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IEEE SIGNAL PROCESSING LETTERS, VOL. 2, NO. 7, JULY 1995
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Application of Multidimensional Wave Digital
Filters to Boundary Value Problems
H. Krau, R. Rabenstein
Abstract | The application of multidimensional wave digital lter principles to the numerical solution of partial dierential equations has been introduced in various recent publications. However, the examples presented so far did not
explicitly consider boundary conditions. This letter shows
how to incorporate boundary values into a multidimensional
wave digital lter. A lossy transmission line of nite extent
with given terminations is chosen as an example.
T
I. Introduction
HE application of multidimensional wave digital lter
(MWDF) theory to the numerical solution of partial
dierential equations (PDEs) has been treated in various
recent publications such as [1], [2], [3]. For a comprehensive
survey of publications see e.g. [5].
In these contributions MWDF principles were applied to
physical problems described by linear PDEs with constant
and nonconstant coecients as well as to nonlinear problems, too. One starts from a given set of PDEs and derives
the structure of a MWDF which is the digital counterpart
of the original PDEs. The procedure is very systematic and
preserves many interesting properties of the PDEs. However, it does not take into account any boundary conditions.
Nonetheless there exist several possibilities for incorporating boundary values in the algorithm. According to [4],
there are two distinct approaches:
The (space{dependent) coecients of the MWDF
structure which describe the behaviour near the boundaries are set to values which force the fulllment of
the boundary conditions.
Wave quantities at discrete locations beyond the boundaries are considered as independent quantities. Their
values are determined through the use of the boundary
conditions.
The latter approach is the more exible and will be outlined
in this letter.
II. Problem Statement
We consider a transmission line of length A terminated
with a voltage source plus source resistance on one side and
a load resistance on the other side as shown in g. 1.
The electrical behaviour of the transmission line is described by the two partial dierential equations
@ u(x; t) + l @ i(x; t) + ri(x; t) = 0
(1)
@x
@t
H. Krau is with the Lehrstuhl fur Nachrichtentechnik, Universitat Erlangen{Nurnberg, Cauerstr. 7, D-91058 Erlangen, Germany,
Email: [email protected]
R. Rabenstein is with the Lehrstuhl fur Nachrichtentechnik, Universitat Erlangen{Nurnberg, Cauerstr. 7, D-91058 Erlangen, Germany,
Email: [email protected]
RS
i(0,t)
i(A,t)
u(0,t)
uS(t)
0
r,l,g,c
u(A,t)
A
RL
x
Fig. 1. Transmission line with specic terminations on both sides
@ i(x; t) + c @ u(x; t) + gu(x; t) = 0
(2)
@x
@t
for voltage u(x; t) and current i(x; t) along the line with
space coordinate x 2 [0; A] and time t 0. The coecients
are given in terms of the physical quantities: l (series inductance), c (shunt capacitance), r (serial resistance), and
g (shunt conductance), which may depend on x.
The terminations at x = 0 and x = A constitute the
boundary conditions to the PDEs (1,2). Their mathematical formulation is
x=0:
x=A:
u(0; t) = uS (t) ? RS i(0; t) ;
u(A; t) = RL i(A; t) :
III. Multidimensional Wave Digital Filter
(MWDF)
(3)
(4)
The key steps in the derivation of a MWDF for the numerical solution of (1,2) consist of the derivation of a multidimensional (MD) Kirchho circuit from the underlying
PDEs and the design of a MWDF for its discrete simulation. In the rst step the PDEs are interpreted as mesh
and node equations of a MD Kirchho circuit whose circuit
elements are an alternate way of expressing the operations
on u(x; t) and i(x; t) given by (1,2). Then in a second
step, principles of multidimensional wave digital ltering
are applied to the MD Kirchho circuit, which serves as
the analog reference circuit as it is usual in WDF practice.
The principles are: transition from voltage and current to
incident and reected voltage waves and application of a
generalized trapezoidal rule of integration [2]. The result
is a MD dierence equation that can be easily cast into a
computer program. Thus a MWDF for the system of PDEs
(1,2) is obtained.
