Modeling and Simulation of High Speed VLSI Interconnects: A Special Issue of Analog Integrated Circuits and Signal Processing An International Journal Vol. 5, No. 1 (1994)

103 Pages·1994·3.17 MB·English

by  Michel Nakhla

MODELING AND SIMULATION OF
HIGH SPEED VLSI INTERCONNECTS
edited by
s.
M. Nakhla
Q. J. Zhang
Carleton University
A Special Issue of
ANALOG INTEGRATED CIRCUITS
AND SIGNAL PROCESSING
An International Journal
VoI. 5, No. 1 (1994)
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Contents
Special Issue: Modeling and Simulation of High Speed VLSI Interconnects
Guest Editors' Introduction ................................... Michel Nakhla and Q.l. Zhang 5
Efficient Transient Analysis of Nonlinearly Loaded Low-Loss Multiconductor Interconnects ....... .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. /. Maio, S. Pignari and F. Canavero 7
A Simplified Synthesis of Transmission Lines with a Tree Structure .......................... .
· ......................................... D. Zhou, S. Su, F. Tsui, D.S. Gao and J.S. Cong 19
An Interconnect Model for Arbitrary Terminations Based on Scattering Parameters .............. .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Giri V. Devarayanadurg and Mani Soma 31
Electromagnetic Analysis of Multiconductor Losses and Disperion in High-Speed Interconnects ....
.............................................................................. Ke UU 47
Circuit Modeling of General Hybrid Uniform Waveguide Structures in Chiral Anisotropic
Inhomogeneous Media . . . . . . . . . . . . . . . . . . . .. Tom Dhaene, Frank Olyslager and Dania De Zutter 57
An Efficient, CAD-Oriented Model for the Characteristic Parameters of Multiconductor Buses in
High-Speed Digital GaAs ICs ............................................ Giovanni Ghione 67
Full-Wave Analysis of Radiation Effect of Microstrip Transmission Lines ...................... .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Guangwen Pan and Jilin Tan 77
Optimizing VLSI Interconnect Model for SPICE Simulation ....................... Juliusz Poltz 87
Statistical Simulation and Optimization of High-Speed VLSI Interconnects ..................... .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Qi-jun Zhang and M.S. Nakhla 95
ISBN 978-1-4613-6171-8 ISBN 978-1-4615-2718-3 (eBook)
DOI 10.1007/978-1-4615-2718-3
Library of Congress Cataloging-in-Publication Data
A c.I.P. Catalogue record for this book is available from the Library of Congress.
Copyright © 1994 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover 1s t edition 1994
AU rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise,
without the prior written permission of the publisher, Springer Science+Business Media, LLC.
Printed on acid-free pap er.
Analog Integrated Circuits and Signal Processing 5, 5-6 (1994)
© 1994 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Guest Editor's Introduction
As signal speeds increase, interconnect effects such as signal delay, distortion, reflections, and crosstalk become
dominant factors limiting the performance of VLSI systems. This problem can be found at chip level, printed cir
cuit board (PCB) level, multichip modules (MCM), and chip packaging. Over the past several years, significant
research activities have addressed several aspects of the problem, including interconnect modeling, simulation,
optimization, and characterization. This special issue of Analog Integrated Circuits and Signal Processing includes
several contributions representing recent advances in this strategic area authored by an international group of re
searchers from the United States, Canada, and Europe. The scope of these papers cover EM level modeling, simula
tion of transmission line networks, optimization and statistical design, and EM radiation effects.
A typical VLSI design at the chip or system level includes a large number of interconnects modeled as distributed
coupled transmission lines and linear and nonlinear lumped components. Simulation of such mixed distributed
and lumped circuits presents a challenging task for conventional circuit simulators. Several alternative simulation
techniques are proposed in this special issue. A combined time and frequency approach for the analysis of low-loss
interconnects with nonlinear terminations is addressed by I. Maio, S. Pignari, and F. Canavero. D. Zhou, et al.
present a closed-form solution for a tree structure of transmission lines. An interconnect macromodel for uniform
and nonuniform lines based on scattering parameters is presented by G. Devarayanadurg and M. Soma.
