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Non-Linear Statistical Static Timing Analysis for Non-Gaussian Variation
Sources
Non-Linear Statistical Static Timing Analysis for Non-Gaussian Variation
Sources
Lerong Cheng1, Jinjun Xiong2,
and Prof. Lei He1
1EE Department, UCLA*2IBM Research Center
Address comments to [email protected]
*Dr. Xiong's work was finished while he was with UCLA
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA
Experimental results
Conclusions and future work
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MotivationMotivation
Gaussian variation sources
Linear delay model, tightness probability [C.V DAC’04] Quadratic delay model, tightness probability [L.Z DAC’05] Quadratic delay model, moment matching [Y.Z DAC’05]
Non-Gaussian variation sources Non-linear delay model, tightness probability [C.V DAC’05] Linear delay model, ICA and moment matching [J.S DAC’06]
Need fast and accurate SSTA for Non-linear Delay model with Non-Gaussian variation sources
not all variation is Gaussian in reality
computationally inefficient
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA
Experimental results
Conclusions and future work
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Delay ModelingDelay Modeling
Delay with variation
Linear delay model
Quadratic delay model
Xis are independent random variables with arbitrary distribution Gaussian or non-Gaussian
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA Max operation Add operation Complexity analysis
Experimental results
Conclusions and future work
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Max OperationMax Operation
Problem formulation: Given
Compute:
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To represent D=max(D1,D2) back to the quadratic form
We can show the following equations hold
mi,k is the kth moment of Xi, which is known from the process characterization
From the joint moments between D and Xis the coefficients ais and bis can be computed by solving the above linear equations
Use random term and constant term to match the first three moments of max(D1, D2)
Reconstruct Using Moment MatchingReconstruct Using Moment Matching
22021 )(),max( rrrr
iiiii XbXaXbXadDDD
3,2,21 )],max([ iiiii mbmaDDXE
)()],[max()],max([ 22,4,3,2,2121
2iiiiiii mmbmamDDEDDXE
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Basic IdeaBasic Idea
Compute the joint PDF of D1 and D2
Compute the moments of max(D1,D2)
Compute the Joint moments of Xi and max(D1,D2)
Reconstruct the quadratic form of max(D1,D2) Keep the exact correlation between max(D1,D2) and Xi Keep the exact first-three moments of max(D1,D2)
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Assume that D1 and D2 are within the ±3σ range The joint PDF of D1 and D2, f(v1, v2)≈0, when v1 and v2 is not in
the ±3σ range
Approximate the Joint PDF of D1 and D2 by the first Kth order Fourier Series within the ±3σ range:
where
,
αij are Fourier coefficients
JPDF by Fourier SeriesJPDF by Fourier Series
13 Dl 23 Dh
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Fourier CoefficientsFourier Coefficients
The Fourier coefficients can be computed as:
Considering outside the range of
where
can be written in the form of . can be pre-computed and store in a 2-dimensional look
up table indexed by c1 and c2
0),( 21 vvf
pqiY ,][ ,pqiYeE
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Assume that all the variation sources have uniform distributions within [-0.5, 0.5] Our method can be applies to arbitrary variation distributions
Maximum order of Fourier Series K=4
JPDF ComparisonJPDF Comparison
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Moments of D=max(D1, D2)Moments of D=max(D1, D2)
The tth order raw moment of D=max(D1,D2) is
Replacing the joint PDF with its Fourier Series:
where
L can be computed using close form formulas
The central moments of D can be computed from the raw moments
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Joint MomentsJoint Moments
Approximate the Joint PDF of Xi, D1, and D2 with Fourier Series:
The Fourier coefficients can be computed in the similar way as
The joint moments between D and Xis are computed as:
Replacing the f with the Fourier Series
where
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PDF Comparison for One Step Max PDF Comparison for One Step Max
Assume that all the variation sources have uniform distributions within [-0.5, 0.5]
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA Max operation Add operation Complexity analysis
Experimental results
Conclusions and future work
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Add OperationAdd Operation
Problem formulation Given D1 and D2, compute D=D1+D2
Just add the correspondent parameters to get the parameters of D
The random terms are computed to match the second and third order moments of D
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Complexity AnalysisComplexity Analysis
Max operation O(nK3)Where n is the number of variation sources and K is the max order
of Fourier Series
Add operation O(n)
Whole SSTA process The number of max and add operations are linear related to the
circuit size
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA
Experimental results
Conclusions and future work
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Experimental SettingExperimental Setting
Variation sources: Gaussian only Non-Gaussian
Uniform Triangle
Comparison cases Linear SSTA with Gaussian variation sources only
Our implementation of [C.V DAC04] Monte Carlo with 100000 samples
Benchmark ISCAS89 with randomly generated variation sensitivity
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PDF ComparisonPDF Comparison
PDF comparison for s5738
Assume all variation sources are Gaussian
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Mean and Variance Comparison for Gaussian Variation SourcesMean and Variance Comparison for Gaussian Variation Sources
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Mean and Variance Comparison for non-Gaussian Variation SourcesMean and Variance Comparison for non-Gaussian Variation Sources
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OutlineOutline
Background and motivation
Delay modeling
Atomic operations for SSTA
Experimental results
Conclusions and future work
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Conclusion and Future WorkConclusion and Future Work
We propose a novel SSTA technique is presented to handle both non-linear delay dependency and non-Gaussian variation sources
The SSTA process are based on look up tables and close form formulas
Our approach predict all timing characteristics of circuit delay with less than 2% error
In the future, we will move on to consider the cross terms of the quadratic delay model
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