Probability density functions for correlators with noisy reference signals

Recently, correlation functions have had to be considered where both the reference waveform, which is usually the desired signal, and the input waveform are masked by different samples of additive noise. In this article, we derive the probability density function for the random variable where begin{equation} beta = sum_{i=1}^k (As_{i,x} + N_{i,x})(Bs_{i,y} + N_{i,y}). end{equation} The and are the signal components, and and are samples of Gaussian noise. Exact expressions involving Bessel and Whittaker functions are given for several cases. Asymptotic expressions allow to be plotted when these exact expressions cannot be obtained or conveniently evaluated.