RANDOM PROCESSES
Couple of random processes
Definitions
Let us consider an experiment whose result is represented by a couple of random processes X(t) and
Y(t) (for example the components of the seismic motion at the base of 2 piers of a viaduct, the wind
velocities registered by 2 anemometers, the dynamic response of a 2-D.O.F. system). The couple of
the processes X(t) and Y(t) is also called a 2-variate random process.
Let us consider the values x  j  t1  and y j  t 2   j  1, 2,... assumed by the sample functions
x  j  t  and y j  t  of X(t) and Y(t) for t  t1 and t  t 2 (Fig. 1). The set of these values constitutes
the couple of random variables X1  X  t1  and Y2  Y  t 2  . They are characterised by the joint
density function of the second order pXY  x1 , t1; y2 , t 2  . From this it is immediate to derive the
marginal density functions of the first order of X  t1  and Y  t 2  :
Fig. 1
1

pX  x1 , t1    pXY  x1 , t1; y2 , t 2  dy2
(1)


pY  y2 , t 2    pXY  x1 , t1; y2 , t 2  dx1
(2)

From the marginal density functions of the first order, it is immediate to derive the statistical
averages of the first order already defined for each random process.
Statistical averages of the second order
The statistical averages of the second order are the joint indexes of X1  X  t1  e Y2  Y  t 2  ; they
can be derived from the joint density function of the second order pXY  x1 , t1; y2 , t 2  .
In particular, the cross–correlation function of X  t  and Y  t  is defined as:
R XY  t1 , t 2   E  X  t1  Y  t 2   




 
x1 y 2 p XY  x1 , t1 ; y 2 , t 2  dx1dy 2
(3)
The cross–covariance function is defined as:
CXY  t1 , t 2   E   X  t1    X  t1    Y  t 2    Y  t 2   






 x1   X  t1    y 2   Y  t 2   p XY  x1 , t1; y 2 , t 2  dx1dy 2
(4)
It derives:
CXY  t1 , t 2   R XY  t1, t 2   X  t1  Y  t 2 
(5)
The normalised cross-covariance function is defined as:
XY  t1 , t 2  
CXY  t1 , t 2 
 X  t1   Y  t 2 
(6)
The prefix “cross-” indicates that the random variables X  t1  and Y  t 2  are extracted from
different random processes (although associated with the same experiment).
Eqs. (3), (4) and (6) involve the following properties:
R XY  t1 , t 2   R YX  t 2 , t1 
CXY  t1 , t 2   CYX  t 2 , t1 
(7)
XY  t1 , t 2   YX  t 2 , t1 
Two random processes X  t  and Y  t  are defined as not correlated if:
R XY  t1, t 2   X (t1 )Y (t 2 ) t1, t 2  R
(8)
2
CXY  t1 , t 2   0 t1 , t 2  R
(9)
Two random processes X  t  and Y  t  are defined as orthogonal if:
R XY  t1 , t 2   0 t1, t 2  R
(10)
Stationary processes
A couple of random processes is defined as weakly stationary if the density functions of the first
order and the joint density functions of the second order are independent of any translation  of the
origin of the axis of time:
pX  x1 , t1   pX  x1 , t1  
(11a)
pY  y2 , t 2   p Y  y 2 , t 2   
(11a)
pXY  x1 , t1; y2 , t 2   pXY  x1, t1  ; y2 , t 2   
(11b)
Assigning   t1 , it is immediate to show that Eq. (11) involves the following properties:
(a) the density function of the first order is independent of t 1 ;
(b) the joint density function of the second order depends on only the time interval  t 2  t1  .
Thus, the statistical averages of the first order are independent of time. The statistical averages of
the second order depend on only the time lag   t 2  t1 :
R XY  t1 , t 2   R XY  
CXY  t1 , t 2   CXY  
XY  t1 , t 2   XY  
The cross-correlation function of two (weakly) stationary processes:
R XY     E  X  t  Y  t    
(12)
has several noteworthy properties (Fig. 2):
1) Setting  = 0 in Eq. (12):
R XY  0   E  X  t  Y  t  
2)
(13)
R XY    CXY    XY  XY    XY  XY . Thus, since XY  1, it follows that:
XY  XY  R XY     XY  XY   R
(14)
3) For  tending to infinite the couple of random variables X  t  , X  t   tends to become not
correlated  XY  0  . Thus:
3
lim R XY      X  Y
(15)
 
4) Setting t  t   , Eq. (11) becomes R XY     E  X  t    Y  t   . Thus it results:
R XY    R YX  
(16)
Fig. 2
The cross-covariance function of two (weakly) stationary processes:
CXY     E X  t    X Y  t      Y 
(17)
has properties analogous to the cross-correlation function (Fig. 3):
CXY  0   E X  t    X Y  t    Y 
(18)
XY  CXY     XY   R
(19)
lim CXY     0
(20)
CXY    CYX  
(21)
 
