How to derive the optimum filter in presence of arbitrary noises, time-domain constraints, and shaped input signals : a new method

The optimum filter constrained to finite width and other key features in the time domain, suitable for nuclear pulse spectrometry as well as for many other applications, is derived by means of a new method. A pattern of typical noise sources, including series and parallel Lorentzian packets, white and fn noises with n positive or negative integer are considered. Such a pattern permits also to treat the N-order fitting to an arbitrary spectral density of noise. The new method features a fast-convergence capability, and, in contrast to other methods, accepts either δ-like or arbitrarily shaped input signals. Flat tops, finite width, and/or zero-area constraints are met by a 'shift, sum and weigh' procedure, which uses as core function the non-flat-topped, non-time-limited minimum-noise weight function. This latter is determined, for all the specified input noises, in closed form. The optimum filter for the measurement of a DC signal is also provided as a by product of the method. A computer program, running in seconds in the Matlab environment, is available, and this demonstrates the effectiveness of the method.