24.2
Delay and Slew Metrics Using the Lognormal Distribution
Charles J. Alpert, Frank Liu, Chandramouli Kashyap, and Anirudh Devgan
IBM Corp., 11501 Burnet Road, Austin, Texas 78758
[alpert, frankliu, vchandra, devgan]@us.ibm.com
ABSTRACT
Elmore [4], as it turns out, also proposed a slew metric in his
seminal paper, namely a constant times “radius of gyration”, i.e.,
the standard deviation of the impulse response. Gupta et al. [5] also
recognized this as a “good measure” for slew. Bakoglu [3]
proposed a first-order slew metric which is a constant times the
Elmore delay. Bakoglu’s metric an be derived by matching the first
moment of the impulse response to the mean of exponential
distribution.
For optimizations like physical synthesis and static timing analysis,
efficient interconnect delay and slew computation is critical. Since
one cannot often afford to run AWE [12], constant time solutions
are required. This work presents the first complete solution to
closed form formulae for both delay and slew. Our metrics are
derived from matching circuit moments to the lognormal
distribution. From a single table, one can easily implement the
metrics for delay and slew for both step and ramp inputs.
Experiments validate the effectiveness of the metrics for nets from
a real industrial design.
We present closed form metrics based on the lognormal
distribution. Unlike [8][9][10], matching to the lognormal
distribution produces closed form formulae. We make the
following contributions:
Categories and Subject Descriptors
•
A simple delay metric LnD (lognormal delay) is derived using
the first two moments of the impulse response. The resulting
delay metric is actually D2M [1] though with a different scaling
constant. Hence, the derivation of LnD actually provides the
first theoretical justification for the highly accurate D2M (and
hence LnD) delay metrics.
• A closed form slew metric LnS (lognormal slew) is presented
from a derivation of the first three circuit moments.
• The LnD and LnS metrics can be extended to ramp inputs using
the PERI method [7]. The results are consolidated into a single
table that makes it easy for the reader to implement the metrics
for both step and ramp inputs.
The effectiveness of the lognormal metrics is demonstrated on nets
from an industrial design.
B.7.2 [Integrated Circuits]: Design Aids - Simulation;
General Terms
Algorithms, Design, Theory
1. INTRODUCTION
Delay computation is key for both performance estimation and
optimization of high performance integrated circuits. Interconnect
(or linear circuit) delay computation is at the core of physical
analysis and physical design tools. One must be able to efficiently
and accurately compute interconnect delay and slew since several
million calculations are required to analyze and optimize a design.
By interpreting the impulse response of a linear circuit as a
probability distribution function (PDF), Elmore [4] proposed using
the mean of the impulse response to approximate the median of the
impulse response under the probability interpretation under a step
excitation. The Elmore delay metric has been incredibly popular
because it is simple, closed-form and easy to evaluate. This metric
was resurrected when Rubenstein et al. [13] published a simple
closed-form formula for computing the mean of the impulse
response of RC interconnect trees. The widely known achilles heel
of the Elmore metric is that it is highly inaccurate when there is a
high degree of resistive shielding.
2. BACKGROUND
Assume that h ( t ) is the impulse response of a node voltage in an
RC circuit. The circuit moments of the impulse response are
k
( –1 ) ∞ k
m k = ------------- ∫ t h ( t ) dt
(1)
k! 0
The circuit moments can be computed directly as functions of the
RC’s in time linear in the size of the circuit, e.g., via path tracing.
The impulse response h ( t ) satisfies the following conditions [13]:
The Elmore metric can be improved by computing and matching
higher order moments of the impulse response via AWE
(Asymptotic Waveform Evaluation) [12]. However, AWE cannot
be expressed by a closed-form formula, instead requiring the
solution of a non-linear equation. A closed form metric is
preferable for both efficiency and implementation simplicity, as
long it is sufficiently accurate. Pileggi [11] has written a survey of
timing metrics for RC trees.
h ( t ) ≥ 0 and
∞
∫0 h ( t ) dt
= 1
(2)
Consequently, the impulse response is a PDF, though there is no
known underlying statistical distribution describing it (which is
why fast and accurate delay and slew computation is so difficult).
The mean of the impulse response is:
µ =
Alpert et al. [1] proposed the D2M metric which is a simple
function of the first two circuit moments. In this work, we present
the first theoretical analysis behind D2M. The PRIMO [8], hGamma [9], and WED [10] metrics are based on matching the
moments of the impulse response to a PDF. The first two match to
the Gamma distribution, while WED matches to the Weibull
distribution. All require some type of table lookup operation.
