The probability density function for the output of an analog cross-correlator with correlated bandpass inputs

The probability density function (pdf) for the output of an analog cross-correlator with correlated bandpass inputs is derived. The pdf is derived by a "direct method" without resorting to the "characteristic function method," which usually requires contour integrations in a complex plane for inversion operations. The correlator consists of bandpass filters, a multiplier, and a zonal low-pass filter. We treat the general situation in which the two inputs are narrow-band signals of unequal power and of different phases. The bandpass input noises are assumed to be correlated and may have different powers. In the Appendix, another derivation for the pdf is given in the special case of equal power correlated noise. This derivation is based on the fact that the correlator output random variable is the difference of two independent noncentral chi-square variables of two degrees of freedom. We show that the two expressions for the pdf (one from the direct method and the other from the characteristic function method) are indeed equivalent. Finally, we discuss two major areas of application.