The Elmore Delay as a Bound for RC Trees with Generalized Input Signals
†
Rohini Gupta, Byron Krauter
*
, Bogdan Tutuianu, John Willis and Lawrence T. Pileggi
§
The University of Texas at Austin
Department of Electrical and Computer Engineering
Austin, Texas 78712
ABSTRACT
The Elmore delay is an extremely popular delay metric, par-
ticularly for RC tree analysis. The widespread usage of this met-
ric is mainly attributable to it being the most accurate delay
measure that is a simple analytical function of the circuit param-
eters. The only drawbacks to this delay metric are the uncertain-
ty as to whether it is an optimistic or a pessimistic estimate, and
the restriction to step response delay estimation.
In this paper, we prove that the Elmore delay is an absolute
upper bound on the 50% delay of an RC tree response. More-
over, we prove that this bound holds for input signals other than
steps, and that the actual delay asymptotically approaches the
Elmore delay as the input signal rise time increases. A lower
bound on the delay is also developed using the Elmore delay and
the second moment of the impulse response. The utility of this
bound is for understanding the accuracy and the limitations of
the Elmore delay metric as we use it for design automation.
I. INTRODUCTION
RC trees are commonly used to model digital logic gates and their
associated interconnect paths at various stages of the design process.
During the early phases of design, simple approximations or delay
bounds are often applied since exact solution of an approximate or
fluctuating circuit model is superfluous.
The omnipresent Elmore delay [6], or first moment of the impulse
response, is the delay approximation of choice for RC trees because
of the ease with which it is calculated. In the original work of 1948,
Elmore attempted to estimate the 50% delay of a monotonic step re-
sponse by the mean of the impulse response. Penfield and Rubinstein
[15] proved that RC tree step responses are indeed monotonic, and
thereby discovered the popular Elmore delay metric for analyzing
gate and interconnect delays. However, because the median of the
impulse response is the exact 50% delay, and Elmore is approximat-
ing the median by the mean, Penfield and Rubinstein developed best
and worst case bounds on the step response waveform [15].
§ Formerly Lawrence T. Pillage.
* Byron Krauter is with IBM, Austin, TX 78758.
† This work was supported in part by the Semiconductor Research Corpora-
tion under contract 94-DJ-343 and the National Science Foundation under
contract MIP-9157263.
These bounds were improved in [18], and later extended to a two
time constant approximation in [3]. Some time later higher order mo-
ment matching techniques were developed for RLC circuits [16] for
which RC trees are an important subset.
But even with higher order approximations with accuracy compa-
rable to SPICE, the Elmore delay remains a popular metric merely
for its simplicity. It is used during logic synthesis to estimate wiring
delays for approximate tree routes. It is used during performance
driven placement and routing because it is the only delay metric
which is easily measured in terms of net widths and lengths. The lim-
itations of this model are the uncertainty as to whether it is an opti-
mistic or a pessimistic estimate, and the restriction to it being an
estimate only for the step response delay.
In this paper we prove that the Elmore delay value is an absolute
upper bound on the 50% delay of an RC tree. Moreover, we demon-
strate that this proof applies not only to the step response, but also to
any input forcing function which has a unimodal derivative (e.g. a
saturated ramp with finite rise time). With a calculation of the mean
and the variance of the impulse response, we also specify an absolute
lower bound on the 50% delay. In addition, we will show that the ex-
act delay approaches the Elmore bound as the variance of the input-
signal derivative increases.
II. RC TREES AND THEIR APPROXIMATIONS
A. Interconnect Models
RC trees, such as the one shown in Fig.1, have been widely used
for modeling equivalent gate and interconnect circuits. For modeling
simplicity, nonlinear drivers are linearized as shown in Fig.1. A great
deal of work has been compiled over the last several years regarding
these linearized gate models [1,8,13,19]. In this paper, however, we
will focus on estimating the linearized RC tree delay.
B. The Elmore Delay
The step and unit impulse responses for the node at capacitor
of the RC tree in Fig.1 is shown in Fig.2. Since the zero-state step re-
sponse is the integral of the impulse response h(t), the 50% point de-
lay of the monotonic step response (nonnegative transfer function) is
the time τ at which . Referring to Fig.2, Elmore pro-
FIGURE 1: A simple RC tree.
V
in
C
4
C
3
C
2
C
1
R
4
R
3
R
2
R
1
+
-
+
-
C
7
C
6
R
7
R
6
R
5
C
5
80Ω
60Ω
60Ω
60Ω
60Ω
60Ω
60Ω
0.5pF
1pF
1pF
1.2pF
1pF
1pF
1.2pF
+
-
C
5
ht() td
0
τ
∫
0.5=
32nd ACM/IEEE Design Automation Conference
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posed to approximate τ by the mean of the h(t) distribution.
Treating the nonnegative impulse response in Fig.2 as a distribu-
tion function, the mean is defined by its first moment, m
1
. Elmore’s
approximation for the unit step response delay, , is:
(1)
where, . This approximation is valid for a symmetri-
cal function, where the mean is equal to the median, however it is
somewhat erroneous for the real impulse response in Fig.2, which is
skewed asymmetrically. It is this skew, however, which will allow us
to bound the delay (τ) by the mean (T
D
).
C. Calculating the Elmore Delay
The Elmore delay is a fitting metric for RC trees because it can be
calculated very efficiently for this circuit topology[15,19]. Two
traversals of the tree, where is the number of nodes in the
tree, yield the Elmore delay for node i:
(2)
where R
ki
is the resistance of the portion of the (unique) path between
the input and node i, that is common with the (unique) path between
the input and node k, and C
k
is the capacitance at node k [18]. The
Elmore delay values at nodes C
1
, C
5
and C
7
for the circuit in Fig.1
are given in column (3) of TABLE 1.
D. The First Moment of the Impulse Response
The Elmore delay has also been used as a dominant time constant
approximation. Consider the RC tree transfer function
(3)
where
and m
q
is the q-th coefficient of the impulse re-
sponse defined as [16]:
(4)
(1) (2) (3) (4) (5) (6) (7)
Node
Actual
delay
(ns)
Elmore
delay, T
D
(ns)
Lower
bound,
T
D
-σ (ns)
Single pole,
T
D
.ln(2)
(ns)
RPH
upper
bound,
t
max
(ns)
RPH
lower
bound,
t
min
(ns)
C
1
0.196 0.55 0 0.383 0.55 0
C
5
0.919 1.2 0.2 0.83 1.32 0.51
C
7
0.45 0.75 0 0.524 1.02 0.054
TABLE 1: Delay bounds for circuit in Fig.1.
FIGURE 2: The unit step and the unit impulse (scaled by 1e-09) response
for the voltage across C
5
in Fig.1.
0.0 1.0 2.0 3.0
0.2
0.4
0.6
0.8
1.0
volts
time (ns)
step response
impulse response
τ
T
D
T
D
T
D
m
1
th t()dt
0
∞
∫
==
ht()dt
0
∞
∫
1=
ON() N
T
D
i
R
ki
C
k
k 1=
N
∑
=
Hs()
1 a
1
sa
2
s
2
…a
n
s
n
++ ++
1b
1
sb
2
s
2
…b
m
s
m
++ ++
-------------------------------------------------------------------- m
0
m
1
sm
2
s
2
…++ +==
mn≥
m
q
1–()
q
q!
---------------
t
q
ht()dt
0
∞
∫
=
These coefficients are related to the moments of a distribution func-
tion h(t) (from distribution theory) by the term. That is,
the n-th moment is defined to be . Hereafter, however,
we shall refer to
m
q
as the q-th moment of h(t).
To understand the connection between the first moment and the
dominant pole, the terms and can be shown [7] to be the sum
of the reciprocal poles and zeros respectively:
(5)
If there are no low frequency zeros, the numerator coefficients, in-
cluding , are small and . Now, if one of the time con-
stants (or poles) is assumed to be the dominant one, i.e.