The necessary steps of this approach have already been
treated in the literature for Maxwell's equations in two and
three space dimensions [2], [3] and simplied to the transmission line equations in [6]. So we just present the MWDF
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IEEE SIGNAL PROCESSING LETTERS, VOL. 2, NO. 7, JULY 1995
in g. 2 that results from the application of the above mentioned principles.
IV. Boundary Conditions
First we want to describe how to realize boundary condition (3) for x = 0. To this end we wish to express x2(0; kT )
in terms of the state quantities x1(0; kT ); x3(0; kT ), and
x4(0; kT ), which may be calculated without any problem.
Note that x1 ; : : :; x4 constitute the outputs of the spatial and temporal shift elements of g. 2. Unless stated
otherwise, all MWDF quantities in the sequel are valid for
(x; t) = (0; kT ).
We start with the series adaptor equations
b5 = a5 ? 5 (a2 + a5 ) ; b6 = a6 ? 6 (a4 + a6 )
Fig. 2. Multidimensional WDF for the simulation of the PDEs (1,2)
The structure is valid with the port resistances of the
3{port adaptors chosen as
R3 = gr22 ;
R1 = r ;
R2 = 2(l T? ) ;
2
R4 = 2(cr2T ? ) ; R5 = R6 = 2hr2 = 2T ;
(5)
where r2 and are positive constants of dimension and
As
2
Vm , respectively, that obey the restrictions l ; cr2
and the coupling of R5 and R6 in (5). Furthermore, h and
T denote the spatial and temporal step sizes.
The desired voltage u and current i at every grid point
(x; t) = (nh; kT ) are obtained from the incident and reected voltage wave quantities a and b of g. 2 by calculating
e.g.
b6
(6)
u = r2 a62?
R6
b5
(7)
i = a52?
R
5
Some remarks are appropriate concerning the operation
of the structure in g. 2. The structure has to be applied separately for every spatial sampling point x =
nh; h = NA , of interest; so N + 1 locations x 2 [0; A]
require N + 1 elements as shown in g. 2. Furthermore, one can see from g. 2 that in order to calculate
wave quantities at (x; t) = (nh; kT ) appropriate quantities at grid points ((n ? 1)h; (k ? 1)T ); (nh; (k ? 1)T ) and
((n + 1)h; (k ? 1)T ) are necessary. In other words, wave
quantities of the former time step located at the same
place and at adjacent points in space contribute to (x; t) =
(nh; kT ).
So far boundary conditions (3,4) have not been considered, whereas it must be stated that the mere application
of the structure of g. 2 at x = 0 and x = A would require
knowledge of wave quantities outside the actual transmission line, e.g. of b5 and b6 for (x; t) = (?h; (k ? 1)T ) to give
a5 and a6 for (x; t) = (0; kT ). The solution of this problem
will be covered next.
(8)
with adaptor coecients 5 and 6 of the left and right
series adaptors, respectively. The two coecients 5 and 6
can be calculated from the corresponding port resistances
of the adaptors as it is usual in WDF practice. Replacing
b5 and b6 in (6,7) by (8) gives
(9)
u = r22R6 (a4 + a6) ; i = 2R5 (a2 + a5 ) :
6
5
A closer look at the structure of g. 2 reveals the coupling
between wave quantitites and outputs of the shift elements
by
a2 = ?x3 ;
a4 = ?x4 ;
1
(10)
a5 = 2 (x1 ? x2 ) ; a6 = ? 21 (x1 + x2) :
Equ. (9) together with (10) yields
u = ? r42R6 (x1 + x2 +2x4 ) ; i = 4R5 (x1 ? x2 ? 2x3) : (11)
6
5
Note, that the state x2 in (11) cannot be computed for
x = 0 from the structure of g. 2, since b5 and b6 are not
available for (?h; (k ? 1)T ). However, we can compute
x2(0; kT ) from the boundary condition (3). By inserting u
and i from (11) into (3) and solving for x2, we get
x2 = ?4R6 uS + (RS 5 ?Rr26 )+x1r ? 2RS 5 x3 ? 2r26 x4 ;
S 5 2 6
(12)
which is the result we were looking for. Equ. (12) enables
us to calculate voltage and current at x = 0 and to satisfy
boundary condition (3) as well. Thus, in every time step
x1; x3, and x4 are calculated for location x = 0 with the
necessary operations of g. 2 and then x2 is determined via
(12) for x = 0.
