Resolution, signal-to-noise ratio, and measurement precision

Abstract

We have studied the problem of simultaneously determining the midpoint position, separation, and relative intensities of a pair of closely spaced point sources. Our objective has been to see how the signal-to-noise ratio affects the measurement precision and to determine the significance of the various optical-system design parameters, and from this to develop a concept of resolution relevant to the generalized resolution problem, as posed by Ronchi over a decade ago. General results have been obtained for a scanning sensor designed to produce a one-dimensional output signal, with additive Gaussian noise. The rms precision with which the position of a single target can be measured has been determined. From this, we have obtained a resolution scale that, when divided by the signal-to-noise (voltage) ratio, gives the noise-limited position-measurement precision. A generalized expression that allows calculations of the resolution scale from the optical-signal transfer function of the system is given. Results in generalized form are developed for the precision with which the separation, midpoint position, and relative intensity of a pair of point sources can be measured. These results are presented in terms of the modulation transfer function of the optics and of the detector, and of the noise power spectrum. Quantitative results have been calculated for a system that consists of diffraction-limited, unobscured-circular-aperture optics and a sharply delineated rectangular detector, with additive white Gaussian noise. The numerical results show that the resolution scale is apparently also a significant descriptor of the two-target measurement capabilities of the instrument. This suggests that the resolution scale ought to be considered the resolution of. an instrument—to the extent that any single number can be used to define the resolution of an instrument.