(6)
then . This dominant time constant approximation is then
used to fit a single pole approximation so that:
(7)
Solving (7) for the 50% point delay effectively scales the Elmore de-
lay approximation by , or about .
We should point out that this dominant time constant delay pre-
diction can be pessimistic at one node while optimistic at another for
the same RC tree. For example, column (5) of TABLE 1 shows the
values of at nodes C
1
, C
5
and C
7
for the circuit in
Fig.1. Notice that, when compared with the actual delay values in
column (1), the response at is optimistically predicted by
while that at is pessimistically predicted. One way
to explain this is by the excessive skew in the distribution for
, which is shown with the step response for this node in Fig.3, as
compared with the skew for the response at
(shown in Fig.2). It
can be expected that using to approximate the median
will be vastly different for these two distributions.
It is difficult to know when a single pole dominates the low fre-
quency behavior of an RC tree. For this reason, Rubinstein and Pen-
field established bounds for the RC step response delay.
E. The Rubinstein, Penfield and Horowitz Bounds
Penfield and Rubinstein were the first to use the Elmore delay to
analyze RC trees [15]. Calculating these bounds requires calculating
two additional terms in addition to the Elmore delay. All of these
terms, however, are obtained with complexity. The upper
and lower bounds, t
max
and t
min
, at the 50% point for our example in
Fig.1 are given in columns (6) and (7) of TABLE 1. Note that
at the loads, C
5
and C
7
, and at the driving
1–()
q
q!⁄
t
n
ht()dt
0
∞
∫
b
1
a
1
b
1
1
p
j
----
j 1=
m
∑
= a
1
1
z
j
----
j 1=
n
∑
=
a
1
T
D
b
1
≅
1
p
d
-----
1
p
j
----
,»
j 12…mjd≠,, , ,=
T
D
1p
d
⁄≅
vt() 1 e
p
d
t–
–=
2()ln 0.7
2()ln T
D
⋅
C
5
2()ln T
D
⋅ C
1
ht()
C
1
C
5
2()ln Mean⋅
FIGURE 3: The unit step and the unit impulse response (scaled by 4e-09)
for the voltage across C
1
in FIGURE 1.
0.0 1.0 2.0 3.0
0.2
0.4
0.6
0.8
1.0
volts
time (ns)
step response
impulse response
ON()
t
max
T
D
> t
max
T
D
=
point, C
1
. Also note the values of t
min
as a lower bound on delay.
In general, one can calculate more moments for the RC tree, and
generate a 2-pole [3] or a q-pole [16] approximation. Higher order
moments are obtained with complexity too. But for certain
applications the Elmore expression is invaluable, and this paper is to-
wards a better understanding of this approximation.
III. THE ELMORE DELAY AS A BOUND
Referring back to Fig.2 and Fig.3, it is apparent that with such an
asymmetrical distribution for the impulse response, the mean would
not coincide with the median. In this section, we will show that these
asymmetric distributions have a “long tail” on the right side of the
mode (roughly the maximum value point). Such distributions are
said to have positive skew. We will prove that the impulse response
for an RC tree is unimodal and positively skewed, then use these two
properties to prove that:
(8)
We will further show that (8) holds for any input that has a unimo-
dal derivative and that the mean becomes a better approximation of
the median as the rise-time of the input-signal increases. Further in
the section, we will also provide a lower bound on the 50% delay for
an RC tree. But first a few definitions:
Definition 1: The mode, M, of a continuous distribution function
f(x) is the maximum point of the distribution[4]. A unique mode
exists only if f is unimodal and is the solution of
(9)
Definition 2: The median, m, of a distribution function f is that
value of the variate which divides the total frequency into two
equal halves[9], i.e.
(10)
Definition 3: The mean, µ, of a distribution function f about the
point is defined by
(11)
Definition 4: A density function h(t) is called unimodal, if and
only if, there exists at least one value t = t
m
such that h(t) is non-
decreasing for t < t
m
and nonincreasing for t > t
m
[17].
