Frame-theoretic analysis and design of oversampled filter banks

We provide a frame-theoretic analysis of oversampled and critically sampled, FIR and IIR, uniform filter banks (FBs). Our analysis is based on a relation between the polyphase matrices and the frame operator. For a given oversampled analysis FB, we parameterize all synthesis FBs providing perfect reconstruction, and we discuss the minimum norm synthesis FB and its approximative construction. We find conditions for a FB to provide a frame expansion. Paraunitary and biorthogonal FBs are shown to correspond to tight and exact frames, respectively. A new procedure for the design of paraunitary FBs is formulated. We show that the frame bounds are related with the eigenvalues of the polyphase matrices and the oversampling factor, and that they determine important numerical properties of the FB.