Fast, Non-Monte-Carlo Transient Noise
Analysis for High-Precision Analog/RF
Circuits by Stochastic Orthogonal
Polynomials
Fang Gong1, Hao Yu2 and Lei He1
1University
of California, Los Angeles, Los Angeles, USA
2Nanyang Technological University, Singapore
Presented by Fang Gong
Motivation
 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) …
V output  V norm inal   V noise
 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.
Existing 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 slide
 Modeling of Thermal Noise
2 kT
ith ( t ) 
ith ( t ) 
 (t )
4 kT    g m  ( t )
R
stationary process with constant
power spectral density (PSD)
 Stochastic differential algebra equation (SDAE)
Stochastic component
deterministic component
A
d
m
q ( x ( t ))  f ( x ( t ), t ) 
dt
g
r
( X ( t ), t )  r ( t )  0
r 1
g r ( X ( t ), t )
 r (t )
noise intensities
Standard noise sources
(White noise)
 Integrate it to build Itô-Integral based SDAE
t
t
A q ( x ( s )) t 0 

m
f ( x ( s ), s ) ds 
W r (t ) 

0
g
r 1 t 0
t0
t
t
r
( X ( t ), t ) dW r ( t )  0
Wiener process
t
r
( s ) ds 
 dW
0
r
(s)
 W ( t n )  W ( t n )  W ( t n 1 ) ~ N (0, h n )
Existing Solution to Itô-Integral based SDAE
 Stochastic Integral scheme for SDAE (e.g. backward
differentiation formula (BDF) with fixed time-step)
q ( xn ) 
A
4
3
q ( x n 1 ) 
1
3
q ( xn 2 )
hn
3
 W n  W n  W n 1 ~ N (0, h n )
r
r

2
r
f ( xn ) 
g
Wn
r
m
r
( x n 1 )
r 1

hn
1
3
g
 W n 1
r
m
r
( xn  2 )
r 1
0
hn
Sampled with Monte Carlo at each time step
 With piecewise linearization along nominal transient
trajectory:
q ( xn )  q ( xn ) 
(0)
xn  xn   xn
(0)
f ( xn )  f ( xn ) 
(0)
Nominal
response
Transient
noise
q
x
  xn  q ( xn )  C n   xn
(0)
(0)
x  xn
f
x
(0)
  xn  f ( xn )  G n   xn
(0)
(0)
x  xn
(0)
New SOP based Solution
 Stochastic Orthogonal Polynomials without Monte Carlo
 ( )    ( )    ( ) 
    ( )
n
0
0
1
 ( )  [1, ,  1,
2
1
i
i
i0
T
]
 Expand random variables with SoP
 W n  W n  W n 1 ~ N (0, h )   W n   0  0   1  1  h   1 (  0 =0)
r
r
r
r
 x n   0 (t n )  0   1 (t n )  1   1 (t n )  1
q(x
(0)
n

)  C n   1 (t n )  1 
(0)
A
q(x
3
4
(0)
n 1
(  0  0)

)  C n 1   1 ( t n )  1 
(0)
q(x
3
1
(0)
n2
h
2

3
 f (x
C n   1 (t n ) 
(0)
 A
4
3
(0)
n
 2
(0)
)  G n  1 (tn ) 1  
 3

C n 1   1 ( t n 1 ) 
(0)
1
3
SoP expansions
h  1
m
g
r 1
r
0
)  C n  2   1 (t n )  1
(0)

3σ boundary in time
domain
h
C n  2   1 (tn  2 )
(0)
h

2
3

G
(0)
n
 1 (tn ) 
2
m
g

3
r 1

V ar (  x n )   1 ( t n )

2
r
0
nominal
response
Experimental 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.
Runtime Comparison on Different Circuits
MC
SoP
Method
CMOS comparator
Inverter
OPAM
Comparator
Oscillator
Time(s)
91.95
4266.64
2226.71
146851.2
Error
0.43%
0.93%
1.78%
1.62%
Time(s)
1.87
52.35
12.72
300.91
Speedup
49X
81X
175X
488X
Conclusion
 A fast non-Monte-Carlo transient noise analysis using ItôIntegral 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!