Optimized filtering for pulse-shape based pile-up rejection applied to $0\nu \beta \beta$ search with ${}^{100}$Mo

Pile-up events, arising from the partial or complete temporal overlap of distinct signals, represent a major challenge in many areas of experimental physics where rare or low-rate processes are targeted. If not properly identified, pile-up can distort reconstructed observables, degrade energy resolution, and generate backgrounds that mimic genuine events of interest. This work presents an algorithm to obtain an optimized digital filter for the discrimination of pile-up events for detectors with known signal response and stationary noise power spectral density. It is developed in the context of the search for neutrinoless double beta decay with cryogenic Li${}_2$${}^{100}$MoO${}_4$ detectors like CUPID, where pile-up induced background from ${}^{100}$Mo $2\nu \beta \beta$ is expected to be the leading background contribution. For this application, the new filter discriminant reduces the pile-up induced background (at 90% efficiency) by 31%, compared to an analysis with a reference method previously presented in Eur. Phys. J. C 83(5), 373 (2023). While the discussion is grounded in cryogenic calorimetric detectors, the concepts and methods described are broadly applicable to a wide class of detector technologies and experimental contexts.