Accurate time-domain semisymbolic analysis

Abstract

The paper deals with a method for accurate semisymbolic time-domain analysis of highly idealized linear lumped circuits. Pulse and step responses can be computed by means of the partial fraction decomposition. The procedure relies on an accurate computation of poles of the transfer function. The well known problem of the QR and QZ algorithms is their poor accuracy in the case of multiple roots. Moreover, the partial fraction decomposition itself is an ill-posed problem for closely-spaced clusters of roots. The method presented in this paper is based on an improved reduction procedure for transforming the generalized eigenproblem into a standard one in combination with an algorithm for computing the Jordan canonical form of inexact matrices.

Full text

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Accurate Time-Domain Semisymbolic Analysis

Zdenek Kolka, Dalibor Biolek, Viera Biolkova
Faculty of Electrical Engineering and Communication
Brno University of Technology
Brno, Czech Republic
kolka@feec.vutbr.cz


Abstract— The paper deals with a method for accurate
semisymbolic time-domain analysis of highly idealized linear
lumped circuits. Pulse and step responses can be computed by
means of the partial fraction decomposition. The procedure relies
on an accurate computation of poles of the transfer function. The
well known problem of the QR and QZ algorithms is their poor
accuracy in the case of multiple roots. Moreover, the partial
fraction decomposition itself is an ill-posed problem for closely-
spaced clusters of roots. The method presented in this paper is
based on an improved reduction procedure for transforming the
generalized eigenproblem into a standard one in combination
with an algorithm for computing the Jordan canonical form of
inexact matrices [1].
Keywords-linear
eigenvalues; pulse and step responses
circuits; inverse Laplace transform;
I.
INTRODUCTION
Pulse and step responses of lumped linear systems can be
expressed semisymbolically, i.e. as a formula which, in
addition to the numerical constants, also contains the symbol of
time t. For a transfer function in rational form the inverse
Laplace transform can be performed using the well-known
partial fraction decomposition. The procedure relies on an
accurate computation of poles of the transfer function and on
reliable identification of pole multiplicity.
The pole-zero analysis is very sensitive to the rounding of
input data as well as to numerical errors of floating-point
operations. The problem appears naturally during the analysis
of large systems and also during the analysis of relatively small
but highly idealized circuits that often lead to multiple
eigenvalues. As shown in [2], computational errors can have
severe consequences for the semisymbolic analysis of time-
domain responses.
One of the most difficult problems is the computation of
multiple roots. In fact, the question whether there exists a
multiple eigenvalue is an ill-posed problem [3]. Using any
numerical precision we always get a cluster of eigenvalues.
The method of secondary root polishing [2] applies a heuristic
procedure to identify multiple roots from closely-spaced
clusters. But the result still depends on the accuracy of
computed roots.
There is an accuracy limit to computing multiple roots [4]:
to calculate an m-fold root to the precision of k correct digits,
the accuracy of the coefficients and the machine precision must
be at least m?k digits. This property naturally leads to the use of
multiprecision arithmetic to maintain the required accuracy, but
at the cost of highly increased computational time [5].
The first step of the time-domain semisymbolic analysis is
the pole-zero analysis, which leads to the generalized
eigenvalue problem. We will use our enhanced procedure [6]
for reducing the matrix pencil of the generalized problem to a
standard one. The reduction uses the Singular Value
Decomposition for explicit rank estimation with the aim of
avoiding spurious roots.
Subsequently, the resulting standard eigenvalue problem is
solved using a new algorithm of Z. Zeng for computing the
Jordan canonical form of inexact matrices [1]. Unlike classical
methods, which treat multiple roots as a collection of single
ones, the algorithm [1] estimates the multiplicity structure first
and then finds the respective eigenvalues using a well-
conditioned iteration scheme. The result is an unprecedented
accuracy that breaks the attainable accuracy barrier.
Section 2 of the paper describes the algorithm and Section 3
gives some numerical examples.
II.
TIME-DOMAIN SEMISYMBOLIC ANALYSIS
A. Basic concept
Let F(s) be a network function of the circuit being
analyzed. The pulse response g(t) and the step response h(t) can
be obtained using the inverse Laplace transform

g(t) = £-1{F(s)}, h(t) = £-1{F(s)/s}. (1)
Let Y(s) be a function given either as Y(s) = F(s) or as
Y(s) = F(s)/s depending on the response to be computed. For
lumped circuits we obtain

