and the pattern of the array, assuming point sources; it is called the array factor. Array theory is largely concerned with synthesizing an array factor to form a specified pattern. In communications most arrays are planar arrays, with the elements being spaced over a plane, but the principles can be understood by considering an array of two elements with equal amplitudes (see Figure 9.7(a)). This has an array factor given by equation 9.12, where Y is given by equation 9.13. E=E1+E2e (9.12) y=8+kdcos0 (9.13) Adaptive Arrays Fig. 9.7 Two-element array and radiation pattern: (a) two elements with equal amplitudes; (b) half-wavelength spacing; (c) 90° phase difference. The pattern for small spacings will be almost omnidirectional, and as the spacing is increased the pattern develops a maximum perpendicular to the axis of the array. At a spacing of half a wavelength, a null appears along the array axis (see Figure 9.7(b)). This is called a broadside array. If a phase difference of 90 degrees exists between the two elements, then the pattern shown in Figure 9.7(c) results. Now the main beam is along the direction of the array, and the array is called the end-fire array. This illustrates one of the prime advantages of the array: By changing the electrical phase it is possible to make the peak beam direction occur in any angular direction. Increasing the spacing above half a wavelength results in the appearance of additional radiation lobes, which are generally undesirable. Consequently, the ideal array spacing is half a wavelength. However, if waveguides or horns are used this is not usually possible because the basic element is greater than half a wavelength in size. Changing the relative amplitudes, phases and spacings can produce a wide variety of patterns SO that it is possible to synthesize almost any specified radia- tion pattern. The array factor for an N-element linear array of equal amplitude is given by equation 9.14. E=NSinCNy/2 (9.14) sin(y/2) This is similar to the pattern of a line source aperture (equati