The Cusp Filter as a Least-Squares Fitter for Tail Pulses

A challenge in the Nab experiment is detecting the low energy proton peak above the background noise of the pedestal. State-of-the-art trapezoid and cusp filters provide robust extractions of the pulse height in order to discriminate against noise fluctuations. It was previously shown that the trapezoid filter is equivalent to a 2-parameter sliding least-squares fit to an exponential tail pulse. We now show that the cusp filter is the response function for a 3-parameter fit which rejects both exponential and constant pedestal backgrounds. We present an intuitive description of the recursive implementation of this fitter in FPGA, which includes the calculation of $chi^2$ for further rejection of background, allowing us to trigger on lower threshold proton pulses.