Source Coding When the Side Information May Be Delayed

TL;DR

This paper investigates source coding with uncertain and potentially delayed side information, providing single-letter characterizations for various delay scenarios in sources modeled as hidden Markov processes, with applications to binary and Gaussian sources.

Contribution

It offers new single-letter characterizations of the rate-distortion region for sources with uncertain delay in side information, extending previous results to more general source models.

Findings

Single-letter characterization for arbitrary known delay at encoder.

Rate-distortion region for delay zero or one with unknown delay.

Extended results including additional information when side info is not delayed.

Abstract

For memoryless sources, delayed side information at the decoder does not improve the rate-distortion function. However, this is not the case for more general sources with memory, as demonstrated by a number of works focusing on the special case of (delayed) feedforward. In this paper, a setting is studied in which the encoder is potentially uncertain about the delay with which measurements of the side information are acquired at the decoder. Assuming a hidden Markov model for the sources, at first, a single-letter characterization is given for the set-up where the side information delay is arbitrary and known at the encoder, and the reconstruction at the destination is required to be (near) lossless. Then, with delay equal to zero or one source symbol, a single-letter characterization is given of the rate-distortion region for the case where side information may be delayed or not, unbeknownst to the encoder. The characterization is further extended to allow for additional information to be sent when the side information is not delayed. Finally, examples for binary and Gaussian sources are provided.