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 lhe@ee.ucla.edu

*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