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Forward Discrete Probability Propagation for Device Performance Characterization under Process Variations 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 1 Outline Forward Discrete Probability Propagation Motivation and Comparison with Monte Carlo Probability Discretization Theory Q, F, B, R and Q-1 Operators Experimental Results Conclusions 2 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 3 SPICE Formula Hierarchy tox 0 Weff Leff T NSUB F Cox k Qdep Level2 Vth Level3 Level4 rout ex: gm=2*k*ID Level0 Level1 ID gm ms Level5 •SPICE formulas are hierarchical; hence can tie physical parameters to circuit parameters •A hierarchical tree representation possible : connectivity graphs 4 Process Variation Model •Recent models attribute process variations to physical parameters •Physical parameters correspond to the lowest level in connectivity graphs •Probability density functions (pdf’s), acquired through a test chip, can be independently input to the lowest level nodes 5 Probability Propagation •Estimation of device parameters at highest level needed to examine effects of process variations •An analytic solution not possible since functions highly nonlinear and Gaussian approximations not accurate in deep sub-micron •A method to propagate pdf’s to highest level necessary GOALS Algebraic tractability : enabling manual applicability Speed : be comparable or outperform Monte Carlo Flexibility : be able to use non-standard densities 6 Monte Carlo for Probability Propagation W L VFB NSUB Cox Vth k ID gm n tox Level0 Level1 Level2 Level3 Level4 •Pick independent samples from distributions of Level0 parameters •Compute functions using these samples until highest level reached •Iterate by repeating the preceding 2 steps 7 •Construct a histogram to approximate the distribution 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 8 Implementing FDPP Analytic operation on continuous distributions difficult; instead work in discrete domain and convert back at the end: •Q (Quantize) : Discretize a pdf to operate on its samples •F (Forward) : Given a function, estimates the distribution of next node in the formula hierarchy using samples •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 •Q-1 (De-Quantize) : Convert a discrete pdf back to continuous 9 domain : interpolation Necessary Operators (Q, F, B, R) on a Connectivity Graph tox 0 Weff Leff T NSUB ms F Cox k Qdep Q Q T NSUB Vth F B ID R PHIf gm rout Q-1 •F, B and R repeated until we acquire the distribution of10a high level parameter (ex. g ) pdf(X) Probability Discretization Theory: QN Operator •can write spdf(X) as : pdf(X) (X ) p (x w ) i1.. N spdf(X) X spdf(X)=(X) i i where : pi : probability for i’th impulse wi : value of i’th impulse X ( X ) QN ( pdf ( X )) •QN band-pass filter pdf(X) and divide into bins N in QN indicates number or bins •Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist 11 Error Analysis for Quantization Operator •If quantizer uniform and small, quantization error random variable Q is uniformly distributed Variance of quantization error: E[Q ] 2 Q 2 /2 / 2 2 q pdf (Q) dq 2 12 12 F Operator •F operator implements a function over spdf’s using deterministic sampling (Y ) F ( ( X1 ),.., ( X r )) Xi, 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 (Y ) p s1 ,.., sr X1 s1 .. p ( y f (w ,.., w )) Xr sr X1 s1 X1 s1 pXs : probabilities of the set of all samples s belonging to X 13 Effect of Non-linear Functions Impulses after F, before B and R •Non-linear nature of functions cause accumulation in certain ranges •De-quantization would not result in a correct shape •Increased number of samples would induce a computational burden 14 Band-pass and re-bin operations needed after F operation Band-pass, Be, Operator ' ( X ) Be ( ( X )) Margin-based Definition: •Eliminate samples having values out of range (6): might cut off tails of bi-modal or long-tailed distributions Novel Error-based Definition: •Eliminate samples having probabilities least likely to occur : eliminates samples in useful range hence offers more computational efficiency p (x w ) (X ) i i:( pi max i ( pi ) ) ( pi ( X )) e i e : error rate •Implementation : eliminate samples with probabilities less 15 than 1/e times the sample with the largest probability Re-bin, RN, Operator Impulses after F Unite into one bin ' ( X ) RN ( ( X )) Resulting spdf(X) •Samples falling into the same bin congregated in one st. w j bi ( X ) p i ( x wi ) where : pi p j i sj •Without R, Q-1 would result in a noisy graph which16 is not a pdf as samples would not be equally separated Error Analysis for Re-bin Operator Distortion caused by representing samples in a bin by a single 2 sample: d (m , w ) (m w ) i j i j mi : center or i’th bin Total distortion: d (m , w ) p i , j:i ( jbi ) i j j 17 Experimental Results (X) for Vth (X) for gm •Impulse representation for threshold voltage and 18 transconductance are obtained through FDPP on the graph Monte Carlo – FDPP Comparison Pdf of Vth Pdf of ID solid : FDPP dotted : Monte Carlo •A close match is observed after interpolation 19 Monte Carlo – FDPP Comparison with a Low Sample Number Pdf of F solid : FDPP with 100 samples noisy : Monte Carlo with 1000 samples Pdf of F solid : FDPP with 100 samples noisy : Monte Carlo with 100000 samples •Monte Carlo inaccurate for moderate number of samples •Indicates FDPP can be manually applied without major 20 accuracy degradation 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 21