Step-response-based characterization of nonuniform perfect-reconstruction filter banks

An algebraic characterization of nonuniform perfect-reconstruction (PR) filter banks with integer decimation factors is derived. The PR property is formulated in the z domain based on the response of the systems to delayed unit step signals. This leads to derivation of a new class of characterizing formulas in terms of the rate conversion factors and the transfer functions of channel filters. We use the formulas to give a set of analytic necessary and sufficient conditions for a nonuniform delay chain filter bank to be a PR system. These conditions are identical to those that characterize exact covering systems of congruence relations using roots of unity. We introduce important results from the mathematics literature on exact covering systems. These results elucidate the admissible factors of decimation in nonuniform PR delay chain systems with maximally distinct decimation factors.