Power spectral representation of nonstationary random processes defined over semi-infinite intervals

Abstract

A new sequence of locally time-averaged power spectra is defined that describes the time evolution of the frequency content of nonstationary random processes and linear system impulse response functions. The exact input-response relations for these spectral sequences are shown to be finite discrete convolutions of the input and system spectral sequences. Expressions for the coefficients of a Laguerre function expansion of the time-varying mean square response are derived from the response spectral sequence, and the time resolution and convergence of the expansion are discussed. Simple formulas are derived for generation of the spectra from recorded sample functions using Fourier transform computational algorithms. The relationship of the spectra to the Laplace transform is also developed. Physical interpretation of the spectra is discussed in detail and related to the uncertainty principle for Fourier transforms. Subject Classification: [43]45.40; [43]60.20; [43]20.50; [43]30.30; [43]40.35.