Then

Denso QR Bar Code barcode library on .netusing barcode encoding for .net vs 2010 control to generate, create qr code image in .net vs 2010 applications.

(5.1.23) (5.1.24)

Qr Bidimensional Barcode scanner on .netUsing Barcode decoder for .net framework Control to read, scan read, scan image in .net framework applications.

. ~H61) ( kJx~+z~ )

.net Vs 2010 Crystal bar code creator in .netusing visual .net crystal todeploy barcode on asp.net web,windows application

For the first three terms,

Attach barcode for .netuse visual .net barcode drawer toinclude bar code for .net

NT 2)1 ~al(Xd) ( ~~

Control qr codes size with visual c#to deploy qrcode and qr barcode data, size, image with visual c#.net barcode sdk

(5.1.25)

QR Code JIS X 0510 barcode library in .netgenerate, create qr-codes none for .net projects

a1(Xd)

Control qr code iso/iec18004 data with vbto connect quick response code and denso qr bar code data, size, image with vb barcode sdk

= -4 H1 (kXd)T

Render barcode on .netusing .net vs 2010 toprint barcode in asp.net web,windows application

(5.1.26)

Visual Studio .NET pdf 417 drawer on .netgenerate, create pdf417 none with .net projects

5 FAST METHODS FOR ROUGH SURFACE SCATTERING

UPC - 13 barcode library for .netusing barcode integrated for .net vs 2010 crystal control to generate, create gs1 - 13 image in .net vs 2010 crystal applications.

If we retain three terms in the Taylor expansion (NT = 3),

Add ansi/aim code 39 on .netuse visual studio .net crystal ansi/aim code 39 drawer toprint 39 barcode on .net

= '""'" (Z(w) - Z(w)(O)) ~ mn mn

Insert cbc for .netusing .net framework crystal touse royal mail barcode for asp.net web,windows application

(5.1.29)

Control data matrix barcodes data in office excelto access datamatrix and data matrix 2d barcode data, size, image with office excel barcode sdk

1.2 Formulation and Computational Procedure

Barcode Pdf417 drawer for .netgenerate, create barcode pdf417 none in .net projects

(5.1.30) We make use of FFT in the calculation of Ym in (5.1.30). For example, for the second term in (5.1.30), -2(j(x m )) L:~=1 al~~d) f(xn)u n , we calculate in d the following manner: (1) pre-multiply Un by f(x n ) to get f(xn)u n (2) calculate L....J

Attach bar code on wordgenerate, create barcode none with office word projects

= Wn

Assign bar code for visual c#.netusing .net vs 2010 touse barcode with asp.net web,windows application

~ - - 2-w by FFT al(xd) n

Code128b barcode library in vb.netuse .net windows forms crystal code128b integrated torender code-128c in vb.net

=(w)

1d Barcode generation for exceluse excel linear barcode integrated togenerate linear in excel

xd (3) post-multiply by -2f(xm )

Control barcode pdf417 image with excel spreadsheetsgenerate, create pdf417 none for excel spreadsheets projects

In the BMIA/CAG method, the Z is decomposed into a sum represented by the Taylor series expansion. Thus

=(w)

= L....J Zm

rn=O

'""'" =(w)

(5.1.31)

=(w)

The m = 0 term corresponds to that of a flat surface. The form of Z m is such that it consists of terms that are products of a diagonal matrix TTl a translationally invariant matrix Zd, and a diagonal matrix T s : (5.1.32) where T s is a function of the coordinates x' of the scattering source, while T r is a function of the coordinates x of the field.

(B) Iteration Based on Updating the Right-Hand Side

Let X(O) and X(n) represent the zeroth-order solution and the nth-order solution, respectively. They obey the equations

=c =(S)-(n+l) -(n) Z X =c

=(s)-(O)

(5.1.33) (5.1.34)

where

d n ) represents the updated right-hand side with

-(0)

= C - L....J Zm X

-(n)

=C- Z

=(w)-(n)

'""'" =(w)-(n)

(5.1.35)

5 FAST METHODS FOR ROUGH SURFACE SCATTERING

Note that for this case of iteration, only Z(s) is kept in the left-hand side of (5.1.34). A residual R(n) can be defined as follows such that its norm provides the stopping criterion for the iterative procedure:

R(n)

-z X(n) + C = _(Z(s) + Z(Wlyx(n) + C

IIR(n)II/IIClI

(5.1.36)

x 100%. From

where the normalized L-2 norm is defined as (5.1.33)-(5.1.36), it follows that

-(0) R =

=(s)-(O) =(w)-(O) X - Z X

+ C = -z

=(w)-(O) " =(w)-(O) X = - L..t Zm X

(5.1.37)

-(n+l)

= -Z X

=(S)-(n+l)

=(W)-(n+l)

+ -C = -(n+l) C

-(n)

(5.1.38)

Thus the residual vector can be computed readily from the updated righthand sides. In the numerical results illustrated in this section, the stopping criterion of the iterative solution is set at 0.1%.

Computational Complexity

For the TE case the matrix is symmetric. The bandwidth b is usually much smaller than the order of the matrix N. To take full advantage of the banded matrix Z(s), a direct banded matrix solver is used to solve (5.1.33) and (5.1.34). The LU decomposition requires O(b 2 N/2) operations, while the backsubstitution only requires O(2bN) operations. The

=(w)_ Z(w) X

product is

computed by the FFT. Therefore, we can evaluate Z X in r N (log N) + sN operations (where r accounts for the number of FFTs and s accounts for the number of pre- and post-multiplications before the FFT). The computational complexity up to the nth-order solutions are O(nb 2 N)+O(nrN log N +nsN).

(C) Solution Based on Complete Impedance Matrix and Conjugate Gradient Method (CGM)

Another iteration approach is to keep the entire impedance matrix on the left-hand side. Then we apply a conjugate gradient method (CGM) to the matrix equation with the matrix decomposition.

( =(S) Z

+ L..t Zm

~ =(W))

(5.1.39)

1.3 Weak Matrix and Unknown Column Vector

For the CGM version, an initial guess of XeD)

0 is chosen. Let N c be

=(s) =(w)

the number of CGM iterations. By decomposing into Z and Zm and the use of FFT in conjunction with CGM, the approach requires O(Nc(bN + rNlog N + sN)).

Memory Requirements

The memory requirement of the strong matrix Z(s) is O(bN). The coefficients am(xd) in the Taylor expansions are translationally invariant. The storage requirement for Zm , m = 0, 1,2, ... , M, is O((M +1)N). The total memory requirement for the algorithm is O(bN + (M + 1)N). In the simulations, the bandwidth b is an adjustable parameter. In the updated right-hand-side approach there is a minimum bandwidth bmin for which the iteration process works. It requires many more iteration steps to converge at b = bmin than at a larger bandwidth. Therefore, in the simulations, b is chosen to be greater than the bmin so as to reduce the number of iteration steps. For the CGM iterative approach discussed above, the bandwidth can be smaller than the one used in the updated right-hand-side approach. This is because the bandwidth in this case depends on the accuracy of the Taylor series expansion. As a result, this approach requires less computer memory, and therefore it is ideal for very large surface lengths. However, it usually takes more iteration steps to converge. In Section 1.5, only the 2500 wavelength surface examples are performed by applying the CGM iterative approach.

=(w)