Probability density functions for correlators with noisy reference signals
Glenn M. Roe, Gerald M. White · IEEE Transactions on Information Theory · 1961
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\betawhere \begin{equation} \beta = \sum_{i=1}^k (As_{i,x} + N_{i,x})(Bs_{i,y} + N_{i,y}). \end{equation} Thes_{i,x}ands_{i,y}are the signal components, andN_{i,x}andN_{i,y}are samples of Gaussian noise. Exact expressions involving Bessel and Whittaker functions are given for several cases. Asymptotic expressions allowW(\beta)to be plotted when these exact expressions cannot be obtained or conveniently evaluated.