Chapter 9
Random Processes
Contents
• Definition of Random Process
• Specifying a Random Process
Definition of Random Process
• Let E be a random experiment specified by the
outcomes  from some sample space S.
• Suppose that to every outcome   S, we assign a
function of time according to some rule:
 X (t ,  )
t  I.
– Realization, sample path, or sample function: the graph
of the function X(t,  ) versus t, for  fixed.
• We can view the outcome of the random experiment as
producing an entire function of time
– For each fixed tk, from the index set I, X(tk,  ) is a
random variable.
Definition of Random Process
(Cont.)
Definition of Random Process
(Cont.)
• Random Process (R.P.) [also, Stochastic Process]
– Be a family (or ensemble) of random variables indexed
by the parameter t, {X(t,  ), t  I}.
– R.P. is typically notated as X(t).
– Discrete-Time r.p.
• The indexed I is a countable set.
– Continuous-Time r.p.
• I is continuous.
Definition of Random Process
(Cont.)
• Example 9.2: Random Sinusoids
– Let  be selected at random from the interval [-1,1].
– Define the continuous-time r.p. X(t,  ) by
X (t ,  )   cos(2t )
   t  .
– The realizations of this r.p. are sinusoids with random
amplitude .
– Random
Amplitude
Definition of Random Process
(Cont.)
• Example 9.2 (Cont.)
– Let  be selected at random from the interval [-, ].
– Let Y(t,  ) = cos(2t + ).
– The realizations of Y(t,  ) are time-shifted versions of
cos(2t).
– Random
Phase
Definition of Random Process
(Cont.)
• Example 9.4
– Let  be selected at random from the interval [-1,1].
– Let X(t,  ) =  cos(2t) - < t < .
– Find the pdf of X0= X(t0,  ).
– If t0 is such that cos(2 t0) = 0,
then X(t0,  ) = 0 for all .
• The pdf of X(t0) is a delta function
of unit weight at x = 0.
– Otherwise, X(t0,  ) is uniformly distributed
in the interval (-cos(2t0), cos(2t0))
since  is uniformly distributed in [-1, 1].
Definition of Random Process
(Cont.)
• Example 9.4 (Cont.)
– Let  be selected at random from the interval [-, ].
– Let Y(t,  ) = cos(2t + ).
– Find the pdf of Y0= Y(t0,  ).
– From Example 4.36, we have
 fY ( y ) 
1
 1 y
2
,
y  1.
– Note that the pdf of Y0= Y(t0,  )
does not depend on t0.
Specifying a Random Process
• Let X1, X2, …, Xk be the k r.v.s obtained by sampling the r.p. X(t,  ) at
time t1, t2, …, tk:
 X1  X (t1 ,  ), X 2  X (t2 ,  ),, X k  X (tk ,  ).
•
Joint CDF of vector r.v. (X1, X2, …, Xk)
 FX1 , X 2 ,, X k ( x1 , x2 ,, xk )  P[ X 1  x1 , X 2  x2 ,, X k  xk ],
for any k and any choice of sampling instants t1 , t2 ,, tk .
• Joint pmf (discrete Xk)
 p X1 , X 2 ,, X k ( x1 , x2 ,, xk )  P[ X1  x1 , X 2  x2 ,, X k  xk ].
• Joint pdf (continuous Xk)
 f X1 , X 2 ,, X n ( x1 , x2 ,, xk ).
Specifying a Random Process (Cont.)
• Mean of a R.P. X(t)
 m (t )  E[ X (t )]   xf

X

X (t )
( x)dx,
where f X (t ) is the pdf of X (t ).
– In general, mX(t) is a function of time.
• Autocorrelation of a R.P. X(t)
– It is defined as the joint moment of X(t1) and X(t2)
 RX (t1 , t2 )  E[ X (t1 ) X (t2 )]  



