PHYSICAL REVIEW E 81, 031105 共2010兲
1 Õ f noise from nonlinear stochastic differential equations
J. Ruseckas* and B. Kaulakys
Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 12, LT-01108 Vilnius, Lithuania
共Received 20 October 2009; published 8 March 2010兲
We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the
power spectral density in any desirably wide range of frequency. Such equations were obtained starting from
the point process models of 1 / f ␤ noise. In this article the power-law behavior of spectrum is derived directly
from the stochastic differential equations, without using the point process models. The analysis reveals that the
power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional
justification of equations, expands the class of equations generating 1 / f ␤ noise, and provides further insights
into the origin of 1 / f ␤ noise.
DOI: 10.1103/PhysRevE.81.031105
PACS number共s兲: 05.40.⫺a, 72.70.⫹m, 89.75.Da
I. INTRODUCTION
Power-law distributions of spectra of signals, including
1 / f noise 共also known as 1 / f fluctuations, flicker noise, and
pink noise兲, as well as scaling behavior in general, are ubiquitous in physics and in many other fields, including natural
phenomena, human activities, traffics in computer networks,
and financial markets. This subject has been a hot research
topic for many decades 共see, e.g., a bibliographic list of papers by Li 关1兴, and a short review in Scholarpedia 关2兴兲.
Despite the numerous models and theories proposed since
its discovery more than 80 years ago 关3,4兴, the intrinsic origin of 1 / f noise still remains an open question. There is no
conventional picture of the phenomenon and the mechanism
leading to 1 / f fluctuations are not often clear. Most of the
models and theories have restricted validity because of the
assumptions specific to the problem under consideration. A
short categorization of the theories and models of 1 / f noise
is presented in the introduction of the paper 关5兴.
Until recently, probably the most general and common
models, theories and explanations of 1 / f noise have been
based on some formal mathematical description such as fractional Brownian motion, the half-integral of the white noise,
or some algorithms for generation of signals with scaled
properties 关6–14兴 and the popular modeling of 1 / f noise as
the superposition of independent elementary processes with
the Lorentzian spectra and a proper distribution of relaxation
times, e.g., a 1 / ␶relax distribution 关15–21兴. The weakness of
the latter approach is that the simulation of 1 / f ␤ noise with
the desirable slope ␤ requires finding the special distributions of parameters of the system under consideration; at
least a wide range of relaxation time constants should be
assumed in order to allow correlation with experiments
关22–28兴.
Nonlinear stochastic differential equation with linear
noise and nonlinear drift, was considered in Ref. 关9兴. It was
found that if the damping is decreasing with increase in the
absolute value of the stochastic variable, then the solution of
such a nonlinear stochastic differential equation 共SDE兲 has
long correlation time. Recently nonlinear SDEs generating
signals with 1 / f noise were obtained in Refs. 关29,30兴 共see
also recent papers 关5,31兴兲, starting from the point process
model of 1 / f noise 关27,32–39兴.
The purpose of this article is to derive the behavior of the
power spectral density directly from the SDE, without using
the point process model. Such a derivation offers additional
justification of the proposed SDE and provides further insights into the origin of 1 / f noise.
II. PROPOSED STOCHASTIC DIFFERENTIAL
EQUATIONS
Starting from the point process model, proposed and analyzed in Refs. 关27,32–38兴, the nonlinear stochastic differential equations are derived 关5,29,30兴. The general expression
for the SDE is
冉 冊
dx = ␴2 ␩ −
␯ 2␩−1
x
dt + ␴x␩dW.
2
共1兲
Here, x is the signal, ␩ is the exponent of the multiplicative
noise, ␯ defines the behavior of stationary probability distribution, and W is a standard Wiener process.
SDE 共1兲 has the simplest form of the multiplicative noise
term, ␴x␩dW. Multiplicative equations with the drift coefficient proportional to the Stratonovich drift correction for
transformation from the Stratonovich to the Itô stochastic
equation 关40兴 generate signals with the power-law distributions 关5兴. Equation 共1兲 is of such type and has probability
distribution of the power-law form P共x兲 ⬃ x−␯. Because of the
divergence of the power-law distribution and the requirement
of the stationarity of the process, the SDE 共1兲 should be
analyzed together with the appropriate restrictions of the diffusion in some finite interval. For simplicity, in this article,
we will adopt reflective boundary conditions at x = xmin and
x = xmax. However, other forms of restrictions are possible.
For example, exponential restriction of the diffusion can be
obtained by introducing additional terms in Eq. 共1兲,
冉
dx = ␴2 ␩ −
冉 冊 冉 冊冊
␯ m xmin
+
2 2 x
m
−
m x
2 xmax
m
x2␩−1dt + ␴x␩dW.
共2兲
*julius.ruseckas@tfai.vu.lt
1539-3755/2010/81共3兲/031105共7兲
Here, m is some parameter.
031105-1
©2010 The American Physical Society
PHYSICAL REVIEW E 81, 031105 共2010兲
J. RUSECKAS AND B. KAULAKYS
Equation 共1兲 with the reflective boundary condition at xmin
and xmax can be rewritten in a form that does not contain
parameters ␴ and xmin. Introducing the scaled stochastic vari2␩−2
t one transforms
able x → x / xmin and scaled time t → ␴2xmin
Eq. 共1兲 to
冉 冊
dx = ␩ −
␯ 2␩−1
x
dt + x␩dW.
2
冉 冊
S共x,t兲 = ␩ −
1 ⳵ 2␩
␯ 2␩−1
x P.
x
P−
2 ⳵x
2
共10兲
At the reflective boundaries xmin = 1 and xmax = ␰ the probability current S共x , t兲 should vanish, and, therefore, the boundary
conditions for Eq. 共8兲 are
共3兲
S共␰,t兲 = 0.
S共1,t兲 = 0,
共11兲
The scaled Eq. 共3兲 has a boundary at x = 1 and at
A. Eigenfunction expansion
xmax
␰=
.
xmin
共4兲
We solve Eq. 共8兲 using the method of eigenfunctions. An
ansatz of the form
Further, we will consider Eq. 共3兲 only. In order to obtain 1 / f ␤
noise we require that the region of diffusion of the stochastic
variable x should be large. Therefore, we assume that ␰ Ⰷ 1.
leads to the equation
III. POWER SPECTRAL DENSITY FROM
THE FOKKER-PLANCK EQUATION
− ␩−
冉 冊
According to Wiener-Khintchine relations, the power
spectral density is
S共f兲 = 2
冕
⬁
C共t兲ei␻tdt = 4
−⬁
冕
⬁
C共t兲cos共␻t兲dt,
共5兲
C共t兲 = 具x共t⬘兲x共t⬘ + t兲典.
共6兲
This average can be written as
冕 冕
dx
dx⬘xx⬘ P0共x兲Px共x⬘,t兩x,0兲,
共7兲
where P0共x兲 is the steady-state probability distribution function and Px共x⬘ , t 兩 x , 0兲 is the transition probability 共the conditional probability that at time t the signal has value x⬘ with
the condition that at time t = 0 the signal had the value x兲.
The transition probability can be obtained from the solution
of the Fokker-Planck equation with the initial condition
Px共x⬘ , 0 兩 x , 0兲 = ␦共x⬘ − x兲.
Therefore, for the calculation of the power spectral density of the signal x we will use the Fokker-Planck equation
instead of stochastic differential Eq. 共3兲. The Fokker-Planck
equation corresponding to the Itô solution of Eq. 共3兲 is
关41,42兴
冉 冊
1 ⳵ 2 2␩
⳵
␯ ⳵ 2␩−1
P+
P=− ␩−
x
x P.
2 ⳵ x2
⳵t
2 ⳵x
共8兲
The steady-state solution of Eq. 共8兲 has the form
P0共x兲 =
冦
冧
␯ − 1 −␯
x , ␯ ⫽ 1,
1 − ␰1−␯
1 −1
␯ = 1.
x ,
ln ␰
1 ⳵ 2 2␩
␯ ⳵ 2␩−1
x
P␭ +
x P␭ = − ␭P␭共x兲,
2 ⳵ x2
2 ⳵x
冕
␰
1
e⌽共x兲 P␭共x兲P␭⬘共x兲dx = ␦␭,␭⬘ ,
⌽共x兲 = − ln P0共x兲.
