6.7 Difference Beam Equalization

We take the difference beam pattern to be given by the derivative with respect to the angle of the sum beam pattern. We will use the sine-angle coordinate u, where u = sin ?, as this simplifies the expressions below, particularly for the difference beam slope, but otherwise does not affect the principles being illustrated. (In this form, the beam shape, plotted against u, remains unchanged as the beam is scanned.) Thus, in this section, we replace sin ? with u, particularly in equations that use (6.28). If w _k( u ₀) is the weight applied to the output of element k to steer in direction ? ₀, where u ₀ = sin ? ₀, then the sum beam gain (array factor) is given, as a function of frequency and angle, by

(6.31)

For narrowband steering we take w _k( u ₀) = exp (-2 ?id _kf ₀ u ₀/ c), so that the signals add in phase in the look direction ? ₀ at the center frequency f ₀. The sum is over all elements and the signal delay ? _k relative to the center of the array is given by d _ku/c [from (6.28)], where the element is at distance d _k from the array centroid, the mean element position, such that the sum of the element positions measured...