National Central University
Department of Mathematics
Numerical ElectroMagnetics
&
Semiconductor Industrial Applications
12 2D-NUFFT & Applications
Ke-Ying Su Ph.D.
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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
Numerical methods
• Finite difference time domain approach (FDTD)
• Spectral domain approach (SDA)
• Finite element method (FEM)
• Integral equation (IE)
• Mode matching technique
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
1) 
1,2) 
Using the 2D discrete fast Fourier
Transform (FFT)
2D-FFT + Using the first few resonant
modes’ current distributions
Nonuniform meshs for mixed potential
integral equation (MPIE)
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New solution
SDA + 2D-NUFFT + Nonuniform meshs
NUFFT : 1D  2D
0.7
The core idea of the 1D-NUFFT:
0.6
0.5
q 1
  k1   ( s j )e
i ( v j    q / 21) 2k / mN f
0.4
rl( sj)
e
iks j
sj = 0.5207
q=8
0.3
0.2
 1
0.1
S j-q/2
S j-1 sj S j
S j+1
S j+q/2
0
-0.1
1
2
3
4
5
l
6
7
8
9
The q+1 nonzero coefficients.
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NUFFT : 1D  2D
The square 2D-NUFFT
Some of these 2D coefficients
approach to zero rapidly.
The nonsquare 2D-NUFFT
The (q+1)2 nonzero coefficients.
coefficients
remove directly
accuracy 
least square error
accuracy 
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II. 2D-NUFFT Algorithm
Dst 
Our aim
M / 2 1
N / 2 1
 G
m M / 2 n  N / 2
mn
e
e
(3.1)
 mn e imx e iny
Y s-q/2
Y s-1
s
Y s+q/2
  rpg ( xt , y s )e i ( vt  p ) 2m / cM e i ( us  g ) 2n / cN
p
Xt-1 Xt Xt+1
Ys
Y s+1
The key step
t
Xt-q/2
imxt inys
(3.2)
g
2D-NUFFT
Dst 
q/2
q/2
 r
pg
p  q / 2 g  q / 2
( xt , ys )H pg ( s, t )
(3.3)
2D-FFT
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Xt+q/2
Regular Fourier Matrix
 mn e imx e iny 
t
s
[r1,  q / 2 ( vt 1) m ( us q / 2 ) n  r0,  q / 2 ( vt 0) m ( us q / 2 ) n  r1,  q / 2 ( vt 1) m ( us q / 2 ) n ]   
 [r1,q / 2 ( vt 1) m ( us  q / 2 ) n  r0,q / 2 ( vt 0 ) m ( us  q / 2 ) n  r1,q / 2 ( vt 1) m ( us  q / 2 ) n ]
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)
Xt-q/2
A: (MN)(q2/2+3q+1)
b : (MN)1
 r(xt,ys) = [A*A]-1[A*b(xt,ys)] = Fr-1Pr
Xt-1 Xt Xt+1
Xt+q/2
Y s-q/2
Y s-1
(3.5)
Ys
Y s+1
Y s+q/2
where Fr is the regular Fourier matrix with size (q2/2+3q+1)2
Fr & Pr : closed forms
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Solution
Extract Fr and Pr from Ff and Pf
where Ff is the regular Fourier matrix with size (q+1)2
1) Define a vector product  as
-q/2
-q/2
[a1, a2, …, am]  [b1, b2, …, bn] =
(3.6)
[a1b1, a2b1, …, amb1, …, a1bn, a2bn, …, ambn]
0
0
q/2
Let Vp and Vg be the (p+1)th and (g+1)th
row of the regular Foruier matrix for 1D problem
p and g = 0, 1, …,q.
 The [g(q+1)+(p+1)]th row of Ff equals VpVg.
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q/2
Let  = ei2/cM and  = ei2/cN,
(3.7)
 (  N / 2 )(  g )   ( N / 2 )(  g )
 (  N / 2 )( 1)   ( N / 2 )( 1)
Vg  [
, ...,
, N,
g
1
1
1
 (  N / 2 )(1)   ( N / 2 )(1)
 (  N / 2 )( q g )   ( N / 2 )( q g )
, ...,
]
1
q g
1
1
(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
Pf ( xt , y s ) g ( q 1)( p 1) 



sin  ( 2 p  d  q  2{xt cM 2 })  

 2c

 

 d  1,1

i
( 2{ xt cM 2 } q  2 p  d )
cM
1 e







sin
(
2
g

d

q

2
{
y
cN
2

})



s
2
c





 d  1,1

i ( 2{ y s cN 2 } q  2 g  d )
cN
1 e




(3.9)
where {x} = x - [x].
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3) Fill Fr and Pr from Ff and Pf
For square grid points:
1
 = [1, 2, …, (q+1)2]
q2+3q/2
q/2
For octagonal grid points:
q 2+q+1
q/2+1
q/2+2
q2+3q/2+1
q2+3q/2+2
 = [q/2, q/2+1, q/2+2, 3q/2, …,
q2+3q/2, q2+3q/2+1, q2+3q/2+2]
q+1
2
(q+1)
Fr(i, j) = Ff( (i),  (j))
Pr(i) = Pf( (i))
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4
31 40 49
76
6
33 42 51
78
The
relation
between
Fr and Ff, and Pr and Pf
5
32 41 50
77
Example: Let q = 8
9
36 45 54
81
Index  = [1, 2, 3, …, 79, 80, 81]
Index  = [4, 5, 6, 12, 13, 14, 15, 16, …, 76, 77, 78]
1 2 3 4 5 6 7 8
81
44
55
66
31
31
32
32
33
33
99
36 45
45 54
54
36
40
40
41
41
42
42
49
49
50
50
51
51
73
76
77
78
Fr
Ff =
81
1 2 3 4 5 6 7 8
28
28 37
37 46
46
1 2 3 4 5 6 7 8
11
1 2 3 4 5 6 7 8
81
81
81
81
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2D-NUFFT
1~3)
Ff, Pf  Fr, Pr
 rr = Fr-1 Pr
4)
Xt-q/2
H pg 