共14兲
共15兲
It should be noted that the restriction of diffusion of the
variable x by xmin and xmax ensures that the eigenvalue spectrum is discrete. Expansion of the transition probability density in a series of the eigenfunctions has the form 关42兴
Px共x⬘,t兩x,0兲 = 兺 P␭共x⬘兲e⌽共x兲 P␭共x兲e−␭t .
␭
共16兲
Substituting Eq. 共16兲 into Eq. 共7兲 we get the autocorrelation
function
C共t兲 = 兺 e−␭tX␭2 .
␭
共17兲
Here,
X␭ =
冕
␰
xP␭共x兲dx
共18兲
1
is the first moment of the stochastic variable x evaluated with
the ␭-th eigenfunction P␭共x兲. Such an expression for the autocorrelation function has been obtained in Ref. 关43兴. Using
Eqs. 共5兲 and 共17兲 we obtain the power spectral density
␭
The boundary conditions for Eq. 共8兲 can be expressed using
the probability current 关42兴
共13兲
where ⌽共x兲 is the potential, associated with Eq. 共8兲,
S共f兲 = 4 兺
共9兲
共12兲
where P␭共x兲 are the eigenfunctions and ␭ ⱖ 0 are the corresponding eigenvalues. The eigenfunctions P␭共x兲 obey the orthonormality relation 关42兴
0
where ␻ = 2␲ f and C共t兲 is the autocorrelation function. For
the stationary process, the autocorrelation function can be
expressed as an average over realizations of the stochastic
process,
C共t兲 =
P共x,t兲 = P␭共x兲e−␭t
␭
X2 .
␭ + ␻2 ␭
2
共19兲
This expression for the power spectral density resembles the
models of 1 / f noise using the sum of the Lorentzian spectra
关15–18,27,28,44,45兴. Here, we see that the Lorentzians can
arise from the single nonlinear stochastic differential equation. Similar expression for the spectrum has been obtained
031105-2
1 / f NOISE FROM NONLINEAR STOCHASTIC…
PHYSICAL REVIEW E 81, 031105 共2010兲
in Ref. 关14兴 where reversible Markov chains on finite state
spaces were considered 关Eq. 共34兲 in Ref. 关14兴 with −␥k,m
playing the role of ␭兴.
A pure 1 / f ␤ power spectrum is physically impossible because the total power would be infinity. It should be noted
that the spectrum of signal x, obeying SDE 共3兲, has 1 / f ␤
behavior only in some intermediate region of frequencies,
f min Ⰶ f Ⰶ f max, whereas for small frequencies f Ⰶ f min the
spectrum is bounded. The behavior of spectrum at frequencies f min Ⰶ f Ⰶ f max is connected with the behavior of the autocorrelation function at times 1 / f max Ⰶ t Ⰶ 1 / f min. Often
1 / f ␤ noise is described by a long-memory process, characterized by S共f兲 ⬃ 1 / f ␤ as f → 0. An Abelian-Tauberian theorem relating regularly varying tails shows that this longrange dependence property is equivalent to similar behavior
of autocorrelation function C共t兲 as t → ⬁ 关46兴. However, this
behavior of the autocorrelation function is not necessary for
obtaining required form of the power spectrum in a finite
interval of the frequencies which does not include zero
关47–49兴.
From Eq. 共19兲, it follows that if the terms with small ␭
dominate the sum, then one obtains 1 / f 2 behavior of the
spectrum for large frequencies f. If the terms with 1 / f ␤ 共with
␤ ⬍ 2兲 are present in Eq. 共19兲, then at sufficiently large frequencies those terms will dominate over the terms with 1 / f 2
behavior. Since the terms with small ␭ lead to 1 / f 2 behavior
of the spectrum, we can expect to obtain 1 / f ␤ spectrum in a
frequency region where the main contribution to the sum in
Eqs. 共17兲 and 共19兲 is from the large values of ␭. Thus we
need to determine the behavior of the eigenfunctions P␭共x兲
for large ␭. The conditions when eigenvalue ␭ can be considered as large will be investigated below.