4
Fig. 3
If X  t  and Y  t  are not correlated then, due to Eq. (9):
CXY    0   R
(22)
If X  t  and Y  t  are orthogonal then, due to Eq. (10):
R XY     0   R
(23)
Bi-variate normal process
Two random stationary processes X(t) and Y(t) have a bi-variate normal distribution if their joint
density function of the second order is given by the relationship:
p XY  x, y;   
1
2
2X Y 1  XY
()

 2  x   2  2   ()  x     y      2  y   2 
 Y
X
X Y XY
X
y
X
y

 exp 

2 2
2
2X Y 1  XY () 




(24)
where  X and  Y are the means of X(t) and Y(t),  2X e  2Y are the variances, XY () is the
normalised cross-covariance function.
5
Cross-power spectral density
Let us consider a couple of stationary random processes with zero mean; in this case the crosscorrelation function RXY() coincides with the cross-covariance function CXY(). The cross-power
spectral density, or more simply the cross-power spectrum, SXY() of X(t) andY(t), is defined as:
SXY   
1 
CXY    e  i d


2
(25)
Unless the factor 1/2, it coincides with the Fourier transform of the cross-covariance function
CXY(). Thus, the cross-covariance function is the inverse Fourier transform (unless the factor 2)
of the cross-power spectral density SXY():

CXY      SXY   ei d

(26)
Also the Eqs. (25) and (26) are known as the Wiener-Khintchine equations.
SXY() exists if CXY() is absolutely integrable:



CXY    d  
Since CXY() is in general a non symmetric function, SXY() is in general a complex function. As
such it can be expressed as:
SXY    SCXY    iSQXY  
(27)
where SCXY ()  Re SXY () is referred to as the co-spectrum and SQXY ()  Im SXY () is referred
to as the quad-spectrum.
Let us rewrite CXY() as:
CXY    
1
1
CXY     CXY      C XY     C XY    
2
2
(28)
where the two terms in the brackets are, respectively, symmetric and anti-symmetric functions of .
Let us execute the Fourier transform of both terms. It follows:
SCXY   
1 
CXY     CXY     e  i d
4  
(29)
SQXY   
1 
CXY     CXY     e  i d


4
(30)
SCXY   
1 
CXY     CXY     cos  d
4  
(31)
SQXY   
1 
CXY     CXY     sin  d
4  
(32)
6
Eqs. (29) and (30) demonstrate that SCXY   is the Fourier transform (unless the factor 4) of the
symmetric part of CXY    ; SQXY   is the Fourier transform (unless the factor 4) of the antisymmetric part of CXY    . Thus, SXY () is real if CXY () is symmetric.
Eqs. (31) and (32) lead to the relationships:
SCXY    SCXY  
(33)
SQXY    SQXY  
(34)
So, SCXY   and SQXY   are, respectively, symmetric and anti-symmetric functions of .
Dealing with SYX   in an analogous way it results:
SCXY    SCYX  
(35)
SQXY    SQYX  
(36)
from which it derives:
SXY    S*YX  
(37)
where S* is the complex conjugate of S.
Coherence, correlation and dependence
The coherence function of two stationary processes X(t) and Y(t) is defined as:
 XY   
SXY  
(38)
SXX   SYY  
The real and the imaginary part of the coherence function are referred to as, respectively, the cocoherence and the quad-coherence:

C
XY

Q
XY
   Re   XY () 
   Im   XY () 
SCXY  
SXX   SYY  
SQXY  
SXX   SYY  
(39)
(40)
Thus:
 XY ()   CXY ()  i QXY ()
(41)
Once introduced the coherence function, the cross-power spectral density assumes the form:
7
SXY ()  SXX ()SYY ()  XY ()
(42)
which shows that the cross-power spectral density is known through the knowledge of the power
spectral densities of each process and through the coherence function.
The coherence function may be interpreted as the frequency domain counter-part of the normalised
cross-correlation function of the processes X(t) and Y(t). It can be proved that:
0   XY ()  1
(43)
In particular, when:
 XY ()  1
(44)
the processes X(t) and Y(t) are referred to as perfectly correlated at the circular frequency . If such
condition is satisfied for any , then the two processes are referred to as totally or identically
coherent. In this case:
SXY    SXX   SYY  
(45)
Instead, when:
 XY ()  0
(46)
the two processes X(t) and Y(t) are referred to as not correlated at the circular frequency . If such
condition is satisfied for any , then the two processes are referred to as totally or identically
incoherent. In this case:
SXY() = 0
(47)
and, consequently, CXY() = 0.
Two stationary processes X(t) and Y(t) are defined as statistically independent if:
pXY  x1 , t1; y2 , t 2   pX  x1, t1  pY  y2 , t 2 
(48)
for any t1 and t2. It is easy to show that two independent random processes are also not correlated.
The opposite is generally not true; however, if X(t) and Y(t) are normal processes, then the not
correlation implies the independence.
8
Vector of random processes
Definitions
Let us consider an experiment whose result is represented by a vector of n random processes
X  t   X1  t  X2  t  ..Xn  t  (for example the components of the seismic motion at the base of n
piers of a viaduct, the wind velocities registered by n anemometers, the dynamic response of a nD.O.F. system). The vector X(t) is also called a n-variate random process.
Let us consider the values x i( j)  t i  (i = 1,2,..n; j = 1,2,..) assumed by the sample functions x i( j)  t  of
T
Xi(t) for t  t i . The set of these values constitutes n random variables Xi  X  t i  (i = 1,2,..n). They
are characterised by the joint density function of order n pX1X2 ..Xn  x1 , t1; x 2 , t 2 ;..x n , t n  = pX  x, t  ,
where x = x1 x 2 ..x n  is the vector of the state variables and t = t1 t2 ..t n  is the vector of the
times in correspondence of which the random variables are extracted.
If X(t) is a n-variate normal random process, its complete probabilistic definition calls for the
knowledge of only the joint density functions of the second order of all the possible couples of
random variables Xi (t i ), X j (t j ) i, j  1, 2,...n  : p Xi X j  x i , t i ; x j , t j  , i, j  1..n and t i , t j  R .
T
T
Stationary processes
If X(t) is a (weakly) stationary random process, the joint desity functions of the second order of all
the possible couples of random variables Xi (t i ), X j (t j ) i, j  1, 2,...n  do not depend on the instants
ti, tj but only on the time lag  = tj - ti: p Xi X j  x i , t i ; x j , t j  = p Xi X j  x i , x j ;   , i, j  1..n and R .
Let us define as mean vector  X of the stationary process X  t  , a vector whose components  Xi (t)
are the means of the processes Xi  t   i  1,..n  :
 X  E  X(t)    X1  X2 ..  Xn 
T
(49)
Let us define as correlation matrix and covariance matrix of X(t), respectively, the matrices whose
i,j-th terms are the functions R Xi X j    and CXi X j    :
 R X1X1 () R X1X2 () .. R X1Xn () 
R
R X2X2 () .. R X2Xn () 
X 2 X1 ( )
T

R X     E  X  t  X  t     




 R Xn X1 () R Xn X2 () .. R Xn Xn () 
(50)
 CX1X1 () CX1X2 () .. CX1Xn () 
 C () C
.. CX2Xn () 
T
X 2 X1
X 2 X 2 ( )



C X     E  X  t    X  X  t      X  

 



CXn X1 () CXn X2 () .. CXn Xn () 
(51)
9
The on-diagonal terms are, respectively, the auto-correlation functions and the auto-covariance
functions of each process; the off-diagonal terms are, respectively, the cross-correlation functions
and the cross-covariance functions of all the possible couples of the processes.
Eqs. (50) and (51) involve:
R X    CX    XTX
(52)
Due to Eqs. (16) and (21):
R X    RTX  
(53)
CX    CTX  
(54)
Thus, the matrices R X and CX are symmetric for   0 :
R X  0  RTX  0
(55)
CX  0  CTX  0
(56)
It can be shown that R X and CX are semi-positive definite for any value of  ( f R  uT R Xu  0,
f C  v T CX v  0 for u, v  0 ).
Let us define as power spectral density matrix of X(t) the matrix whose i,j-th term is the function
SXi X j   :
 SX1X1 () SX1X2 () .. SX1Xn () 
S () S
.. SX2Xn () 
1 
X 2 X1
X 2 X 2 ()
 i

S X   
CX    e d 


2 


SXn X1 () SXn X2 () .. SXn Xn () 
(57)
 CX1X1 () CX1X2 () .. CX1Xn () 
 C () C
.. CX2Xn () 

X 2 X1
X 2 X 2 ( )
i

CX      SX   e d 





CXn X1 () CXn X2 () .. CXn Xn () 
(58)
The on-diagonal terms of the spectral matrix are the power spectral densities of each process; the
off-diagonal terms are the cross-power spectral densities of all the possible couples of the processes.
Due to Eq. (37) S X   is a Hermitian matrix. It can be proved that it is also semi-positive definite
for any value of . From Eq. (58) it results:
10
 2X1
CX1X2 (0) .. C X1Xn (0) 


CX2 X1 (0)
X2 2
.. CX2Xn (0) 


 = CX  0   S X   d 





X2 n 
CXn X1 (0) CXn X2 (0) ..
(59)
The on-diagonal terms are the variances of each random process; the off-diagonal terms are the
cross-covariance functions (i.e. the cross-correlation functions for a zero mean process) of all the
possible couples of processes, at the time lag  = 0.
n-variate normal process
A random stationary process X(t) has a normal n-variate distribution if all the couples of the random
processes that constitute X(t) have a joint density function of order 2 given by the Eq. (24).
A particular linear transformation
Let Y(t) be a n-variate process linked with the n-variate process X(t) by the relationship:
Y t   AX t 
It can be shown that:
μ Y  Aμ X
CY    ACX    AT
RY    AR X   AT
SY    ASX   AT
11