OPTIMAL POLAR QUANTIZATION OF COMPLEX VARIABLES WITH CIRCULARLY SYMMETRIC DENSITIES
DOI:
https://doi.org/10.47839/ijc.3.3.312Keywords:
Quantization, polar quantization, iterative process, extended distortion function, convexity function, optimization, integer optimizationAbstract
In this paper we consider quantization of complex variables and mean-square error (MSE). The best polar quantizer is Wilson’s unrestricted polar quantizer (UPQ) [1]. The MSE minimization is constrained only by the total number of quantization points, N. Our method is different from Wilson’s algorithm [1] that has predetermined number of points Mi at each magnitude level i, 1?i?L, which makes it impractical for large number of points. In our approach, we consider MSE as a function of the vector M= L i i M ? ? 1 ) (whose elements are numbers of phase quantization levels at each magnitude level. The Wilson's method finds the optimal quantizer in such a way that the decision and reconstruction levels r, m are iterative found for each combination M, while the optimal combination is found by searching all combinations. Wilson's algorithm cannot be applied for middle and great N. The asymptotic analysis of the polar quantizers with circular symmetric densities is given in [2]. This analysis is approximate and cannot be applied for any number of points and for great N, which will be shown in this paper. We define the extension of the MSE over RL (denoted by MSE(P)). We prove the convexity of this function and show an efficient way to find M= L i i M ? ? 1 ) (by Popt. Our algorithm consists of two main iterative processes. The first iterative process finds Popt, ropt, mopt with ? accuracy, while the second iterative process determines Mopt, mopt, ropt using Popt as the starting value. This paper eliminates incompleteness from [1] and [2]. We also give an example of the quantizer construction for a Gaussian source. The authors see their work as a contribution in knowing the best possible solution in these classes of problems and also a possibility of applying the technique exposed inhere on other classes of problems and on larger dimensions.References
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