The approximation of power spectra by phase matching

We consider the problem of approximating high order multivariable power spectra by lower order power spectra using the phase matching approach of Jonckheere, Helton and Harshavadhana [1,2,3]. The motivation for the phase matching approach is the identification [1,2,3] of the Hankel operator associated with the "phase function" as Akaike\´s canonical correlation operator [4]. Thus the phase matching approach can also be viewed as an algorithm for the approximate realization of a stochastic process by canonical variables, which is desirable for the implementation of an approximate Kalman filter of the power spectrum.