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

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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. X. Li, P. Gopalakrishnan and L. Pileggi , CMU J. Le , Extreme DA. Overview. - PowerPoint PPT Presentation

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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

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OverviewOverview

Introduction Asymptotic Probability EXtraction (APEX) Implementation of APEX Numerical examples Conclusion

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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!

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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

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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

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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

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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.

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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

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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]

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pEpE

pEpEpE

MomentMatching

pdf(p)

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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

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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

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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 !

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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

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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

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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

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OverviewOverview

Introduction Asymptotic Probability EXtraction (APEX) Implementation of APEX Numerical examples Conclusion

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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