Universal Statistical Behavior of Wiener-Szego Polynomials Associated to Gaussian and Non Gaussian Stochastic Signals

Abstract

The starting point is the autocorrelation function of the signal. We assume the latter to be the sum of a random component (noise) plus d trigonometric functions with incommensurate frequencies. The time is descretized and the signal has a finite length N. The autocorrelation is thus itself a set of N random numbers C_i, i = 0,…(N - 1); - periodic boundary conditions are assumed. Considered as Fourier coefficients, the C_i define a random measure dμ with support on the unit. One can then build by a recurrence relation the family of polynomials orthogonal with respect to this measure. These Szego polynomials are simply related to the transfer functions of a family of Wierner-Levinson filters. The n complex random zeros of the random orthogonal polynomial of degree n are the focus point of the method, (i) all the zeros are within the unit circle [1]. When n and N increase, they all have their mean distance D to the unit circle going to zero [2]. (ii) d pairs of complex conjugate zeros can be associated to the d trigonometric components of the signal [3]. In the mean their distance (D) to the unit circle goes to zeros like n ^–2 [4]. Asymptotically, the modulus of their phases are equal to the frequencies of the d trigonometric components [2]. (iii) the (n – 2d) remaining zeros can be associated to the noise. The bulk of these “noisy zeros” is further away from the unit circle [3]. In the mean, the distance to the latter is decreasing like n ^– 1 [4]. (iv) if only noise is present the mean positions of the n zeros are regularly spaced on an inner circle [5]. This exactly regular spacing or crystallization is the signature of the noise. It is universal with respect to the probability law of the noise and to the length of the signal, provided lnN.