Timing resolution, dead time and pileup

Near a threshold crossing, a small voltage fluctuation shifts the crossing time by voltage noise divided by local pulse slope. This linearization assumes the slope is sufficiently well defined and the noise is small. Pulse-height changes also shift a fixed-threshold time, producing time walk; amplitude corrections or constant-fraction discrimination can reduce that effect but do not automatically remove electronic noise or fluctuations in the physical signal formation process.

Independent rms timing contributions add as variances, not as widths directly. An ideal uniformly quantized time bin contributes its width divided by the square root of twelve if the quantization error is uniformly distributed. Clock jitter and calibration residuals must be separately specified. Two equal independent detector timestamps give a difference width larger by the square root of two; correlated clock fluctuations can cancel or add differently, so that inference cannot be made without checking correlations.

Let the incident event stream be stationary Poisson, and let a constant independent intrinsic efficiency thin it into a Poisson stream of triggerable arrivals. Call that latter rate r. In the ideal nonparalyzable model, each accepted event blocks the detector for a fixed dead time, while arrivals during the blocked interval are lost without extending it. A mean accepted cycle consists of the dead interval followed by a mean waiting time for the next arrival.

In the ideal paralyzable model, every arrival restarts recovery, including those not recorded. An event is recorded only if the preceding interval of one dead-time length contains no other arrival, giving an exponential live probability. The recorded rate can decrease at high input rate, unlike the monotonic saturating nonparalyzable curve. Real instruments can mix recovery mechanisms, have multiple channels or use buffering, so a rate correction needs a measured response model rather than an assumed device label.

Pileup occurs when arrivals close enough in time produce overlapping signals that cannot be separated by the chosen reconstruction. For a specified forward overlap window, the Poisson probability of at least one additional triggerable arrival follows from the zero-arrival probability. The window is a reconstruction parameter and need not equal recovery dead time. Some overlapping pulses are rejected, some shift measured energy, and some are resolved by waveform fitting; none of those outcomes is determined solely by the arrival probability.

Dead time also changes the statistical structure of the recorded stream. Its output is generally not a Poisson process, so correcting the mean rate does not justify assigning a square-root-of-count uncertainty without further analysis. Intrinsic efficiency, live fraction and selection acceptance likewise should not be multiplied blindly when their definitions or event correlations overlap. The binomial lab illustrates independent fixed-trial efficiency statistics only; it intentionally does not implement dead time, temporal pileup or correlated counting errors.