Stationary Random Process
The concept of stationarity plays an important role in solving
practical problems
involving random processes. Just like time-invariance is an important characteristics of
many deterministic systems, stationarity describes certain time-invariant property of a
random process. Stationarity also leads to frequency-domain description of a random
process.
Strict-sense Stationary Process
A random process  X (t ) is called strict-sense stationary (SSS) if its probability
structure is invariant with time. In terms of the joint distribution function,  X (t ) is
called SSS if
FX (t1 ), X (t2 )..., X ( tn ) ( x1 , x2 ..., xn )  FX ( t1 t0 ), X ( t2 t0 )..., X ( tn t0 ) ( x1 , x2 ..., xn )
n  N , t0  and for all choices of sample points t1 , t2 ,...tn .
Thus the joint distribution functions of any set of random variables X ( t1 ), X ( t 2 ), ..., X (t n )
does not depend on the placement of the origin of the time axis. This requirement is a
very strict. Less strict form of stationarity may be defined.
Particularly,
if
FX (t1 ), X (t2 )..... X (tn ) ( x1 , x2 .....xn )  FX (t1 t0 ), X (t2 t0 )..... X ( tn t0 ) ( x1 , x2 .....xn ) for n  1, 2,.., k ,
 X (t ) is called kth order stationary.

If  X (t ) is stationary up to order 1
FX (t1 ) ( x1 )  FX (t1 t0 ) ( x1 ) t 0  T
Let us assume t0  t1 . Then
FX (t1 ) ( x1 )  FX (0) ( x1 ) which is independent of time.
As a consequence
EX (t1 )  EX (0)   X (0)  constant

If  X (t ) is stationary up to order 2
then
FX (t1 ), X (t2 ) ( x1 , x 2 .)  FX (t1 t0 ), X (t2 t0 ) ( x1 , x 2 )
Put t 0  t 2
FX (t1 ), X (t2 ) ( x1 , x2 )  FX (t1 t2 ), X (0) ( x1 , x2 )
This implies that the second-order distribution depends only on the time-lag t1  t2 .
As a consequence, for such a process
RX (t1 , t2 )  E ( X (t1 ) X (t2 ))
 

 xx
1 2
 
f X (0) X ( t1 t2 ) ( x1 , x2 ) dx1 dx2
= RX (t1  t2 )
Similarly,
CX (t1 , t2 )= C X (t1  t2 )
Therefore, the autocorrelation function of a SSS process depends only on the time lag
t1  t2 .
We can also define the joint stationarity of two random processes. Two processes
 X (t ) and Y (t ) are called jointly strict-sense stationary
if their joint probability
distributions of any order is invariant under the translation of time. A complex process
Z (t )  X (t )  jY (t ) is called SSS if  X (t ) and Y (t ) are jointly SSS.
Example An iid process is SSS. This is because, n,
FX (t1 ), X (t2 )..., X (tn ) ( x1 , x2 ..., xn )  FX (t1 ) ( x1 ) FX (t2 ) ( x2 )...FX (tn ) ( xn )
 FX ( x1 ) FX ( x2 )...FX ( xn )
 FX (t1 t0 ), X (t2 t0 )..., X (tn t0 ) ( x1 , x2 ..., xn )
Example The Poisson process is {N (t ), t  0} not stationary, because
EN (t )   t
which varies with time.
Wide-sense stationary process
It is very difficult to test whether a process is SSS or not. A subclass of the SSS process
called the wide sense stationary process is extremely important from practical point of
view.
A random process { X (t )} is called wide sense stationary process (WSS) if
EX (t )   X  constant
and
EX (t1 ) X (t2 )  RX (t1  t2 ) is a function of time lag t1  t2 .
(Equivalently, Cov( X (t1 ) X (t2 ))  C X (t1  t2 ) is a function of time lag t1  t2 )
Remark
(1) For a WSS process { X (t )},
 EX 2 (t )  RX (0)  constant
var( X (t )=EX 2 (t )  ( EX (t )) 2  constant
C X (t1 , t2 )  EX (t1 ) X (t2 )  EX (t1 ) EX (t2 )
 RX (t2  t1 )   X2
 C X (t1 , t2 ) is a function of lag (t2  t1 ).
(2) An SSS process is always WSS, but the converse is not always true.
Example: Sinusoid with random phase
Consider the random process  X (t ) given by
X (t )  A cos(w0 t   ) where A and w0 are constants and  is uniformly distributed
between 0 and 2 .
 This is the model of the carrier wave (sinusoid of fixed frequency) used to
analyse the noise performance of many receivers.
Note that
 1
0    2

f  ( )   2
0 otherwise
By applying the rule for the transformation of a random variable, we get
1

-A  x  A

f X (t ) ( x)   A2  x 2
0 otherwise

which is independent of t. Hence
Note that
 X (t ) is first-order stationary.
EX (t )  EA cos( w0 t   )
2

