Variable-Length Resolvability for Mixed Sources and its Application to Variable-Length Source Coding

TL;DR

This paper derives formulas for the minimum length of uniform random numbers needed to approximate mixed source outputs within a specified error, extending to second-order analysis and applications in variable-length source coding.

Contribution

It introduces a new variant of variable-length $oldsymbol{ extit{ extdelta}}$-channel resolvability for mixed sources and derives single-letter formulas for both first and second-order cases.

Findings

Established a general formula for $ extdelta$-resolvability for channels.

Derived a single-letter formula for mixed memoryless sources.

Extended results to second-order asymptotics and source coding applications.

Abstract

In the problem of variable-length $\delta$-channel resolvability, the channel output is approximated by encoding a variable-length uniform random number under the constraint that the variational distance between the target and approximated distributions should be within a given constant $\delta$ asymptotically. In this paper, we assume that the given channel input is a mixed source whose components may be general sources. To analyze the minimum achievable length rate of the uniform random number, called the $\delta$-resolvability, we introduce a variant problem of the variable-length $\delta$-channel resolvability. A general formula for the $\delta$-resolvability in this variant problem is established for a general channel. When the channel is an identity mapping, it is shown that the $\delta$-resolvability in the original and variant problems coincide. This relation leads to a direct derivation of a single-letter formula for the $\delta$-resolvability when the given source is a mixed memoryless source. We extend the result to the second-order case. As a byproduct, we obtain the first-order and second-order formulas for fixed-to-variable length source coding allowing error probability up to $\delta$.