I feel sorry that the manuscript is written in a hurry and most content of the synthesis of the linear arrays is similar to that discussed in C.A. Balanis book as pointed out by reviewers. However, I still post this manuscript on this server as I indeed derived it from the very beginning and I wish some part of them may be of usage for others, especially the 2-D cases and some of my considerations on this issue. The direct synthesis results are very good initial values for further optimization with a method in which the properties of the entire functions and the number of degrees of the freedom of far fields are made full use of.

In this paper, the current distribution is expanded with Fourier series. Analysis shows that only a limited number of the harmonic components of the current can generate propagation fields and contribute to the far-field pattern. Most importantly, it is found that this part of the current can be directly obtained from the far-field without utilizing any optimization algorithm. The degree of freedom of the far-field can be exactly counted for a linear source or a rectangular planar source. Numerical examples verify that the method is very efficient for synthesizing large antenna arrays.