  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.
