![Table 1. Technology parameters and 3 variations [1]](/figures/table-1-technology-parameters-and-3-variations-1-2k6nh6dy.webp)
Asymptotic Probability Extraction for Non-Normal Distributions of Circuit
Performance
Xin Li, Jiayong Le, Padmini Gopalakrishnan and Lawrence T. Pileggi
Dept. of ECE, Carnegie Mellon University
5000 Forbes Avenue, Pittsburgh, PA 15213, USA
{xinli, jiayongl, pgopalak, pileggi}@ece.cmu.edu
Abstract
While process variations are becoming more significant with
each new IC technology generation, they are often modeled via
linear regression models so that the resulting performance
variations can be captured via Normal distributions. Nonlinear
(e.g. quadratic) response surface models can be utilized to capture
larger scale process variations; however, such models result in
non-Normal distributions for circuit performance which are
difficult to capture since the distribution model is unknown. In
this paper we propose an asymptotic probability extraction
method, APEX, for estimating the unknown random distribution
when using nonlinear response surface modeling. APEX first uses
a novel binomial moment evaluation to efficiently compute the
high order moments of the unknown distribution, and then applies
moment matching to approximate the characteristic function of
the random circuit performance by an efficient rational function.
A simple statistical timing example and an analog circuit example
demonstrate that APEX can provide better accuracy than Monte
Carlo simulation with 10
4
samples and achieve orders of
magnitude more efficiency. We also show the error incurred by
the popular Normal modeling assumption using standard IC
technologies.
1. Introduction
As IC technologies are scaled to the deep sub-micron region,
process variations are becoming critical and significantly impact
the overall performance of a circuit. Table 1 shows some typical
process parameters and their 3ı variations as technologies are
scaled from 0.25 µm to 70 nm. These large-scale variations
introduce uncertainties in circuit behavior, thereby making IC
design increasingly difficult. Low product yield or unnecessary
over-design cannot be avoided if these process variations are not
accurately modeled and analyzed within the IC design flow.
Table 1. Technology parameters and 3ı variations [1]
Leff (nm) Tox (nm) Vth (mV) W (µm) H (µm)
250r80 5.0r0.40 500r50 0.80r0.20 1.2r0.3
180r60 4.5r0.36 450r45 0.65r0.17 1.0r0.3
130r45 4.0r0.39 400r40 0.50r0.14 0.9r0.27
100r40 3.5r0.42 350r40 0.40r0.12 0.8r0.27
70r33 3.0r0.48 300r40 0.30r0.10 0.7r0.25
During the past decade, various statistical analysis techniques
[1]-[7] have been proposed and utilized in many applications such
as statistical timing analysis, mismatch analysis, yield
optimization, etc. The objective of these techniques is to model
the probability distribution of the circuit performance under
random process variations. The author in [1] applies linear
regression to approximate a given circuit performance f (e.g.
delay, gain, etc.) as a function of the process variations (e.g. Vth,
Tox, etc.), and assumes that all random variations are normally
distributed. As such, the performance f is also a Normal
distribution, since the linear combination of normally distributed
random variables still has a Normal distribution [8].
The linear regression model is efficient and accurate when the
process variations are sufficiently small. However, the large-scale
variations in the deep sub-micron technologies, which can reach
r35% as shown in Table 1, suggest applying high order regression
models in order to guarantee high approximation accuracy [4]-[7].
Using a high order response surface model, however, brings about
new challenges due to the nonlinear mapping between the process
variations and the circuit performance f. The distribution of f is no
longer Normal, as is the case for the linear regression model. The
authors in [3]-[5] utilize Monte Carlo simulation to evaluate the
probability distribution of f, which is computationally expensive.
Note that the computation cost for this probability extraction is
crucial, especially when the extraction procedure is an inner loop
within the optimization flow.
In this paper, we propose a novel Asymptotic Probability
EXtraction approach, APEX, for estimating the unknown
PDF/CDF functions using nonlinear response surface modeling.
Given a circuit performance f (e.g. a digital circuit path delay or
the performance parameter of an analog/RF circuit), the response
surface modeling approximates f as a polynomial function of the
process parameters (e.g. Vth, Tox, etc.). Since the process
parameters are modeled as random variables, the circuit
performance f is a function of these random variables, which is
also a random variable. APEX applies moment matching to
approximate the characteristic function of f (i.e. the Fourier
transform of the probability density function [8]) by a rational
function H. We conceptually consider H to be of the form of the
transfer function of a linear time-invariant (LTI) system, and the
probability distribution function (PDF) and the cumulative
distribution function (CDF) of f are approximated by the impulse
response and the step response of the LTI system H, respectively.
