On Simplified Order-One Factorizations of Paraunitary Filterbanks

This paper revisits the order-one factorization of causal finite impulse response (FIR) paraunitary filterbanks (PU FBs). The basic form of the factorization was proposed by Vaidyanathan et al. in 1987, which is a cascade of general unitary matrices separated by diagonal delay matrices with arbitrary number of delay elements. Recently, Gao et al. have proved the completeness of this factorization and developed a more efficient structure that only uses approximately half number of free parameters. In this paper, by briefly analyzing Gao et al.’s derivation, we first point out that Gao et al.’s factorization contains redundant free parameters. Two simplified structures of Vaidyanathan’s factorization are then developed, i.e., a post-filtering-based structure and a prefiltering-based structure. Our simplification relies on consecutive removal of extra degrees of freedom in adjacent stages, which is accomplished through the C-S decomposition of a general unitary matrix. Since the conventional C-S decomposition leads to a redundant representation, a new C-S decomposition is developed to minimize the number of free parameters by further incorporating the Givens rotation factorization. The proposed structures can maintain the completeness and the minimality of the original lattice. Compared with Gao et al.’s factorization, our derivations are much simpler, while the resulting structures contain fewer free parameters and less implementation cost. Besides, these new factorizations indicate that for a PU FB with a given filter length, the symmetric-delay factorization offers the largest degrees of design freedom. Several design examples are presented to confirm the validity of the theory.