Stochastic Analog Circuit Behavior Modeling by Point g y Estimation Method Fang Gong 1 , Hao Yu 2 , Lei He 1 1 Univ. of California, Los Angeles 2 Nanyang Technological University, Singapore Nanyang Technological University, Singapore
O tli Outline Backgrounds B k d Existing Methods and Limitations Proposed Algorithms Experimental Results Experimental Results Conclusions
IC Technology Scaling Feature size keeps scaling down to 45nm and below 90nm 65nm 45nm Shrinking Sh i ki Feature Sizes Large process variation lead to circuit failures and yield problem . L i ti l d t d i it f il i ld bl * Data Source: Dr. Ralf Sommer, DATE 2006, COM BTS DAT DF AMF;
Statistical Problems in IC Technology Statistical methods were proposed to address variation problems Focus on performance probability distribution extraction in p p y this work Fixed Fixed Random Random Parameter Value Distribution Space Design g Process Parameters Parameters Mapping? Unknown Performance Circuit Distribution Performance Space p How to model the stochastic circuit behavior (performance)?
Leakage Power Distribution An example ISCAS-85 benchmark circuit: all threshold voltages (Vth) of MOSFETs have variations that follow Normal distribution . The leakage power distribution follow lognormal distribution. *Courtesy by Fernandes, R.; Vemuri, R.; , ICCD 2009. pp.451 458, 4 7 Oct. 2009 pp 451-458 4-7 Oct 2009 It i d It is desired to extract the arbitrary (usually non-normal) distribution of i d t t t th bit ( ll l) di t ib ti f performance exactly.
Problem Formulation Given : random variables in parameter space a set of (normal) random variables { ε 1 , ε 2 , ε 3 , ...} to model process a set o ( o a ) a do a ab es { ε 1 , ε 2 , ε 3 , } to ode p ocess variation sources. Goal : extract the arbitrary probability distribution of performance Goal : extract the arbitrary probability distribution of performance f( ε 1 , ε 2 , ε 3 , ...) in performance space . Variable process performance variation mapping Performance Space Parameter Space
O tli Outline Backgrounds B k d Existing Methods and Limitations Proposed Algorithms Experimental Results Experimental Results Conclusions
Monte Carlo simulation Monte Carlo simulation is the most straight-forward g method. Device SPICE variation Monte Carlo Analysis Analysis Performance Domain Parameter Domain However, it is highly time-consuming!
Response Surface Model (RSM) Approximate circuit performance (e.g. delay) as an analytical pp p ( g y) y function of all process variations (e.g. V TH , etc ) Synthesize analytical function of performance as random variations. Results in a multi dimensional model fitting problem. Results in a multi-dimensional model fitting problem. f Response surface model can be used to Estimate performance variability p y Identify critical variation sources Δ x 1 Extract worst-case performance corner Δ x 2 Etc. Etc f p 0 1 1 N N
Flow Chart of APEX* Synthesize analytical function f p of performance using RSM 0 1 1 N N Calculate moments Calculate the probability distribution function (PDF) of ( ) performance based on RSM h(t) can be h(t) can be used to estimate pdf(f) sed to estimate df(f) *Xin Li, Jiayong Le, Padmini Gopalakrishnan and Lawrence Pileggi, " Asymptotic probability extraction for non-Normal distributions of circuit performance ," IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 2-9, 2004.
Li Limitation of APEX it ti f APEX RSM based method is time-consuming to get the analytical function of RSM based method is time consuming to get the analytical function of performance. It has exponential complexity with the number of variable parameters n and order of polynomial function q . ) q f x x ( , , , x ) ( x x x 1 2 n 1 1 2 2 n n e.g., for 10,000 variables, APEX requires 10,000 simulations for linear function, and 100 millions simulations for quadratic function. RSM based high-order moments calculation has high complexity the number of terms in f k increases exponentially with the order of moments. f k i th b f t i ti ll ith th d f t k k q f ( , x x , , x ) ( x x x ) n n n 1 2 1 1 2 2
Contribution of Our Work Step 1: Calculate High Order Moments of Performance APEX Proposed Method Find analytical function of performance using RSM Fi d l ti l f ti f f i RSM A few samplings at selected points. f p 0 1 1 N N Calculate high order moments Calculate high order moments Calculate moments by Point Estimation Method k k m ( f pdf ( f )) df f Step 2: Extract the PDF of performance k k M M ( 1 ) ( 1 ) a k 1 k k k k b t r m m ( ( f f pdf pdf ( ( f f )) )) df df m m ( ( t t h h ( ( t t )) )) dt dt h h ( ( t t ) ) a a e e pdf pdf ( ( f f ) ) r f t r k 1 k ! k ! b r 1 r 1 r Our contribution: We do NOT need to use analytical formula in RSM; We do NOT need to use analytical formula in RSM; Calculate high-order moments efficiently using Point Estimation Method;
O tli Outline Backgrounds B k d Existing Methods and Limitations Proposed Algorithms Experimental Results Experimental Results Conclusions
Moments via Point Estimation Point Estimation: approximate high order moments with a weighted sum of sampling values of f(x). are estimating points of random variable are estimating points of random variable. P j are corresponding weights. k -th moment of f(x) can be estimated with PDF x 1 x 2 x 3 Existing work in mechanical area* only provide empirical analytical formulae for x j and P j for first four moments . l ti l f l f d f fi t f t Question – how can we accurately and efficiently calculate the higher order moments of f(x) ? calculate the higher order moments of f(x) ? * Y.-G. Zhao and T. Ono, "New point estimation for probability moments," Journal of Engineering Mechanics, vol. 126, no. 4, pp. 433-436, 2000.
Calculate moments of performance Theorem in Probability: assume x and f(x) are both continuous random variables, then: Flow Chart to calculate high order moments of performance: Flow Chart to calculate high order moments of performance: Step 5: extract performance distribution pdf(f) pdf(x) of parameters is known Step 1: calculate moments of parameters Step 4: calculate moments of performance m m k k k k k k m ( f pdf ( x )) dx P ( f ( x )) m ( x pdf ( x )) dx P ( x ) f j j x j j j 1 j 1 Step 3: run simulation at estimating points x j Step 2: calculate the estimating points x j and weights P j and get performance samplings f( x j ) Step 2 is the most important step in this process.
Estimating Points x j and Weights P j With moment matching method, With t t hi th d and P j can be calculated by d b l l t d b x j m k P 1 m j x j 1 m 1 P x P x E x E x ( ) ( ) m m j j j j x j 1 m 2 2 2 P x E x ( ) m j j x j 1 m 2 m 1 2 m 1 2 m 1 P x E x ( ) m j j x j 1 can be calculated exactly with pdf(x). k m k m ( 0,...,2 1) x Assume residues a j = P j and poles b j = 1/ x j k m ( 1) k P f x ( ) j j k ! j 1 system matrix is well-structured (Vandermonde matrix); nonlinear system can solved with deterministic method.
Extension to Multiple Parameters p Model moments with multiple parameters as a linear combination p p of moments with single parameter. • f(x 1 ,x 2 ,…,x n ) is the function with multiple parameters. • f(x i ) is the function where xi is the single parameter. • g i is the weight for moments of f(x i ) • c is a scaling constant.
Error Estimation We use approximation with q+1 moments as the exact value, when investigating PDF extracted with q moments. Wh When moments decrease progressively lized) k k f k k m ( ( f pdf df ( ( f f )) )) df df Moment (norma f 0 < f < 1 Magnitude of M M Order of Moment Other cases can be handled after shift (f<0), reciprocal (f>1) or scaling operations of performance merits.
O tli Outline Backgrounds B k d Existing Methods and Limitations Proposed Algorithms Experimental Results Experimental Results Conclusions
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