Proof shows feedback filters alone achieve Gaussian channel capacity
A Hardy-Space Proof of the Filter-Only Gaussian Feedback-Capacity Formula
Information Theory
Summary
This paper solves a gap in a previous mathematical proof about how to send data over noisy channels with feedback. The authors confirm that using only certain filters without extra noise components can achieve the best possible communication rate. They use advanced math to link the problem to functions called Schur functions and show how to build equivalent filters. Their work helps finalize a formula for maximum data rates in Gaussian noise channels with feedback.
What this means in practice
- •For communication system engineers: Design feedback filters that alone achieve maximum data rates over Gaussian noise channels without needing separate noise components.
- •For signal processing teams: Develop methods for creating equivalent power-constrained filters based on spectral factorization to optimize feedback communication systems.
A theory result. No direct application yet.
Authors
Jun Su, Guangyue Han
Abstract
The feedback capacity of power-constrained channels with additive stationary Gaussian noise was formulated by Kim in \cite{kim2010feedback} 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 \cite{derpich2022comments} identified a gap in the original proof, leaving the original derivation of the filter-only capacity formula incomplete. In this paper, we establish this reduction at the level of capacity suprema. For the noise spectrum bounded away from zero, canonical spectral factorization represents the independent component as the defect of a Schur function from innerness. Finite Blaschke products that preserve its value at the origin produce filters with exactly the same output spectrum and convergent input powers. A power-backoff argument then enforces the original power constraint. Finally, an interleaved AR(1) example relates the reduction to the capacity-achieving SK(2) construction. The general approximation theorem does not require or establish attainment by a single filter.