Oversampled linear-phase perfect reconstruction filterbanks: theory, lattice structure and parameterization

The paper presents the theory, lattice structure, and parameterization for a general class of P-channel oversampled linear-phase perfect reconstruction filterbanks (OLPPRFBs) - systems with sampling factor M (P≥M) and filter length of L=KM (K≥1) each. For these OLPPRFBs, the necessary existence conditions on the number of symmetric filters, n_β, and antisymmetric filters, n_α, (i.e., symmetry polarity) are first investigated. VLSI-friendly lattice structures are then developed for two types of OLPPRFBs, type I system (n_β=n_α) and type II system (n_β≠n_α). The completeness and minimality of each type of lattice are also analyzed. Compared with existing work, the proposed lattices are the most general and efficient ones for OLPPRFBs. Besides, through the lattice structures, the sufficiency of the existence conditions is also verified. Next, lifting-based structures are proposed to parameterize a left invertible matrix and all of its left inverses, which leads to unconstrained optimization as well as robust implementation of OLPPRFBs. Finally, several design examples are presented to confirm the validity of the theory and demonstrate the versatility of synthesis filterbanks in the oversampled system.