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RANDOM PROCESSES
In practical problems we deal with time varying waveforms whose value at a time is
random in nature. For example, the speech waveform, the signal received by
communication receiver or the daily record of stock-market data represents random
variables that change with time. How do we characterize such data? Such data are
characterized as random or stochastic processes. This lecture covers the fundamentals of
random processes..
Random processes
Recall that a random variable maps each sample point in the sample space to a point in
the real line. A random process maps each sample point to a waveform.
Consider a probability space {S , F , P}. A random process can be defined on {S , F , P} as
an indexed family of random variables {X (s, t ), s  S,t } where  is an index set which
may be discrete or continuous usually denoting time. Thus a random process is a function
of the sample point  and index variable t and may be written as X (t ,  ).
Remark

For a fixed t ( t 0 ), X (t 0 ,  ) is a random variable.

For a fixed  (  0 ), X (t ,  0 ) is a single realization of the random process and
is a deterministic function.

For a fixed  ( 0 ) and a fixed t ( t 0 ), X (t , 0 ) is a single number.

When both t and  are varying we have the random process X (t ,  ).
The random process { X ( s, t ), s  S , t  T } is normally denoted by { X (t )}. Following
figure illustrates a random procee.
A random process is illustrated below.
X (t , s3 )
S
X (t , s2 )
s3
s2
s1
X (t , s1 )
t
Figure Random Process
( To Be animated)
Example Consider a sinusoidal signal X (t )  A cos t where A is a binary random
variable with probability mass functions pA (1)  p and pA (1)  1  p.
Clearly, { X (t ), t } is a random process with two possible realizations X1 (t )  cos t
and X 2 (t )   cos t. At a particular time t0 X (t0 ) is a random variable with two values
cos  t0 and  cos  t0 .
Continuous-time vs. discrete-time process
If the index set  is continuous, { X (t ), t } is called a continuous-time process.
Example Suppose X (t )  A cos(w0 t   ) where A and w0 are constants and  is
uniformly distributed between 0 and 2 . X (t ) is an example of a continuous-time
process.
4 realizations of the process is illustrated below.
(TO BE ANIMATED)
  0.8373
  0.9320
  1.6924
  1.8636
If the index set  is a countable set, { X (t ), t  } is called a discrete-time process.
Such a random process can be represented as  X [n], n  Z  and called a random sequence.
Sometimes the notation  X n , n  0 is used to describe a random sequence indexed by the
set of positive integers.
We can define a discrete-time random process on discrete points of time. Particularly,
we can get a discrete-time random process  X [n], n  Z  by sampling a continuous-time
process { X (t ), t  } at a uniform interval T such that X [n]  X (nT ).
The discrete-time random process is more important in practical implementations.
Advanced statistical signal processing techniques have been developed to process this
type of signals.
Example Suppose X n   2cos(0 n  Y ) where  0 is a constant and Y is a random
variable uniformly distributed between  and -  .
X n is an example of a discrete-time process.
  0.4623
  1.9003
  0.9720
Continuous-state vs. discrete-state process:
The value of a random process X (t ) is at any time t can be described from its probabilistic
model.
The state is the value taken by X (t ) at a time t, and the set of all such states is called the
state space. A random process is discrete-state if the state-space is finite or countable. It
also means that the corresponding sample space is also finite countable. Other-wise the
random process is called continuous state.
Example Consider the random sequence { X n , n  0} generated by repeated tossing of a
fair coin where we assign 1 to Head and 0 to Tail.
Clearly X n can take only two values- 0 and 1. Hence { X n , n  0} is a discrete-time twostate process.
How to describe a random process?
As we have observed above that X (t ) at a specific time t is a random variable and can be
described by its probability distribution function FX (t ) ( x)  P( X (t )  x). This distribution
function is called the first-order probability distribution function. We can similarly
define the first-order probability density function f X (t ) ( x) 
dFX (t ) ( x)
dx
.
To describe { X (t ), t  } we have to use joint distribution function of the random
variables at all possible values of t . For any positive integer n , X (t1 ), X (t 2 ),..... X (t n )
represents
n
jointly distributed random variables. Thus a random process
{ X (t ), t  } can thus be described by specifying the n-th order joint distribution
function
FX (t1 ), X (t2 )..... X (tn ) ( x1 , x2 .....xn )  P( X (t1 )  x1 , X (t2 )  x2 ..... X (tn )  xn ),  n  1 and tn  
or th the n-th order joint density function
f X (t1 ), X (t2 )..... X (tn ) ( x1 , x2 .....xn ) 
n
FX (t1 ), X (t2 )..... X (tn ) ( x1 , x2 .....xn )
x1x2 ...xn
If { X (t ), t  } is a discrete-state random process, then it can be also specified by the
collection of n-th order joint probability mass function
p X (t1 ), X (t2 )..... X ( tn ) ( x1 , x2 .....xn )  P( X (t1 )  x1 , X (t2 )  x2 ..... X (tn )  xn ),  n  1 and tn  
If the random process is continuous-state, it can be specified by
Moments of a random process
We defined the moments of a random variable and joint moments of random variables.
We can define all the possible moments and joint moments of a random process
{ X (t ), t  }. Particularly, following moments are important.