Definition 5: Coefficient of skewness for a distribution function
is given by , where , and and are
the second and third central moments of the distribution func-
tion respectively[4].
L
EMMA 1: The impulse response h(t) at any node of an RC tree
is a unimodal, positive density function.
Proof: The proof is by induction. For a general RC circuit, it is well-
known that the poles and zeros of the driving point admittance, Y(s),
are simple, interlaced and are located on the negative real axis of the
s-plane[20]. Furthermore, the residues at the poles of Y(s) are real
and negative[20]. Therefore, in Fig.4(a), if v
in
(t) is an impulse input,
(12)
ON()
Mode Median Mean≤≤
f′x()
xd
d
fx() 0 f″x(),
x
2
d
d
fx() 0<== =
fx()xd
∞–
m
∫
fx()xd
m
∞
∫
1
2
---==
xa=
µ xa–()fx()xd
∞–
∞
∫
=
γµ
3
σ
3
⁄= σµ
2
= µ
2
µ
3
Is() Ys()= and, i t() k
i
e
p
i
t–
i
∑
–=
Now, looking at the first downstream node in the RC tree, in
Fig.4(b), clearly, from Fig.4(a). Therefore,
(13)
If v
1
(t) is an impulse, then v
2
(t) is the impulse response at node 2:
(14)
Following Definition 4, it is clear that h
2
(t) is unimodal.
Now consider Fig.5 which shows node k and everything “down-
stream” of it. To complete our induction argument, we assume that
h
k
(t) is unimodal, and
(15)
If v
k
(t) is an impulse, then h
k,k+1
(t) is the transfer function at node
k+1 w.r.t. input at node k. This has the same form as in (14) and is
unimodal. Thus, the transfer function at node k+1 w.r.t. node 1,
h
k+1
(t), is given by:
(16)
where is the convolution operator. Since the convolution of two
unimodal density functions is also a unimodal density function [17],
we have that h
k+1
(t) is also unimodal. Thus, h(t) at any node of an
RC tree is a unimodal function. That h(t) is a positive density func-
tion has been shown in [18].
L
EMMA 2: For the impulse response h(t) at any node of an RC
tree, the coefficient of skewness, γ, is always nonnegative.
Proof: The proof follows an induction-based argument. Following
Definition 5, we need to show that for h(t) at any node of an RC tree,
and . First we show that the coefficient of skewness,
γ, is positive at the first node of an RC tree, and then use the additive
property of central moments over convolution to motivate our induc-
tion argument.
In Fig.6(a), consider a general RC tree for which the first three mo-
ments of the driving point admittance, Y
1
(s) at node 1, can be used to
synthesize a π-model as shown in Fig.6(b)[12]. Note that this π-mod-
v
in
(t)
Y
1
i(t)
1
FIGURE 4: (a) Input node of an RC tree with admittance Y
1
at node 1.
(b)Admittance Y
2
of an RC tree at the first capacitor node - node 2.
(a)
v
1
(t)
+
-
v
in
(t)
R
1
Y
2
i
12
(t)
2
1
(b)
v
2
(t)
+
-
i
12
t() it()=
v
2
t() v
1
t() R
1
i
12
t()–=
h
2
t() δ t() R
1
k
i
e
p
i
t–
i
∑
+=
v
k
(t)
R
k
Y
k+1
i
k,k+1
(t)
k+1
k
FIGURE 5: Admittance Y
k+1
of an RC tree at an arbitrary node k+1.
V
k+1
(t)
+
-
v
k 1+
t() v
k
t() R
k
i
kk 1+,
t()–=
h
k 1+
t() h
kk 1+,
t() h
k
t()•=
•
µ
3
0≥µ
2
0≥
FIGURE 6: (a) Driving point admittance Y
1
(s) of an RC tree at the first
capacitor node. (b)A reduced order π-model for the Y
1
(s) in Fig.6(a).
v
in
(t)
R
1
(a)
1
Y
1
(s)
C
1
R
2
R
1
C
2
1
2
(b)
v
in
(t)
el exactly matches the first three moments of the driving point admit-
tance of the original RC circuit. The π-model parameters are:
(17)
where, m
1
(Y
1
), m
2
(Y
1
), m
3
(Y
1
) are the first three moments of Y
1
(s).
