A Mellin transform approach to wavelet analysis

• Wavelet transform is represented by Riesz integrals and mother wavelet fractional moments.
• Wavelet analysis of linear systems is pursued by wavelet transform fractional representation.
• Fractional representation provides response wavelet transform by a single matrix at all scales.
• Wavelet transform fractional representation proves robust and computationally efficient.

The paper proposes a fractional calculus approach to continuous wavelet analysis. Upon introducing a Mellin transform expression of the mother wavelet, it is shown that the wavelet transform of an arbitrary function f(t) can be given a fractional representation involving a suitable number of Riesz integrals of f(t), and corresponding fractional moments of the mother wavelet. This result serves as a basis for an original approach to wavelet analysis of linear systems under arbitrary excitations. In particular, using the proposed fractional representation for the wavelet transform of the excitation, it is found that the wavelet transform of the response can readily be computed by a Mellin transform expression, with fractional moments obtained from a set of algebraic equations whose coefficient matrix applies for any scale a of the wavelet transform. Robustness and computationally efficiency of the proposed approach are shown in the paper.