2007 IEEE International Conference on Signal Processing and Communications (ICSPC 2007), 24-27 November 2007, Dubai, United Arab Emirates SOME RESULTS ON STOCHASTIC RESONANCE IN ONE-BIT QUANTIZERS Farzad Talebi, Vahid TabatabaVakili and Mehdi Adibi Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844-13114, Iran ftalebigee.iust.ac.ir, vakiligiust.ac.ir, mehdi_adibigee.iust.ac.ir ABSTRACT Stochastic Resonance (SR) is a phenomenon that occurs in certain non-linear systems in which noise can be used to improve the system response [1]. A recent approach to quantizer design has employed Stochastic Resonance by randomizing the threshold of the quantizer (Stochastic Thresholding). Stochastic Thresholding has shown a good performance in detection of weak signals buried in heavy noise even in -50dB SNR. In fact, Stochastic Thresholding helps ML estimation by some Fisher Information gain given in the quantizer. Driven by this motivation, in this paper we obtain new results on Stochastic Resonance in one-bit quantizers. Also, some discussions on threshold Probability Distribution Function (PDF) especially on the relation of the mean and the variance of PDF with a constant signal which we intend to estimate are implemented. Index Terms- One-bit Quantizers, Stochastic Thresholding, Fisher Information, Weibull distribution. constant analysis are presented which show how the obtained Fisher Information gain is affected by the mean and the variance of the threshold PDF and in section 5, the results are briefly discussed. 2. STOCHASTIC THRESHOLDING IN A ONE- BIT QUANTIZER A one-bit quantizer is a non-linear system defined by an input- output relation, given by: + I y(n) = d , if x(n) > y(n) , if x(n) < y(n) (1) in which x(n), y(n) and y(n) are input signal, output signal, and the threshold respectively, and n represents time. We assume that x(n) is a constant signal with the value a which is corrupted by additive white Gaussian noise: x(n) = a +i1(n) (2) 1. INTRODUCTION A One-bit quantizer is a platform for studying the Stochastic Resonance in quantizers due to its simple structure and analysis [2,3,4]. For more convenience, a constant signal which is corrupted by additive white Gaussian noise is used as the input signal for analyzing the efficacy of the quantizer using Fisher Information. Fisher information sets a lower bound to the efficacy of any conceivable unbiased ML estimator of the desired signal [4]. In [3] it has been shown that with Stochastic Thresholding of two- and tri-level quantizers with a Rayleigh PDF, a Fisher Information gain can be obtained which helps ML estimation. Fisher Information gain is the ratio of the output Fisher Information to the input one. In this paper, more results on Stochastic Thresholding are obtained using a two-parameter Weibull p.d.f which gives us the ability of mean-constant and variance-constant analysis of the quantizer. The mean-constant analysis shows that the less the variance of the threshold PDF, the more the obtained Fisher Information gain. Also, in case of variance-constant analysis (where we kept the standard deviation constant), the nearer the mean of the threshold PDF to the desired signal level, the more the obtained Fisher Information gain. This paper is organized as follows: In section 2, Stochastic Thresholding in one-bit quantizers is introduced. In section 3, calculation method of Fisher Information for one-bit quantizers are presented. In section 4, the mean-constant and the variance- The PDF of zero mean additive white Gaussian noise is: 2 2 (3) In Stochastic Thresholding y(n) becomes a stochastic sequence with some PDFs such as the Rayleigh distribution: 2 f7u= exp(- U) s 2s u.O 2 (4) For quantization of each sample a new threshold is created with the given PDF. The parameter of distribution, s, can be selected by some criteria. A simple criterion is discussed in the next section. 3. FISHER INFORMATION CALCULATION IN A ONE-BIT QUANTIZER We use Fisher Information gain as a measure of the performance of a one-bit quantizer. Fisher Information gain is the ratio of the output Fisher Information to the input one. The inverse of Fisher Information sets a lower bound for the This work was supported by Young Researchers Club, No 5, Malek St., Shariati St., Tehran, Iran, P. 0. Box 15655-461, Tel.:+98 21 884 47678, +98 21 884 10965; http://www.yrc.ir. 1-4244-1236-6/07/$25.00 © 2007 IEEE 329