1
⳵
S␭共z兲 = 共␩ − 1兲z共␯−␩兲/共␩−1兲 u␭共z兲.
2
⳵z
Therefore, the boundary conditions for Eq. 共21兲, according to
Eq. 共11兲 are u␭⬘ 共1兲 = 0 and u␭⬘ 共␰1−␩兲 = 0. Here, u␭⬘ 共z兲 is the derivative of the function u␭共z兲. The orthonormality relation
共14兲 yields the orthonormality relation for functions u␭共z兲,
1 − ␰1−␯
共␯ − 1兲共1 − ␩兲
P␭共x兲 = x−␯u␭共x1−␩兲.
共20兲
The functions u␭共z兲 with z = x1−␩ obey the equation
1 d
d2
u␭共z兲 = − ␳2u␭共z兲,
2 u␭共z兲 − 共2␣ − 1兲
dz
z dz
共21兲
where the coefficients ␣ and ␳ are
␣=1+
␯−1
,
2共1 − ␩兲
␳=
冑2␭
兩␩ − 1兩
共22兲
.
The area of diffusion of the variable z = x1−␩ is restricted by
the minimum and maximum values zmin and zmax,
zmin =
再
冎
␰1−␩ , ␩ ⬎ 1,
1,
␩ ⬍ 1,
zmax =
再
1,
␰
1−␩
冎
␩ ⬎ 1,
, ␩ ⬍ 1.
␰1−␩
1
z共␩−␯兲/共1−␩兲u␭共z兲u␭⬘共z兲dz = ␦␭,␭⬘ .
The expression 共18兲 for the first moment X␭ of the stochastic
variable x evaluated with the ␭-th eigenfunction becomes
1
X␭ =
1−␩
冕
␰1−␩
z1+␩−␯/1−␩u␭共z兲dz.
共26兲
1
C. Solution of the equation for eigenfunctions
The solutions of Eq. 共21兲 are 关50兴
u␭共z兲 = z␣关c1J␣共␳z兲 + c2Y ␣共␳z兲兴,
共27兲
where J␣共z兲 and Y ␣共z兲 are the Bessel functions of the first
and second kind, respectively. The coefficients c1 and c2
needs to be determined from the boundary and normalization
conditions for function u␭共z兲. The asymptotic expression for
the function u␭共z兲 is
u␭共z兲 ⬇ c␭z␣−1/2␳−1/2 cos共␳z + a兲,
␳z Ⰷ 1.
共28兲
Here, a is a constant to be determined from the boundary
conditions and c␭ is the constant to be determined from the
normalization 共25兲.
The behavior of the power spectral density in Eq. 共19兲 as
1 / f ␤ can be only due to terms with large ␭. Therefore, we
will consider the values of ␭ for which at least the product
␳zmax is large, ␳zmax Ⰷ 1. The first moment of the variable x
in the expression for the autocorrelation function 共17兲 is expressed via integral 共26兲. If the condition ␳z Ⰷ 1 is satisfied
for all z then the function u␭共z兲 has frequent oscillations in
all the region of the integration, and the integral is almost
zero. Consequently, the biggest contribution to the sum in
Eq. 共17兲 makes the terms corresponding to those values of ␭,
for which the condition ␳z Ⰷ 1 is not satisfied for all values
of z between zmin and zmax. Therefore, we will restrict the
values of ␭ by the condition ␳zmin Ⰶ 1. Thus we will consider
eigenvalues ␭ satisfying the conditions
共23兲
The probability current S␭共x兲, Eq. 共10兲, rewritten in terms of
functions u␭, is
冕
共25兲
B. Eigenfunctions of the Fokker-Planck equation
For ␩ ⫽ 1, it is convenient to solve Eq. 共13兲 by writing the
eigenfunctions P␭共x兲 in the form
共24兲
1/zmax Ⰶ ␳ Ⰶ 1/zmin .
共29兲
Explicitly, we have conditions 1 Ⰶ ␳ Ⰶ ␰␩−1 if ␩ ⬎ 1 and
1 / ␰1−␩ Ⰶ ␳ Ⰶ 1 if ␩ ⬍ 1.
