The Trapezoid Filter as a Least Squares Template Fitter

Finite Impulse Response (FIR) filters offer a powerful, flexible approach to maximizing the sensitivity in digital spectroscopy.  For example, the trapezoid filter [1] removes the exponential background from a tail pulse and integrates the decaying charge collected over time to account for ballistic deficit, yielding a robust pulse energy amid uncertainties in the arrival time.  Higher order cusp filters can also subtract constant background and reduce the energy uncertainty by preferentially weighting samples closer to the rising edge of the pulse.  In addition to these classical filters, the new algorithm for performing least squares fits to template waveforms [2] is also implemented as an FIR filter.  Conversely, we show that any FIR filter can be interpreted as a least squares fit, and that the trapezoid filter aptly arises from a particularly simple fit to the exponential tail pulse which it analyzes so well.

[1] V. Jordanov and G. Knoll, Nucl. Instrum. Methods, A345, 337345, 1994.

[2] A. P. Jezghani, L. J. Broussard, and C. B. Crawford, arXiv:2012.05937 [physics.ins-det]