Stochastic Orthogonal Polynomials (SoPs)– Any stochastic random variable can be represented by stochastic orthogonal polynomials.

–Gaussian distribution can be described with Hermite Polynomials:

– Orthogonal Property:

Statistical Modeling and Simulation for VLSI Circuits and Systems

Student: Fang Gong (fang08@ee.ucla.edu) Advisor: Lei HeEDA Lab (http://eda.ee.ucla.edu), Electrical Engineering Department, UCLA

• Fang Gong, Hao Yu, Lei He, “PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation”, ACM/IEEE 46th Annual Design Automation Conference (DAC09), 2009

• Fang Gong, Hao Yu, Yiyu Shi, Daesoo Kim, Junyan Ren, Lei He, “QuickYield: An Efficient Global-Search Based Parametric Yield Estimation With Performance Constraints”, ACM/IEEE 47th Annual Design Automation Conference (DAC10), 2010

• Fang Gong, Hao Yu, Lei He, “Stochastic Analog Circuit Behavior Modeling by Point Estimation Method”, International Symposium on Physical Design (ISPD'11), 2011.

• Fang Gong, Hao Yu, Lei He, "Fast Non-Monte-Carlo Transient Noise Analysis for High-Precision Analog/RF Circuits by Stochastic Orthogonal Polynomials",

ACM/IEEE 48th Annual Design Automation Conference (DAC11), 2011

• Collaborators: Dr. Hao Yu, Dr. Yiyu Shi, Dr. Junyan Ren, Mr. Daesoo Kim.

Introduction

References & Collaborators

Random Device Noise [DAC11]–Significant impact on nanometer high-precision analogue/RF circuits.

CMOS PLL phase noise and jitter.– Noise-sensitive circuits: ADCs, PLLS, etc.–Thermal noise, flicker noise, shot noise, etc.– Traditional transient verification is difficult

nonlinear transient noise analysis cannot be achieved unknown how to analyze flicker noise

Noise-free (nominal) response

Statistical Modeling of Performance Distribution [ISPD11]– It is desired to extract the arbitrary distribution of performance merit

Such as oscillator period, voltage discharge, etc. Monte Carlo method is usually used very time-consuming!

–Try to estimate unknown behavioral distribution in performance domain under known stochastic variations in parameter domain Find link between them!

– Existing method:Assume performance merits follow Gaussian distribution. not realistic!Response Surface Model (RSM): approximate circuit performance as an

analytical polynomial function of all process variations

Contribution on High Order Moments Calculation– approximate high order moments with a weighted sum of sampling values of f(x) without analytical function efficiently and accurately.

– can be extended to multiple parameter cases with linear complexity.

Noise Models [DAC11]– Thermal Noise: noise-free element and a Gaussian white noise current source in parallel.

– Flicker Noise: modeled by a noise current in parallel with the MOSFET. Power spectrum density of flicker noise in MOSFET

Synthesize Flicker Noise in Time Domain– Model with Summation of Lorentzian spectra:

Numeric Experiment

Stochastic Analog Circuit Behavior Modeling under Process Variations

Fast Non-Monte-Carlo Transient Noise Analysis

Extract Behavioral Distribution pdf(f) Using RSM [ISPD11]

Limitations of RSM based Method: synthesis of analytical function becomes highly difficult for large scale problems. calculation of high order moments is too complicated or prohibitive

AbstractAs semiconductor industry enters into the 65nm and below, large process variations and device noise become inevitable and hence pose a serious threat to both design and manufacturing of high-precision VLSI systems and circuits. Therefore, stochastic modeling and simulation has become the frontier research topic in recent years in combating such variation effects. To this end, we propose accurate stochastic models of those uncertainties and further develop highly efficient algorithms using statistical simulation techniques, for example, to extract the variable capacitance in parallel, estimate parametric yield, approximate the arbitrary distribution of circuit behavior, and perform efficient transient noise analysis.

Parallel Variational Capacitance Extraction

Yield boundary is the projection of intersection boundary.

Many times of circuit simulations are required to locate one point local search.

