Reliability Function for Streaming Over a DMC With Feedback

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 progressively 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 whose priors are close to the capacity-achieving distribution, and the encoder determines the group index of the actual sequence of symbols arrived so far and applies randomization to exactly 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 in terms of the reliability function. For streaming over a symmetric binary-input DMC, we propose a one-phase instantaneous small-enough difference (SED) code that not only achieves the JSCC reliability function, but also, thanks to its single-phase time-invariant coding rule, can be used to stabilize an unstable linear system over a noisy channel. For equiprobably distributed source symbols, we design low complexity algorithms to implement both the instantaneous encoding phase and the instantaneous SED code. The algorithms group the source sequences into sets we call types, which enable the encoder and the decoder to track the priors and the posteriors of source sequences jointly, leading to a log-linear complexity in time. While the reliability function is derived for non-degenerate DMCs, i.e., DMCs whose transition probability matrix has all positive entries, for degenerate DMCs, we design a code with instantaneous encoding that achieves zero error for all rates below Shannon’s joint source-channel coding limit.