Numerical ElectroMagnetics&

Semiconductor Industrial Applications

Ke-Ying Su Ph.D.

National Central University

Department of Mathematics

12 2D-NUFFT & Applications

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Part II : 2D-NUFFT

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Outline : 2D-NUFFT

1. Introduction

2. 2D-NUFFT algorithm

3. Approach

4. Results and discussions

5. Conclusion

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I. Introduction

• Finite difference time domain approach (FDTD)

• Spectral domain approach (SDA)

• Finite element method (FEM)

• Integral equation (IE)

• Mode matching technique

Numerical methods

Widely used technique

Method of Moment (MoM)

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Disadvantages

1) Slow convergence of Green’s functions

2) A large number of basis functions

Used solutions

Using the 2D discrete fast Fourier Transform (FFT)

1)

2D-FFT + Using the first few resonant modes’ current distributions

1,2) Nonuniform meshs for mixed potential integral equation (MPIE)

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NUFFT : 1D 2D

The q+1 nonzero coefficients.

The core idea of the 1D-NUFFT:

SDA + 2D-NUFFT + Nonuniform meshs

New solution

fjj mNkqviq

jkiks ese /2)12/(

1

1

1 )(

1 2 3 4 5 6 7 8 9-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

l

rl(

s j)

sj = 0.5207

q = 8

Sjsj Sj+1Sj-1 Sj+q/2Sj-q/2

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The (q+1)2 nonzero coefficients.

The square 2D-NUFFT

Some of these 2D coefficients approach to zero rapidly.

NUFFT : 1D 2D

coefficientsremove directly

least square error accuracy accuracy

The nonsquare 2D-NUFFT

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Our aim

12/

2/

12/

2/

M

Mm

N

Nn

inyimxmnst

st eeGD

2D-NUFFT

2/

2/

2/

2/

),(),(q

qppg

q

qgstpgst tsHyxrD

2D-FFT

II. 2D-NUFFT Algorithm

p

cNnguicMmpvi

gstpg

inyimxmn

st

st

eeyxr

ee

/2)(/2)(),(

Xt-1

Ys+q/2

Ys-q/2

Ys

Ys+1

Ys-1

Xt-q/2 Xt+q/2Xt+1Xt

The key step

(3.1)

(3.2)

(3.3)

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][

][)2/()1(

2/,1)2/()0(

2/,0)2/()1(

2/,1

)2/()1(2/,1

)2/()0(2/,0

)2/()1(2/,1

nqumvq

nqumvq

nqumvq

nqumvq

nqumvq

nqumvq

inyimxmn

ststst

ststst

st

rrr

rrr

ee

Regular Fourier Matrix

For a given (xt,ys), for m = -M/2,…,M/2-1 and n = -N/2,…, N/2-1

Ar(xt,ys) = b(xt,ys)

r(xt,ys) = [A*A]-1[A*b(xt,ys)] = Fr-1Pr

where Fr is the regular Fourier matrix with size (q2/2+3q+1)2

Fr & Pr : closed forms

Xt-1

Ys+q/2

Ys-q/2

Ys

Ys+1

Ys-1

Xt-q/2 Xt+q/2Xt+1Xt

A: (MN)(q2/2+3q+1)b : (MN)1

(3.5)

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-q/2

q/2

-q/2

0

0 q/2

Solution

Extract Fr and Pr from Ff and Pfwhere Ff is the regular Fourier matrix with size (q+1)2

[a1, a2, …, am] [b1, b2, …, bn] =

[a1b1, a2b1, …, amb1, …, a1bn, a2bn, …, ambn]

1) Define a vector product as

Let Vp and Vg be the (p+1)th and (g+1)th row of the regular Foruier matrix for 1D problemp and g = 0, 1, …,q.

The [g(q+1)+(p+1)]th row of Ff equals VpVg.

(3.6)

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]1

...,,1

,,1

...,,1

[

))(2/())(2/(

1

)1)(2/()1)(2/(

1

)1)(2/()1)(2/())(2/())(2/(

gq

gqNgqNNN

NN

g

gNgN

g NV

Let = ei2/cM and = ei2/cN,

(3.7)

(3.8)

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2) Choose mn = cos(m/cM)cos(n/cN).

The [g(q+1) +(p+1)]th element of Pf

1,1 )2}2{2(

1,1 )2}2{2(

)1()1(

1

})2{22(2

sin

1

})2{22(2

sin

),(

d dgqcNycN

i

s

d dpqcMxcM

i

t

pqgstf

s

t

e

cNyqdgc

e

cMxqdpc

yxP

where {x} = x - [x].

