Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance

1. Asymptotic Probability Extraction for Non-Normal Distributions of Circuit PerformanceBy: Sedigheh Hashemi201C-Spring2009

2. Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance X. Li, P. Gopalakrishnan and L. Pileggi, CMU J. Le, Extreme DA

3. Overview • Introduction • Asymptotic Probability EXtraction (APEX) • Implementation of APEX • Numerical examples • Conclusion

4. 0.35 μm 0.18 μm 90nm IC Technology Scaling Feature Size Scale Down Process Variations (3σ / Nominal) [Nassif 01] Process variation is becoming relatively larger!

5. Statistical methods have been proposed to address various statistical problems We focus on analysis problem in this work RSM Modeling MOR etc. Fixed Value Random Distribution Timing Process Parameters Design Parameters Analysis Yield etc. Unknown Distribution Gate Sizing Synthesis Design Centering Circuit Performance etc. Statistical Problems in IC

6. 0 6σi Δy3 Δx3 Δx1 Δy1 Δx2 Δy2 Modeling Process Variations • Assumption • Process variations Δxi satisfy Normal distributions N(0,σi) • Principle component analysis (PCA) • Δxi can be decomposed into independent Δyi ~ N(0,1)

7. p p Δy1 Δy1 Δy2 Δy2 Linear RSM is Not Sufficiently Accurate p p Δy1 Δy1 Δy2 Δy2 Response Surface Model

8. Response Surface Model • A low noise amplifier example designed in IBM 0.25 μm process Normal Distribution Δyi Nonlinear Transform Non-Normal Distribution p Regression Modeling Error for LNA

9. Impulse Response Moment Matching • Key idea • Conceptually consider PDF as the impulse response of an LTI system Nonlinear Transform Normal Distribution Unknown PDF Match Moments LTI System Impulse Excitation

10. Impulse response Moments Match the first 2M moments ai & bi can be solved by using the algorithm in [Pillage 90] [Pillage 90]: Asymptotic waveform evaluation for timing analysis, IEEE TCAD, 1990. Moment Matching Impulse Excitation Impulse Response

11. Connection to Probability Theory • Φ(ω) is called characteristic function in probability theory • We actually match the first 2M terms of Taylor expansion at ω = 0 System Theory Probability Theory

12. Proposition 1 Proposition 2 Typical characteristic functions are "low-pass filters" A low-pass system is determined by its behavior in low-freq band (ω = 0) Taylor expansion is accurate around expansion point (ω = 0) Moment matching is efficient in approximating low-pass systems [Celik 02] Connection to Probability Theory Characteristic Function for Typical Random Distributions [Celik 02]: IC Interconnect Analysis, Kluwer Academic Publishers, 2002

13. Δyi The Classical Moment Problem [T. Stieltjes 1894] RSM Moment Matching Probability Extraction pdf(p) pdf(p)

14. Different • APEX • Efficiently compute high order moments • Efficiently approximate the unknown PDF/CDF APEX Asymptotic Probability Extraction • Classical moment problem • Existence & uniqueness of the solution • Find complete bases to expand PDF function space

15. # of Terms Exponentially Increase!!! k Direct Moment Evaluation • If Δy1, Δy2,... are independent standard Normal distribution N(0,1) • Require computing symbolic expression for pk(Y)

16. Quad Model Diagonalization Binomial Moment Evaluation Recursive Moment Evaluation Binomial Moment Evaluation • Key idea • Recursively compute high order moments • Derived from eigenvalue decomposition & statistical independence theory

17. Δy3 u3 u1 Δy1 Δy2 u2 Δz3 Δz1 Δz2 Step 1 – Model Diagonalization Δzi are independent N(0,1) since eigenvectors U are orthogonal !

18. Binomial Series (k+1) Terms Binomial Series (k+1) Terms Binomial Series (k+1) Terms Binomial Series (k+1) Terms Step 2 – Moment Evaluation • NOT compute symbolic expression for pk(Y) • Achieve more than106x speedup compared with direct evaluation

19. Overview • Introduction • Asymptotic Probability EXtraction (APEX) • Implementation of APEX • PDF/CDF shifting • Reverse PDF/CDF evaluation • Numerical examples • Conclusion

20. PDF/CDF Shifting • PDF/CDF shifting is required in two cases • Over-shifting results in large approximation error • The challenging problem is to accurately determine ξ ξ ξ pdf(p) pdf(p) 0 p 0 p Mean μ Mean μ Case 1 – Not Causal Case 2 – Large Delay

21. Exact ξ doesn't exist since pdf(p) is unbounded Define a bound ξ such that the probability P(p ≤ μ-ξ) is sufficiently small Propose a generalized Chebyshev inequality to estimate ξ using central moments ξ p Mean μ ξ p Mean μ PDF/CDF Shifting

22. Final value theorem of Laplace transform Moment matching is accurate for estimating upper bound Use flipped pdf(-p) for estimating lower bound Accurate for Estimating Upper Bound pdf(p) 0 p Flipped pdf(-p) p 0 Accurate for Estimating Lower Bound Reverse PDF/CDF Evaluation

23. Overview • Introduction • Asymptotic Probability EXtraction (APEX) • Implementation of APEX • Numerical examples • Conclusion

24. ST 0.13 μm process 6 principal random factors MOSFET variations No intra-die variation No interconnect variation Linear delay modeling error 4.48% Quadratic delay modeling error 1.10% (4x smaller) ISCAS'89 S27 Longest Path in ISCAS'89 S27

25. Δyi ISCAS'89 S27 • Binomial moment evaluation achieves more than 106x speedup Moment Evaluation Computation Time for Moment Evaluation

26. Numerical oscillation for low order approximation Increasing approx. order provides better accuracy Typical approx. order is 7 ~ 10 ISCAS'89 S27 Delay Cumulative Distribution Function for Delay

27. ISCAS'89 S27 • APEX is the most accurate approach • APEX achieves more than 200x speedup compared with MC 104 runs • APEX: 0.18 seconds • MC 104 runs: 43.44 seconds Comparison on Estimation Error

28. IBM 0.25 μm process 8 principal random factors MOSFET & RCL variations No mismatches Low Noise Amplifier Regression Modeling Error for LNA Circuit Schematic for LNA

29. Low Noise Amplifier • APEX is the most accurate approach • APEX achieves more than 200x speedup compared with MC 104 runs • APEX: 1.29 seconds • MC 104 runs: 334.37 seconds Comparison on Estimation Error

30. IBM 0.25 μm process 49 principal random factors MOSFET variations from design kit Include mismatches Operational Amplifier Circuit Schematic for OpAmp Regression Modeling Error for OpAmp

31. Operational Amplifier • APEX achieve more than 100x speedup compared with MC 104 runs Comparison on Estimation Error

32. Application of APEX • APEX can be incorporated into statistical analysis/synthesis tools • E.g. robust analog design [Li 04] Optimization Engine Unsized Topology Optimized Circuit Size Simulation Engine APEX [Li 04]: Robust analog/RF circuit design with projection-based posynomial modeling, IEEE ICCAD, 2004

33. Conclusion • APEX applies moment matching for PDF/CDF extraction • Propose a binomial moment evaluation for computing high order moments • Moments are efficiently matched to a pole/residue formulation • Solve several implementation issues of APEX • PDF/CDF shifting using generalized Chebyshev inequality • Reverse PDF/CDF Evaluation • APEX can be incorporated into statistical analysis/synthesis tools • Statistical timing analysis • Yield optimization