Rate–Distortion Functions for Gamma-Type Sources Under Absolute-Log Distortion Measure

Abstract

When the information source is a continuous distribution and the rate–distortion function is strictly larger than the Shannon lower bound, the explicit evaluation of the rate–distortion function is not straightforward. We evaluate the rate–distortion function for an independent identically distributed gamma source with respect to the absolute-log distortion measure. The logarithmic transformation reduces this rate–distortion problem to that under the absolute distortion measure. Extending the explicit evaluation of the rate–distortion function for the Gaussian sources, we obtain the parametric form of the rate–distortion function. We show that the optimal distribution of reconstruction consists of a continuous component enclosed by left and right discrete components, and the left discrete component vanishes when the acceptable distortion is small. We further extend the result for a wider class of source distributions.