Statistical Treatment of Beam Position Monitor Data

We review beam position monitors adopting the perspective of an analogue-to- digital converter in a sampling data acquisition system. From a statistical treatment of independent data samples we derive basic formulae of position uncertainty for beam position monitors. Uncertainty estimates only rely on a few simple model parameters and have been calculated for two "practical" signal shapes, a square pulse and a triangular pulse. The analysis has been carried out for three approaches: the established signal integration and root-sum-square ap- proaches, and a least-square fit for the models of direct proportion and straight-line. The latter approach has not been reported in the literature so far. The straight-line fit provides the most robust estimator since it does not require baseline restoration, it is immune to signal offsets, and its standard deviation is smallest. Consequently, of the analysed estimators it promises the highest fidelity of results. The fit approach represents a simple, natural way to analyse beam position monitor data which can be easily implemented in hardware and offers potential for new applications.