Feedback capacity of OU-colored AWGN channels

Abstract

As a very important part of Shannon’s well-known original work on information theory, continuous-time channels, especially Gaussian channels, have become a significant focus of study in information theory. Despite extensive research efforts, apart from the additive white Gaussian noise (AWGN) channel, the derivation of a “computable” and “explicit” formula of feedback capacity for stationary continuous-time additive colored Gaussian noise (ACGN) channels has been a long-standing open problem that is of fundamental importance in information theory.

In this thesis, we consider a special case of OU-colored AWGN channels where the channel input, after going through an AWGN channel, may be further corrupted by an Ornstein-Uhlenbeck (OU) noise. Specifically, the channel is given by y(t) = x(t) + z(t), where the channel input {x(t)} satisfies a power constraint P and the noise {z(t)} is a stationary Gaussian process defined by z(t) = w(t) + λu(t; κ), where {w(t)} is a white Gaussian process and {u(t; κ)} is the Ornstein-Uhlenbeck process.

We derive an explicit formula of the feedback capacity for such a channel, which, besides the AWGN channel, has been the only nontrivial stationary ACGN channel with known closed-form feedback capacity. More specifically, we show that the feedback capacity of this channel is equal to the unique positive root of the third-order polynomial equation P(x + κ) ^2 = 2x(x + | κ + λ | )^2 when −2κ < λ < 0 and is equal to P/2 otherwise. Among many others, this result shows that, as opposed to a discrete-time ACGN channel, feedback may not increase the capacity of a continuous-time ACGN channel even if the noise process is colored, and moreover, at least in some cases, the continuous-time Schalkwijk-Kailath coding scheme achieves the feedback capacity for such a channel. By applying this optimal coding scheme, we also derive an asymptotic characterization of the mutual information rate for a special family of ACGN channels. We further conduct a thorough analysis of the effect of feedback on the capacity of an OU-colored AWGN channel. More specifically, we characterize when the feedback capacity equals or doubles the non-feedback capacity for such a channel; moreover, we give two counterexamples to disprove continuous-time versions of some classical results and conjectures for discrete-time ACGN channels, including a counterexample to the continuous-time version of the well-known Cover’s 2P conjecture.