A stochastic (random) process may be nonstationary if its stochastic features vary with a shift of time. For the most practical processes, although more or less nonstationary, the generalized harmonics representation still makes sense; so does the spectral density function which is now defined as being ‘‘evolutionary’’ in view of the time dependency. The present paper briefly reviews this spectral description for a nonstationary process and further models it as the output from a white‐noise excited time‐variant shaping filter. With this model the nonstationary processes X(t) and Y(t) are denoted as ∫∞−∞Ax,t(ω)ejωt dW(ω) and ∫∞−∞Ay,t(ω)ejωt dW(ω), respectively, where Ax,t(ω) and Ay,t(ω) are the so‐called modulation functions (MFs) and dW(ω) is a random variable which retains the orthogonality. Previous papers have investigated and shown some advantage of using such a modulation function (MF) description in solving many practical nonstationary problems which hinged on the concept of the evolutionary auto/cross‐spectral density (EASD/ECSD). In this paper an attempt is made to apply further this MF model to describe the time‐varying auto/cross‐covariance functions (ACVF/CCVF) for the nonstationary processes and it is found that they are closely related to the relevant MFs but not to the EASD/ECSD in the Fourier transform sense, as has been summarized in the well‐known Wiener–Khintchine (W–K) relationship for the stationary processes. The new relationship has effectively generalized the W–K theorem in a special way, which has been proven efficient and accurate to both the synthetic signals such as the uniformly amplitude‐modulated process, the uniformly frequency‐modulated process and random‐phase process, and to the practical signals such as the nonstationary acoustic processes.
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November 1984
November 01 1984
Spectral model and time‐varying covariance functions for the nonstationary processes
Y. H. Tsao
Y. H. Tsao
Intercole Systems Ltd., Avenger Close, Chandlers Ford, Eastleigh, Hants, S05 3YU, England
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J. Acoust. Soc. Am. 76, 1422–1426 (1984)
Article history
Received:
November 29 1982
Accepted:
April 24 1984
Citation
Y. H. Tsao; Spectral model and time‐varying covariance functions for the nonstationary processes. J. Acoust. Soc. Am. 1 November 1984; 76 (5): 1422–1426. https://doi.org/10.1121/1.391625
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