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An attempt is made to deduce rules for designing circular arrays which produce desired directional patterns. The paper considers circular arrays consisting of equally spaced radiators distributed on the whole circumference or on a coherent part of it. First the general relation between the desired far-field amplitude distribution and the complex input amplitudes of the individual radiators is deduced. The results can be written as a linear system of equations. Its order depends only on the ratio of diameter to wavelength. This form permits a clear and simple numerical treatment of the task by means of a digital computer. Additionally, practical hints are given for the realization of the array feed function. There by it is found that a problem of the directional pattern synthesis can be reduced to a synthesis of linear networks. In a second step the actual optimizing problem is dealt with. It is intended to find, for a given ratio of circular array diameter to wavelength, the design for which the maximum array gain is obtainable with simultaneous maximum attenuation of side-lobes. The result shows that there exists a theoretical limit for the obtainable attenuation of the dominating side-lobe, a limitation which is independent of the ratio of circular array diameter to wavelength. In the investigated case this boundary is at Â¿26-43 dB. The array gain increases in a nearly linear mannerwith the indicated ratio and there seems to be no theoretical limit to any desired increase of the gain. All considerations are valid for transmission and for reception. The correctness of the treatment has been confirmed by a laboratory experiment.