Decomposition of the instantaneous spectrum of a random system

ARTICLE IN PRESS Signal Processing 90 (2010) 860–865

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Decomposition of the instantaneous spectrum of a random system Lorenzo Galleani Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o

abstract

Article history: Received 29 January 2009 Received in revised form 21 August 2009 Accepted 6 September 2009 Available online 11 September 2009

We characterize the nature of the instantaneous spectrum of a random system. We define the instantaneous spectrum as the Wigner spectrum of the state of the system, and we apply a method which transforms the system in the domain of the Wigner spectrum. By using this approach we prove that the instantaneous spectrum is made by three components. Two components decay with time, while a third stationary component is reached for large times. & 2009 Elsevier B.V. All rights reserved.

Keywords: Random systems Transient Time–frequency Wigner spectrum

1. Introduction

oscillator driven by white noise

Many physical systems can be modeled by the random differential equation dn xðtÞ dn1 xðtÞ dxðtÞ an þ    þ a1 þ a0 xðtÞ ¼ f ðtÞ n þ an1 n1 dt dt dt

ð1Þ

where f ðtÞ is a white Gaussian noise with zero mean and autocorrelation function Rf ðtÞ ¼ E½f ðtÞf  ðt þ tÞ ¼ N0 dðtÞ

ð2Þ

dt

2

þ 2m

dxðtÞ þ o20 xðtÞ ¼ f ðtÞ dt

ð3Þ

where m is the mass of the particle and b the friction. The solution xðtÞ to the Langevin equation is the Ornstein–Uhlenbeck process [2]. Eq. (3) governs also an RC circuit whose input is white Gaussian noise [3]. By setting n ¼ 2 in Eq. (1) we instead obtain the classical harmonic E-mail addresses: [email protected], [email protected]. 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.09.003

ð4Þ

This equation is, for instance, the simplest model for a building subjected to random vibrations. One of the most important ways to characterize random systems described by Eq. (1) is frequency analysis. The power spectrum Sx ðoÞ of the solution xðtÞ can be written as a function of the power spectrum Sf ðoÞ of the input noise [4] Sx ðoÞ ¼ jHðoÞj2 Sf ðoÞ

The operator E½ is the expectation value, and the star sign indicates complex conjugation. By setting n ¼ 1 in Eq. (1) we obtain the Langevin equation [1] dxðtÞ þ bxðtÞ ¼ f ðtÞ m dt

d2 xðtÞ

ð5Þ

where HðoÞ is the transfer function, defined as HðoÞ ¼

1 an ðioÞn þ    þ a1 io þ a0

ð6Þ

and the power spectrum is defined as the Fourier transform of the autocorrelation function Z þ1 1 Sx ðoÞ ¼ pffiffiffiffiffiffi Rx ðtÞeito dt ð7Þ 2p 1 Suppose now that at time t ¼ 0 we turn on the system described by Eq. (1). We assume that the system is stable, or, equivalently, that the roots l1 ; . . . ; ln of the equation n

an l þ    þ a1 l þ a0 ¼ 0

ð8Þ

ARTICLE IN PRESS L. Galleani / Signal Processing 90 (2010) 860–865

have negative real parts, that is Rflk go0, for k ¼ 1; . . . ; n. Given this assumption, for t40 the system undergoes a transient behavior, which eventually reaches a stationary phase for t-1. In the stationary phase the power spectrum is given by Eq. (5), while in the transient phase we expect the frequencies to change with time. We define the transient spectrum to be the time-varying spectrum of the solution xðtÞ for all times. Contrary to the frequency spectrum, which is ordinarily defined through the Fourier transform, there are infinite ways to define a time-varying spectrum. Historically, the field of time–frequency analysis has developed and investigated several definitions of time-varying spectra [5]. Here we use the Wigner spectrum [6] Z þ1 1 W x ðt; oÞ ¼ E½x ðt  t=2Þxðt þ t=2Þeito dt ð9Þ 2p 1

861

decays to zero as time increases. The third component is the time–frequency spectrum generated by the initial conditions. As an example, we apply our method to the Langevin equation, and we characterize the nature of its instantaneous spectrum. 2. Transformation to the Wigner domain The random system described by Eq. (1) can be transformed to the Wigner spectrum domain in an exact analytic way [9–11]. First Eq. (1) is rewritten in polynomial form PðDÞxðtÞ ¼ f ðtÞ

