1 the university of mississippi department of electrical engineering center of applied...

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The University of Mississippi Department of Electrical EngineeringC

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Atef Z. Elsherbeni [email protected]

The University of Mississippi

Electromagnetic Scattering From Chiral Media

Mohamed H. Al Sharkawy [email protected]

The University of Mississippi

Veysel [email protected]

Syracuse University

Ercument [email protected] University

Samir [email protected]

wKuwait University

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Outline Objectives Properties of Chiral material Example of Chiral Objects Problem Geometry Solution Techniques

FDTD Solution

Boundary Value Solution

Iterative Solution Verifications Numerical Results and Applications

Conclusion

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Objectives

Techniques for the scattering from arbitrary shaped two-dimensional chiral, dielectric and conducting scatterers.

RCS reduction and field focusing using composite scatterers.

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Properties of Chiral Material Unlike dielectric or conducting cylinders,

chiral scatterers produce both co-polarized and cross-polarized scattered fields.

A chiral medium is therefore characterized by right-hand circularly polarized waves (RCP) and left-hand circularly polarized waves (LCP).

Coating with chiral material can reduce or enhance the radar cross-section of targets.

A short metallic helix as a chiral object and its enantiomorphism.

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A sample of chiral material manufactured by a Finnish company. The sample measures 15 cm in diameters.

A closer view of the individual helices and their orientation.

Example of Chiral Objects

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Chiral Media Parameters

2/ 1c i i i

xxkk 21

k

c

or c is the chiral admittance

k is the wave number, depending on the chirality material

is the chirality parameter

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Constitutive Relation for the Chiral Material

,c cD E j B B H j E ejt is assumed

Waves in a chiral medium can be expressed as a superposition of RCP (R) and LCP (L) waves

1, cE R L H j R L

Maxwell’s equations in chiral medium:

2

c

c c

E j H E M

H j E H J

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Time Domain Solution Technique

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Multiple Frequency FDTD Formulation for Chiral Media

Using Maxwell’s equations and applying the 2nd order central difference approximation for the ejt time harmonic variation , we get

0 01, 2

0 0

0 01, 2

0 0

( , , )

( , , )

n xscat x x x

x x x

n xscat x x x

x x x

jtE i j k M N

jtH i j k M N

Where Mx and Nx are defined in terms of field components at previous time.

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Multiple Frequency FDTD Formulation for Chiral Media

0 0 0.5 0.5, , , ,

0.5 0.5 1, , , ,

0 0

1( , , ) ( , , ) ( , , 1) ( , , )

( )1( , 1, ) ( , , ) ( , , ) ( , , )

n n n nxx scat x scat x scat y scat y

n n n no xscat z scat z inc x inc x

i

jM H i j k E i j k E i j k E i j k

t t z

E i j k E i j k H i j k H i j ky t

jE

t

1 0.5 0.5, , , ,( , , ) ( , , ) ( , , ) ( , , )n n m n n

nc x inc x x inc x scat xi j k E i j k H i j k H i j k

0 0 0.5 0.5, , , ,

0.5 0.5 1, , , ,

0 0

1( , , ) ( , , ) ( , , ) ( , , 1)

( )1( , , ) ( , 1, ) ( , , ) ( , , )

n n n nxx scat x scat x scat y scat y

n n n noscat z scat z inc x inc x

i

jN E i j k H i j k H i j k H i j k

t t z

H i j k H i j k E i j k E i j ky t

jH

t

1 0.5 0.5, , , ,( , , ) ( , , ) ( , , ) ( , , )n n e n n

nc x inc x inc x scat xi j k H i j k E i j k E i j k

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Reflection and Transmission from a One Dimensional Chiral Slab of r = 2 and = 0.3

0 0, 0 0,

incEtranE

refE

x

z

0.1m

, ,r r

Co-Polarized Field.

Cr-Polarized Field.

