Direct Synthesis of Far-Field Patterns

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 useage for others, especially the 2-D cases and some of my considerations on this issue. I have searched journal papers but not books before I began to write this issue. I should have to.     

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.