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Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications by Antonio Napolitano

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4.6 The Frequency-Smoothed Cross-Periodogram

It is well known that the power spectrum of WSS processes can be consistently estimated by the frequency-smoothed periodogram, provided that some mixing assumptions regulating the memory of the process are satisfied (Parzen 1957), (Brillinger 1981), (Thomson 1982). Moreover, for ACS processes, the frequency-smoothed cyclic periodogram at a given cycle frequency is a mean-square consistent and asymptotically Normal estimator of the cyclic spectrum at that cycle frequency (Alekseev 1988), (Hurd 1989a), (Hurd 1991), (Hurd and Gerr 1991), (Hurd and Leimgkow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Gerr and Allen 1994), (Dehay and Leimgkow 1996), (Sadler and Dandawatacute; 1998). In both cases of WSS and ACS processes, the frequency-smoothing procedure consists in considering frequency averages of the cross-periodogram along the support lines (with unity slope) of the Loève bifrequency spectrum. This technique was extended in (Allen and Hobbs 1992), (Lii and Rosenblatt 1998), and (Lii and Rosenblatt 2002) to the case of SC processes with known support lines with not necessarily unit slope.

By following the above idea, in this section the cross-periodogram frequency-smoothed along a known given support curve is proposed as ...

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