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IEEE Transactions on Information Theory

Lower Bounds on the Error Probability of Block Codes Based on Improvements on de Caen's Inequality

A. Cohen, Neri Merhav

Published Feb 1, 2004• 41 citations
Publisher

Abstract

New lower bounds on the error probability of block codes with maximum-likelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen's lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binary-symmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin (1998) and by Keren and Litsyn (2001). Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caen-based bounds.

DOI
10.1109/tit.2003.822577

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•Jan 1, 2022•0 citations
New lower bounds on the error probability of block codes with maximum-likelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen's lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binary-symmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin (1998) and by Keren and Litsyn (2001). Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caen-based bounds.
A novel technique for deriving lower bounds on the error probability when communicating one of M signals over a communication channel is proposed. At the basis of the technique stands an improvement on a recent lower bound on the probability of a union of events by de Caen (1997). The new bound includes a function which can be optimized in order to achieve the tightest results. By applying this bound to the problem of lower bounding the error probability, while suggesting an appropriate optimization function, in the spirit of the relevant channel model and type of the code, new bounds on the error probability can be derived. In this paper we apply the new bound to the problem of lower bounding the error probability of binary linear codes over the binary symmetric channel (BSC). The resulting bound improves on the latest bound appearing in the current literature, by Keren and Litsyn (2000).
A lower bound on the error rate of linear binary block codes(under maximum likelihood decoding) over BSC channels is proposed.According to the principle of the maximum likelihood(ML) decoding algorithm,the decoding error probability is firstly converted into the joint probability of the error events,and the judge rule of the redundant error events is deduced based on the optimization rule of the improved Dawson-Sankoff bound.Moreover,the calculation expression about lower bound of the error probability solely depends on the Hamming weight enumerator function of the code and the crossover probability of the channel.The simulation results applying to various LDPC codes show that the new lower bound outperforms those generic lower bounds and the sphere packing bound.Its computational complexity is also lower.
Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>
This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in NNN-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d.\ random coding are derived and compared to the exact calculations.
With the development of communication technology, modern communication networks need to achieve ultra-high reliability to meet practical demands. Forward error correction coding has been widely used in real-world communication systems for providing reliable data transmission over noisy channels. In this thesis, we investigate the efficient performance evaluation methods for block codes over memoryless channels. Performance bounds are powerful tools to provide insights on the error performance of the coded systems. In many practical scenarios where performance bounds are not applicable (e.g., due to the unavailability of the relevant coding parameters under a given decoder), the Monte Carlo simulation is still popular despite its inefficiency, especially in the low error probability regime. As a variance-reduction technique, importance sampling (IS) can significantly reduce the sample size needed for reliable estimation. We firstly examine the problem of efficient performance evaluation of linear block codes over binary symmetric channels (BSCs). Contrary to the conventional wisdom, we prove that for BSCs, all bounds based on Gallager’s first bounding technique, including the famous union bound, are not asymptotically tight for all possible choices of the Gallager region. By proposing the so-called input demodulated-output weight enumerating function (IDWEF) of a code, asymptotically tight maximum-likelihood decoding upper and lower bounds for BSCs are derived. For the cases that coding parameters are absent or suboptimal decoders are applied, we propose an efficient IS estimator by deriving the optimal IS distribution of the Hamming weight of the error vector. In addition, the asymptotic percentage saving of the proposed IS estimator over the state-of-the-art IS estimator is characterized in terms of the sample size. Its accuracy in predicting the efficiency of the proposed IS estimator is verified by extensive computer simulation. We further examine the IS estimator design and efficiency analysis problems for the additive white generalized Gaussian noise (AWGGN) channel. The generalized Gaussian distribution is an effective model to describe real-world systems with impulsive noise. It can be specialized to the Laplace and Gaussian distributions depending on the choices of the decay rates of the heavy tail. By deriving the optimal IS distribution on the one-dimensional space mapped from the observation space, we present a general framework for designing IS estimators for linear block codes over a memoryless continuous channel. We apply the framework to the AWGGN channel and propose a noise-magnitude-domain IS (NM-IS) estimator. As an efficiency measure, the asymptotic IS gain of the proposed estimator is derived in a multiple integral as the signal-to-noise ratio (SNR) tends to infinity. Specifically, we reduce the gains of both Laplace and Gaussian noise cases to a one-dimensional integral. In addition, by limiting the use of the union bound to an L1-norm sphere with an optimized radius, we derive the sphere bound for the additive white Laplace noise channel. Simulation results show the efficiency of the proposed IS estimator in terms of the sample size and verify the accuracy of the prediction of the derived IS gain. Besides, we specialize the proposed framework to further improve the efficiency of the IS simulation based on the side information of the modulation. For the AWGGN channel with binary phase-shift keying (BPSK) modulation, we propose an enhanced NM-IS estimator, which outperforms the NM-IS estimator in efficiency. For the additive white Gaussian noise channel with M-PSK modulation, we propose an angular domain IS estimator and provide its efficiency analysis in terms of the asymptotic IS gain. Furthermore, in the performance evaluation of communication systems, it is often seen that researchers are more interested in the required SNR at a target error probability Pe than the other way around. The conventional solution to this Pe-targeted SNR estimation problem involves interpolation of simulated error probabilities at multiple handpicked SNRs for estimating the required SNR at the target error probability. We propose a Pe-targeted iterative IS estimator that solves the problem algorithmically. The proposed method can achieve the same efficiency (in terms of sample size) as estimating the error probability at the required SNR and allow the confidence interval of the SNR estimate to be obtained.