4 is obtained. (9.24) This is simply the product of the element pattern and the array factor. The array factor is generally more directive than the element pattern SO that it dominates properties such as main beamwidth and sidelobe levels. The element pattern will generally reduce far-out sidelobe levels, including any grating lobes that might be present. Nulls A null of the radiation pattern is here taken to be synonymous with a 0-that is, no trans- mission or reception at a particular angle. A null can also mean a minimum of the pattern, but in the present analysis exact Os can be achieved. The analysis and synthesis of nulls can be achieved using Schelkunoff's representation. Consider the complex variable given by equation 9.25. z=exp(jkdcos0) (9.25) A O varies from 0 to IT; Z moves along a locus in the Argand diagram, as is shown in Figure 9.9. The modulus of Z is 1, while its phase (argument) is as in equation 9.26. 121=27dcast (9.26) This therefore varies between -2nd/A and +2nd/A. The locus of Z is thus an arc of the unit circle to the Argand diagram. For a large value of d, Z might move several times around the unit circle as O varies from 0 to TT. If d = A/2, the locus of Z as O varies from 0 to R just closes to form a complete circle. If d < /2, the locus of Z is a small arc of the unit circle. Now consider the polynomial representation. Any polynomial can be expressed as a product of linear factors. The polynomial of equation 9.16 will have N = 1 roots Z1, Z2, Zn-2, and can be written as in equation 9.27. Adaptive Arrays Im(z) O = 12 Re(z) Fig. 9.9 Argand diagram for Z = exp(jkd cos 0). F(z) = (9.27) Since only the relative radiation pattern is important, set an-1 giving equation 9.28. (9.28) The roots of Z1, Z2, and SO on (of F(z) = 0), are complex numbers, but do not necessarily lie on the unit circle. If they do lie on the unit circle, and d is such that that part of the unit circle is traversed by Z as O varies from 0 to TT, a null will be present in the radiation pattern. Because these are N - 1 r