5464 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012 Fast Fourier Transform Based Iterative Method for Electromagnetic Scattering From 1D Flat Surfaces Dung Trinh-Xuan, Patrick Bradley, and Conor Brennan Abstract—An efficient iterative method is proposed for computing the electromagnetic fields scattered from a one dimensional (1D) flat surface. The new iterative method is based on a similar implementation to the Con- jugate Gradient Fast Fourier Transform (CG-FFT), where acceleration of the matrix-vector multiplications is achieved using fast Fourier transforms (FFT). However, the iterative method proposed is not based on Krylov sub- space expansions and is shown to converge faster than GMRES-FFT and CGNE-FFT while maintaining the computational complexity and memory usage of those methods. Analysis is presented deriving an explicit conver- gence criterion. Index Terms—Electric field integral equation, fast Fourier transform (FFT), method of moments (MoM), wave scattering. I. INTRODUCTION Efficient computation of electromagnetic wave scattering from sur- faces has a wide range of applications in microwave circuits, rough surface scattering and antenna applications. Typically, the relevant in- tegral equation (IE) is discretized using the method of moments (MoM) resulting in a system of dense linear equations. The efficient numer- ical solution of these equations is a key topic in computational elec- tromagnetics. Different techniques have been developed for the solu- tion of such systems of equations including iterative methods: Forward backward method (FBM) [1], GMRES [2], [3] etc. In addition accel- eration methods such as the fast multipole method (FMM) have been developed to expedite each iteration step. For problems where the un- knowns are arranged on a regular grid methods such as the CG-FFT which exploit the cyclical nature of the basis function interactions can significantly reduce the computational cost of matrix-vector multipli- cations [4], [5]. Moreover, the use of the pre-corrected FFT [6], [7] allows the CG-FFT to be applied to scattering problems involving ar- bitrary surface shape. However, the performance of CG-FFT schemes can suffer from slow convergence or stagnation. In this paper, we pro- pose a new iterative approach which is shown to converge more rapidly than CG-FFT while requiring the same memory usage. II. FORMULATION The work presented in this paper examines scattering from a one di- mensional flat perfectly conducting surface, although the extension to dielectric surfaces is possible. A time dependence of is assumed and suppressed in what follows. The scatterer is illuminated by an in- cident wave and the scattered electric field can be formulated using the electric field integral equation [8]. This can be solved using the method of moments (MoM) with basis and testing functions, re- sulting in the following linear system (1) Manuscript received May 18, 2011; revised January 27, 2012; accepted June 18, 2012. Date of publication July 13, 2012; date of current version October 26, 2012. The authors are with the RF modelling and simulation group, The RINCE Institute, School of Electronic Engineering, Dublin City University, Ire- land (e-mail: xuan.trinh2@mail.dcu.ie; bradleyp@eeng.dcu.ie; brennanc@ eeng.dcu.ie). Digital Object Identifier 10.1109/TAP.2012.2208609 In what follows we assume the use of pulse basis functions and Dirac- delta testing functions (point matching). For a 1D flat surface is a symmetric Toeplitz matrix [8] and the linear system can be written as . . . . . . . . . . . . . . . . . . . . . (2) where . A. Proposed Method In order to apply the FFT to the impedance matrix, one must first embed within a circulant matrix. To achieve this, (2) is extended from a system of equations into a system of equations by appending further unknowns to . As a result, (1) is embedded into a system of equations which has the form of circulant convolution: (3) where (4) (5) (6) The original unknowns have now been embedded inside a system of linear equations. In general the values obtained by solving (3) will not equal those obtained by solving (2). They will only match if one chooses to extend the right hand side vector with values that force to equal zero. This is achieved using the iterative technique outlined later in this section. The iterative process involves sequentially updating and and at each step forcing to be zero for . The advantage of expanding the linear system in this fashion is to facilitate the use of the FFT to speed up the matrix-vector multiplication as in the CG-FFT [4], [8]. As (3) is a circulant discrete convolution of length , the discrete convolution theorem states that it is equivalent to (7) where the symbol denotes component by component multiplication of two vectors and are the Discrete Fourier Transforms (DFT) for the sequence of length of . This can be efficiently com- puted using the FFT [8]. (8) (9) (10) Hence can be obtained using component-wise division (11) 0018-926X/$31.00 © 2012 IEEE