REFERENCES

1. R. Tayrani, A monolithic X-band class-E power amplifier, IEEE GaAsIC Symp Tech Dig (2001), 205–208.

2. R. Wang, M. Cole, L.D. Hou, P. Chu, C.D. Chang, T.A. Midford, andT. Cisco, A 55% efficiency 5 W PHEMT X-band MMIC high poweramplifier, IEEE GaAs IC Symp Tech Dig (1996), 111–114.

3. A. Bessemoulin, R. Quay, S. Ramberger, H. Massler, and M. Schlecht-weg, A 4-W X-band compact coplanar high-power amplifier MMICwith 18-dB gain and 25% PAE, IEEE J Solid State Circuits 38 (2003),1433–1437.

4. S.L.G. Chu, A. Platzker, M. Borkowski, R. Mallavarpu, M. Snow, A.Bowlby, D. Teeter, T. Kazior, and K. Alavi, A 7.4 to 8.4 GHz highefficiency PHEMT three-stage power amplifier, IEEE MTT-S Dig(2000), 947–950.

5. M. Cardullo, C. Page, D. Teeter, and A. Platzker, High efficiencyX-Ku band MMIC power amplifiers, IEEE MTT-S Dig (1996), 145–148.

6. B. Kraemer, R. Basset, P. Chye, D. Day, and J. Wei, Power PHEMTmodule delivers 12 watts, 40% P.A.E. over the 8.5 to 10.5 GHz band,IEEE MTT-S Dig (1994), 683–686.

7. A.P. de Hek, P.A.H. Hunneman, M. Demmler, and A. Hulsmann, Acompact broadband high efficient X-band 9-watt PHEMT MMIChigh-power amplifier for phased array radar applications, GaAs99Conference (1999), 276–280.

8. W. Bosch, J.G.E. Mayock, M.F. O’Keefe, and J. McMonagle, Lowcost X-band power amplifier MMIC fabricated on a 0.25 �m GaAspHEMT process, IEEE Radar Conf (2005), 22–26.

9. United monolithic semiconductor (UMS), X-band GaInP HBT 10 Whigh power amplifier including on-chip bias control circuit, IEEEMTT-S Dig (2003), 855–858.

10. C.K. Chu, H.K. Huang, H.Z. Liu, R.J. Chiu, C.H. Lin, C.C. Wang,M.P. Houng, Y.H. Wang, C.C. Hsu, W. Wu, C.L. Wu, and C.S. Chang,A fully matched high linearity 2W PHEMT MMIC power amplifier for3.5 GHz applications, IEEE Microwave Wireless Components Lett 15(2005), 667–669.

11. S.C. Cripps, A theory for the prediction of GaAs FET load-pull powercontours, IEEE MTT-S Dig (1983), 221–223.

12. S.C. Cripps, RF power amplifiers for wireless communications, ArtechHouse, Boston.

13. H.Z. Liu, C.C. Wang, Y.H. Wang, J.W. Huang, C.H. Chang, W. Wu,C.L. Wu, and C.S. Chang, A four-stage Ku-band 1 watt PHEMTMMIC power amplifier, IEEE GaAs IC Symp Tech Dig (2002),33–36.

14. C.K. Chu, H.K. Huang, H.Z. Liu, R.J. Chiu, C. H. Lin, C.C. Wang,Y.H. Wang, C.C. Hsu, W. Wu, C.L. Wu, and C.S. Chang, Fullymatched 8 W X-band PHEMT MMIC high power amplifier, IEEEGaAs IC Symp Tech Dig (2004), 137–140.

© 2006 Wiley Periodicals, Inc.

