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The design of "thinned" planar array antennas is considered in which the density of elements located within the aperture is made proportional to the amplitude of the aperture illumination of a conventional "filled" array. Density tapering permits good sidelobe performance from arrays of equally radiating elements. The selection of the element locations is performed statistically by utilizing the amplitude taper as the probability density function for specifying the location of elements. The statistical design procedures and the theoretical prediction of performance are given. Application to a 50 wavelength diameter planar aperture is discussed and the results compared to conventional amplitude-taper designs. Examples of computed patterns are shown for density tapers modeled after 25, 30, 35, and 40 db circular Taylor distributions. The properties of a planar array of 10,000 elements are examined for "natural" thinning and for 70 per cent and 90 per cent of the elements removed. The sidelobes are determined more by the number of remaining elements than by the model amplitude taper. Statistically designed density-tapered arrays are useful when the number of elements is large and when it is not practical to employ an amplitude taper to achieve low sidelobes.