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Scattering from a Chiral Sphere using the FDTD at 1 GHz with r = 4

7.2r cm

0 0, incE

x

z

, ,r r

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Scattering from a Chiral Sphere using the FDTD at four different frequencies of r = 4 and = 0.5

7.2r cm

0 0, incE

x

z

, ,r r

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Scattering from a Chiral Sphere using the FDTD at four different frequencies of r = 4 & = 0.5

7.2r cm

0 0, incE

x

z

, ,r r

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Frequency Domain Solution Techniques

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Problem Geometry Conducting Cylinder

Dielectric Cylinder

Chiral CylinderyM

xMM

1x

y

0

'

i'

j

'

j

yj

xjyi

xi

y1

x1

dij

'

i

ji

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Plane Wave Excitation-Incident E & H

E-polarized incident wave TMz

The corresponding component of the H-polarized incident wave

'0 0 0 0

'00 0

'cos( ) cos( )0

'cos( )0 0

( , )

( )

i i i i

ii i

jk jkincz i i

jnjk nn i

n

E E e e

E e j J k e

''00 0cos( ) '0

00

( , ) ( ) ii i jnjkinc ni i i n i

n

EH e j J k e

j

0

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Scattered and Internal Co-Polarized Fields

Cin , Ain and Bin are unknown coefficients.

0

'

(2)0 0

0

(2)00

0

' '0

( , ) ( )

( , ) [ ( ) ( )]

( , ) ( )

( , ) ( ) ( )

i

i

i

i

jnszi i i in n i

n

jnczi i i in n i in n i

n

jnsi i i in n i

n

jnci i i in n i in n i

nci

E E C H k e

E E A J k B J k e

EH C H k e

j

EH A J k B J k e

j

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Scattered and Internal Cross-Polarized Fields

Din , Ain and Bin are unknown coefficients.

0

(2)00

0

0

(2) '0 0

' '0

( , ) ( )

( , ) [ ( ) ( )]

( , ) ( )

( , ) ( ) ( )

i

i

i

i

jnszi i i in n i

n

jnczi i i in n i in n i

nci

jnsi i i in n i

n

jnci i i in n i in n i

n

EH j D H k e

EH j A J k B J k e

E E D H k e

E E A J k B J k e

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Boundary Value Solution(BVS)

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Boundary Conditions at the Surface of One Chiral Cylinder (Cylinder “i”, i = ai )

To solve for the unknown coefficients, the boundary conditions must be applied on cylinder i

czi

M

g

szg

inczi EEE

1

ci

M

g

sg

inci HHH

1

czi

M

g

szg HH

1

ci

M

g

sg EE

1

These four equations are repeated for all M cylinders.

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(2)0 0

[ ( ) ](2)0 0 0

( , ) ( )

( , ) ( ) ( )

g

i ig

jnszg g g gn n g

n

j m m nszg i i gn m i m n ig

n m

E E C H k e

E E C J k H k d e

Transformation of Scattered Fields

The scattered field components from the gth cylinder in terms of the local coordinates of the ith cylinder

x

y

0

'

i'g

'g

yg

xgyi

xi

dig

'

i

gi

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Transformation of Scattered Fields

The scattered field components from the gth cylinder in terms of the local coordinates of the ith cylinder

0(2)

0 0

0 [ ( ) ]

' (2)0 00

[ ( ) ]0

(2)0 00

[ ( ) ]0

( ) ( )( , )

( ) ( )( , )

( ) ( )( , )

i ig

i ig

i ig

m i m n igszg i i gn j m m n

n m

m i m n igsg i i gn j m m n

n m

m i m n igszg i i gn j m m n

m

J k H k dE E C

e

J k H k dEH C

j e

J k H k dEH j D

e

' (2)0 0

0 [ ( ) ]

( ) ( )( , )

i ig

n

m i m n igsg i i gn j m m n

n m

J k H k dE E D

e

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Determining the External Coefficients Cgn and Dgn

M

g n

niggn

niggn

i RDSCV1

M

g n

niggn

niggn

i RDSCV1

)cos(0'

0'

0''

0

)(

)(

)(

)(

iijk

i

i

i

ii

i eakJ

akJ

akJ

akJvV

From the application of the boundary conditions on all M cylinders

Where

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nigS

nigR

0 0 ngi ,

'

'

J

hv

J

hi

'

'

J

hv

J

hi

'

'

