General solution of certain matrix equations arising in filter design applications

In this work we present the explicit expression of all rectangular Toeplitz matrices B , C which verify the equation BB H + CC H = a I for some a > 0 . This matrix equation arises in some signal processing problems. For instance, it appears when designing the even and odd components of paraunitar...

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Bibliographic Details
Published in:Linear algebra and its applications 2010-04, Vol.432 (8), p.2077-2088
Main Author: DOMINGUEZ JIMENEZ, M. Elena
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Filter design
Linear and multilinear algebra, matrix theory
Mathematics
Paraunitary filters
Sciences and techniques of general use
Spectral factorization
Toeplitz matrices
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Summary:In this work we present the explicit expression of all rectangular Toeplitz matrices B , C which verify the equation BB H + CC H = a I for some a > 0 . This matrix equation arises in some signal processing problems. For instance, it appears when designing the even and odd components of paraunitary filters, which are widely used for signal compression and denoising purposes. We also point out the relationship between the above matrix equation and the polynomial Bézout equation | B ( z ) | 2 + | C ( z ) | 2 = a > 0 for | z | = 1 . By exploiting this fact, our results also yield a constructive method for the parameterization of all solutions B ( z ) , C ( z ) . The main advantage of our approach is that B and C are built without need of spectral factorization. Besides these theoretical advances, in order to illustrate the effectiveness of our approach, some examples of paraunitary filters design are finally given.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2009.09.023