Capacity Analysis for Continuous Alphabet Channels with Side Information, Part I: A General Framework

TL;DR

This paper develops a comprehensive analytical framework for determining the capacity of continuous alphabet channels with receiver side information, addressing the challenges of extending finite alphabet results to continuous cases.

Contribution

It introduces new necessary and sufficient conditions for mutual information properties, enabling analysis of capacity and capacity-achieving measures in continuous channels.

Findings

Established conditions for weak* continuity and strict concavity of mutual information.

Proved existence and uniqueness of capacity-achieving measures.

Derived criteria for capacity value and optimal measures.

Abstract

Capacity analysis for channels with side information at the receiver has been an active area of interest. This problem is well investigated for the case of finite alphabet channels. However, the results are not easily generalizable to the case of continuous alphabet channels due to analytic difficulties inherent with continuous alphabets. In the first part of this two-part paper, we address an analytical framework for capacity analysis of continuous alphabet channels with side information at the receiver. For this purpose, we establish novel necessary and sufficient conditions for weak* continuity and strict concavity of the mutual information. These conditions are used in investigating the existence and uniqueness of the capacity-achieving measures. Furthermore, we derive necessary and sufficient conditions that characterize the capacity value and the capacity-achieving measure for continuous alphabet channels with side information at the receiver.