J

h

J

hvi

J

hv

J

hi'

'ngi ,

'

J

Hv

J

H ni

n

'

J

Hv

J

H ni

n

'

J

H

J

Hv nn

i

J

Hv

J

H ni

n'

gi

Solving for the External Coefficients Cgn and Dgn

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Solving for the Internal Coefficients Agn and Bgn

'

'

'

'

'

'

'

'

lli

llili

lli

llili

lli

lil

lli

ilil

JJv

hhCv

JJv

hhDv

JJv

JvJ

JJv

JvJA

'

'

'

'

'

'

'

'

lil

llili

lli

llili

lli

lli

lil

lilil

JvJ

hhDv

JJv

hhCv

JJv

JJv

JvJ

JvJB

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Used Variables:

)(

)(

0

0)2(

i

i

akJ

akHh

)(

)(

0'

0)'2(

'

i

i

akJ

akHh

)(

)(

0 i

i

akJ

akJJ

)(

)(

0'

''

i

i

akJ

akJJ

)(

)(

0 i

i

akJ

akJJ

)(

)(

0'

''

i

i

akJ

akJJ

)cos( 0''

0 iijki eVignj

ignn edkHH)(

0)2( )(

20 1

rrci

iv

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Iterative Solution(IS)

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Algorithm for Iterative Solution

• Stage 1: Apply the boundary condition on the surface of each cylinder, as the incident field is due to an external source.

• Stage 2: The incident field on each cylinder is produced by the scattered field from all other cylinders.

inc s czi zi ziE E E

inc s ci i iH H H

s czi ziH H

s ci iE E

1

Ms s czg zi zi

gg i

E E E

1

Ms s cg i i

gg i

H H H

1

Ms s cg i i

gg i

E E E

1

Ms s czg zi zi

gg i

H H H

0 0 0 0, ,C D A and B

1 0 1 0

1 0 1 0

( ), ( )

( ) ( )

C C D D

A A and B B

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Algorithm for Iterative Solution

' '0 0

(2) (2) '0 0I

' '0 0

(2) ' (2)0 0

( ) ( ) ( ) ( )

'( ) ( ) ( ) ( )C

( ) ( ) ( ) ( )

( ) ( ) '( ) ( )

i i i i i

i i i i i

i i i i i

i i i i i

J k a J k a v J k a J k a

H k a J k a v H k a J k a


v H k a J k a H k a J k a

' '0 0

(2) (2) '0 0I

' '0 0

(2) ' (2)0 0

( ) ( ) ( ) ( )

'( ) ( ) ( ) ( )D

( ) ( ) ( ) ( )

( ) ( ) '( ) ( )

i i i i i

i i i i i

i i i i i

i i i i i

v J k a J k a J k a J k a




(2) (2) '0 0

(2) (2) '0 0I

(2) (2) '0 0

(2) ' (2)0 0

'( ) ( ) ( ) ( )

'( ) ( ) ( ) ( )X

'( ) ( ) ( ) ( )

( ) ( ) '( ) ( )

i i i i i

i i i i i

i i i i i

i i i i i





( ) ( )I 0 (2) I 0 (2)0 0

1 11

I

C ( ) D ( )

X

ig ig

M Mj n j n

gn n ig gn n igg n g ng i g i

i

C H k d e D H k d e

C

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Matrix Formulation

1 1p p pC TC C RC D

1 1p p pD TD C RD D

1

p

p pi

pM

C

C C

C

1, 1,

2,1 2,1 2,

,

,1 ,

0

0

0

j M

M

i j

M M j

T T

T T T

T T

T T

1,1 1,, ,

,,

,

,1 ,, ,

ni j i j

ni j

i j

m m ni j i j

T T

TT

T T

(1, )

(1, )

(1,2 1)

(1,2 1)i

j

i M

j M

m N

n N

where

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Bistatic Scattering Cross Section of Five Perfectly Conducting Cylinders

Each cylinder has radius = 0.1 and their centers are separated by 0.5 due to a TM plane wave incident at 0=180

y

x

* Atef Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Science, vol. 29, pp. 1023-1033, July-August 1994.

