Comments on “Feedback Capacity of Stationary Gaussian Channels”
Milan S. Derpich, Jan Stubbe Østergaard · IEEE Transactions on Information Theory · 2022
In the above article[1], the author characterizes the capacity of power-constrained channels with additive stationary Gaussian noise. Specifically, inSection III, the feedback capacity is shown to correspond with the solution to a variational optimization problem. Section IV gives structural properties of the solution for 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$S_{V}$equal to zero. These structural properties are further developed in its sections V and VI for first- and$m$-th-order autoregressive 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 ofCorollary 4.4in[1](which claims that the choice$S_{V} = 0$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 asremarks 4.5and 5.2, and Lemma 6.1 in[1]are invalid, because they rely upon the optimality of choosing$S_{V} =0$.