A proof of the odd-symmetry of the phases for minimum weight perturbation phase-only null synthesis

It is proved that if phase-only weight control is used to impose nulls in a real pattern of a linear array of isotropic equispaced elements while minimizing the sum of the squares of the absolute weight perturbations, then the phase perturbations are odd-symmetric with respect to a phase reference at the array center. Equivalently, the perturbed pattern is real.