Advent of nonregularity in photon-pulse delay estimation (Corresp.)

The mean-square error of the delay estimate of a photon pulse is known to decrease as
Q^{-1}
if the pulse envelope is smooth, but as
Q^{-2}
if it has sharp edges, thereby belonging to "nonregular" estimation cases (
Q
denotes the expected photon count). The transition from the
Q^{-1}
to the
Q^{-2}
law is investigated for trapezoidal pulse models, and is found to occur in the region where the standard deviation of the error is on the order of the width of the pulse slopes. Thus for values of
Q
below this region, practical pulses are effectively rectangular, and their estimation problem may be qualified as "nonregular."