 array with a phase gradient superimposed. Using equation 9.33 gives equation 9.40. (9.40) Adaptive Arrays The peak of this radiation pattern is at 0=0ml where COS Om = -q/kd. Equation 9.40 can be written as in equation 9.41. in(1/2Nkd(cos0-cos0m)) (9.41) Grating lobes will appear if the denominator of equation 9.41 becomes 0 for some O = Om. This will occur if equation 9.42 is true by comparison with equation 9.19. (9.42) Let the deflection angle from broadside (0=90°) be a, SO that 0 = 90° + a. Equation 9.42 then becomes as in equation 9.43. sinam-sina== (9.43) The extreme values of sin a are +1. If we consider am > 0, the magnitude of the left-hand side of equation 9.43 has a maximum value of sin am + 1. Thus, with the inequalities equations 9.44 and 9.45, grating lobes will not appear. >>1+sinam (9.44) (9.45) For example, if the maximum beam-steering angle is 30° from broadside, these inequali- ties require d < 2/32. If the beam can be steered by 90°, to "end fire," d < 1/22 is needed to avoid grating lobes. Adaptive Null Steering Communications antenna systems are susceptible to degradation in performance due to interference received in their main beam or sidelobes. The interference might be deliberate jamming or signals using the same frequency in a cellular system, originating from a reuse cell. In these cases the angle of arrival of the interfering signal needs to be determined and null formed in the antenna pattern in that direction. An antenna array allows these functions to be performed, with up to N - 1 nulls for an N-element array. To independently control the angular location of the nulls, it is necessary to control the phase and amplitude of signals being fed to each array element. Both the angle-of-arrival determination and the control of the element excitations are difficult to perform accurately, which limits the performance of an "open-loop" system. For this reason adaptive "closed-loop" systems are often used, which can allow for statistical errors due to receiver noise and array tolerances. I