- Email: [email protected]
Indoor field strength prediction based on FMM
THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volume 13, Issue 3, September 2006
WANG Zheng-bin, ZHANG Ye-rong
Indoor field strength prediction based on FMM CLC number T N O l 1.92
Abstract In this article, the fast multipole method 0 is applied to a n w the field shength of an ultra-wideband (UWB) signal. Small coverage of UWB communication systems and high efficiency of the algorithm make it possible to calculate the amplitude of elechic field accwdtely. A homogeneous dielectric body and a multilayered dielau-ic object are studied. The computational results obtained by applying the Fh4M agree well with those obtained by applying the method of moments (MOM). Keywords indoor field strength prediction, fast multipole method FMM, multilevel fast multipole algorithm (MLFMA)
I
Article ID 1005-8885(2006)03-0020-04
Document A
Introduction
With the popularization of personal communication service (PCS), indoor radio propagation has attracted more attention, especially for ultra-wideband (UWB) signals. Ultra-wideband communication systems have very high data rates (at least 100 Mbps in ten meters). Hence they are suitable for application in wireless networks of houses and offices. Indoor field strength prediction is very important for the design of piconet. Due to the ultra-wideband, UWB signals experience different propagation impairments than the traditional “narrowband” signals, the impairment being attributed to smaller coverage and more complicated environment. The field strength varies rapidly according to the material of obstruction and the layout of furniture I11. In the domain of narrowband and wideband communications, many researchers have studied the field strength prediction, and have put forth several models since the 1950s. In general, they can be divided into two types. One is by stochastic analysis, such as the Stochastic Model, the Empiric Model, and the Regression Model. The other is the deterministic style, such as the ray-tracing method and the physical optics (PO) 12, 31. Undoubtedly, a deterministic model can provide the most Received date: 2005-1 1-08 WANG Zheng-bin i 1 7 1 College of Mathemlics and Physics, Nanjing University of Posts and Telecommunications, Nanjing 2 10003, China b a i l : [email protected] ZHANG Ye-rong College of Communication and Inlormation Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
accurate field distribution. One deterministic model often used is the ray-tracing method. Compared with the method of moments (MOM), it computes faster. However, the calculated results will have serious errors when there are electrically small obstacles in the room. Therefore, some researchers have combined the ray-tracing method with other low-frequency methods, such as finite-element method(FEM), finite-difference time-domain(FDTD), and the MOM. These hybrid methods improve the accuracy of the solution and also are applicable to large and arbitrarily shaped structures. However, the hybrid methods have the same drawback, in that they are almost case sensitive [3].The regions containing electrically small features, which will be analyzed by a low-frequency method, must be identified and separated. This preprocess is very difficult to realize automatically. In this article, another deterministic method, that is, solving the volume integral equation (VIE)by the fast multipole method (FMM) is presented. Initially, taking into account the small coverage of UWB signals, the low-frequency method, namely, the MOM, which belongs to the deterministic style is introduced. As a room is electrically large, the number of unknown variables increases rapidly with an increase in the number of objects. It has to be accelerated by fast solvers. Acceleration techniques, such as the conjugate gradient fast Fourier transform method (CGFFT) and the multilevel fast multipole algorithm (MLFMA), are often used. Comparing the two methods, the former is not efficient in solving problems involving objects that have sparse dielectric distribution, while the latter is more optional to be performed 141. The basic concept of the FMM and its multilevel implementation has been reported elsewhere[3-15]. In this article, it is employed to calculate radar cross section(RCS) and the field strength of a multilayer dielectric object, which can be regarded as the wall or the ceiling. Then it is compared with the MOM.The results verify the feasibility. The discussion is in the frequency domain, and the time factor exp(-im) has been suppressed from the equations.
2 Formulation [3-151 2.1 TheVIE Consider a dielectric cube residing in a homogeneous background medium of infinite extent with ~ ~ as,
p
~
21
WANG Zheng-Bin, et al.: Indoor field strength prediction based on FMM
No.3
permittivity and permeability parameters, respectively. The dielectric has position-dependent permittivity E, . To simplify
function. A basic idea in the fast multipole method is to express the matrix element 2,by different representations according to
the formulation, this study assumes that the permeability of the material is a constant u = ub .
the distance c, ; if 5, is small, Z , is called a near-neighbor element. Otherwise, Ze is called a far-field matrix.
