Phase inversion and collapse of cross-spectral function

Cross-spectral analysis is a mathematical tool for extracting the power spectral density of a correlated signal from two time series in the presence of uncorrelated interfering signals. A set of conditions is demonstrated and explained where the detection of the desired signal using cross-spectral analysis fails partially or entirely in the presence of a second uncorrelated signal. Not understanding when and how this effect occurs can lead to dramatic under-reporting of the desired signal. Theoretical and simulated demonstrations of this effect are presented.