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Conventionally, posterior matching is investigated in channel coding andblock encoding contexts -- the source symbols are equiprobably distributed andare entirely known by the encoder before the transmission. In this paper, weconsider a streaming source, whose symbols arrive at the encoder at a sequenceof 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 operatesduring the symbol arriving period and achieves the JSCC reliability functionfor streaming when followed by a block encoding scheme that achieves the JSCCreliability function for a classical source whose symbols are fully accessiblebefore the transmission. During the instantaneous encoding phase, the evolvingmessage alphabet is partitioned into groups, and the encoder determines theindex of the group that contains the symbols arrived so far and appliesrandomization to match the distribution of the transmitted index to thecapacity-achieving one. Surprisingly, the JSCC reliability function forstreaming is equal to that for a fully accessible source, implying that theknowledge of the entire symbol sequence before the transmission offers noadvantage regarding the reliability function. For streaming over a symmetric2-input DMC, we propose an instantaneous small-enough difference (SED) codethat not only achieves the JSCC reliability function but also can be used tostabilize an unstable linear system over a noisy channel. We design lowcomplexity algorithms to implement both the instantaneous encoding phase andthe instantaneous SED code. While the reliability function is derived fornon-degenerate DMCs, for degenerate DMCs we design a code with instantaneousencoding that achieves zero error for all rates below Shannon's JSCC limit.

Reliability function for streaming over a DMC with feedback