Non-Gaussian Statistical Timing Analysis Using Second Order
Polynomial Fitting
Non-Gaussian Statistical Timing Analysis Using Second Order
Polynomial Fitting
Lerong Cheng1, Jinjun Xiong2,
and Lei He1
1EE Department, UCLA*2IBM Research Center
Address comments to [email protected]
*Dr. Xiong's work was finished while he was with UCLA. This work was partially sponsored by NSF and Actel.
OutlineOutline
Background and motivation
Second order polynomial fitting for max operation
Quadratic SSTA
Experiment results
Conclusions and future work
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] Non-linear delay model, Fourier Series [Cheng DAC’06]
Need fast and accurate SSTA for Non-linear Delay model with Non-Gaussian variation sources
Background and MotivationBackground and Motivation
OutlineOutline
Background and motivation
Second order polynomial fitting for max operation
Quadratic SSTA
Experiment results
Conclusions and future work
Linear Fitting: Tightness ProbabilityLinear Fitting: Tightness Probability
Previously, max operation is approximated by tightness probability [Chandu DAC04]
where
Tightness probability approximation is a linear fitting Tightness probability is hard to obtain when A and B are non-
Gaussian random variables
Max operation is a non-linear operation Linear fitting is not accurate enough
Need a more accurate and efficient non-linear approximation
Non-linear fitting of Max: Second Order Polynomial FittingNon-linear fitting of Max: Second Order Polynomial Fitting
Using second order polynomial instead of linear function to approximate the max operation
where and V is a random variable with any arbitrary distribution
Fitting coefficients are computed by matching the mean of the max operation while minimizing the square error between h and max(V,0) within the 3σrange
where
How to obtain these Fitting Coefficients?
Mean of the Max OperationMean of the Max Operation
When V is a non-Gaussian random variable, it is hard to compute E[max(V,0)]
Two step solution We approximate the non-Gaussian random variable V as a
quadratic function of a standard Gaussian random variable W by matching the first 3 moments [Zhang’ISPD05]
c2, c1, and c0 can be computed by close form formulas Use E[max(g(W), 0)] to approximate E[max(V,0)]
Compute Fitting CoefficientsCompute Fitting Coefficients
Recall the constraint that matching the mean of the max operation we have
The constrained optimization equation can be written as the unconstrained optimization equation:
where
Expanding the integral, the square error can be represented as quadratic form of
can be computed easily by letting partial derivative of SE to be 0
Comparison between Second Order Fitting and Linear FittingComparison between Second Order Fitting and Linear Fitting
Assume V~N(0.7, 1)
-2
-1
0
1
2
3
4
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
max(V,0)Linear FitSecond-Order Fit
0
0.2
0.4
0.6
0.8
1
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
OutlineOutline
Background and motivation
Second order polynomial fitting for max operation
Quadratic SSTA
Experiment results
Conclusions and future work
Quadratic Delay ModelQuadratic Delay Model
Delay is quadratic function of variation sources
Xi’s are independent random variables with arbitrary distribution Xi’s are with zero mean and unit standard deviation R is local random variation, which is modeled as a Gaussian
random variable
Atomic Operations for SSTAAtomic Operations for SSTA
Two atomic operations for block based SSTA, max and add Given
Compute
Max Operation FlowMax Operation Flow
Compute mean, variance, and
skewness of Dp=D1-D2
Obtain the fitting coefficients Θ for
max(Dp,0)
Reconstruct Dm=max(D1,D2) to
quadratic delay form
Moments of DpMoments of Dp
Quadratic form of Dp
where
First three moments of DP
Because Dp is in quadratic form of variation sources Xi’s, the moments of Dp can be computed by close form formulas
With the first three moments, it is easy to obtain the fitting coefficients Θ for max(Dp,0)=h(Dp,Θ)
Reconstruct Dm to Quadratic FormReconstruct Dm to Quadratic Form
Fitting result of Dm
Dm is a 4th order polynomial of Xi’s
Moment Matching [Zhan DAC2005]
Joint moments between Dm and Xi’s can be computed by close form formulas
Add OperationAdd Operation
Just add the correspondent parameters to get the parameters of Ds
Computational ComplexityComputational Complexity
The computational complexity of one step max operation is O(n3), where n is the number of variation sources
The computational complexity of one step add operation is O(n2)
The complexity measured as the total number of max and add operations of the SSTA is linear with respect to the circuit size
Semi-Quadratic SSTASemi-Quadratic SSTA
Effect of the crossing terms in the quadratic model is weak [Zhan DAC2005], ignoring crossing terms will not affects the accuracy too much
Simplified delay model without crossing terms (semi-quadratic delay model)
Where
The SSTA flow for the semi-quadratric delay model is similar to that of the quadratic delay model, but much simpler The computational complexity of both max and add operation for
semi-quadratic SSTA is O(n)
OutlineOutline
Background and motivation
Second order polynomial fitting for max operation
Quadratic SSTA
Experiment results
Conclusions
Experiment SettingExperiment Setting
ISCAS89 benchmark set
65nm technology
Two types of variation sources, both with skew-normal distribution Leff Vth
Three types of variation Inter-die variation Intra-die spatial variation (grid based model) Intra-die random variation
Three comparison cases Linear SSTA [Chandu DAC2004] Nonlinear SSTA using Fourier Series [Cheng DAC2007] 100,000 sample Monte-Carlo simulation
PDF Comparison for s15850PDF Comparison for s15850
Error and Run Time ComparisonError and Run Time Comparison
For quadratic SSTA, the error of mean, standard deviation, and skewness is within 1%, 1%, and 5%, respectively.
For Semi-Quadratic SSTA, the error of mean, standard deviation, and skewness is within 1%, 2%, and 25% error. Semi-quadratic SSTA ignores crossing terms which affects skewness
Semi-quadratic SSTA results similar error as Fourier SSTA, but 20X faster
Semi-quadratic SSTA is more accurate than linear SSTA with similar run time
OutlineOutline
Background and motivation
Second order polynomial fitting for max operation
Quadratic SSTA
Experiment results
Conclusions
ConclusionConclusion
A new second order polynomial fitting of the max operation is proposed
All the SSTA operations are based on close form formulas
The quadratic SSTA predicts the error of mean, standard deviation, and skewnss withing 1%, 1%, and 5% error, respectively
The semi-quadric SSTA has similar accuracy as the SSTA with Fourier Series, but 20X faster