1
 A cos( w t   ) 2 d
0
0
 0 which is a constant
and
RX (t1 , t2 )  EX (t1 ) X (t2 )
 EA cos( w0 t1   ) A cos( w0 t2   )
2
A
E[c os( w0 t1    w0 t2   )  c os( w0 t1    w0 t2   )]
2
A2

E[c os( w0 (t1  t2 )  2 )  c os( w0 (t1  t2 )]
2
A2

c os( w0 (t1  t2 ) which is a function of the lag t1  t2 .
2

Hence
 X (t ) is wide-sense stationary.
Example: Sinusoid with random amplitude
Consider the random process  X (t ) given by
X (t )  A cos(w0 t   ) where  and w0 are constants and A is a random variable. Here,
EX (t )  EA  cos( w0t   )
which is independent of time only if EA  0.
RX (t1 , t2 )  EX (t1 ) X (t2 )
 EA cos( w0t1   ) A cos(w0t2   )
 EA2  cos( w0t1   ) cos(w0t2   )
1
 EA2  [c os( w0 (t1  t2 )  2 )  c os( w0 (t1  t2 )]
2
which will not be function of (t1  t2 ) only.
Example: Random binary wave
Consider a binary random process  X (t ) consisting of a sequence of random pulses of
duration T with the following features:
1
 Pulse amplitude AK is a random variable with two values p AK (1)  and
2
1
p AK ( 1) 
2


Pulse amplitudes at different pulse durations are independent of each other.
The start time of the pulse sequence can be any value between 0 to T. Thus the
random start time D (Delay) is uniformly distributed between 0 and T.
A realization of the random binary wave is shown in Fig. above. Such waveforms are
used in binary munication- a pulse of amplitude 1is used to transmit ‘1’ and a pulse of
amplitude -1 is used to transmit ‘0’.
The random process X (t) can be written as,
X (t ) 

 A rect
n 
n
(t  nT  D)
T
For any t,
1
1
 (1)( )  0
2
2
1
1
EX 2 (t )  12   (1) 2  1
2
2
EX (t )  1 
Thus mean and variance of the process are constants.
To find the autocorrelation function RX (t1 , t2 ) let us consider the case 0  t1  t1    T .
Depending on the delay D, the points t1 and t2 may lie on one or two pulse intervals.
Case 1:
X (t1 ) X (t2 )  1
Case 2:
X (t1 ) X (t1 )  (1)(1)  1
Case 3:
X (t1 ) X (t2 )  1
Thus,
0  D  t1
1
X (t1 ) X (t2 )  
1
or t2  D  T
t1  D  t2
 RX (t1 , t2 )  EEX (t1 ) X (t2 ) | D
 E ( X (t1 ) X (t2 ) | P(0  D  t1 or t2  D  T ))  E ( X (t1 ) X (t 2 )) | t1  D  t 2 ).P (t1  D  t 2 )
1
1 t t
 t t 
 1  1  2 1   (1   1  ) 2 1
T 
2
2 T

t t
 1 2 1
T
We also have, RX (t2 , t1 )  EX (t2 ) X (t1 )  EX (t1 ) X (t2 )  RX (t1 , t2 )
So that RX (t1 , t2 )  1 
t2  t1
T
t2  t1  T
For t2  t1  T , t1 and t2 are at different pulse intervals.
 EX (t1 ) X (t2 )  EX (t1 ) EX (t2 )  0
Thus the autocorrelation function for the random binary waveform depends on
  t2  t1 , and we can write
RX ( )  1 
1
T
 T
The plot of RX ( ) is shown below.
RX ( )
-T1
T1

Example Gaussian Random Process
Consider the Gaussian process
 X (t ) discussed earlier. For any positive integer
n, X (t1 ), X (t 2 ),..... X (t n ) is jointly Gaussian with the joint density function given by
f X (t1 ), X (t2 )... X (tn ) ( x1 , x2 ,...xn ) 

1
 X'CX1X
e 2
2

n
det(CX )
where CX  E ( X  μ X )( X  μ X ) '
and μ X  E ( X)   E ( X1 ), E ( X 2 )......E ( X n )  '.
If  X (t ) is WSS, then
 EX (t1 )    X 

  
 EX (t2 )    X 
  .. 
μ X  ..

  
..
 .. 
 EX (t )    
n 

 X
 X (t )  

X   X (t1 )   X  
 1
  X (t2 )   X   X (t2 )   X  

 

 ..
 
CX  E  ..



 
 ..
 ..
 




  X (tn )   X   X (tn )   X  


C X (t2  t1 )... C X (tn  t1 ) 
C X (0)


C (t  t )
C X (0)... C X (t2  tn ) 
 X 2 1




C X (0) 
C X (tn  t1 ) C X (tn  t1 )...
We see that f X (t1 ), X (t2 )... X (tn ) ( x1 , x2 ,...xn ) depends on the time-lags. Thus, for a Gaussian
random process, WSS implies strict sense stationarity, because this process is completely
described by the mean and the autocorrelation functions.