Xt+q/2
Y s-q/2
(3.5)
Y s-1
Ys
Y s+1
Y s+q/2
2D-FFT:
M / 2 1
Xt-1 Xt Xt+1
N / 2 1
1 i ( vt  p ) 2m / cM i ( us  g ) 2n / cN
G

e
 mn mne
m M / 2 n  N / 2
(3.4)
If M = N = 210 and c = 2, then a 2D-FFT with size cMcN uses
3.02 seconds (CPU:1.6GHz).
5)
Dst   rpg ( xt , ys )H pg ( s, t )
p
(3.3)
g
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III. Approach
z
The Green’s functions
c
b
substrate thickness t
box dimension abc
kxm = m/a, kyn = n/b
t
0
y
a
x
~
Gxx ( x, x' , y, y ' )   Gxx cos( k xm x' ) sin (k yn y ' ) cos( k xm x ) sin( k yn y )
m
n
m
n
~
Gxy ( x, x' , y, y ' )   Gxy sin( k xm x' ) cos(k yn y ' ) cos( k xm x ) sin( k yn y )
m n
~
G yy ( x, x' , y, y ' )   G yy sin( k xm x' ) cos(k yn y ' ) sin( k xm x ) cos( k yn y )
~
where Gxx is the spectral domain Green’s function
(3.10)
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Spectral domain Green’s functions
z
c
b
t
0
y
a
x
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Solution procedure
Asymmetric rooftop functions and the nonuniform meshs
J(x, y) = axJx(x, y) + byJy(x, y)
(3.11)
source
J s(x,y)
J x(x,y)
Jy(x,y)
J y(x,y)
J L(x,y)
load terminal
y
J L(x,y)
x
load terminal
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Asymmetric rooftop function
J x,y)
y y
y y
x  x 
x  x  x 
Jx = Jxx(x, x)Jxy(y, y)
(3.12)
cos( k xm x )  cos( k xm ( x  x 1 )) cos( k xm x )  cos( k xm ( x  x 2 ))
~
J xx 

2
2
x 1k xm
x 2 k xm
(3.14a)
cos( k yn ( y  y / 2)) cos( k yn ( y  y / 2))
~
J xy 

k yn
k yn
(3.14b)
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Galerkin’s procedure
Final MoM matrix
~
~
~
Z xx (d , e)   G xx (m, n ) J xd (m, n )J xe (m, n )
m
(3.15)
n
Trigonometric identities
~
 M / 21 N / 21 Guv ( m, n )
sin( k xm ( xd  xe )) cos( k yn ( y d  y e ))
  
g
h
m M / 2 n  N / 2 k xm k yn
(3.16)

~
 M / 21 N / 21 G
uv ( m, n )
  
cos( k xm ( xd  xe )) sin( k yn ( y d  y e ))
g
h
m M / 2 n  N / 2 k xm k yn
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Procedure for evaluating the MoM matrix
Nonuniform
mesh
scheme
~
G xx
3
k xm
k yn2
~
G xx
k xm4 k yn2
~
G yx
2
k xm
k yn3
~
G yx
3
k xm
k yn3
~
G yy
2
k xm
k yn4
2D-NUFFT
2D-NUFFT
2D-NUFFT
MoM
2D-NUFFT
2D-NUFFT
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III. Numerical Results
a
L4
Hairpin resonator
b
GPOF
g2
w1
w2
L3
w2
y
w2
x
c
b
GPOF
g1
L2
z
L1
y
z
y
t
0
c
a
x
L5
t
L5
er
er = 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.
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Table 3.1
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 3.2.1
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-q/2
-q/2
The measured and calculated S parameters
of the hairpin resonator.
0
10
0
|S 11| and | S 21| (dB)
0
q/2
|S11|
Measurement
-20
-40
-60
-80
2D-NUFFT, q = 4
2D-NUFFT, q = 6
2D-NUFFT, q = 8
|S21|
2
2.2
2.4
2.6
Frequency (GHz)
2.8
3
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q/2
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
y
d
x
L4
L2
e
s
gd
L3 L 4
er = 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.
L1
L1
L5
Comparison of Analyses of The Ingerdigital capacitor
with Uniform and Nonuniform Grids
Table 3.2.2
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The measured and calculated S parameters of
the interdigital capacitor.
|S 11| and | S 21| (dB)
0
|S11|
-4
-8
|S21|
-12
2D-NUFFT
Measurement
-16
-20
1 2
3
4 5 6 7 8 9 10 11 12
Frequency (GHz)
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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
a
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, er = 10.8. All
dimensions are in
mm.
L12
L1
L6
b
y
L5
L11
L10
L8
L7
x
L4
L13
L3
L2
L1
L9
z
y of Analyses of LThe
Comparison
Wideband filter with
14
L15
L15
c
Uniform and Nonuniform Grids
t
er
Table 3.2.3
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The measured and calculated S
parameters of the wideband filter.
0
|S 11| and | S 21| (dB)
-10
-20
|S21|
|S11|
-30
-40
2D-NUFFT
Measurement [37]
-50
-60
2
4
10 12
6
8
Frequency (GHz)
14
16
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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 END
Thank You for your
Participation !
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