∞
∫0 t ⋅ h ( t ) dt
(3)
Elmore [4] showed that µ = – m 1 and therefore approximated the
median (the desired delay) by the mean of the impulse response.
We let ED = µ = – m 1 denote the Elmore delay. The kth central
moment is given by
µk =
2
∞
∫0 ( t – µ ) h ( t ) dt
k
(4)
The variance ( σ ) and the skewness ( γ ) of the impulse response
can be expressed in terms of the central moments and also the
circuit moments [5]:
While delay metrics are fairly well-studied, few metrics have been
proposed to compute slew. As ultra deep sub-micron effects
continue to wreak havoc on signal integrity, computing slew
efficiently and accurately has become increasingly critical.
2
2
σ = µ 2 = 2m 2 – m 1
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3
– 6m 3 + 6m 1 m 2 – 2m 1
-----------------------------------------------------2 3⁄2
( 2m 2 – m 1 )
(5)
µ3
γ = ------ =
(6)
3
σ
The key idea behind our delay and slew metrics is to match the
mean, variance and skewness of the impulse response to those of
the lognormal distribution.
382
3. LOGNORMAL DELAY METRIC
M
1 ln t – M 2
– ---  ------------------
2
2
m1
2
(18)
-  e k 2 ln ( 2m2 ⁄ m1 ) – e –k 2 ln ( 2m2 ⁄ m1 )
LnS 12 = ------------
2m 2 
We call this metric LnS 12 for Lognormal Slew matching the first
and second moments. Empirically, LnS 12 is fairly accurate at the
near-end, but tends to underestimate slew by about 20% at the farend. One reason is that it ignores the skewness of the impulse
response. While the variance describes how wide the waveform
spreads (the essential notion of slew), skewness reflects its degree
of asymmetry. At the far-end when the impulse response flattens,
skewness becomes more important than the mean.
2 S 
1
(7)
P ( t ) = ---------------- e
tS 2π
where M > 0 and S > 0 are the scale and shape parameters,
respectively. Its cumulative density function (CDF) is given by
ln t – M
1
(8)
D ( t ) = ---  1 + erf  ------------------  .



2
S 2
The expected value (or mean) and the variance are respectively
given by
2
M+S ⁄2
2M + S
2
S
2
E(t) = e
(e – 1) .
and V ( t ) = e
(9)
One can match two common properties of the lognormal
distribution and the circuit’s impulse response. Recall that the
mean
and variance
of the impulse response are µ = – m 1 and
2
2
σ = 2m 2 – m 1 , respectively. Using Equation (9) to match the
mean and variance yields the system
2
M+S ⁄2
2
2M + S
, 2m 2 – m 1 = e
–m1 = e
Solving this system for M and S yields
2
S
Hence, we now derive an alternative slew metric by matching
variance and skewness. The skewness of the lognormal
distribution is given by
S
2
(10)
1⁄3
 γ + 4 + γ 2
1 2
where z =  y – --- + 1 and y =  ----------------------------
(21)

y
2


Next2 we match the2 variance by setting V ( t ) in Equation (9) equal
to σ = 2m 2 – m 1 , yielding:
2
2M + S
S
2
2
(22)
2
2
2m 2 – m 1
2m 2 – m 1
M = ln --------------------------=
ln
----------------------2
2
S
S
z(z – 1)
e (e – 1)
(23)
Substituting these values back into Equation (17) yields the LnS 23
metric (since it matches the second and third circuit moments):
≥ 0 , it follows (as in [1]) that
Using the fact that 2m 2 –
LnD ≤ – m 1 = ED .Thus, the LnD metric is bounded above by the
Elmore delay and is also always nonnegative.
2
2m 2 – m 1 k 2 ln ( z ) – k 2 ln ( z )
(24)
–e
)
----------------------- ( e
z(z – 1)
Note that the formula for LnS 23 looks like the standard deviation
σ times a constant that depends only on the skewness of the
distribution. One can verify that for a single RC network, γ = 2 .
In this case, the constant term times σ evaluates to 2.206 , which
is very close to ln ( 9 ) = 2.197 , the correct value for when there is
only one pole.