...
)(
)(
~
P
)(
)(
)(
10
1
???
?
??
?
?
i
sdd
ps
s
K
sQ
sP
sY
r
m
i
i
, (2)
where pi is generally a complex pole with multiplicity mi, r is
the number of distinct poles, and K is a real multiplying
coefficient. The fraction on the right-hand side of (2) is proper,
This work has been supported by the Czech Science Foundation under
grant No. GA 102/08/0784 and by the Czech Ministry of Education under
research project OC09016, which is a part of the COST Action IC0803
RFCSET. The research leading to these results has received funding from the
European Community's Seventh Framework Programme (FP7/2007-2013)
under grant agreement No. 230126.
2010 XIth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design (SM2ACD)
978-1-4244-6815-7/10/$26.00 ©2010 IEEE

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i.e. the degree of the numerator is lower than the degree of the
denominator,

))(
~
P
deg(
1
sm
r
i
i?
?
?
. (3)
The numerator
long division of P(s) by Q(s). The real quotients di of the long
division directly lead to the Dirac pulses in the output response.
The fractional part of (2) can be decomposed into a sum of
partial fractions. Due to the linearity of the Laplace transform
the time-domain response will be a sum of inverse transforms
of individual partial fractions.
For a pole pi we obtain mi partial fractions in the form
)(
~sP
can be obtained as a remainder after the

??
??
i
i
m
i
m
p
i
i
s
c
ps
c
ps
c
?
?
??
?
?
...
2
21
. (4)
The coefficients ci are generally complex. An accurate
algorithm for the expansion, which does not require the
evaluation of polynomial coefficients, can be found in [7].
The inverse Laplace transform of one partial fraction from
(4) for a real pole is

??
??
tp
j
j
j
i
j
p
i
et
j
c
s
c
1
!1
ˆ
?
?
?
?
,
0
?
t
. (5)
For a complex conjugated pair
fractions (4) appear in conjugated pairs. The inverse transform
of one pair leads to
iii
jp
? ? ??
,
iii
jp
? ? ??

??
?
?
? )
j
arg( cos
!1
2
j
ˆ
?
)()(
1
i
t
j
j
j
i
j
p
j
i
j
p
ctet
c
?
s
c
s
c
i
?
?
?
?
?
?
, (6)
The numerical computation of multiple roots is an ill-posed
problem. Using any numerical precision we always get a
cluster of eigenvalues. A typical situation is illustrated in Fig. 1
for a zero eigenvalue with multiplicity 5 computed using the
QZ method [8].
-0.10 0.1
-0.1
-0.05
0
0.05
0.1

Figure 1. The typical effect of finite precision on multiple eigenvalue.
(circle – 5-fold eigenvalue, asterisk – numerically determined cluster)
The partial fraction decomposition is numerically unstable
for such closely-spaced clusters. Let us assume that the
eigenvalue algorithm found a cluster of single or conjugated
poles instead of a multiple one. Then, the Heaviside method
can be used to find the coefficients of expansion

?
?
i
?
?
n
ii
i
pspQ
pP
sQ
sP
1
) )( ( '
)(
~
)(
)(
~
, (7)
where pi are distinct poles (roots of Q), and Q’ is the derivative
of Q with respect to the Laplace operator s. According to the
Gauss-Lucas theorem [9] the roots of the polynomial derivative
lie within the convex hull of the original roots of Q, i.e. within
the cluster around the exact multiple eigenvalue. Thus the
value of Q’(pi) is given by numerical errors of the eigenvalue
algorithm, which is essentially an arbitrary number.
To overcome the problem, various polishing methods have
been proposed. In principle, they heuristically identify closely-
spaced clusters as one multiple eigenvalue. Nevertheless, the
result always depends on the (poor) accuracy of computed
roots.
The next chapter describes briefly an application of a novel
algorithm [1], which estimates the multiplicity structure first
and then finds the respective eigenvalues using a well-
conditioned iteration scheme. As a result, the partial fraction
decomposition is well-conditioned too.
B. Accurate analysis of multiple roots
The pole-zero analysis leads to the generalized eigenvalue
problem. As the available algorithm [1] is only for standard
eigenvalue problems, the first step begins with transforming the
matrices.
Let (A, B) be a matrix pencil whose eigenvalues are zeros
or poles of a network function. The procedure for obtaining
(A, B) from the circuit matrix can be found in [10]. Let us write

