Unitary FIR filter banks and symmetry

Recently, unitary FIR filter banks with linear-phase have been completely parameterized by exploiting the eigenstructure of the exchange matrix. This correspondence gives new characterizations of polyphase matrices of filter banks with various types of symmetry on the filters. Using matrix extensions of the well-known hyperbolic and orthogonal lattices, we give a alternative proof for the parameterization of linear-phase unitary filter banks. A complete parameterization of FIR unitary filter banks with each of the different types of symmetries considered (not just linear-phase) is also given. These results can also be used to generate non-unitary filter banks with symmetries, though no completeness results can be obtained. In some cases implicit, and in others explicit parameterization of wavelet tight frames associated with these filter banks are also given. This paper only considers filter banks with an even number of channels. A similar theory can be developed if the number of channels is an odd integer