From distribution theory [4], central moments of a distribu-
tion function are given by:
(18)
where, and m
k
is the k-th moment of the distribution func-
tion. Thus, for the impulse response of a circuit, we can express the
central moments µ
n
as a function of the circuit moments m
k
:
(19)
It can be shown that[7] the moments m
0
through m
3
of the impulse
response H
1
(s) at node 1 in Fig.6(a) are a function only of the mo-
ments m
0
through m
3
of the driving point admittance Y
1
(s) at node 1.
Therefore, the π-model in Fig.6(b) provides the exact moments m
0
through m
3
of the impulse response H
1
(s).
For the RC circuit in Fig.6(b), we have [16],
(20)
(21)
where, m
k
(p)
denotes the k-th moment at node p.
Thus, for the impulse response h
1
(t) at node 1 in Fig.6(b), from
(20) and (21) and Definition 2, we have .
Next consider Fig.7 which shows node k and its “downstream”
part of the tree. To complete our induction argument, we assume that
at node k, we have and for h
k
(t), and hence, . If
v
k
(t) is an impulse, then h
k,k+1
(t) is the transfer function at node k+1
w.r.t. the input at node k. This has the same form as in Fig.6(a) for
which the above argument shows that and (from (20)
and (21)). Now, the transfer function at node k+1 w.r.t. node 1,
h
k+1
(t), is given by:
(22)
From [4,7], when , we have the property that the second and
third central moments add under convolution. Thus,
(23)
Thus, for h
k+1
(t), from Definition 5, at all nodes.
This proves that for every node in an RC tree, the coefficient of skew-
R
2
m
3
Y
1
()()
2
m
2
Y
1
()()
3
------------------------------–=
C
1
m
1
Y
1
()
m
2
Y
1
()()
2
m
3
Y
1
()
------------------------------–=
C
2
m
2
Y
1
()()
2
m
3
Y
1
()
------------------------------=
µ
n
µ
n
n
k
m
k
η–()
nk–
k 0=
n
∑
=
η m
1
=
µ
3
6m
3
–6m
1
m
2
2m
1
3
–+=
µ
2
2m
2
m
1
2
and–=
µ
2
1()
2m
2
1()
m
1
1()
2
–=
R
1
2
C
1
2
C
2
2
+
2R
1
2
C
1
C
2
2R
1
R
2
C
2
2
++ 0≥=
µ
3
1()
6m
3
1()
–6m
1
1()
m
2
1()
2m
1
1()
3
–+=
6R
1
R
2
C
2
2
R
1
C
1
C
2
+()R
2
C
2
+()2R
1
C
1
C
2
+()()
3
+0≥=
γ0≥
µ
3
0≥µ
2
0≥γ0≥
µ
3
0≥µ
2
0≥
h
k1+
t() h
kk 1+,
t() h
k
t()•=
m
0
1=
µ
3
h
k 1+
()µ
3
h
kk 1+,
()µ
3
h
k
()+0≥=
µ
2
h
k1+
()µ
2
h
kk 1+,
()µ
2
h
k
()+0≥=
γ0≥
ness,.
Now, in the following we use the above two lemmas to prove that the
Elmore delay is indeed a bound on the 50% delay for an RC tree.
T
HEOREM: For the impulse response h(t) at any node in an RC
tree,
(24)
Proof: For a unimodal “skewed” distribution function, the mean,
median, mode inequality states that these three quantities occur ei-
ther in alphabetical order or the reverse alphabetical order [9], i.e. ei-
ther or . From
Lemma 1 and Lemma 2, we have that each node in an RC tree has a
unimodal distribution function for which . We now prove, by
contradiction, that for an RC tree, .