The resulting probability distribution function can then be used to
characterize and/or optimize the statistical performance of analog
and digital circuits under process variations.
APEX extends existing moment matching methods via three
important new contributions which significantly reduce the
computation cost and improve the approximation accuracy for this
particular application. Firstly, a key operation required by APEX
is to compute the high order moments, which is extremely
expensive when using traditional techniques. In APEX, we
propose a binomial evaluation scheme to recursively compute the
high order moments for a given quadratic response surface model.
The binominal moment evaluation is derived from statistical
independence theory and principal component analysis (PCA)
methods. It can achieve more than 10
6
x speedup compared with
direct moment evaluation.
Secondly, APEX approximates the unknown probability
distribution function by the impulse response of an LTI system.
Directly applying such an approximation to any circuit
performance with negative values is infeasible, since it results in
0-7803-8702-3/04/$20.00 ©2004 IEEE.
2
an LTI system that is non-causal. To overcome this difficulty,
APEX applies a generalized Chebyshev inequality for PDF/CDF
shifting.
Lastly, the best-case performance (e.g. the 1% point on CDF)
and the worst-case performance (e.g. the 99% point on CDF) are
two important metrics to be evaluated. Direct moment matching
cannot capture the 1% point value accurately since the moment
matching approximation is most accurate for low frequency
components (corresponding to the final values of CDF), and least
accurate for high frequency components (corresponding to the
initial values of CDF). To address this problem, a reverse
evaluation technique is proposed in this paper to produce an
accurate estimation of the 1% point.
The remainder of the paper is organized as follows. In Section
2 we review the background on response surface modeling. Then
we propose our APEX approach in Section 3. We discuss several
implementation issues, including the high order moment
evaluation, PDF/CDF shifting and reverse PDF/CDF evaluation,
in Section 4. The efficacy of APEX is demonstrated by several
circuit examples in Section 5. Finally, we conclude in Section 6.
2. Background
Given a circuit topology, the circuit performance (e.g. gain,
delay) is a function of the design parameters (e.g. bias current,
transistor sizes) and the process parameters (e.g. V
TH
, T
OX
). The
design parameters are optimized and fixed during the design
process; however, the process parameters must be modeled as
random variables to account for any uncertain variations. Given a
set of fixed design parameters, the circuit performance f can be
approximated by a linear regression model [1]:
XBXfXf
T
'
ˆ
(1)
where
>@
T
N
xxxX ,,,
21
denotes the process parameters,
X
is the mean value of
X,
X
X
X
' represents the process
variations,
N
R
B
ˆ
stands for the linear model coefficients and N
is the total number of these random variations.
The process variations in (1), i.e. ǻ
X, are often approximated
by zero-mean Normal distributions
*
. In addition, correlated
process variations can be expressed in terms of independent
factors using principal component analysis (PCA) [9]. Given a set
of normally distributed random variables ǻ
X and their symmetric,
positive semi-definite correlation matrix
R, PCA decomposes R
as:
T
VVR 6 (2)
where
N
diag
O
O
O
,,,
21
6 contains the eigenvalues of R, and
>@
N
VVVV ,,,
21
contains the corresponding eigenvectors that
are orthonormal, i.e.
IVV
T
(I is the identity matrix). Based on
Ȉ and V, PCA defines a set of new random variables:
XVY
T
'6 '
5.0
(3)
It is easy to verify that the random variables ǻ
Y are independent
and satisfy the Normal distribution
1,0N
(i.e. zero mean and
unit standard deviation).
The factors extracted from PCA can be interpreted as
coordinate rotations of the space defined by the original random
variables. In addition, if the magnitude of the eigenvalues
i
O
deceases quickly, it is possible to use a small number of principal
components to approximate the original N-dimensional space.
*
If a process parameter ȗ satisfies a log-Normal distribution, it can also be
transformed to a Normal distribution by taking the logarithmic operator,
i.e. ln(ȗ) is normally distributed.
More details on PCA can be found in [9].
Substituting (3) into (1) yields:
YBCYf
T
' ' (4)
where
XfC and BVB
T
ˆ
5.0
6 . The linear regression
model in (4) is accurate when the process variations are small.
