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