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International Conference on Communications

On computing the performance of linear block codes in nonindependent channel errors

Hans‐Jürgen Zepernick

Published Dec 23, 2002• 3 citations
Publisher

Abstract

We present an analytical method to compute the exact performance of algebraically decoded linear block codes in nonindependent channel errors. Performance is computed by rating decoding events. The block error distribution and the coset probabilities are of fundamental interest. A recursion in matrix notation is taken to compute the block error distribution. By using an automata model of the channel-decoder-cascade, coset probabilities can be computed directly in an original domain. Alternatively, the analysis can be performed in a spectral domain having a simpler structure. The relationship between both domains is given by a Walsh-Hadamard transform. Aspects of implementation are discussed and examples are presented to illustrate the efficiency of the method.

DOI
10.1109/icc.1996.541359

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We present an analytical method to compute the exact performance of algebraically decoded linear block codes in nonindependent channel errors. Performance is computed by rating decoding events. The block error distribution and the coset probabilities are of fundamental interest. A recursion in matrix notation is taken to compute the block error distribution. By using an automata model of the channel-decoder-cascade, coset probabilities can be computed directly in an original domain. Alternatively, the analysis can be performed in a spectral domain having a simpler structure. The relationship between both domains is given by a Walsh-Hadamard transform. Aspects of implementation are discussed and examples are presented to illustrate the efficiency of the method.
A coset representation of the bit error probabilities for binary linear block codes and arbitrary symmetric memoryless channels is derived. Such a representation leads to simple upper bounds on the bit error probabilities for code components expressed via 2n code spectra, where n is the code length, and to a "bounded distance decoding" algorithm. The performance of the algorithm is illustrated for the (23,12) Golay code and a particular channel having non-binary output alphabet where it attains a sub-optimum performance.
An important measure of the performance of error detecting codes is the probability of undetected error. Extensive study on the subject has yielded results which allow for the computation of the probability of undetected error for many codes on the binary symmetric channel (BSC). However, little is known about code performance in more complicated channel models. The Gilbert channel is a two-state, three-parameter model with memory which simulates the effects of burst noise. In this thesis, we investigate methods to compute the probability of undetected error of binary linear block codes on this channel. We examine an approach to approximate code performance based on the P(m,n) distribution which is the probability of m errors in a block of n bits and the weight distribution of the code. For the Gilbert channel, P(m,n) can in principle be calculated from the channel parameters. In practice however, existing methodologies suffer from rather excessive computational requirements, particularly when n is larger than one thousand or so. We have developed an efficient method to calculate P(m,n) for reasonable channel parameters. This allows the probability of undetected error for many codes to be readily estimated. For certain channel and code parameters, the approximation method described above may not be sufficiently accurate. Exact analytical results are difficult to obtain, however; because unlike the BSC, the probability of a particular error pattern on the Gilbert channel depends not just on the number of 1's in the pattern. Nevertheless, by appropriately exploiting certain symmetries present on the Gilbert channel, we can acquire some useful results. We have derived the probability of undetected error for the single parity check code. We have also obtained a formula for summing over a cyclic class of vectors and shown that reciprocal generator polynomials generate cyclic codes which have the same probability of undetected error on the Gilbert channel. The Monte Carlo simulation technique is often used when exact analysis is difficult. In a simulation study of CRC codes, we are able to observe several interesting qualitative results with just a reasonable amount of computational effort. We find that as on the BSC, on the Gilbert channel the probability of undetected error does not always increase with worsening channel conditions. Also, the CRC-CCITT code appears to maintain its superiority in terms of error detection performance over the CRC-ANSI code on the Gilbert channel, and perhaps most significantly, for some ranges of channel parameters, the probability of undetected error estimated using BSC results with the effective bit error rate can be quite inaccurate.
The performance of optimum linear block codes has been examined by computer simluation, and the error distribution at the decoder output, the block-error probability and the bit-error probability have been determined for the codes when they are used on a binary symmetric channel with random errors. A table showing the distribution of errors in the decoder output is presented, and it is shown how the table can be used to determine exact values for the bit-error probability and the block-error probability. The exact values of bit-error probability computed from the table are compared with values obtained using a known theoretical approximation to the bit-error probability, and some conclusions are drawn as to the general usefulness of the theoretical approximation. Also, block-error probabilities are computed from the information presented in the table and these values are compared with the estimates of block-error probability obtained from minimumdistance considerations.
The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.
A method for estimating the block error probability of Reed-Muller codes on a binary erasure channel and a binary symmetric channel under a permutation decoding algorithm is proposed. This method is based on an analysis of the block error probability under a recursive decoding algorithm for a fixed number of input erasures or errors. Simulation results demonstrate that the proposed method allows to effectively predict the block error rate performance of Reed-Muller codes under permutation decoding with a different number of permutations used.