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Maximum Likelihood Decoding of Codes on the Asymmetric Z-channel

Susanna Spinsante, Ángela I. Barbero

Published Jan 1, 2005• 3 citations

Abstract

AbstractThe aim of this paper is to extend some basic concepts related to the MaximumLikelihood decoding of codes on the Z-channel, which is a particular, but very important,example of an asymmetric channel. We study distance properties of linear codes overthe Z-channel, in order to de ne a suitable metric for the implementation of a MaximumLikelihood decoder on the channel. A combinatorial expression for an upper bound on theprobability of incorrect Maximum Likelihood decoding is also provided, and comparisonsare given to evaluate the tightness of the bound, when a Hamming code, a Turbo codeand an LDPC code are decoded on the asymmetric channel. 1 Introduction Errors induced by communications on a noisy channel or storage medium can be reduced to anydesired level by proper encoding of the information, as long as the information rate is less thanthe capacity of the channel. This fact was shown in a landmark paper by Shannon, in 1948 [1];since that time, much work has been devoted to the problem of devising ecient encoding anddecoding methods for error control in a noisy environment.In a block-coded system, the source output u represents a k-bit message, the encoder outputc represents an n-symbol codeword, the demodulator output r represents the correspondingbinary received n-tuple and the decoder output u^ represents the k-bit estimate of the encodedmessage. The decoder must produce an estimate u^ of the information sequence u based onthe received sequence r. Equivalently, since there is a one-to-one correspondence between theinformation sequence u and the codeword c, the decoder can produce an estimate ^c of thecodeword c. Obviously, u^ = u if and only if ^c = c. The strategy adopted for choosing an

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AbstractThe aim of this paper is to extend some basic concepts related to the MaximumLikelihood decoding of codes on the Z-channel, which is a particular, but very important,example of an asymmetric channel. We study distance properties of linear codes overthe Z-channel, in order to de ne a suitable metric for the implementation of a MaximumLikelihood decoder on the channel. A combinatorial expression for an upper bound on theprobability of incorrect Maximum Likelihood decoding is also provided, and comparisonsare given to evaluate the tightness of the bound, when a Hamming code, a Turbo codeand an LDPC code are decoded on the asymmetric channel. 1 Introduction Errors induced by communications on a noisy channel or storage medium can be reduced to anydesired level by proper encoding of the information, as long as the information rate is less thanthe capacity of the channel. This fact was shown in a landmark paper by Shannon, in 1948 [1];since that time, much work has been devoted to the problem of devising ecient encoding anddecoding methods for error control in a noisy environment.In a block-coded system, the source output u represents a k-bit message, the encoder outputc represents an n-symbol codeword, the demodulator output r represents the correspondingbinary received n-tuple and the decoder output u^ represents the k-bit estimate of the encodedmessage. The decoder must produce an estimate u^ of the information sequence u based onthe received sequence r. Equivalently, since there is a one-to-one correspondence between theinformation sequence u and the codeword c, the decoder can produce an estimate ^c of thecodeword c. Obviously, u^ = u if and only if ^c = c. The strategy adopted for choosing an

The aim of this paper is to extend some basic concepts related to the Maximum Likelihood decoding of codes on the Z-channel, which is a particular, but very important, example of an asymmetric channel. We study distance properties of linear codes over the Z-channel, in order to define a suitable metric for the implementation of a Maximum Likelihood decoder on the channel. A combinatorial expression for an approximation on the probability of incorrect Maximum Likelihood decoding is also provided, and comparisons are given to evaluate the tightness of the estimation, when a Hamming code, a Turbo code and an LDPC code are used for communicating over the Z-channel.

We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.

Most of the basic concepts of algebraic coding theory are derived for the memoryless binary symmetric channel. These concepts do not necessarily hold for time-varying channels or for channels with memory. However, errors in real-life channels seem to occur in bursts rather than independently, suggesting that these channels exhibit some statistical dependence or memory. Nonetheless, the same algebraic codes are still commonly used in current communication systems that employ interleaving to spread channel error bursts over the set of received codewords to make the channel appear memoryless to the block decoder. This method suffers from immediate shortcomings as it fails to exploit the channel’s memory while adding delay to the system. We study optimal maximum likelihood block decoding of binary codes sent over several binary additive channels with infinite and finite memory. We derive conditions on general binary codes and channels parameters under which maximum likelihood and minimum distance decoding are equivalent. The channels considered in this work are the infinite and finite memory Polya contagion channels [1],

Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial focuses on the performance evaluation of linear codes under optimal maximum-likelihood (ML) decoding. Though the ML decoding algorithm is prohibitively complex for most practical codes, their performance analysis under ML decoding allows to predict their performance without resorting to computer simulations. It also provides a benchmark for testing the sub-optimality of iterative (or other practical) decoding algorithms. This analysis also establishes the goodness of linear codes (or ensembles), determined by the gap between their achievable rates under optimal ML decoding and information theoretical limits. In Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial, upper and lower bounds on the error probability of linear codes under ML decoding are surveyed and applied to codes and ensembles of codes on graphs. For upper bounds, we discuss various bounds where focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, and also consider sphere-packing bounds with their recent improvements targeting codes of moderate block lengths. Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial is a comprehensive introduction to this important topic for students, practitioners and researchers working in communications and information theory.

It is shown that the relatively unknown Dorsch decoder may be extended to produce a decoder that is capable of maximum-likelihood decoding. The extension involves a technique for any linear (n, k) code that ensures that n−k less reliable, soft decisions of each received vector may be treated as erasures in determining candidate codewords. These codewords are derived from low information weight codewords and it is shown that an upper bound of this information weight may be calculated from each received vector in order to guarantee that the decoder will achieve maximum-likelihood decoding. Using the cross-correlation function, it is shown that the most likely codeword may be derived from a partial correlation function of these low information weight codewords, which leads to an efficient fast decoder. For a practical implementation, this decoder may be further simplified into a concatenation of a hard-decision decoder and a partial correlation decoder with insignificant performance degradation. Results are presented for some powerful, known codes, including a GF(4) non-binary BCH code. It is shown that maximum-likelihood decoding is realised for a high percentage of decoded codewords and that performance close to the sphere packing bound is attainable for codeword lengths up to 1000 bits.

We derive bounds on the error probability of ML decoded LDPC (low density parity check) codes, for any binary-input symmetric-output channel. For appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. The lower and upper bounds coincide asymptotically, indicating a polynomially decreasing ensemble averaged error probability. For ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing. Furthermore, the error exponent can be set arbitrarily close to the standard random coding exponent.