(3.9)

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= [1, 2, …, (q+1)2]

= [q/2, q/2+1, q/2+2, 3q/2, …, q2+3q/2, q2+3q/2+1, q2+3q/2+2]

Fr(i, j) = Ff( (i), (j))

Pr(i) = Pf( (i))

For square grid points:

For octagonal grid points:

q+1

1 q +q+1

(q+1)

q/2

q/2+1

q/2+2

2

2

2q +3q/2

q +3q/2+12

q +3q/2+22

3) Fill Fr and Pr from Ff and Pf

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1

4

5

6

9

28

36

32

33

31

37

45

41

42

40

46

54

50

51

49

73

81

77

78

76

734637281

5445369

31

33

325

6

4

5041

5142

4940

81

77

78

76

Ff =

1 2 3 4 5 876 81

76

54

32

181

8

rF

Ff =

21 3 54 6 7 8

812

87

56

34

81

1The relation between Fr and Ff, and Pr and Pf

Example: Let q = 8

1

4

5

6

9

28

36

32

33

31

37

45

41

42

40

46

54

50

51

49

73

81

77

78

76

734637281

5445369

31

33

325

6

4

5041

5142

4940

81

77

78

76

Index = [4, 5, 6, 12, 13, 14, 15, 16, …, 76, 77, 78]

Ff =

1 2 3 4 5 876 81

76

54

32

181

8

rF

Ff =

21 3 54 6 7 8

812

87

56

34

81

1

Index = [1, 2, 3, …, 79, 80, 81]

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1~3) Ff, Pf Fr, Pr

4)

cNnguicMmpviM

Mm

N

Nnmnmnpg

st eeGH /2)(/2)(12/

2/

12/

2/

1

2D-FFT:

5) p

pgg

stpgst tsHyxrD ),(),(

rr = Fr-1 Pr

2D-NUFFTXt-1

Ys+q/2

Ys-q/2

Ys

Ys+1

Ys-1

Xt-q/2 Xt+q/2Xt+1Xt

If M = N = 210 and c = 2, then a 2D-FFT with size cMcN uses 3.02 seconds (CPU:1.6GHz).

(3.5)

(3.4)

(3.3)

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III. Approach

substrate thickness t

box dimension

abc

The Green’s functions

a0

t

cb

y

z

xa0

t

cb

y

z

x

)sin()cos()'(sin)'cos(~

)',,',( ykxkykxkGyyxxG ynxmynm n

xmxxxx

)sin()cos()'(cos)'sin(~

)',,',( ykxkykxkGyyxxG ynxmynm n

xmxyxy

)cos()sin()'(cos)'sin(~

)',,',( ykxkykxkGyyxxG ynxmynm n

xmyyyy

kxm = m/a, kyn = n/b

is the spectral domain Green’s function xxG~

where(3.10)

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Spectral domain Green’s functions

a0

t

cb

y

z

xa0

t

cb

y

z

x

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Solution procedure

x

y

Js(x,y)

JL(x,y) JL(x,y)

Jx(x,y)Jy(x,y)

Jy(x,y)

Asymmetric rooftop functions and the nonuniform meshs

source

J(x, y) = axJx(x, y) + byJy(x, y)

load terminalload terminal

(3.11)

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Asymmetric rooftop function

Jx = Jxx(x, x)Jxy(y, y)

22

22

1

1 ))(cos()cos())(cos()cos(~

xm

xmxm

xm

xmxmxx kx

xxkxk

kx

xxkxkJ

yn

yn

yn

ynxy k

yyk

k

yykJ

))2/(cos())2/(cos(~

xx xx x

y yy y

Jx,y)

(3.12)

(3.14b)

(3.14a)

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Galerkin’s procedure

Final MoM matrix

),(~

),(~

),(~

),( nmJnmJnmGedZ xem n

xdxxxx

12/

2/

12/

2/

12/

2/

12/

2/

))(sin())(cos(),(

~

))(cos())(sin(),(

~

M

Mm

N

Nnedynedxmh

yngxm

uv

M

Mm

N

Nnedynedxmh

yngxm

uv

yykxxkkk

nmG

yykxxkkk

nmG

Trigonometric identities

(3.15)

(3.16)

