The expressions for the scanned array pattern indicate that for constant frequency f0, either phase shifters or time delay units form a beam with peak at the scan parameters (u0, v0) and that the shape of that pattern only depends on the displacement (u — u0, v — v0) and not the scan parameters. The pattern is displaced with scan and otherwise remains unchanged. The array beam width is therefore constant in (u, v) space for any given azimuth angle ф, but in terms of the 0 dependence, the beam width broadens as the array is scanned from zenith (0 = 0) to the horizon. For a large array and scan angle 00 not too near the horizon, this beam width is given in terms of the beam width 0B at broadside as
The beam width along the scan plane 0 thus broadens like sec 00 as the array is scanned in 0.
Accompanying this beam width increase is a decrease in array directivity so that the directivity is given in terms of the broadside directivity DB as where the sum is taken symmetrically about the array center. The coefficients an are the array element excitation and are given from orthogonality as
In this expression the integral is taken over the periodic distance in u space, namely half way to the two nearest grating lobes for a broadside beam. Used in this way, the technique gives the best mean square approximation to the desired pattern. This feature is lost if spacings are less than half wavelength, although the technique is still useful.
A second technique that has found extensive application is the “Woodward” synthesis method (4). This approach uses an orthogonal set ofpencil beams to synthesize the desired pattern. The technique has important practical utility because the constituant orthogonal beams are naturally formed by a Butler (5) matrix or other multiple-beam system.
Figure 10. Low sidelobe sum and difference pattern synthesis. (a) Taylor sum pattern with -30 dB (n = 6) pattern. (b) Bayliss difference pattern with -30 dB (n = 6) pattern.
Other techniques for periodic arrays are based on the polynomial structure of the far-field patterns. These include the method of Schelkunov (6), the Dolph-Chebyshev method (7), and others. Among the most successful and used methods are the pencil beam synthesis technique of Taylor (8) and the associated monopulse syntheses technique of Bayliss (9). These techniques are derived
as improvements to the equal ripple method of Dolph — Chebyshev, and result in more realizable aperture distributions, improved gain and other advantages. Figure 10(a-b) shows the array factors for 32 element arrays with 30 dB Taylor and Bayliss distributions. Note that the first sidelobe in both cases is very close to -30 dB with respect to the pattern maximum. In general, the discretizing of continuous distributions introduces errors in the synthesized pattern, and these are more significant for small arrays or for arrays that are forced to have very low sidelobes. Space precludes giving a detailed description ofthese procedures, but they are described in detail in a number of references. Usually discretizing the continuous distribution is not a problem, but when it is, there are a number of iterative techniques to converge to the original desired pattern. Notable among these is the work of Elliott (10).
Finally, in addition to these classic synthesis procedures, there have been many iterative numerical solutions to the synthesis problem. These have, in general, been shown to be efficient and useful. One successful iterative procedure was introduced by Orchard (11) that allows for complete power pattern design, even to the extent ofcontrolling each pattern ripple or sidelobe level. Other recently developed methods have used simulated annealing or genetic algorithms (12).