where m2 = meter squared units introduced by the double integration of the unit impulses. Thus, the infinite series of point sources on the antenna transforms into an infinite array of impulses or grating lobes in the far field. These far-field impulses exist only at values of KXg = m2n/Sx and KYg = nlir/Sy. Using the definitions associated with Eq. 7, grating lobes occur whenever For ground locations on the Xg and Yg axes, these expressions for grating lobe locations reduce to the one-dimensional form, sin0 = nK/Sx, given in Eq. 3. Using SPS reference system parameters, Sx = 5y = 10.4328 m, X - 0.1225 m, Ro = 36,000 km, in Eq. 15, the grating lobes occur on-axis with angular spacing of 0 = 0.673° which corresponds to a ground spacing of 423 km. One way to intuitively predict the off-axis lobes is to recognize that the array of point sources acts like a diffraction grating at many polar angles, since many sets of parallel grating lines are implicit in the array. Referring back to Fig. 1, it is seen that a line of evenly-spaced points, properly phased to transmit in the direction 0 = 0°, will also add in phase when 0 = sin'^nX/^j.). Extrapolating to the two-dimensional SPS array of point sources as shown in Fig. 4, a rotation of the antenna axes by 45° allows the phase centers to be intersected by sets of parallel lines with slope SY/SX = 1. Grating lobes occur along the 45° ground axis orthogonal to these parallel lines, at increments determined by the spacing between the lines. Since the new lines are more closely spaced, = Sx/X^, the increment between grating lobes is larger, 0^145< = sin-1(nX/545=). The 45° grating lobes are spaced at intervals of 599 km (423 x V2), fitting into a rectangular grid pattern. The grating lobe pattern for phase control at the subarray level is given by Eq. 14 and shown in Fig. 5. The off-axis lobes occur at linear combinations of the axial grating lobe locations. The amplitudes of the off-axis grating lobes are determined by the subarray characteristics as will be discused later. Circular Antenna Function (f2) The second antenna illumination to consider is the circular antenna of radius RT with an electric field distribution Ea<rn>. The electric field at the ground, as determined by Eqs. 6 and 7, is given by where is symmetric with respect to </>a. The Fourier transform is changed to the Fourier Bessel transform by using the Bessel function identity
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