For our contradiction argument, let hold
for any node, α, in an RC tree. In a symmetrical distribution, for
which the coefficient of skewness, γ, is exactly zero, the mean, the
median and the mode coincide [9,11]. Thus a natural measure of
skewness for an asymmetrical distribution is the deviation of the
mean from the median, or the mean from the mode. Thus,
(25)
where, . Thus, at the node α, since
holds, the skewness, . But, from
Lemma-2, we have that the coefficient of skewness, . Thus, at
α, either or we have a contradiction. In the former case,
, i.e. the distribution is symmetric and
the mean and median coincide. And in the latter case,
.
Since the choice of the node α is arbitrary, the proof is com-
plete.
We should note at this point that the Elmore Delay, T
D
, or the
mean, µ, of the impulse response approaches the 50% delay point at
nodes further downstream from the source in an RC tree, as dis-
cussed further in Section IV.
A. A Lower Bound on Delay
COROLLARY 1: A lower bound on the 50% delay for an RC
tree is given by
(26)
where is the mean and .
Proof: Consider a positively skewed impulse response h(t), with
mean at . We define another function H(t) as:
(27)
With a simple change in the x-coordinate such that , we
have such that its mean is at in the new coordinate sys-
tem. Then, using the following inequality from [4](page 256):
v
k
(t)
R
k
Y
k+1
(s)
k+1
k
FIGURE 7: Admittance Y
k+1
of an RC tree at an arbitrary node k+1.
γ 0≥
Mode Median Mean≤≤
Mean Median Mode≤≤ Mode Median Mean≤≤
γ0≥
Mode Median Mean≤≤
Mean Median Mode≤≤
Sk
Mean Median–
σ
--------------------------------------=
σµ
2
=
Mean Median Mode≤≤ Sk 0≤
γ 0≥
Skew 0=
Mean Median Mode==
Mode Median Mean≤≤
max µσ0,–()
µσµ
2
=
tµ=
Ht() h ζ()ζd
∞–
t
∫
=
τ t µ–=
hτ() τ 0=
(28)
For , equations (27) and (28) show that
(29)
Equation (29) states that in the new coordinate system, is
less than the median. Thus, in the original coordinate system for h(t)
we have that .
When , since the RC tree system is causal and relaxed[2]
and for , we have , and hence
. This completes the proof.
Referring back to the example in Fig.1 and the delay bounds in
TABLE 1, the lower bound at C
1
equals t
min
[18], whereas at
C
5
and C
7
, t
min
is a tighter lower bound than . However, as
observed in Section III, the Elmore delay upper bound, µ, becomes a
tighter upper bound (as compared to the 50% bounds in [18]) at the
leaf-nodes of an RC tree as is evident at C
5
and C
7
in TABLE 1.
B. Approximating the Output Signal Transition Time
Another measure of practical importance for RC circuits, other
than the 50% delay point, is the rise-time, T
R
, which may be defined
as the 10 to 90 percent transition time[6]. A good measure of T
R
is
(30)
where is the second central moment of the distribution function
. Elmore proposed this value, which he terms the radius of gy-
ration, as a rise-time measure for step-responses [6].
IV. GENERAL INPUT SIGNALS
It has been shown above that the Elmore delay is an upper bound
on the 50% step response delay. However, when using the Elmore
delay to estimate RC interconnect delays, the signal from a digital
gate is generally modeled by a saturated ramp. Of course, several
models have been developed to characterize the switching gate by a
linear resistor and a voltage step for compatibility with the Elmore-
step-response model [1,8,13,19] but at the expense of accuracy. One
recent work attempts to model high-speed CMOS gates with linear
resistors for efficiency, but time varying voltage sources to capture
the high-frequency phenomena such as resistance shielding and ef-
fective capacitance [5]. Most timing analyzers characterize gate and
output signal transition time empirically as a function of load, and
then drive the RC tree interconnect model with a voltage that repre-
sents this transition time. For these reasons we extend the Elmore
bound to consider a non-zero input signal transition time.
