Theoretical performance of a complex cross-correlator with Gaussian signals

Analytical expressions are obtained for the joint probability density function (PDF) of the in-phase (X) and quadrature (Y) output signals of a complex cross-correlator when the inputs are narrowband partially-correlated Gaussian signals. The PDF and cumulative distribution function (CDF) for the modulus (X² +Y²)^1/2 are then derived, and applied to the problem of detecting a Gaussian noise jammer in the presence of uncorrelated Gaussian noise. Results are presented in the form of curves showing the value of input correlation coefficient required to achieve a specified detection probability versus the number of samples integrated, for various false alarm probabilities