  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 D-invariant, order-increasing
group lifting structures, it is shown that the associated lifting cascade group
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 S
is the semidirect product of C by the diagonal gain scaling matrix group 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, W, that is in fact equal to its own scaled lifting group,
W=S_W, the results of this paper characterize the group-theoretic structure of
W up to isomorphism. Although the half-sample symmetric class H does not form a
group, it can be partitioned into cosets of its lifting cascade group, C_H, or,
alternatively, into cosets of its scaled lifting group, S_H. Homomorphic
comparisons reveal that scaled lifting groups covered by the results in this
paper have a structure analogous to a "noncommutative vector space."
