For complex-valued processes, both covariance and conjugate covariance are needed for a complete second-order wide-sense characterization (Picinbono 1996; Picinbono and Bondon 1997; Schreier and Scharf 2003a,2003b).
In this section, results analogous to those stated in Theorems 2.4.7 and 2.4.13 are presented for the conjugate covariance of the cyclic cross-correlogram.
By reasoning as for Theorem 2.4.7, we get the following result.
Theorem 3.7.1 Conjugate Covariance of the Cyclic Cross-Correlogram. Let y(t) and x(t) be zero-mean wide-sense jointly GACS stochastic processes with cross-correlation function (2.31c). Under Assumptions 2.4.2–2.4.5, the conjugate covariance of the cyclic cross-correlogram is given by
(3.132)
where
(3.134)
with
In (3.133)–(3.135), for notational simplicity, , , , and .
By reasoning as for Theorem 2.4.13, we get the following result.
Theorem 3.7.2 Asymptotic Conjugate Covariance of the Cyclic Cross-Correlogram. Let y(t) and x(t) be zero-mean wide-sense jointly GACS stochastic processes with cross-correlation function (2.31c). Under Assumptions 2.4.2–2.4.5, and 2.4.8, the asymptotic conjugate covariance of the cyclic cross-correlogram is given by
where
with .
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