However, the large-scale variations that are expected for
nanoscale technologies suggest that applying quadratic response
surface models might be required to provide sufficient accuracy:
YAYYBCYf
TT
''' ' (5)
In (5), RC is the constant term,
N
R
B
represents the linear
coefficients and
NN
R
A
u
denotes the quadratic coefficients.
Without loss of generality, we assume that
A is symmetric in this
paper, since any asymmetric quadratic form can be easily
converted to an equivalent symmetric form [10].
3. Asymptotic Probability Extraction
Given the quadratic response surface model in (5), the
objective of probability extraction is to estimate the unknown
probability density function
fpdf and cumulative distribution
function
fcdf for performance f. Instead of running expensive
Monte Carlo simulations, APEX tries to find an M-th order LTI
system
H whose impulse response
th and step response
ts are
the optimal approximations for the
fpdf
and
fcdf
respectively. Here, the variable
t in
th and
ts corresponds to
the variable
f in
fpdf and
fcdf . The optimal approximation
is determined by matching the first 2M moments between
th
and
fpdf for an M-th order approximation. We first describe
the mathematical formulation of APEX in Section 3.1. Then, in
Section 3.2 we will link APEX to traditional probability theory
and explain why it can be used to efficiently approximate
PDF/CDF functions.
3.1 Mathematical Formulation
We define the time moments [11] for a given circuit
performance
f whose probability density function is
fpdf
as
follows:
³
f
f
dffpdff
k
m
k
k
k
!
1
(6)
In (6), the definition of time moments is identical to the traditional
definition of moments in probability theory except for the scaling
factor
!1 k
k
.
Similarly, time moments can be defined for an LTI system
H
[11]. Given an M-th order LTI system whose transfer function and
impulse response are:
°
¯
°
®
t
¦
¦
00
0
1
1
t
tea
thand
bs
a
sH
M
i
tb
i
M
i
i
i
i
(7)
The time moments of
H are expressed as [11]:
¦
³
f
f
M
i
k
i
i
k
k
k
b
a
dttht
k
m
1
1
!
1
(8)
In (7), the poles
^
`
Mib
i
,,2,1,
and residues
^`
Mia
i
,,2,1,
are the 2M unknowns that need to be
determined. Matching the first 2M moments in (6) and (8) yields
the following 2M nonlinear equations:
3
12
22
2
2
2
1
1
1
22
2
2
2
1
1
0
2
2
1
1
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
M
M
M
M
MM
M
M
M
M
m
b
a
b
a
b
a
m
b
a
b
a
b
a
m
b
a
b
a
b
a
(9)
The nonlinear equations in (9) can be solved using the algorithm
proposed in [11]. Once the poles
i
b
and residues
i
a
have been
determined, the probability density function
fpdf
is optimally
approximated by
th in (7), and the cumulative distribution
function
fcdf is optimally approximated by the step response:
°
¯
°
®
t
¦
³
00
01
1
0
t
te
b
a
dhts
M
i
tb
i
i
t
i
WW
(10)
It is worth noting that many implementation issues must be
considered to make our proposed approach, APEX, feasible and
efficient. For example, the impulse response of a causal LTI
system is only nonzero for
0tt , but a PDF in practical
applications can be nonzero for
0df . In section 4, we will
propose several schemes to address these problems.
The aforementioned moment-matching method was
previously applied for IC interconnect order reduction [11], [12]
and is related to the Padé approximation in linear control theory
[13]. In the following section, we will explain why this moment-
matching approach is an efficient way to approximate PDF/CDF
functions.
3.2 Connection to Probability Theory
In probability theory, given a random variable f whose
probability density function is
fpdf , the characteristic function
is defined as the Fourier transform of
fpdf [8].
³
¦
³
f
f
f
f
f
) df
k
fj
fpdfdfefpdf
k
k
fj
0
!
Z
Z
Z
(11)
Substituting (6) into (11) yields:
¦
f
)
0k
k
k
jm
ZZ
(12)
This implies an important fact: the time moments defined in (6)
are related to the Taylor expansion of the characteristic function at
the expansion point 0
Z
. Matching the first 2M moments in (9)
is equivalent to matching the first 2M Taylor expansion
coefficients between the original characteristic function
Z
)
and the approximated rational function
sH .
To explain why the moment-matching approach is efficient,
we first need to show two important properties of the
characteristic function [8]:
Property 1: A characteristic function has maximal magnitude at
0
Z
, i.e.
10 )d)
Z
.
Property 2: A characteristic function
0o)
Z
when fo
Z
.
