asymptotic probability extraction for non-normal distributions of circuit performance
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1
Asymptotic Probability Extraction for Non-Normal Distributions of Circuit
Performance
By: Sedigheh Hashemi
201C-Spring2009
Asymptotic Probability Extraction for Non-Normal Distributions of Circuit
Performance
By: Sedigheh Hashemi
201C-Spring2009
Asymptotic Probability Extraction for Non-Normal Distributions of
Circuit Performance
Asymptotic Probability Extraction for Non-Normal Distributions of
Circuit Performance
X. Li, P. Gopalakrishnan and L. Pileggi, CMUJ. Le, Extreme DA
3
OverviewOverview
Introduction Asymptotic Probability EXtraction (APEX) Implementation of APEX Numerical examples Conclusion
4
IC Technology ScalingIC Technology Scaling
Feature SizeScale Down
0.35 μm 0.18 μm 90nm
Year Leff (nm) W L Tox Vth H
1997 250 25.0% 32.0% 8.0% 10.0% 25.0% 22.2%
1999 180 26.2% 33.3% 8.0% 10.0% 30.0% 24.0%
2002 130 28.0% 34.6% 9.8% 10.0% 30.0% 27.3%
2005 100 30.0% 40.0% 12.0% 11.4% 33.8% 31.7%
2006 70 33.3% 47.1% 16.0% 13.3% 35.7% 33.3%
Process Variations (3σ / Nominal) [Nassif 01]
Process variation is becoming relatively larger!
5
Statistical Problems in ICStatistical Problems in IC
Statistical methods have been proposed to address various statistical problems
We focus on analysis problem in this work
ModelingModeling
RSM
MOR
etc.
AnalysisAnalysis
Timing
Yield
etc.
SynthesisSynthesis
Gate Sizing
Design Centering
etc.
DesignParameters
ProcessParameters
RandomDistribution
FixedValue
CircuitPerformance
Unknown Distribution
6
Modeling Process VariationsModeling Process Variations
Assumption Process variations Δxi satisfy Normal distributions N(0,σi)
Principle component analysis (PCA) Δxi can be decomposed into independent Δyi ~ N(0,1)
3
2
1
3
2
1
y
y
y
x
x
x
Δx1
Δx3
Δx2Δy1
Δy3
Δy2
06σi
7
Response Surface ModelResponse Surface Model
NN yyYpYp 110
Δy1
Δy2
Δy1
p
Δy2
p
Δy1
Δy2
p
Δy2
Δy1
p
YAYYBCYp
yyy
yyYpYp
TT
NN
...
21122111
110
Linear RSM is Not Sufficiently Accurate
8
Response Surface ModelResponse Surface Model
A low noise amplifier example designed in IBM 0.25 μm process
Performance Linear Quadratic
F0 1.76% 0.14%
S11 6.40% 1.32%
S12 3.44% 0.61%
S21 2.94% 0.34%
S22 5.56% 3.47%
NF 2.38% 0.23%
IIP3 4.49% 0.91%
Power 3.79% 0.70%
Regression Modeling Error for LNA
NormalDistribution Δyi
NonlinearTransformNonlinearTransform
Non-Normal Distribution p
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Moment MatchingMoment Matching
Key idea Conceptually consider PDF as the impulse response of an LTI system
MatchMoments
ImpulseExcitation
LTISystem
LTISystem
ImpulseResponse
UnknownPDF
NonlinearTransformNonlinearTransform
NormalDistribution
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Moment MatchingMoment Matching
Impulse response
Moments
Match the first 2M moments
ImpulseExcitation
ImpulseResponse
M
1i i
i
bs
a
00
01
t
teath
M
i
tbi
i
dPPpdfPm
b
akdtthtime
kk
M
iki
ik
k
1
!moment t
111
12222
221
1
1222
221
1
02
2
1
1
MMQ
MMM
M
M
M
M
mb
a
b
a
b
a
mb
a
b
a
b
a
mb
a
b
a
b
a
ai & bi can be solved by using the algorithm in [Pillage 90]
[Pillage 90]: Asymptotic waveform evaluation for timing analysis, IEEE TCAD, 1990.
