Minimum Expected Distortion in Gaussian Source Coding with Fading Side Information

TL;DR

This paper analyzes the optimal rate allocation for Gaussian source coding with fading side information, showing that single-layer encoding is optimal under certain fading distributions like Rayleigh.

Contribution

It derives the minimum expected distortion for Gaussian source coding with fading side information and proves the optimality of single-layer encoding for continuous, quasiconcave fading distributions.

Findings

Single-layer rate allocation is optimal for continuous, quasiconcave fading distributions.

Under Rayleigh fading, the optimal strategy targets the least favorable state.

The problem is formulated as a convex optimization with linearly many variables.

Abstract

An encoder, subject to a rate constraint, wishes to describe a Gaussian source under squared error distortion. The decoder, besides receiving the encoder's description, also observes side information consisting of uncompressed source symbol subject to slow fading and noise. The decoder knows the fading realization but the encoder knows only its distribution. The rate-distortion function that simultaneously satisfies the distortion constraints for all fading states was derived by Heegard and Berger. A layered encoding strategy is considered in which each codeword layer targets a given fading state. When the side-information channel has two discrete fading states, the expected distortion is minimized by optimally allocating the encoding rate between the two codeword layers. For multiple fading states, the minimum expected distortion is formulated as the solution of a convex optimization problem with linearly many variables and constraints. Through a limiting process on the primal and dual solutions, it is shown that single-layer rate allocation is optimal when the fading probability density function is continuous and quasiconcave (e.g., Rayleigh, Rician, Nakagami, and log-normal). In particular, under Rayleigh fading, the optimal single codeword layer targets the least favorable state as if the side information was absent.