Fig. 1 shows the characteristic functions for several typical
random distributions. The above two properties imply an
interesting fact: namely, given a random variable
f, its
characteristic function decays as
Ȧ increases. Therefore, the
optimally approximated
sH in (7) is a low pass system. It is
well-known that a Taylor expansion is accurate around the
expansion point. Since a low-pass system is mainly determined by
its behavior in the low-frequency band (around 0
Z
), it can be
accurately approximated by matching the first several Taylor
coefficients at 0
Z
, i.e. the moments. This conclusion has been
verified in other applications (e.g. IC interconnect order reduction
[11], [12]) and it provides the theoretical background to explain
why moment-matching works well for the PDF/CDF evaluations
that we will demonstrate in Section 5.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Z
|
)
(
Z
)|
normal
cauchy
chi-square
gamma
Fig. 1.
Characteristic function for typical distributions.
4. Implementation of APEX
Our proposed APEX approach is made practically feasible by
applying several novel algorithms, including: 1) a binomial
scheme for high order moment computation, 2) a generalized
Chebyshev inequality for PDF/CDF shifting and 3) a reverse
evaluation technique for best-case/worst-case analysis. In this
section, we describe the mathematical formulation of each of
these algorithms.
4.1 Binomial Moment Evaluation
A key operation required in APEX is the computation of the
high order time moments defined in (6) for a given random
variable
f. Such a moment evaluation is equivalent to computing
the expectation of
^
`
12,,1,0, Mkf
k
. Given the quadratic
response surface model in (5),
k
f is a high order polynomial in
ǻ
Y:
¦
''' '
i
N
i
k
Niii
yyycYf
DDD
21
21
(13)
where
i
y' is the i-th element in the vector ǻY,
i
c is the
coefficient of the i-th product term and
ij
D
is the positive integer
exponent. Since the random variables ǻ
Y are independent after
PCA analysis, we have:
¦
'''
i
Ni
k
Niii
yEyEyEcfE
DDD
21
21
(14)
where
xE
stands for the expectation. In addition, remember that
each random variable
i
y' has a Normal distribution
1,0N ,
which yields [8]:
°
¯
°
®
'
,6,4,2131
,5,3,10
01
kk
k
k
yE
k
i
(15)
Substituting (15) into (14), the expectation of
k
f can be
determined.
The above computation scheme is called
direct moment
4
evaluation in this paper. The key disadvantage of the direct
moment evaluation is that, as
k increases, the total number of the
product terms in (14) will increase exponentially, thereby quickly
making their computation infeasible. To overcome this difficulty,
we propose a novel
binomial moment evaluation scheme that
consists of two steps: quadratic model diagonalization and
moment evaluation. The binomial moment evaluation scheme
recursively computes the high order moments, instead of
explicitly constructing the high order polynomials
k
f in (14).
A. Quadratic Model Diagonalization
The first step in binomial moment evaluation is to remove the
cross product terms in the quadratic response surface model (5),
thereby yielding a much simpler, but equivalent, quadratic model.
According to matrix theory [10], any symmetric matrix
NN
R
A
u
can be diagonalized as:
T
UUA / (16)
where
N
diag
VVV
,,,
21
/ contains the eigenvalues of A
and
>@
N
UUUU ,,,
21
is an orthogonal matrix (i.e. IUU
T
)
containing the eigenvectors. Define new random variables
'Z as
follows:
YUZ
T
' ' (17)
Substituting (17) into (5) yields:
¦
''
'/'' '
N
i
iiii
TT
zzqC
ZZZQCZf
1
2
V
(18)
where
i
z' is the i-th element in the vector ǻZ and
>@
BUqqqQ
T
T
N
,,,
21
. Equation (18) implies that there is
no cross product term in the quadratic model after diagonalization.
In addition, the following theorem guarantees that the random
variables ǻ
Z defined in (17) are still independent and satisfy the
Normal distribution
1,0N .
Theorem 1: Given a set of independent random variables ǻY with
the Normal distribution
1,0N and an orthogonal matrix U, the
random variables ǻZ defined in (17) are independent and satisfy
the Normal distribution
1,0N
.
Proof: Since the random variables ǻZ are linear combinations of
normally distributed random variables ǻ
Y, they are normally
distributed. The correlation matrix for ǻ
Z is given by:
UYYEUUYYUEZZE
TTTTT
'' '' '' (19)
Remember that ǻ
Y is a set of independent random variables with a
Normal distribution
1,0N , i.e.
