FIR oversampled filter banks and frames in ℓ2(Z)

Perfect reconstruction FIR filter banks implement a particular class of signal expansions in ℓ²(Z). These expansions are studied in this paper. Necessary and sufficient conditions on an FIR filter bank to implement a frame or a tight frame decomposition are given, as well as the necessary and sufficient condition for a feasibility of perfect reconstruction using FIR filters. Complete parameterizations of FIR filter banks satisfying these conditions are given. Further, we study the condition under which the minimal dual frame to the frame associated to an FIR filter bank is also FIR, and give a parameterization of a class of filter banks having this property. We then concentrate on the least constrained class, namely nonsubsampled filter banks, for which these frame conditions have particular forms.