A volume integral equation [3] is formulated by representing the total electric field in the dielectric region as the sum of the scattered field and the incident field, i.e.,
2.3
E(F)= EhC(7) +iqubJVg(7,r~)~,(7~)~~-
For far-field interaction, the first term of RHS of Eq. (3) is ignored, and an alternative form of the integral operator is used, which is based on the dyadic Green's function, then [ 5 ]
Application of 3DV- FMM
Z, =iqu, J, J , , ~ ( i ~ ( i , ~ ~ ) ~ ( i ~ ) d ~ ~ d(4)~ -.
where V is the dielectric region, Em" is the incident field in the absence of the cube, E is the total electric field, 7, is the induced volume current in the dielectric body, g(7,T') = exp(-ikbR)/(4@) is the three-dimensional scalar
t,(F) , $(TI) denote the testing function and the basis
function, respectively. The addition theorem can be used to expand the scalar Green's function into a multipole expression, and
Green's function for the background media, R = IF- F'l is the distance from a source point 7' to a field point 7 , and k, is the wavenumber of the background material.
As the equivalent volume current
7,
in Q. (1) is related
to the total electric field by J = j GE(J;) - E , ]&(F)
(2)
vector from source to field.
and
r, is the
<.are the centers of the
groups that the source and field belong to, respectively, and - r , ,= r, -rmm. The transform factor is, L
There is only one unknown vector function E in Q. (1). 2.2
The integral is defined on the Ewald sphere.
T~.(F-,.k^)=Ci'(2l+l)h:"(kI&-%
I)P,(k^Fmm,)
(6)
l=O
Discretizationof the VIE
&(''(.) is the sphere Hankel function of the first kind, and
The MOM has been applied to solve the VIE to convert Q. (1) into a matrix equation. The dielectric region is partitioned into eight smaller cubic cells. Each subcube is then recursively subdivided into smaller cubes until the length of the edge of the finest cube is about 1/25-1/30 of the wavelength. Cubes at all levels are indexed. Nonempty cubes are found by further sorting. Only nonempty cubes are recorded using tree-structure data at all levels. TO solve the integral equation, the unknown vector J, in Q. (1) is represented by
P,:(.) is the legendre function of order I . The number of modes L can be obtained from some semi-empirical formulas; the one used here is L=kd+aln(x+kd) [12], adepends on the accuracy of the requirement. Using the -
identity of
-
(7- g )= (7- ~
- -& - 1 G t, (r -)(F
(7m-G
t, (T)(F - &-)e
(7
) -i&) . , )j=(Fl)=
e5- T,(FI)(F- ii)e*Df z (7)
Substituting Eqs. (7) and (5) into Q. (4), the impedance matrix element corresponding to well-separated testing and basis functions can be written as
where p f ) ,p F ) , p t ) are pulse functions in directions parallel -Pn(y) = Pn(') = I , respectively to i , f , Z , that is., p?) -
Fed\
, and
p : ) = p t ) = p f ) = O , FeAV"
.
Then the
(9)
integral Eq. (1) can be converted into a matrix equation that can be solved for the expansion coefficients a , . The matrix elements are given by (3)
is the dyadic Green's
3 Numerical results and discussion To validate the algorithm, the bistatic RCS of a dielectric cube illuminated by a plane wave is first calculated. The
22
2006
The Journal of CHUFT
kequency of the incident wave is 4.2 GHz, and the background media is free space. The cube is subdivided into 512 small cubes, and the finest edge is A l 2 5 . From Figs. 1 and 2, it is seen that the bistatic RCS calculated by the two methods coincide very well. However, the expended time per iterative step for the FMM is just about 40 % of that for the Conj-MOM.
