Design of Successively Refinable Unrestricted Polar Quantizer

Abstract

This paper addresses the design of two-stage successively refinable unrestricted polar quantizers for bivariate circularly symmetric sources in the entropy-constrained and fixed-rate cases. The proposed solutions are globally optimal when the thresholds of the magnitude quantizers are confined to finite discretizations of the interval $\left[0,\infty\right)$ . The algorithm developed for the entropy-constrained case involves a series of stages, including solving the minimum-weight path problem for multiple node pairs in certain weighted directed acyclic graphs. The asymptotical time complexity is ${O}({K}_{1}{K}_{2}^{2}{P}_{max})$ , where ${K}_{1}$ and ${K}_{2}$ are the sizes of the sets of possible magnitude thresholds of the coarse and refined unrestricted polar quantizers (UPQs), respectively, while ${P}_{max}$ is an upper bound on the number of phase levels in any phase quantizer of the coarse UPQ. The solution algorithm for the fixed-rate case is based on solving a succession of dynamic programming problems for multiple coarse quantizer bins. The time complexity in the fixed-rate case amounts to ${O}({K}_{1}{K}_{2}{N}^{2}{N}_{1})$ , where ${N}_{1}$ is the number of cells of the coarse UPQ and $N$ is the ratio between the number of bins of the fine and coarse UPQs. The extensive experimental results on a bivariate circularly symmetric Gaussian source show the effectiveness of the proposed schemes.