EM modeling of transmission line structures is discussed in two papers, one by K. Wu and one by T. Dhaene,
et al. Closed-form expressions of the line parameters based on quasi-TEM approximation are presented by G. Ghione.
G. Pan and J. Tan describe a method for full-wave analysis of radiation effect of high-speed interconnects.
The importance of matching the frequency range used for EM modeling with the frequency spectrum used during
circuit simulation is discussed by J. Poltz.
In the paper by Q.J. Zhang and M. Nakhla the statistical variations of the physical parameters of the interconnects
are considered. Techniques for yield estimation and design centering are presented.
The guest editors would like to acknowledge the efforts of the reviewers for their constructive suggestions and
detailed comments. These reviewers are E. Chiprout, G. Costache, S. Daijavad, R. Goulette, N. Jain, K.
Kalaichelvan, S. Manney, R. Mittra, M. Ney, L. Pillage, J. Poltz, T.K. Sarkar, J. Song, and R. Wang.
Michel Nakhla
Q.J. Zhang
6 Nakhla and Zhang
Michel S. Nakhla (S'73-M'76-SM'88) received the RSc. degree in Qi-jun Zhang received the REng. degree from the East China
electronics and communications from Cairo University, Egypt, in Engineering Institute, Nanjing, China, in 1982, and the Ph.D. degree
1967 and the M.A.Sc. and Ph.D. degrees in electrical engineering from McMaster University, Hamilton, Canada, in 1987, both in elec
from Waterloo University, Ontario, Canada, in 1973 and 1975, respec trical engineering. He was with the Institute of Systems Engineer
tively. During 1975, he was a postdoctoral fellow at the University ing, Tianjin University, Tianjin, China, from 1982 to 1983. He was
of Toronto, Ontario, Canada. In 1976 he joined Bell-Northern Re a research engineer with Optimization Systems Associates Inc.,
search, Ottawa, Canada, as a member of the scientific staff, where Dundas, Ontario, Canada, from 1988 to 1990. During 1989 and 1990
he became manager of the simulation group in 1980 and manager he was also an assistant professor (part-time) of electrical and com
of the computer-aided engineering group in 1983. In 1988, he joined puter engineering in McMaster University. He joined the Depart
Carleton University, Ottawa, Canada, where he is currently a professor ment of Electronics, Carleton University, Ottawa, Canada, in 1990,
in the Department of Electronics. His research interests include where he is presently an assistant professor. His professional interests
computer-aided design of VLSI and communication systems, high include all aspects of circuit CAD with emphasis on large-scale sim
frequency interconnects and synthesis of analog circuits. Dr. Nakhla ulation and optimization, statistical design and modeling, parameter
was the recipient of the Bell-Northern Research Outstanding Con extraction, sensitivity analysis, and optimization of microwave cir
tribution Patent Award in 1984 and in 1985. Currently he is the holder cuits and high-speed VLSI interconnections. He is a contributor to
of the Computer-Aided Engineering Industrial Chair established at Analog Methods for Computer-Aided Analysis and Diagnosis, (Marcel
Carleton University by Bell-Northern Research and the Natural Sci Dekker, 1988). Dr. Zhang is the holder of the Junior Industrial Chair
ences and Engineering Research Council of Canada. in CAE established at Carleton University by Bell-Northern Research
and the Natural Sciences and Engineering Research Council of
Canada.
Analog Integrated Circuits and Signal Processing 5, 7-17 (1994)
© 1994 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Efficient Transient Analysis of Nonlinearly Loaded Low-Loss
Multiconductor Interconnects
I. MAIO, S. PIGNARI AND F. CANAVERO
Politecnico di Torino, Dipanimento di Elettronica, Corso Duca degli Abruzzi 24, 1-10129 Torino, Italy
Received January 4, 1993; Revised May 10, 1993.
Abstract. The combined time-and frequency-domain analysis of nonlinearly loaded low-loss interconnects is ad
dressed. We show that a variety of interconnects commonly employed in different technological applications are
characterized by transfer functions, whose impulse responses have a fast initial-time structure (due to the skin
effect) and a slow long-time part (due to ohmic losses). The dependence of the impulse response structure on
the line parameters is discussed, along with the exact analytical solutions valid for the skin effect and ohmic losses,
separately. A piecewise linear approximation of the transient functions with nonuniform sampling is proposed as
an effective method to obtain high accuracy at low computational costs. Various numerical examples are used to
validate the effectiveness of the proposed representation, and to show that a matched characterization of the line
must be adopted in order to avoid numerical artifacts.
