On Zero Delay Source-Channel Coding

TL;DR

This paper investigates the properties and optimality conditions of zero-delay source-channel coding, extending classical results beyond Gaussian assumptions and exploring the linearity of mappings under various conditions.

Contribution

It derives necessary and sufficient conditions for linearity of optimal mappings in non-Gaussian settings and characterizes the uniqueness of Gaussian pairs for linearity at multiple CSNRs.

Findings

Optimal mappings are linear only for Gaussian source-channel pairs at multiple CSNRs.

Asymptotic linearity occurs at low CSNR for Gaussian channels and high CSNR for Gaussian sources.

Numerical results show significant improvements over prior methods.

Abstract

In this paper, we study the zero-delay source-channel coding problem, and specifically the problem of obtaining the vector transformations that optimally map between the m-dimensional source space and the k-dimensional channel space, under a given transmission power constraint and for the mean square error distortion. We first study the functional properties of this problem and show that the objective is concave in the source and noise densities and convex in the density of the input to the channel. We then derive the necessary conditions for optimality of the encoder and decoder mappings. A well known result in information theory pertains to the linearity of optimal encoding and decoding mappings in the scalar Gaussian source and channel setting, at all channel signal-to-noise ratios (CSNRs). In this paper, we study this result more generally, beyond the Gaussian source and channel, and derive the necessary and sufficient condition for linearity of optimal mappings, given a noise (or source) distribution, and a specified power constraint. We also prove that the Gaussian source-channel pair is unique in the sense that it is the only source-channel pair for which the optimal mappings are linear at more than one CSNR values. Moreover, we show the asymptotic linearity of optimal mappings for low CSNR if the channel is Gaussian regardless of the source and, at the other extreme, for high CSNR if the source is Gaussian, regardless of the channel. Our numerical results show strict improvement over prior methods. The numerical approach is extended to the scenario of source-channel coding with decoder side information. The resulting encoding mappings are shown to be continuous relatives of, and in fact subsume as special case, the Wyner-Ziv mappings encountered in digital distributed source coding systems.