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Exact Expressions in Source and Channel Coding Problems Using Integral Representations

机译：使用积分表示的源和通道编码问题中的精确表达式

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摘要

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).
机译：我们探索对数和幂函数的已知积分表示，并证明它们对信息理论分析的有用性。我们获得了几个源和通道编码问题的紧凑，易于计算的精确公式，这些问题涉及期望值和正随机变量的对数的更高矩以及非负随机变量的阶数ρ> 0（或iid正随机变量）。这些积分表示法可用于各种应用，包括计算由于干扰导致的通道输入和输出之间的互信息衰减，通用无损数据压缩，香农和雷尼熵评估以及对人体的遍历容量评估。具有随机参数的单输入多输出（SIMO）高斯信道（发送器和接收器均已知）。对数函数及其变体的积分表示形式有望作为流行（但非严格）复制方法的严格替代方法（至少在某些情况下）。

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