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

2 2006-4-13

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

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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.

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Correlated and Uncorrelated (independent) Variation

Gate A delay Gate A delay Gate B delay Gate B delay Gate A delay Gate A delay Gate B delay Gate B delay

Correlated Correlated Uncorrelated Uncorrelated

Gate A Gate A Gate B Gate B

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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~Wa/Wb) operation may be non-linear

Distribution parameters have errors

Confidence interval analysis

measures the fidelity of the given distribution parameters

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Confidence Interval Analysis (QC for Data)

Suppose gate length L follows an arbitrary distribution with true mean

value of M. We get estimation of M from the channel length measurement on N gates: L1,L2,…,LN as

The true mean M is deterministic, but the estimation has limited

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

The 97% confidence interval for the gate length mean estimation is:

chance to have the true mean value fall inside the interval is 97%

N L L L M

N /

) ( ˆ

2 1

+ + + = L

M ˆ

97% Confidence Interval

M ˆ

L

M

M ˆ

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Non-Gaussian Gate and Interconnect Delay and Quadratic Gaussian Approximation

• 20

Inverter D elay Distribution Delay [ps] Probability Density Spice Monte Carlo Canonical Mode Quadratic Mode

p.d.f. Comparison Delay Probability Density Monte Carlo Canonical Quadratic

Interconnect Gate

Gate delay are functions of gate length, Vt, … Interconnect delay are functions of width, thickness,…

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Points to make

Gate and Interconnect delay may be non-gaussian and also depend

• n 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

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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

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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

First three statistical moments (mean, variation, skewness)

and correlations are exactly matched

Fully compatible with the our proposed quadratic SSTA

c bX aX Y + + =

2

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SSTA Delay Model

Canonical Gaussian delay model Quadratic Gaussian delay model Problems

Process parameters Y1, Y2, … are assumed to be Gaussian

random variables

Process parameter distributions are known without any

uncertainties

∑

+ + =

i i iY

R D β α µ

∑ ∑ ∑

Γ + + + =

i j j i ij i i i

Y Y Y R D β α µ

“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

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Moment Properties

The mean, variation, and skewness of the

measured data set (Y1,Y2,L,YN) and modeling function (Y=aX2+bX+c) satisfy:

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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|< σY/1.414:

Solutions for b and c can be found as:

Y=aX2+bX+c

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Covariance Matched As Well

Given data sets (Yα,1,Yα,2,L,Yα,N) and (Yβ,1,Yβ,2,L,Yβ,N)

covariance between parameters Yα and Yβ , cov(Yα,Yβ) can bet estimated as

Given quadratic models (Yα¼aα Xα

2+bα Xα+cα) and

(Yβ¼aβ Xβ

2+bβ Xβ+cβ), correlation between Xα and Xβ, ρX=

cov(Xα, Xβ) can be solved analytically as follows:

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Conditions for Real a, b, c

Exact moment matching

is possible only when Y's skewness

Most real parameter

variations is within this constraint

• 500

500 1000 1500 2000 0.5 1 1.5 2 2.5 x 10

• 3

Delay[ps] Probability Density

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Confidence Interval

3.365 2.571 2.015 5 3.747 2.776 2.132 4 4.541 3.182 2.353 3 6.965 4.303 2.920 2 31.82 12.71 6.314 1 0.01(98%)

0.025 (95%)

0.05(90%)

df p

N S df p t X N S df p t X p

• p,df

t N- df p ) , ( ) , ( : Interval Confidence ) 2 1 ( ) ( :

• n

Distributi t s Student' 1 : freedom

• f

Degree : Level Confidence + ≤ ≤ − = µ

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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!)

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Data Analysis Process for Non-Gaussian Mean Estimate

======= ========== ========== ========== ========== + ≤ ≤ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ====== ========== ========== ========== ==========

+ +

k S df p t X k S df p t X S X X X X X m k mk N X X X X X X X

XG XG XG XG XG XG Gk G G G X N X m m X m m X m

Gk G G G

) , ( ) , ( : Interval Confidence and : Deviation Stabdard and Average ) Number Law (Large : istributed Normally D size with sections and : Sectioning

3 2 1 3 1 2 2 1 1

3 2 1

µ L 3 2 1 K 4 43 4 42 1 43 42 1 4 3 4 2 1

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== ========== ========== ========== ========== ========== + ≤ ≤ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ == ========== ========== ========== ========== ==========

+ +

k S df p t S k S df p t S S S S S S S X X X X X X X

SG G SG G SG G Gk G G G S N S m m S m m S m

Gk G G G

) , ( ) , ( : Interval Confidence and : Deviation Stabdard and Average : istributed Normally D : Sectioning

3 2 1 3 1 2 2 1 1

3 2 1

σ L 3 2 1 K 4 43 4 42 1 43 42 1 4 3 4 2 1

Data Analysis Process for Non-Gaussian Variance Estimate

20 2006-4-13

Experimental Result: Moment Matching for Quadratic Parameter Modeling

Skewness = 0 Skewness = 0.75

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Experimental Result: Estimation Error v.s. Data Points

Error is reduced in the rate of 1/sqrt(N)

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Conclusions

We propose to model Gaussian and non-

Gaussian process parameters using quadratic format of Gaussian random variables

Three moments and covariance are matched Fully compatible with the existing SSTA method We develop Jackknife-method-based confidence

interval analysis to estimate the modeling error inherited from the limited resources

Generally speaking, to improve one digit of

accuracy, 100X experiments will be required

Error reduces as 1/sqrt(N)….