Minimum-mean-square-error estimation of photon pulse delay (Corresp.)

The minimum-mean-square-error (rose) estimation of the delay of a coherent pulse of photons by a direct-detection receiver is shown to depend on the shape of the pulse envelope. Whereas with smooth envelopes the minimum mse decreases only as the expected energy in the pulse, with sharp-edged envelopes it decreases as its square, provided the optimum estimator is used. Occurrence of photoelectrons due to additive noise increases the rose by an additional term, proportional to the expected intensity of the noise. It approximately equals the photon-limited term at a signal-to-noise power ratio of 17 dB in the case of rectangular envelopes.