Algorithms for designing wavelets to match a specified signal

Algorithms for designing a mother wavelet psi (x) such that it matches a si gnal of interest and such that the family of wavelets {2(-(j/2))psi (2(-j)x - k)} forms an orthonormal Riesz basis of L-2(R) are developed, The algori thms are based on a closed form solution for finding the scaling function s pectrum from the wavelet spectrum, Many applications of signal representati on, adaptive coding and pattern recognition ri:quire wavelets that are matc hed to a signal of interest. Most current design techniques, however, do no t design the wavelet directly, They either build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing m ultiresolution analysis or design a scaling function that generates a multi resolution analysis with some desired properties, In this paper, two sets o f equations are developed that allow us to design the wavelet directly from the signal of interest. Both sets impose bandlimitedness, resulting id clo sed form solutions. The first set derives expressions for continuous matche d wavelet spectrum amplitudes. The second set of equations provides a direc t discrete algorithm for calculating close approximations to the optimal co mplex wavelet spectrum. The discrete solution for the matched wavelet spect rum amplitude is identical to that of the continuous solution at the sample d frequencies, An interesting byproduct of this work is the result that Mey er's spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary Specific examples are given which dem onstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals.