Orthogonality Conditions for Non-Dyadic Wavelet Analysis

The conventional dyadic multiresolution analysis constructs a succession of frequency intervals in the form of (<i>π</i> / 2^<i> j</i>, <i>π</i> / 2^{<i> j</i> - 1}); <i>j</i> = 1, 2, . . . , <i>n</i> of which the bandwidths are halved repeatedly in the descent from high frequencies to low frequencies. Whereas this scheme provides an excellent framework for encoding and transmitting signals with a high degree of data compression, it is less appropriate to the purposes of statistical data analysis.<br>       A non-dyadic mixed-radix wavelet analysis is described that allows the wave bands to be defined more flexibly than in the case of a conventional dyadic analysis. The wavelets that form the basis vectors for the wave bands are derived from the Fourier transforms of a variety of functions that specify the frequency responses of the filters corresponding to the sequences of wavelet coefficients.