This expression, which is the Fourier transform integral of the taper function Ea(Xa, K„) across the antenna aperture, gives the corresponding electromagnetic field on the ground. The taper and ground pattern are related as a Fourier transform pair. It is assumed that the antenna aperture is nearly equiphase, and the propagated wave is linearly polarized. Convolution and Multiplication One method for predicting the electric field produced by the SPS antenna is to break the aperture distribution into simpler components and use the convolution principle to get the composite pattern. The actual SPS array may be thought of as the product of an infinite two-dimensional array of point sources located at the center of each subarray or power module [Fig. 3(a)] and a circular antenna function [Fig. 3(b)], This antenna function is equal in diameter to the actual array and can be shaped according to the illumination taper. Two tapers will later be considered: a uniform illumination and a 10 dB Gaussian taper. From the convolution principle, the ground pattern of the product (truncated, tapered) array will be the convolution of the two individual ground patterns and can be written A multiplication process in the antenna implies convolution of the ground patterns as shown in Figs. 3(a), (b), and (c). Since each point source in the phase array represents a single subarray, the subarray aperture function denoted by f3, must be multiplied with each source [Fig. 3(d)], Convolution in the antenna associates each point (phase center) of the array with a complete subarray function, resulting in an array of subarrays. When two functions are convolved in the antenna, the composite ground pattern (Fourier transform) is the product of the two separate ground patterns, and can be written where f3 is the subarray function and./'Ii2 is the circular array of point sources. The composite SPS antenna pattern (main beam plus grating lobes) is obtained by combining Eqs. 8 and 9, i.e., multiplying the subarray pattern by the array or grating lobe pattern, and is written The composite main beam plus graing lobe pattern is given in Fig. 3(e). The grating lobes are actually replicas of the main beam multiplied by the subarray pattern. It will be shown that the grating lobes normally occur at nulls of the subarray pattern and hence are greatly attenuated. The impact of grating lobes on the SPS
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