7. ANALYSIS AND PROCESSING OF RANDOM
PROCESSES
Basic Calculus of Random processes (Mean Square Calculus)
Stochastic continuity
A random process
X(t )
is said to be continuous in mean square or mean square continuous if:
 lim

0
E


X
 t
  

X
  
2


0
We can write the expression under the limes as:
E




E
X
R
X

X
 t
 t
2


 t



, t





X

 
 
2

X

2
2

 t


R
X

 t
  
 
, t



X
2
R
X
  
 
The random process
X(t )
is continuous if, and only if, its autocorrelation function
R
X
(t
1
,t
2
) is continuous.
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Note: The mean square continuity of does not imply that the sample functions of are continuous, e.g. Poisson process is mean square continuous, but sample functions of the Poisson process have a countably infinite number of discontinuities.
For the Poisson process X(t ) with the rate

> 0 , we have:
E


X
 t
  

X
  
2

   

    
2



0
0
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If
X (t )
is WSS process, then it is mean square continuous only, and only if, its autocorrelation function
R
X
( continuous for

= 0
.

) is
For WSS process we can write the previous expression as:


E


X
R
X
2 R
X
 t
 t


 






2
( t
R
X
X

 

 
)


2



2 R
X
 t
   t


R
X
 
 lim

0
E


X
 t
  

X
  
2


 lim

0

R
X
 

R x
  

0
Stochastic Processes – Analysis and Processing
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If
X (t )
is a mean square continuous process, then its mean is also continuous (but not mean square continuous):
 lim

0

X
( t
 
)
 
X
( t ) which can be written as
 lim

0
E

X
 t
 
 

E
 lim

)
X
 t
 




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Stochastic derivatives
The random process
X (t )
has a mean square derivative X

(t )
if l

.
i .
m

0
X
 t
  

X
 


X

( t ) where l.i.m
denotes limit in the mean (square), that is
 lim

0
E







X
 t
  

X
 


X

( t )


 2





0
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The mean and the autocorrelation function of
X

(t )
are given by:
E

X
   
 d dt
E

X
  
  
X
 
R
X
 t
1 t
2

E

X
    
1 2


 2
R
X
 
1
 t
1
 t
2
2
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Stochastic integrals
A mean square integral of a random process
X (t )
is defined by:
Y t
1  

X
  d
 t
0
  l .
 t i i .
m

0 where t
0
 t
1

Stochastic Processes – Analysis and Processing
  t and
 t i
7-7
 i
X
  i
 t i
 t i

1
 t i

 t
 t
0

X
  d

R
Y
 t
1
, t
2


E

 t
 t
1
0
X
  d
 t
2
X
  d
 t
0




0 t t
1   2 t t
0
E

X
     d
 d

7-8 Stochastic Processes – Analysis and Processing

0 t t
1   2 t t
0
R
X
 
,
  d
 d


Power spectral densities
The power spectral density (or just power spectrum
)
S
X
(

) of a continuous time random process
X(t )
is defined as the
Fourier transform of
R
X
(

)
:
S
X



 
R
X
  e
 j
 d

R
X

1
2






S
X
  e
 j
 d

These two relations are known as the Wiener-Khinchin relations .
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White Noise
A continuous time white noise process
W(t )
is a WSS zeromean random process, with the autocorrelation function given by:
R
W
 
 
2

( t )
The power spectrum of the white noise is obtained to be:
S
W
 
2


 
   e
 j
 d
  
2
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Response of a Continuous Time Linear Time-Invariant (CT LTI)
System to Random Inputs
X(t )
CT LTI
System
Y(t )
If input to the system
X(t )
is a random process,, then the output
Y(t )
will also be a random process, given by:
Y
 

 h
   t
   d

 

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Also:
R
Y t
1
,
2
E

Y
 

 h
 
E

X
 t
    d

 

 

 
   h
    
X t
1
 
, t
2
   d
 d

If the input
X(t )
is WSS process, then:
E Y t
 
X



 h
  d
  
X
H
   

 h
  e



 j
 t dt
R
Y




 
   h
     
X
     d
 d

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The power spectrum of
Y(t )
is given by:
S
Y
 


R
Y
  e
 
  j
 d
 
H
 
2
S
X
 
When the autocorrelation function of the output
R
Y
(

)
is required, than it is easier to obtain it from the power spectrum
S
Y
(

)
:
R
Y

2
1






S
Y
  e
 j
 d







H
 
2
S
X
  e
 j
 d

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30
Stochastic Periodicity
A continuous time random process
X(t )
is mean square periodic , with period
T
, if
E


X
 t

T


X
  
2


0
If
X(t )
is WSS, then
X(t )
is mean square periodic if, and only if, its autocorrelation function is periodic with period
T
:
R
X
  
T


R
X
 
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