BAC
ss
??
)(
(8)
Our goal is to transform in two steps the matrix pencil into
a matrix Q which has (approximately) the same eigenvalues.
The first step converts the matrix C to form (9) using a
procedure similar to the Gaussian elimination, and the second
step reduces the matrix C(I) to form (10) by the pivotal
condensation. I is the identity matrix.

??
?
?
??
?
?
?
?
2221
I
s
12 11
A
?
(I)
)(
A
AIA
C
s
s
, (9)

QC
s
?
)(
(II)
. (10)
Although the eigenvalues of C can be well-defined,
performing the conversion in finite precision often produces
rounding errors, which lead to additional spurious eigenvalues
of very big absolute values. To eliminate the effect, we
compute beforehand the rank of B by the Singular Value
Decomposition, which is known to be the most reliable
algorithm for this task [11]. Details of the reduction procedure
can be found in our paper [6].
2010 XIth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design (SM2ACD)

Page 3

The reduction algorithm takes the linear matrix pencil and
transforms it into a single matrix Q, whose eigenvalues
approximate the finite eigenvalues of the original problem.
This matrix is then passed
ApproxiJordanForm for computing the Jordan canonical form
of inexact matrices, which is a part of the ApaTools package
[12]. First, the procedure estimates the eigenvalue multiplicity
structure (or equivalently finds the Jordan blocks) and then
computes the eigenvalues, using a well-conditioned iteration
process [1].
to the algorithm
III.
NUMERICAL EXAMPLES
The proposed method of semisymbolic time-domain
analysis was tested in Matlab (v7.8). Figure 2 shows a
benchmark circuit - two-stage filter designed using the FIR-BL
(BiLinear) approximation [13]. Considering ideal operational
amplifiers and the following equalities
C11 = C21 = C31 = C12 = C22= C32 = C,
R11 = R21 = R31 = R41 = R21 = R22 = R32 = R42 = R,
R51 = R52 = 4 R21 = 4 R22
the transfer function of each stage is

??
2 , 1
?
,)(
2
0
2
zi
2
?
?
?
?
i
s
s
asK
ii
?
(11)
where
i
i
i
R
R
a
6
4
?
,
R
R
RC
i
zi
6
1
?
?
,
2 , 1
?
i
,
RC
2
1
0??
. (12)
The transfer function has two conjugate zeros and one
quadruple real pole

4
4 , 3 , 2 , 1
p
5
4 , 3
z
5
2 , 1
z
109 6179775280. 5
?
101 5405087109 . 2
j
104 9895302776 . 7
j
??
???
???
(13)
Additionally, the matrix pencil of the denominator of the
voltage transfer function is defective, i.e. the eigenspace
dimension is only one whereas the pole is fourfold.
Table I shows the results of benchmark analysis of poles for
different numbers n of sections obtained using the QZ
algorithm of Matlab and the reduction method with
approximate Jordan decomposition (RedJord), see Section II.
The exact pole value was computed using (12).
In the ideal case, for n sections we obtain a 2n-fold pole. As
the QZ algorithm provides only a cluster of poles, the error
shown is the maximum relative distance from –?0,
max
??
. The RedJord algorithm found the pole
multiplicity correctly in all cases, providing accuracy
comparable to the machine precision (eps) of double variables.
Deviations in accuracy of the RedJord algorithm in Table I are
0
ip
in the order of machine eps. The QZ algorithm lost its precision
quickly for higher multiplicities.
R11
890
R12
890
R21
890
R31
890
R41
890
C11
10n
C21
10n
R61
45k
Input
0
0
0
0
0
0
0
C31
10n
R51
3.56k
R22
890
R32
890
R42
890
C12
10n
C22
10n
R62
4.55k
0
0
0
0
0
0
C32
10n
R52
3.56k
Output
0