50 100 150 200 250 300 350-10

-5

0

5

10

15

(degrees)

/ 0

(d

B)

BVSBVSITS

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Echo-Width of a Homogenous Chiral Strip

2, 3, 0.0005,

frequency =300 MHzr r c

(degrees)

Ech

o W

idth

(dB

/m)

* Michael S. Kluskens and Edward H. Newman, “Scattering by a Chiral Cylinder of Arbitrary Cross Section,” IEEE Trans. Antennas Propagate., vol. 38, pp. 1448-1455, Sept. 1990.

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Echo-Width of an Inhomogeneous Chiral Strip

2, 3, 0.0005,

frequency =300 MHzr r c

(degrees)

Ech

o W

idth

(dB

/m)

* Michael S. Kluskens and Edward H. Newman, “Scattering by a Chiral Cylinder of Arbitrary Cross Section,” IEEE Trans. Antennas Propagate., vol. 38, pp. 1448-1455, Sept. 1990.

0.0005c 0.0005c

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= 0.041

Co-Polarized

X-Polarized

Dielectric 0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

r = 5

r = 0.1

d = 0.75

Normalized Scattered Field Using BVS and IS

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-30

-20

-10

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

-30

-20

-10

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0 = 0.041

Co-Polarized

X-Polarized

Dielectric

F. RCSmax_Diel = 18.3 dB

F. RCSmax_Chi_Co = 11.8 dB

F. RCSmax_Chi_X = -1.35 dB

B. RCSmax_Diel = 17 dB

B. RCSmax_Chi_Co = -20 dB

B. RCSmax_Chi_X = -8 dB

r = 5

r = 0.1

d = 0.75

RCS Radiation Pattern in dB Using BVS and IS

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Back-ward Scattered field for different Chiral admittance values

For-ward Scattered field for different Chiral admittance values

2 3 4 5 6 7 8 9 10

x 10-3

-20

-15

-10

-5

0

5

10

15

20

Ba

ckw

ard

-Sca

ttere

r (d

B)

c

Backward-DielectricBackward-Co-PolarizedBackward-X-Polarized

2 3 4 5 6 7 8 9 10

x 10-3

-10

-5

0

5

10

15

20

25

Fo

rwa

rd-S

catte

rer

(dB

)

c

Forward-DielectricForwrd-Co-PolarizedForwrd-X-Polarized

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= 0.00573

Co-Polarized

X-Polarized

Dielectric

F. RCSmax_Diel = 20 dB

F. RCSmax_Chi_Co = 26 dB

F. RCSmax_Chi_X = -5 dB

B. RCSmax_Diel = 15.5 dB

B. RCSmax_Chi_Co = 2 dB


RCS Radiation Pattern in dB for a TMz

Plane Wavec = - 0.00573

--- --

c = + 0.00573

c = - 0.00573

c = + 0.00573--

---

---

- --

c = + 0.00573

c = - 0.00573

-40 -30 -20 -10

0 10 20 30

30

210

60

240

90

270

120

300

150

330

180 0

-40 -30

-20 -10

0 10 20

30

30

210

60

240

90

270

120

300

150

330

180 0

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= 0.002445

Co-Polarized

X-Polarized

Dielectric

F. RCSmax_Diel = 20 dB

F. RCSmax_Chi_Co = 25 dB

F. RCSmax_Chi_X = -5 dB

B. RCSmax_Diel = 15.5 dB

B. RCSmax_Chi_Co = 0 dB


RCS Radiation Pattern in dB for a TEz

Plane Wavec = - 0.002445

--- --

c = + 0.002445

c = - 0.002445

c = + 0.002445--

---

---

- --

c = + 0.002445

c = - 0.002445

-40 -30 -20 -10

0 10 20 30

30

210

60

240

90

270

120

300

150

330

180 0

-40

-30

-20

-10 0 10 20 30

30

210

60

240

90

270

120

300

150

330

180 0

40


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rch

(C

AE

SR

)

Conclusions

Techniques are developed for time and frequency domain analysis of chiral material.

Application of chiral material is demonstrated for:

designing anti-reflection composite structures controlling or altering the RCS of scatterers.

