Stationary gaussian channel capacity achieved with optimal feedback scheme
Feedback Capacity of Stationary Gaussian Channels: An Optimal Schalkwijk-Kailath Scheme
Information Theory
Summary
This paper solves a problem about how to send messages over noisy communication channels when you get perfect feedback. The authors prove that a certain straightforward way of encoding messages, based on an older technique called the Schalkwijk-Kailath scheme, indeed achieves the best possible transmission rate. They also show that you don’t need to waste power on sending parts of the message unrelated to feedback. This strengthens understanding of communication limits for channels that have certain types of noise.
What this means in practice
- •For communication engineers: Design feedback-based coding systems for channels with colored Gaussian noise to approach capacity using explicit constructive schemes.
- •For signal processing developers: Implement optimal encoding methods in noisy channels where feedback is available, improving robustness and efficiency in signal transmission.
Authors
David Fay, Oron Sabag
Abstract
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.