Real-Time Coding With Limited Lookahead

A real-time coding system with lookahead consists of a memoryless source, a memoryless channel, an encoder, which encodes the source symbols sequentially with knowledge of future source symbols up to a fixed finite lookahead
$d$
, with or without feedback of the past channel output symbols and a decoder, which sequentially constructs the source symbols using the channel output. The objective is to minimize the expected per-symbol distortion. For a fixed finite lookahead
$d\geq 1$
, we invoke the theory of controlled Markov chains to obtain an average cost optimality equation (ACOE), the solution of which, denoted by
$D(d)$
, is the minimum expected per-symbol distortion. With increasing
$d$
,
$D(d)$
bridges the gap between causal encoding,
$d=0$
, where symbol-by-symbol encoding–decoding is optimal and the infinite lookahead case,
$d=\infty$
, where Shannon Theoretic arguments show that separation is optimal. We extend the analysis to a system with finite-state decoders, with or without noise-free feedback. For a Bernoulli source and binary symmetric channel, under Hamming loss, we compute the optimal distortion for various source and channel parameters, and thus obtain computable bounds on
$D(d)$
. We also identify regions of source and channel parameters where symbol-by-symbol encoding–decoding is suboptimal. Finally, we demonstrate the wide applicability of our approach by applying it in additional coding scenarios, such as the case where the sequential decoder can take cost-constrained actions affecting the quality or availability of side information about the source.