41


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End of Presentation

42


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0 0.01 0.02 0.03 0.04 0.050

50

100

150

200

250

300

350

Chiral Admittance, c

Ch

iral I

mp

ed

na

ce

Parameters1Parameters2

Intrinsic Impedance and Wave Number Versus Chiral Admittance

0r 0

0

Parameters 1: 2, 3, 1 , 307.812rr d

r

m

21 ( / ) /i i ci i i cik k

2/[ (1 ( / ) )]ci i i i i ci

0r 0

0

Parameters 2: 4, 8, 1 , 266.5736rr d

r

m

0 0.01 0.02 0.03 0.04 0.05-5

0

5

10

15

20

25

30

Chiral Admittance, c

Wa

ve N

um

be

r (d

B)

k +k -k

Parameters 1

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) Parametric Study for a vertical Inhomogeneous Strip

0.05 0.1 0.15 0.2 0.25 0.3

-40

-30

-20

-10

0

10

20

Ba

ckw

ard

-Sca

ttere

r (d

B)

Radius

Backward-DielectricBackward-Co-PolarizedBackward-X-Polarized

Back-ward Scattered field for different Radius values

0.05 0.1 0.15 0.2 0.25 0.3

-40

-30

-20

-10

0

10

20

Fo

rwa

rd-S

catte

rer

(dB

)

Radius

Forward-DielectricForwrd-Co-PolarizedForwrd-X-Polarized

For-ward Scattered field for different Radius values

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Bistatic Echo Width of a Circular Chiral Cylinder Illuminated by a TMz Plane Wave

mrmme

rr

1.0,1

,0.0tan, 0.0tan

,0005.0,5.1,4

0

* Majeed A. Al-Kanhal and Ercument Arvas, “Electromagnetic Scattering from a Chiral Cylinder of Arbitrary Cross Section,” IEEE Trans. Antennas Propagate., vol. 44, pp. 1041-1048, July 1996.

0 50 100 150-18

-16

-14

-12

-10

-8

-6

-4

(degrees)

/

0 (

dB)

BVSIter. Soln.MOM* Soln.BVSIter. Soln.MOM* Soln.

CO-POLARIZED ( Ez )

CROSS POLARIZED ( E )

x

y

inczE

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Internal H-field along y = 0 of a Circular Chiral Cylinder Excited by a TEz Wave

0

2 , 3 , 0.002,

0.15 , tan 0.05,

tan 0.05 , 1

r r

e

m

r m

m

-0.1 -0.05 0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

x(m)

|H/H

inc|

BVSIter. Soln.MOM* Soln.BVSIter. Soln.MOM* Soln. 0) (achiral Hz

zHx

y

izH

* Majeed A. Al-Kanhal and Ercument Arvas, “Electromagnetic Scattering from a Chiral Cylinder of Arbitrary Cross Section,” IEEE Trans. Antennas Propagate., vol. 44, pp. 1041-1048, July 1996.

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|Ez| of a TMz Plane Wave within a Homogeneous Dielectric Cylinder

-0.15 -0.1 -0.05 0 0.05 0.1 0.150.4

0.6

0.8

1

1.2

1.4

1.6

x(m)

Ez-m

agni

tude

BVSMOM* Soln.Iter. Soln.

x

y

inczE

A cylinder of circumference 1.0 and r = 3

* Andrew F. Peterson, Scott L. Ray and Raj Mittra, “Computational Methods for Electromagnetics,” IEEE Antennas Propagate.,© 1998 by the Institute of Electrical and Electronics Engineers, Inc.

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|Hz/Hinc| of a TEz Plane Wave within a Circular, Homogeneous Dielectric Cylinder

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

1.3

1.4

1.5

1.6

Distance along center cut

|Hz/H

inc|

BVSMOM* Soln.Iter. Soln.

y

x

izH

A cylinder of circumference 1.0 and r=2.56

* Andrew F. Peterson, Scott L. Ray and Raj Mittra, “Computational Methods for Electromagnetics,” IEEE Antennas Propagate.,© 1998 by the Institute of Electrical and Electronics Engineers, Inc.

1 the university of mississippi department of electrical engineering center of applied...

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