Optimal codes improve noisy feedback communication with linear methods
KKTCode: Asymptotically Optimal Linear Codes for Noisy Feedback AWGN Channels
Information Theory
Summary
Sending messages over noisy channels can be tricky, especially when feedback from the receiver is also noisy. Previous work mostly looked at simple feedback methods, which don’t reach the best possible performance. The authors studied a method where both sender and receiver use active feedback coding, improving the communication quality. Their design matches known theoretical limits and includes simpler methods as special cases, showing it can be computed efficiently. This advances understanding of how to communicate optimally over noisy channels with feedback.
What this means in practice
- •For communication system engineers: Design communication protocols that achieve near-optimal data transmission rates over noisy channels with feedback coding at both sender and receiver.
- •For signal processing developers: Implement efficient linear feedback coding algorithms with logarithmic complexity for adaptive noise reduction in communication devices.
Authors
Hongjae Nam, Jonggyu Jang, Vishrant Tripathi, David J. Love
Abstract
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.