

• Presents an introduction to BICM, placing it in the context of other coded modulation schemes

• Offers explanations of the functioning principles and design alternatives

• Provides a unique approach, focusing on communication theory aspects

• Shows examples and case studies to illustrate analysis and design of BICM

• Adopts a tutorial approach, explaining the problems in simple terms and presenting solutions using accessible mathematical tools

Leszek Szczecinski University of Quebec, Canada
Alex Alvarado University College London, UK

Inhalt

Dedication iii

Preface ix

Acknowledgements xi

Acronyms xvi

1 Introduction 1

1.1 Coded Modulation 1

1.2 The Road Towards BICM 4

1.3 Why Everybody Loves BICM 5

1.4 Outline of the Contents 8

References 9

2 Preliminaries 13

2.1 Notation Convention 13

2.2 Linear Modulation 16

2.3 Coded modulation 18

2.4 Channel Models 21

2.5 The Mapper 23

2.6 Codes and Encoding 34

2.7 The Interleaver 43

2.8 Bibliographical Notes 54

References 56

3 Decoding 61

3.1 Optimal Decoding 61

3.2 BICM Decoder 64

3.3 L-values 66

3.4 Hard-decision Decoding 78

3.5 Bibliographical Notes 80

References 81

4 Information-Theoretic Elements 83

4.1 Mutual Information and Channel Capacity 83

4.2 Coded Modulation 86

4.3 Bit-interleaved Coded Modulation 94

4.4 BICM, CM and MLC: A Comparison 111

4.5 Numerical Calculation of Mutual Information 117

4.6 Bibliographical Notes 121

References 122

5 Probability Density Functions of L-values 125

5.1 Introduction and Motivation 125

5.2 PDF of L-values for 1D Constellations 135

5.3 PDF of L-values for 2D Constellations 140

5.4 Fading Channels 155

5.5 Gaussian Approximations 160

5.6 Bibliographical Notes 172

References 174

6 Performance Evaluation 177

6.1 Uncoded Transmission 177

6.2 Coded Transmission 200

6.3 PEP Evaluation 225

6.4 Performance of BICM via Gaussian Approximations 238

6.5 Bibliographical Notes 247

References 249

7 Correction of L-values 253

7.1 Mismatched Decoding and L-values Correction 253

7.2 Optimal Correction of L-values 258

7.3 Suboptimal Correction of L-values 263

7.4 Bibliographical Notes 278

References 279

8 Interleaver Design 281

8.1 UEP in BICM and M-Interleavers 281

8.2 Exploiting UEP in MPAM Constellations 291

8.3 Bibliographical Notes 299

References 301

9 BICM Receivers for Trellis Codes 303

9.1 BICM with Trivial Interleavers 303

9.2 Code Design for BICM-T 312

9.3 Equivalent Labelings for Trellis Encoders 317

9.4 Bibliographical Notes 325

References 326

Index 328

Textauszug

Chapter 1
Introduction

1.1 Coded Modulation

The main challenge in the design of communication systems is to reliably transmit digital information (very often, bits generated by the source) over a medium which we call the communication channel or simply the channel. This is done by mapping a sequence of bits to a sequence of symbols . These symbols are then used to vary (modulate) parameters of the continuous-time waveforms (such as amplitude, phase, and/or frequency), which are sent over the channel every seconds, i.e., at a symbol rate . The transmission rate of the system is thus equal to

1.1

where the bandwidth occupied by the waveforms is directly proportional to the symbol rate . Depending on the channel and the frequency used to carry the information, the waveforms may be electromagnetic, acoustic, optical, etc.

Throughout this book we will make abstraction of the actual waveforms and instead, consider a discrete-time model where the sequence of symbols is transmitted through the channel resulting in a sequence of received symbols . In this discrete-time model, both the transmitted and received symbol at each time instant are -dimensional column vectors. We also assume that linear modulation is used, that the transmitted waveforms satisfy the Nyquist condition, and that the channel is memoryless. Therefore, assuming perfect time/frequency synchronization, it is enough to model the relationship between the transmitted and received signals at time , i.e.,

1.2

In (1.2), we use to model the channel attenuation (gain) and to model an unknown interfering signal (most often the noise). Using this model, we analyze the transmission rate (also known as spectral efficiency):

1.3

which is independent of , thus allowing us to make abstraction of the bandwidth of the waveforms used for transmission. Clearly, and are related via

1.4

The process of mapping the information bits to the symbols is known as coding and are called codewords. We will often relate to well-known results stemming from the works of Shannon [1, 2] which defined the fundamental limits for reliable communication over the channel. Modeling the transmitted and received symbols and as random vectors and with distributions and , the rate is upper bounded by the mutual information (MI) . As long as , the probability of decoding error (i.e., choosing the wrong information sequence) can be made arbitrarily small when goes to infinity. The maximum achievable rate , called the channel capacity, is obtained by maximizing over the distribution of the symbols , and it represents the ultimate transmission rate for the channel. In the case when is modeled as a Gaussian vector, the probability density function (PDF) that maximizes the MI is also Gaussian, which is one of the most popular results establishing limits for a reliable transmission.

