International Journal of Antennas and Propagation

Volume 2017, Article ID 2939702, 6 pages

https://doi.org/10.1155/2017/2939702

## A Novel Scheme with FG-FFT for Analysis of Electromagnetic Scattering from Large Objects

^{1}Industrial Center, Nanjing Institute of Technology, Nanjing 211167, China^{2}State Key Laboratory of Millimeter Waves, Nanjing 210096, China^{3}College of Information Engineering, Yancheng Institute of Technology, Yancheng 224051, China^{4}Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University, Hefei 230601, China

Correspondence should be addressed to Jia-Ye Xie; nc.ude.tijn@eixeyaij

Received 27 April 2017; Accepted 17 August 2017; Published 28 September 2017

Academic Editor: Angelo Liseno

Copyright © 2017 Jia-Ye Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The replacement values to the singularity of fitting Green’s function are intensively researched in Fitting Green’s function Fast Fourier Transformation (FG-FFT). As is shown in the research, the accuracy of fitting Green’s function of near element is affected by different replacement values. The experiments show that an appropriate replacement value can improve the accuracy of fitting Green’s function of near element, and it is called the optimal replacement value in this paper. In the case of application of the proposed scheme to FG-FFT, the number of the near correction elements is significantly reduced. Therefore, the optimal replacement scheme can dramatically reduce the memory requirement and computing time in FG-FFT. The experiments show that, compared with traditional random replacement schemes, the optimal replacement scheme can reduce the number of the near correction elements by about 55%60% and greatly improve the computational efficiency of FG-FFT.

#### 1. Introduction

Method of Moments (MoM) [1] is a popular tool for the analysis of electromagnetic scattering and radiation by an object. However, the scope of its application is limited due to the high complexity. A number of methods, usually called fast algorithms, are proposed to reduce the computation complexity of the MoM. The traditional fast algorithms of MoM include three categories: methods based on the addition theorem [2–4], methods based on Fast Fourier Transformation (the FFT-based methods) [5–7], and the methods based on matrix-compression algorithm [8–10]. Due to its high universality, the FFT-based methods are widely used and studied in depth [11, 12].

Because FG-FFT is more concise and accurate in the FFT-based methods, this method is further studied and developed in recently years [13–15]. The memory requirement and computational complexity of the FFT-based methods are and for surface integral equation, respectively. It is well known that the FFT-based methods still have some problems for electrically large targets: the fine grid spacing can reduce the number of the near correction matrix elements but increase the burden of FFT. Instead, the coarse grid spacing can significantly reduce the burden of FFT but dramatically increase the number of the near correction matrix elements. In order to alleviate this contradiction, in this paper, the numerical calculation method of near element is intensively researched.

The vast experimental data is obtained to find out a phenomenon: in near field, when the singularity of fitting Green’s function is replaced with different replacement values, the accuracy of the fitting results shows significant difference. An optimal replacement value can reduce the fitting errors of the near elements. The near element refers to the intersecting element of the expansion box of both the basis function and the testing function under the uniform Cartesian grid.

As is shown in this paper, when the optimal replacement scheme is used in the FG-FFT, the near elements can keep a relatively high fitting accuracy within a certain range. Within this range the near element need not be corrected by MoM and these near elements belong to far matrix, which means that the novel scheme can dramatically reduce the number of the near correction matrix elements. The memory requirement and the filling time of the near matrix as well as the time of iterative computation are also reduced.

#### 2. Formulation

##### 2.1. FFT-Based Methods

Considering an arbitrarily shaped 3D perfect electrical conductor (PEC) object illuminated by an incident plane wave , the electric field integral equation (EFIE) on the surface of the object can be given as follows:where is the equivalent surface electric current, and are the wave impedance and the wave number in the free space, respectively, and denotes Green’s function in the free space, which can be written as

After the application of the MoM, the EFIE can be converted into a matrix equationwhere , , and represent the MoM-Matrix, the current coefficients vector, and the excitation vector, respectively.

In the FFT-based methods, can be split into two parts where is the far matrix and can be rewritten aswhere and are the coefficient transformation matrices; is a triple Toeplitz matrix related to Green’s function. The matrix-vector product of can be sped up by FFT in the FFT-based methods. It should be noted that includes not only the far field elements, but also the near field elements. is called the far matrix because of the approximation for the far field elements and the inaccurate for the near field elements. In order to ensure the accuracy of near field elements, there must be corrections. Hence, is the near correction sparse matrix. The every element of the must be directly calculated and stored, due to which the majority of memory requirement and computational time are consumed for .

##### 2.2. FG-FFT

In order to obtain the projection coefficients, the following overdetermined equation (6) is solved in FG-FFT [7] where is the expansion box of and are a group of sample points. is the grid nodes of . The projection coefficients are to be determined (see Figure 1).