KKTCode：用于带噪反馈AWGN信道的渐近最优线性码
KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels
- Elmore Family School of Electrical and Computer Engineering, Purdue University（普渡大学埃尔莫电气与计算机工程学院）
- Chungnam National University（忠南国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对带噪反馈AWGN信道，提出主动编码的KKT最优线性反馈设计，渐近达到Elias-Butman逆界，并恢复被动解作为特例，解决了60年未决的最优性问题。
AI中文摘要：
带加性高斯白噪声（AWGN）且具有带噪输出反馈的信道中，最优因果线性反馈方案的设计问题已悬而未决超过60年。先前的工作集中于受限的策略类别，尤其是被动（未编码）带噪输出反馈，其中仅发射机执行反馈编码。然而，被动带噪输出反馈从根本上缺乏达到信息论性能极限所需的自由度。在本文中，我们考虑主动（编码）带噪输出反馈设置，其中发射机和接收机均执行反馈编码。我们随后开发了一种构造性的KKT最优主动线性反馈设计，该设计渐近地达到Elias-Butman SNR逆界，从而在整个因果线性方案类别中确立了MSE/SNR最优性。此外，我们证明了最优被动反馈解作为主动设计的一个特例被恢复。该被动解具有几何Toeplitz（GT）结构，并具有Chance-Love（CL）风格的一次性多项式表征，且能以O(log T)复杂度计算。因此，我们的结果为因果线性反馈编码下带噪输出反馈的长期最优性问题提供了肯定答案，且我们的数值结果支持理论发现。
英文摘要：
The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.