Local Search is inefficient, especially for nonlinear circuits.

Fast Yield Estimation considering Process Variations

With stochastic modeling, random process variation can be integrated into our parallel Fast Multi-pole Method, and different variations can be considered by updating the nominal system incrementally.

Contribution in QuickYield [DAC10]Augmenting DAE system with performance constraint

Locating the yield boundary with global search

Up to hundreds faster than Monte Carlo method and up to 4.7X than state-of-the-art method.

Experiments:

3-stage ring oscillator.

Consider MOSFETs channel width variations.

Period should be bounded by [Tmin, Tmax].

Process Variation– Variation Sources:

Optical Proximity Effects Chemical Etching Chemical-Mechanical Planarization Polishing

– What we design is NOT what we got

Small SizeLarge Variation

90nm 65nm 45nm

90nm NAND gate

LLL

WWW

ΔW

ΔL

Process Variations

Results with Process Variation– Circuit Behavior Variation [ISPD11]

Static Timing Analysis: delay variation Signal Integrity Analysis: parasitic RLC variation Analog Mismatch Effect

– Serious yield loss issues [DAC10] process variation will dominate yield loss Yield enhancement should consider process variations

IBM 90nm: Vt variationIBM 90nm: Vt variation

μ: mean valueσ: standard deviation

Noisy response

Potential Coefficient0 1 0 1( , )ij d wP M d d w w

Solve with GMRES

Build Spectral pre-conditioner

Evaluate the MVP (Pxq) with FMM in parallel

Calculate Cij with the charge distribution.

Geometric Moments0 1 0 1( , )d wM d d w w

Incrementally update

preconditioner

Geometry Info Process Variation

0 0( , )d w( , )d w

Capacitance Extraction Procedure [DAC09]Discretize metal surface into small panels. Form linear system by collocation. Results in dense potential coefficients. Solve by iterative GMRES

▪ Contribution in proposed PiCAP: Develop one Parallel Fast Multi-pole Method (FMM) to evaluate Matrix-Vector-Product (MVP) with linear complexity.Handle different variation sources incrementally with novel precondition method.

P q vTable 1: Accuracy and Runtime (s) Comparison

between Monte Carlo and PiCAP.

Table 2: Total Runtime (seconds) Comparison

ƒm(γp)=ƒworst

Framework of Existing Methods Existing Method QuickYield

Tmin

Tmax

Method Yield Time (s) Speedup

MC (5000) 0.62658 44073.8 1X

YENSS* (10 points) 0.6482 317 139X

QuickYield (10 points) 0.6463 84.9 519X

SPICE Monte Carlo

Analysis

Device variation

Parameter Domain Performance Domain

NNpf 110

Synthesize analytical function of performance using RSM

NNpf 110

Calculate time moments

Match with the time moment of a LTI system

h(t) can be used to estimate pdf(f)

Experiment Results

– consider 6-T SRAM Cell and discharge behavior during reading.

– all threshold voltages of MOSFETs are independent variable.

– Proposed method (PEM) can provide high accuracy as Monte Carlo and existing method called APEX.

– On average, PEM can achieve up to 181X speedup over MC and up to 15X speedup over APEX with similar accuracy.

( )2

Fm

ox

K Cg t g

C WL kT

1

2hm mR C

NMC Transient Noise Analysis [DAC11]– Expand all random variables with SoPs;

–Take inner-product with SoPs due to orthogonal property;

– Obtain the SoP expansion of noise at each time-step.

• W: channel width• L: channel length• Cox: gate oxide capacitance per unit area• KF: flicker noise coefficient, process-dependent constant

(0) (0) (0)1 1 1 1 2 1 2

(0)1

1 1 21 1

4 1( ) ( ) ( ) 23 3 ( )

32 1

( ) ( ) ( ) 03 3

k k kn n n n n n

kn n

m mk

k n r n r nk r r

C t C t C tA G t

h

T t g x g x

Flicker Noise Modeling Ring-oscillator

• achieve 488X speedup over MC with 0.5% error;

• can be 6.8X faster than existing method.

• provide high accuracy in the entire range.