Spectral analysis of the convolution and filtering of non-stationary stochastic processes

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A unified theory is presented that deals with the spectral description, the convolution, and the filtering on non-stationary stochastic processes. Two time-dependent spectrum definitions, the physical spectrum and the instantaneous spectrum, are discussed in detail. The physical spectrum is an easily interpreted, non-negative, frequency-time decomposition of the expected energy of a process; it is a natural extension to non-stationary processes of the usual stationary process power spectrum defined as lim⁡T→∞E[1T|∫t−T/2t+ϒ/2x(u)e−12πfudu|2]. The instantaneous spectrum is a partially negative frequency-time decomposition of the expected energy of a process; it is a simple generalization of the stationary process power spectrum defined as the Fourier transform of the (ensemble average) autocorrelation function. Simple, exact, linear-system input-response relations for the instantaneous spectrum are derived for both deterministic and stochastic systems. A band-limited white definition based on the instantaneous spectrum is provided for non-stationary processes. For any desired time (or frequency) resolution, we show that the physical spectrum can be generated from the instantaneous spectrum by appropriately smoothing the instantaneous spectrum in the frequency-time plane. We also derive relationships that show how to generate the physical spectrum from (ensemble average) autocorrelation functions taken either in the time domain or the frequency domain. A discussion of non-stationary white processes is provided. Frequency-time duality is stressed throughout. Applications of the work to active sonar, passive sonar and structural vibrations are discussed.