Spectral relationships for non-stationary, stochastic processes

Abstract

Various spectral relationships are developed for energy-bounded and power-bounded functions in terms of a running-parameter (time) Fourier Transform. The resultant spectral densities and auto-(or cross-) correlation functions are initially generalized for N dimensions, and then specialized in terms of the more familiar one-dimensional relationships. From each relationship a parametric (temporal) average is developed; for the temporally averaged power (time-variant) spectral density the relationship which accrues is a well-known one. When the (temporal) function is considered as a member of a non-stationary, stochastic process, several of the resultant ensemble averaged relationships are also well-known. Moreover, when a wide-sense (second-order) stationary stochastic process is considered as a special case of a non-stationary stochastic process, the ensemble averaged, power spectral density-auto-correlation relationship is the well-known Wiener-Khintchine theorem. Furthermore, the running-parameter (time) Fourier Transform is expressed in a Bôchner form, and the complementary inverse transform is related to Wiener's Generalized Harmonic Theorem. Finally, an indication is given as to how the relationships may be extended to general force-fields (such as distance, velocity, acceleration, force, etc.), and analytic representations in terms of Hermitian products. The exposition does much to clarify the essential difference between a correlation function defined for stochastic processes (or other power-bounded functions) and a correlation function defined for aperiodic signals (or other energy-bounded functions); also, the respective relationships of these two different correlation functions to running variable (time), power spectral density, and energy spectral density.