Cyclic LTI systems and the paraunitary interpolation problem

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. Since the frequency domain is a uniform discrete grid, there is more freedom in theoretical and design aspects. The basics of cyclic (L) multirate systems and filter banks have already appeared in the literature, and important differences between the cyclic and noncyclic cases are known. Since there is a strong connection between paraunitary filter banks and orthonormal wavelets, some deeper questions pertaining to cyclic (L) paraunitary matrices are addressed in this paper. It is shown that cyclic (L) paraunitary matrices do not in general have noncyclic paraunitary FIR interpolants, though IIR interpolants can always be constructed. It is shown, as a consequence, that cyclic paraunitary systems cannot in general be factored into degree one nonrecursive paraunitary building blocks. The connection to unitariness of the cyclic state space realization is also addressed.