Minimum expected distortion in Gaussian source coding with uncertain side information

We consider a layered approach to source coding with side information received over an uncertain channel that minimizes expected distortion. Specifically, we assume a Gaussian source encoder whereby the decoder receives a compressed version of the symbol at a given rate, as well as an uncompressed version over a separate side-information channel with slow fading and noise. The decoder knows the realization of the slow fading but the encoder knows only its distribution. We consider a layered encoding strategy with a base layer describing the source assuming worst-case fading on the side-information channel, and subsequent layers describing the source under better fading conditions. Optimization of the layering scheme utilizes the Heegard-Berger rate-distortion function that describes the rate required to meet a different distortion constraint for each fading state. When the side-information channel has two discrete fading states, we obtain closed-form expressions for the optimal rate allocation between the fading states and the resulting minimum expected distortion. For multiple fading states, the minimum expected distortion is formulated as the solution of a convex optimization problem. Under discretized Rayleigh fading, we show that the optimal rate allocation puts almost all rate into the base layer associated with the worst-case fading. This implies that uncertain side information yields little performance benefit over no side information. Moreover, as the source coding rate increases, the benefit of uncertain side-information decreases.