031105-3
PHYSICAL REVIEW E 81, 031105 共2010兲
J. RUSECKAS AND B. KAULAKYS
The derivative of the function u␭共z兲, Eq. 共27兲, is
u␭⬘ 共z兲 = ␳z␣关c1J␣−1共␳z兲 + c2Y ␣−1共␳z兲兴.
共30兲
Since we consider the case ␳zmin Ⰶ 1, then, using Eq. 共30兲,
instead of the boundary condition u␭⬘ 共zmin兲 = 0 we can approximately take the condition
lim 关c1J␣−1共y兲 + c2Y ␣−1共y兲兴 = 0.
y→0
If ␣ ⬎ 1, then we get c2 = 0; if ␣ ⬍ 1 then c2 = −c1 tan共␲␣兲.
Using those values of the coefficient c2, we obtain the solutions of Eq. 共21兲
u␭共z兲 ⬇
再
冎
c␭z␣J−␣共␳z兲, ␣ ⬍ 1,
␣
c␭z J␣共␳z兲,
␣ ⬎ 1.
␤=1+
X␭ ⬇
c␭2
1 − ␰1−␯
共␯ − 1兲共1 − ␩兲
1−␯
=
冕
␰1−␩
1
1−␰
共␯ − 1兲 兩1 − ␩兩␳2
c␭2
␳zmax
␳zmin
2
yJ⫾
␣共y兲dy ⬇ 1.
Taking into account the condition ␳zmin Ⰶ 1 and replacing the
lower limit of integration by 0, we obtain that the integral is
approximately equal to
冕
␳zmax
0
2
yJ⫾
␣共y兲dy ⬇
冑
兩1 − ␩兩 ␯ − 1
␲␳ .
zmax 1 − ␰1−␯
共32兲
The expression 共17兲 for the autocorrelation function contains the first moment X␭ of the variable x, expressed as an
integral of the function u␭, Eq. 共26兲. Using Eq. 共31兲, we get
=
Here,
冕
␰1−␩
1
c␭
兩1 − ␩兩␳␤
冕
␳zmin
y ␤−1J⫾␣共y兲dy.
J⫾共␣−1兲共␳zmax兲 = 0.
c␭ 1
.
兩1 − ␩兩 ␳␤
共34兲
共35兲
共36兲
Now we are ready to estimate the power spectral density.
Since ␳zmax Ⰷ 1, from the boundary condition u␭⬘ 共zmax兲
= 0 using the asymptotic expression 共28兲 for the function
u␭共z兲 we obtain the condition sin共␳zmax + a兲 = 0 and ␳zmax
= ␲n − a. Then
共1 − ␩兲2
2
2zmax
共␲n − a兲2 .
共37兲
Equation 共37兲 shows that the density of eigenvalues D共␭兲 is
proportional to 1 / 冑␭. Since the parameter a does not depend
on ␭, it follows that the density of eigenvalues and, consequently, the autocorrelation function do not depend on the
parameter a.
In order to estimate the sum in expression 共17兲 for the
autocorrelation function, we replace summation by the integration,
z␤−1J⫾␣共␳z兲dz
␳zmax
冊
y ␤−2J⫾共␣−1兲共y兲dy .
Therefore, the first term in the expression 共34兲 for X␭ is zero.
If ␤ ⬍ 25 , taking into account that ␳zmax Ⰷ 1, we can extend
the upper limit of integration to +⬁. We get that the integral
for X␭ approximately does not depend on the upper limit of
integration ␳zmax.
Therefore, the first moment X␭ of the variable x is proportional to the expression
␭n ⬇
A. Estimation of the first moment X␭ of the stochastic variable
x
c␭
1−␩
␳zmin
B. Power spectral density
IV. CALCULATION OF THE POWER
SPECTRAL DENSITY
X␭ ⬇
␳zmax
Using expression 共31兲 for the function u␭共z兲, the boundary
conditions u␭⬘ 共zmin兲 = 0 and u␭⬘ 共zmax兲 = 0 leads to
␳zmax
.
