Beta-Beta边界：黄金公式的半块长度模拟
Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula
摘要：
众所周知，两个随机变量之间的互信息可以表示为依赖于辅助分布的两个相对熵的差，这种关系有时被称为黄金公式。本文讨论了这种关系的有限块长推广。该扩展由两个元素组成：1）Polyanskiy和Verdú（2014）的有限块长度信道编码逆界，其涉及两个Neyman-Pearson $β$函数的比率（β-β逆界）;以及2）新的β-β信道编码可扩展性界，再次表示为两个Neyman-Pearson $β$函数的比率。为了证明黄金公式的有限块长扩展的有用性，使用β-β可扩展性和逆界来获得Verdú（2002）宽斜率近似的有限块长扩展。证明与后者的推导类似，用β-β界代替黄金公式。β-β（可扩展性）界限也被证明是有用的情况下，实现能力的输出分布不是一个产品分布，由于，例如，对码本的成本约束或结构约束，例如正交性或恒定组成。作为一个例子，该界被用来表征的加性指数噪声信道的信道色散，并获得有限块长度的可扩展性界（迄今为止最紧的）的多输入多输出瑞利衰落信道与完美的信道状态信息在接收机。
英文摘要：
It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verdú (2014), which involves the ratio of two Neyman-Pearson $β$ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $β$ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú's (2002) wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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