Power spectrum reduction by optimal Hankel norm approximation of the phase of the outer spectral factor

Abstract

A new approach to stochastic processes power spectrum reduction, based on Akaike's canonical correlation between the process past and future, is developed. As a first result, the correlation between the canonical past and the canonical future is given its precise functional-analytic interpretation as the Hankel operator generated lay the all-pass function characterizing the phase of the process outer spectral factor. The canonical correlation coefficients turn out to he nothing else but the singular values of this Hankel operator. Next, the all-pass function characterizing the phase of the outer spectral factor is reduced using the optimal Hankel-norm procedure of Adamjan, Arov, and Krein. The reduction of the all-pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through. This involves Bode's relations between the gain and the phase of an outer function, Nehari's extension, and the winding number of the all-pass phase function which is itself shown to be related to the definition of the past and the future. Finally, the net result is a reduced-order, outer spectral factor that matches the high-order spectral factor, fairly well.