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In this paper the spherical wave expansion technique is used to analyze the general properties of paraboloidal antenna feed systems. As a result, fundamental aperture efficiency and noise temperature limitations are established as quantitative functions of antenna wavelength size. A boundary-value solution is found for synthesis of ideal subreflector shapes in a Cassegrain-type feed system. The resulting surface is found to reduce the classical hyperboloid in the limit of zero wavelength. Applications of this synthesis technique to high performance feed systems and subreflector matching are discussed. Finally, an interesting quantitative cross-check between vector spherical wave and vector diffraction theories is obtained.