LnS 23 =
4. LOGNORMAL SLEW METRIC
To derive a slew metric for say 10/90 slew, one can use the same
distribution matching as for the delay metric and then compute the
90% and 10% delay points. Note that these formulae can be
generalized for other measurement points. Let t lo and t hi be the
10/90 delay points. From Equation (8), the 10% and 90% delay
points can be found by setting D ( t lo ) = 0.1 D ( t hi ) = 0.9 , i.e.,
ln t lo – M  
ln t hi – M  
1
1
0.1 = ---  1 + erf  ---------------------, 0.9 = ---  1 + erf  ---------------------(14)






2
S 2
S 2 
2
Empirically, LnS 12 is better suited for the near-end and LnS 23 is
better suited for the far-end. One might think that LnS 23 would
also be more appropriate at the near-end, where the skewness of
the impulse response is large. However, the large skewness
actually reflects the long “tail” of the waveform, which is often
well beyond even the 90% delay point. For these cases, using
skewness instead of the mean can introduce large error. For the
near end nodes, LnS 12 remains a better choice.
ln t lo – M 
ln t hi – M 
= – 0.8 and erf  ---------------------= 0.8 .
or erf  ---------------------(15)
 S 2 
 S 2 
Let k be such that erf ( k ) = 0.8 ( k ≈ 0.9062 ). Since
erf ( x ) = – erf ( – x ) , solving for t lo and t hi yields:
t lo = e
and t hi = e
So the formula for slew becomes
2
e
( e – 1 ) = 2m 2 – m 1
Solving for M yields:
2
m1
M + Sk 2
(20)
Setting G ( t ) = γ and solving for S yields two complex roots and
one real root. The real root is given by S = ln ( z )
2
LnD = e
= m 1 ⁄ 2m 2
(12)
Thus, the delay function is a simple function of the first two circuit
moments. Observe
that this metric is actually similar to the metric
2
[2] D2M = m 1 ln 2 ⁄ m 2 . In fact, observe that
D2M
ln 2
------------ = -------------- ≈ 0.9802 ,
(13)
LnD
1⁄ 2
i.e., D2M and LnD are actually the same delay metric, except for a
two percent constant difference. D2M was derived empirically and
shown to be accurate. By matching to the lognormal distribution,
we provide the first theoretical justification of the D2M metric.
M – Sk 2
2
3
2
2
S
µ
– 6m 3 + 6m 1 m 2 – 2m 1
γ = -----3- = -----------------------------------------------------.
3
2 3⁄2
σ
( 2m 2 – m 1 )
(11)
M = ln ( m 1 ⁄ 2m 2 ) and S = ln ( 2m 2 ⁄ m 1 )
M
The median of the lognormal distribution is given by e . One can
verify this by setting D ( t ) = 0.5 in Equation (8) and solving for
t . Thus, when matching the impulse response the median becomes
our 50% delay metric:
ln ( m 1 ⁄ 2m )
2
G(t ) = e – 1(2 + e )
(19)
while the skewness γ for the impulse response is given by [5]
2
(e – 1) .
–S k 2
Sk 2
LnS = t hi – t lo = e ( e
–e
)
(17)
The only part that remains is to match moments to derive values
for the lognormal distribution parameters M and S . Using the
values from Equation (11) derived from matching the mean and
variance yields the following slew metric:
The lognormal distribution t ∼ Logn ( M , S ) is a two-parameter
continuous distribution in which the logarithm of the input variable
has a Gaussian distribution. The lognormal distribution is wellsuited to match the impulse response since both are unimodal and
have nonnegative skewness. The lognormal PDF is given by
To know which metric to use, one must be able to identify whether
a sink is near-end or far-end. For this purpose, we use the ratio
r = m 1 ⁄ m 2 . This ratio r is typically much smaller than one for
a near-end sink and is provably greater than one for a far-end sink.
We seek to have a smooth tradeoff between the near-end and farend via a linear combination of the two. When r ≥ 1 , we use
(16)
383
LnS = LnS 23 , and when r ≤ 0.35 we use LnS = LnS 12 . For all
other values, we trade off between the two using a linear
combination. The linear combination for 0.35 < r < 1 is given by
20
20
7
LnS =  ------ r – ------ LnS 23 + ------ ( 1 – r )LnS 12
(25)
 13
13
13
Table 1 consolidates all the formulae into a single table in which
each is given as an explicit function of the first three circuit
moments and the whole swing input slew T .
