Radio Brightness Measurement of Incoherent Sources: Second-Order Brightness Spatial Statistics

Abstract

Radio brightness measurements of an incoherent source, in front of a constant-temperature background, depart from the true source brightness because of finite antenna beamwidth, internal receiver noise, and background noise. Noisy measured brightness samples may be used to estimate the measured and the true brightness, at and between the measured points. Such estimates are called "interpolation" and "restoration," respectively. Optimum linear estimation - interpolation or restoration - requires only the second-order brightness statistics of the true and measured brightness samples, and of the corresponding noise samples. We derive these correlation functions in terms of the antenna parameters, subject to the assumptions that the source brightness varies rapidly compared to the antenna beamwidth, and that the background temperature is constant. We apply these results to calculating the interpolation and restoration errors for optimum linear processing of a finite number of noisy measured brightness samples, as functions of the antenna illumination and the signal-to-noise ratio. Analytic results are given for a square array of four measured points. For this case the interpolation and restoration errors increase rapidly as the separation between measured points increases beyond critical sampling, which for an antenna with Gaussian illumination approximates the 3 db half-beamwidth.