The problem of the synthesis of optimal continuous aperture sources to optimally realize the satellite multibeam coverage of Earth is stated and solved. The design approach relies on a far-field representation which exploits at best the degrees of freedom arising from the geometrical structure of the well-known four-colors colour coverage map. The overall synthesis is stated as a convex-programming problem wherein the fast achievement of the (unique) globally optimal solution is guaranteed. The introduced tools allow stating the ultimate theoretical radiation performances achievable by any circular-aperture antenna of fixed size and, at the same time, can be exploited as a reference in the synthesis of isophoric direct-radiating arrays. Numerical examples concerning a mission scenario recently proposed by the European Space Agency are provided.
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