Presented by Fang Gong

1University of California, Los Angeles, Los Angeles, USA2Nanyang Technological University, Singapore

Fang Gong1, Hao Yu2 and Lei He1

Fast, Non-Monte-Carlo Transient Noise Analysis for High-Precision Analog/RF

Circuits by Stochastic Orthogonal Polynomials

MotivationMotivation Device noise can not be neglected for high-precision

analog circuit anymore! Signal-to-noise ratio (SNR) is reduced; Has large impact on noise-sensitive circuits: PLLs (phase noise

and jitter), ADCs (BER) …

Device Noise Sources: Thermal Noise: random thermal motion of the charge carriers in

a conductor; Flicker Noise (1/f Noise): random trapping and de-trapping of

charge carriers in the traps located at the gate oxide interface.

output norminal noiseV V V

Existing WorkExisting Work Monte Carlo method

Model the thermal noises as stochastic current sources attached to noise-free device components.

Sample the stochastic current sources to generate many traces.

Non-Monte-Carlo method: [A. Demir, 1994] Decouple the noisy system into a stochastic differential equation

(SDE) and an algebraic constraint. Use perturbation analysis and covariance matrix to solve for

variance of transient noise in time domain.

Examples of Commercial tools: Transient noise analysis in HSPICE (Synopsys) AFS transient noise analysis (Berkeley Design Automation), …

SDAE based Noise Analysis- primer slideSDAE based Noise Analysis- primer slide

2( ) ( )th

kTi t t

R

( ) 4 ( )th mi t kT g t

stationary process with constant power spectral density (PSD)

Stochastic differential algebra equation (SDAE) noise intensities

Standard noise sources (White noise)

( ( ), )rg X t t

( )r t

Stochastic componentdeterministic component

1

( ( )) ( ( ), ) ( ( ), ) ( ) 0m

r rr

dA q x t f x t t g X t t tdt

Integrate it to build Itô-Integral based SDAE

010 0

( ( )) ( ( ), ) ( ( ), ) ( ) 0t tm

t

r rtrt t

Aq x s f x s s ds g X t t dW t

1

0 0

( ) ( ) ( ) ( ) ( ) ( ) ~ (0, )t t

r r r n n n nW t s ds dW s W t W t W t N h

Wiener process

Modeling of Thermal Noise

Existing Solution to Itô-Integral based SDAEExisting Solution to Itô-Integral based SDAE Stochastic Integral scheme for SDAE (e.g. backward differentiation formula (BDF) with fixed time-step)

With piecewise linearization along nominal transient trajectory:

(0)

(0)

(0) (0) (0)

(0) (0) (0)

( ) ( ) ( )

( ) ( ) ( )

n

n

n n n n n nx x

n n n n n nx x

qq x q x x q x C x

x

ff x f x x f x G x

x

(0)n n nx x x

Transient noise

Nominal response

1 21

1 21 1

1

4 1( ) ( ) ( ) 2 13 3 ( ) ( ) ( ) 0

3 3

~ (0, )

r rm mn n nn n

n r n r nr rn n n

r r rn n n n

q x q x q x W WA f x g x g x

h h h

W W W N h

Sampled with Monte Carlo at each time step

New SOP based SolutionNew SOP based Solution

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

(0) (0) 11 1

1

4 1( ) ( ) ( ) ( ) ( ) ( )

3 3

2 2 ( ) ( ) 0

3 3

n n n n n n n n n

m

n n n rr

q x C t q x C t q x C tA

hh

f x G t gh

(0) (0) (0)

1 1 1 1 2 1 2

(0)1

1

4 1( ) ( ) ( )

3 3

2 2 ( ) 0

3 3

n n n n n n

m

n n rr

C t C t C tA

h

G t g

2

1 ( ) ( )n nVar x t

3σ boundary in time domain

nominal response

Stochastic Orthogonal Polynomials without Monte Carlo

0 0 1 10

( ) ( ) ( ) ( )n

i ii

2( ) [1, , 1, ]T

1 0 0 1 1 1 0~ (0, ) ( =0)r r r rn n n nW W W N h W h

0 0 1 1 1 1 0( ) ( ) ( ) ( 0)n n n nx t t t SoP expansions

Expand random variables with SoP

Experimental ResultsExperimental Results

Experiment Settings Consider both thermal and flicker noise for all MOSFETs. Resistors only have thermal noise.

Accuracy and efficiency validity SoP expansion method can achieve up to 488X speedup with

0.5% error in time domain, when compared with MC.

CMOS comparator

Inverter OPAM Comparator Oscillator

MC Time(s) 91.95 4266.64 2226.71 146851.2

SoP Method

Error 0.43% 0.93% 1.78% 1.62%

Time(s) 1.87 52.35 12.72 300.91

Speedup 49X 81X 175X 488X

Runtime Comparison on Different Circuits

ConclusionConclusion

A fast non-Monte-Carlo transient noise analysis using Itô-Integral based SDAE and stochastic orthogonal polynomials (SoPs)

The first solution of SDAE by SoPs Expand all random variables with SoPs Apply inner-product with SoPs to expansions (orthogonal

property) Obtain the SoP expansion of transient noise at each time-step

To learn more come to poster session!To learn more come to poster session!