Gröbner bases and wavelet design

AbstractIn this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases

Biorthogonal multivariate filter banks from centrally symmetric matrices

AbstractWe provide a practical characterization of block centrally symmetric and anti-symmetric matrices which arise in the construction of multivariate filter banks and use these matrices for the construction of biorthogonal filter banks with linear phase

Lattice factorization and design of Perfect Reconstruction Filter Banks with any Length yielding linear phase

Publication in the conference proceedings of EUSIPCO, Florence, Italy, 200

Characterizations of rectangular (para)-unitary rational Functions

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters. (iii) Through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.Comment: Improved versio