IYYE
T
'' , and matrix U is
orthogonal, i.e.
IUU
T
. Thus, we have:
IUIUUYYEUZZE
TTTT
'' '' (20)
Equation (20) implies that the random variables in ǻ
Z are
uncorrelated. In addition, uncorrelated random variables with
Normal distributions are also independent [8]. Ŷ
B. Moment Evaluation
We now demonstrate the use of the simplified quadratic
model (18) for fast moment evaluation. Based on (18), we define
a set of new random variables:
¦¦
''
''
l
i
iiii
l
i
il
iiiii
zzqCgCh
zzqg
1
2
1
2
V
V
(21)
Comparing (21) with (18), it is obvious that when
Nl ,
fh
N
. Instead of computing the high order moments of f
directly, the proposed binomial moment evaluation scheme
successively computes the moments of
l
h , as shown in Fig. 2.
1.
Start from Ch
0
and compute
kk
ChE
0
for each
12,,1,0 Mk . Set 1 l .
2. For each 12,,1,0 Mk , compute:
¦
'
¸
¸
¹
·
¨
¨
©
§
»
¼
º
«
¬
ª
''
k
i
ik
l
ik
l
i
l
k
llll
k
l
zEq
i
k
zzqEgE
0
22
VV
(22)
>@
¦
¸
¸
¹
·
¨
¨
©
§
k
i
ik
l
i
l
k
ll
k
l
gEhE
i
k
ghEhE
0
11
(23)
3. If Nl , then go to Step 4. Otherwise, 1 ll and return
Step 2.
4. For each 12,,1,0 Mk , we have
k
N
k
hEfE
.
Fig. 2. Binomial moment evaluation algorithm.
Step 2 in Fig. 2 is the key operation required by the binomial
moment evaluation algorithm. In Step 2, both (22) and (23) utilize
the binomial theorem to get the binomial series. Therefore, we
refer to this algorithm as
binomial moment evaluation in this
paper.
In (22), the expectation
ik
l
zE
'
2
can be easily evaluated
using the closed-form expression (15), since
l
z'
is normally
distributed
1,0N . Equation (23) utilizes the property that
1l
h
and
l
g
are independent, because
1l
h
is a function of
^`
1,,2,1, ' liz
i
,
l
g is a function of
l
z' and all
i
z' are
mutually independent. Therefore,
ik
l
i
l
ik
l
i
l
gEhEghE
11
,
where the values of
i
l
hE
1
and
ik
l
gE
have already been
computed in previous steps.
The main advantage of the binomial moment evaluation is
that, unlike the direct moment evaluation in (14),
it does not
explicitly construct the high order polynomial
k
f . Therefore,
unlike direct moment evaluation, where the total number of the
product terms will exponentially increase, both
k
l
gE in (22) and
k
l
hE in (23) contain at most 2M product terms. Since
12,,1,0
Mk and Nl ,,1,0 for an M-th order APEX
approximation with
N independent random variations, the total
number of
k
l
gE
and
k
l
hE
that need to be computed is
MNO . In addition, the matrix diagonalization in (16) only
needs to be computed once and has a complexity of
3
NO .
Therefore, the computational complexity of the proposed
algorithm is
32
NONMO . In most circuit applications, N is
small (around 5~100) after PCA analysis, and selecting
10~7
M provides sufficient accuracy for moment matching.
With these typical values for
M and N, the proposed binomial
moment evaluation is extremely fast, as we will demonstrate with
numerical examples in Section 5.
It should be noted that as long as the circuit performance f is
represented by the quadratic model in (5) and the process
variations are normally distributed, binomial moment evaluation
provides the
exact high order moment values (except for
numerical errors). There is no further assumption or
5
approximation made by the algorithm.
In summary, binomial moment evaluation utilizes statistical
independence theory and principal component analysis (PCA) to
efficiently compute high order moment values, which are required
in moment matching for probability extraction.
4.2 PDF/CDF Shifting
Mean µ
ȟ
fpdf
f
0
Case 1 ņ
Not Causal
Case 2 ņ
Large Delay
Fig. 3.
Illustration for PDF/CDF shifting.
APEX approximates the unknown PDF
fpdf
as the
impulse response
th of an LTI system. The impulse response of
a causal system is only nonzero for
0tt , but a PDF in practical
applications can be nonzero for
0
df
. In such cases, we need to
right-shift the unknown
fpdf by
0
f and use the impulse
response
th to approximate the shifted PDF
0
ffpdf
, as
shown in Fig. 3 (case 1).