6. EXPERIMENTAL RESULTS
To verify the effectiveness of the lognormal-based delay and slew
metrics, we extracted 432 routed nets containing 2244 sinks from
an industrial ASIC part in 0.18 micron technology. The nets were
chosen by a filtering process that required the maximum sink delay
to be at least 10 ps and for the ratio of the closest sink to the
furthest sink in the net to be less than 0.25. This ensures that each
net has at least one “near-end” sink. We classify the 2244 total
sinks into three categories:
5. EXTENSIONS FOR RAMP INPUTS
Recently, Kashyap et al. [7] proposed the PERI (Probability
distribution function Extension for Ramp Inputs) technique for
extending any step-input based delay and slew metric to ramp
inputs. We briefly review the corresponding formulae and show
how to apply them to LnD and LnS.
•
1187 far-end sinks have delay greater or equal to 75% of the
maximum delay to the furthest sink in the net,
• 670 mid-end sinks which have delay between 25% and 75% of
the maximum delay, and
• 367 near-end sinks which have delay less than or equal to 25%
of the maximum delay.
For each sink we compute delay and slew according to SPICE and
measure the relative error of the appropriate metric to the SPICE
result. We average the absolute values over of the errors and report
the average relative error over all sinks in the following tables. We
hook up a driver to the source of each RC network, where the
driver is a voltage source (excited with a step response) followed
by a resistor.
Inputs: whole swing input slew T , r = m 1 ⁄ m 2
Step
Delay
2
LnD = m 1 ⁄ 2m 2
LnD ( T ) = αLnD – m 1 ( 1 – α )
Ramp
Delay
5⁄2
2m 2 – m 1


where α =  ---------------------------------------------
 2m – m 2 + T 2 ⁄ 12
2
2
Step
Slew
r ≤ 0.35
LnS 12
where S =
2
1
m1
kS
-(e
= ------------2m 2
2
– e –k S
2)
2
ln ( 2m 2 ⁄ m 1 ) and k = 0.9062 .
2
m1
Step
Slew
r≥1
2m 2 –
k 2 ln ( z )
– e – k 2 ln ( z ) )
----------------------- ( e
z(z – 1)
2 1⁄3
2
+ 4+γ 
 y – 1--- + 1 y =  γ--------------------------z
=
-
where
,
,

y
2


3
– 6m 3 + 6m 1 m 2 – 2m 1
k = 0.9062 , and γ = -----------------------------------------------------2 3⁄2
( 2m 2 – m 1 )
Step Slew
0.35 < r
and r < 1
20
20
7
LnS =  ------ r – ------ LnS 23 + ------ ( 1 – r )LnS 12
 13
13
13
LnS 23 =
Ramp
Slew
2
Sinks
near
mid
far
total
Sinks
near
mid
far
total
2 2
LnS + ( 0.8 ) T
LnS ( T ) =
Table 1 Summary of proposed lognormal based slew and delay
metrics for step and ramp inputs. All formulae are explicit
functions of the whole swing input slew T and the first three
circuit moments.
Sinks
near
mid
far
total
For delay, PERI uses a parameter α to reflect the degree of
significance of the ramp on the step metric. As the input slew
approaches infinity, the delay asymptotically approaches the
Elmore delay. Hence, the value of α is chosen such that α = 1
when the full swing input slew T is zero (thereby yielding the step
delay metric) and α → 0 when T → ∞ (yielding the Elmore
delay). The lognormal formula LnD ( T ) is given by
LnD ( T ) = αLnD + ( 1 – α )ED where
2


2m 2 – m 1
α =  --------------------------------------------
2
2
 2m – m + T ⁄ 12
2
Table 2 Delay comparisons for the lognormal metric.
6.1 Experiments for Delay
5⁄2
.
LnD is a constant factor of two percent larger than D2M, and D2M
was shown to be quite effective compared to previous delay
metrics in [1]. However, those experiments were performed on
randomly generated networks. To demonstrate the effectiveness
for nets from a real design, we revisit the comparisons to the best
of the remaining known closed-form metrics. We compare to the
Elmore delay (Elm), the Kahng-Muddu [6] (KM) metric that uses
two circuit moments, and the Tutuianu et al. [14] metric (1NR) that
performs a single Newton-Raphson iteration.1 The results are
summarized in Table 2. We make the following observations:
(26)
1
Note that the formula for α achieves the desired asymptotic
behavior. The formula derives from the assumption that the
Pearson Skewness Coefficient for the PDF for step response
matches that of the ramp response.