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schememesh

Nonuniform

kxm kyn2

3kxm

4

3kyn

kxm2 kyn

3

2kyn3kxm

4kxm2kyn

Gxx~

~Gxx

~Gyx

~Gyx

~Gyy

2D-NUFFT

2D-NUFFT

2D-NUFFT

2D-NUFFT

2D-NUFFT

MoM

Procedure for evaluating the MoM matrix

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III. Numerical Results

Hairpin resonator

a0

t

cb

y

z

xa0

t

cb

y

z

x

r = 10.2, L1 = 0.7, L2 = 1.01, L3 = 2.74, L4 = 8, L5 = 6, w1 = 1, w2 = 1.19, g1

= 0.2 and g2 = 0.8. All dimensions are in mm.

g1

L4 1L

L3

L2

g2 w1

w2

w2

w2

x

y

GPOF GPOF

a

b

yc

t

z

r

L5 L5

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Comparison of CPU Time and L2 error of One Call of

the 2D-NUFFT in Analysis of a Hairpin Resonator

Comparison of Analyses of The Hairpin Resonator with Uniform and Nonuniform Grids

Table 3.1

Table 3.2.1

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2 2.2 2.4 2.6 2.8 3

10

-80

-60

-40

-20

0

Frequency (GHz)

|S11

| an d

|S21

| (d B

) Measurement 11

21|S |

|S |

2D-NUFFT, q = 42D-NUFFT, q = 62D-NUFFT, q = 8

The measured and calculated S parameters of the hairpin resonator.

-q/2

q/2

-q/2

0

0 q/2

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Normalized magnitudes of the current distribution on the hairpin resonator at 2.473 GHz.

(a) |Jx(x,y)|

(b) |Jy(x,y)|

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Normalized magnitudes of the current distribution on the hairpin resonator at 2.397 GHz.

(c) |Jx(x,y)|

(d) |Jy(x,y)|

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Interdigital capacitor

L4 se

d2Ld g

4L3L

y

x

L1 L1

L5

Comparison of Analyses of The Ingerdigital capacitor with Uniform and Nonuniform Grids

r = 10.2, L1 = 8, L2 = 1.6, L3

= 0.8, L4 = 1.2, L5 = 7.9, d =

0.4, e = 0.4, g = 0.2 and s = 0.2. The thickness of substrate is 1.27. All dimensions are in mm.

Table 3.2.2

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1 2 3 4 5 6 7 8 9 10 11 12-20

-16

-12

-8

-4

0

2D-NUFFT

Frequency (GHz)

|S11

| an d

|S2

1| (d

B)

21

|S |11

Measurement

|S |

The measured and calculated S parameters of the interdigital capacitor.

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Normalized magnitudes of the current distribution on the interdigital capacitor at 5 GHz.

(a) |Jx(x,y)|

(b) |Jy(x,y)|

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Wideband filter

Comparison of Analyses of The Wideband filter with Uniform and Nonuniform Grids

a

b

y

x

c

t r

L

zy

15 L15

L1

1L

L9

L10

L12

L13

L14

2L

L4 L3

5LL11

7L

LL8

6

L1 = 8, L2 = 0.56, L3 =

0.576, L4 = 0.69, L5 =

0.3605, L6 = 0.125, L7

= 0.125, L8 = 0.125,

L9 = 5.19, L10 = 4.88,

L11 = 0.38, L12 = 2.06,

L13 = 1.9, L14 = 7.75,

L15 = 11.3, t = 0.635,

r = 10.8. All

dimensions are in mm.

Table 3.2.3

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-50

-40

-30

-20

-10

0

2 4 6 8 10 12 14 16Frequency (GHz)

|S11

| an d

|S21

| (d B

)

21|S |

Measurement [37]-60

2D-NUFFT

|S |11

The measured and calculated S parameters of the wideband filter.

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Normalized magnitudes of

the current distribution at

6 GHz

(a) |Jx(x,y)|

(b) |Jy(x,y)|

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VI. Conclusion

• A 2D-NUFFT algorithm with octagonal interpolated coefficients are used to enhance the Computation.

• The octagonal 2D-NUFFT uses less CPU time than the square 2D-NUFFT.

• The L2 error of the octagonal 2D-NUFFT is the same

as that of square 2D-NUFFT.

• The scattering parameters of the hairpin resonator, an interdigital capacitor and a wideband filter are calculated and validated by measurements.

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THE ENDThank You for your Participation !