Analytical modeling of pulse-pileup distortion using the true pulse shape; applications to Fermi-GBM

Pulse-pileup affects most photon counting systems and occurs when photon detections occur faster than the detector's registration and recovery time. At high input rates, shaped pulses interfere and the source spectrum, as well as intensity information, get distorted. For instruments using bipolar pulse shaping there are two aspects to consider: `peak' and `tail' pileup effects, which raise and lower the measured energy, respectively. Peak effects have been extensively modeled in the past. Tail effects have garnered less attention due to the increased complexity: bipolar tails mean the tail pulse-height measurement depends on events in more than one time interval. We leverage previous work to derive an accurate, semi-analytical prediction for peak and tail pileup, up to high orders. We use the true pulse shape from the detectors of the Fermi Gamma-ray Burst Monitor. The measured spectrum is calculated by writing exposure time as a state-space expansion of overlapping pileup states and is valid up to very high rates. This expansion models losses due to fixed and extendable deadtime by averaging overlap configurations. Additionally, the model correctly predicts energy-dependent losses due to tail subtraction (sub-threshold) effects. We discuss pileup losses in terms of the true rate of photon detections versus the recorded count rate.