980 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 10, OCTOBER 2013 Design of Asymptotically Optimal Unrestricted Polar Quantizer for Gaussian Source Zoran Perić, Member, IEEE, and Jelena Nikolić, Member, IEEE Abstract—In this letter, in order to outperform the existing method for unrestricted polar quantizer (UPQ) design in terms of signal to quantization noise ratio, the asymptotic approximations of Rayleigh distributed function are applied to all magnitude regions of the UPQ, except to the last one. Given the constraint, the UPQ is designed to provide the minimum of the asymptotic mean-squared error distortion for the Gaussian source of unit variance. The effects of this constraint are studied for different bit rates . The accuracy of the derived formulas is assessed and the reasonable accuracy is observed for bit/sample. Index Terms—Asymptotic approximations, Gaussian source, un- restricted polar quantization. I. INTRODUCTION I N polar quantization, vector is quantized in terms of its magnitude and phase , where [1]. Assume that and are indepen- dent and identically distributed Gaussian random variables with mean zero and unit variance. The magnitude and phase are then independent random variables, the magnitude being Rayleigh distributed, , and the phase uniformly dis- tributed, [1], [2]. Prompted by the circular sym- metry of , in this letter we focus on polar quantizers. For such an assumed source we design an asymptotically optimal nonuniform UPQ. In order to outperform the UPQ from [1] in terms of SQNR (signal to quantization noise ratio) we set the constraint in the application of the asymptotic approximations from [1]. An -point UPQ partitions the plane into magnitude regions, each possibly having a different number of phase regions [1]. In this letter, as in [1], the -level nonuniform scalar quantizer is used for magnitude quantization and uniform scalar quantizers with levels, , are used for phase quantization. In particular, we begin with the UPQ, , defined by: 1) a compressor function ; 2) magnitude decision levels , with denoting the support region threshold; 3) magnitude reconstruc- tion levels ; 4) phase decision levels ; 5) phase reconstruction levels ; 6) quantiza- tion cells Manuscript received April 10, 2013; revised July 16, 2013; accepted July 22, 2013. Date of publication July 24, 2013; date of current version August 21, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Chao Tian. The authors are with the Faculty of Electronic Engineering, University of Niš, Niš, Serbia (e-mail: zoran.peric@elfak.ni.ac.rs; jelena.nikolic@elfak.ni.ac.rs). Digital Object Identifier 10.1109/LSP.2013.2274759 ; 7) the quantization rule . We proceed with its optimization to provide the minimum of the asymptotic MSE (mean-squared error) distortion for the Gaussian source of unit variance. As in [3], we assume that , and that does not vary just a little over the last magnitude region as it was assumed in [1]. We discuss the effects of this constraint for different bit rates . In addition, we address the advantages of the proposed method for UPQ design over the ones from [4] and [5]. In [4], Wilson assumed a set of meeting the constraint that the sum of , is . Then he solved iteratively the system of equations resulting in optimal and for the given set of . For a given , by exam- ining all observed combinations he ended up with the one mini- mizing the MSE distortion of the optimal UPQ . Wilson pointed out that the optimization of is cumbersome for moderate and large due to the consideration of many combi- nations of which sum to . For that reason, the analysis in [4] was restricted to . The lack of the Wilson’s method has been overcome some- what in [5], where an iterative method for near optimal UPQ design has been proposed. Although the UPQ from [5] outper- forms the one from [1] in terms of SQNR, due to a much higher design complexity the application of the method from [5] re- mains restricted. In this letter, the goal is not only to outperform the UPQ given in [1] in terms of SQNR, but also to propose a method for a simple design of asymptotically optimal UPQ for any . II. ASYMPTOTICALLY OPTIMAL UPQ DESIGN The MSE distortion per dimension of an UPQ is given by (1) [1], shown at the bottom of the next page. Similarly as in [6] and [7], denotes an inner distortion, and denotes an outer distortion, where, in our case, and are distortions from and , respectively. of our can be approximated using and the asymptotic approximations , [1], [3], as: (2) where denotes the number of magnitude levels in and (3) 1070-9908 © 2013 IEEE