 x (t )  Mean of the random process at t  E ( X (t )

RX (t1 , t2 ) = autocorrelation function of the process at times t1 , t2  E ( X (t 1 ) X (t2 ))
Note that
RX (t1 , t2 ) = RX (t2 , t1 , ) and
RX (t , t )  EX 2 (t )  sec ond moment or mean - square value at time t.

The autocovariance function CX (t1 , t2 ) of the random process at time
t1 and t2 is defined by
C X (t1 , t2 )  E ( X (t 1 )   X (t1 ))( X (t2 )   X (t2 ))
=RX (t1 , t2 )   X (t1 )  X (t2 )
C X (t , t )  E ( X (t )   X (t )) 2  variance of the process at time t .
These moments give partial information about the process.
The ratio  X (t1 , t2 ) 
C X (t1 , t2 )
is called the correlation coefficient.
C X (t1 , t1 ) C X (t2 , t2 )
The autocorrelation function and the autocovariance functions are widely used
to
characterize a class of random process called the wide-sense stationary process.
We can also define higher-order moments
R X (t1 , t 2 , t 3 )  E ( X (t 1), X (t 2 ), X (t 3 )) = Triple correlation function at t1 , t2 , t3 etc.
The above definitions are easily extended to a random sequence { X n , n  0}.
Example
(a) Gaussian Random Process
n jointly
For any positive integer n, X (t1 ), X (t 2 ),..... X (t n ) represent
variables.
These
n
random
variables
define
a
random
random
vector
X  [ X (t1 ), X (t2 ),..... X (tn )]'. The process X (t ) is called Gaussian if the random vector
[ X (t1 ), X (t2 ),..... X (tn )]' is jointly Gaussian with the joint density function given by
f X (t1 ), X (t2 )... X (tn ) ( x1 , x2 ,..., xn ) 

1
 X'CX1 X
e 2
2

where CX  E ( X  μ X )( X  μ X )'
and μ X  E ( X)   E ( X 1 ), E ( X 2 )......E ( X n )  '.
n
det(CX )
The Gaussian Random Process is completely specified by the autocovariance matrix
C X and hence by the mean vector μ X and the autocorrelation matrix R X  EXX ' .
(b) Bernoulli Random Process
A Bernoulli process is a discrete-time random process consisting of a sequence of
independent and identically distributed Bernoulli random variables. Thus the discrete –
time random process { X n , n  0} is Bernoulli process if
P{ X n  1}  p and
P{ X n  0}  1  p
Example
Consider the random sequence { X n , n  0} generated by repeated tossing of a fair coin
where we assign 1 to Head and 0 to Tail. Here { X n , n  0} is a Bernoulli process where
each random variable X n is a Bernoulli random variable with
1
and
2
1
p X (0)  P{ X n  0} 
2
p X (1)  P{ X n  1} 
(c) A sinusoid with a random phase
X (t )  A cos(w0 t   ) where A and w0 are constants and  is uniformly distributed
between 0 and 2 . Thus
1
f  ( ) 
2
X (t ) at a particular t is a random variable and it can be shown that
1


f X ( t ) ( x )    A2  x 2
0

xA
otherwise
The pdf is sketched in the Fig. below:
The mean and autocorrelation of X (t ) :
 X ( t )  EX (t )
 EA cos( w0t   )

