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Time-Varying Spectra and Linear Transformation

01 September 1971

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R. M. Loynes 1 recently established a list of desirable properties for the spectrum of nonstationary processes. The elementary properties include: (i) a nonstationary spectrum should be rigorously defined, (ii) it should describe in some sense the energy distribution over frequency and time, and (Hi) it reduces to the ordinary spectrum when the process is stationary (Loynes' properties Al, A2, and A5). Both Page's instantaneous power spectrum 2 and M. B. Priestley's evolutionary spectrum 3 satisfy these basic requirements. Another spectrum definition based on the notions of two-dimensional spectra arising in the consideration of harmonizable processes 4,5 also satisfy these basic requirements with some qualifications. However, from the practical point of view, especially when the filtering and convolution of a random process is involved the most important requirement is the existence of simple transform relationships for linear systems. A spectrum should transform simply and reasonably when the process is transformed linearly (property A3, Loynes). In other words, input-output relationships are required so that a knowledge of the spectrum of a process determines the spectrum of the transformed process. In addition, these 2365