5.8 Conjugate Covariance

For complex-valued processes, both covariance and conjugate covariance are needed for a complete second-order characterization (Picinbono 1996), (Picinbono and Bondon 1997), (Schreier and Scharf 2003a), (Schreier and Scharf 2003b).

In this section, results analogous to those stated in Lemma 4.4.7 and Theorems 4.6.4, and 4.7.7 are presented for the conjugate covariance of the bifrequency cross-periodogram and the frequency-smoothed cross periodogram.

Lemma 5.8.1 Conjugate Covariance of the Bifrequency Cross-Periodogram. Let and be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency spectra and cross-spectra (4.95). Under Assumptions 4.4.2, 4.4.3, and 4.4.5 (with Δf = 1/T), the conjugate covariance of the bifrequency cross-periodogram (4.93) is given by

(5.157)

with

(5.158)

(5.159)

(5.160)

where, for notation simplicity, , , , and .

Theorem 5.8.2 Conjugate Covariance of the Frequency-Smoothed Cross-Periodogram. Let and be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency spectra and cross-spectra (4.95). Under Assumptions 4.4.2, 4.4.3, 4.4.5 (with Δf = 1/T), and 4.6.2, the conjugate covariance of the frequency-smoothed cross-periodogram (4.147) is given by

(5.161)

where

(5.162)

(5.163)

(5.164)

Theorem 5.8.3 Asymptotic Conjugate Covariance of the Frequency-Smoothed Cross-Periodogram. Let and be second-order harmonizable zero-mean singularly and jointly SC stochastic processes with bifrequency spectra and cross-spectra (4.95). Under Assumptions 4.4.2, 4.4.3, 4.4.5 (with Δf = 1/T), and 4.6.2, the asymptotic (T→ ∞, Δf → 0, with TΔf→ ∞) conjugate covariance of the frequency-smoothed cross-periodogram (4.147) is given by

(5.165)

where

(5.166)

(5.167)

with

(5.168)

(5.169)

(5.170)

(5.171)