1. Introduction of convolution operation [1-9]. Besides, a basic feature
of this class of methods is that the terms of the solution
The study of multiconductor interconnects with non equations retain an evident physical meaning, so that
linear loads is of key importance for the analysis of their interpretation offers a way to check the final
complex electronic systems. Fast digital and analog cir results. The physical interpretation can also lead to sim
cuits, at any level of integration, offer a wide choice ple equivalent circuits [2] that can be solved by a stan
of examples of single and multiconductor transmission dard circuit simulator like SPICE, thus exploiting it as
lines connecting nonlinear devices. The decreasing rise a nonlinear solver.
time of signal waveforms emphasizes the importance In this paper, we concentrate on the transient analysis
of the propagation effects, and the signal corruption of low-loss interconnects defined as those for which
caused by parasitic phenomena as losses and skin effect RDC.£ < 2ZLC, where RDC is the line ohmic resistance
is now a relevant issue for many applications. As a per unit length, .£ is the line length, and ZLC is the
result, in the last years a growing interest has developed characteristic impedance of an associated ideal line
for precise and fast (possibly capable of a CAD imple without losses. A large number of interconnects appear
mentation) analysis methods to predict the behavior of ing in common technological applications satisfy the
lossy and dispersive multiconductor lines in nonlinear above condition (e.g., see Table 1 of [10] for typical
circuits. numerical values of RDcand .£). The low-loss assump
Among the possible analysis techniques, a great deal tion implies the influence of the skin losses in the line
of work was recently devoted to the development of transfer function: in Section 3, we show that the behav
mixed approaches that have some remarkable features. ior of the corresponding impulse responses present a
These methods exploit the natural formulation of both fast short-time structure, together with the usual long
linear and nonlinear electric systems; in fact, they adopt time shape of the curve, due to ohmic losses. Functions
a frequency-domain characterization for the line (which of this kind are difficult to accurately compute through
is the basic linear element of the system) and the time a numerical inversion of their frequency characteriza
domain characteristic equations for the loads (where tion: the relevance of the accurate approximation of the
the nonlinearities are), and combine the two by means line transient response is related to the ability to
8 Maio, Pignari and Canavero
describe such effects as the far end crosstalk on the loads are assumed to be multiport networks described
nonexcited lines. In order to provide a precise represen by the nonlinear vector characteristics -ip = gp(vp -
tation of the line impulse response over long time inter ep), where p = 1, 2 denotes the right and left line
vals with the minimal amount of data, a nonuniformly ends, respectively; ep is the vector of the voltage
sampled piecewise linear approximation of the func sources connected at the pth line end. The transmis
tions involved in the transient is proposed. The numeri sion line is characterized by its impedance matrix
cal examples discussed in Section 4 provide a validation z<fl(w) and its mode propagation functions hi(W)},
of our computation procedure. i = 1, ... , N [11].
Throughout the paper, lowercase letters are time A set of transient equations, based on the line scat
domain variables and uppercase letters indicate their tering parameters defined with respect to a reference
Zp,
counterparts in the frequency domain, e.g., impedance matrix can be obtained by introducing
if-I the voltage wave vectors ap and bp:
x(t) ;=> X(w) , vp = ap + bp
if
ip = y~ * (ap - bp), p = 1, 2 (1)
and if denotes the Fourier transform operation. Also,
where y~(t) is the transient expression of the reference
the boldface character is used for the collections of ele
*
admittance, and denotes the convolution operator. The
ments, and the superscript D distinguishes matrices
voltage waves at the two line ends are related through
from vectors, i.e., a indicates a time-varying vector and
the transient scattering equations
ZD is a matrix in the frequency domain.
bl = sI1r * a, + sTIr * ~
b2 = STIr * a, + sI1r * a2 (2)
2. Analysis Method
sI1r
where s~r and are the N X N transient scattering
The problem of analyzing the transient behavior of a matrices of the line. The equations describing the
multiconductor transmission line terminated on non reflections from the two terminal circuits that load the
linear loads is summarized in figure 1, where, for sim line are obtained by means of the load characteristics
plicity, a three-conductor line (two wires and a common
-y~ * (ap - bp) = gp(ap + bp - ep), p = 1, 2
voltage reference) is represented. Although figure 1 is
(3)
drawn for a three-conductor line, our formulation is
general. The unknowns of the problem are the vectors The problem of characterizing low-loss lines is best
Vb V2, ib i2, that contain the voltages and currents at described in the simple case of a symmetric line with
both line ends for each of the N wires. The terminal N = 2, which allows a modal decomposition into even
a2~
b2~
e
21
..
