Two-dimensional quantization of bivariate circularly symmetric densities

The problem of quantizing a two-dimensional random variable whose bivariate density has circular symmetry is considered in detail. Two quantization methods are considered, leading to polar and rectangular representations. A simple necessary and sufficient condition is derived to determine which of these two quantization schemes is best. If polar quantization is deemed best, the question arises as to the ratio of the number of phase quantizer levels to that or magnitude quantizer levels when the product of these numbers is fixed. A simple expression is derived for this ratio that depends only upon the magnitude distribution. Several examples of common circularly symmetric bivariate densities are worked out in detail using these expressions.