A Simple Proof for a Spectral Factorization Theorem |
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Authors: | CAINES, P. E. GERENCSER, LASZLO |
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Affiliation: | Department of Electrical Engineering, McGill University, Montreal and the Canadian Institute for Advanced Research Department of Electrical Engineering, McGill University Montreal |
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Abstract: | It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H2 (D) and (ii) Z1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary. |
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