11
Connection to Probability TheoryConnection to Probability Theory
Φ(ω) is called characteristic function in probability theory
We actually match the first 2M terms of Taylor expansion at ω = 0
HDomainFrequency
ppdfthDomainTime
""
""
System Theory Probability Theory
00 !! kk
k
k
kk
pj mk
jdpppdfp
k
jdpeppdf
0 !k
kpj
k
pje
12
Connection to Probability TheoryConnection to Probability Theory
Proposition 1 Proposition 2 Typical characteristic functions are
"low-pass filters"
A low-pass system is determined by its behavior in low-freq band (ω = 0)
Taylor expansion is accurate around expansion point (ω = 0)
Moment matching is efficient in approximating low-pass systems [Celik 02]
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
|
()|
normal
cauchy
chi-square gamma
10
0lim
Characteristic Function for Typical Random Distributions
[Celik 02]: IC Interconnect Analysis, Kluwer Academic Publishers, 2002
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The Classical Moment ProblemThe Classical Moment Problem
Δyi
pdf(p)
ProbabilityExtraction
RSM
21122111
110
yyy
yyYpYp NN
[T. Stieltjes 1894]
54
32
pEpE
pEpEpE
MomentMatching
pdf(p)
14
APEX Asymptotic Probability ExtractionAPEX Asymptotic Probability Extraction
2
2
2
2
t
Ξ
t
ΘΦ
Classical moment problem Existence & uniqueness of the solution Find complete bases to expand PDF function space
APEX Efficiently compute high order moments Efficiently approximate the unknown PDF/CDF
Different
15
Direct Moment EvaluationDirect Moment Evaluation
If Δy1, Δy2,... are independent standard Normal distribution N(0,1)
Require computing symbolic expression for pk(Y)
21122111110 yyyyyYpYp NN
,4,2131
,3,10
5353 22211
22211
kk
kyE
yEyEyEyEyyyyE
ki
kp
yyyyy
yyyyyp
yyyyp
22
221
22
21
212211
2
22211
2510
3069
53
k
# of Terms
Exponentially Increase!!!
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Binomial Moment EvaluationBinomial Moment Evaluation
Key idea Recursively compute high order moments Derived from eigenvalue decomposition & statistical independence theory
1,0~ NywhereCYBYAYp iTT
Quad ModelDiagonalizationQuad Model
Diagonalization
A
Recursive MomentEvaluation
Recursive MomentEvaluation
2pEpE
Binomial Moment
Evaluation
17
Step 1 – Model DiagonalizationStep 1 – Model Diagonalization
TT
TT
AAwhereUUA
CYBYAYp
ΔYUΔZ T
Czqz
CZQZZp
iiiii
TT
2
Δy1
Δy3
Δy2
u1
u3
u2
Δz1
Δz3
Δz2
IUYYUEZZE TTT
Δzi are independent N(0,1) since eigenvectors U are orthogonal !
