This paper presents the complete parameterizations of the class of M-channel perfect reconstruction filter banks (PRFBs) having a prescribed admissible scaling filter H/sub 0/(z). We show that in the biorthogonal case, such filter banks (BOFBs) are conveniently parameterized by the recently reported M-channel lifting factorization, which appropriately confines the degrees of freedom within the remaining M - 1 filters while guaranteeing H/sub 0/(z) as prescribed. On the other hand, for the orthogonal case, such filter banks (PUFBs) with a prescribed scaling filter H/sub 0/(z) of a certain length can be completed by the order-one factorization of an M /spl times/ 1 lossless system proposed in this paper - the constraint on McMillian degree of the resulting PUFBs previously present in the literature has been relaxed, and PUFBs with better performance are thus obtained. Design examples for both classes are given.