Zero-Phase Filter Bank and Wavelet Code r Matrices: Properties, Triangular Decompositions, and a Fast Algorithm

• Authors: T. G. Marshall
• Venue: Multidimensional Systems and Signal Processing
• Year: 1997

Theclass of digital filter banks (DFB‘s) and wavelets composed ofzero-phase filters is particularly useful for image processingbecause of the desirable filter responses and the possibilityof using the McClellan transformation for two-dimensional design.In this paper unimodular polyphase matrices with the identitymatrix for Smith canonical form are introduced, and then decomposedto a product of unit upper and lower triangular or block-triangularmatrices which define ladder structures. A fundamental approachto obtaining suitable unimodular matrices for one and two dimensionsis to focus on the shift (translation) operators, as is donein the harmonic analysis discipline. Several matrix shift operatorsof different dimensions are introduced and their properties andapplications are presented, most notable of which is that theMcClellan transformation can be effected by a simple substitutionof a 2\times 2 circulant matrix for the polynomialvariable, w = (z + z^{-1})/2. Unimodular matrixgroups and pertinent subgroups are identified, and these areobserved to be subgroups of the special linear group over polynomials,SL(k[w]). A class of coiflet-like wavelets containingthe well-known wavelet, based on the Burt and Adelson filter,is decomposed by these methods and is seen to require only 3/2multiplications/sample if a scaling property, introduced herein,is satisfied. Making use of certain paraunitary wavelets, coiflets,that are closely comparable to the zero-phase wavelets of thisclass, it is seen that, in these cases, the zero-phase ladderalgorithm is twice as fast as the paraunitary lattice algorithm.