Optimum Filter for Determination of the Position of an Arbitrary Waveform in the Presence of Noise

Abstract

This paper presents a general approach to solving the problem of defining the time of arrival of a signal in the noise, or the similar one of defining the position of a peak in a histogram. In particular, it solves the problem of a signal in white noise, giving the well know result that the optimum filter before a zero crossing discriminator should have an impulse response function equal to the derivative of the waveform to be observed. It then solves the problem of estimating the position of a peak in a histogram in which the number of counts in each bin has a variance equal to the number of counts in that same bin. The resulting optimum filter is the derivative divided by the function itself. Examples of applying the results are given and the un iqueness of the solution to the set of non-linear simultaneous equations resulting from the problem is demonstrated.