Comments on "Feedback Capacity of Stationary Gaussian Channels".

The 2010 paper “Feedback Capacity of Stationary Gaussian Channels” [1] characterizes the capacity of power-constrained channels with additive stationary Gaussian noise. Specifically, in its Section III the feedback capacity is shown to correspond with the solution to a variational optimization problem. Its Section IV gives structural properties of the solution of this optimization problem. Among these properties it stands out the capacity-achieving optimality of using a short-length input V with arbitrarily small variance, and thus a power spectral density SV equal to zero. These structural properties are further developed in its sections V and VI for first- and m-th-order auto regressive moving average channel noise, respectively, also showing the capacity-achieving optimality of the Schalkwijk-Kailath scheme for this scenario. In this note we show that the proof of Corollary 4.4 in [1] (which claims that the choice SV = 0 is capacity achieving) is erroneous. As a consequence the proofs of theorems 4.1, 4.6, 5.3 and 6.1, propositions 4.7 and 5.1, as well as remarks 4.5 and 5.2, and Lemma 6.1 in [1] are invalid, because they rely upon the optimality of choosing SV = 0.