ð15Þ

where PðDÞ ¼ an Dn þ    þ a1 D þ a0

ð16Þ

which is basically the expectation value of the Wigner distribution [7] Z þ1 1 Wx ðt; oÞ ¼ x ðt  t=2Þxðt þ t=2Þeito dt ð10Þ 2p 1

and D ¼ d=dt. Then, by applying the transformation method, Eq. (15) is transformed to the Wigner spectrum domain as [9]

The Wigner spectrum is directly connected to the twotime autocorrelation function Z þ1 1 W x ðt; oÞ ¼ Rx ðt þ t=2; t  t=2Þeito dt ð11Þ 2p 1

where the star sign indicates complex conjugation of the coefficients a0 ; a1 ; . . . ; an , and A and B are the operators

where

Eq. (17) is a partial differential equation in time and frequency, whose forcing term is the Wigner spectrum of the original forcing term f ðtÞ, and whose solution is the Wigner spectrum of the solution xðtÞ. By expanding the term P ðAÞPðBÞ we can rewrite Eq. (17) as



Rx ðt1 ; t2 Þ ¼ E½xðt1 Þx ðt2 Þ

ð12Þ

The variance of the random process xðtÞ can be recovered from its Wigner spectrum. We first note that Z þ1 W x ðt; oÞ do ¼ E½jxðtÞj2  ð13Þ

P  ðAÞPðBÞW x ðt; oÞ ¼ W f ðt; oÞ

A¼

b2n

1

and, if the process xðtÞ has zero mean, we immediately obtain its variance Z þ1 s2x ðtÞ ¼ W x ðt; oÞ do ð14Þ

B¼

1@ þ io 2 @t

ð18Þ

@2n @ W x ðt; oÞ þ    þ b1 W x ðt; oÞ þ b0 W x ðt; oÞ ¼ W f ðt; oÞ @t @t2n

ð19Þ where the coefficients b0 ; b1 ; . . . ; b2n are a function of a0 ; a1 ; . . . ; an and of the frequency o. It can be proved that the coefficient b0 is given by [12]

1

Therefore, the larger the area of the Wigner spectrum at time t, the bigger the variance of the process at that time. We point out that, as happens in general with many time–frequency distributions, the Wigner spectrum can take negative values, but since this phenomenon is strictly local, it is accepted as a minor drawback. In addition, the Wigner spectrum has many properties that make it an ideal tool for the characterization of random signals and systems. In [8], for example, the Wigner spectrum is applied to the harmonic oscillator given in Eq. (4), both with constant and time-varying coefficients. The analysis is carried out assuming the transient to be over, and the nonstationarities are generated by the time-varying nature of the system parameters. The results show that the Wigner spectrum describes the dynamic of the system in a very effective way. In this article we investigate the nature of the transient spectrum, and we show that it can be always decomposed in three components. The first component is a stationary spectrum, which does not vary with time, and which equals, up to a constant, the classical power spectrum. The second component is a spectrum which exponentially

1@  io; 2 @t

ð17Þ

b0 ¼

1 jHðoÞj2

ð20Þ

Since Eq. (19) does not contain any derivative with respect to o, we can solve it as an ordinary differential equation. In particular, by using Eq. (9), the Wigner spectrum of the white Gaussian noise f ðtÞ is readily obtained W f ðt; oÞ ¼

N0 uðtÞ 2p

where uðtÞ is the Heaviside step function ( 1; tZ0 uðtÞ ¼ 0; to0

ð21Þ

ð22Þ

Substitution in Eq. (19) gives the system equation in the Wigner spectrum domain b2n

@2n @ N0 W x ðt; oÞ þ    þ b1 W x ðt; oÞ þ b0 W x ðt; oÞ ¼ uðtÞ @t 2p @t2n

ð23Þ When Eq. (1) is solved with a set of non-zero initial conditions x0 ¼ xð0Þ;

x1 ¼ xð1Þ ð0Þ; . . . ; xn1 ¼ xðn1Þ ð0Þ

ð24Þ

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L. Galleani / Signal Processing 90 (2010) 860–865

where k

xðkÞ ðtÞ ¼

d xðtÞ dt

ð25Þ

k

we have to add a term to the solution of Eq. (23). We first indicate by xC ðtÞ the complete solution to Eq. (1) when the initial conditions are taken into account, and by xI ðtÞ the solution to the homogeneous equation PðDÞxI ðtÞ ¼ 0