Figure 2. Idealized two-stage active filter.
TABLE I.
MAXIMUM RELATIVE ERROR OF COMPUTED POLES
n
QZ
RedJord
n
QZ
RedJord
1 2 3
2.6 ? 10-16
3.9 ? 10-16
1.6 ? 10-8
1.2 ? 10-15
4.5 ? 10-5
1.8 ? 10-15
4 5 6
4.0 ? 10-3
0.0
1.8 ? 10-2
3.9 ? 10-16
2.1 ? 10-2
0.0

Figure 3 shows the ability of RedJord to separate nearby
multiple roots. The circuit in Fig. 2 was analyzed with different
values of capacitors for each section
C11 = C21 = C31 = C1 ,
C12 = C22= C32 = C2 .
The value of C1 was firmly set to 10 nF, while the value of
C2 was swept in close vicinity of C1 from 0.9985 to 1.0015 of
the nominal value of 10nF. The poles are then

1
2 , 1
p
2
1
RC
??
,
2
4 . 3
p
2
1
RC
??
. (14)
Theoretically, it is only for C2 = 10nF that the circuit has a
quadruple pole. For other values it has two double poles. The
horizontal axis in Fig. 3 is the exact distance between p1,2 and
p3,4 during the sweep of C2 computed using (14). The vertical
axis shows the position of numerically determined p1,2 and p3,4
relatively to the theoretical value of p1,2 (14). If the distance
between poles is smaller than a certain value, which can be
2010 XIth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design (SM2ACD)

Page 4

specified by the user, the nearby double poles are identified as
one quadruple pole.
-100-50050100
-100
-50
0
50
100
p3,4 - p1,2 [rad/s]
pi - p1,2 [rad/s]

Figure 3. Analysis of two double poles by RedJord.
(asterisk–double pole, dot–quadruple pole)
The pulse response of the circuit in Fig. 2 in case of two
double real poles p1 and p2 will be in semisymbolic form
tptptptp
tecec tecectdtg
2211
2 , 2 1 , 22 , 11 , 11
)()(
??????
, (15)
where ????? ??? ???? ???????????? ???? ci,j are the coefficients of
partial fraction expansion. Table II shows poles computed for
C1 = 10 nF and C2 = 9.994985 nF (the first point of Fig. 3) and
the coefficients of partial fraction decomposition computed
using algorithm [7]. The exact values of ci,j were obtained from
analytical form of (15) using Maple. The difference between
the exact response and the response computed with RedJord
was 346, i.e. 0.18% at the peak, while the difference for the QZ
response was 1.7% apart from the fact that the structure of
poles was wrong. The occurrence of a complex pair in results
of the QZ algorithm leads to oscillating terms in the pulse
response (15), i.e. there will be a complex pair of coefficients
c1,2. All the results were computed with standard IEEE 754
double-precision arithmetic.
TABLE II.
COMPUTED POLES AND PFD COEFFICIENTS
RedJord
poles coefficients
-5.617977528093498?104 (double)

-5.626417153818045?104 (double)

c1,1 = -1.4463383375688?1017
c1,2 = 6.10280499938053?1018
c2,1 = 1.44633833756655?1017
c2,2 = 6.10374996265219?1018
QZ (Matlab)
poles coefficients
-5.626417153820619?104 (double)

-5.617977528089886?104
± j 9.427184263548398?10-4
c1,1 = 1.44633833402522?1017
c1,2 = 6.10374995294316?1018
c2,1 = -7.23169167013737?1016
? j 3.23681218990624?1021

IV. CONCLUSIONS
The advantage of the method presented if well-defined
behavior in the proximity of transfer functions with multiple
roots. Numerical experiments have shown the numerical
accuracy to be maintained even for defective circuit matrices
with high multiplicity roots.
0 0.10.2
t [ms]
0.3 0.4
0
0.5
1
1.5
2x 10
5
g(t)

Figure 4. Pulse response of circuit in Fig. 2 for C1 = 10 nF and
C2 = 9.994985 nF (without the Dirac pulse).
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2010 XIth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design (SM2ACD)