An Analysis of Phase Noise
and Fokker-Planck Equations
Hao-Min Zhou
School of Mathematics
Georgia Institute of Technology
Joint work with Shui-Nee Chow
International conference of random dynamical
systems, Tianjin, China, June 8-12, 2009
Partially Supported by NSF
Outline
• Introduction and motivation
• Moving coordinate transforms
• Phase noise equations and
Fokker-Planck equations
• Example: van der Pol
oscillators and ACD
• Conclusion
Introduction and Motivation
• A orbital stable periodic solution (limit cycle)  (with period T ) of a
differential system
 du1
 dt  f (u1 , u 2 )
 du
2

 g (u1 , u 2 )
 dt
• Phase noise is caused by perturbations, which are unavoidable in
practice: the solution doesn’t return to the starting point after a
period  .
• Phase noise usually persists, may become large.
• Phase noise is important in many areas including circuit design,
and optics.
Oscillators
• Phase noise in nonlinear
electric oscillators:
• Small noise can lead to
dramatic spectral changes
• Many undesired problems
associated with phase noise,
such as interchannel
interference and jitter.
Analog to Digital Converter (ADC)
• ADC is essential for wireless
communications.
• Input: wave (amplitude, frequency).
Output: digit computed in real-time,
during one single period (number of
spikes).
• Effect of the noise in the transmission
system.
5
7
correct output
5
7
5
8
wrong output
Bit Error Rate (BER) : ratio of received bits that are in error, relative to the amount
of bits received. BER expressed in log scale (dB).
ADC Example
A piecewise linear ADC model is
 x'   [ f (t ) - y ]

 y '  1 [ x - g ( y )],


y0
 y,

g ( y)  Ky,
i  y  0
  y , y  i 

The input is an analog signal, i.e. f (t )  sin t
The output is the number of spikes in a period, which realizes the
conversion of analog signals to digital ones.
Our goals
• Establish a framework to rigorously analyze
phase noise from both dynamic system and
probability perspectives.
• Develop numerical schemes to compute phase
noise, which are useful tools for system design.
• Estimate Shannon entropy curves to evaluate the
performance of practical systems
Approaches
• Traditional nonlinear analysis based on linearization
is invalid: decompose the perturbed solution ( x(t ), y(t ))
 x(t )  u1 (t )  w1 (t )

 y (t )  u2 (t )  w2 (t )
where (u1 (t ), u2 (t )) is the unperturbed solution and (w1 (t ), w2 (t ))
is the deviation, then the error satisfies
dw
 A(t ) w(t )  B (u (t ))b(t )
dt
•The system is self-sustained, and A(t ) must have one
as its eigenvalue.
• The deviation w(t ) can grow to infinitely large (even
amplitude error remains small for stable systems, but
phase error can be large)
Approaches
• A conjecture: decompose perturbations into two (orthogonal)
components, one along the tangent, one along normal direction,
perturbations along tangent generates purely phase noise and
normal component causes only amplitude deviation, Hajimiri-Lee
(’97).
•This conjecture is not valid, Demir-Roychowdhury (’98).
Perturbation orthogonal to the orbit can also cause phase
deviation.
Approaches
• Large literature is available for individual systems,
such as pumped lasers by Lax (’67), but lack of
general theory for phase noise.
• Two appealing approaches:
1. Model the perturbed systems by SDE’s and derive the
associated Fokker-Planck equations, then use asymptotic
analysis to estimate the leading contributions of transition
probability distribution function , i.e. in Limketkai (’05), the
leading term  is approximated by a gaussian:
 ( x,  , )  ( , )Ce
 Ex2
where  ( , ) satisfy a diffusion PDE
  A  B  0,
and A, B, C , E are coefficients obtained in asymptotic expansions
Approaches
2. Decompose oscillator response into phase and magnitude
components and obtain equations for the phase error, for
examples: Kartner (’90), Hajimiri-Lee (’98),Demir-MehrotraRoychowdhury (’00), i.e.
x(t )  u (t   (t ))  y (t )
where  (t ) is defined by a SDE depending on the largest
eigenvalue 1 and eigenfunction v1 of state transition matrix in
Floquet theory:
d (t )
 v1T (t   (t )) B(u (t   (t )))dWt
dt
 (t ) may grow to infinitely large even for small perturbations
Moving Orthogonal Systems
• A moving orthogonal coordinate systems along 
• Consider solutions ( x(t ), y (t )) of the perturbed systems
 dx
 dt  f ( x, y )  h( x, y, t )
 dy
  g ( x, y )  k ( x, y , t )
 dt
h( x, y, t ), k ( x, y, t ) are small perturbations
Equations for the new variables
• Solutions of the perturbed system can be represented by
 x(t )   u1 ( (t ))  
 y (t )  u ( (t ))   z ( (t ))  (t )

  2

denoted by
 (t )   x(t ) 
:
   y (t )

(
t
)

 



• Two components u ( (t )) and z ( (t )) are not orthogonal, which is
different from the usual orthogonal decompositions.
• For small perturbations, this transform is invertible and both
forward and inverse transforms are smooth.
Equations for the new variables
• The new phase  (t ) and amplitude deviation  (t ) satisfy (Hale (’67))
 d s
 dt  r ( f ( f  h )  g ( g  k ))
 d 1

