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The Wiener-Khinchin Theorem for Non-wide Sense stationary Random Processes

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arxiv 0904.0602 v1 pith:2ARY4PQP submitted 2009-04-03 math.ST stat.TH

The Wiener-Khinchin Theorem for Non-wide Sense stationary Random Processes

classification math.ST stat.TH
keywords randomtheorembandlimitednesscertaingeneralized-psdnon-wideprocessessense
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We extend the Wiener-Khinchin theorem to non-wide sense stationary (WSS) random processes, i.e. we prove that, under certain assumptions, the power spectral density (PSD) of any random process is equal to the Fourier transform of the time-averaged autocorrelation function. We use the theorem to show that bandlimitedness of the PSD implies bandlimitedness of the generalized-PSD for a certain class of non-WSS signals. This fact allows us to apply the Nyquist criterion derived by Gardner for the generalized-PSD.

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    A positive dynamic spectrum S(f,t) generalizes the stationary power spectrum by defining Gramian closed-form noise covariances in Fourier and Wilson-Daubechies wavelet bases for gravitational wave data.