Minimum-noise centroid-finding filters in the presence of coloured noise and time-domain constraints

Optimization of time-of-flight measurements is often made more challenging by the aleatoriety of the observed detector current signal. In fact the shape of the current signal seen at the collecting electrode may vary depending upon different statistical physical effects, such as thermal diffusion of the charge cloud, angled trajectories of the ionizing particles, etc. As a result the risetime of the induced signal is slowed down and the current waveform shape tends to be Gaussian rather than delta-like. In these cases the pulse centroid rather than a threshold height should be used as a reference point for timing. In this paper we show that the minimum-noise centroid-finding filter for timing measurements, may be derived by means of assessed optimization techniques which permit to impose arbitrary time-domain constraints in the optimal-filter impulse response. Namely the centroid-finding peculiarity is obtained by constraining the impulse response to show a "constant-slope crossover" as large as the maximum signal width is. Such a "constant-slope crossover" may be seen as the counterpart of the well known "flat top" used for ballistic-deficit rejection in pulse-height measurements. In addition the finite-width constraint may be used in order to diminish the pileup effects in high-rate operation.