A Remez Exchange Algorithm for Orthonormal Wavelets

Compactly supported orthonormal wavelets are obtained from two-band paraunitary FIR filter bank solutions, with the additional “flatness” constraint that the lowpass filter should have K zeroes at half the sampling frequency. This constraint is set to obtain “regular” wavelets. However, it is somewhat in contradiction with the usual requirement for good frequency selectivity, since it is well known that maximally flat filters (yielding Daubechies wavelets) have poor frequency selectivity. An efficient procedure for designing maximally frequency selective filter banks under a given flatness constraint is described in this paper. Classical Remez exchange algorithms, based on the alternation theorem, can no longer be used in this case. Linear programming techniques are capable of setting up constraints of this type, but require high memory storage and computation time. First, a variation of the alternation theorem adapted to this new situation is derived. Then, a modified Remez exchange algorithm for the design of “wavelet” filters is derived in the spirit of the Parks-McClellan algorithm. The efficiency of the algorithm is greatly improved as compared to linear programming techniques, and optimum filters are generally obtained after 3 or 4 iterations. A MATLAB listing is provided.