Correlation matrix for quadrature components of two cross-correlated stationary narrow-band Gaussian processes

Collecting the work of several previous authors concerning the cross-correlation functions of stationary narrow-band Gaussian processes with their Hilbert transforms, a brief derivation is given for the correlation matrix for the quadrature components of two cross-correlated stationary narrow-band Gaussian processes. The elements of the matrix are represented in terms of the auto-correlation functions R
N
(τ) and R
N2
(τ) of the two Gaussian processes, the cross-correlation function R
12
(τ), and the Hilbert transforms of these functions. Alternative representations are given in terms of the power density spectra S
1
(ω) and S
2
(ω) and the cross density spectrum S
12
(ω) of the two processes. The sixteen matrix elements are found to consist of a maximum of six independent functions.