1. 1 Rasit Onur Topaloglu and Alex Orailoglu {rtopalog|alex}@cse.ucsd.edu University of California, San Diego Computer Science and Engineering Department 9500 Gilman Dr., La Jolla, CA, 92093 Forward Discrete Probability Propagation for Device Performance Characterization under Process Variations

2. 2 Outline Forward Discrete Probability Propagation Probability Discretization Theory Q, F, B, R and Q -1 Operators Experimental Results Conclusions Motivation and Comparison with Monte Carlo

3. 3 Motivation Process variations have become dominant even at the device level in deep sub-micron technologies To reduce design iterations, there is a need to accurately estimate the effects of process variations on device performance Current tools/methods not quite suitable for this problem due to accuracy & speed bottlenecks Most simulators use SPICE formulas

4. 4 SPICE Formula Hierarchy ex: g m =  2*k*I D Level1 Level4 Level0 TL eff W eff t ox Level2 Level3 00 NSUB  ms C ox FF k Q dep V th IDID gmgm r out Level5 SPICE formulas are hierarchical; hence can tie physical parameters to circuit parameters A hierarchical tree representation possible : connectivity graphs

5. 5 Process Variation Model Physical parameters correspond to the lowest level in connectivitygraphs Recent models attribute process variations to physical parameters Probability density functions (pdf’s), acquired through a test chip, can be independently input to the lowest level nodes

6. 6 Probability Propagation Estimation of device parameters at highest level needed to examine effects of process variations An analytic solution not possible since functions highly non- linear and Gaussian approximations not accurate in deep sub-micron GOALS Algebraic tractability : enabling manual applicability Speed : be comparable or outperform Monte Carlo A method to propagate pdf’s to highest level necessary Flexibility : be able to use non-standard densities

7. 7 Monte Carlo for Probability Propagation gmgm Level1 Level4 Level0  n n V FB NSUB LW V th C ox t ox IDID k Level2 Level3 Pick independent samples from distributions of Level0 parameters Compute functions using these samples until highest level reached Iterate by repeating the preceding 2 steps Construct a histogram to approximate the distribution

8. 8 Shortcomings of Monte Carlo Not manually applicable due to large number of iterations and random sampling Limited to standard distributions : Random number generators in CAD tools only provide certain distributions Accuracy : May miss points that are less likely to occur due to random sampling; a large number of iterations necessary which is quite costly for simulators

9. 9 Implementing FDPP F (Forward) : Given a function, estimates the distribution of next node in the formula hierarchy using samples Q (Quantize) : Discretize a pdf to operate on its samples Analytic operation on continuous distributions difficult; instead work in discrete domain and convert back at the end: Q -1 (De-Quantize) : Convert a discrete pdf back to continuous domain : interpolation B (Band-pass) : Used to decrement number of samples using a threshold on sample probabilities R (Re-bin) : Used to decrement number of samples by combining close samples together

10. 10 T NSUB PHIf Necessary Operators (Q, F, B, R) on a Connectivity Graph QQ F B F, B and R repeated until we acquire the distribution of a high level parameter (ex. g m ) R Q -1 TL eff W eff t ox 00 NSUB  ms C ox FF k V th IDID gmgm r out Q dep

11. 11 pdf(X) Probability Discretization Theory: Q N Operator Q N band-pass filter pdf(X) and divide into bins N in Q N indicates number or bins spdf(X)=  (X) X pdf(X) spdf(X) X can write spdf(X) as : where : p i : probability for i’th impulse w i : value of i’th impulse Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist

12. 12 Error Analysis for Quantization Operator Variance of quantization error: If quantizer uniform and  small, quantization error random variable Q is uniformly distributed

13. 13 F Operator F operator implements a function over spdf’s using deterministic sampling X i, Y : random variables Corresponding function in connectivity graph applied to deterministic combination of impulses Heights of impulses (probabilities) multiplied Probabilities are normalized to 1 at the end p X s : probabilities of the set of all samples s belonging to X

14. 14 Effect of Non-linear Functions Non-linear nature of functions cause accumulation in certain ranges Band-pass and re-bin operations needed after F operation Impulses after F, before B and R De-quantization would not result in a correct shape Increased number of samples would induce a computational burden

15. 15 Band-pass, B e, Operator Eliminate samples having values out of range (6  ): might cut off tails of bi-modal or long-tailed distributions Margin-based Definition: Novel Error-based Definition: Eliminate samples having probabilities least likely to occur : eliminates samples in useful range hence offers more computational efficiency e : error rate Implementation : eliminate samples with probabilities less than 1/e times the sample with the largest probability

16. 16 Re-bin, R N, Operator Impulses after F Resulting spdf(X) Unite into one  bin Samples falling into the same bin congregated in one where : Without R, Q -1 would result in a noisy graph which is not a pdf as samples would not be equally separated

17. 17 Error Analysis for Re-bin Operator Total distortion: Distortion caused by representing samples in a bin by a single sample: m i : center or i’th bin

18. 18 Experimental Results Impulse representation for threshold voltage and transconductance are obtained through FDPP on the graph  (X) for gm  (X) for V th

19. 19 A close match is observed after interpolation Monte Carlo – FDPP Comparison solid : FDPP dotted : Monte Carlo Pdf of V th Pdf of I D

20. 20 Monte Carlo – FDPP Comparison with a Low Sample Number Monte Carlo inaccurate for moderate number of samples Indicates FDPP can be manually applied without major accuracy degradation solid : FDPP with 100 samples Pdf of  F noisy : Monte Carlo with 1000 samples solid : FDPP with 100 samples noisy : Monte Carlo with 100000 samples

21. 21 Conclusions Forward Discrete Probability Propagation is introduced as an alternative to Monte Carlo based methods FDPP should be preferred when low probability samples need to be accounted for without significantly increasing the number of iterations FDPP provides an algebraic intuition due to deterministic sampling and manual applicability due to using less number of samples FDPP can account for non-standard pdf’s where Monte Carlo-based methods would substantially fail in terms of accuracy