A wavelet-based spectrum for non-stationary processes

This paper introduces a wavelet-based time-dependent spectrum for arbitrary non-stationary processes, for which no exact spectrum is defined. In analogy to the spectrum already built by one of the authors (Spanos and Failla, 2004) for oscillatory processes (Priestley, 1981), the proposed spectrum is cast as a series expansion involving the square moduli of the Fourier transforms of the wavelets at a number of scales, to be chosen according to the frequency content of the process. The coefficients of the series are computed by a set of integral equations, each involving the mean square value of the wavelet transform at one of the selected scales. Simple manipulations show that the proposed wavelet-based spectrum represents indeed an approximate value for time-dependent Fourier power spectral densities, defined over scale-dependent time intervals. Numerical results, assessed in terms of statistics depending on the spectral moments, prove satisfactory for typical earthquake processes