平稳高斯信道的反馈容量：一种最优的Schalkwijk-Kailath方案

中文摘要 AI 辅助

我们考虑具有加性有色高斯噪声和无噪声反馈的信道。Kim的开创性工作推导了反馈容量的平稳变分刻画，并进一步断言达到容量的平稳输入不必包含与反馈无关的高斯分量。这些结果导致了基于Schalkwijk--Kailath（SK）细化原理的简单编码方案的构造，并证明了该方案能够达到容量。最近的一篇注记指出了在反馈无关分量去除断言的证明中存在一个缺口，从而使得SK方案的最优性以及依赖于它的后续结果不完整。在本文中，我们证明了对于具有有理功率谱密度的平稳高斯噪声信道，该分量去除断言成立。我们的证明使用了反馈容量凸优化公式的扰动分析，并且确实表明每个优化器都将零功率分配给反馈无关分量。利用这一更强的性质，我们从任意优化器构造了一个显式的SK编码方案，该方案以双重指数衰减的最大错误概率实现反馈容量以下的每个速率。

英文摘要

We consider channels with additive colored Gaussian noise and noiseless feedback. Kim's seminal work derived a stationary variational characterization of feedback capacity and further asserted that the capacity-achieving stationary input need not contain a feedback-independent Gaussian component. These results led to the construction of a simple coding scheme, based on the Schalkwijk--Kailath (SK) refinement principle, which was shown to be capacity-achieving. A recent note identified a gap in the proof of the feedback-independent component-removal assertion, thereby leaving the optimality of the SK scheme and subsequent results that rely on it incomplete. In this paper, we prove the component-removal assertion for channels with stationary Gaussian noise that has a rational power spectral density. Our proof uses a perturbation analysis of a convex optimization formulation of feedback capacity and, indeed, shows that every optimizer assigns zero power to the feedback-independent component. Using this stronger property, we construct from any optimizer an explicit SK coding scheme that achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

发表机构

• Rachel and Selim Benin School of Computer Science and Engineering, The Hebrew University of Jerusalem（耶路撒冷希伯来大学雷切尔和塞利姆·贝宁计算机科学与工程学）

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