1. Slide 1 Bayesian Model Fusion: Large-Scale Performance Modeling of Analog and Mixed- Signal Circuits by Reusing Early-Stage Data Fa Wang*, Wangyang Zhang*, Shupeng Sun*, Xin Li*, Chenjie Gu ┼ *ECE Dept. Carnegie Mellon University, Pittsburgh, PA 15213 ┼ Intel Corp. Hillsboro, OR 97124

2. Slide 2 Outline Background Bayesian Model Fusion Experiment Results Conclusion

3. Slide 3 Process Variations and Performance Modeling Statistical performance modeling: approximate circuit performance as an analytical function of process variations Performance model is a powerful tool for efficient circuit analysis:  Yield estimation  Corner extraction  Sensitivity analysis Small Size Large Variation 65nm45nm32nm f:circuit performance of interest (e.g. read delay of SRAM) ∆X:a vector of random variables to model process variations g i (∆X):basis functions (e.g., linear or quadratic polynomials) α i :model coefficients

4. Slide 4 Solving Performance Model: Least Squares Fitting (LSF) Determine performance model Total of M basis Total of K MC samples Basis 1Basis 2 Basis M Basis functions Model coefficients LSF  A set of sampling points are collected  Model coefficients are solved from the following linear equation The problem is required to be over-determined in order to be solvable (i.e. K > M)

5. Slide 5 Challenge: High Dimensionality High dimensionality becomes a challenge in performance modeling  Large number of independent random variables must be used to describe variations in each transistor  Increased number of transistors in circuits Example: a commercial 32nm CMOS process  ~40 random variables to model mismatches of a single transistor Due to high dimensionality (i.e. large # of basis functions), it’s unrealistic to apply LSF (which requires # of MC samples> # of basis functions) CircuitTransistor #Random variable # Operational amplifier~ 50~ 2000 SRAM critical path~ 10K~ 400K

6. Slide 6 To handle the high dimensionality problem, sparsity feature of circuits has been explored [1] Sparsity means that the circuit performance variability is only dominated by a few random variables Example: In SRAM critical path, many Vth mismatches of transistors are not important Performance model has a sparse profile:  Most of coefficients are zero or close to zero Sparsity [1] X. Li, "Finding deterministic solution from underdetermined equation: large-scale performance modeling of analog/RF circuits," TCAD, vol. 29, no. 11, pp. 1661-1668, Nov. 2010 Basis functions Model coefficients Performance

7. Slide 7 Sparse Regression Sparse regression algorithm is an efficient performance modeling algorithm that utilizes the sparsity feature Sparse regression is better than LSF because it requires less number of samples by using sparsity feature Efficiency of performance modeling can be further improved, by considering additional information from design flow (will be discussed in detail later)

8. Slide 8 Outline Background Bayesian Model Fusion Experiment Results Conclusion

9. Slide 9 Bayesian Model Fusion (BMF): Overview Key idea: BMF facilitates late stage performance modeling by reusing data collected in the early stage Early stage data Late stage data Performance modeling Traditional BMF Performance modeling Proposed

10. Slide 10 Analog and mixed-signal (AMS) circuit design spans multiple stages AMS Circuit Design Flow Design cycle for analog and mixed-signal circuits Schematic design stage Layout design stage Circuit modeling Performance modeling … … Performance modeling … Early stage Late stage

11. Slide 11 Correlation in AMS Design Flow Leads to correlation among different stages Comparator: Schematic stage Layout stage One important fact in AMS design flow is that different stages share the same circuit topology and functionality

12. Slide 12 Correlation in Performance Models Correlation: f E (∆X) and f L (∆X) are “likely” to be similar α E1 α E2 α E3 α E4 … α L1  α E1 α L2  α E2 α L3  α E3 α L4  α E4 … f E (∆X) f L (∆X) g 1 (∆X)g 2 (∆X)g 3 (∆X)g 4 (∆X) f E (∆X):early-stage performance model f L (∆X):late-stage performance model α Ei, α Li :model coefficients g i (∆X):basis functions

