Fast Fourier Transform Based Iterative

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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 ef ficient 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 F ast F ourier Transform (CG-FFT), where acceleration of

the matrix-vector multiplications is achieved using fast Fouriertransforms

(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. I NTRODUCTION

Ef ficient computation of electromagnetic wave scattering from sur-

faces has a wide range of applications in microwave circuits, roughsurface 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 ef ficient 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 themethod 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 June18, 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 RINCEInstitute, School of Electronic Engineering, Dublin City University, Ire-land (e-mail: [email protected]; [email protected]; [email protected]).

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 extendedfrom

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 withvalues 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 canbe ef ficiently com-

puted using the FFT [8].

(8)

(9)

(10)

Hence can be obtained using component-wise division

(11)

0018-926X/$31.00 © 2012 IEEE



IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012 5465

TABLE I NEW ITERATIVE METHOD FOR ELECTROMAGNETIC SCATTERING FROM

PERFECTLY ELECTRICALLY CONDUCTING FLAT SURFACES

To start the new iterative method as given in Table I, the additional

components on the RHS are initialized with zeros, i.e., for

. Next, is ef ficiently calculated using the FFT

as shown in (11). The zero in brackets indicates that this is the zeroth

iteration of the solution. Inside the “for” loop, the vector components

are set to zero and then the revised is

used to update , by using (7), Subsequently, are reset to the

incident field values for and this revised

is then used to calculate , by using (11). The process continues to

form the iterative chain as

depicted in Table I.

B. Complexity Analysis

As outlined in Table I, the proposed method requires 4 FFT, 1 array

multiplication and 1 array division operations for each iteration. In

addition, it requires 3 FFT and 1 array division for the initialization

and finalization. In terms of memory usage, the new approach requires

cells to store the entries of in the

frequency and spatial domains. In short, the computational complexity

of the proposed approach is equivalent to that of the CG-FFT approach

[4], requiring operations per iteration, while the memory

costs are . The complexity of the GMRES-FFT is similar to the

CG-FFT in terms of the number of FFTs needed, but requires other op-

erations which slows it further.

C. Convergence AnalysisIn this section we derive a requirement on the structure of that

ensures that the solution of the proposed approach converges to the

solution of (1). Firstly, we write (3) in its matrix form

(12)

where is the impedance matrix is a symmetric Toeplitz

matrix whose first row is and is the right

hand side vector of (1) respectively. and are the addi-

tional components added to the solution vector and right hand side

vector to enable the use of the FFT such that

and . Because the matrix on the LHS of (12) is a

circulant matrix, we obtain and . This is a system

of equations where we must choose to ensure that

which will force to equal the desired solution from (1). The cor-

rect choice of will thus yield

(13)

The iterative method uses the FFT to solve the more general (12),

the solution of which can be written as:

(14)

where . One entry in the inverse matrix is

unused in the subsequent calculation and is marked as a cross sign.

The proposed iterative approach will converge if

(15)

From Table I, and from (14), it follows that

(16)

where

(17)

(18)

The proposed method uses (11) to ef ficiently solve for (and thus

). At this point the components of are set

to zero and (7) is used to update (and thus ). From (12) we see

that

(19)

is then used to calculate and the iterations continue in thismanner. Using (14)

(20)

After iterations we obtain

(21)



5466 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012

As approaches infinity, then

(22)

The Neumann series converges to provided that

and so, assuming this with

(23)

This analysis shows that the proposed iterative method converges pro-

vided that the spectral radius of is smaller than unity.

(24)

III. R ESULTS

In this section, we evaluate the accuracy and convergence of the pro-

posed method. A comparison of the new method with two Krylov sub-

space based methods is also performed in terms of relative residual

error and processing time. Finally, the field obtained using a single it-

eration of the new approach is compared to the direct matrix inversion

(DMI) to demonstrate the performance of the new approach.

A. Investigation of Convergence Versus Problem Size

We have shown that the new approach converges if the spectral ra-

dius of is smaller than unity, . In this section,we investigate the relationship between the spectral radius and the size

of the scatterer, as well as discretisation size.

The spectral radius of was computed for different discreti-

sation sizes:

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Fast Fourier Transform Based Iterative