
Generalized Daubechies Wavelets 
Cédric Vonesch
Section Sytèmes de Communication, EPFL 
Diploma project September 2004 
Abstract 
We present a generalization of the Daubechies wavelet family. The context is that of a nonstationary multiresolution analysis i.e., a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. The constraints that we impose on these scaling functions are: (1) orthogonality with respect to translation, (2) reproduction of a given set of exponential polynomials, and (3) minimal support. These design requirements lead to the construction of a general family of compactlysupported, orthonormal waveletlike bases of L2. If the exponential parameters are all zero, then one recovers Daubechies wavelets, which are orthogonal to the polynomials of degree (N  1) where N is the order (vanishingmoment property). A fast filterbank implementation of the generalized wavelet transform follows naturally; it is similar to Mallat is algorithm, except that the filters are now scaledependent. The new transforms offer increased flexibility and are tunable to the spectral characteristics of a wide class of signals. 


