On Hierarchical Joint Source-Channel Coding With Causal Side Information at the Decoders

Abstract

Consider a process, {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> }- <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> , producing independent copies of a triplet of jointly distributed random variables (RVs). The {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } part of the process - the source, is observed at the encoder, and is supposed to be reproduced at two decoders, decoder 1 and decoder 2 , where the {Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } and the {Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } parts of the process are observed, respectively in a causal manner. The communication between the encoder and the decoders is carried out across two memoryless channels in two successive communication stages. In the first stage, the compressed transmission is available to both decoders, but only decoder 1 reconstructs the source (according to the received data-stream and its causal side information {Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> }0. In the second stage, the second decoder reconstructs the source according to {Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } and the transmissions of the encoder in both stages. It is desired to find the necessary and sufficient conditions on communication between the encoder and decoders, so that the distortions incurred (in each round) will not exceed given thresholds. We first derive a single-letter characterization of achievable rates for a pure source-coding problem with successive refinement and causal side information at the decoders. Then, for a joint source-channel coding setting, we prove a separation theorem, asserting that in the limit of long blocks, no optimality is lost by first applying lossy successive-refinement source coding, regardless of the channels, and then applying good channel codes to each one of the resulting bitstreams, regardless of the source