␲
Here, we assumed that ␳zmax Ⰷ 1. Therefore, the normalization constant c␭ is
c␭ ⬇
冕
J⫾共␣−1兲共␳zmin兲 = 0,
2
zJ⫾
␣共␳z兲dz
冕
冉
c␭
⫿y ␤−1J⫾共␣−1兲共y兲兩␳␳zzmax
min
兩1 − ␩兩␳␤
⫾ 共␤ + ␣ − 2兲
D. Normalization
Taking ␭ = ␭⬘ from Eq. 共25兲 we get the normalization condition. Using Eq. 共31兲, we have
共33兲
and “+” sign is for ␣ ⬎ 1, while “−” is for ␣ ⬍ 1.
If ⫾␣ + ␤ ⬎ 0, taking into account that ␳zmin Ⰶ 1, we can
replace the lower limit of the integration by 0, X␭
c
⬇ 兩1−␭␩兩 ␳1␤ 兰0␳zmaxy ␤−1J⫾␣共y兲dy. We get that the integral in the
expression for X␭ approximately does not depend on the
lower limit of integration ␳zmin. We can integrate the integral
by parts and use the properties of the Bessel functions to
obtain
共31兲
From approximate solution 共31兲, using asymptotic expression for the Bessel functions, we can determine the parameter a in the asymptotic expression 共28兲. We obtain that the
parameter a depends on ␣ and does not depend on ␳.
␯−3
2共␩ − 1兲
C共t兲 ⬇
冕
e−␭tX␭2 D共␭兲d␭
共38兲
Such a replacement is valid when ␳zmax Ⰷ 1. Using the approximate expressions 共32兲 and 共36兲, we get the expression
for the autocorrelation function
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1 / f NOISE FROM NONLINEAR STOCHASTIC…
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10
PHYSICAL REVIEW E 81, 031105 共2010兲
102
0
101
0
-4
10
P(x) 10-8
S(f)
10-1
-2
10
10-3
-12
10
10-4
(a)
10-16
-1
10
-5
0
1
10
10
10
2
3
10
10
(b)
x
C共t兲 ⬃
冕
−2
zmin
−2
zmax
10
4
10-3 10-2 10-1 100 101 102 103 104
f
␭−␤e−␭td␭
S共f兲 ⬃
−2
−2
= t␤−1关⌫共1 − ␤,zmax
t兲 − ⌫共1 − ␤,zmin
t兲兴.
共39兲
Here, ⌫共a , z兲 = 兰z⬁ta−1e−tdt is the incomplete Gamma function.
2
2
Ⰶ t Ⰶ zmax
we have the following lowest powers in
When zmin
the expansion of the approximate expression 共39兲 for the
autocorrelation function in the power series of t:
C共t兲 ⬃
冦
2共␤−1兲
zmax
tz2共␤−2兲
− max ,
␤−1
␤−2
␤⬎2
2
−2
zmax
+ 共␥ − 1兲t + t ln共zmax
t兲, ␤ = 2
2共␤−1兲
zmax
+ t␤−1⌫共1 − ␤兲,
␤−1
1⬍␤⬍2.
−2
− ␥ − ln共zmax
t兲,
␤=1
1
t
FIG. 1. 共Color online兲 Probability distribution
function P共x兲 共left兲 and power spectral density
S共f兲 共right兲 for the stochastic process defined by
the stochastic differential Eq. 共44兲. Dashed green
lines are analytical expression 共9兲 for the steadystate distribution function P0共x兲 on the left and
the slope 1 / f on the right. Parameters used are
xmin = 1, xmax = 102, and ␴ = 1.
1−␤ ⌫共1
␤⬍1
− ␤兲,
共40兲
Here, ␥ ⬇ 0.577216 is the Euler’s constant. Similar first
terms in the expansion of the autocorrelation function in the
power series of time t has been obtained in Ref. 关5兴.
Similarly, when ␳zmax Ⰷ 1, replacing in Eq. 共19兲 the summation by the integration, we obtain the power spectral density
S共f兲 ⬇ 4
冕
␭
X2 D共␭兲d␭.
