Noisy Reference Research Articles

Remarks on Birdsall's “Detection of a Signal Specified Exactly with a Noisy Stored Reference Signal”

The existence of an optimum decision procedure for a signal specified exactly with reference to a noisy stored reference signal is re-examined. It is shown that, within the framework of the theory of signal detectability, no such procedure exists.

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta</tex> where \begin{equation} \beta = \sum_{i=1}^k (As_{i,x} + N_{i,x})(Bs_{i,y} + N_{i,y}). \end{equation} The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s_{i,x}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s_{i,y}</tex> are the signal components, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{i,x}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{i,y}</tex> are samples of Gaussian noise. Exact expressions involving Bessel and Whittaker functions are given for several cases. Asymptotic expressions allow <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(\beta)</tex> to be plotted when these exact expressions cannot be obtained or conveniently evaluated.