Cramér–Rao lower bounds on the performance of charge-coupled-device optical position estimators

The problem of optically estimating an object’s position by using a charge-coupled device (CCD) array composed of square pixels Δx on a side is analyzed. The object’s image spot at the CCD is assumed to have a Gaussian intensity profile with a 1/e point at a radial distance of 2σs from the peak, and the CCD noise is modeled as Poisson-distributed, dark-current shot noise. A two-dimensional Cramér–Rao bound is developed and used to determine a lower limit for the mean-squared error of any unbiased position estimator, and the maximum-likelihood estimator is also derived. For the one-dimensional position-estimation problem the lower bound is shown to be minimum for a pixel-to-image size ratio Δx/σs of between 1 and 2 over a wide range of signal-to-noise ratios. Similarly for the two-dimensional problem, the optimum ratio is shown to lie between 1.5 and 2.5. As is customary in direct detection systems, it is also observed that the lower bound is a function of both the signal power and noise power separately and not just of their ratio. Finally, the maximum-likelihood estimator is shown to be independent of the signal and noise powers at high signal-to-noise ratios.