Reliability function for streaming over a DMC with feedback

TL;DR

This paper derives the reliability function for streaming over a DMC with feedback, introducing a novel instantaneous encoding scheme that matches the reliability of full knowledge scenarios and has applications in system stabilization.

Contribution

It introduces a new instantaneous encoding phase for streaming over DMCs with feedback, matching the reliability function of fully known sources, and proposes practical low-complexity algorithms.

Findings

The JSCC reliability function for streaming equals that for fully accessible sources.

The proposed SED code achieves the JSCC reliability function and stabilizes unstable systems.

Codes with zero error below Shannon's JSCC limit are designed for degenerate DMCs.

Abstract

Conventionally, posterior matching is investigated in channel coding and block encoding contexts -- the source symbols are equiprobably distributed and are entirely known by the encoder before the transmission. In this paper, we consider a streaming source, whose symbols arrive at the encoder at a sequence of deterministic times. We derive the joint source-channel coding (JSCC) reliability function for streaming over a discrete memoryless channel (DMC) with feedback. We propose a novel instantaneous encoding phase that operates during the symbol arriving period and achieves the JSCC reliability function for streaming when followed by a block encoding scheme that achieves the JSCC reliability function for a classical source whose symbols are fully accessible before the transmission. During the instantaneous encoding phase, the evolving message alphabet is partitioned into groups, and the encoder determines the index of the group that contains the symbols arrived so far and applies randomization to match the distribution of the transmitted index to the capacity-achieving one. Surprisingly, the JSCC reliability function for streaming is equal to that for a fully accessible source, implying that the knowledge of the entire symbol sequence before the transmission offers no advantage regarding the reliability function. For streaming over a symmetric 2-input DMC, we propose an instantaneous small-enough difference (SED) code that not only achieves the JSCC reliability function but also can be used to stabilize an unstable linear system over a noisy channel. We design low complexity algorithms to implement both the instantaneous encoding phase and the instantaneous SED code. While the reliability function is derived for non-degenerate DMCs, for degenerate DMCs we design a code with instantaneous encoding that achieves zero error for all rates below Shannon's JSCC limit.