The achievability proof is typically based on random-coding arguments, where, to create the codewords which form the codebook , the symbols are generated from the distribution . At the receiver's side, the decoder decides in favor of the most likely sequence from the codebook, i.e., it uses the maximum likelihood (ML) decoding rule:

1.5

When grows, however, the suggested encoding and decoding cannot be used as practical means for the construction of the coding scheme. First, because storing the codewords results in excessive memory requirements, and second, because an exhaustive enumeration over codewords in the set in (1.5) would be prohibitively complex. In practice, the codebook is not randomly generated, but instead, obtained using an appropriately defined deterministic algorithm taking information bits as input. The structure of the code should simplify the decoding but also the encoding, i.e., the transmitter can generate the codewords on the fly (the codewords do not need to be stored).

To make the encoding and decoding practical, some “structure” has to be imposed on the codewords. The first constraint typically imposed is that the transmitted symbols are taken from a discrete predefined set , called a constellation. Moreover, the structure of the code should also simplify the decoding, as only the codewords complying with the constraints of the imposed structure are to be considered.

Consider the simple case of (i.e., is in fact a scalar), when the constellation is given by , i.e., a -ary pulse amplitude modulation (PAM) (2PAM) constellation, and when the received symbol is corrupted by additive white Gaussian noise (AWGN), i.e., where is a zero-mean Gaussian random variable with variance . We show in Fig. 1.1 the MI for this case and compare it with the capacity which relies on the optimization of the distribution of . This figure indicates that for , both values are practically identical. This allows us to focus on the design of binary codes with rate that can operate reliably without bothering about the theoretical suboptimality of the chosen constellation. This has been in fact the focus of research for many years, resulting in various binary codes being developed and used in practice. Initially, convolutional codes (CCs) received quite a lot of attention, but more recently, the focus has been on “capacity-approaching” codes such as turbo codes (TCs) or low-density parity-check (LDPC) codes. Their performance is deemed to be very “close” to the limits defined by the MI, even though the decoding does not rely on the ML rule in (1.5).

Figure 1.1 Channel capacity and the MI for 2PAM

One particular problem becomes evident when analyzing Fig 1.1: the MI curve for 2PAM saturates at , and thus, it is impossible to transmit at rates . From (1.4) and , we conclude that if we want to increase the transmission rate , we need to increase the rate , and therefore, the transmission bandwidth. This is usually called bandwidth expansion and might be unacceptable in many cases, including modern wireless communication systems with stringent constraints on the available frequency bands.

The solution to the problem of increasing the transmission rate without bandwidth expansion is to use a high-order constellation and move the upper bound on from to . Combining coding with nonbinary modulation (i.e., high-order constellations) is often referred to as coded modulation (CM), to emphasize that not only coding but also the mapping from the code bits to the constellation symbols is important. The core problem in CM design is to choose the appropriate coding scheme that generates symbols from the constellation and results in reliable transmission at rate . On the practical side, we also need a CM which is easy to implement and which—because of the ever-increasing importance of wireless communications—allows us to adjust the coding rate to the channel state, usually known as adaptive modulation and coding (AMC).

A well-known CM is the so-called trellis-coded modulation (TCM), where convolutional encoders (CENCs) are carefully combined with a high-order constellation . However, in the past decades, a new CM became prevalent and is the focus of this work: bit-interleaved coded modulation (BICM). The key component in BICM is a (suboptimal) two-step decoding process. First, logarithmic likelihood ratios (LLRs, also known as L-values) are calculated, and then a soft-input binary decoder is used. In the next section, we give a brief outline of the historical developments in the area of CM, with a particular emphasis on BICM.

1.2 The Road Toward BICM

The area of CM has been explored for many years. In what follows, we show some of the milestones in this area, which culminated with the introduction of BICM. In Fig. 1.2, we show a timeline of the CM developments, with emphasis on BICM.

Figure 1.2 Timeline of the CM developments with emphasis on BICM

The early works on CM in the 1970s include those by de Buda [27, 28], Massey [29], Miyakawa et al. [30], Anderson and Taylor [31], and Aulin [32]. The first breakthroughs for coding for came with Ungerboeck and Csajka's TCM [5, 6, 33, 34] and Imai and Hirakawa's multilevel coding (MLC) [9, 35]. For a detailed historical overview of the early works on CM, we refer the reader to [36, Section 1.2] and [37, pp. 952–953]. Also, good summaries of the efforts made over the years to approach Shannon's limit can be found in [38, 39].

BICM's birth is attributed to the paper by Zehavi [10]. More particularly, Zehavi compared BICM based on CCs to TCM and showed that while BICM loses to TCM in terms of Euclidean distance (ED), it wins in terms of diversity. This fundamental observation spurred interest in BICM.

BICM was then analyzed in [17] by Caire et al., where achievable rates for BICM were shown to be very close to those reached by CM, provided that a Gray labeling was used. Gray codes were then conjectured to be the optimal binary labeling for BICM. These information-theoretic arguments were important because at the time [10] appeared, capacity-approaching codes (TCs) invented by Berrou et al. [7] were already being used together with binary modulation. Furthermore, CM inspired by the...

Produkt-Details

• Erscheint lt. Verlag: 15.12.2014
• Reihe/Serie: IEEE Press
• Reihe/Serie: Wiley - IEEE
• Reihe/Serie: Wiley - IEEE
• Sprache: englisch
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Alex Alvarado • bicm • BICM transmission • Bit-Interleaved Coded Modulation Fundamentals • Drahtlose Kommunikation • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Iterative receiver structures • Leszek Szczecinski • L-values • Mobile & Wireless Communications • Modulation • Signal Processing • Signalverarbeitung
• ISBN-10: 1-118-69402-3 / 1118694023
• ISBN-13: 978-1-118-69402-2 / 9781118694022