Design of Optimal Entropy-Constrained Unrestricted Polar Quantizer for Bivariate Circularly Symmetric Sources

Abstract

This paper proposes an algorithm for the design of entropy-constrained unrestricted polar quantizer (ECUPQ) for bivariate circularly symmetric sources. The algorithm is globally optimal for the class of ECUPQs with magnitude quantizer thresholds confined to a finite set. The optimization problem is formulated as the minimization of a weighted sum of distortion and entropy, and the proposed solution is based on modeling the problem as a minimum-weight path problem in a certain weighted directed acyclic graph. Each graph edge corresponds to a possible magnitude quantizer bin and computing its weight involves solving another optimization problem. We develop a fast strategy for evaluating all edge weights, leading to a $O(K^{2}+\textit {KP}_{\mathrm{ max}})$ time solution algorithm, where $K$ is the size of the set of possible magnitude thresholds and $P_{\mathrm{ max}}$ is the maximum number of phase levels. The practical performance of the proposed algorithm is assessed for a bivariate circularly symmetric Gaussian source, at rates ranging from 0.5 to 6 bits/sample. Our results demonstrate that the proposed approach achieves performance very close to the asymptotically optimal ECUPQ at all rates, while at low rates it significantly outperforms all previous UPQ schemes. Notably, peak improvement of 0.755 dB can be achieved for rates below 2.5.