In addition, even if the unknown PDF
fpdf is zero for all
0
df , it can be far away from the origin, as shown in Fig. 3 (case
2). As such, the corresponding impulse response
th presents a
large delay in time domain, which cannot be accurately captured
by a low-order approximation. In such cases, we need to left-shift
the unknown
fpdf by
0
f and use the impulse response
th
to approximate the shifted PDF
0
ffpdf .
The above analysis implies that it is crucial to determine the
correct value of
0
f for PDF/CDF shifting. Over-shifting the
unknown PDF to either left or right side can increase the
approximation error. In this paper, process variations are modeled
as Normal distributions, which are unbounded and distributed
over
ff, . Therefore, any circuit performance f represented
by the quadratic model in (5) is also unbounded. It is impossible
to completely shift
fpdf to the positive axis.
However, since
f is a random variable,
fpdf
can be left-
shifted by
0
f (
0
f is negative in case of right-shifting) such that
the probability
0
0
d ffP is sufficiently small. As shown in
Fig. 3, the PDF/CDF shifting problem can be stated as follows:
find the value
ȟ and left-shift
fpdf
by
[
P
0
f , where µ is
the mean value of
f, such that the probability
0
0
d ffP is not
greater than a given error tolerance
İ. In addition, we want to
select the value
ȟ to be as small as possible, i.e. find the smallest ȟ
satisfying
H
dd 0
0
ffP . A small ȟ results in a small time-
domain delay in
th and, therefore, high approximation accuracy
for
fpdf . To estimate ȟ, we need the following theorem.
Theorem 2: Given a random variable f, for any 0!
[
and
,6,4,2 k ,
>@
k
k
fE
fP
[
P
[P
dt
(24)
where µ is the mean value of f.
Proof: For any ,6,4,2 k , we have
>@
k
k
k
k
f
k
k
f
fE
dffpdf
f
dffpdf
f
dffpdffP
[
P
[
P
[
P
[P
[P[P
d
d t
³
³³
f
f
tt
(25)
Note that the above proof is not restricted to any special
probability distribution. Ŷ
Based on (24), if the unknown PDF
fpdf is left-shifted by
[P
0
f , we have:
>@
k
k
fE
fP
fPfPffP
[
P
[P
[P[P
dtd
t d d 00
0
(26)
where ,6,4,2
k . Therefore, one sufficient condition for
H
dd 0
0
ffP is:
>@
,6,4,2 d
k
fE
k
k
H
[
P
(27)
which is equivalent to:
>@
,6,4,2
1
°
¿
°
¾
½
°
¯
°
®
t k
fE
k
k
H
P
[
(28)
Equation (28) estimates
ȟ using high order central moments. In an
M-th order approximation, after the high order expectations
k
fE
12,,1,0
Mk are computed by the binomial
moment evaluation algorithm in Fig. 2, the central moments can
be easily calculated using the binomial theorem:
>@
¦
¸
¸
¹
·
¨
¨
©
§
k
i
ik
i
k
fE
i
k
fE
0
PP
(29)
where
fE
P
. Then, using (28), an estimated ȟ is computed for
each 22,,4,2
Mk , which is denoted as
k
[
. The minimal
value of all these
k
[
values is utilized as the final ȟ for PDF/CDF
shifting, since we aim to find the smallest
ȟ to achieve high
approximation accuracy for
fpdf .
It is worth mentioning that when 2
k , equation (24) is the
well-known Chebyshev inequality [8]. We have generalized the
2nd order Chebyshev inequality to higher orders and, therefore
refer to (24) as the
generalized Chebyshev inequality. In practical
applications we find that high order moments provide a much
tighter (i.e. smaller) estimation of
ȟ, as is demonstrated by the
numerical examples in Section 5.
In summary, the proposed generalized Chebyshev inequality
(24) provides an effective way to estimate the boundary for
PDF/CDF shifting. As such, the major part of the unknown
PDF/CDF can be moved to the positive axis, which is then
accurately approximated by the impulse/step response of a causal
LTI system.
4.3 Reverse PDF/CDF Evaluation
In many practical applications, such as robust circuit
optimization [4], [14], the best-case performance (e.g. the 1%
point on CDF) and the worst-case performance (e.g. the 99%
point on CDF) are two important metrics to be evaluated. As
discussed in Section 3.2, APEX matches the first 2M Taylor
expansion coefficients between the original characteristic function
Z
) and the approximated rational function. Remember that the
6