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

2009/04/03 by Wei Lü, Lu, Wei, Namrata Vaswani +1 · 1 citation
Decision Sciences · Engineering · Environmental Science · #Advanced Statistical Process Monitoring #FOS: Mathematics #Fault Detection and Control Systems #Soil Geostatistics and Mapping #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.0904.0602

openalex publication_date 2009/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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