Non-Gaussian Parameter Modeling for SSTA with Confidential Interval - PowerPoint PPT Presentation

Non-Gaussian Parameter Modeling for SSTA with Confidential Interval Analysis Lizheng Zhang, Jun Shao Charlie Chungping Chen University of Wisconsin and National Taiwan University

General Variation Sources � Inter- and Intra-die Process Variation-W, L, … .. � PVT Variations: � Power Fluctuation: IR-drop, … . � Non-uniform Temperature Distribution � Vt Variation � Crosstalk: capacitive, inductive, and substrate coupling Courtesy from Intel Courtesy from Intel 2 2006-4-13

Technology Parameter Variations Parameter variation as a percentage of its nominal value becomes larger and larger � � Variations do not scale down or scale down slower than the nominal values Parameters may be correlated: there are three types of correlations � � (1) Physical dependency; (2) Spatial correlation; (3) topological correlation Circuit performance is a function of technology parameters � � Becomes a random variable and needs statistical method to compute “Models of process variations in device and interconnect” Duane Boning, MIT & Sani Nassif, IBM ARL. 3 2006-4-13

Correlated and Uncorrelated (independent) Variation Gate A Gate A Gate B Gate B Gate B delay Gate B delay Gate B delay Gate B delay Gate A delay Gate A delay Gate A delay Gate A delay Correlated Uncorrelated Correlated Uncorrelated 4 2006-4-13

Process Parameter Variations � Process parameter variations can be ( given by some poor guys ) represented by statistical distributions � Those distributions are characterized by several distribution parameters � Distribution parameters can be estimated from physical measurements on test chips � Limited budget makes it infeasible to have infinite number of test chips � The distribution may be non-Gaussian � The interested variable (say Elmore Delay, RC~W a /W b ) operation may be non-linear � Distribution parameters have errors � Confidence interval analysis measures the fidelity of the given 5 2006-4-13 distribution parameters

Confidence Interval Analysis (QC for Data) ˆ L M M 97% Confidence Interval � Suppose gate length L follows an arbitrary distribution with true mean value of M . We get estimation of M from the channel length = + + + ˆ measurement on N gates: L 1 ,L 2 ,…,L N as L M L L L N ( N / ) 1 2 ˆ � The true mean M is deterministic, but the estimation has limited M accuracy due to limited test chips � The gate length L may not be Gaussian, but the distribution of the estimation can be close to Gaussian ˆ M � The 97% confidence interval for the gate length mean estimation is: � chance to have the true mean value fall inside the interval is 97% 6 2006-4-13

Non-Gaussian Gate and Interconnect Delay and Quadratic Gaussian Approximation � Gate delay are functions of gate length, Vt, … � Interconnect delay are functions of width, thickness,… p.d.f. Comparison Inverter D elay Distribution 0.025 0.05 Monte Carlo Spice Monte Carlo Canonical Canonical Mode 0.045 Quadratic Quadratic Mode 0.02 0.04 0.035 Probability Density Probability Density 0.015 0.03 0.025 0.01 0.02 0.015 0.005 0.01 0.005 0 0 100 150 200 250 300 350 -20 0 20 40 60 80 100 Delay Delay [ps] 7 2006-4-13 Interconnect Gate

Points to make � Gate and Interconnect delay may be non-gaussian and also depend on width, length, vt, vdd, temperatures,….etc � Delay calculation operation may be non-linear � Due to finite test resources, confidence interval need to be carefully considered � We propose to use quadratic gaussian model to match first three moments (mean, variation, and skewness) and correlations � We provide systematic way to fit the measurement data � We provide confidence interval analysis at the same time 8 2006-4-13

Literature Review of SSTA � In literature, there are two categories of SSTA approaches � Path-based SSTA: can consider false paths but with exponential complexity in the worst case � Michael Orshansky (DAC’02): statistical bounding � Kwang-Ting Cheng (DAC’02): delay testing oriented � Block-base SSTA: has linear complexity but is difficult to consider false paths � Chandu Visweswariah (DAC’04): parametric delay and direct linear MAX operation � Sachin S. Sapatnekar (ICCAD’03): parametric delay and projection- based linear MAX operation � Larry Pileggi (DAC’04): parametric delay and table-lookup based MAX operation � David Blauuw (TCAD’03): PDF-based approach � C. Chen (DAC’ 05): Quadratic SSTA 9 2006-4-13