A. The Elmore Delay Upper Bound
COROLLARY 2: For an RC circuit with a monotonically
increasing, piecewise-smooth input u(t) such that is a
unimodal symmetric function, holds
for the output response y(t) at any node.
Proof: The output response y(t) at any node of an RC tree in response
to an input u(t) is given in the Laplace domain by
(31)
where H(s) is the impulse response of the circuit at that node. Using
H τ()
σ
2
σ
2
τ
2
+
------------------
τ 0<,≤
τσ–=
Hσ–() hζ()ζd
∞–
σ–
∫
σ
2
σ
2
σ–()
2
+
-----------------------------
≤
1
2
---==
τσ–=
µσ– Median≤
µσ≤
ut() 0= t 0< Median 0≥
µσ–0Median≤≤
µσ–
µσ–
T
R
σµ
2
==
µ
2
y′t()
u′ t()
Mode Median Mean≤≤
Ys() Hs() Us()⋅=
the property of addition under convolution, it can be shown that[7]:
(32)
From Lemma 2, we know that and .
From hypothesis, we also have,
(33)
From (32) and (33), thus, and , and
from Definition 5, . For y(t),.
C
OROLLARY 3: For a finite sized RC circuit with a monotoni-
cally increasing, piecewise-smooth input u(t) with rise-time t
r
,
such that is a unimodal symmetric function, as ,
, i.e. .
Proof: From hypothesis, we have that is a symmetric func-
tion, . Also, since ,
(34)
Since the circuit is finite sized, . Thus, for the out-
put response at any node of an RC tree,
(35)
Since , . Thus, as
,.
It is noteworthy here that since , i.e. is a
symmetric function, its mean and median coincide. Further,
(36)
where is the mean. Thus, it can be shown that [7]:
(37)
i.e., the area between the input and the output response equals the
Elmore Delay, T
D
[10].
B. Delay Curves
The estimation of the 50% delay by the Elmore delay as a function
of the rise-time of the input signal, as stated in Corollary 3, is shown
in Fig.9 for the RC tree example in Fig.1. As the rise-time of the in-
put signal increases, the delay asymptotically approaches T
D
.
It was observed in Section III that as one moves away from the
source, T
D
(i.e. the mean, µ) is a better approximation of the net de-
lay. The proof for Lemma 2 in Section III uses the additive property
of the central moments under convolution. Referring to equation
(23), for any node k, . Furthermore, using
equations (20) and (21), it can be shown that and
form decreasing and hence convergent sequences[7].
µ
3
y′ t()()µ
3
ht()()µ
3
u′t()()+=
µ
2
y′t()()µ
2
ht()()µ
2
u′t()()+=
µ
2
ht()()0≥µ
3
ht()()0≥
µ
2
u′t()()0and µ
3
u′ t()()≥ 0=
µ
2
y′t()()0≥µ
3
y′t()()0≥
γy′t()()0≥ Median Mean≤∴
u′ t() t
r
∞→
T
D
50% Delay→ Mean Median→
FIGURE 8: Input signal u(t) with rise-time, t
r
and its derivative, u’(t).
t
r
1
1/ t
r
u (t)
u’(t)
u′ t()
µ
3
u′ t()()0=∴µ
2
u′t()()t
r
∝
t
r
∞→⇒µ
2
u′t()()∞→
µ
3
ht()()∞<
yt() ht() ut()•=
γ y′ t()()
µ
3
y′t()()
µ
2
y′t()()()
32⁄
------------------------------------------
0→= as t
r
∞→
γ Mean Median–∝γ0Mean Median→⇒→
t
r
∞→ T
D
50% Delay→
µ
3
u′ t()()0= u′t()
µ y′ t()()µu′t()()– T
D
=
ty′ t() td
0
∞
∫
tu′ t() td
0
∞
∫
–⇒ T
D
=
µ .()
1 yt()–[]td
0
∞
∫
1ut()–[]td
0
∞
∫
–⇒ T
D
=
µ
2
h
k
()µ
3
h
k
(), 0≥
µ
2
h
kk 1+,
()
µ
3
h
kk 1+,
()