For slew metrics, PERI proposes a root-mean square relationship
between the step slew. This gives a slew formula of
LnS ( T ) =
2
2 2
LnS + ( 0.8 ) T .
Driver Resistance = 0 Ohms
Average % Relative Error
Standard Deviation %
LnD Elm KM 1NR LnD Elm KM 1NR
57.78 339.9 142.2 543.5 40.85 201.1 105.9 2637
18.18 84.33 20.3 208.4 12.06 30.37 15.86 1307
1.38 34.64 1.37 0.40 0.61 3.87 1.35 0.56
18.37 115.2 37.44 176.8 29.85 153.9 75.22 1427
Driver Resistance = 100 Ohms
LnD Elm KM 1NR LnD Elm KM 1NR
108.0 520.7 248.2 335.3 69.83 308.5 165.0 86.92
16.91 80.93 18.52 120.3 11.10 27.8 14.53 422.6
1.51 34.38 1.62 0.40 0.62 3.73 1.34 0.25
26.32 140.8 53.65 99.03 50.89 230.1 119.4 262.3
Driver Resistance = 200 Ohms
LnD Elm KM 1NR LnD Elm KM 1NR
125.0 459.5 221.6 275.1 66.50 228.6 221.6 53.38
14.29 74.49 15.09 96.04 9.07 22.85 15.09 465.5
1.54 34.26 1.67 0.39 0.59 3.47 1.67 0.22
28.05 124.5 46.21 78.79 54.81 188.6 46.21 271.0
(27)
384
•
Like Elmore, LnD tends to overestimate delay, though LnD’s
relative error is much smaller than Elmore. While Elmore
delays are on average four to six times more than SPICE at the
near-end, LnD overestimates close to a factor of two.
• At the far-end, LnD is quite accurate, averaging roughly 1.5%
error over all inputs with very low standard deviation.
• 1NR is the most accurate metric at the far-end but is inadequate for the near-end, frequently yielding negative delays.
• The KM metric, while more accurate than Elmore, is not as
accurate as LnD, especially at the near-end.
Overall, LnD has by far the smallest relative error and standard
deviation of the four metrics.
•
•
•
6.2 Experiments for Slew
There are only two closed form slew metrics in the literature. For
10/90 slew, Bakoglu [3] proposed using ln ( 9 ) times the RC delay
(we use Elmore). We call this first order approximation Bak. The
other closed form metric, as noted by [4] and [5] is simply
σ = µ 2 times a constant. Elmore suggested a constant of
1 ⁄ 2π , which is not particularly accurate. A better choice is a
constant of ln ( 9 ) since it is exact for a single RC network and the
constant that results when matching the variance of the impulse
response to an exponential distribution. We call this metric 2CM
for the second central moment. Finally, although not a closed-form
metric, we also compare to WED since it enables sampling at the
10% and 90% delay points. The results are shown in Table 3. We
observe the following:
Sinks
near
mid
far
total
Sinks
near
mid
far
total
Sinks
near
mid
far
total
7. Conclusions
We have used the lognormal distribution to derive closed form
formulae for both delay and slew that can be used with either a step
or ramp input. We have made it easy for anyone to implement by
consolidating all the formulae in Table 1. Our experiments
demonstrate significant accuracy advantages over previous
approaches. In future work, we seek to implement these metrics
within a static timing analyzer and a physical synthesis engine.
REFERENCES
[1] C. J. Alpert, A. Devgan, and C. Kashyap, “RC Delay Metrics
for Performance Optimization”, IEEE Trans. on ComputerAided Design,20(5), pp. 571-582, 2001.
[2] C. J. Alpert, A. Devgan, and C. Kashyap, “A Two Moment
RC Delay Metric for Performance Optimization”, International Symposium on Physical Design, 2000, pp. 69-74.
[3] H. B. Bakoglu, Circuits, Interconnects, and Packaging for
VLSI. Addison-Wesley Publishing Company, 1990.
[4] W. C. Elmore, “The Transient Response of Damped Linear
Network with Particular Regard to Wideband Amplifiers”, J.
Applied Physics, 19, 1948, pp. 55-63.
[5] R. Gupta, B. Tutuianu, and L. T. Pileggi, “The Elmore Delay
as a Bound for RC Trees with Generalized Input Signals”,
IEEE Trans. on CAD, 16(1), pp. 95-104, 1997.
