# Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance

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Asymptotic Probability Extraction for Non-Normal Distributions of Circuit PerformanceX. Li, P. Gopalakrishnan and L. Pileggi, CMUJ. Le, Extreme DA

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OverviewIntroductionAsymptotic Probability EXtraction (APEX)Implementation of APEXNumerical examplesConclusion

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IC Technology ScalingFeature SizeScale DownProcess Variations (3 / Nominal) [Nassif 01]Process variation is becoming relatively larger!

YearLeff (nm)WLToxVthH199725025.0%32.0%8.0%10.0%25.0%22.2%199918026.2%33.3%8.0%10.0%30.0%24.0%200213028.0%34.6%9.8%10.0%30.0%27.3%200510030.0%40.0%12.0%11.4%33.8%31.7%20067033.3%47.1%16.0%13.3%35.7%33.3%

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Statistical Problems in ICStatistical methods have been proposed to address various statistical problemsWe focus on analysis problem in this work

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Modeling Process VariationsAssumptionProcess variations xi satisfy Normal distributions N(0,i)

Principle component analysis (PCA)xi can be decomposed into independent yi ~ N(0,1)

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Response Surface Model

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Response Surface ModelA low noise amplifier example designed in IBM 0.25 m process

Regression Modeling Error for LNANormalDistribution yiNonlinearTransformNon-Normal Distribution p

PerformanceLinearQuadraticF01.76%0.14%S116.40%1.32%S123.44%0.61%S212.94%0.34%S225.56%3.47%NF2.38%0.23%IIP34.49%0.91%Power3.79%0.70%

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Moment MatchingKey ideaConceptually consider PDF as the impulse response of an LTI systemMatchMomentsImpulseExcitationLTISystemUnknownPDFNonlinearTransformNormalDistribution

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Moment Matching

Impulse response

Moments

Match the first 2M momentsImpulseExcitationImpulseResponse

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Connection to Probability Theory

() is called characteristic function in probability theory

We actually match the first 2M terms of Taylor expansion at = 0System TheoryProbability Theory

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Connection to Probability TheoryProposition 1Proposition 2Typical characteristic functions are "low-pass filters"

A low-pass system is determined by its behavior in low-freq band ( = 0)Taylor expansion is accurate around expansion point ( = 0)

Moment matching is efficient in approximating low-pass systems [Celik 02]Characteristic Function for Typical Random Distributions[Celik 02]: IC Interconnect Analysis, Kluwer Academic Publishers, 2002

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The Classical Moment Problempdf(p)ProbabilityExtractionRSM[T. Stieltjes 1894]MomentMatchingpdf(p)

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APEX Asymptotic Probability ExtractionClassical moment problemExistence & uniqueness of the solutionFind complete bases to expand PDF function space

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Direct Moment Evaluation

If y1, y2,... are independent standard Normal distribution N(0,1)

Require computing symbolic expression for pk(Y)

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Binomial Moment EvaluationKey ideaRecursively compute high order momentsDerived from eigenvalue decomposition & statistical independence theory

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Step 1 Model Diagonalizationzi are independent N(0,1) since eigenvectors U are orthogonal !

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Step 2 Moment Evaluation

NOT compute symbolic expression for pk(Y)Achieve more than 106x speedup compared with direct evaluation

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OverviewIntroductionAsymptotic Probability EXtraction (APEX)Implementation of APEXPDF/CDF shiftingReverse PDF/CDF evaluationNumerical examples Conclusion

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PDF/CDF shifting is required in two casesOver-shifting results in large approximation errorThe challenging problem is to accurately determine

PDF/CDF Shiftingpdf(p)Mean p0pdf(p)Mean p0Case 1 Not CausalCase 2 Large Delay

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PDF/CDF ShiftingExact doesn't exist since pdf(p) is unbounded

Define a bound such that the probability P(p -) is sufficiently small

Propose a generalized Chebyshev inequality to estimate using central moments

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Reverse PDF/CDF EvaluationFinal value theorem of Laplace transform

Moment matching is accurate for estimating upper bound

Use flipped pdf(-p) for estimating lower bound

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OverviewIntroductionAsymptotic Probability EXtraction (APEX)Implementation of APEXNumerical examples Conclusion

