New Design of Orthogonal Filter Banks Using the Cayley Transform

It is a challenging task to design orthogonal filter banks, especially multidimensional (MD) ones. In the one-dimensional (ID) two-channel finite impulse response (FIR) filter bank case, several design methods exist. Among them; designs based on spectral factorizations (by Smith and Barnwell) and designs based on lattice factorizations (by Vaidynanathan and Hoang) are the most effective and widely used. The ID two-channel infinite impulse response (IIR) filter banks and associated wavelets were considered by Herley and Vetterli. All of these design methods are based on spectral factorization. Since in multiple dimensions, there is no factorization theorem, traditional ID design methods fail to generalize. Tensor products can be used to construct MD orthogonal filter banks from ID orthogonal filter banks, yielding separable filter banks. In contrast to separable filter banks, nonseparable filter banks are designed directly, and result in more freedom and better frequency selectivity. In the FIR case, Kovačević and Vetterli designed specific two-dimensional and three-dimensional nonseparable FIR orthogonal filter banks. In the IIR case, there are few design results (if any) for MD orthogonal IIR filter banks. To design orthogonal filter banks, we must design paraunitary matrices, which leads to solving sets of nonlinear equations. The Cayley transform establishes a one-to-one mapping between paraunitary matrices and para-skew-Hermitian matrices. In contrast to nonlinear equations, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve. We present the complete characterization of both paraunitary FIR matrices and paraunitary IIR matrices in the Cayley domain. We also propose efficient design methods for MD orthogonal filter banks and corresponding methods to impose the vanishing-moment condition.