+
I
i21
V21
g2
e
22
.. .. ..
+
I
i22
V22
Fig. 1. Nonlinearly loaded three-conductor transmission line (N = 2). The relevant quantitites for our formulation are indicated. The voltage
sources epq, p, q = I, 2, account for the system excitation and the functions gp describe the (possibly nonlinear) loads.
Efficient Transient Analysis 9
and odd modes. The extension to N > 2 is straightfor pulse responses obtained via the inverse Fourier trans
ward, although the structures without a symmetry plane form of He and Hm respectively. In the following, to
require to take into account frequency-dependent modal simplify the notation, the suffixes e, 0 denoting the even
vectors. In the N = 2 even-odd case, the modal quan and odd modes are dropped, wherever the discussion
tities Xm (e.g., voltages, or currents, or voltage waves) and formulas apply to both modes.
are related to the corresponding physical quantities x by
J
[xeJ
x = = EOx EO = _1 [1 1 (4) 3. Impulse Responses of Low-Loss Lines
m Xo ' -J2 1 -1
The features of the impulse response h(t) are deter
EO being the self-orthogonal matrix of the mode vec
x;; mined by the characteristics of the functions R(w),
tors. Similarly, the modal matrix parameters (e.g.,
C(w), L(w), and G(w), that represent the frequency
scattering or impedance matrices) are related to the
x;; dependent per-unit-Iength parameters of the real inter
corresponding physical parameters xO by =
connection lines. In practice, the hypothesis of a small
EOxo(EO) -1.
dispersion allows us to employ a frequency-independent
In the modal representation, the characteristic impe
zg approximation for the per-unit-Iength capacitance. On
dance is diagonal, i.e., = diag{ZoeCw), ZOo(w)},
and the scattering matrices defined for Z? = zfl are the contrary, the values of the per-unit-Iength resistance
and inductance may be greatly influenced by the skin
sRrm = 0° effect, i.e., by the magnetic flux penetration inside the
conductors [12]. In particular, we adopt the following
Sflrm = diag{HeCw), Ho(w)} approximation for R(w):
= diag{exp( -'Ye£), exp( -'Yo£)} (5) RDC + jRo Fw,
R(w) = { (7)
where 0° is the null matrix, He' Ho are the line trans (1 + j)RoFw,
fer functions of the two modes, l'e ' 'Yo are their propa
gation functions, and £ is the line length. The mode where RDC is the DC resistance of the line and Ro =
propagation functions are expressed in terms of the line -J KRDC is the skin parameter (the value of the coeffi
parameters, i.e., cient K depends on the transverse conductor geometry
[13]). The onset of the skin effect takes place when the
'Yj(w) = aj(w) + j{3j(w)
penetration depth becomes comparable with the con
ductor size, and the imaginary part of R takes into
= -J [Rj + jwLd[Gj + jwCj]
account of the variation of the conductor internal
=jW. -JLj Cj Jl -.j wR-Lj j -•j wG-Cjj -w-R2LjGj- Cj j , indTuchtea nqcuea.n tity G parameterizes the effect of dielectric
losses and is usually assumed to be proportional to w
i = e, 0 (6) [14]. It should be remarked, however, that although a
conductivity function of the form G ex w yields the
where a and {3 represent the attenuation and phase func
qualitative behavior of usual G, it does not preserve
tions, and
the hermitian property of the transfer function H. The
R[~J = Rll ± R12, L[~ J = Lll ± L12 pinr o[b1l5e]m, owfh tehree c hita riasc tsehroizwatni otnh oatf Gan h aesm bpeiernic aadl dmreossdeedl
closely describing many kinds of dielectric materials
G[~J = Gll ± G12, C[~J = Cll ± C12 over a wide frequency range is
G(w) = GoCjw)" , Go = const., 0 < p < 1 (8)
where Rpq' Gpq' Lpq' Cpq (p, q = 1, 2) are the ele
ments of the line resistance, conductance, inductance The effects of the parameters of H(w) on the features
and capacitance matrices, respectively [11]. of impulse response h(t) can be explained through the
In the modal representation, the transient scattering attenuation curve a(w), which determines the ampli
parameters corresponding to (5) are s~rm = 0° and tude of the transfer function. Figure 2 shows two ex
sg" = diag{he(t), ho(t)}, where he and ho are the im- amples of the attenuation function a(w) in log-log
10 Maio, Pignari and Canavero
region of the log(a)-curves is a£ = RDC£12 ZLC;
105 ! thus, if RDC£ < 2ZLC, H is actually influenced by the
104 ' skin effect. In practice, the bandwidth of the ohmic
region in the case of on-chip connections is so wide
103
a
b that real signals can hardly experience any high
102
/' C frequency effect. However, other interconnects of low
E.,
101 loss type exist, so that the skin effect can be experienced
'" by realistic signals that travel on such buses.
