Approximate difference pattern response of the weighted multibeam array

An approximate relationship for the amplitude difference pattern response as a function of angle relative to the boresight plane of an incident electric field for an amplitude-weighted Butler matrix has been given. The expression is valid between the crossover angles of the beams adjacent to the beam chosen as the reference. Idealized difference patterns of a weighted Butler array have been used for comparison and examples indicate that over the angular region of interest, the expression is in agreement within 0.8 percent for a 32-element weighted array and better than 1 percent for an eight-element weighted array.