Indirect Rate-Distortion Function of a Binary i.i.d Source

TL;DR

This paper derives the indirect rate-distortion function for a Bernoulli source observed through a noisy binary symmetric channel, providing explicit formulas and bounds that quantify the impact of observation noise on compression efficiency.

Contribution

It presents a novel explicit expression and an upper bound for the indirect rate-distortion function of a Bernoulli source with noisy observations, extending classic rate-distortion results.

Findings

Explicit formula for the indirect rate-distortion function.

Closed-form upper bound on the rate-distortion function.

Observation noise reduces the benefit of increased bit-rate.

Abstract

The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a binary symmetric channel so that the channel crossover probability controls the amount of information available about the source realization at the encoder. We use classic results in rate-distortion theory to compute an expression of the rate-distortion function for this model, where the Bernoulli source is not necessarily symmetric. The indirect rate-distortion function is given in terms of a solution to a simple equation. In addition, we derive an upper bound on the indirect rate-distortion function which is given in a closed. These expressions capture precisely the expected behavior that the noisier the observations, the smaller the return from increasing bit-rate to reduce distortion.