10°
d From the above discussion, it is evident that the im
10-'
pulse response of a low-loss line with low values of
W-2 RDCIL = RDCvlZLC (v = 1/.../LC) contains both a fast
time structure, due to the high frequency part of the
10-3
103 105 10' 10" 1013 transfer function, and slow time components, due to
f [Hz] the low frequency ohmic part of the transfer function.
Fig. 2. Line attenuation function vs. frequency, in log-log scale. In order to gain further insight into the structure of h(t) ,
Curves a and b are computed for parameter values typical of on it is useful to study the exact analytic impulse responses
chip and thick-ftlm interconnects technologies, respectively. Curves corresponding to a low-and a high-frequency approxi
c and d are the approximations of curve b, for skin effect (R = mation of the transfer function H(w). The low-frequency
(1 + j)Ro ~) and ohmic (R = RDC) losses, respectively. approximation corresponds to the propagation function
scale: curve a is for RDC = 19.5 kO/m (typical for on of a line with pure DC ohmic losses, i.e., with R =
chip lines), and curve b is for RDC = 55 O/m (typical RDC, G = 0, for all frequencies in (6):
value of thick-film lines) [10]; in both curves, Go = J
2 X 10-13 Flm, and /I = 1 are adopted. /fr = ar + }{3r =} ~ 1 -} ~~~;
The characteristic shape of the attenuation curve (see
curves a and b of figure 2) can be described in terms
The high-frequency approximation is obtained from the
of different physical phenomena occurring on the line.
high-frequency representation of (6), with G = 0:
At low frequencies, where the DC losses prevail, the
argument of the square root of (6) is dominated by the
RDClwL term and log(a) has a constant slope; then the /fs = as + J' {3 s = J. -w + (1 + J. ) 2RZo 'IIW
V LC
attenuation curve saturates, because RDC becomes
smaller than wL, and fmally increases again, due to the In figure 2, the curves c and d represent the attenuation
--./w dependence of the skin effect. Lastly, the change of functions ar and as, respectively, for the same RDC
slope of loge a) is caused by the growth of the conductiv and Ro values of curve b. Similar plots also hold for
ity losses that overcome the skin losses: this happens the propagation parameter {3.
when the term GjC becomes larger than Rj--./wL. Ac The exact impulse response corresponding to the
tually, the sharp knee in the curves at the onset of the low-frequency approximation /fr is (see [16])
skin effect is caused by the approximation assumed for
R(w). The real behavior of the curves in this region
depends on the line transverse geometry [11], and is
smoother than shown. However, since we are interested + ~I: e -a(t-7)
only in the main features of the attenuation, we neglect
2 .Ja(t - r)(a(t - r) + 0
this effect.
