UNIVERSAL STATISTICAL BEHAVIOR OF THE COMPLEX ZEROS OF WIENER TRANSFER-FUNCTIONS

To N real random variables the sample autocorrelation coefficients, wh ich are also the N Fourier coefficients of a measure on the unit circl e are associated. The polynomials orthogonal with respect to this meas ure define the transfer functions of the Wiener-Levinson predictors. W e show that the statistics of the zeros of those random polynomials ex hibits a universal law of crystallization on a circle of radius [1 - ( lnN)/2n], n being the order of the predictor. These results are suppor ted by extensive computer experiments and backed by a theoretical scal ing argument in the asymptotic domain In N << n << N. These results ar e independent of the nature of the noise and robust for signals of fin ite length N.