A CRANK–NICHOLSON-BASEDUNCONDITIONALLY STABLE TIME-DOMAIN ALGORITHM FOR 2D AND 3DPROBLEMS

Xin Xie,1 George Pan,2 and Stephen Hall31 Software Engineer, Cadence Design Systems, Inc., 2655 SeelyAvenue, San Jose, CA 951342 Department of Electrical Engineering, Arizona State University,Tempe, AZ 85287-72063 Sr. Staff Engineer, Intel Corporation, 2111 N.E. 25th Avenue,Hillsboro, OR 97124-5961

Received 16 June 2006

ABSTRACT: It has been shown that both ADI-FDTD and CN-FDTDare unconditionally stable. While the ADI is a second-order approxima-

tion, CN is only in the first order. However, analytical expressions re-veal that the CN-FDTD has much smaller truncation errors and is moreaccurate than the ADI-FDTD. Nonetheless, it is more difficult to imple-ment the CN than the ADI, especially for 3D problems. In this paper,we present an unconditionally stable time-domain method, CNRG-TD,which is based upon the Crank–Nicholson scheme and implemented withthe Ritz–Galerkin procedure. We provide a physically meaningful stabil-ity proof, without resorting to tedious symbolic derivations. Numericalexamples of the new method demonstrate high precision and high effi-ciency. In a 2D capacitance problem, we have enlarged the time step,�t, 400 times of the CFL limit, yet preserved good accuracy. In the 3Dantenna case, we use the time step, �t, 7.6 times larger that that of theADI-FDTD i.e., more than 38 times of the CFL limit, with excellentagreement of the benchmark solution. © 2006 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 261–265, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22101

Key words: Maxwell’s equations; FDTD; Courant–Friedrich–Levy con-dition; Rayleigh–Ritz procedure

1. INTRODUCTION

The simplicity of the FDTD method in mathematics and its ver-satility in handling boundaries and material inhomogeneity haveproven to be its great advantages. However, the standard FDTD issubject to the Courant–Friedrich–Levy (CFL) condition [1]. Wepresent an unconditionally stable time-domain method, CNRG-TD, which is based upon the Crank–Nicholson scheme and im-plemented with the Ritz–Galerkin procedure. Analytical expres-sions show that the CN algorithm has much smaller truncationerrors and is more accurate than the ADI-FDTD [2]. Numericalexamples of the new method demonstrate high precision and highefficiency. In a 2D capacitance problem, we have enlarged the timestep, �t 400 times of the CFL limit, yet preserved good accuracy.In the 3D antenna case, we use the time step, �t, 7.6 times largerthan that of the ADI-FDTD, i.e., more than 38 times of the CFLlimit, with excellent agreement of the benchmark solution.

2. CN-FDTD UPDATING EQUATIONS

For ease of reference, the Crank–Nicholson scheme [3] is brieflyderived here. The Maxwell equations in a lossless medium

�

�t� � � 0

1

�� �

�1

�� � 0 �� � �0 �

1

�I

0 0� J�t�

: � RT� � LJ�t�, ��0� � �0, (1)

where matrix RT represents the two Maxwell’s curl operators andis called the Maxwellian matrix, and the composite field vector

� � �Ex, Ey, Ez, Hx, Hy, Hz�T.

The field vector �t��t and �t may be expressed in the Taylorseries below

�t��t � �t � �t�

�t�t �

�t2

2

�2

�t2�t, (2)

�t � �t��t � �t�

�t�t��t �

�t2

2

�2

�t2�t��t, (3)

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 2, February 2007 261

leaving a truncation error of order �t3. Subtracting Eq. (3) fromEq. (2) and substituting for Eq. (1), and solving for �t��t, thefollowing equation is obtained

�t��t � � I ��t

2RT �

�t2

4RT

2��1� I ��t

2RT �

�t2

4� RT

2��t

� � I ��t

2RT �

�t2

4RT

2��1�� I ��t

2RT��t

2LJt

� �I ��t

2RT��t

2LJt��t �

�t2

4L��

�tJt �

�

176tJt��t��. (4)

3. CN-FDTD STABILITY

In practice, one may truncate Eq. (4) to the first-order approxima-tion. The truncated Eq. (4) is numerically stable regardless of thevalue of �t because the magnitude of each eigenvalue of the matrix

P � � I ��t

2RT��1� I �

�t

2RT� (5)

is no greater than one. To prove the statement above, we first claimthat for 3D cases, the 6 � 6 Maxwellian matrix RT can bediagonalized into

RT � �1, (6)

where T is the transformation matrix, whose columns are the sixlinearly independent eigenvectors of RT and T is a diagonal matrixof the three distinct eigenvalues of RT, each repeats twice. Becauseof repeated eigenvalues, one would consider the Jordan canonicalform of RT and naturally question the validity of Eq. (6). Thediagonalization has been verified numerically. More rigorously, itholds owing to the fact that under proper normalization, RT isskew-symmetric as seen later in this section in Eq. (10). Thespectral theorem in linear algebra states that any real or complexskew-symmetric matrix can be diagonalized [4].

Substituting Eq. (6) into Eq. (5), and noticing the independenceof the eigenvectors, we arrive at

P � � I ��t

2��1� I �

�t

2��1, (7)

which gives the eigenvalues of the matrix P as

pi �

1 ��t

2�i

1 ��t

2�i

, i � 1, 2,· · ·, n, (8)

where �i, i � 1, 2, . . . , n are the eigenvalues of RT, n � 6 for 3Dcases and n � 3 for 2D (TE(z)) problems. From Eq. (8), themagnitude of each eigenvalue of P is no greater than one if Re(�i) 0, i � 1, 2, . . . , n that is always true if Maxwell’s equationsrepresent a stable system. More specifically, from Eq. (8) themagnitude of each eigenvalue of P is exactly equal to one if �i ispurely imaginary or zero. This is true when the Maxwellian matrixRT can be scaled to a skew-symmetric (i.e., antisymmetric) [5]matrix below. Let

X�t� � ��E�t�, Y�t� � ��H�t�. (9)

In terms of the fields X(t) and Y(t), Maxwell’s curl Eq. (1) read

�

�t�X�t�Y�t�� ��

1

��� �

Y�t�

��1

��� �

X�t�

��� � � 0

J�t�

���

� � 0 �1

��� �

1

��1

��� �

1

��0 � �X�t�

Y�t�� � � 0J�t�

���. (10)

Since a scaling or normalization process does not change physics,the stability of the scaled quantities X and Y guarantees stability ofthe original field quantities E and H.

From matrix theory [6], when a matrix is real skew-symmetric(or complex skew-Hermitian to be more general), its eigenvaluesare all pure imaginary. Here we quote the corresponding eigen-values from an example of a 3D rectangular hollow cavity below.

�i � �� 1.410543510440750e � 009i1.410543510440750e � 009i

� 1.410543510440750e � 009i1.410543510440750e � 009i

00

� (11)

From Eq. (8), we obtain the eigenvalues of P as

pi � �0.96377299807382 � 0.26672384254844i0.96377299807382 � 0.26672384254844i0.96377299807382 � 0.26672384254844i0.96377299807382 � 0.26672384254844i

11

� (12)

The results agree with the values computed by Matlab, up to everydecimal digit. We have rigorously proven that every eigenvalue ofthe CN matrix P equals one in magnitude, which means that theCN iteration procedure is on the border between stable and unsta-ble.

4. RITZ–GALERKIN PROCEDURE OF THE CNFORMULATION

From the truncation error analysis, CN-FDTD is superior to theADI-FDTD. However, in the CN scheme, �n�1 is approximatedby �n in the crossing operator in order to obtain a two-substepseparated system of equations, whose system matrix is tridiagonal,similar to that in the ADI-FDTD [7]. But the approximation cannotbe accurate when �t is large. Here, we implement the CN schemeby the FEM formulation. Considering a general lossy mediumhere, consequently the matrix RT is replaced by RT�, which is

R�T � � �1

�

1

�� �

�1

�� � 0 � (13)

where is electric conductivity. Upon elimination of the magneticfields, we arrive at the curl–curl equation

� �

��t�2 �

2�t�

1

2� � � 1

�� � �En�1 � � 2�

��t�2 �1

2�

� �1

�� � �En � � �

��t�2 �

2�tEn�1 (14)

262 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 2, February 2007 DOI 10.1002/mop

for the electric field E. For the boundary condition, we assume thefirst-order absorbing boundary condition [8], given by

n � � 1

�� � E � Yc

�

�t n � n � E� � 0 r � S, (15)

where Yc � ��/�, which is the intrinsic admittance, and nrepresents the outward unit vector normal to the boundary S. Thecorresponding weak-form solution of the boundary-value problemdescribed above can be obtained as

V

� 1

��t�2�Ni�r� � En�1�r� �1

2�tNi�r� � En�1�r� �

1

2���

� Ni�r�� � � � En�1�r��� dV �s

�1

2Yc n � Ni�r�� �

�

�t n

� En�1�r��� dS �V

� 2

��t�2�Ni�r� � En�r� �1

2� � � Ni�r��.

� � En�r��� dV �V

� 1

��t�2�Ni�r� � En�1�r�

�1

2�tNi�r� � En�1�r�� dV �

s

�1

2Yc n � Ni�r��

��

�t n � En�1�r��� dS, (16)

where the Ni(r) denote vector testing functions. Both zeroth- andhigher-order vector basis functions can be used here. Expandingthe electric field as

E�r� � �j�1

N

uj�t�Nj�r� (17)

with N denoting the total number of unknowns, and substitutingEq. (17) into Eq. (16), we obtain

� 1

��t�2 T� �1

2�t T� �

1

2 S�� �u�n�1 � � 2

��t�2 T� �1

2 S�� �u�n

� � 1

��t�2 T� �1

2�t�T�� �u�n�1. (18)

In the equation above, {u} � [u1, u2, . . . , uN]T; [T] � [R] � [Q];[T[, [R], [Q], and [S] are square matrices given as following

Tij � V

�Ni�r� � Nj�r� dV, (19)

Rij � V

Ni�r� � Nj�r� dV, (20)

Qij � S

Yc n � Ni�r�� � n � Nj�r�� dS, (21)

Sij � V

1

� � � Ni�r�� � � � Nj�r�� dV. (22)

Clearly, if initial values {u}0 and {u}1 are given, we can solve Eq.(18) for {u}2, then {u}3 and so on. A convenient choice is to use{u}0 � 0 and {u}1 with the source distribution. This approach isan implicit scheme because it requires solving a matrix equationcontaining [S] in each time step. It is unconditionally stable,regardless of the choice of time step.

5. NUMERICAL SIMULATIONS

Example 1: A simple parallel-plate capacitor is excited by alow-frequency ideal voltage source as presented in Ref. 2. Weconsider the 2D TEz case, demonstrated in Figure 1, to permitdirect comparisons of CNRG-TD and ADI-FDTD results in acomputationally efficient manner. The configuration consists oftwo 2-m wide parallel conducting plates in free space separated bya distance of 0.2 m. It is fed by an ideal voltage source at thecenter. The source waveform is a 750 kHz (the period T � 1.33�s) raised cosine, which is held constant after reaching its maxi-mum value of 1 V,

V�t� � 1 � cos�2�ft� if t T

2;

1 if t �T

2.

The wavelength in the free space at 750 kHz is 400 m. Thus, thisproblem was designed with the cell size (�x � �y � 0.2 m) tocapture the subwavelength geometrical features of this structure.

The Yee FDTD solution is computed with �x � �y � 0.2 m,corresponding to a grid sampling rate of 2000 points per wave-length for the given voltage source. The Courant limit on the timestep is 0.471 ns. Different time step �t are chosen to resolve thetransient variation of the source excitation of a duration of T/2 �0.667 �s (not shown in the text). The Yee FDTD solution (�t �0.377 ns, 0.8 time of the courant limit) serves as our benchmark.The solid line in Figure 2 shows the computed Ey fields along ahorizontal cut through the parallel plate structure in the steady stateby the Yee FDTD solution. In the region between two conductingplates (4 m x 6 m), the field amplitude is constant. Outside theplates the field amplitude decays very rapidly to zero.

Figure 1 A 2D TEz model of a parallel plate capacitor excited by anideal voltage source. The grid is terminated with perfect magnetic con-ducting boundaries [2]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 2, February 2007 263

For the CNRG-TD scheme, we have also used �x � �y �0.2 m. However, because of the unconditional stability of thisscheme, we can increase the time step much larger than Courantlimit. Figure 2 compares the results from several differentCNRG-TD simulations with the ADI-FDTD solution. The resultsfrom CNRG-TD simulation with time steps �t 40–400 timesgreater than that CFL limit are in excellent agreement with ourbenchmark data, while the data in Ref. 2 showed only 270 times ofthe Courant limit. The computation time is reduced from 252.19 sof the benchmark to 9.647 s for the CNRG-TD. Similar results areobtained with the ADI-FDTD simulation only when the time stepis the same as that used in the Yee FDTD simulation. We observethat the accuracy of the ADI-FDTD scheme degrades quickly withmodest increments in the time step beyond the Courant limit.

Example 2: As a 3D problem, we choose an example in Ref. 9.Figure 3 shows the geometry of a �/4 monopole antenna mountednear a thin dielectric wall. The 160 mm long monopole antenna ismounted near the 6 mm thick, 640 mm wide, and 176 mm highdielectric wall on an infinite ground plane. The distance from theantenna to the wall surface is 124 mm. Observation point A is at a124 mm distance from the opposite side of dielectric wall. Thefirst-order absorbing boundary condition is set on the outer sur-faces, except the bottom ground plane. The PEC boundary condi-tions are set for the antenna and the ground plane. The spatialdiscretization employs a nonuniform grid. Because of the spacelimit, the detail is omitted. A continuous sinusoidal wave of the Ez

component at frequency 460 MHz is excited at the bottom of theantenna. The Ez component at observation points A is our output,which is normalized by the value of the excitation amplitude. Theelectric characteristics of the dielectric wall are set as r � 4.0, �r

� 1.0, and � 0.0 S/m. The results from CNRG-TD and ADI-FDTD are shown in Figure 4. The time step for CNRG-TD is �tCN

� 250 ps, while for ADI-FDTD, the time step is �tADI � 33.07 ps,that is 10 times as large as the Yee FDTD, 3.307 ps. Here theminimum size of cells is 2 � 8 � 80 mm3 and the CFL conditionis �t 6.4657 ps. The CNRG results are in excellent agreementwith the benchmark data, and also agree well with the results inRef. 9. In our observation, the l2 error will increase dramaticallywhen the time step in ADI-FDTD is larger than 10 times ofCourant limit, as shown in Table 1. Table 1 tabulates the l2 errorusing the following expression

Error��2� � 100 ��Ez � Ez�2

�Ez�2, � f�2 � ��i�1

N �fi�2

N. (23)

where Ez is the result from CNRG-TD with the time step 10 ps asour benchmark, Ez is the result from ADI-FDTD or CNRG-TDwith the different time step.

6. CONCLUSION

In this paper we proved the stability of the Crank–Nicholsonalgorithm without resorting to lengthy and tedious symbolic der-ivations. In fact, all eigenvalues of the CN iteration matrix are incomplex conjugate pairs and every eigenvalue is exactly equal toone in magnitude. This is due to the fact that the Maxwellianmatrix RT (under proper normalization) is skew-symmetric and the

Figure 2 Electric fields calculated by the CNRG-TD scheme, YeeFDTD and ADI-FDTD scheme, with different time steps normalized by theCFL limit

Figure 3 Geometry of a monopole antenna near a dielectric wall

Figure 4 Normalized electric field at point A behind wall of r � 4.0, �r

� 1.0, and � 0.0 S/m

TABLE 1 Error of Ez in ADI-FDTD and CNRG-TD withDifferent Time Steps

Time Step (ps) Times of CFL Error (%)

CNRG-TD 10 1.55 N/A (benchmark)CNRG-TD 100 15.47 0.43CNRG-TD 250 38.67 1.21ADI-FDTD 33.07 5.11 1.76ADI-FDTD 82.68 12.79 5.43

264 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 2, February 2007 DOI 10.1002/mop

CN matrix is related to RT by the Cayley transformation of Eq. (7).We have also developed the CNRG-TD algorithm which is free ofthe CFL condition. The new method utilizes the Crank–Nicholsonformulation to achieve the unconditional stability with much lowertruncation errors than the ADI-FDTD possesses. The new methodis versatile and easy to implement in combination with the tradi-tional FDTD. Numerical examples of a 3D case strongly supportthe novel approach in terms of accuracy, CPU time, and memoryconsumption.

ACKNOWLEDGMENTS

The authors thank H. Heck of Intel Co., S. Ogurtsov, and Z. Huangof Arizona State University for helpful discussions. This work issponsored in part by a Grant, DWJ-0084 from the Intel Co.,Hillsboro, OR 97124.

REFERENCES

1. A. Taflove, Computational electrodynamics: The finite-difference time-domain method, 2nd ed., Artech House, Boston, MA, 2000.

2. S. Gonzalez-Garcia, T.W. Lee, and S.C. Hagness, On the accuracy ofthe AD-FDTD method, IEEE Antennas Wireless propag Lett 1 (2002),31–34.

3. E.J. Davison, A high-order Crank–Nicholson technique for solvingdifferential equations, Comput J 10 (1967), 195–197.

4. H. Eves, Elementary matrix theory, Dover, New York, 1980.5. H. De Raedt, K. Michielsen, J.S. Kole, and M.T. Figge, One-step

finite-difference time-domain algorithm to solve the Maxwell equations,Phys Rev E 67 (2003), 056706.

6. J. Ortaga, Matrix theory, 2nd ed., Plenum, New York, 1987, p. 34.7. G. Sun and C.W. Trueman, Unconditionally-stable FDTD method based

on Crank–Nicolson scheme for solving three-dimensional Maxwellequations, IEEE Electron Lett 40 (2004), 122–125.

8. G.W. Pan, J. Tan, and B. K. Gilbert, Full wave edge element basedanalysis of 3D metal-dielectric structures for high clock rate digital andmicrowave applications, IEE Proc Microwave Antennas Propag 147(2000), 391–397.

9. T. Namiki, 3D ADI-FDTD method-unconditionally stable time-domainalgorithm for solving full vector Maxwell’s equations, IEEE TransMicrowave Theory Tech, 48 (2000), 1743–1748.

© 2006 Wiley Periodicals, Inc.

INTEGRAL EQUATION-BASEDOVERLAPPED DOMAINDECOMPOSITION METHOD FOR THEANALYSIS OF ELECTROMAGNETICSCATTERING OF 3D CONDUCTINGOBJECTS

Wei-Dong Li, Wei Hong, and Hou-Xing ZhouState Key Laboratory of Millimeter Waves, Department of RadioEngineering, Southeast University, Nanjing 210096, People’sRepublic of China

Received 24 June 2006

ABSTRACT: An integral equation-based overlapped domain decompo-sition method (IE-ODDM) is presented for the analysis of three-dimen-sional (3D) electromagnetic scattering problems. Compared with theforward and backward buffer region iterative method proposed by Bren-nan et al., the convergence of the IE-ODDM is faster due to the edge-effect of the current in each subdomain being effectively depressed.Moreover, the iterative sequence among the subdomains of IE-ODDM ismore flexible, which means that asynchronous iteration and even paral-lel computing are doable. Simulated RCS results of several convex and

concave 3D conducting objects verify the speedup and validity of thismethod. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49:265–274, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22110

Key words: domain decomposition method; iterative method; electro-magnetic scattering; three-dimensional objects; RCS

1. INTRODUCTION

The electric/magnetic field integral equations (IEs) provide a fullwave formulation to the problems of electromagnetic (EM) scat-tering. It is known that a large and dense linear system will beresulted in the process of solving the IEs based on the method ofmoments (MoM), and that the computational complexity of O�N3�and the memory requirement of O�N2� will be the bottleneck forelectrically large EM scattering problems if we directly solve sucha huge matrix equation. Hence, reducing the unknowns is a goodway to improve the efficiency of MoM and to deal with muchlarger objects.

Besides the MoM, some physically inspired iterative methodswere also applied to electrically large EM scattering problems.Among them, a current marching algorithm for 2D scatteringproblems was given in Ref. 1, the method of ordered interactionsin Ref. 2, the forward/backward method in Refs. 3 and 4, and thegeneral forward–backward method in Ref. 5. A 3D version for

Figure 1 Illustration of the FBBR method and IE-ODDM: (a) originaldomain decomposition, (b) forward iteration of FBBR in Ref. 7, (c)backward iteration of FBBR in Ref. 7, (d) overlapped domain decompo-sition method (ODDM) in this article

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 2, February 2007 265