18
k
i
ikik
i
ikikk gEhEi
kgh
i
kEghEhE
031
031312
Binomial Series (k+1) Terms
k
i
ikik
i
ikikk gEgEi
kgg
i
kEggEhE
021
021211
Binomial Series (k+1) Terms
k
i
ikl
ikl
il
k
i
ikl
ikl
il
k
llllkl zEq
i
kzq
i
kEzqzEgE
0
2
0
22
Binomial Series (k+1) Terms
Step 2 – Moment EvaluationStep 2 – Moment Evaluation
NOT compute symbolic expression for pk(Y) Achieve more than 106x speedup compared with direct evaluation
Czqzzqzzqzp 3323322
22211
211
1g 2g 3g
1h
2h
3h
k
i
ikik
i
ikikkk ChEi
kCh
i
kEChEhEpE
02
0223
Binomial Series (k+1) Terms
19
OverviewOverview
Introduction Asymptotic Probability EXtraction (APEX) Implementation of APEX
PDF/CDF shifting Reverse PDF/CDF evaluation
Numerical examples Conclusion
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PDF/CDF shifting is required in two cases Over-shifting results in large approximation error The challenging problem is to accurately determine ξ
PDF/CDF ShiftingPDF/CDF Shifting
pdf(p)
Mean μ
ξ
p0
pdf(p)
Mean μ
ξ
p0
Case 1 – Not Causal Case 2 – Large Delay
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PDF/CDF ShiftingPDF/CDF Shifting
Exact ξ doesn't exist since pdf(p) is unbounded
Define a bound ξ such that the probability P(p ≤ μ-ξ) is sufficiently small
Propose a generalized Chebyshev inequality to estimate ξ using central moments
ξ
pMean μ
ξ
p
εξμpP
Mean μ
,6,4,2
1
kpE kk
22
Reverse PDF/CDF EvaluationReverse PDF/CDF Evaluation
Final value theorem of Laplace transform
Moment matching is accurate for estimating upper bound
Use flipped pdf(-p) for estimating lower bound
pdf(p)
p0
Accurate for Estimating Upper Bound
ssppdfsp
0
limlim
Flipped pdf(-p)
p0
Accurate for Estimating Lower Bound
23
OverviewOverview
Introduction Asymptotic Probability EXtraction (APEX) Implementation of APEX Numerical examples Conclusion
24
ISCAS'89 S27ISCAS'89 S27
ST 0.13 μm process 6 principal random factors
MOSFET variations No intra-die variation No interconnect variation
Linear delay modeling error 4.48%
Quadratic delay modeling error 1.10% (4x smaller)
Longest Path in ISCAS'89 S27
25
ISCAS'89 S27ISCAS'89 S27
Binomial moment evaluation achieves more than 106x speedup
Moment Order
Direct Binomial
# of Terms Time (Sec.) Time (Sec.)
1 28 1.00 10-2 0.01
3 924 3.02 100 0.01
5 8008 2.33 102 0.01
6 18564 1.57 103 0.01
7 38760 8.43 103 0.02
8 74613 3.73 104 0.02
15 — — 0.04
20 — — 0.07Computation Time for Moment Evaluation
Δyi
21122111
110
yyy
yyYpYp NN
MomentEvaluation
54
32
pEpE
pEpEpE
26
ISCAS'89 S27ISCAS'89 S27
Numerical oscillation for low order approximation
Increasing approx. order provides better accuracy
Typical approx. order is 7 ~ 100.2 0.3 0.4 0.5 0.6 0.7
0
0.2
0.4
0.6
0.8
1
Delay (ns)
Cu
mu
lati
ve
Dis
trib
uti
on
Fu
nc
tio
n
Approx Order = 4Approx Order = 8Exact
Cumulative Distribution Function for Delay
Delay
27
ISCAS'89 S27ISCAS'89 S27
APEX is the most accurate approach APEX achieves more than 200x speedup compared with MC 104 runs
APEX: 0.18 seconds MC 104 runs: 43.44 seconds
Linear Legendre APEX
1% Point 1.43% 0.87% 0.04%
10% Point 4.63% 0.02% 0.01%
25% Point 5.76% 0.12% 0.03%