␭2 + ␻2 ␭
共41兲
Equation, similar to Eq. 共41兲 has been obtained in Ref. 关8兴 by
considering a relaxing linear system driven by white noise
关Eq. 共27兲 in Ref. 关8兴兴. Similar equation also has been obtained in Ref. 关14兴 where reversible Markov chains on finite
state spaces were considered. In both Refs. 关8,14兴 the power
spectral density is expressed as a sum or an integral over the
eigenvalues of a matrix describing transitions in the system.
Using the approximate expressions 共32兲 and 共36兲, we get
the equation
S共f兲 ⬃
冕
−2
zmin
−2
zmax
1
␭
␤−1
1
d␭.
␭ + ␻2
2
共42兲
−2
−2
When zmax
Ⰶ ␻ Ⰶ zmin
then the leading term in the expansion
of the approximate expression 共42兲 for the power spectral
density in the power series of ␻ is
再
冎
␻−␤ , ␤ ⬍ 2,
␻−2 , ␤ ⱖ 2.
共43兲
The second term in the expansion is proportional to ␻−2. In
the case of ␤ ⬍ 2, the term with ␻−␤ becomes larger than the
−2
Ⰶ ␻. Therefore, we obtain 1 / f ␤
term with ␻−2 when zmax
spectrum in the frequency interval 1 Ⰶ ␻ Ⰶ ␰2共␩−1兲 if ␩ ⬎ 1
and the frequency interval 1 / ␰2共1−␩兲 Ⰶ ␻ Ⰶ 1 if ␩ ⬍ 1. It
should be noted that time t and frequency ␻ in our analysis
are dimensionless.
Equation 共41兲 shows that the shape of the power spectrum
depends on the behavior of the eigenfunctions and the eigenvalues in terms of the function X␭2 D共␭兲. This function
X␭2 D共␭兲 should be proportional to ␭−␤ in order to obtain 1 / f ␤
behavior. Similar condition has been obtained in Ref. 关14兴.
Equation 共13兲 for discrete time process and the unnumbered
equation after Eq. 共34兲 for a continuous time process in Ref.
关14兴 are analogous to the condition X␭2 D共␭兲 ⬃ ␭−␤ since the
density of eigenvalues in Ref. 关14兴 is proportional to
1 / 兩␥⬘共x兲兩. Equations 共39兲 and 共42兲 for the power spectral
density and autocorrelation function resembles those obtained from the sum of Lorentzian signals with appropriate
weights in Ref. 关27兴.
V. NUMERICAL EXAMPLES
If ␯ = 3, we get that ␤ = 1 and stochastic differential Eq. 共1兲
should give signal exhibiting 1 / f noise. We will solve numerically two cases: ␩ = 25 ⬎ 1 and ␩ = − 21 ⬍ 1. For the numerical solution, we use Euler-Marujama approximation,
transforming differential equations to difference equations.
Equation 共44兲 with ␩ = 5 / 2 was solved using variable step of
integration, solution Eq. 共45兲 with ␩ = −1 / 2 was performed
using a fixed step of integration.
When ␩ = 25 and ␯ = 3 then Eq. 共1兲 is dx = ␴2x4dt
+ ␴x5/2dW. Using exponential restriction of the diffusion region, we have the equation
冉
dx = ␴2 1 +
冊
1 xmin 1 x
−
x4dt + ␴x5/2dW.
2 x
2 xmax
共44兲
The equation was solved using the variable step of integration, ⌬tk = ␬2 / x3k , with ␬ Ⰶ 1 being a small parameter. The
steady-state probability distribution function P0共x兲 and the
power spectral density S共f兲 are presented in Fig. 1. We see a
good agreement of the numerical results with the analytical
expressions. The 1 / f interval in the power spectral density in
Fig. 1 is approximately between f min ⬇ 2 ⫻ 10−1 and f max
⬇ 2 ⫻ 102. The width of this region is much narrower than
031105-5
PHYSICAL REVIEW E 81, 031105 共2010兲
J. RUSECKAS AND B. KAULAKYS
107
10
6
0
10
5
10
-2
10
P(x)
S(f)
-4
10
104
103
102
101
-6
10
100
(a)
1
10
x
100
(b)
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
FIG. 2. 共Color online兲 Probability distribution
function P共x兲 共left兲 and power spectral density
S共f兲 共right兲 for the stochastic process defined by
the stochastic differential Eq. 共45兲. Dashed green
lines are analytical expression 共9兲 for the steadystate distribution function P0共x兲 on the left and
the slope 1 / f on the right. Parameters used are
xmin = 1, xmax = 102, and ␴ = 1.
