# Sec11_NUFFT_I ```National Central University
Department of Mathematics
Numerical ElectroMagnetics
&amp;
Semiconductor Industrial Applications
11 NUFFT &amp; Applications
Ke-Ying Su Ph.D.
Part I : 1D-NUFFT
r
h3
s w
s
2N2
2N2-1
2
2N2+1
w
w
1
2
h1 r
0
w
s
21
21
h2 r
h2 r
3
s
11
w
11
s
1N1-1
w
1N1
s
s
t
1N1+1
1
a
1
h1 r
0
2
s
2
w
N
s
N+1
1
a
3/67
Outline : 1D-NUFFT
1. Introduction
2. NUFFT algorithm
3. Approach
4. Incorporating the NUFFT into
the analysis
5. Results and discussions
6. Conclusion
4/67
I. Introduction
Numerical methods
• Finite difference time
domain approach (FDTD)
• Finite element method
(FEM)
r
h3
3
s w
h2 r
2
h1 r
s
11
w
s
21
21
w
11
s
2N2
2N2-1
s
1N1-1
2N2+1
w
1N1
s
1N1+1
1
a
0
Analytical formulations
• Spectral domain approach (SDA)
• Singular integral equation (SIE)
• Electric-field integral equation (EFIE)
5/67
• Easy formulation in the form of algebraic
equations
• A rigorous full-wave solution for uniform
planar structures
r
h2
2
w
w
1
s
t
1
r
h1
0
2
s
2
w
N
s
N +1
1
a
6/67
1) Green’s function :
The spectral series has a poor convergence
 Asymptotic Extraction Technique
2) Galerkin’s procedure:
Both of Green’s function and Electric field are in
space domain or spectral dimain
SDA : number of operations  N2
Green’s function is in spectral domain and
Electric field is in space domain
 NUFFT : number of operations  N
7/67
II. NUFFT Algorithm
dj 
N f / 2 1
f e
iks j
k
k  N f / 2
(2.1)
for j = 1, 2, …, Nd, sj  [-, ],
and sj's are nonuniform; Nd  Nf
Idea: approximate each eiks in terms of values
at the nearest q + 1 equispaced nodes
j
k e
S j-q/2
iks j
q 1
    ( s j )e
S j-1 sj S j
i ( v j    q / 2 1) 2k / mN f
(2.2)
 1
S j+1
S j+q/2
where
Sj+t = (vj+t) 2/mNf
vj = [sjmNf /2]
8/67
The regular Fourier matrix
k e
iks j
q 1
    ( s j )e
i ( v j    q / 2 1) 2k / mN f
(2.2)
 1
for a given sj and
for k = -Nf/2, …, Nf/2-1
 ei ( v j q / 2) 2 (  N f / 2) / mN f  ei ( v j q / 2) 2 (  N f / 2) / mN f    1( s j )    N / 2ei(  N / 2) s 



 



 




 

i ( N / 21) s

ei ( v j q / 2 ) 2 ( N f / 21) / mN f  ei ( v j q / 2) 2 ( N f / 21) / mN f   q1( s j )  N / 21e



f
j
f
f
j
f
 Ar(sj) = v(sj)
where A : Nf(q+1)
 r(sj) = [A*A]-1[A*v(sj)] = F-1P
(2.3)
closed forms
where F is the regular Fourier matrix with size (q+1)2
9/67
Closed forms
 n
i

 i mn
m
e
e
, n
 n

i 2

Fn  
mN f
 1 e
 Nf ,
n

(2.4)
Choose k = cos(k/mNf)
 2  q  2 j  3 
 2  q  2 j  1 
sin 
  sin 
 
2m
2m




P ( s j )  i

i
2   q  2 j  3
2   q  2 j 1
1 e
i
mN f
where j = sjmNf/2  vj

1 e
i
mN f
(2.5)

