Real excitation coefficients suffice for sidelobe control in a linear array

Minimax design of a linear antenna array with arbitrary fixed elements leads to the following mathematical problem. minimize max complex subject to where and are real numbers. It is proven that this problem has a solution with real excitation coefficients . In the antenna application this shows that there is no need to allow phasing at the individual elements of the array; amplitude control alone will achieve all the sidelobe reduction possible. An analogous result can be proved for a more general complex approximation problem.