مرکز منطقه ای اطلاع رساني علوم و فناوري - A theorem on cross correlation between noisy channels

Abstract :

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 $Z = [{1 + (\\sigma _1^2/\\sigma _0^2)} {(1 + (\\sigma _2^2/\\sigma _0^2)}]^{-frac{1}{2}}$ .