50% Point 6.24% 0.05% 0.02%
75% Point 5.77% 0.03% 0.02%
90% Point 4.53% 0.16% 0.03%
99% Point 0.18% 0.78% 0.09%
Comparison on Estimation Error
28
Low Noise AmplifierLow Noise Amplifier
IBM 0.25 μm process 8 principal random factors
MOSFET & RCL variations No mismatches
Circuit Schematic for LNA
Performance Linear Quadratic
F0 1.76% 0.14%
S11 6.40% 1.32%
S12 3.44% 0.61%
S21 2.94% 0.34%
S22 5.56% 3.47%
NF 2.38% 0.23%
IIP3 4.49% 0.91%
Power 3.79% 0.70%
Regression Modeling Error for LNA
29
Low Noise AmplifierLow Noise Amplifier
APEX is the most accurate approach APEX achieves more than 200x speedup compared with MC 104 runs
APEX: 1.29 seconds MC 104 runs: 334.37 seconds
PerformanceCorner Linear Legendre APEX
1% 99% 1% 99% 1% 99% 1% 99%
F0 15.8% 20.1% 1.11% 1.10% 0.20% 0.55% 0.06% 0.05%
S11 45.4% 51.5% 5.78% 1.40% 2.94% 3.28% 0.09% 0.08%
S12 38.9% 44.6% 3.88% 1.16% 0.39% 0.27% 0.14% 0.28%
S21 60.3% 51.6% 2.91% 4.69% 0.37% 0.01% 0.17% 0.19%
S22 23.1% 36.0% 1.01% 5.61% 1.11% 0.84% 0.07% 0.19%
NF 51.9% 72.8% 3.70% 3.52% 0.34% 0.37% 0.06% 0.12%
IIP3 54.6% 59.7% 5.02% 5.93% 0.29% 0.43% 0.33% 0.26%
Power 16.6% 42.5% 0.01% 1.24% 0.92% 0.93% 0.09% 0.02%
Comparison on Estimation Error
30
Operational AmplifierOperational Amplifier
IBM 0.25 μm process 49 principal random factors
MOSFET variations from design kit Include mismatches
Circuit Schematic for OpAmp
Performance Linear Quadratic
Gain 3.92% 1.57%
Offset 21.80% 7.49%
UGF 1.14% 0.45%
GM 0.96% 0.52%
PM 1.11% 0.41%
SR (P) 0.82% 0.66%
SR (N) 1.27% 0.44%
SW (P) 0.38% 0.16%
SW (N) 0.36% 0.12%
Power 1.00% 0.64%
Regression Modeling Error for OpAmp
31
Operational AmplifierOperational Amplifier
APEX achieve more than 100x speedup compared with MC 104 runs
PerformanceLinear Legendre APEX
1% 99% 1% 99% 1% 99%
Gain 22.7% 10.4% 22.0% 81.7% 1.45% 0.32%
Offset 11.5% 74.7% 222% 159% 0.58% 3.20%
UGF 3.78% 4.30% 0.39% 0.33% 0.03% 0.18%
GM 2.72% 2.46% 0.37% 0.20% 0.08% 0.04%
PM 4.41% 3.79% 0.40% 0.52% 0.13% 0.02%
SR (P) 0.81% 0.97% 0.35% 0.34% 0.11% 0.07%
SR (N) 3.83% 4.31% 0.24% 0.27% 0.13% 0.24%
SW (P) 0.13% 0.03% 0.37% 0.37% 0.16% 0.06%
SW (N) 0.06% 0.03% 0.34% 0.43% 0.09% 0.01%
Power 0.69% 0.65% 0.35% 0.41% 0.11% 0.00%
Comparison on Estimation Error
32
Application of APEXApplication of APEX
APEX can be incorporated into statistical analysis/synthesis tools E.g. robust analog design [Li 04]
OptimizationEngine
OptimizationEngine
UnsizedTopology
OptimizedCircuit Size
SimulationEngine
SimulationEngine
APEXAPEX
[Li 04]: Robust analog/RF circuit design with projection-based posynomial modeling, IEEE ICCAD, 2004
33
ConclusionConclusion
APEX applies moment matching for PDF/CDF extraction Propose a binomial moment evaluation for computing high order moments Moments are efficiently matched to a pole/residue formulation
Solve several implementation issues of APEX PDF/CDF shifting using generalized Chebyshev inequality Reverse PDF/CDF Evaluation
APEX can be incorporated into statistical analysis/synthesis tools Statistical timing analysis Yield optimization
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