Group-Theoretic Structure of Linear Phase Multirate Filter Banks
Christopher M. Brislawn · IEEE Transactions on Information Theory · 2013
Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For${\cal {D}}$-invariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group${\cal {C}}$is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group${\cal {S}}$is the semidirect product of${\cal {C}}$by the diagonal gain scaling matrix group${\cal {D}}$. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group,${\cal {W}}$, that is in fact equal to its own scaled lifting group,${\cal {W}}={\cal {S}}_{{\cal {W}}}$, the results of this paper characterize the group-theoretic structure of${\cal {W}}$up to isomorphism. Although the half-sample symmetric class${\Fraktur {H}}$does not form a group, it can be partitioned into cosets of its lifting cascade group,${\cal {C}}_{\Fraktur {H}}$, or, alternatively, into cosets of its scaled lifting group,${\cal {S}}_{\Fraktur {H}}$. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a “noncommutative vector space.”