Design of orthonormal symmetric wavelet filters using real allpass filters

In this paper, a class of real-valued orthonormal symmetric wavelet "lters is constructed by using allpass "lters, and a new method for designing the allpass-based wavelet "lters with the given degrees of #atness is proposed. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez exchange algorithm and considering the #atness condition. Therefore, a set of "lter coe$cients can be easily computed by solving the eigenvalue problem, and the optimal solution in the minimax sense is obtained through a few iterations. Furthermore, the design of the maximally #at allpass-based wavelet "lters is also included as a speci"c case, but it has a closed-form solution that is the same as in Selesnick (IEEE Trans. Signal Process. 46 (4) (April 1998) 1138}1141) so that the iteration procedure is not needed. Finally, some examples are designed to investigate the "lter characteristics, and it is shown that the number of delay elements strongly in#uences the "lter magnitude responses. ( 2000 Elsevier Science B.V. All rights reserved.