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
|T(u)|
w_{k}
complex
u_{0}\leq|u|\lequ_{1}
subject to
T(0) = 1
where
T(u) = \Sigma\min{k=1}\max{N} w_{k} \exp (-id_{k}u)
and
d_{k}
are real numbers. It is proven that this problem has a solution with real excitation coefficients
w_{k}
. 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.