A fast implementation of wavelet transform for m-band filter banks

An orthogonal m-band discrete wavelet transform has an O(m(2)) complexity. In this paper we present a fast implementation of such a discrete wavelet transform. In an orthonormal m-band wavelet system, the vanishing moments (which corresponds to the approximation order and smoothness) and orthogonality conditions are imposed on the scaling filter (or lowpass filter) only. Given a scaling filter, one can design the other m-l wavelet filters (or highpass filters). It's well-known that there are infinitely many solutions in such designing procedure. Here we choose one specific type of solutions and implement the corresponding wavelet transform in a scheme which has complexity O(m). Thus for any scaling filter, one can always construct a full orthogonal nz-band wavelet matrix with an O(m) discrete wavelet transform.