In a similar way as described above, the expression
(13)
x1 = (RL5 ? r26R)x2+ +2RrL5 x3 ? 2r26 x4
L 5 2 6
is obtained, which guarantees computability and validity
of boundary condition (4) for x = A.
V. Example
As an example we consider a coaxial cable [7] of length
A = 10 km and with line parameters r = 46:8 m
=m, l =
KRAU AND RABENSTEIN: APPLICATION OF MULTIDIMENSIONAL WAVE DIGITAL FILTERS TO BOUNDARY VALUE PROBLEMS
5:6 H=m, g = 0 and c = 120 pF=m. The voltage source
has a bell-shaped form
uS (t) = sin2 t
[?1 (t) ? ?1 (t ? )]
p
and its source resistance RS = l=c = 216 is equal to
the characteristic impedance for high frequencies. The load
resistance RL = 104 RS implies that the line is terminated
by an approximately open circuit at x = A.
Simulating the line's behaviour with step sizesp h =
A=32 = 312:5 m and T = h=v0 = 8:1 s; v0 = 1= lc according to g. 2 and equ. (12,13) yields the voltage distribution shown in g. 3.
u(x,t) in V
0.5
0.4
0.3
0.2
0.1
0
0
10
8
0.2
6
0.4
4
0.6
2
0.8
t in ms
1
0
x in km
Fig. 3. Voltage simulation result
Besides the damping, which the pulse undergoes while
travelling across the line, one can clearly see the eect of
the two boundary conditions on the voltage distribution:
almost reection{free termination at the left end and open{
circuit termination at the right end.
VI. Summary
At rst, we have shown how the application of MWDF
principles results in a discrete algorithm for the simulation of a transmission line. Then it has been outlined how
boundary conditions imposed by the terminations of the
transmission line can be incorporated into the MWDF algorithm. This method also applies to the MWDF simulation
of other problems: Whenever the structure of a MWDF
calls for wave quantities beyond the boundaries, the boundary conditions are used instead, in order to compute the
state quantities at the boundaries. As the simulation result for our example shows, the proposed method works
excellently.
References
[1] A. Fettweis, G. Nitsche: Transformation Approach to Numerically Integrating PDEs by Means of WDF Principles, Multidimensional Systems and Signal Processing, 2, 1991, pp. 127-159
[2] A. Fettweis, G. Nitsche: Numerical Integration of Partial Differential Equations Using Principles of Multidimensional Wave
Digital Filters, Journal of VLSI Signal Processing, 3, 1991, pp.
7-24
3
[3] A. Fettweis: Multidimensional Wave Digital Filters for DiscreteTime Modelling of Maxwell's Equations, Int. Journal for Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 5,
1992, pp. 183-201
[4] A. Fettweis: Private Communication, November 1993
[5] A. Fettweis: Multidimensional Wave-digital Principles: From Filtering to Numerical Integration, in: Proceedings Int. Conf. Acoustics, Speech, and Signal Processing, Adelaide, Australia, April
1994, pp. VI-173{VI-181
[6] H. Krau, R. Rabenstein, M. Gerken: Simulation of Wave Propagation by Multidimensional Digital Filters, submitted for publication
[7] H. Mokhtari, A. Nyeck, C. Tosser{Roussey, A. Tosser-Roussey:
Finite dierence method and Pspice simulation applied to the
coaxial cable in a linear temperature gradient, IEE Proceedings{
A, Vol. 139, No. 1, Jan. 1992, pp. 39-41