 ( g ( f  h )  f ( g  k ))
 dt r
where notations are
r  ( f  g ),
2
2
2
s  (r  w ) ,
 f  f (u1 ( ), u2 ( ))
,

 g  g (u1 ( ), u2 ( ))
Evaluate on the
unperturbed orbit
1
fg ' gf '
w
r2
 f  f ( x( ,  ), y ( ,  ))

 g  g ( x( ,  ), y ( ,  ))

h  h( x( ,  ), y ( ,  ), t )
k  k ( x( ,  ), y ( ,  ), t )
Evaluate on the perturbed orbit
Stochastic Perturbations
• Perturbations in oscillators are random, which are often modeled by
dX  ( f ( X , Y )  h( X , Y , t )) dt  a( X , Y )dWt1

2
 dY  ( g ( X , Y )  k ( X , Y , t )) dt  b( X , Y )dWt
Where Wt1 , Wt 2 are independent Brownian motions.
• The transform becomes
 X (t )  (t )   u1 ((t ))  
 Y (t )     (t )   u ((t ))  z ((t )) (t ),

 
  2

• Theorem 1: if  X (t ), Y (t ) stay close to  , then (t ), (t ) remain as Ito
processes and satisfy
d  1dt  1dWt1   2 dWt 2

1
2
d



dt


dW


dW
2
1
t
2
t

Stochastic Perturbations
• The coefficients are

1



s



fa
1

r

s
s
wsfg 2
 ( f ( f  h )  g ( g  k )  (( fa ) 2  ( gb ) 2 ) 
(b  a 2 ))  2  s gb

r
2r
2r
r
, 
1
1
ws
  1  ga
 2  ( g ( f  h )  f ( g  k )  (( fa ) 2  ( gb ) 2 )
r

r
2r
1



fb
2

r

• Theorem 2: the transition probability p( ,  , t ) of (t ), (t ) satisfies
the Fokker-Planck equation
1
pt  (1 p)  ( 2 p)   ((( 12   22 ) p)  2(( 1 1   2 2 ) p)  (( 12   22 ) p)  )
2
with initial condition
p t 0   (  0 ) ( )
Stochastic Perturbations
• For a general problem in R n



 
dX  ( f ( X )  h( X , t ))dt  a( X )dWt
The solution
 can also be transformed
 into
n( n 1)
X (t )  u ((t ))  z ((t ))  (t ) where z  R


• Theorem 3: if X (t ) stay close to  , then (t ),  (t )  remain as Ito processes and
 
 d  1dt   dWt

  
d   2 dt  dWt ,
satisfy


n 1
n
( n 1)n
where 1 ,  2  R ,   R ,   R
can be determined similarly.


• Theorem 4: the transition probability p ( ,  , t ) of (t ),  (t )  satisfies
the Fokker-Planck equation

1
pt    (p )    (  ( T p )),
2
where

 
1 
    ,    
 2 
 

van der Pol Oscillators
• Unperturbed van der Pol Oscillators are often described by
q   (1  q 2 )q  q  0
introduce new variable
the equation becomes
v  q
q  v


2

v


q


(
1

v
)v

• In practice, noise enters the system, which is model by
q   (1  q 2 )q  q  dWt  0
by introducing the new variable Y  dX , the system becomes
dX  Y


2
dY


X


(
1

Y
)Y  dWt



• Both and are positive small constant numbers, it is interesting to
study the case    eventually.
van der Pol Oscillators
Assume    are small (in oscillators, the periodic orbits are stable,
and perturbations of amplitude will remain small, i.e.  is small). The
leading term system is
2 4

d  dt   (
 ) cos dWt

3 3

2
2

d  ((1  4 sin ) sin )dt   sin dWt
By the method of averaging for stochastic equations, it is equivalent to
2 4

 ) cos dWt
d  dt  (

3 3

 d  dt   sin dWt
The corresponding Fokker-Planck equation is
1
3 3 2 2
3 3
pt   p  (p)   (((
  )  cos 2  ) p)  ((
  ) 2 sin( 2 ) p)  (( 2 sin 2  ) p)  )
2
2 4
2 4
van der Pol Oscillators
Two interesting observations (made by engineers,
Hajimiri-Lee(’98), Limketkai(’05 ) ):
1. Impuse noise in current at the peak
of current (zero voltage),
  0,    , sin   0, cos   1,
Perturbation has no impact on amplitude,
and maximum impact on phase noise.
2. Impose noise in current at the peak
of voltage (zero current),


3 cos   0, sin   1,
,
,
2
2
Noise has no impact on phase, and
maximum impact on amplitude error.
van der Pol Oscillators
The dynamic of amplitude error can be approximated by
d  dt   sin tdWt
which leads to the following properties if the initial  is small:
• The mean: E ( )  0 . t
• The variance: E (2 )   2  e 2 (t  s ) sin 2 sds
0
  2 ((1  e  2t ) 
2
1  2t
1
1
1
1
e
(
 2 ) 1 ( cos 2t  2 sin 2t ))
2
2 
2

as t  
• It is a Gaussian variable.
The amplitude error also satisfies:
sup ( s)  sup Y ( s )   2 log t
0 s t
0 s t
where
dY (t )  Y (t )dt  dWt

This implies that if t  e  , then sup  ( s )   for any given  .
2
0 s t
Conclusion
• A general framework, based on a moving orthogonal
coordinate system, has been established to rigorously
study the phase and amplitude noise.
• Both dynamic equations and Fokker-Planck equations
for the phase noise are derived.
• The general theory has been applied to the van der Pol
oscillators. Derived equations can explain some
interesting observations in practice.