Some properties of generalized factorable 2-D FIR filters

Factorable M-dimensional filters are interesting because they can be implemented efficiently: their computational complexity is O(Mn) instead of O(n^M) (as in the case of generic non-factorable filters). Unfortunately, the passband support of a factorable filter can assume only very simple shapes (parallelepipeds with edges pairwise parallel to the axes), which are not adequate for most applications. In a recent paper, Chen and Vaidyanathan (1991, 1993) proposed a new class of non-factorable M-dimensional filters, whose passband support can be any parallelepiped, which can be realized with complexity O(Mn). In addition, they are designed starting from 1-D prototypes, which makes for a very simple design procedure. In this paper, we show that such filters belong to the class of generalized factorable (GF) filters (whose formal definition we introduce here), and derive some properties of theirs relative to the 2-D case. Our review includes issues such as the relation between minimax frequency response parameters and filter size (which is nontrivial in the multidimensional case), symmetries, 2-D step response, and frequency response constraints