Abstract
We derive finite-length bounds for two problems with Markov chains: source coding with side-information where the source and side-information are a joint Markov chain and channel coding for channels with Markovian conditional additive noise. For this purpose, we point out two important aspects of finite-length analysis that must be argued when finite-length bounds are proposed. The first is the asymptotic tightness, and the other is the efficient computability of the bound. Then, we derive finite-length upper and lower bounds for the coding length in both settings such that their computational complexity is low. We argue the first of the above-mentioned aspects by deriving the large deviation bounds, the moderate deviation bounds, and second-order bounds for these two topics and show that these finite-length bounds achieve the asymptotic optimality in these senses. Several kinds of information measures for transition matrices are introduced for the purpose of this discussion.
Highlights
In recent years, finite-length analyses for coding problems have been attracting considerable attention [1]
The check marks indicate that the tight asymptotic bounds can be obtained from those bounds
Under Assumption 1, we can derive the bound of the Markov case only for that special choice of QY, while under Assumption 2, we can derive the bound of the Markov case for the optimal choice of QY
Summary
Finite-length analyses for coding problems have been attracting considerable attention [1]. This paper focuses on finite-length analyses for two representative coding problems: One is source coding with side-information for Markov sources, i.e., the Markov–Slepian–Wolf problem on the system Xn with full side-information Yn at the decoder, where only the decoder observes the side-information and the source and the side-information are a joint Markov chain. The main purpose of this paper is finite-length analyses, we present a unified approach we developed to investigate these topics including asymptotic analyses. Since this discussion is spread across a number of subtopics, we explain them separately in the Introduction.
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