For array beampattern synthesis, it is possible to simplify the model and reduce the computational load by formulating it to be a Quadratic Programming (QP) problem. The QP method is conceptually simple and imposes no restriction on array geometry. In the QP method, a key component is the template function which describes the desired beampattern as a deterministic function of direction. In this paper, the template functions in the form of Hypergeometric Function corresponding to Legendre arrays and Dolph-Chebyshev arrays, namely Legendre Hypergeometric Function (LHF) and Dolph-Chebyshev Hypergeometric Function (DCHF), are derived and the synthesis procedures are also presented. The simulation results show that the DCHF and LHF work in the QP method and provide the exactly synthesized beampattern. Moreover, another synthesis method using a Gegenbauer polynomial to synthesize the beampattern of a Uniform Linear Array, is proposed. This method gives rise to the Gegenbauer arrays. Gegenbauer arrays are very generalized and Legendre arrays and Dolph-Chebyshev arrays are considered as its special cases. Using the Gegenbauer array synthesis method, one is able to further adjust the beam efficiency and the directivity when the Side-Lobe Level and the element number are specified. As well as Dolph-Chebyshev arrays and Legendre arrays, its template function for the QP method, Gegenbauer Hypergeometric Function, is also derived.
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