ð26Þ

solved with the initial conditions given by Eq. (24). By using the linearity of Eq. (15), it is xC ðtÞ ¼ xðtÞ þ xI ðtÞ

ð27Þ

where xðtÞ is the solution to Eq. (15) with zero initial conditions, which can be written as [13] Z t xðtÞ ¼ hðt  t 0 Þf ðt 0 Þ dt 0 ð28Þ 0

where hðtÞ is the impulse response, obtained by solving the equation PðDÞhðtÞ ¼ 0

ð29Þ

with the initial conditions [14] hð0Þ ¼ hð1Þ ð0Þ ¼    ¼ hðn2Þ ð0Þ ¼ 0;

hðn1Þ ð0Þ ¼

1 an

ð30Þ

For simplicity, we consider the poles l1 ; . . . ; ln , defined by Eq. (8), to be distinct, which implies that the solution to Eq. (26) can be written in general as xI ðtÞ ¼ uðtÞ

n X

ck elk t

ð31Þ

k¼1

By setting the initial conditions of Eq. (24) on xI ðtÞ, we obtain

Kc ¼ x0 where 2 3 c1 6 c2 7 6 7 c ¼ 6 7; 4 ^ 5 cn 2

1 6 l 6 1 K¼6 6 ^ 4

ln1 1

ð32Þ 2

x0

3

7 6 6 x1 7 7 x0 ¼ 6 6 ^ 7; 5 4 xn1 3  1 ... ln 7 7 7 ^ 7 5 n1    ln

At t ¼ 0 we randomly apply an initial displacement and velocity to the mass, which correspond to setting the initial conditions x0 ¼ xð0Þ and x1 ¼ xð1Þ ð0Þ, respectively. At t40 the mass is free to oscillate, and it is only subjected to the random force f ðtÞ. If, for instance, the harmonic oscillator is used to model a one-story building, then the driving force f ðtÞ can represent the ambient noise generated by the traffic nearby. The mechanism we adopt to set the initial displacement and velocity of the system at t ¼ 0, does not depend in general, to the random force that drives the oscillation for t40. This physical independence is modeled by the statistical independence of the initial conditions x0 ; . . . ; xn1 and of the driving force f ðtÞ. As a consequence, the Wigner spectrum of Eq. (27) becomes W C ðt; oÞ ¼ W x ðt; oÞ þ W I ðt; oÞ

where W C ðt; oÞ and W I ðt; oÞ are the Wigner spectra of xC and xI ðtÞ, respectively. To obtain Eq. (35), we make use of the orthogonality between xðtÞ and xI ðtÞ, which is based on the relationship E½cxðtÞ ¼ 0

ð36Þ

To prove this relationship, we first replace Eqs. (28) and (34) Z t hðt  t 0 ÞE½x0 f ðt 0 Þ dt0 ð37Þ E½cxðtÞ ¼ K1 0

and then we note that E½x0 f ðtÞ ¼ E½x0 E½f ðtÞ ¼ 0

ð38Þ

which holds since the initial conditions x0 and the forcing term f ðtÞ are statistically independent, and f ðtÞ has zero mean. The Wigner distribution of xI ðtÞ from Eq. (31) can be evaluated in an exact analytic form W I ðt; oÞ ¼

1

p þ

uðtÞ

n X

E½jck2 je2ak t

k¼1

2

p

uðtÞ

sin½2tðbk  oÞ bk  o

n1 X n X

1

2 2 k¼1 l¼kþ1 ðak  al Þ þ ðbk þ bl  2oÞ

 ½E½Rfck gRfcl g þ Ifck gIfcl g  ½ðe2al t cos2ðbl  oÞt

ð33Þ

 e2ak t cos2ðbk  oÞtÞðal  ak Þ þ ðe2al t sin2ðbl  oÞt þ e2ak t

Therefore c ¼ K1 x0

ð35Þ

 sin2ðbk  oÞtÞðbk þ bl  2oÞ ð34Þ

þ E½Rfck gIfcl g  Rfcl gIfck g

Since the poles are generally complex, then also the coefficients c1 ; . . . ; cn are, in general, complex. We take the initial conditions x0 ; . . . ; xn1 to be random, and we assume that they are statistically independent from the forcing term f ðtÞ. To explain the meaning of this assumption, we consider the case of a mass–spring oscillator subjected to viscous damping, which is described by Eq. (4). In this system, the quantity xðtÞ represents the displacement of the mass with respect to the reference position x ¼ 0. When the mass is in the reference position, the restoring force of the spring is zero.

