The relations between the field distribution across an aperture, its autocorrelation function and the angular power spectrum of the radiated field are used to analyse the fields produced by particular aperture distributions. The effect of illuminating only a small part of the total area of a large aperture is considered, and some of the average properties resulting from a randomly distributed illumination are derived. The case of linear arrays with a statistically small number of elements is also considered. The method outlined may also be extended to cylindrical or spherical arrays.