A theorem on cross correlation between noisy channels

Two channels carry noise waveforms, N_0 (t) + N_1 ( t) and N_0 (t) + N_2 (t) , where N_0 (t) is a common narrow-band Gaussian noise and N_1 (t) and N_2 (t) are independent narrow-band Gaussian noises associated with each channel. The outputs of each channel are sent through detectors whose outputs, F(x, y) , are identical homogeneous functions of the components, x and y , of their inputs N , where N(t) = x(t) \cos \omega_0 t + y(t) \sin \omega_0 t . Let R_{12}(\tau) be the normalized cross-correlation function of the two detector outputs. It is shown that to determine R_{12}(\tau) it suffices to know the normalized auto-correlation function R_0(\tau) of the output of a single such detector when the input is N_0(t) ; i.e., if R_0(\tau) = G(\sigma_0^2, p(\tau)) where p(\tau) and \sigma_0 are the normalized auto-correlation function and rms of either component of N_0 , then it is shown that R_{12}(\tau) = G(\sigma_0^2, Z_{\rho}(\tau)) where Z = [\{1 + (\sigma_1^2/\sigma_0^2)\} \{(1 + (\sigma_2^2/\sigma_0^2)\}]^{-\frac{1}{2}} .