On a Source-Coding Problem with Two Channels and Three Receivers

TLDR

This work studies the transmission of a memoryless unit‑variance Gaussian source to three receivers over two parallel channels, where one receiver observes both outputs, motivated by scenarios requiring high‑quality source reproduction at a single receiver over two unreliable links whose failures the encoder cannot detect. The study seeks to determine the fundamental limits on reconstruction fidelity when both links are operational while ensuring graceful degradation when either link fails. An explicit expression for the jointly achievable mean‑squared‑error distortions is derived, revealing an inverse trade‑off: approaching optimal performance with both links active forces large distortion during failures, whereas achieving near‑ideal performance during failures yields sub‑optimal distortion when both links are available.

Abstract

This paper treats the problem of communicating a memoryless unit-variance Gaussian source to three receivers. Two channels are available, each with a separate receiver. A third receiver has the outputs of both channels available. We obtain an expression for the simultaneously achievable distortions (mean-squared error). This problem applies to the following situation: Assume that high-quality reproduction of a source is desired at a single receiver which is connected to the source over a pair of links operating in parallel. Further assume that the links are unreliable in that either may fail, and that the source encoder is unaware of the failures. One can then ask how “robust” a system designed for this situation can be. That is, what are the limits on the fidelity achievable when both links are functioning if graceful degradation is required during the failure of either link? An inverse relation between performance in the two modes is obtained in the sense that, as performance in the presence of both links approaches its theoretical optimum, average distortion during failures becomes large. Conversely, if near-ideal performance during link failures is desired, then the distortion achieved when both links operate is far from its optimum value.