发表机构
The University of Hong Kong; Fudan University（香港大学; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Hardy空间框架下，通过谱分解与Blaschke乘积及功率回退论证，严格证明了反馈容量公式中可省略独立平稳高斯分量，从而补全了仅滤波器容量公式的推导。
AI 中文摘要
具有加性平稳高斯噪声的功率受限信道的反馈容量，由Kim在\cite{kim2010feedback}中表述为对独立平稳高斯分量的频谱和严格因果反馈滤波器上的无限维优化。Kim进一步断言，独立平稳分量可以被省略。然而，Derpich和Østergaard在\cite{derpich2022comments}中指出原始证明中存在一个漏洞，使得仅滤波器容量公式的原始推导不完整。在本文中，我们在容量上确界层面建立了这一约简。对于远离零的噪声频谱，典型谱分解将独立分量表示为Schur函数相对于内函数性质的缺陷。在原点保持其值的有限Blaschke乘积产生具有完全相同输出频谱和收敛输入功率的滤波器。然后，功率回退论证强制执行原始功率约束。最后，一个交错的AR(1)示例将该约简与达到容量的SK(2)构造联系起来。一般逼近定理不要求也不建立由单个滤波器达到最优。
英文摘要
The feedback capacity of power-constrained channels with additive stationary Gaussian noise was formulated by Kim in [Kim, 2010] as an infinite-dimensional optimization over the spectrum of an independent stationary Gaussian component and a strictly causal feedback filter. Kim further asserted that the independent stationary component can be omitted. However, Derpich and Østergaard [Derpich and Østergaard, 2022] identified a gap in the original proof, leaving the original derivation of the filter-only capacity formula incomplete. In this paper, we establish the filter-only formula at the level of capacity suprema for noise spectra satisfying the Paley-Wiener condition. In particular, we show that polynomial feedback filters suffice in the capacity supremum, without a positive lower bound on the noise spectrum or a rationality assumption. Under a positive essential lower bound, a complementary approximation preserves the output spectrum and rate exactly, while its input power converges to that of the original pair. An interleaved AR(1) example connects these results with the SK(2) coding scheme. The argument does not assert attainment by a single filter.
