### ASA 127th Meeting M.I.T. 1994 June 6-10

## 4aSA7. Theory and application of harmonic wavelets.

**David E. Newl
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*Dept. of Eng., Univ. of Cambridge, Trumpington St., Cambridge CB2 1PZ, UK
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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).