A Simple Tight Bound on Error Probability of Block Codes with Application to Turbo Codes
A Simple Tight Bound on Error Probability of Block Codes with Application to Turbo Codes
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发表时间：
1999-07
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通讯作者：
D. Divsalar
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作者：
D. Divsalar
本文以封闭形式导出了分组码译码错误概率的一个简单界。这个边界是基于Gallager开发的边界技术。我们得到了分组码的字错误概率和比特错误概率的上界。这个界限很简单，因为它在最终版本中不需要任何集成或优化。该界限是紧的，因为它适用于非常接近香农容量极限的信噪比（SNR）。绑定只使用代码的权重分布。非随机码的界是严格的比原来的Gallager界和它的新版本由Sason和Shamai和Viterbi和Viterbi。它也比最近由Viterbi和Viterbi提出的更简单的界更紧，比Duman和Salehi提出的需要两个参数优化的界更简单。对于长块，它可以很好地与涉及积分和参数优化的更复杂的边界竞争，例如由Poltyrev定义的切向球，由Sason和Shamai详细阐述，并由Viterbi和Viterbi研究，以及由Dolinar，Ekroot和Pollara定义的几何形状。我们还得到了一个封闭形式的表达的最小信噪比阈值，可以作为一个紧的上限上的最大似然能力的非随机码。我们还表明，我们的边界的最小SNR阈值是相同的切向球界Poltyrev。我们将这个简单的约束应用于Turbo码。
A simple bound on the probability of decoding error for block codes is derived in closed form. This bound is based on the bounding techniques developed by Gallager. We obtained an upper bound both on the word-error probability and the bit-error probability of block codes. The bound is simple, since it does not require any integration or optimization in its final version. The bound is tight since it works for signal-to-noise ratios (SNRs) very close to the Shannon capacity limit. The bound uses only the weight distribution of the code. The bound for nonrandom codes is tighter than the original Gallager bound and its new versions derived by Sason and Shamai and by Viterbi and Viterbi. It also is tighter than the recent simpler bound by Viterbi and Viterbi and simpler than the bound by Duman and Salehi, which requires two-parameter optimization. For long blocks, it competes well with more complex bounds that involve integration and parameter optimization, such as the tangential sphere bound by Poltyrev, elaborated by Sason and Shamai, and investigated by Viterbi and Viterbi, and the geometry bound by Dolinar, Ekroot, and Pollara. We also obtained a closed-form expression for the minimum SNR threshold that can serve as a tight upper bound on maximum-likelihood capacity of nonrandom codes. We also have shown that this minimum SNR threshold of our bound is the same as for the tangential sphere bound of Poltyrev. We applied this simple bound to turbo-like codes.