Perfect reconstruction FIR filter banks: lapped transforms, pseudo QMFs and paraunitary matrices

Perfect-reconstruction FIR (finite-impulse-response) filter banks are analyzed in both the z-transform and time domains. A condition for equal-length analysis and synthesis filters is given in terms of orthogonality constraints on overlapping parts of the filters. If the further condition that the filters themselves form an orthonormal basis set is met, one obtains a paraunitary (in the z-transform domain) or unitary solution (in the time domain). For the restricted length case of L=2N, solutions are shown to exist (like the lapped orthogonal transform or the pseudo-QMF modulated filters) that allow perfect reconstruction and lend themselves to a fast algorithm implementation. Therefore, there is a large class of computationally efficient perfect-reconstruction FIR filter banks where analysis and synthesis have an identical frequency behavior.<>