Phase-only beam broadening in large transmit arrays using complex-weight gradient descent

Day recently obtained phase-only element weights for a large transmit array of arbitrary geometry by optimizing complex weights using an objective function that penalized amplitude variation. He used gradient descent modified with a classic SVD technique to create point and sector nulls as desired. Here Day's approach is extended to allow for main-beam broadening as might be needed to illuminate multiple receive beams. This is done by adding a second objective term, one penalizing variation in array-factor amplitude across a grid of beamspace points covering the desired broad main beam. The tension between the two objective terms turns out to be challenging to manage, and at this preliminary stage of algorithm development, parameter scheduling across a sequence of stages of the optimization is required to obtain good solutions.