Discrete biorthogonal wavelet transforms as block circulant matrices Original Research Article

A complete characterization of banded block circulant matrices that have banded inverse is derived by factorizations similar to those used for orthogonal matrices of this kind. The nonorthogonal part of these matrices is shown to be related to block companion type matrices that are nilpotent. It also follows from this characterization that the width of the band of the inverse is bounded in terms of the number and size of the blocks generating the block circulant matrix. Matrices of this type appear in the description of the action of perfect reconstruction filter banks as well as in applications of discrete biorthogonal higher multiplicity wavelet transforms.