The problem of the optimal power-pattern synthesis of circularly symmetric shaped beams is addressed and solved. In particular, an existence criterion is provided to test a priori (i.e., without solving the overall synthesis problem) if a non-superdirective source of given size can radiate or not a circularly-symmetric power pattern lying in a given arbitrary mask. In the affirmative case, the proposed approach guarantees the achievement of the optimal solution, i.e., that of maximum directivity or minimum size, by using convex optimization techniques. In general, a number of equivalent source distributions (i.e., radiating the same power pattern) do exist, which can be explicitly determined in a deterministic way. Numerical examples concerning applications of actual interest support the given theory and confirm the effectiveness of the developed synthesis procedure.
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