 ½ðe2al t cos2ðbl  oÞt  e2ak t  cos2ðbk  oÞtÞðbk þ bl  2oÞ  ðe2al t sin2ðbl  oÞt þ e2ak t sin2ðbk  oÞtÞðal  ak Þ where

ð39Þ

lk ¼ ak þ ibk

ð40Þ

Summarizing, the Wigner spectrum W C ðt; oÞ of the complete solution is the sum of the Wigner spectrum

ARTICLE IN PRESS L. Galleani / Signal Processing 90 (2010) 860–865

W x ðt; oÞ obtained by solving Eq. (19) with zero initial conditions, and of the Wigner spectrum W I ðt; oÞ corresponding to the initial conditions given by Eq. (39).

Laplace transform lim W D ðs; oÞ ¼ limsW D ðs; oÞ

t-1

¼

The transient spectrum of a random system can be decomposed into the sum of two parts, if the initial conditions are zero, or three parts, if the initial conditions are non-zero. We discuss the two cases separately. Zero initial conditions: when the initial conditions are zero, the transient spectrum of the random system described by Eq. (1) can be always written as the sum of two parts

¼0

ð41Þ

where W S ðt; oÞ is the stationary spectrum, and W D ðt; oÞ the decaying spectrum. To prove such decomposition, we write the transient spectrum by taking the Laplace transform1 of Eq. (23) W x ðs; oÞ ¼

N0 1 2p s½b2n s2n þ    þ b1 s þ b0 

1 ¼ jHðoÞj2 b0

ð43Þ

ð44Þ

where b0 has been substituted from Eq. (20). The stationary spectrum is therefore given by W S ðs; oÞ ¼

N0 1 jHðoÞj2 2p s

ð45Þ

which in the time domain corresponds to 1 W S ðt; oÞ ¼ pffiffiffiffiffiffi uðtÞSx ðoÞ 2p

ð46Þ

where we have used Eq. (5). (We note that, to keep the notation simple, we use W S ðs; oÞ to indicate the Laplace transform of the Wigner spectrum W S ðt; oÞ on the time variable only.) Therefore the stationary spectrum is constant with time, for t40. The decaying spectrum is given by W D ðt; oÞ ¼

N0 NðsÞ 2p b2n s2n þ    þ b1 s þ b0

ð49Þ ð50Þ

This result holds if b0 a0, which is always true. From Eq. (20), in fact, we see that b0 -0 only when jHðoÞj-1, which never happens, since all the poles l1 ; . . . ; ln have negative real part. Moreover, the final value theorem can be applied because the poles of the rational function Q ðsÞ ¼

NðsÞ b2n s2n þ    þ b1 s þ b0

ð51Þ

have negative real parts [12]. Non-zero initial conditions: when the initial conditions are non-zero, an extra term must be added to the decomposition of the transient spectrum. The Wigner spectrum of the complete solution can be obtained by substituting Eq. (41) into Eq. (35) W C ðt; oÞ ¼ W S ðt; oÞ þ W D ðt; oÞ þ W I ðt; oÞ

where C is a constant and NðsÞ is a polynomial in the complex variable s. The first term corresponds to the stationary spectrum, while the second term is the decaying spectrum. By using the method of partial fractions, we obtain C¼

ð48Þ

ð42Þ

This rational function can be decomposed in the sum of two terms 1 C NðsÞ ¼ þ s b2n s2n þ    þ b1 s þ b0 sðb2n s2n þ    þ b1 s þ b0 Þ

s-0

N0 NðsÞ lims 2p s-0 b2n s2n þ b2n1 s2n1 þ    þ b1 s þ b0

3. Decomposition of the transient spectrum

W x ðt; oÞ ¼ W S ðt; oÞ þ W D ðt; oÞ

863

ð47Þ

The decaying spectrum goes to zero as time increases, as it can be seen by using the final value theorem of the 1 We define the Laplace transform of a function xðtÞ by XðsÞ ¼ xðtÞest dt.

