Cross-correlation and cross-power spectral density representation by complex spectral moments

A new approach to provide a complete characterization of normal multivariate stochastic vector processes is presented in this paper. Such proposed method is based on the evaluation of the complex spectral moments of the processes. These quantities are strictly related to the Mellin transform and they are the generalization of the integer-order spectral Moments introduced by Vanmarcke. The knowledge of the complex spectral moments permits to obtain the power spectral densities and their cross counterpart by a complex series expansions. Moreover, with just the aid of some mathematical properties the complex fractional moments permit to obtain also the correlation and cross-correlation functions, providing a complete characterization of the multivariate stochastic vector processes. Some numerical applications are reported in order to show the capabilities of this method. In particular, the examples regard two dimensional linear oscillators forced by Gaussian white noise, the characterization of the wind velocity field, and the stochastic response analysis of vibro-impact system under Gaussian white noise.