Polar quantization revisited

In polar quantization, a vector (X₁,X₂) is quantized in terms of its magnitude and phase. This is natural for circularly symmetric densities such as IID Gaussian. A number of polar schemes have been proposed, analyzed and optimized. One of the best is unrestricted polar quantization (UPQ), in which the phase step size depends on the quantized magnitude. Prior analysis has shown that when optimized, its SNR is only 0.17 dB less than optimal 2-dimensional VQ. We show that UPQ can be very simply analyzed and optimized using the VQ version of Bennett´s integral. Indeed, it is shown that one may optimize UPQ based solely on the criteria that it have the point density of an optimal VQ, and square rather than hexagonal cells. We also analyze and optimize power law polar quantization, in which the number of phase levels is proportional to a power of the quantized magnitude