On Daubechies Wavelet Based Time Domain Scheme Youri V Tretiakov and George W P a n D e p a r t m e n t of Electrical Engineering Arizona State University Abstmct Recently the waveletGalerkin time domain WGTD approach was pre posed by Cheong et al which signifleantly simplified the multiresolution time domain technique MRTD of Krurnphoh Katehi et al In this paper we provide a rigorous anal ysis of the WGTD employing positive Sam pling functions and their hiorthogonal dual This biotho......
On
Daubechies Wavelet Based
Time Domain Scheme
Youri
V.
Tretiakov
and
George
W.
P a n
D e p a r t m e n t
of
Electrical Engineering, A r i z o n a
State
University
Abstmct-
Recently, the wavelet-Galerkin
time domain (WGTD) approach
was
pre-
posed by Cheong
et
al.,
which signifleantly
simplified the multi-resolution time domain
technique (MRTD) of Krurnphoh, Katehi
et
al.
In this paper,
we
provide
a
rigorous anal-
ysis of the WGTD, employing positive Sam-
pling functions and their hiorthogonal dual.
This biothogonal system has exact interpo-
lation
properties
and
demonstratea
superi-
ority over the FDTD in terms of memory
and speed. Numerical examples and com-
pariaons t o the traditional FDTD results
are
provided.
I.
INTRODUCTION
The multiresolution time domain tech-
nique (MRTD) was proposed in [I] using
the Battle-Lemarie wavelets in space. In
the MRTD, the standard Galerkin proce-
dure converts the two vector curl equations
into a system consisting of six updating
equations similar to the traditional FDTD
scheme of Yee
[Z].
For instance, one of the
six equations is
For
i
<
0
the coefficients
{a,)
are given by
the symmetry relation
a,
=
-a-1-,.
To
improve MRTD, the Wavelet-Galerkin
time-domain (WGTD) scheme was pro-
posed in
131,
employing Daubechies scaling
functions of order two
(Dz)
with the
141
support
[0,3].
In this paper the positive
sampling
basis and
its
hiorthogonal dual
tesing fuctions are constructed, employing
the Daubechies
D2
scaling functions. Ow-
ing
to the exact sampling property of the
basis functions and their biorthogonal test-
ing functions, the error estimate can be eas-
ily conducted.
11.
W G T D SCHEME
BASED
UPON
SHIFTED
DAUBECHIES SCALING
Dz
FUNCTIONS
It was remarked by mathematicians
151,
that shifted Daubechies
Dz
scaling func-
tions has approximate sampling properties,
namely
d(k
+
MI)
6r,o
(3)
where
MI
=
J_',"
z+(z)dz.
Chong
et
al.
immediately recognized
these approximate sampling properties, and
constructed the Wavelet-Galerkin time do-
main (WGTD) algorithm
131.
To make use
of the shifted interpolation property
(3)
we
recall the following expansion functions
L.-1
--
Az
i=-L,
~1L+1/2~,1,2,......+i+i/.]
c
(1)
where
L,
denotes the effective support size
of the basis function
$(z).
It has been
found numerically
[l]
that coefficients
{a,)
for
i
2
9
and
i
5
-10
are negligible, that
is
L.
9.
The coefficient
{ a i }
represents
the weight of the contributions to a node
by its neighbors and for
i
2
0
is calculated
numerically from
Plotted in Fig.1 are functions defined in
(4)
with
Az
=
1
and
m
=
0,1,2.
Due to
finite support
of
Daubechies scaling func-
tions, the number of nonzero coefficients
{a,}
is
also
finite. One can easily verify that
a
#
0
for
-3
5
i
5
2
and hence
L.
=
3
in
i
(1).
The numerical values of the coefficients
{ai}
have been tabulated in
[3].
From the above discussion it follows that
the use of Dauhechies scaling functions is
more computationally efficient than that
of
0-7803-7070-8/01/$10.00 0 2 0 0 1
IEEE
810
the above-defined sampling function
SmL(z)
Sm(k)
=
L , n .
(9)
The support of the sampling function
S( )
,z
is
[m
-
1,
+CO).
Notice that
a
sam-
is not orthogonal
pling function
{S,(z)}
with respect to its shift
Fig.
1 .
Daubechies shifted
scaling
functions
( N
=
2)
for
rn
=
0 , 1 , 2 .
Therefore the biorthogonal testing func-
tions
Q,(z)
were introduced
in
[GI,
such
that
Qn(x)
=
the Batt,le-Lemarie scaling functions. An-
other advantage
is
the interpolation prop-
crty
(3).