  A cos( w0t   )

1
d
2
0
RX (t1 , t2 )  EA cos( w0t1   ) A cos( w0t2   )
 A2 E cos( w0t1   ) cos( w0t2   )
A2
E (cos( w0 (t1  t2 ))  cos( w0 (t1  t2  2 )))
2
A2
A2 
1

cos( w0 (t1  t2 )) 
d
 cos( w0 (t1  t2  2 ))
2
2 
2
A2

cos( w0 (t1  t2 ))
2

Two or More Random Processes
In practical situations we deal with two or more random processes. We often deal with
the input and output processes of a system. To describe two or more random processes
we have to use the joint distribution functions and the joint moments.
Consider two random processes { X (t ), t  } and {Y (t ), t  }. For any positive integers
n and m , X (t1 ), X (t2 ),..... X (tn ), Y (t1/ ), Y (t2/ ),.....Y (tm/ ) represent m  n jointly distributed
random variables. Thus these two random processes can be described by the
(n  m)th order joint distribution function
FX (t ), X (t
1
/
/
/
2 )..... X ( tn ),Y ( t1 ),Y ( t2 ),.....Y ( t m
)
( x1 , x2 .....xn , y1 , y2 ..... ym )
 P( X (t1 )  x1 , X (t2 )  x2 ..... X (tn )  xn , Y (t1/ )  y1 , Y (t 2/ )  y2 .....Y (tm/ )  ym )
or the corresponding (n  m)th order joint density function
f X (t ), X (t
1
2 )..... X
( tn ),Y ( t1/ ),Y ( t2/ ),.....Y ( tm/ )
( x1 , x2 .....xn , y1 , y2 ..... ym )
 2n

F
/
/
/ ( x1 , x2 ..... xn , y1 , y2 ..... ym )
x1x2 ...xn y1y2 ...ym X (t1 ), X (t2 )..... X (tn ),Y ( t1 ),Y (t2 ),.....Y ( t m )
Two random processes can be partially described by the joint moments:
Cross  correlation function of the processes at times t1 , t2

RXY (t1 , t2 )  E ( X (t 1 )Y (t2 ))  E ( X (t 1 )Y (t2 ))
Cross  cov ariance function of the processes at times t1 , t2
C XY (t1 , t2 )  E ( X (t 1 )   X (t1 ))(Y (t2 )  Y (t2 ))

 RXY (t1 , t2 )   X (t1 ) Y (t2 )
.
Cross-correlation coefficient

 XY (t1 , t2 ) 
C XY (t1 , t2 )
C X (t1 , t1 ) CY (t2 , t2 )
On the basis of the above definitions, we can study the degree of dependence between
two random processes
 Independent processes: Two random processes { X (t ), t  } and {Y (t ), t  }.
are called independent if
 Uncorrelated processes: Two random processes { X (t ), t  } and {Y (t ), t  }.
are called uncorrelated if
CXY (t1 , t2 )  0  t1 , t2  
This also implies that for such two processes
RXY (t1 , t2 )   X (t1 ) Y (t2 )
.
 Orthogonal processes: Two random processes { X (t ), t  } and {Y (t ), t  }.
are called orthogonal if
R XY (t1 , t2 )  0  t1 , t2  
Example Suppose X (t )  A cos(w0 t   ) and Y (t )  A sin( w0t   ) where A and w0 are
constants and  is uniformly distributed between 0 and 2 .