Synthesis of unequally-spaced arrays using the fractional Fourier series

This paper introduces an algebraic method to synthesize the radiation pattern of unequally-spaced arrays. In this method, first, the characteristic matrix, containing the sampled data of the desired pattern, is established. Then, the fractional Fourier series is calculated using the distinct eigenvalues and eigenvectors of the characteristic matrix. In the following, the location of the array elements is estimated using the eigenvalues by considering the mutual coupling effect. The magnitude and phase of the excitation currents are computed using the least square method. Also, it is proved that the number of elements can be reduced using the eigenvalue decomposition and differential spectrum method. Several important patterns are investigated to verify the performance of the proposed method, and then a comprehensive discussion is expressed for all cases. It is shown that using the proposed method, a greater average spacing is achieved which allows bettering mitigate the phenomenon of mutual coupling. Furthermore, the various results show that the proposed method offers a better approximation of the desired array factor especially for beam-shaped patterns. This method also has a noble ability to reduce the number of array elements compared to a generic reference array, while still retaining a good ability to approximate the desired pattern fairly faithfully.