David E. Newl
Dept. of Eng., Univ. of Cambridge, Trumpington St., Cambridge CB2 1PZ, UK
Wavelet analysis allows a signal f(x) to be decomposed into a family of orthogonal functions whose members have different scales and different positions along the x axis. There has been a great deal of research into the theory of dilation wavelets, which arise from the recursive solution of a special class of difference equation, and which cannot be expressed in the form of mathematical functions. In contrast, the author's harmonic wavelengths have a simple structure that can be written in terms of harmonic functions. They are concentrated locally and are orthogonal to their own discrete translations and dilations. Their frequency bandwidth can be chosen arbitrarily. Various applications will be demonstrated, including the analysis of music and the extraction of wave velocity data from impulse response records. Results are shown in the form of wavelet maps which are similar in concept to sonograms and show the distribution of mean-square over position and frequency. There is a quick algorithm for computing the harmonic wavelet transform. This is implemented by the FFT and is faster than the usual algorithm for computing dilation wavelet transforms. It can be optimized in order to make the harmonic wavelet transform invariant to position (which the dilation wavelet transform is not).