Support region estimation of the product polar companded quantizer for Gaussian source

This paper describes two approaches to optimization of the key design parameters, the support region threshold and the number of magnitude representation levels, of product polar companded quantizer (PPCQ) for Gaussian source of unit variance. The first approach is based on the exact performance analysis of PPCQ and on distortion optimization with respect to the key design parameters. Due to the optimization problem complexity we encountered with the first approach, some suitable approximations are introduced with the second one. As a result, much simpler asymptotic closed-form formula for distortion of PPCQ is derived as a function of the support region threshold. Although with this approach the closed-form formula for the support region threshold cannot be derived, the results of this approach indicate the useful support region threshold form. By combining the results of both approaches we propose, the worthy closed-form formulas for the support region threshold and signal to quantization noise ratio of a nearly optimal PPCQ are provided. Moreover, from the results of both analyses the lower and upper bound expressions for the number of magnitude representation levels are provided. The analysis presented in the paper is useful for designing PPCQ and is of great theoretical and practical importance.