Characterization of nonuniform perfect-reconstruction filterbanks using unit-step signal

An algebraic characterization of nonuniform perfect reconstruction (PR) filterbanks with integer decimation factors is presented. The PR property is formulated in the z domain based on the response of the linear multirate systems to the delayed unit-step signals. This leads to a unique class of characterizing formulas that are necessary and sufficient conditions for the PR property and free from the complex roots of unity. Two related characterizations of nonuniform PR systems, in the form of necessary conditions, are also developed based on these formulas. As a concrete example, the results are then used to derive necessary and sufficient conditions for PR nonuniform delay chain filterbanks. The conditions show that nonuniform delay chain filterbanks are the signal processing realizations of the mathematical notion of the exact covering systems of congruence relations. Important results from the mathematics literature on the exact covering systems are introduced. The results elucidate the admissible factors of decimation for the nonuniform PR delay chain systems in settings with maximally distinct decimation factors. A simple test of the PR property for delay chain systems is also presented. The test is based on the divisibility of certain polynomials by the cyclotomic polynomials. Finally, multirate systems based on the Beatty sequences, which are the irrational generalization of the exact covering systems, are briefly discussed.