." 0
~
50
100 eidegree
150
200
parameters, the merits of FMM will be more outstanding. But the distribution of inner field from the two algorithms seems to be different to a certain degree. The source of error is mainly attributed to the grouping in FMM.The relative error can be defined as:
Fig. 3 A multilayered dielectric object enclosed by a cube
Fk. 1 Bktiitic RCS ( 8 direction) cafcuiated by MOM, MoMconjugate iterative, and 3DV-FMM
MOM 0 Conj-MOM f
FMM
I
50
0
100
150
200
pldeqree
0
50
100
150
Fig. 4 Bistatic RCS of @ direction
200
@degree
Fig. 2 Bistatic RCS ( fJ7 direction) calculated by MOM,
7
-Conj-MOM
MoMconjugate iterative, and 3DV-FMM
The bistatic RCS of a multiplayer dielectric obstruct (Fig. 3) is then calculated, which can be regarded as the ceiling or the wall. Initially, it is enclosed within a large cube and is subdivided into smaller cubes as above. Then nonempty cubes are found by sorting. Only nonempty cubes are recorded using tree-structure data, which also saves much memory. The = 3.0, simulation parameters are , I = 3.1 GHz, E,, = 5.0, E,, = 4.0, E , = ~ 5.0. The thickness of the object is 4A/3 , and
the edge of the finest cube is 1113. From the graph (Figs. 4 and 5) it can be Seen that the scattering field of the multilayered dielectric obstruct calculated by the two methods also coincide very well. The itaative steps o€ FMM are 17, much lesser than that of MOM. With the increasing of unknown
-100 0
50
100 pldeqree
150
200
Q. 5 Bistatic RCS of 0 direction dth where J , ( i ) , J,(i) is the equivalent current of the ith cube b m MOM and FMM,respectively. The computational results
N0.3
WANG Zheng-Bin, et al.: Indoor field strength prediction based on FMM
show that the relative error of the inner field is about 10 dB.
4 Summary FMM based on VIE is described for the analysis of indoor field strength prediction. It reduces the memory and CPU time requirement significantlyin solving the VIE. As a result of the small coverage of U W B signals and high efficiency of the fast algorithm, indoor propagation environment can be simulated and the field strength analyzed. Compared with other field strength prediction models, the method has three merits: I) It is convenient to model 3D structures that contain fine and electrically small features. 2) It gives accurate solution. 3) Once given the partition grid of the object, it does not require human intervention. Only few small models were designed because of the restricted workstation’s capability, but it is undoubtedly feasible to apply MLFMA to the scattering analysis of UWB signals or other communication signals. The optimization of the procedure and acceleration of the algorithm is the subsequent key point.
References 1. Xiao Shang-hui, Jiang Yi. Signal design for ultra-wide-band comm- unication based on optimal method. Journal of Chongqing University of Posts and Telecommunications: Natural Seience, 2005,17(5):562 - 565(in Chinese) 2. Zhang Yu, Luo Han-wen. Research on the theory of indoor field
23
the Wave Equation. IEEE Antennas and propagation Magazine, 1993,35(3):7 - 12 8. Chen Xiao-guang, Jin Ya-qiu. The fast multipole method of three dimensional electromagnetic waves volume integral equation (3DV-FMM). Journal of Electronics, 2000, 22(6): 1007 - 1015(in Chinese) 9. Lu Cai-cheng, Chew Wengcho. A multilevel algotirhm for solving a boundary integral equation of wave scattering. Microwave and Oplical TeChuOlOgy Letters., 1994,7(10): 466 - 470 10. Dame E. The fast multipole method: numerical implementation. Journal of Computational. Physics, 2000,160(1): 195 - 240 11. Chao Hsueh-yung, Lin Che, Kandasamy Pirapaharan, et al., Fast field calculation by a multilevel fast multipole algorithm for large complex radiators and scatterers, in IEEE ,2003 12. Zhang Yao-jiang, Gong Zhong-lin, Zhou Le-zhu. Guidelines of parameter settings in 3D fast multipole method. Proceedings of 2nd International Conference on Microwave and Millimeter Wave Technology,Sep 14-16, 2000, Beijing, China. Piseataway, Nj, USA EEE, 2000: 387 - 390 13. Wang Jo JH, Dubberley J R. Computation of fields in an arbitrarily shaped heterogeneous dielectric or biological body by an iterative conjugate gradient method. IEEE Transactions on Microwave Theory and Techniques, 1989,37(7): 1119 - 1125 14. Kottmann J P, Martin 0 J F. Accurate solution of the volume integral equation for high-permittivity scatterers. IEEE Transactions on Antennas and Propagation, 2000,48(11): 1719 1726 15. Sertel K, Volakis J L. Method of moments solution of volume integral equations using parametric Geometry modeling. Radio Science., 2002,37(1): 101- 107
strength prediction and engineering applications. Communication Technology, 2002(11): 27 - 29(in Chinese) 3. Lu Cai-cheng. Indoor radio wave propagation modeling by
Biographies: WANG Zheng-bin, male, from
multilevel fast multipole algorithm. Microwave and Optical Techaology Letters., 2001,29(3): 168 - 175
Nanjmg University of Posts and Telecommuni-
4. Song Ji-ming, Lu Cai-cheng, Chew Weng-cho. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation, 1997,45(10):1488 - 1493 5. Lu Cai-cheng. A fast algorithm based on volume integral equation for analysis of arbitrary shaped delechic radomes. IEEE Transactions on Antennas and Propagation, 2003,51(3): 606 - 612 6. Lu Cai-cheng, Song Ji-ming, Chew Weng-cho. A multilevel fast multipole algorithm for solving 3D volume integral equations of electromagnetic scattering. Proceedings of IEEE Antennas and propagation. Society International Sgmposium: Vol 4, Jan 20-25, 2000, Salt Lake City, UT, USA. Piscataway, NJ, USA: IEEE, 2000: 1864 - 1867 7. Coifman R, Rokhlin V,Wandzura S. The fast multipole method for
Jjangsu Province. Received M. S. degree from cations, interested in the research on wireless
communication and EMC.
ZHANG
Ye-rong,
male,
from
Anhui
Province. Ph. D., the vice director of Key laboratory on wireless communications and EMC. His research interests are primarily in the areas of wireless communication and EMC.
WANG Zheng-bin, ZHANG Ye-rong
Indoor field strength prediction based on FMM CLC number T N O l 1.92
Abstract In this article, the fast multipole method 0 is applied to a n w the field shength of an ultra-wideband (UWB) signal. Small coverage of UWB communication systems and high efficiency of the algorithm make it possible to calculate the amplitude of elechic field accwdtely. A homogeneous dielectric body and a multilayered dielau-ic object are studied. The computational results obtained by applying the Fh4M agree well with those obtained by applying the method of moments (MOM). Keywords indoor field strength prediction, fast multipole method FMM, multilevel fast multipole algorithm (MLFMA)
I
Article ID 1005-8885(2006)03-0020-04
Document A
Introduction
With the popularization of personal communication service (PCS), indoor radio propagation has attracted more attention, especially for ultra-wideband (UWB) signals. Ultra-wideband communication systems have very high data rates (at least 100 Mbps in ten meters). Hence they are suitable for application in wireless networks of houses and offices. Indoor field strength prediction is very important for the design of piconet. Due to the ultra-wideband, UWB signals experience different propagation impairments than the traditional “narrowband” signals, the impairment being attributed to smaller coverage and more complicated environment. The field strength varies rapidly according to the material of obstruction and the layout of furniture I11. In the domain of narrowband and wideband communications, many researchers have studied the field strength prediction, and have put forth several models since the 1950s. In general, they can be divided into two types. One is by stochastic analysis, such as the Stochastic Model, the Empiric Model, and the Regression Model. The other is the deterministic style, such as the ray-tracing method and the physical optics (PO) 12, 31. Undoubtedly, a deterministic model can provide the most Received date: 2005-1 1-08 WANG Zheng-bin i 1 7 1 College of Mathemlics and Physics, Nanjing University of Posts and Telecommunications, Nanjing 2 10003, China b a i l : [email protected] ZHANG Ye-rong College of Communication and Inlormation Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
accurate field distribution. One deterministic model often used is the ray-tracing method. Compared with the method of moments (MOM), it computes faster. However, the calculated results will have serious errors when there are electrically small obstacles in the room. Therefore, some researchers have combined the ray-tracing method with other low-frequency methods, such as finite-element method(FEM), finite-difference time-domain(FDTD), and the MOM. These hybrid methods improve the accuracy of the solution and also are applicable to large and arbitrarily shaped structures. However, the hybrid methods have the same drawback, in that they are almost case sensitive [3].The regions containing electrically small features, which will be analyzed by a low-frequency method, must be identified and separated. This preprocess is very difficult to realize automatically. In this article, another deterministic method, that is, solving the volume integral equation (VIE)by the fast multipole method (FMM) is presented. Initially, taking into account the small coverage of UWB signals, the low-frequency method, namely, the MOM, which belongs to the deterministic style is introduced. As a room is electrically large, the number of unknown variables increases rapidly with an increase in the number of objects. It has to be accelerated by fast solvers. Acceleration techniques, such as the conjugate gradient fast Fourier transform method (CGFFT) and the multilevel fast multipole algorithm (MLFMA), are often used. Comparing the two methods, the former is not efficient in solving problems involving objects that have sparse dielectric distribution, while the latter is more optional to be performed 141. The basic concept of the FMM and its multilevel implementation has been reported elsewhere[3-15]. In this article, it is employed to calculate radar cross section(RCS) and the field strength of a multilayer dielectric object, which can be regarded as the wall or the ceiling. Then it is compared with the MOM.The results verify the feasibility. The discussion is in the frequency domain, and the time factor exp(-im) has been suppressed from the equations.
2 Formulation [3-151 2.1 TheVIE Consider a dielectric cube residing in a homogeneous background medium of infinite extent with ~ ~ as,
p
~
21
WANG Zheng-Bin, et al.: Indoor field strength prediction based on FMM
No.3
permittivity and permeability parameters, respectively. The dielectric has position-dependent permittivity E, . To simplify
function. A basic idea in the fast multipole method is to express the matrix element 2,by different representations according to
the formulation, this study assumes that the permeability of the material is a constant u = ub .
the distance c, ; if 5, is small, Z , is called a near-neighbor element. Otherwise, Ze is called a far-field matrix.
A volume integral equation [3] is formulated by representing the total electric field in the dielectric region as the sum of the scattered field and the incident field, i.e.,
2.3
E(F)= EhC(7) +iqubJVg(7,r~)~,(7~)~~-
For far-field interaction, the first term of RHS of Eq. (3) is ignored, and an alternative form of the integral operator is used, which is based on the dyadic Green's function, then [ 5 ]
Application of 3DV- FMM
Z, =iqu, J, J , , ~ ( i ~ ( i , ~ ~ ) ~ ( i ~ ) d ~ ~ d(4)~ -.
where V is the dielectric region, Em" is the incident field in the absence of the cube, E is the total electric field, 7, is the induced volume current in the dielectric body, g(7,T') = exp(-ikbR)/(4@) is the three-dimensional scalar
t,(F) , $(TI) denote the testing function and the basis
function, respectively. The addition theorem can be used to expand the scalar Green's function into a multipole expression, and
Green's function for the background media, R = IF- F'l is the distance from a source point 7' to a field point 7 , and k, is the wavenumber of the background material.
As the equivalent volume current
7,
in Q. (1) is related
to the total electric field by J = j GE(J;) - E , ]&(F)
(2)
vector from source to field.
and
r, is the
<.are the centers of the
groups that the source and field belong to, respectively, and - r , ,= r, -rmm. The transform factor is, L
There is only one unknown vector function E in Q. (1). 2.2
The integral is defined on the Ewald sphere.
T~.(F-,.k^)=Ci'(2l+l)h:"(kI&-%
I)P,(k^Fmm,)
(6)
l=O
Discretizationof the VIE
&(''(.) is the sphere Hankel function of the first kind, and
The MOM has been applied to solve the VIE to convert Q. (1) into a matrix equation. The dielectric region is partitioned into eight smaller cubic cells. Each subcube is then recursively subdivided into smaller cubes until the length of the edge of the finest cube is about 1/25-1/30 of the wavelength. Cubes at all levels are indexed. Nonempty cubes are found by further sorting. Only nonempty cubes are recorded using tree-structure data at all levels. TO solve the integral equation, the unknown vector J, in Q. (1) is represented by
P,:(.) is the legendre function of order I . The number of modes L can be obtained from some semi-empirical formulas; the one used here is L=kd+aln(x+kd) [12], adepends on the accuracy of the requirement. Using the -
identity of
-
(7- g )= (7- ~
- -& - 1 G t, (r -)(F
(7m-G
t, (T)(F - &-)e
(7
) -i&) . , )j=(Fl)=
e5- T,(FI)(F- ii)e*Df z (7)
Substituting Eqs. (7) and (5) into Q. (4), the impedance matrix element corresponding to well-separated testing and basis functions can be written as
where p f ) ,p F ) , p t ) are pulse functions in directions parallel -Pn(y) = Pn(') = I , respectively to i , f , Z , that is., p?) -
Fed\
, and
p : ) = p t ) = p f ) = O , FeAV"
.
Then the
(9)
integral Eq. (1) can be converted into a matrix equation that can be solved for the expansion coefficients a , . The matrix elements are given by (3)
is the dyadic Green's
3 Numerical results and discussion To validate the algorithm, the bistatic RCS of a dielectric cube illuminated by a plane wave is first calculated. The
22
2006
The Journal of CHUFT
kequency of the incident wave is 4.2 GHz, and the background media is free space. The cube is subdivided into 512 small cubes, and the finest edge is A l 2 5 . From Figs. 1 and 2, it is seen that the bistatic RCS calculated by the two methods coincide very well. However, the expended time per iterative step for the FMM is just about 40 % of that for the Conj-MOM.
." 0
~
50
100 eidegree
150
200
parameters, the merits of FMM will be more outstanding. But the distribution of inner field from the two algorithms seems to be different to a certain degree. The source of error is mainly attributed to the grouping in FMM.The relative error can be defined as:
Fig. 3 A multilayered dielectric object enclosed by a cube
Fk. 1 Bktiitic RCS ( 8 direction) cafcuiated by MOM, MoMconjugate iterative, and 3DV-FMM
MOM 0 Conj-MOM f
FMM
I
50
0
100
150
200
pldeqree
0
50
100
150
Fig. 4 Bistatic RCS of @ direction
200
@degree
Fig. 2 Bistatic RCS ( fJ7 direction) calculated by MOM,
7
-Conj-MOM
MoMconjugate iterative, and 3DV-FMM
The bistatic RCS of a multiplayer dielectric obstruct (Fig. 3) is then calculated, which can be regarded as the ceiling or the wall. Initially, it is enclosed within a large cube and is subdivided into smaller cubes as above. Then nonempty cubes are found by sorting. Only nonempty cubes are recorded using tree-structure data, which also saves much memory. The = 3.0, simulation parameters are , I = 3.1 GHz, E,, = 5.0, E,, = 4.0, E , = ~ 5.0. The thickness of the object is 4A/3 , and
the edge of the finest cube is 1113. From the graph (Figs. 4 and 5) it can be Seen that the scattering field of the multilayered dielectric obstruct calculated by the two methods also coincide very well. The itaative steps o€ FMM are 17, much lesser than that of MOM. With the increasing of unknown
-100 0
50
100 pldeqree
150
200
Q. 5 Bistatic RCS of 0 direction dth where J , ( i ) , J,(i) is the equivalent current of the ith cube b m MOM and FMM,respectively. The computational results
N0.3
WANG Zheng-Bin, et al.: Indoor field strength prediction based on FMM
show that the relative error of the inner field is about 10 dB.
4 Summary FMM based on VIE is described for the analysis of indoor field strength prediction. It reduces the memory and CPU time requirement significantlyin solving the VIE. As a result of the small coverage of U W B signals and high efficiency of the fast algorithm, indoor propagation environment can be simulated and the field strength analyzed. Compared with other field strength prediction models, the method has three merits: I) It is convenient to model 3D structures that contain fine and electrically small features. 2) It gives accurate solution. 3) Once given the partition grid of the object, it does not require human intervention. Only few small models were designed because of the restricted workstation’s capability, but it is undoubtedly feasible to apply MLFMA to the scattering analysis of UWB signals or other communication signals. The optimization of the procedure and acceleration of the algorithm is the subsequent key point.
References 1. Xiao Shang-hui, Jiang Yi. Signal design for ultra-wide-band comm- unication based on optimal method. Journal of Chongqing University of Posts and Telecommunications: Natural Seience, 2005,17(5):562 - 565(in Chinese) 2. Zhang Yu, Luo Han-wen. Research on the theory of indoor field
23
the Wave Equation. IEEE Antennas and propagation Magazine, 1993,35(3):7 - 12 8. Chen Xiao-guang, Jin Ya-qiu. The fast multipole method of three dimensional electromagnetic waves volume integral equation (3DV-FMM). Journal of Electronics, 2000, 22(6): 1007 - 1015(in Chinese) 9. Lu Cai-cheng, Chew Wengcho. A multilevel algotirhm for solving a boundary integral equation of wave scattering. Microwave and Oplical TeChuOlOgy Letters., 1994,7(10): 466 - 470 10. Dame E. The fast multipole method: numerical implementation. Journal of Computational. Physics, 2000,160(1): 195 - 240 11. Chao Hsueh-yung, Lin Che, Kandasamy Pirapaharan, et al., Fast field calculation by a multilevel fast multipole algorithm for large complex radiators and scatterers, in IEEE ,2003 12. Zhang Yao-jiang, Gong Zhong-lin, Zhou Le-zhu. Guidelines of parameter settings in 3D fast multipole method. Proceedings of 2nd International Conference on Microwave and Millimeter Wave Technology,Sep 14-16, 2000, Beijing, China. Piseataway, Nj, USA EEE, 2000: 387 - 390 13. Wang Jo JH, Dubberley J R. Computation of fields in an arbitrarily shaped heterogeneous dielectric or biological body by an iterative conjugate gradient method. IEEE Transactions on Microwave Theory and Techniques, 1989,37(7): 1119 - 1125 14. Kottmann J P, Martin 0 J F. Accurate solution of the volume integral equation for high-permittivity scatterers. IEEE Transactions on Antennas and Propagation, 2000,48(11): 1719 1726 15. Sertel K, Volakis J L. Method of moments solution of volume integral equations using parametric Geometry modeling. Radio Science., 2002,37(1): 101- 107
strength prediction and engineering applications. Communication Technology, 2002(11): 27 - 29(in Chinese) 3. Lu Cai-cheng. Indoor radio wave propagation modeling by
Biographies: WANG Zheng-bin, male, from
multilevel fast multipole algorithm. Microwave and Optical Techaology Letters., 2001,29(3): 168 - 175
Nanjmg University of Posts and Telecommuni-
4. Song Ji-ming, Lu Cai-cheng, Chew Weng-cho. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation, 1997,45(10):1488 - 1493 5. Lu Cai-cheng. A fast algorithm based on volume integral equation for analysis of arbitrary shaped delechic radomes. IEEE Transactions on Antennas and Propagation, 2003,51(3): 606 - 612 6. Lu Cai-cheng, Song Ji-ming, Chew Weng-cho. A multilevel fast multipole algorithm for solving 3D volume integral equations of electromagnetic scattering. Proceedings of IEEE Antennas and propagation. Society International Sgmposium: Vol 4, Jan 20-25, 2000, Salt Lake City, UT, USA. Piscataway, NJ, USA: IEEE, 2000: 1864 - 1867 7. Coifman R, Rokhlin V,Wandzura S. The fast multipole method for
Jjangsu Province. Received M. S. degree from cations, interested in the research on wireless
communication and EMC.
ZHANG
Ye-rong,
male,
from
Anhui
Province. Ph. D., the vice director of Key laboratory on wireless communications and EMC. His research interests are primarily in the areas of wireless communication and EMC.