Source Coding When the Side Information May Be 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 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, which is available at the encoder, are acquired at the decoder. Assuming a hidden Markov model for the source sequences, 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 asymptotically lossless. Then, with delay equal to zero or one source symbol, a singleletter characterization of the rate-distortion region is given for the case where, unbeknownst to the encoder, the side information may be delayed or not, and additional information can be received by the decoder when the side information is not delayed. Finally, examples for binary and Gaussian sources are provided. Index TermsRate-distortion function, Hidden Markov Model, Markov Gaussian process, multiplexing, strictly causal side information, causal conditioning.O. Simeone is with

An achievability proof for the lossy coding of Markov sources with feed-forward

“…In the following we present an achievability scheme based on the scheme proposed in [7] and later developed in [10]. We will see later that the argument given in [10] needs refinement in our case.…”

“…The analysis given in [10] is not applicable in our case since Y ) be the reproduction sequence and assume that the Hamming distortion between the two sequences is normalized, that is,…”

“…In this work, we adopt the idea given in [7] and developed in [10] and propose a constructive coding scheme for mary Markov sources which achieves the feed-forward rate distortion function and is conceptually simpler than the one given in [2], [3]. This paper is organized as follows.…”

Rate-distortion functions for source coding when side information with unknown delay may be present

“…There are some related works [7], [8] to our coding problem. In [7], a lossy source coding problem with delayed causal side information is considered, while delayed noncausal side information is considered in our setting.…”

“…In [7], a lossy source coding problem with delayed causal side information is considered, while delayed noncausal side information is considered in our setting. In [8], the lossy source coding problem with feedforward is considered, in which the delayed source sequence is available at the decoder.…”