Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

A code trellis is a graphical representation of a code, block or convolutional, in which every path represents a codeword (or a code sequence for a convolutional code). This representation makes it possible to implement Maximum Likelihood Decoding (MLD) of a code with reduced decoding complexity. The most well known trellis-based MLD algorithm is the Viterbi algorithm. The trellis representation was first introduced and used for convolutional codes [23]. This representation, together with the Viterbi decoding algorithm, has resulted in a wide range of applications of convolutional codes for error control in digital communications over the last two decades. There are two major reasons for this inactive period of research in this area. First, most coding theorists at that time believed that block codes did not have simple trellis structure like convolutional codes and maximum likelihood decoding of linear block codes using the Viterbi algorithm was practically impossible, except for very short block codes. Second, since almost all of the linear block codes are constructed algebraically or based on finite geometries, it was the belief of many coding theorists that algebraic decoding was the only way to decode these codes. These two reasons seriously hindered the development of efficient soft-decision decoding methods for linear block codes and their applications to error control in digital communications. This led to a general belief that block codes are inferior to convolutional codes and hence, that they were not useful. Chapter 2 gives a brief review of linear block codes. The goal is to provide the essential background material for the development of trellis structure and trellis-based decoding algorithms for linear block codes in the later chapters. Chapters 3 through 6 present the fundamental concepts, finite-state machine model, state space formulation, basic structural properties, state labeling, construction procedures, complexity, minimality, and sectionalization of trellises. Chapter 7 discusses trellis decomposition and subtrellises for low-weight codewords. Chapter 8 first presents well known methods for constructing long powerful codes from short component codes or component codes of smaller dimensions, and then provides methods for constructing their trellises which include Shannon and Cartesian product techniques. Chapter 9 deals with convolutional codes, puncturing, zero-tail termination and tail-biting.Chapters 10 through 13 present various trellis-based decoding algorithms, old and new. Chapter 10 first discusses the application of the well known Viterbi decoding algorithm to linear block codes, optimum sectionalization of a code trellis to minimize computation complexity, and design issues for IC (integrated circuit) implementation of a Viterbi decoder. Then it presents a new decoding algorithm for convolutional codes, named Differential Trellis Decoding (DTD) algorithm. Chapter 12 presents a suboptimum reliability-based iterative decoding algorithm with a low-weight trellis search for the most likely codeword. This decoding algorithm provides a good trade-off between error performance and decoding complexity. All the decoding algorithms presented in Chapters 10 through 12 are devised to minimize word error probability. Chapter 13 presents decoding algorithms that minimize bit error probability and provide the corresponding soft (reliability) information at the output of the decoder. Decoding algorithms presented are the MAP (maximum a posteriori probability) decoding algorithm and the Soft-Output Viterbi Algorithm (SOVA) algorithm. Finally, the minimization of bit error probability in trellis-based MLD is discussed

Turbo Decoding and Detection for Wireless Applications

A historical perspective of turbo coding and turbo transceivers inspired by the generic turbo principles is provided, as it evolved from Shannon’s visionary predictions. More specifically, we commence by discussing the turbo principles, which have been shown to be capable of performing close to Shannon’s capacity limit. We continue by reviewing the classic maximum a posteriori probability decoder. These discussions are followed by studying the effect of a range of system parameters in a systematic fashion, in order to gauge their performance ramifications. In the second part of this treatise, we focus our attention on the family of iterative receivers designed for wireless communication systems, which were partly inspired by the invention of turbo codes. More specifically, the family of iteratively detected joint coding and modulation schemes, turbo equalization, concatenated spacetime and channel coding arrangements, as well as multi-user detection and three-stage multimedia systems are highlighted

Self-concatenated code design and its application in power-efficient cooperative communications

In this tutorial, we have focused on the design of binary self-concatenated coding schemes with the help of EXtrinsic Information Transfer (EXIT) charts and Union bound analysis. The design methodology of future iteratively decoded self-concatenated aided cooperative communication schemes is presented. In doing so, we will identify the most important milestones in the area of channel coding, concatenated coding schemes and cooperative communication systems till date and suggest future research directions

Codes for QPSK modulation with invariance under 90 degrees rotation

The new rate 1/2 nonlinear convolutional codes for quadrature phase shift keying (QPSK) modulation allow the achievement of full 90 degree rotational invariance of coded QPSK signal sequences at no significant loss in real coding gains when compared to linear codes. For mobile communication systems operating in a fading environment with frequent periods of low signal-to-noise ratio and the possibility of losses of carrier phase synchronization in the receiver, the invariance to 90 degree ambiguous demodulation should be a significant advantage

A Belief Propagation Based Framework for Soft Multiple-Symbol Differential Detection

Soft noncoherent detection, which relies on calculating the \textit{a posteriori} probabilities (APPs) of the bits transmitted with no channel estimation, is imperative for achieving excellent detection performance in high-dimensional wireless communications. In this paper, a high-performance belief propagation (BP)-based soft multiple-symbol differential detection (MSDD) framework, dubbed BP-MSDD, is proposed with its illustrative application in differential space-time block-code (DSTBC)-aided ultra-wideband impulse radio (UWB-IR) systems. Firstly, we revisit the signal sampling with the aid of a trellis structure and decompose the trellis into multiple subtrellises. Furthermore, we derive an APP calculation algorithm, in which the forward-and-backward message passing mechanism of BP operates on the subtrellises. The proposed BP-MSDD is capable of significantly outperforming the conventional hard-decision MSDDs. However, the computational complexity of the BP-MSDD increases exponentially with the number of MSDD trellis states. To circumvent this excessive complexity for practical implementations, we reformulate the BP-MSDD, and additionally propose a Viterbi algorithm (VA)-based hard-decision MSDD (VA-HMSDD) and a VA-based soft-decision MSDD (VA-SMSDD). Moreover, both the proposed BP-MSDD and VA-SMSDD can be exploited in conjunction with soft channel decoding to obtain powerful iterative detection and decoding based receivers. Simulation results demonstrate the effectiveness of the proposed algorithms in DSTBC-aided UWB-IR systems.Comment: 14 pages, 12 figures, 3 tables, accepted to appear on IEEE Transactions on Wireless Communications, Aug. 201