[6] A. B. Kahng and S. Muddu, “An Analytical Delay Model for
RLC Interconnects”, IEEE Trans. on Computer-Aided
Design, 16(12), 1997, pp. 1507-1514.
[7] C. Kashyap, C. J. Alpert, F. Liu, and A. Devgan, “PERI: A
Technique for Extending Delay and Slew Metrics to Ramp
Input”, ACM Symposium on Physical Design, 2003.
[8] R. Kay and L. Pileggi, “PRIMO: Probability Interpretation of
Moments for Delay Calculation”, IEEE/ACM Design Automation Conference, 1998, pp. 463-468.
[9] T. Lin, E. Acar, and L. Pileggi, “h-gamma: An RC Delay Metric Based on a Gamma Distribution Approximation to the
Homogeneous Response”, IEEE/ACM International Conference on Computer-Aided Design, 1998, pp. 19-25.
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Circuits based on the Weibull Distribution”, IEEE/ACM Intl.
Conference on Computer-Aided Design, 2002, pp. 620-624.
[11] L. T. Pileggi, “Timing Metrics for Physical Design of Deep
Submicron Technologies”, International Symposium on Physical Design, 1998, pp. 28-33.
[12] L. T. Pillage and R. A. Rohrer, “Asymptotic Waveform Evaluation for Timing Analysis”, IEEE Transactions on Computer
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[13] J. Rubenstein, P. Penfield, and M. A. Horowitz, “Signal Delay
in RC Tree Networks”, IEE Trans. CAD-2, July 1983.
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Delay Approximation Based on the First Three Moments of
the Impulse Response”, IEEE/ACM DAC, 1996, pp. 611-616.
Driver Resistance = 0 Ohms
Average % Relative Error
Standard Deviation %
LnS Bak 2CM WED LnS Bak 2CM WED
98.88 141.6 1734 421.1 61.98 95.38 1362 334.7
5.38 19.04 38.64 9.86 5.18 7.77 33.71 5.44
0.51 12.50 2.75 3.24 0.63 6.02 3.70 2.45
18.74 36.43 319.1 79.39 46.00 64.16 881.6 214.5
Driver Resistance = 100 Ohms
LnS Bak 2CM WED LnS Bak 2CM WED
35.66 29.80 304.3 56.20 32.02 32.13 200.9 59.89
5.92 20.32 51.94 11.47 4.96 8.30 29.28 6.06
0.49 12.67 2.70 3.17 0.53 6.02 3.52 2.19
5.68 15.72 47.47 11.00 16.60 14.33 124.8 28.15
Driver Resistance = 200 Ohms
LnS Bak 2CM WED LnS Bak 2CM WED
10.89 22.45 175.0 16.69 36.97 39.09 221.5 68.14
6.42 24.09 59.75 12.10 3.13 6.61 28.96 5.13
0.45 12.43 2.44 3.04 0.40 6.01 3.08 1.93
1.85 14.97 17.14 5.12 5.75 9.29 44.85 10.23
Table 3 Slew comparisons for the lognormal metric.
We observe the following:
•
The 2CM metric is actually quite accurate at the far end (averaging 2-3% relative error). However, it is completely inadequate at the near-end, grossly overestimating slew.
1
We do not explicitly compare to PRIMO [8], WED [10], or hgamma [9] since they require table-lookups and are not truly
closed form. All three converge well at the far-end, though PRIMO grossly underestimates delay at the near-end, while WED
tends to underestimate delay by 30-40% at the near-end. h-gamma is clearly the best performing and most complicated of the table-lookup based delay methods.
WED is also fairly accurate at both the mid- and far-end, but
very inaccurate at the near-end (especially for drivers with low
resistance) though not to the degree of 2CM. Indeed, since
WED samples the actual 10% and 90% delay points, its accuracy is highly sensitive to where the 90% point lies on the tail
of the highly-skewed impulse response curve.
For a first order approximation, the Bakoglu metric is actually
quite nice, having accuracy similar to that of Elmore for delay.
It has better accuracy than both 2CM and WED at the nearend, though is quite a bit worse at the far-end.
Finally, the LnS metric clearly dominates the other three metrics in terms of relative error and standard-deviation. LnS is on
average within 1% of the optimal solution at the far-end and is
also within a factor of two of optimal at the near-end where
slew measurements are highly sensitive.
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