The aim of this paper is the presentation of the designing of asymptotic unrestricted polar quantizers with square cells, since it is known that square cells give minimal moment of inertia, which leads to minimization of distortion. Until now, polar quantizers with square cells have been designed only for the optimal companding function and their performances have been analyzed only for the stationary variance. In this paper, the design is done in a general manner, i.e. it is valid for any companding function and performances are analyzed in the wide range of variances. After that, this general design is applied for the logarithmic μ-law companding function. It is important that expressions for the numbers of magnitude and phase levels are obtained in the closed form, which simplifies the design. It is shown that the proposed polar quantizer has better performances (more than 3 dB higher the maximal and the average SQNR (signal-to-quantization noise ratio)) than the corresponding scalar quantizer with μ-law. Simulation is performed, and it is shown that theoretical and simulation results are matched very well. It is shown that the proposed polar quantizer can be used both for stationary and non-stationary signals, choosing the appropriate value of μ. This quantizer can be widely used, for all signals with Gaussian distribution. 
The general design of asymptotic unrestricted polar quantizers with square cells