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 RN(τ) and RN2(τ) of the two Gaussian processes, the cross-correlation function R12(τ), and the Hilbert transforms of these functions. Alternative representations are given in terms of the power density spectra S1(ω) and S2(ω) and the cross density spectrum S12(ω) of the two processes. The sixteen matrix elements are found to consist of a maximum of six independent functions.