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A set of radiation pattern functions, suitable for synthesis of radiation patterns from circular aperture horn antennas, is obtained by assuming an aperture distribution consisting of the fields of cylindrical waveguide modes. A technique is presented for using a linear combination of the radiation pattern functions to approximate a desired radiation pattern. Linear combinations of the radiation pattern functions resulting in maximum secondary gain, when used to illuminate a paraboloidal antenna, are obtained empirically. Using spherical wave theory, maximum performance theoretically obtainable from an antenna is derived as a function of the aperture size of the feed system; the feed efficiency resulting from these theoretical limits on performance is compared to the feed efficiency of patterns obtainable from circular aperture horn antennas, and to experimental results of attempts to realize optimum circular aperture horn patterns.