Non-Gaussian Parameter � Blindly using Gaussian variables to fit the process parameters may induce huge errors � Correlation between non-Gaussian variables may be hard to manipulated directly, so direct non-Gaussian modeling may not be efficient � We propose to use quadratic form of a standard Gaussian variable X to model the non-Gaussian parameter Y with distribution parameters of a b c = + + 2 Y aX bX c � First three statistical moments (mean, variation, skewness) and correlations are exactly matched � Fully compatible with the our proposed quadratic SSTA 10 2006-4-13

SSTA Delay Model ∑ = µ + α + β D R i Y � Canonical Gaussian delay model i � Quadratic Gaussian delay model i ∑ ∑ ∑ = µ + α + β + Γ D R Y Y Y i i ij i j i i j � Problems � Process parameters Y 1 , Y 2 , … are assumed to be Gaussian random variables � Process parameter distributions are known without any uncertainties “ Correlation-Preserved Non-Gaussian Statistical Timing Analysis with Quadratic Timing Model ” Lizheng Zhang, Weijen Chen, Yuhen Hu, John A. Gubner, Charlie Chung-Ping Chen , DAC’05 11 2006-4-13

Moment Properties � The mean, variation, and skewness of the measured data set ( Y 1 ,Y 2 , L ,Y N ) and modeling function ( Y = aX 2 +bX+c ) satisfy: 12 2006-4-13

Moment Matching Process � a will be one and the only one of the following three values whichever is real and within the range of |a| < σ Y /1.414 : � Solutions for b and c can be found as: Y = aX 2 +bX+c 13 2006-4-13

Covariance Matched As Well � Given data sets ( Y α ,1 ,Y α ,2 , L ,Y α ,N ) and ( Y β ,1 ,Y β ,2 , L ,Y β ,N ) covariance between parameters Y α and Y β , cov ( Y α ,Y β ) can bet estimated as 2 +b α X α +c α ) and � Given quadratic models ( Y α ¼ a α X α 2 +b β X β +c β ) , correlation between X α and X β , ρ X = ( Y β ¼ a β X β cov ( X α , X β ) can be solved analytically as follows : 14 2006-4-13

Conditions for Real a, b, c � Exact moment matching -3 2.5 x 10 is possible only when 2 Probability Density Y 's skewness 1.5 1 � Most real parameter 0.5 variations is within this constraint 0 -500 0 500 1000 1500 2000 Delay[ps] 15 2006-4-13

Confidence Interval p Confidence Level : = df N- Degree of freedom : 1 t p,df Student' s t Distributi on : ( ) S S − ≤ µ ≤ + - p X t p df X t p df ( 1 2 ) Confidence Interval : ( , ) ( , ) N N p 0.05(90%) 0.01(98%) 0.025 (95%) df 1 6.314 12.71 31.82 2 2.920 4.303 6.965 3 2.353 3.182 4.541 4 2.132 2.776 3.747 5 2.015 2.571 3.365 16 2006-4-13

Confidence Intervals Analysis � Jackknife Method are used to estimate the confidence interval of coefficients a, b, c θ = a, b , or c (One digit accuracy require 100 data points!) 17 2006-4-13

Data Analysis Process for Non-Gaussian Mean Estimate ========== ========== ========== ========== ====== ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ K X X X X X X X Sectioning : + + 1 4 2 4 3 1 42 43 1 42 4 43 4 1 2 3 m m m m m N 1 1 2 2 1 3 X X X X G 1 G 2 G 3 Gk − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − = − − − − − − − − N mk k m and sections with size − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − L X X X X Normally D istributed : (Large Number Law ) G 1 G 2 G 3 Gk − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − X S Average and Stabdard Deviation : and XG XG − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − S S − ≤ µ ≤ + XG XG X t p df X t p df Confidence Interval : ( , ) ( , ) XG XG k k ========== ========== ========== ========== ======= 18 2006-4-13