A Simple Proof for a Spectral Factorization Theorem
Peter E. Caines, László Gerencsér · IMA Journal of Mathematical Control and Information · 1991
It is shown using simple methods that the transform Z(z), z ε C, of the coefficient sequence of the Wold decomposition of any full-rank wide-sense stationary purely non-deterministic stochastic process satisfies (i) Z( z )ε H2 ( D ) and (ii) Z −1 ( z ) ε H ( D ). Further it is shown that all spectral factors satisfying (i) and (ii) are equal up to right multiplication by orthogonal matrices, and that among these the normalized (Z(0) = I ) spectral factors are equal to the transform of the Wold decomposition. An elementary proof of Youla's Theorem is then given together with a simple proof that the rows of a Cholesky factor of a banded block Toeplitz matrix converge to the coefficients of a stable matrix polynomial.