Group-Theoretic Structure of Linear Phase Multirate Filter Banks

Abstract

Unique lifting factorization results for group lifting structures are used to characterize the grouptheoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, 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."

On the Group-Theoretic Structure of Lifted Filter Banks

“…Let S = (D, U, L, B) be a group lifting structure with lifting cascade group C and scaled lifting group S. The following theorem has the same hypotheses as those of Theorem 4, but rather than invoking the unique factorization theorem the argument in [3] proves Theorem 9 directly from the hypotheses. A similar characterization is possible for HS filter banks.…”

“…The intuition behind Theorem 7 (below) is the identification of irreducible group lifting factorizations over U and L with the group of reduced words over the alphabet U ∪ L, which is the canonical realization of U * L. The reduced word construction of U * L is a somewhat technical chore when done rigorously, and it would be a messy affair at best to write down and verify an isomorphism between the group of reduced words over U ∪ L and a lifting cascade group in one-to-one correspondence with a collection of irreducible group lifting factorizations. For this reason the proof presented in [3] avoids the details of the reduced word construction and instead uses uniqueness of irreducible group lifting factorizations to show that C satisfies the categorical definition of a coproduct. Lemma 6 ensures that all D-invariant, order-increasing group lifting structures satisfy the hypotheses of the following theorem, whose proof consists of showing that C satisfies the universal mapping property in Figure 6.…”

“…The present paper is an expository overview of recent research [4,1,2,3]. It is targeted at a mathematical audience that has at least a passing familiarity with elementary group theory and with the connections between wavelet transforms and multirate filter banks.…”

“…[3], Theorem 2). Let C ≡ U ∪ L be a lifting cascade group over nontrivial lifting matrix groups U and L. C is a free group (necessarily on two generators) if and only if U and L are infinite cyclic groups and C ∼ = U * L.…”

“…[3], Theorem 3). If S is a D-invariant, order-increasing group lifting structure then S is the internal semidirect product of C by D:S = D C.This result can be combined with Theorem 7 to yield a complete group-theoretic description of the group of unimodular WS filter banks,W = S W = DC W .Corollary 2 ([3], Corollary 2).…”