Harmonizable Nonstationary Processes

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 842-867, September 2024.
Abstract.Harmonizable processes can be represented by sums of harmonics with random coefficients, which are correlated rather than uncorrelated as for weakly stationary processes. Harmonizable processes are characterized in the second moment sense by their generalized spectral density functions. It is shown that harmonizable processes admit spectral representations and can be band limited and/or narrow band; samples of harmonizable Gaussian processes can be generated by algorithms similar to those used to generate samples of stationary Gaussian processes; accurate finite dimensional (FD) surrogates, i.e., deterministic functions of time and finite sets of random variables, can be constructed for harmonizable processes; and, under mild conditions, a broad range of nonstationary processes are harmonizable. Numerical illustrations, including various nonstationary processes and outputs of linear systems to random inputs, are presented to demonstrate the versatility of harmonizable processes.