From
this property it follows that
E$(.
p ) $ ( n
-
p )
-
P
(11)
This will safe time when one has to sam-
ple field coniponcnts for further numerical
computations.
111.
TIIE
WGTD
SCHEME
BASED
ON
BIORTHOGONAL SAhlPLINC
BASE
Due to the finite support of the Daubechies
scaling functions we can simplify expression
(11)
to the following compact form
Qn(2)
=
$(~-n+2)$(2)+$(~-~~+1)$(1).
(13)
To
construct Daubechies based biort,hog- From
(13)
it follows immediately that
onal sampling basis we use the following
ex-
Qn(z)
is supported on
[71
-
2 , n
21.
We plotted the sampling function
S ( x )
=
prcssion in
[6]
for
a
positive sampling func-
So(z)
and biorthogonal function
Q(z)
=
tion
BO(.)
in Fig.
1.
+
wlierc
$(z)
is
t,he Daubcchies scaling func-
t,ion, not necessary just
Dz.
However,
i n
rest
of
the paper
w e
shall rcstrict the
Daubcchies scaling functions to only
Dz.
~ 1 1
parameter
U
is
equal to -I/&. w e
e
$(2)
=
notice here bhat
$(l)
=
(see
[4]
dct,ails) and thus we can rewrite
Cor
(6)
in
t,he form
e, e
(7)
'
I
.
S
~
,
,
0
,
>
.
z
I
4
5
I
Fig.
2.
Sampling
function
S(z)
and
biorthogonal
function
Q(z)
We define the shifted versions of the sam-
pling function
S(z)
as
LL(z)
=
S(z
-
m).
For
the WGTD scheme we use the follow-
ing hasis
for
expansion
s( )
,z
=S
(2
-
m)
(14)
(8)
and the biorthogonal testing functions
I t can be shown analytically that the
follow-
ing cxact interpolating property
holds
for
81
1
After applying the standard Galerkin pro-
cedure, The two vector curl Maxwell's equa-
tions becomes six updating expressions,
analogous to (1) with coefficients
{ a , }
given
by
ai
=
1
:
Q - i
a s
(
l / z ( z
)
dx,
x
~
(16)
Fig.
3.
Two-dimesianal
parallel
plate resonator.
f u ~ ~ o n t o ~ ~ ~ ~ i ~ n ~ ~ t e o ~ u ~ ~ o r ~ ~ ~ l ~
testing function
Q(z),
the number of the
nonzero coefficients{a,} is small,
as
in the
E,
=
0
and
H,
=
0.
The dimesions are
case of WGTD technique with the shifted
a
=
2
m,
b
=
1
m and the time step
Daubechies basis. In fact, we obtained from
At
=
sec. The electric field values
(16) that for
i
2
3
all coefficients
{ a i }
are
E,
were sampled during the time period
exactly zeros. This is due to the specific
T,
=
216At
and the fast Fourier transform
supports of the functions
S(z)
and
&(z).
It (FFT) was performed to obtain the spec-
was
found numerically that
for
i
5
-4
all trum
of
the sampled field
E,.
Illustrated
coefficients
{a,}
in
(16)
are
negligibly
small.
in Fig.
4
are the numericall results ob-
For
-3
5
i
5
2, we have verified analyticaly tained with 15
x
7
=
105 Yee cells for both
that the following identity is true
FDTD and WGTD techniques, along with
analytical values. It can
be
seen clearly that
WGTD provides better agreement with an-
alytical solutions, though it is slower than
the FDTD approach. The computational
time is
8.93
s
for FDTD method and
39.89
s
for the WGTD.
To achieve the WGTD accuracy, we de-
This means that
L.
=
3
and
{ a , )
are ex-
cresed the size of the cell in the FDTD. As
actly the same
as
in
[3].
We conclude that the numerical results a result,
40
x
20
=
800 Yee cells for FDTD
will be exactly the same for technique in demonstrated the precision
of
the WGTD
the previous section and tecnique in this with
105
cells. The results are shown in Fig.
sections, although the sampling function
5.
The computational time for the FDTD
S,(x)
obeys the exact interpolation prop- increased to
66.36
sec due to increased num-
erty
(9).
The advantage of using the ber
of
Yce cells.
As
can be seen in the fig-
biorthogonal sampling system
is
that the ure, both methods give almost the same re-
error introduced in the truncation process sults for the eigenfrequency, but the WGTD
can be explicitly identified. Hence, the er- approach here is more efficient in terms of
ror bound can be estimated easily. In con- computational time and computer memory.
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