(2.6)
10/67
The coefficients r
e
iks j

1
k
q 1

 1
r(sj) =
 ( s j )e
i ( v j    q / 2 1) 2k / mN f
(2.2)
0.7
0.6
F-1P
0.5
sj = 0.5207
q=8
rl( sj)
0.4
0.3
S j-q/2
S j-1 sj S j
S j+1
S j+q/2
0.2
0.1
0
where Sj+t = (vj+t) 2/mNf
-0.1
1
2
3
4
5
l
6
7
8
The q+1 nonzero coefficients.
11/67
9
1D-NUFFT
dj 
N f / 2 1
 fk e
iks j
(2.1)
S j-q/2
S j-1 sj S j
S j+1
S j+q/2
k  N f / 2
+
e

iks j

1
k
q 1

 1

( s j )e
i ( v j    q / 2 1) 2k / mN f
 N k / 21
1 i ( v j    q / 2 1) 2k / mN f 
d j     ( s j )   ( f k k ) e

 1
k  N k / 2

(2.2)
q 1
(2.7)
FFT
12/67
III. Approach
The spectral domain electric fields
~
~
~
~
 E z (  , n ) G zz (  , n ) G zx (  , n )   J z ( n ) 
 ~
~
~
  ~

E
(

,
n
)
J
(
n
)
G
(

,
n
)
G
(

,
n
)
xx
 x
  xz
 x 
y
r
h3
3
s w
r
h2
2
h1  r
s
11
w
s
21
21
w
11
s
2N 2
2N 2-1
s
1N 1-1
2N 2+1
w
1N1
s
1N1+1
1
a
0
The spectral domain Green’s functions
Asymptotic extraction technique
~
~
~ re
Gij (  , n)  Gij (  , n)  Gij (  , n)
~
~
Gij (  , n)  Cij (  )  Fij (n)
(2.8)
where  : propagation constant
i, j = z or x
13/67
x
Asymptotic parts
If the observation points and the source points are
• at the same interface : (n = n /a)
Gzz
10
Gzx
90
Gxx
2500
0
60
2000
-10
30
1500
-20
0
1000
-30
-30
500
-40
-60
-50
-60
-1.25
-0.75
-0.25
n
0.25
0.75
1.25
-90
-1.25
4
x 10
1
~
Fzz (n )   n
0
-0.75
-0.25
n
0.25
0.75
~
Fzx (n )   n0
1.25
4
x 10
-500
-1.25
-0.75
-0.25
n
0.25
0.75
1.25
4
x 10
~
Fxx (n)   n
(2.9)
• at different interfaces :
1
~
~
~
Fzz (n)  e nh2  n , Fzx (n)  e nh2  n0 , Fxx (n)  e nh2  n
14/67
(2.10)
Current basis functions
N  N b 1
J z ( x)    aip
Tp ( X i )
i 1 p  0
1 X
2
i
Tp (x)
1
p
p
p
p
p
(2.11a)
0.5
=0
=1
=2
=3
=4
0
N  N b 1
J x ( x)    bip 1  X i2U p ( X i )
(2.11b) -0.5
i 1 p  0
wi
wi
 2( x  x i )
,
x


x

x

i
i
 w
2
2
i
Xi  
 0,
otherwise .

(2.11c)
-1
-1
-0.5
0
0.5
1
0.5
1
Up (x)
5
p
p
p
p
p
4
3
2
=0
=1
=2
=3
=4
1
0
-1
Chebyshev polynomials
Bessel functions
-2
-3
transform
-4
-1
-0.5
0
15/67
Expansion E-field
~
Spectral domain E
Space domain E
transform
~
~
 ~


 E z (  , x )
Gzz (  , n ) Gzx (  , n ) J z (n )  in x
 ~
~
e

   ~
 E x (  , x ) n  Gxz (  , n ) Gxx (  , n )  J x (n )
~
~
~ re
Gij (  , n)  Gij (  , n)  Gij (  , n)
 E z (  , x)   E (  , x)   E (  , x) 


 E (  , x)  
E
(

,
x
)
E
(

,
x
)
 x
 
 


z

x
re
z
re
x
y
r
h3
3
s w
r
h2
2
h1  r
s
11
w
s
21
21
w
11
s
2N 2
2N 2-1
s
1N 1-1
2N 2+1
w
1N1
s
1N1+1
1
a
0
(2.12a)
16/67
x
Expansion E-field


 aipC zz (  ) Ezzip
( x)  bipC zx (  ) Ezxip
( x) 
 Ez (  , x)

 
   


a
C
(

)
E
(
x
)

b
C
(

)
E
(
x
)
E
(

,
x
)

xzip
ip xx
xxip
 x
 i p  ip xz
~
~
~
 E zre (  , x)  G zzre (  , n) Gzxre (  , n)  J z (n)  i n x
 ~
e
 re
    ~ re
~ re
 E x (  , x) n G xz (  , n) Gxx (  , n)  J x (n)
if the observation fields and the currents are
at the same interface :
E

xtip
( x)   Bp ( n wi / 2)e
n
where t = z or x
j n ( x  xi )
E ( x)  

ztip
n
B p ( n wi / 2)
n
(2.12b)
(2.12c)
e j n ( x xi )
(2.13)
closed forms
at different interfaces : numerical calculations
17/67
Unknown coefficients aip’s and bip’s
Galerkin’s procedure

strip i
E z (  , x)
Tp ( X i )
1  X i2
dx  0
2
E
(

,
x
)
1

X
i U p ( X i ) dx  0
 x
(2.14a)
(2.14b)
strip i
for i = 1, …, N, and p = 0, 1, …, Nb – 1

a matrix of 2NNb2NNb
18/67
IV. Incorporating the NUFFT
into the analysis
~
~
 ~


 E z (  , x )
Gzz (  , n ) Gzx (  , n ) J z (n )  in x
 ~
~
e

   ~
 E x (  , x ) n  Gxz (  , n ) Gxx (  , n )  J x (n )
N  N b 1
N  N b 1
i 1 p  0
i 1 p  0
Ez ( x)    aip Ezzip ( x)   bip Ezxip ( x)
y
r
h3
3
s w
N  N b 1
Ex ( x)    aip Exzip ( x)   bip Exxip ( x)
i 1 p  0
r
h2
N  N b 1
2
h1  r
0
s
11
w
s
21
21
w
11
s
2N 2
2N 2-1
s
1N 1-1
2N 2+1
w
1N1
s
1N1+1
1
a
i 1 p  0
(2.15)
19/67
x
Let
Eztip ( x) 
N g 1
d
q 0
T (X j)
jq q
Extip ( x) 
N g 1
h
q 0
Uq ( X j )
jq
(2.16)
where t = z or x
Then
1
d jq 
2 Eztip ( x)Tq ( X j )
 
1
k 
2k  1
,
2N g
1 X
2
j
dX j 
2
Ng
N g 1
E
k 0
ztip
( x jk ) cos q k
k  0, 1, ..., N g  1
where xjk = xj + (wj/2)cosk
(2.17)
20/67
Number of operations for MoM
: Ns(2NNb)2
the proposed method : 2NNb[mNflog2(mNf)]
NUFFT
~
~
 ~


 E z (  , x )
Gzz (  , n ) Gzx (  , n ) J z (n )  in x
 ~
~
e

   ~
 E x (  , x ) n  Gxz (  , n ) Gxx (  , n )  J x (n )
r
h3
3
s
1
h2
r
h1
r
0
w s w
1
2
2
s
N-1
w
N
s
N+1
2
1
a
21/67
Finite metallization
thickness
Mixed spectral domain
approach (MSDA)
N+2
h2 r
1
M N +2,b

N+1
M N +2,b
M N +2,b

h2

2
w
w
1
s
t
1
h1 r
0
2
2
s
2
Mt
w
N
s
1
1
N+1
2
N+1
Mt
Mt
2
N+1
Mb
1
Mb
2
M0,t
1
M 0,t
t
N+1
Mb
1
a
2
N+1
0
M 0,t
22/67
h1
V. Numerical Results
Validity Check
r
h2
2
r
h1
s1 w1 s2w2
s8 w8 s9
1
0
Table 2.1
Convergence Analysis and Comparison of the CPU time for a Quasi-TEM
Mode of an Eight-Line Microstrip Structure Obtained by the Traditional SDA
and the Proposed Method.
(Nb = 4 )
The result of HFSS is 2.6061 (33 seconds).
r1 = r2 = 8.2, r3 = 1, a = 40, h1 + h2 = 1.8, h3 = 5.4, w1 through w8 be 0.26, 0.22, 0.18, 0.14, 0.16, 0.2, 0.24 and 0.28, and s1 through
23/67 s9 be
18.495, 0.25, 0.21, 0.17, 0.15, 0.19, 0.23, 0.27, and 18.355. All dimensions are in mm
a
h3
Validity Check
r
3
h2
r
h1
r
s1 w s2
2
1
0
Table 2.2
Validity Check of the Modal Solutions Obtained by the Proposed Method.
Structure in Fig.1(a): r1 = r3 = 1, r2 = 8.2, a = 18, w = 1.8, s1 = s2 = 8.1,
h1 = h2 = 1.8 and h3 = 5.4, all in mm.
r1 = r3 = 1, r2 = 8.2, a = 18, w = 1.8, s1 = s2 = 8.1, h1 = h2 = 1.8 and h3 = 5.4, all in mm.
24/67
a
Modal Propagation Characteristics
3
Single-Line
Structure
2.5
h2 = 1.5 mm
h2 = 1.27 mm
h2 = 1.1 mm
/k0
2
1.5

1
0.5
0
/k0
-0.5
h
w
r
-1
d
a
-1.5
-2
10
w
s
12.5 a
r
w
h
15 d 17.5
20
Frequency (GHz)
22.5
a = 12.7 mm, w = 1.27 mm, h1 = 0 mm, h3 = 11.43 mm, s1 = s2 = (a – w)/2, r = 8.875.
25
25/67
Eight-Line Structure
2.24
2.75
1
1
s
2
r
w
8
h1
h2
1
a
2.6
3
2.45
)
/k0 (
2.19
/k (
)
2.215
7
0
w s w
4
2.165
2.14
1
2.3
2
5
10
Frequency (GHz)
5
6
7
15
Structural parameters are identical to those of Table 2.1.
8
2.15
20
26/67
Eight-Line Structure : normalized currents
at 10 GHz
1
4
1.5
5
1.5
1
0.5
0.5
0.5
0
x (m m )
1
-1
-1
5
4
z6
-1
-2
0
x (m m )
-1.5
-2
2
-1
1
2
x 10
0
x (m m )
1
-1.5
-2
2
5
8
x 10
5
4
3
2
4
2
1
2
1
1
6
0
-1
0
-2
-4
-2
-6
-3
0
1
2
2
x 10
1
2
6
-1
-3
x (m m )
1
0
-2
-1
0
x (m m )
8
3
-4
-2
-1
7
z7
J (x)/I
0
z5
J (x)/I
1
1
1
-1
1
6
2
-3
-2
0
J (x)/I
x 10
1
-1
x (m m )
5
z4
z3
-0.5
-1
x 10
0
-0.5
5
3
0
-0.5
-1.5
-2
2
J (x)/I
J (x)/I
0
1
-1
1.5
z8
0
-2
5
1
z2
z1
J (x)/I
J (x)/I
5
x 10
1
1
1
10
x 10
4
J (x)/I
x 10
3
1
15
2
-8
-2
-1
0
x (m m )
1
2
-4
-2
-1
0
x (m m )
27/67
Suspended Four-Line Structure : mode 1(odd) &amp; 2
(even)
2.16
s = 0.1mm
s = 0.2mm
s = 0.3mm
s = 0.4mm
2.155
/k 0
2.15
mode 1
2.145
2.14
r3
h3
2.135
mode 2
s1 w1 s2 w2
h2
r2
h1
r1
w4 s5
a
0
2.13
20
0
4
8
12
16
20
Frequency (GHz)
h1 = h2 = 1 mm, h3 = 18 mm, r1 = r3 = 1, r2 = 8.2, w1 = w2 = w3 = w4 = 0.2 mm, s1 = s5 = 10 mm, s2 = s3 = s4 = s.
28/67
Suspended Four-Line Structure : mode 3 (odd) &amp; 4 (even)
2.15
2.05
mode 3
s = 0.1mm
s = 0.2mm
s = 0.3mm
s = 0.4mm
1.85
mode 4
/k 0
/k 0
1.95
1.75
r3
h3
1.65
s1 w1 s2 w2
r2
h2
r1
h1
a
0
1.55
0
4
8
12
w4 s5
16
20
Frequency (GHz)
29/67
Dual-level Eight-Line Structure
3.2
2.925
r3
/k 0
h3
h2 r2
3
s21 w21
s23 w24 s25
s11 w11 s13 w14 s15
5
h1  r1
2.65
1
2
4
a
0
6
2.375
7
2.1
8
1
5
10
15
Frequency (GHz)
20
r1 = 10.2, r2 = 8.2, r3 = 1, a = 40, h1 = 1.27, h2 = 0.53, h3 = 5.4, w11 through w14 are 0.22, 0.14, 0.2, and 0.28, w21 through w24 are 0.26, 0.18,
0.16, and 0.24, s11 through s15 are 19.005, 0.56, 0.5, 0.74, and 18.355, and s21 through s25 are 18.495, 0.68, 0.46, 0.62, and 18.905.
All
30/67
dimensions are in mm.
Dual-level Two-Line Structure
3.5
Mode 2
/k0
3.3
3.1
2.9
h3
Mode 1
2.7

h2 = 0.1905 mm
h2 = 0.127 mm
h2 = 0.0635 mm
h2
r3
s21 w21 s22
 s11
w11 s12
r2
h1  r1
a
0
0
20
40
60
Frequency (GHz)
80
a = 25.4 mm, h1 = w11 = w21 = 0.127 mm, h3 = 25.146 mm, s11 = s22 = 12.895 mm,
s12 = s21 = 12.378 mm, r1 =r2 = 12, and r3 = 1.
100
31/67
Coupled lines with finite metallization
Effective Dielectric Constant
11
Measurement 
10
Even mode
h2 r
9
s1
t
t/h1 = 0.01, 0.044, 0.08,
2
h1  r
w1
s2
w2
s3
1
a
0
0.122, 0.18, 0.25
8
7
Odd mode
6
5
10
15
20
Frequency (GHz)
r1 = 12.5, r2 = 1, w1 = w2 = s2, h1 = 0.6 mm, h2 = 10 mm, and s1 = s3 = 6 mm.
25
32/67
Coupled lines with finite metallization : @ 5 GHz
Table 2.3
Convergence Analysis and Comparison of the CPU time for an
Odd Mode of a Pair of Coupled Lines with t/h1 = 0.01 Obtained
by the MSDA and the Proposed Method.
33/67
VI. Conclusion
• NUFFT and asymptotic extraction technique are
used to enhance the computation.
• Very high efficiency is obtained for shielded
single and multiple coupled microstrips.
• The results have good convergence.
• Mode solutions with varying substrate heights,
microstrips at different dielectric interfaces or
finite metallization thickness are investigated
and presented.
34/67
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