Some results on Gaussian feedback channels

Abstract

This thesis mainly concerns Gaussian feedback channel, which is a common and basic channel in information theory. There are two main branches in this thesis. The first part discusses the Gaussian feedback capacity, especially the ARMA(k) Gaussian feedback capacity. For the non-feedback case, we have an explicit approach in calculating the capacity called water-filling method. However, there is no such elegant result for the feedback scenario, except when the noise is white. In [4], Kim gave necessary and sufficient conditions of the optimal filter achieving feedback capacity of the Gaussian channel with colored noise using variational formulation. Furthermore, Kim also studied one of the specific colored Gaussian channel - the ARMA(k) Gaussian channel, i.e., the additive Gaussian channel where the noise is a k-th order auto-regressive moving average Gaussian process, and provided the necessary and sufficient conditions of the optimal filter. Applying the results by Kim and a surprisingly simple method of changing variable, we are able to obtain the different necessary and sufficient conditions of the optimal filter achieving feedback capacity of the Gaussian channel with colored noise. Moreover, this new method can be utilized for the ARMA(k) Gaussian channel and we can obtain a computable method of studying the ARMA(k) Gaussian feedback capacity. More specifically, the ARMA(k) Gaussian feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations, which has only finitely many solutions for the cases k = 1; 2 and possibly beyond. Another part talks about the I-MMSE relation, or equivalently, the relationship between mutual information and minimum mean-square error (MMSE) for Gaussian feedback channel. Guo, Shamai and Verdu [45] first discussed this relationship and showed that the derivative of the input-output mutual information with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE achieved by optimal estimation of the input given the output. Many extensions about this result exist, but most of them are for non-feedback case. In [46], Han and Song considered the mutual information as the function of differential entropy of the output distribution and proved the I-MMSE relation for the feedback case. Applying this method, we further calculate the second-order derivative of mutual information for feedback Gaussian channel, which can be used to studied the properties of mutual information and MMSE in the future.