In this paper, the sphere bound (SB) is revisited within a general bounding framework based on nested Gallager regions. We show that the SB proposed by Herzberg et al. is equivalent to the SB proposed by Kasami et al.. Interestingly, this general framework provides a new way to evaluate the performance of binary linear codes over additive white Gaussian noise (AWGN) channels. This simulation approach relies on only the geometry structure of the code itself. Therefore, simulation results can be used to evaluate not only the performance over AWGN channels but also that over other symmetrical real channels. Numerical results obtained with the new performance evaluation approach are presented.
In this paper, the sphere bound (SB) is revisited within a general bounding framework based on nested Gallager regions. We show that the SB proposed by Herzberg et al. is equivalent to the SB proposed by Kasami et al.. Interestingly, this general framework provides a new way to evaluate the performance of binary linear codes over additive white Gaussian noise (AWGN) channels. This simulation approach relies on only the geometry structure of the code itself. Therefore, simulation results can be used to evaluate not only the performance over AWGN channels but also that over other symmetrical real channels. Numerical results obtained with the new performance evaluation approach are presented.
In this paper, the sphere bound (SB) is revisited within a general bounding framework based on nested Gallager regions. The equivalence is revealed between the SB proposed by Herzberg and Poltyrev and the SB proposed by Kasami et al., whereas the latter was rarely cited in the literatures. Interestingly and importantly, the derivation of the SB based on nested Gallager regions suggests us a new simulation approach to performance evaluation of binary linear codes over additive white Gaussian noise (AWGN) channels. In order for the performance evaluation, the proposed approach decouples the geometrical structure of the code from the noise statistics. The former specifies the conditional error probabilities, which are independent of signal-to-noise ratios (SNRs) and can be simulated and estimated efficiently, while the latter determines the probabilities of those conditions, which involve SNRs and can be calculated numerically. Numerical results show that the proposed simulation approach matches well with the traditional simulation approach in the high error rate region but is able to evaluate efficiently the performance in the extremely low error rate region.
In this paper, nested Gallager regions with a single parameter is introduced to exploit Gallager's first bounding technique (GFBT). We present a necessary and sufficient condition on the optimal parameter. We also present a sufficient condition (with a simple geometrical explanation) under which the optimal parameter does not depend on the signal-to-noise ratio (SNR). With this general framework, three existing upper bounds are revisited, including the tangential bound (TB) of Berlekamp, the sphere bound (SB) of Herzberg and Poltyrev, and the tangential-sphere bound (TSB) of Poltyrev. This paper also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in literature.
The theoretical performance of binary codes for the additive white Gaussian noise (AWGN) channel is discussed. In particular, Gallager’s coding theorem for binary codes is treated extensively. By assuming a binomial weight distribution for linear codes, it is shown that the decoder error probability performance of some of the best, known linear, binary codes is the same as the average performance of the ensemble of all randomly chosen, binary nonlinear codes having the same length and dimension. Assuming a binomial weight distribution, an upper bound is determined for the erasure performance of any code, and it is shown that this can be translated into an upper bound for code performance in the AWGN channel. Finally, theoretical bounds on the construction of error-correction codes are discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
An efficient numerical approach to the exact ensemble distance spectrum of Gallager codes has been developed by evaluating powers of polynomials. With the exact ensemble distance spectrum of Gallager codes, tangential sphere upper bounds on their maximum likelihood (ML) decoding performance over binary input AWGN channels are investigated. Numerical results indicate improved bounds have been obtained, better than Sason and Shamai's results (which are based on Gallager's upper bound on the ensemble distance spectrum), especially in the error floor region. Furthermore, some critical properties of Gallager codes, including typical minimum distance and the performance tradeoff in the waterfall and error floor regions, have been considered.
The performance of maximum-likelihood (ML) decoded binary linear block codes over the additive white Gaussian noise (AWGN) channel is addressed via the tangential sphere bound (TSB) and two of its recent improved versions. The correspondence is focused on the derivation of the error exponents of these bounds. Although it was shown that some recent improvements of the TSB tighten this bound for finite-length codes, it is demonstrated in this correspondence that their error exponents coincide. For an arbitrary ensemble of binary linear block codes, the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum