Universal statistical behavior of the complex zeros of Wiener transfer functions

To N real random variables the sample autocorrelation coefficients, which are also the N Fourier coefficients of a measure on the unit circle are associated. The polynomials orthogonal with respect to this measure define the transfer functions of the Wiener-Levinson predictors. We show that the statistics of the zeros of those random polynomials exhibits a universal law of crystallization on a circle of radius [1 - (lnN)/2n], n being the order of the predictor. These results are supported by extensive computer experiments and backed by a theoretical scaling argument in the asymptotic domain In N << n << N. These results are independent of the nature of the noise and robust for signals of finite length N.