R þ1 0

ð52Þ

where W I ðt; oÞ is the initial condition spectrum given in Eq. (39). It is the interaction between the stationary spectrum, the decaying spectrum and the spectrum of the initial conditions that determines the transient spectrum at any given time. 4. Example: the Langevin equation We consider the Langevin equation, Eq. (3), that we report here for convenience dxðtÞ þ bxðtÞ ¼ f ðtÞ dt

ð53Þ

where we take m ¼ 1 for simplicity. By using Eq. (5) we obtain the power spectrum N0 1 Sx ðoÞ ¼ pffiffiffiffiffiffi 2 2p b þ o2

ð54Þ

In Fig. 1 we show the power spectrum, which has the typical Cauchy distribution. The equation for the Wigner spectrum can be obtained by using the method given in Section 2 1 @2 W x ðt; oÞ @W x ðt; oÞ N0 2 þb uðtÞ þ ðb þ o2 ÞW x ðt; oÞ ¼ 2p 4 @t @t2

ð55Þ where we take zero initial conditions. This equation can be solved, for example, with the Laplace transform [15] W x ðt; oÞ ¼

  N0 1 N0 1 b uðtÞ  e2bt cos2ot þ sin2ot uðtÞ 2p b2 þ o2 2p b2 þ o2 o

ð56Þ In Fig. 2 we represent the transient spectrum W x ðt; oÞ. We see that the transient spectrum is zero at t ¼ 0, and, as time increases, it approaches the power spectrum shown in Fig. 1. The first term of Eq. (56) is the stationary

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0.35 0.3 0.15 WS (t, )

Sx ()

0.25 0.2

0.1 0.05

0.15

−10 0

0.1

−5

1

t

−10 −8

−6

−4

−2

0 

2

4

6

8

10

Fig. 1. Power spectrum Sx ðoÞ of the Langevin equation.

5

4

5



10

0.15

−10 1

−5 2 t

3

0 4

5

0.1 0.05 0 0



−10 −5

1

2 t

5 10

Fig. 2. The transient spectrum Wx ðt; oÞ of the Langevin equation when the initial conditions are zero.

0 3

5

4



0.1 0.05

WD (t, )

Wx (t, )

3

Fig. 3. Stationary spectrum W S ðt; oÞ of the Langevin equation. Up to a constant, the stationary spectrum equals, at any given time, the power spectrum Sx ðoÞ shown in Fig. 1.

0.15

0 0

0

2

0.05

5 10

Fig. 4. Decaying spectrum W D ðt; oÞ of the Langevin equation. We represent W D ðt; oÞ since it is mainly positive. The decaying spectrum goes to zero has time increases.

spectrum W S ðt; oÞ ¼

N0 1 uðtÞ 2p b2 þ o2

ð57Þ

which is shown in Fig. 3. We recognize it because it is not a function of time for t40. Up to a constant, the stationary spectrum is equal to the classical power spectrum Sx ðoÞ shown in Fig. 1. The stationary spectrum in Eq. (57) is related to the power spectrum given in Eq. (54) by 1 W S ðt; oÞ ¼ pffiffiffiffiffiffi uðtÞSx ðoÞ 2p

ð58Þ

which explains why the transient spectrum is zero at t ¼ 0. The decaying spectrum approaches zero as time increases. If we consider an initial condition x0 ¼ xð0Þ

ð61Þ

then c1 ¼ x0 in Eq. (39) [14], and the transient W C ðt; oÞ of the complete solution xC ðtÞ is following Eq. (35), by adding the transient W x ðt; oÞ, Eq. (56), to the initial condition W I ðt; oÞ, obtained from Eq. (39) N0 1 N0 1 uðtÞ  e2bt 2p b2 þ o2 2p b2 þ o2   b  cos2ot þ sin2ot uðtÞ

which confirms the general result of Eq. (46). The second term in Eq. (56) is the decaying spectrum   N0 1 b W D ðt; oÞ ¼  e2bt cos2ot þ sin2ot uðtÞ 2 2 p b þ o2 o ð59Þ

