Fast communication: Simple proofs of root locations of two symmetric linear prediction models

Authors:
Carlo Magi;Tom Bäckström;Paavo Alku
Affiliations:
Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology (TKK), P.O. Box 3000, FI-02015 TKK, Finland;Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology (TKK), P.O. Box 3000, FI-02015 TKK, Finland;Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology (TKK), P.O. Box 3000, FI-02015 TKK, Finland
Venue:
Signal Processing
Year:
2008

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Abstract

This paper gives simple proofs of the root locations of two linear predictive methods: the symmetric linear prediction model and the eigenfilter model corresponding to the minimal or maximal simple eigenvalues of an autocorrelation matrix. The roots of both symmetric models are proved to lie on the unit circle. Differently from previous proofs, the approach used in the present study also shows, based on the properties of the autocorrelation sequence, that the root angles of the symmetric linear prediction model are limited to occur within a certain interval. Moreover, eigenfilters corresponding to the minimum or maximum eigenvalue of an autocorrelation matrix that have multiplicity greater than unity are also studied. It turns out that it is possible to characterise the whole space spanned by the eigenvectors corresponding to the multiple eigenvalues by a single symmetric/antisymmetric eigenvector of the principal diagonal sub-block of the autocorrelation matrix having all the roots on the unit circle.