NonLinear Statistical Static Timing Analysis for NonGaussian Variation Sources

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Non-Linear Statistical Static Timing Analysis for
Non-Gaussian Variation Sources

• Lerong Cheng1, Jinjun Xiong2,
• and Lei He1
• 1EE Department, UCLA
• 2IBM Research Center
• Address comments to lhe_at_ee.ucla.edu
• Dr. Xiong's work was finished while he was with
  UCLA. This work was partially sponsored by NSF
  and Actel.

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Outline

• Background and motivation
• Atomic operations for SSTA
• Experimental results
• Conclusions and future work

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Motivation

• Gaussian variation sources
• Linear delay model, tightness probability C.V
  DAC04
• Quadratic delay model, tightness probability L.Z
  DAC05
• Quadratic delay model, moment matching Y.Z
  DAC05
• Non-Gaussian variation sources
• Non-linear delay model, tightness probability
  C.V DAC05
• Linear delay model, ICA and moment matching J.S
  DAC06
• Need fast and accurate SSTA for Non-linear Delay
  model with Non-Gaussian variation sources

? not all variation (e.g., via resistance) is
Gaussian
? computationally inefficient
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Outline

• Background and motivation
• Delay modeling
• Atomic operations for SSTA
• Delay Modeling
• Max operation
• Add operation
• Experimental results
• Conclusions and future work

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Delay Modeling

• Delay with variation
• Linear delay model
• Quadratic delay model
• Effect of the crossing terms is weak Zhan
  DAc05, so we assume no crossing terms in this
  work
• Xis are independent random variables with
  arbitrary distribution
• Gaussian or non-Gaussian
• Xr is the local random variation

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Outline

• Background and motivation
• Delay modeling
• Atomic operations for SSTA
• Delay Modeling
• Max operation
• Add operation
• Experimental results
• Conclusions and future work

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Max Operation

• Problem formulation
• Given
• Compute

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Reconstruct Using Moment Matching

• ai and bi can be computed by applying the moment
  matching technique
• mi,k is the kth moment of Xi, which is known from
  the process characterization
• If we know joint moments between D and Xi, we can
  obtain coefficients ai and bi by solving the
  above linear equations

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Matching the First Three Moments of Max(D1, D2)

• d0, ar, and br are computed to match the first
  three moment of Dmax(D1,D2)
• Where U1, U2, and U3 are first three central
  moments of max(D1,D2), and
• When U1, U2, and U3 are known, d0, ar, and br can
  be computed by solving the above equations

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How to Compute the Joint Moments and Central
Moments

• In order to compute Dmax(D1,D2), we need to
  know
• First three central moments of Dmax(D1,D2)
• U1, U2, and U3
• Joint moments between Xi and Dmax(D1,D2)

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JPDF by Fourier Series

• Assume that D1 and D2 are within the 3s range
• When v1 and v2 is not in the 3s range, the joint
  PDF of D1 and D2, f(v1, v2)0
• Approximate the Joint PDF of D1 and D2 by the
  first Kth order Fourier Series within the 3s
  range
• where
• ,
• apq are Fourier coefficients

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Fourier 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

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Validation of JPDF Approximation

• 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 K4

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Moments of Dmax(D1, D2)

• The tth order moment of Dmax(D1,D2) is
• where
• L can be computed using close form formulas
• The central moments of D can be computed from the
  moments

Replacing the joint PDF with its Fourier Series
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How to Compute the Joint Moments and Central
Moments

• In order to compute Dmax(D1,D2), we need to
  know
• First three central moments of Dmax(D1,D2)
• U1, U2, and U3
• Joint moments between Xi and Dmax(D1,D2)

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Joint Moments

• Approximate the Joint PDF of Xi, D1, and D2 with
  Fourier Series
• The Fourier coefficients can be
  computed in the similar way as

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Joint Moments Contd

• The joint moments between D and Xis are computed
  as
• where
• J can be computed by close form formula

Replacing f with the Fourier Series
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Complexity Analysis

• The computational complexity of one step max
  operation is O(nK3),
• where n is the number of variation sources and K
  is the max order of Fourier Series
• K4 can give very accurate estimation

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Validation of One Step Max Operation

• Assume that all the variation sources have
  uniform distributions within -0.5, 0.5, K4

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Outline

• Background and motivation
• Delay modeling
• Atomic operations for SSTA
• Delay Model
• Max operation
• Add operation
• Experimental results
• Conclusions and future work

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Add Operation

• Problem formulation
• Given D1 and D2, compute DD1D2
• Just add the correspondent parameters to get the
  parameters of D
• The computational complexity of one step add
  operation is O(n),
• where n is the number of variation sources
• The complexity measured as the total number of
  max and add operations of the SSTA is linear
  w.r.t the circuit size

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Outline

• Background and motivation
• Delay modeling
• Atomic operations for SSTA
• Experimental results
• Conclusions and future work

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Experimental Setting

• For Gaussian variation sources, we compare with
• Linear SSTA (our implementation of C.V DAC04)
• Monte Carlo with 100,000 samples
• For non-Gaussian
• We consider both uniform and triangle
  distributions
• We compare with Monte Carlo with 100,000 samples
• Benchmark
• ISCAS89 with randomly generated variation
  sensitivity

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Validation of SSTA with GaussianVariation Sources

• PDF comparison for s5738
• Assume all variation sources are Gaussian

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Mean and Variance Comparison for Gaussian
Variation Sources
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Validation of SSTA with non-Gaussian Variation
Sources
(a)
(b)

• PDF comparison for s5738
• (a) Uniform distribution
• (b) Triangle distribution

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Mean and Variance Comparison for non-Gaussian
Variation Sources
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Outline

• Background and motivation
• Delay modeling
• Atomic operations for SSTA
• Experimental results
• Conclusions and future work

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Conclusion and Future Work

• A novel SSTA technique is presented to handle
  both non-linear delay dependency and non-Gaussian
  variation sources
• All SSTA operations are based on look up tables
  and close form formulas
• Our approach predicts all timing characteristics
  of circuit delay with less than 2 error
• In the future, we will consider the cross terms
  of the quadratic delay model

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Thank you