f
In summary, we derived the behavior of the power spectral density from the nonlinear stochastic differential equation. In Refs. 关29,30兴 only the values of the exponent of the
multiplicative noise ␩ greater than 1 has been used. Here, we
showed that it is possible to obtain 1 / f ␤ noise from the same
nonlinear SDE for ␩ ⬍ 1, as well. The analysis reveals that
the power spectrum may be represented as a sum of the
Lorentzian spectra with the coefficients proportional to the
squared first moments of the stochastic variable evaluated
with the appropriate eigenfunctions of the corresponding
Fokker-Planck equation. Nonlinear SDE, corresponding to a
particular case of Eq. 共1兲 with ␩ = 0, i.e., with linear noise
and nonlinear drift, was considered in Ref. 关9兴. It was found
that if the damping is decreasing with increase of 兩x兩, then the
solution of such a nonlinear SDE has long correlation time.
As Eq. 共41兲 shows, the shape of the power spectrum depends on the behavior of the eigenfunctions and the eigenvalues in terms of the function X␭2 D共␭兲, where D共␭兲 is the
density of eigenvalues. The SDE 共3兲 considered in this article
gives the density of eigenvalues D共␭兲 proportional to 1 / 冑␭.
One obtains 1 / f ␤ behavior of the power spectrum when this
function X␭2 D共␭兲 is proportional to ␭−␤ for a wide range of
eigenvalues ␭, as is the case for SDE 共3兲. Similar condition
has been obtained in Ref. 关14兴.
One of the reasons for the appearance of the 1 / f ␤ spectrum is the scaling property of the stochastic differential Eq.
共1兲: changing the stochastic variable from x to x⬘ = ax
changes the time-scale of the equation to t⬘ = a2共1−␩兲t, leaving
the form of the equation invariant. From this property it follows that it is possible to eliminate the eigenvalue ␭ in Eq.
共13兲 by changing the variable from x to z = ␭1/2共1−␩兲x. The
dependence of the eigenfunction on eigenvalue ␭ then enters
only via the boundary conditions. Such scaling properties
were used estimating the norm of the eigenfunction and the
first moment X␭ of the stochastic variable x evaluated with
the ␭-th eigenfunction. Other factor in obtaining the powerlaw spectrum is wide range of the region of diffusion of the
stochastic variable x.
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When ␩ = −1 / 2 and ␯ = 3 then Eq. 共1兲 is
dx = − 2
␴2
␴
2 dt + 冑 dW.
x
x
共45兲
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= 100. The equation was solved with a constant step of integration. The steady-state probability distribution function
P0共x兲 and the power spectral density S共f兲 are presented in
Fig. 2. The 1 / f interval in the power spectral density in Fig.
2 is approximately between f min ⬇ 10−6 and f max ⬇ 2 ⫻ 10−4.
The width of this region is much narrower than the width of
the region 10−6 Ⰶ 2␲ f Ⰶ 1 共␰ = 102兲 predicted in the previous
section.
Numerical solution of the equations confirms the presence
of the frequency region for which the power spectral density
has 1 / f ␤ dependence. The width of this region can be increased by increasing the ratio between minimum and maximum values of the stochastic variable x. In addition, the
region in the power spectral density with the power-law behavior depends on the exponent ␩: if ␩ = 1 then this width
is zero; the width increases with increasing the difference
兩␩ − 1兩. However, the estimation of the width of the region,
obtained in the previous section, is too broad, the width obtained in numerical solutions is narrower. Such a discrepancy
can be explained as the result of various approximations,
made in the derivation.
VI. DISCUSSION
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1 / f NOISE FROM NONLINEAR STOCHASTIC…
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
关25兴
关26兴
关27兴
关28兴
关29兴
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