13. Slide 13 Early stage performance model Very few late stage data Early stage data Bayesian inference (Proposed) Late stage performance model The Proposed Algorithm Flow Early stageLate stage Likelihood Prior

14. Slide 14 Prior Prior is a distribution that describes the uncertainty of parameters based on early stage data, before late stage data is taken into account In our work, information in early design stage is encoded in prior, which describes the uncertainty of late stage model coefficients Prior distribution pdf( α L,m ) α L,m2 α L,m1 Higher Probability Lower Probability

15. Slide 15 Prior Magnitude information of early-stage model coefficients is encoded in prior  Magnitude information here describes whether the absolute value of coefficient is relatively large or small  Small (or zero) coefficients information represents sparsity profile, which is essential for performance model [1] Define prior distribution as a zero-mean Gaussian distribution  Key idea of encoding: the shape of prior is related to magnitude information Prior distribution [1] X. Li, "Finding deterministic solution from underdetermined equation: large-scale performance modeling of analog/RF circuits," TCAD, vol. 29, no. 11, pp. 1661-1668, Nov. 2010

16. Slide 16 Likelihood Likelihood is a function of parameters, which evaluates how parameters fit with data Late stage information is encoded in likelihood function  Specifically, late stage performance function information is encoded in likelihood function  In our work, likelihood function describes how well model coefficients fit with late stage data Likelihood likelihood( α L,m ) α L,m2 α L,m1 Better fitWorse fit

17. Slide 17 However, if we determine model coefficients solely based on likelihood, we may have over-fitting problem  In our case, # of samples in late stage is smaller than # of model coefficients in late stage Bayesian’s theorem Maximum-a-posteriori (MAP) estimation: Maximum-A-Posteriori Estimation Prior Likelihood Posterior MAP estimation of α L Prior distribution pdf( α L ) Likelihood Posterior likelihood( α L )

18. Slide 18 Outline Background Bayesian Model Fusion Experiment Results Conclusion

19. Slide 19 SRAM Example Example 1: CMOS SRAM  Designed in a commercial 32nm SOI  61572 independent random process parameters are considered  Read delay is considered as performance  Linear performance model is fitted  Experiments run on a 2.5GHz Linux server with 16GB memory

20. Slide 20 Modeling Error Two different methods are compared:  The proposed method (BMF)  Orthogonal Matching Pursuit (OMP) Modeling error 4x

21. Slide 21 Modeling Time Speed-up BMF requires 4x less samples to achieve similar accuracy as OMP in SRAM  4x runtime speed-up to build performance model OMP (Traditional) BMF (Proposed) Post-layout samples400100 Read delay error1.02%0.99% Simulation cost (Hour)38.779.69 Fitting cost (Second)3.562.11 Total modeling cost (Hour)38.779.69

22. Slide 22 RO Example Example 2: CMOS ring oscillator  Designed in a commercial 32nm SOI  7177 independent random process parameters are considered  Power, frequency and phase noise are considered as performance  Linear performance model is fitted  Experiments run on a 2.5GHz Linux server with 16GB memory

23. Slide 23 Modeling Error Modeling error is measured for power, frequency and phase noise Power Frequency Phase noise 9x

24. Slide 24 Modeling Time Speed-up BMF requires 9x less samples to achieve similar accuracy as OMP in RO  9x runtime speed-up to build performance model OMP (Traditional) BMF (Proposed) Post-layout samples900100 Power error0.77%0.72% Frequency error0.65%0.54% Phase noise error0.12% Simulation cost (Hour)12.581.40 Fitting cost (Second)5.751.69 Total modeling cost (Hour)12.581.40

25. Slide 25 Conclusion The proposed BMF method facilitates efficient high- dimensional performance modeling at late stage by reusing early stage data BMF achieves more than 4x runtime speedup over traditional OMP method on SRAM and RO test cases BMF can be used for commercial applications such as macro- modeling based verification