Exact Expressions in Source and Channel Coding Problems Using Integral Representations

N. Merhav, I. Sason
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Abstract

We explore known integral representations of the logarithmic and power functions, and demonstrate their usefulness for information-theoretic analyses. We obtain compact, easily–computable exact formulas for several source and channel coding problems that involve expectations and higher moments of the logarithm of a positive random variable and the moment of order ρ>0 of a non-negative random variable (or the sum of i.i.d. positive random variables). These integral representations are used in a variety of applications, including the calculation of the degradation in mutual information between the channel input and output as a result of jamming, universal lossless data compression, Shannon and Rényi entropy evaluations, and the ergodic capacity evaluation of the single-input, multiple–output (SIMO) Gaussian channel with random parameters (known to both transmitter and receiver). The integral representation of the logarithmic function and its variants are anticipated to serve as a rigorous alternative to the popular (but non–rigorous) replica method (at least in some situations).
用积分表示的源和信道编码问题中的精确表达式
我们探索已知的对数函数和幂函数的积分表示,并证明它们对信息论分析的有用性。我们获得了几个源和信道编码问题的紧凑,易于计算的精确公式,这些问题涉及正随机变量的对数的期望和更高矩以及非负随机变量的ρ>0阶矩(或iid个正随机变量的和)。这些积分表示用于各种应用,包括信道输入和输出之间互信息因干扰而退化的计算,通用无损数据压缩,Shannon和rsamunyi熵评估,以及具有随机参数(发射器和接收器都知道)的单输入多输出(SIMO)高斯信道的遍历容量评估。对数函数及其变体的积分表示预计将作为流行的(但不严格的)复制方法的严格替代方法(至少在某些情况下)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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