Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Markov Chains and Stochastic Stability

Information Theory and Reliable Communication

Two simplified versions of the belief propagation algorithm for fast iterative decoding of low-density parity check codes on the additive white Gaussian noise channel are proposed. Both versions are implemented with real additions only, which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed. Also, these two algorithms do not require any knowledge about the channel characteristics. Both algorithms yield a good performance-complexity trade-off and can be efficiently implemented in software as well as in hardware, with possibly quantized received values.

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Reduced complexity iterative decoding of low-density parity check codes based on belief propagation

In recent years there has been much interest in robustness issues in general and in robust signal processing schemes in particular. Robust schemes are useful in situations where imprecise a priori knowledge of input characteristics makes the sensitivity of performance to deviations from assumed conditions an important factor in the design of good signal processing schemes. In this survey we discuss the minimax approach for the design of robust methods for signal processing. This has proven to be a very useful approach because it leads to constructive procedures for designing robust schemes. Our emphasis is on the contributions which have been made in robust signal processing, although key results of other robust statistical procedures are also considered. Most of the results we survey have been obtained in the past fifteen years, although some interesting earlier ideas for minimax signal processing are also mentioned. This survey is organized into five main parts, which deal separately with robust linear filters for signal estimation, robust linear filters for signal detection and related applications, nonlinear methods for robust signal detection, nonlinear methods for robust estimation, and robust data quantization. The interrelationships among many of these results are also discussed in the survey.

Robust techniques for signal processing: A survey

Developments in the theory of linear least-squares estimation in the last thirty years or so are outlined. Particular attention is paid to early mathematica[ work in the field and to more modern developments showing some of the many connections between least-squares filtering and other fields.

A view of three decades of linear filtering theory

Maximum-likelihood soft-decision decoding of linear block codes is addressed. A binary multiple-check generalization of the Wagner rule is presented, and two methods for its implementation, one of which resembles the suboptimal Forney-Chase algorithms, are described. Besides efficient soft decoding of small codes, the generalized rule enables utilization of subspaces of a wide variety, thereby yielding maximum-likelihood decoders with substantially reduced computational complexity for some larger binary codes. More sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the (24, 12) Golay code, yielding a computational gain factor of about 2 with respect to previous methods. A ternary single-check version of the Wagner rule is applied for efficient soft decoding of the (12, 6) ternary Golay code. >

Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes

An approach for efficient utilization of fast Hadamard transform in decoding binary linear block codes is presented. Computational gain is obtained by employing various types of concurring codewords, and memory reduction is also achieved by appropriately selecting rows for the generator matrix. The availability of these codewords in general, and particularly in some of the most frequently encountered codes, is discussed.

Optimal soft decision block decoders based on fast Hadamard transform

Efficient code-search maximum-likelihood decoding algorithms, based on reliability information, are presented for binary Linear block codes. The codewords examined are obtained via encoding. The information set utilized for encoding comprises the positions of those columns of a generator matrix G of the code which, for a given received sequence, constitute the most reliable basis for the column space of G. Substantially reduced computational complexity of decoding is achieved by exploiting the ordering of the positions within this information set. The search procedures do not require memory; the codeword to be examined is constructed from the previously examined codeword according to a fixed rule. Consequently, the search algorithms are applicable to codes of relatively large size. They are also conveniently modifiable to achieve efficient nearly optimum decoding of particularly large codes.

Reliability-based code-search algorithms for maximum-likelihood decoding of block codes

A method whereby a substantially reduced family of error patterns, called survivors, may be created for maximum likelihood soft decoding is introduced. The family of survivor depends on the received word. A decoder based on this approach first forms the survivors, then scores them. Rather than obtaining the survivors by online elimination of error patterns, the use of predetermined lists that represent all families of survivors is proposed. >

Reduced lists of error patterns for maximum likelihood soft decoding

An efficient algorithm is presented for maximum-likelihood soft-decision decoding of the Leech lattice. The superiority of this decoder with respect to both computational and memory complexities is demonstrated in comparison with previously published decoding methods. Gain factors in the range of 2-10 are achieved. The authors conclude with some more advanced ideas for achieving a further reduction of the algorithm complexity based on a generalization of the Wagner decoding method to two parity constraints. A comparison with the complexity of some trellis-coded modulation schemes is discussed. The decoding algorithm presented seems to achieve a computational complexity comparable to that of the equivalent trellis codes. >

J. Snyders

Papers

Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes

Optimal soft decision block decoders based on fast Hadamard transform

Reliability-based code-search algorithms for maximum-likelihood decoding of block codes

Reduced lists of error patterns for maximum likelihood soft decoding

Fast decoding of the Leech lattice