Iterative Methods for Designing Orthogonal and Biorthogonal Two-channel FIR Filter Banks with Regularities

Efficient iterative methods are described for designing orthogonal and biorthogonal two-channel perfect-reconstruction FIR filter banks in such a way that for the analysis and the synthesis lowpass filter the number of zeros at z = −1 are fixed and the energies in the given filter stopband regions are minimized. The regularity of the analysis and the synthesis filter banks resulting when using only half of the tree structure (only the low-pass branch is split into two branches) are roughly proportional to the number of fixed zeros at z = −1 (vanishing moments) in the analysis and the synthesis filter, respectively. The frequency selectivity of these banks, in turn, is reciprocally related to the energies in the filter stopband regions. These two parameters are contradictory. By increasing the number of fixed zeros, the frequency selectivity of the overall filter bank is decreasing and vice versa. Since the selection of these two contradictory parameters depend on the application for which the filter bank is designed, it is necessary to find compromise solutions between them for every particular case. Using the proposed methods with different design requirements enables us to generate, for both orthogonal and biorthogonal filter banks, all possible combination between the maximally flat filter banks (maximum number of vanishing moments in the analysis and the synthesis filter) and standard frequency selective filter banks (no regularity requirements). Comparing the proposed method with some existing methods, for a given number of fixed zeros, filter banks with increased regularity and decreased stopband energies are obtained. The efficiency and flexibility of the proposed synthesis techniques are illustrated by means of several examples.