doi:10.3850/978-981-08-7619-7_P024

ABSTRACT

In this paper wavelet analysis is cast within the theoretical framework of fractional calculus. Specifically it is shown that, based on basic formulas of fractional calculus, the continuous wavelet transform convolution integral can be reverted to a series expansion involving the wavelet-transformed function sampled at different positions, along with coefficients that do not depend on the wavelet function parameters, that is scale and shift, but only on the fractional moments of the Fourier transform of the mother wavelet. The latter coefficients, therefore, depend only the selected wavelet basis and shall be computed once, for any scale and shift, regardless of the function to be wavelet transformed. In this manner, a significant reduction of computational effort can be achieved as compared to standard numerical computation of the wavelet transform convolution integral. Results apply for arbitrary wavelet bases.

Keywords: Wavelet transform, Fractional calculus, Non-stationary signals.

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