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ISCAS'89 S27ST 0.13 m process6 principal random factorsMOSFET variationsNo intra-die variationNo interconnect variationLinear delay modeling error4.48%Quadratic delay modeling error1.10% (4x smaller)Longest Path in ISCAS'89 S27

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ISCAS'89 S27Binomial moment evaluation achieves more than 106x speedupComputation Time for Moment EvaluationMomentEvaluation

Moment OrderDirectBinomial# of TermsTime (Sec.)Time (Sec.)1281.00 10-20.0139243.02 1000.01580082.33 1020.016185641.57 1030.017387608.43 1030.028746133.73 1040.02150.04200.07

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ISCAS'89 S27Numerical oscillation for low order approximation

Increasing approx. order provides better accuracy

Typical approx. order is 7 ~ 10Cumulative Distribution Function for DelayDelay

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ISCAS'89 S27APEX is the most accurate approachAPEX achieves more than 200x speedup compared with MC 104 runsAPEX:0.18 secondsMC 104 runs:43.44 secondsComparison on Estimation Error

LinearLegendreAPEX1% Point1.43%0.87%0.04%10% Point4.63%0.02%0.01%25% Point5.76%0.12%0.03%50% Point6.24%0.05%0.02%75% Point5.77%0.03%0.02%90% Point4.53%0.16%0.03%99% Point0.18%0.78%0.09%

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Low Noise AmplifierIBM 0.25 m process8 principal random factorsMOSFET & RCL variationsNo mismatchesCircuit Schematic for LNARegression Modeling Error for LNA

PerformanceLinearQuadraticF01.76%0.14%S116.40%1.32%S123.44%0.61%S212.94%0.34%S225.56%3.47%NF2.38%0.23%IIP34.49%0.91%Power3.79%0.70%

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Low Noise AmplifierAPEX is the most accurate approachAPEX achieves more than 200x speedup compared with MC 104 runsAPEX:1.29 secondsMC 104 runs:334.37 secondsComparison on Estimation Error

PerformanceCornerLinearLegendreAPEX1%99%1%99%1%99%1%99%F015.8%20.1%1.11%1.10%0.20%0.55%0.06%0.05%S1145.4%51.5%5.78%1.40%2.94%3.28%0.09%0.08%S1238.9%44.6%3.88%1.16%0.39%0.27%0.14%0.28%S2160.3%51.6%2.91%4.69%0.37%0.01%0.17%0.19%S2223.1%36.0%1.01%5.61%1.11%0.84%0.07%0.19%NF51.9%72.8%3.70%3.52%0.34%0.37%0.06%0.12%IIP354.6%59.7%5.02%5.93%0.29%0.43%0.33%0.26%Power16.6%42.5%0.01%1.24%0.92%0.93%0.09%0.02%

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Operational AmplifierIBM 0.25 m process49 principal random factorsMOSFET variations from design kitInclude mismatchesCircuit Schematic for OpAmpRegression Modeling Error for OpAmp

PerformanceLinearQuadraticGain3.92%1.57%Offset21.80%7.49%UGF1.14%0.45%GM0.96%0.52%PM1.11%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%Power1.00%0.64%

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Operational AmplifierAPEX achieve more than 100x speedup compared with MC 104 runsComparison on Estimation Error

PerformanceLinearLegendreAPEX1%99%1%99%1%99%Gain22.7%10.4%22.0%81.7%1.45%0.32%Offset11.5%74.7%222%159%0.58%3.20%UGF3.78%4.30%0.39%0.33%0.03%0.18%GM2.72%2.46%0.37%0.20%0.08%0.04%PM4.41%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%Power0.69%0.65%0.35%0.41%0.11%0.00%

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Application of APEXAPEX can be incorporated into statistical analysis/synthesis toolsE.g. robust analog design [Li 04]OptimizationEngineUnsizedTopologyOptimizedCircuit SizeSimulationEngineAPEX[Li 04]: Robust analog/RF circuit design with projection-based posynomial modeling, IEEE ICCAD, 2004

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ConclusionAPEX applies moment matching for PDF/CDF extractionPropose a binomial moment evaluation for computing high order momentsMoments are efficiently matched to a pole/residue formulation

Solve several implementation issues of APEXPDF/CDF shifting using generalized Chebyshev inequalityReverse PDF/CDF Evaluation

APEX can be incorporated into statistical analysis/synthesis toolsStatistical timing analysisYield optimization