New design of orthogonal filter banks using the Cayley transform

It is a challenging task to design orthogonal fllter banks, especially multidimensional (MD) ones. In the one- dimensional (1D) two-channel flnite impulse response (FIR) fllter bank case, several design methods exist. Among them, designs based on spectral factorizations (by Smith and Barnwell) and designs based on lattice factorizations (by Vaidyanathan and Hoang) are the most efiective and widely used. The 1D two-channel inflnite impulse response (IIR) fllter 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 1D design methods fail to generalize. Tensor products can be used to construct MD orthogonal fllter banks from 1D orthogonal fllter banks, yielding separable fllter banks. In contrast to separable fllter banks, nonseparable fllter banks are designed directly, and result in more freedom and better frequency selectivity. In the FIR case, Kovacevic and Vetterli designed speciflc two-dimensional and three-dimensional nonseparable FIR orthogonal fllter banks. In the IIR case, there are few design results (if any) for MD orthogonal IIR fllter banks. To design orthogonal fllter 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 e-cient design methods for MD orthogonal fllter banks and corresponding methods to impose the vanishing-moment condition.