206 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 3a/2013 Jelena NIKOLIĆ, Zoran PERIĆ, Danijela ALEKSIĆ Faculty of Electronic Engineering, Niš Otimization of μ-law companding Quantizer for Laplacian source using Muller’s method Abstract. The motivation of this paper is based on the fact that a straightforward solution to optimization of the widely used µ-law companding quantizer has not been proposed so far. We deal with this problem for the case of a Laplacian source and we apply Muller’s method for the optimization of the quantizer in question. Particularly, we use minimal distortion criteria and we apply Muller’s method in order to determine the key parameter of the µ-law companding quantizer. The optimization method we propose is very general and it is easily modified for non-Laplacian sources. It can be applied for speech compession, because speech signals are well modeled by Laplacian sources. Streszczenie. W artykule opisano sposób optymalizacji kwantyzatora kompansji u-law. Do tego celu zastosowano metodę Muller’a oraz kryterium minimalnych zakłóceń. Proponowana optymalizacja jest ogólna i może być zmodyfikowana dla źródeł nielaplasjanowych. (Wykorzystanie metody Muller’a do optymalizacji kwantyzatora kompansji u-law dla źródła laplasjanowego). Keywords: µ-law companding quantizer, Muller’s method, optimization, speech compession Słowa kluczowe: kwantyzator kompansji u-law, metoda Muller’a, optymalizacja, kompresja mowy. Introduction Although the two modified logarithmic compressor characteristics obtained by the piecewise linear approximation to the A-law and the µ-law characteristics have become widely used as a design guideline for nonuniform quantization of speech signals in digital telephony [1], the fundamental question of how to provide a simple manner to optimize parameters of these two quantizers has long been open for signals with Laplacian and Gaussian probability density function (PDF). More specifically, it has remained undefined how to provide the straightforward approach to solving the complex system of nonlinear equations, i.e., how to determine the parameters of the quantizers in question that minimize the mean- squared error (MSE) distortion. In this paper, as in [2, 3, 4], we assume Laplacian PDF of the input signal and we focus on the robust µ-law companding quantization which gives an almost constant signal to quantization noise ratio (SQNR) in a wide range of variances. µ-law companding quantizer is preferable for use when the input signal’s variance changes with time in a wide range, as it is the case with speech signals [4]. As reported in [5], one of the reasons of often considering the Laplacian source is that the first approximation to the long- time-averaged PDF of speech amplitudes is provided by the Laplacian PDF. The optimization problem observed in [3, 4] has been solved by numerical optimization of the compression factor µ and the support region of the µ-law companding quantizer under the constraint that compression factor µ has an integer value. In this paper, we go one step further in the optimization. Namely, in order to reduce the search time of an optimal solution to the system of two nonlinear equations, we do not set the constraint on the integer values of the compression factor µ, but instead we apply Muller's method that provide simple and fast determining the optimal compression factor from the range of real values. The rest of this paper is organized as follows. Section 2 provides a detailed description of the proposed simple solution to the problem of optimizing the µ-law companding quantizer designed for the Laplacian source of unit variance. The achieved numerical results are the topics addressed in Section 3. Finally, Section 4 is devoted to the conclusions which summarize the contribution achieved in the paper. Optimization of μ-law companding quantizer for Laplacian PDF An N-level scalar quantizer Q is defined by mapping Q: R  Y [6, 7], where R is a set of real numbers, and   R y y y y Y N   ,..., , , 3 2 1 is a set of representation levels that makes the code book of size │Y│ = N. Every N-level scalar quantizer partitions the set of real numbers into N cells R i = (t i-1 , t i ], i = 1, …, N, where t i , i = 0, 1, …, N are decision thresholds and where it holds that Q(x) = y i , x  R i . Companding technique, we consider in this paper, defines the following steps: compress the input signal x by applying the compressor function c(x); apply the uniform quantizer on the compressed signal Q u (c(x)); expand the quantized version of the compressed signal using an inverse compressor function c -1 (Q u (c(x))). For a µ-law companding quantizer, denoted by Q μ , compression is done using the µ-law compressor function c μ (x): [−x max , x max ] → [−x max , x max ] [6, 7]: (1)    x x x μ μ x (x) c sgn max 1 ln 1 ln max             , max x x  , where the parameter μ is the compression factor and x max is the µ-law companding quantizer’s support region threshold. For the assumed Laplacian PDF p(x) [6]: (2)             2 - exp 2 1 x x p , the expression for the total distortion of the µ-law companding quantizer is given by [8]: (3)                                    max 2 max 2 2 max 2 2 2 2 2 exp 1 2 1 3 1 ln x x x N Q D . Let us aasume that the µ-law companding quantizer is designed for the unit variance, σ 2 = 1. Then, the expression for the total distortion becomes: (4)       max max 2 2 max 2 2 2 exp 1 2 3 1 ln x x x N Q D                   . By setting the first derivate of the distortion given by (4) to zero with respect to µ, we obtain: