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Rasit Onur Topaloglu and Alex Orailoglu {rtopalog|alex}@cse.ucsd.eduUniversity of California, San Diego

Computer Science and Engineering Department9500 Gilman Dr., La Jolla, CA, 92093

Forward Discrete Probability Propagation for Device Performance Characterization under

Process Variations

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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

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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

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SPICE Formula Hierarchy

ex: gm=2*k*ID

Level1

Level4

Level0TLeffWefftox

Level2

Level3

0 NSUBms

CoxF

k Qdep

Vth

ID

gm rout Level5

•SPICE formulas are hierarchical; hence can tie physical parameters to circuit parameters

•A hierarchical tree representation possible : connectivity graphs

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Process Variation Model

•Physical parameters correspond to the lowest level in connectivity graphs

•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

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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

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Monte Carlo for Probability Propagation

gm

Level1

Level4

Level0 nVFBNSUBLW

Vth Cox

tox

ID

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

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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

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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

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T NSUB

PHIf

Necessary Operators (Q, F, B, R) on a Connectivity Graph

Q Q

F

B

•F, B and R repeated until we acquire the distribution of a high level parameter (ex. gm)

R

Q-1

TLeffWefftox 0 NSUBms

CoxF

k

Vth

ID

gmrout

Qdep

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pdf(X)

Probability Discretization Theory: QN Operator

•QN band-pass filter pdf(X) and divide into bins

))(()( XpdfQX N

N in QN indicates number or bins

spdf(X)=(X)X

pdf(

X)

spdf

(X)

X

Ni

ii wxpX..1

)()(

•can write spdf(X) as :

where :

pi : probability for i’th impulse

wi : value of i’th impulse

•Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist

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Error Analysis for Quantization Operator

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2dqQpdfqQEQ )(][

2/

2/

222

Variance of quantization error:

•If quantizer uniform and small, quantization error random variable Q is uniformly distributed

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F Operator •F operator implements a function over spdf’s using

deterministic sampling

))(),..,(()( 1 rXXFY Xi, Y : random variables

r

r

rss

Xs

Xs

Xs

Xs wwfyppY

,..,1

1

1

1

1

1

1)),..,((..)(

•Corresponding function in connectivity graph applied to deterministic combination of impulses•Heights of impulses (probabilities) multiplied•Probabilities are normalized to 1 at the end

pXs : probabilities of the set of all samples s belonging to X

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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

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))(()(' XBX e

Band-pass, Be, 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

))(()

)(max(:

)()(Xp

e

ppi

ii

iii

i

wxpX

e : error rate

•Implementation : eliminate samples with probabilities less than 1/e times the sample with the largest probability

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Re-bin, RN, Operator ))(()(' XRX N

Impulses after F Resulting spdf(X)Unite into one bin

•Samples falling into the same bin congregated in one

i

ii wxpX )()( ijs

ji bwstppj

.where :

•Without R, Q-1 would result in a noisy graph which is not a pdf as samples would not be equally separated

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Error Analysis for Re-bin Operator

jbjiji

ji pwmdi

)(:,

),(Total distortion:

2)(),( jiji wmwmd Distortion caused by representing samples in a bin by a single sample:

mi : center or i’th bin

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Experimental Results

•Impulse representation for threshold voltage and transconductance are obtained through FDPP on the graph

(X) for gm(X) for Vth

19•A close match is observed after interpolation

Monte Carlo – FDPP Comparison

solid : FDPP dotted : Monte Carlo

Pdf of VthPdf of ID

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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 FPdf of F

noisy : Monte Carlo with 1000 samples

solid : FDPP with 100 samples

noisy : Monte Carlo with 100000 samples

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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