W C ðt; oÞ ¼

which is shown in Fig. 4. We see from Eq. (59) that

We note that Eq. (53) has one real pole only

W D ð0; oÞ ¼ W S ð0; oÞ

ð60Þ

spectrum obtained, spectrum spectrum

o

þ

l ¼ b

1

p

x20 uðtÞe2bt

sin2ot

o

ð62Þ

ð63Þ

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WI (t, )

L. Galleani / Signal Processing 90 (2010) 860–865

0.2 0.15 0.1 0.05 0 −0.05 0

−10 1

−5 2 t

3

0 4

5 5



10

WC (t, )

Fig. 5. Wigner spectrum W I ðt; oÞ of the initial conditions of the Langevin equation.

corresponds to the power spectrum obtained from the classical theory of frequency analysis. The understanding of the nature of the transient spectrum is a key problem, firstly because classical frequency analysis is limited to the system in stationary phase, while little knowledge is available when the system exhibits a nonstationary behavior, a common situation in real world systems. The variation with time of the frequency spectrum can instead be investigated with the approach developed in the paper. Moreover, transients are common phenomena in systems, and they happen, for example, when a mechanical or an electrical device is turned on. The corresponding transient is an abrupt phenomenon, which sometimes generate sudden breakdowns. Our method can potentially be the basis for a design technique that can force the instantaneous spectrum to have smooth variations with time, thus minimizing possible system breakdowns.

Acknowledgments

0.25 0.2 0.15 0.1 0.05 0 0

865

−10 1

−5 2 t

0 3

4

5

This work was supported by the PRIN 2007 program. The author thanks the anonymous reviewers for the useful comments. References



5 10

Fig. 6. The transient spectrum W C ðt; oÞ of the complete solution xC ðtÞ to the Langevin equation.

In Fig. 5 we show the initial condition spectrum. At t ¼ 0 the initial condition spectrum is zero, as it can be verified from Eq. (62), while for increasing time the spectrum approaches zero. The Wigner spectrum of the complete solution W C ðt; oÞ is shown in Fig. 6. 5. Conclusions The instantaneous spectrum of the transient of a random system is always made by three parts. First, a stationary component, which equals, up to a constant, the power spectrum of the system. Second, a component which decays exponentially with time. Third, a component due to the initial condition of the system. Also this third component decays with time, and it is not present if the initial conditions are zero. Therefore, the instantaneous spectrum of the transient is the sum of a stationary component, plus two components which decay with time. This fact implies that for large times the instantaneous spectrum approaches the stationary spectrum, which

[1] L. Cohen, The history of noise, IEEE Signal Process. Mag. 22 (2005) 20–45. [2] M.C. Wang, G.E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Mod. Phys. 17 (1945) 323–342. [3] L.O. Chua, C.A. Desoer, E.S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, New York, 1987. [4] A. Papoulis, S.U. Pillai, Probability Random Variables and Stochastic Processes, McGraw-Hill, New York, 2002. [5] L. Cohen, Time–Frequency Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1995. [6] W.D. Mark, Spectral analysis of the convolution and filtering of nonstationary stochastic processes, J. Sound Vib. 11 (1970) 19–63. [7] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749–759. [8] L. Galleani, L. Cohen, Wigner distribution for random systems, J. Mod. Opt. 49 (2002) 2657–2665. [9] L. Galleani, L. Cohen, Direct time–frequency characterization of linear systems governed by differential equations, Signal Process. Lett. 11 (2004) 721–724. [10] L. Galleani, L. Cohen, The Wigner distribution for classical systems, Phys. Lett. A 302 (2002) 149–155. [11] L. Galleani, L. Cohen, Nonstationary stochastic differential equations, in: S. Marshall, G. Sicuranza (Eds.), Advances in Nonlinear Signal and Image Processing, Hindawi, 2006. [12] L. Galleani, Transient analysis of random systems, in: SPIE Fluctuations and Noise, 20–24 May 2007, Florence, Italy. [13] G. Birkhoff, G.C. Rota, Ordinary Differential Equations, Wiley, New York, 1989. [14] A.H. Zemanian, Distribution Theory and Transform Analysis, Dover, New York, 1987. [15] L. Galleani, L. Cohen, Time–frequency characterization of random systems, in: SPIE’s 48th Annual Meeting, 3–8 August 2003, San Diego, CA, USA.