1. Non-Gaussian Statistical Timing Analysis Using Second Order Polynomial Fitting Lerong Cheng 1, Jinjun Xiong 2, and Lei He 1 1 EE Department, UCLA *2 IBM Research Center Address comments to lhe@ee.ucla.edulhe@ee.ucla.edu * Dr. Xiong's work was finished while he was with UCLA. This work was partially sponsored by NSF and Actel.

2. Outline Background and motivation Second order polynomial fitting for max operation Quadratic SSTA Experiment results Conclusions and future work

3. 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 Motivation

4. Outline Background and motivation Second order polynomial fitting for max operation Quadratic SSTA Experiment results Conclusions and future work

5. Linear 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

6. Non-linear fitting of Max: Second Order Polynomial Fitting Using second order polynomial instead of linear function to approximate the max operation whereand 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?

7. Mean 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] c 2, c 1, and c 0 can be computed by close form formulas  Use E[max(g(W), 0)] to approximate E[max(V,0)]

8. Compute 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

9. Comparison between Second Order Fitting and Linear Fitting Assume V~N(0.7, 1)

10. Outline Background and motivation Second order polynomial fitting for max operation Quadratic SSTA Experiment results Conclusions and future work

11. Quadratic Delay Model Delay is quadratic function of variation sources  X i ’s are independent random variables with arbitrary distribution  X i ’s are with zero mean and unit standard deviation  R is local random variation, which is modeled as a Gaussian random variable

12. Atomic Operations for SSTA Two atomic operations for block based SSTA, max and add  Given  Compute

13. Max Operation Flow Compute mean, variance, and skewness of D p =D 1 -D 2 Obtain the fitting coefficients Θ for max(D p,0) Reconstruct D m =max(D 1,D 2 ) to quadratic delay form

14. Moments of D p Quadratic form of Dp where First three moments of D P  Because D p is in quadratic form of variation sources X i ’s, the moments of D p can be computed by close form formulas With the first three moments, it is easy to obtain the fitting coefficients Θ for max(D p,0)=h(D p,Θ)

15. Reconstruct D m to Quadratic Form Fitting result of D m  Dm is a 4 th order polynomial of X i ’s Moment Matching [Zhan DAC2005]  Joint moments between D m and X i ’s can be computed by close form formulas

16. Add Operation Just add the correspondent parameters to get the parameters of D s

17. Computational Complexity The computational complexity of one step max operation is O(n 3 ), where n is the number of variation sources The computational complexity of one step add operation is O(n 2 ) The complexity measured as the total number of max and add operations of the SSTA is linear with respect to the circuit size

18. Semi-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)

19. Outline Background and motivation Second order polynomial fitting for max operation Quadratic SSTA Experiment results Conclusions

20. Experiment 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

21. PDF Comparison for s15850

22. Error 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

23. Outline Background and motivation Second order polynomial fitting for max operation Quadratic SSTA Experiment results Conclusions

24. Conclusion 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