A comparison of curves a and b of figure 2 allows us
+ ~»J
to visualize the effect of the RDC value on the attenua 1,(.Ja(t - r)(a(t - r) (9)
tion behavior. The frequency band dominated by ohmic
losses grows with the value of the ratio RDCIL. It is where the following normalization parameters have
worth noticing that the attenuation curve influences H been introduced: r = £Iv, that represents the line delay,
only up to the frequencies where the overall attenuation a = RDcvl2ZLc and ~ = RDC£IZLC, that are loss
a£ is of the order of a few units, since at higher fre parameters; /1(.) is the modified Bessel function of
quencies the amplitude of Hbecomes negligibly small. order 1. We indicate with hms the continuous part of
As a reference, the attenuation value in the plateau hr in (9), i.e.,
Efficient Transient Analysis 11
e-X
hmix, ~) = (J h (.J x(x + ~) ) (10)
.J x(x + ~)
where x = aCt - 7) represents a normalized time. For
low-loss lines (i.e., ~ < 2), hrns is weakly dependent
on ~, and can be approximated by hmsQc, ~ = 0),
0
whose behavior is shown in figure 3. The durations of .iJ
"" 0.1
the fast part of hms is estimated to be Wr = 6/(J, and
is independent of the line length '£, because of the defi
nition of (J. Usual values of Wr for macroscopic low 0.05
loss interconnects (excluding on-chip interconnects) full
in the range 10 ns-l JJS [10]. Additionally, since the area 0
0 2 3 4 5 6 7 8 10
of the continuous and the delta components of hr are
Y
W2) exp{ -V2} and exp{ -~12} respectively, the rela
tive importance of the continuous component decreases Fig. 4. Impulse response hs vs. the normalized time y. Crosses (x)
for decreasing values of ~. indicate the first 15 nonuniformly spaced samples used in this paper
to represent the function. The time interval over which the samples
The impulse response corresponding to the high
are taken extends up to y = 400.
frequency approximation 1's is (see [11])
effect and evolves as hs. For very low losses (e.g.,
hit) = 1. 1 e -f//(t-7) (11) ~ < 0.2), the contribution of the long-time tail hrns to
1J .J 7r[(t - 7)/1JJ3 the total pulse area becomes negligible, and hence h(t)
becomes adequately represented by hs alone. The im
where 1J = ~K'£/8ZLC' The function hs is shown in
pulse response of these lines is determined by the high
figure 4, versus the normalized time y = (t - 7)/1J.
The pulse area is J/x, hiy(t» dt = 1, and hs decreases frequency losses alone, and distortion arises only for
those signals fast enough to experience such losses.
so slowly that the 94 % of its area is reached only for
The previous considerations on the structure of the
y "'" 400. The estimated duration of hs is Ws = 10'1),
impulse response h(t) evidence that a nonmatched char
proportional to ,£ 2, but much smaller than Wr: com
acterization (i.e., the use of scattering parameters de
mon values of Ws for macroscopic low-loss intercon
fined for ZrD -:;c Z;;) must be avoided in the analysis
nects fall in the range 10 ps-l ns [10].
of low-loss lines. In fact, the line impulse responses
The impulse response of a real low-loss line includes
of nonmatched characterizations contain echoes whose
the features of both functions hr and hs. Its long-time
fast parts are due to the hs component of h(t) and are
part is due to the low-frequency ohmic part of Hand
evolves as hr with a duration independent of '£, while always much shorter than 7, as one can verify for
~ < 2, K E [2 X 10-8,4 X 10-8] (depending on the
its short-time part is caused by the high-frequency skin
line transverse geometry), and for the usual values of
v and ZLC' Impulse responses of this kind look like a
"comb" function and can be hardly obtained via a
numerical inversion of the transfer functions Spqnn( W ).
0.4
In order to have an accurate short- and long-time
representation of h(t), the usual uniformly sampled
~ 0.3 staircase representation is not suitable. In fact, a sam
5' pling pitch adequate to the initial fast part of the func
~
.1
0.2 tion implies that the number of samples needed to
describe the function on the whole time interval of inter
est is too large, thus lengthening the computation of the
convolutions required by the solution of the transient
equations (2), (3). In this paper, a nonuniformly spaced
o 2 3 4 5 6 7 8 9 10 piecewise linear representation of h and of all time func
x tions involved in the transient problem is exploited in
Fig. 3. Nonsingular part hrns of the impulse response hr of equation order to obtain an accurate representation with high
(9), vs. the normalized time x, computed for ~ = O. numerical efficiency. An example of a nonuniform