A Hardy-Space Proof of the Filter-Only Gaussian Feedback-Capacity Formula

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.