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 [Kim, 2010] 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 {\O}stergaard [Derpich and {\O}stergaard, 2022] identified a gap in the original proof, leaving the original derivation of the filter-only capacity formula incomplete. In this paper, we establish the filter-only formula at the level of capacity suprema for noise spectra satisfying the Paley-Wiener condition. In particular, we show that polynomial feedback filters suffice in the capacity supremum, without a positive lower bound on the noise spectrum or a rationality assumption. Under a positive essential lower bound, a complementary approximation preserves the output spectrum and rate exactly, while its input power converges to that of the original pair. An interleaved AR(1) example connects these results with the SK(2) coding scheme. The argument does not assert attainment by a single filter.