 
xyf X (t1 ), X (t2 ) ( x, y )dxdy,
where f X (t1 ), X (t2 ) ( x, y ) is the second - order pdf of X (t ).
– In general, the autocorrelation is a function of t1 and t2.
Specifying a Random Process (Cont.)
• Autocovariance of a R.P. X(t)
 C X (t1 , t2 )  E[X (t1 )  mX (t1 )X (t2 )  mX (t2 )]
 RX (t1 , t2 )  mX (t1 )mX (t2 ).
• Variance of a R.P. X(t)
 var[ X (t )]  E[ X (t )  mX (t ) ]  C X (t , t ).
2
• Correlation coefficient of a R.P. X(t)
  X (t1 , t2 ) 
C X (t1 , t2 )
.
C X (t1 , t1 ) C X (t2 , t2 )
Specifying a Random Process (Cont.)
• Example 9.9
– Let X(t) = A cos(2t ).
• A is some r.v.
– Mean of a R.P. X(t) [See Eq. (4.30)]
 mX (t )  E[ X (t )]  E[ A cos(2t )]  E[ A] cos(2t ).
– Autocorrelation of a R.P. X(t)
 R X (t1 , t 2 )  E[ X (t1 ) X (t 2 )]  E[ A cos( 2t1 ) A cos( 2t 2 )]
 E[ A2 ] cos( 2t1 ) cos( 2t 2 ).
– Autocovariance of a R.P. X(t)
 C X (t1 , t 2 )  R X (t1 , t 2 )  m X (t1 )m X (t 2 )


 E[ A2 ]  E[ A]2 cos( 2t1 ) cos( 2t 2 ).
Trigonometric Identities
•
•
•
•
•
•
•
sin(A+B)= sinAcosB + cosAsinB
sin(A-B) = sinAcosB – cosAsinB
sinAcosB = ½ {sin(A+B)+sin(A-B)}
cos(A-B) = cosAcosB + sinAsinB
cos(A+B) = cosAcosB – sinAsinB
cosAcosB = ½ {cos(A-B)+cos(A+B)}
sinAsinB = ½ {cos(A-B) – cos(A+B}
Specifying a Random Process (Cont.)
• Example 9.10
– Let X(t) = cos(t + ).
•  is uniformly distributed in the interval (-, ).
– Mean of a R.P. X(t)
 mX (t )  E[ X (t )]  E[cos(t  )] 
1 
 cos(t   )d  0.
2 
– Autocorrelation and Autocovariance of a R.P. X(t)
 C X ( t1 ,t 2 )  RX ( t1 ,t 2 )  m X ( t1 )m X ( t 2 )  RX ( t1 ,t 2 )  E [cos( t1   ) cos( t 2   )]
1 
cos( t1   ) cos( t 2   )d
2 
1  1
cos( ( t1  t2 ))  cos( ( t1  t2 )  2 )d

2  2
1
 cos( ( t1  t 2 )).
2

Specifying a Random Process:
Multiple Random Process
• Independent
– The processes X(t) and Y(t) are independent, if the vector r.v.s
X=(X(t1), …, X(tk)) and (Y(t1’), …, Y(tj’)) are independent
for all k, j, and all choices of t1, t2, …, tk and t1’, t2’, …, tj’
 FX,Y ( x1 ,, xk ; y1 ,, y j )  FX ( x1 ,, xk ) FY ( y1 ,, y j )
• Cross-correlation of X(t) and Y(t)
 RX ,Y (t1 , t 2 )  E[ X (t1 )Y (t 2 )]
• X(t) and Y(t) are orthogonal (i.e., Orthogonal R.P.s)
if RX,Y(t1, t2) = 0 for all t1 and t2.
Specifying a Random Process:
Multiple Random Process (Cont.)
• Cross-covariance of X(t) and Y(t)
 C X ,Y (t1 , t 2 )  EX (t1 )  m X (t1 )Y (t 2 )  mY (t 2 )
 R X ,Y (t1 , t 2 )  m X (t1 )mY (t 2 )
• X(t) and Y(t) are uncorrelated (i.e., Uncorrelated R.P.s)
if CX,Y(t1, t2) = 0 for all t1 and t2.
Specifying a Random Process:
Multiple Random Process (Cont.)
• Example 9.11
– Let X(t) = cos(t + ) and Y(t) = sin(t + ).
•  is uniformly distributed in the interval (-, ).
– Find the cross-covariance of X(t) and Y(t)
 C X ,Y (t1 , t 2 )  E[X (t1 )  m X (t1 )Y (t 2 )  mY (t 2 )]
 R X ,Y (t1 , t 2 )  m X (t1 )mY (t 2 )
X(t) & Y(t) are not
uncorrelated r.p.s!
But, X(t1) & X(t2) are
uncorrelated r.v.s for t1
and t2 such that
w(t1-t2)=k for any k
 R X ,Y (t1 , t 2 )
 m X (t1 )  0 and mY (t 2 )  0
 E[cos(t1  ) sin(t 2  )]
 E[ 12 sin( (t1  t 2 ))  12 sin( (t1  t 2 )  2)]
  12 sin( (t1  t 2 )).
Specifying a Random Process:
Multiple Random Process (Cont.)
• Example 9.12
– Suppose we observe a process Y(t), which consists of a
desired signal X(t) plus noise N(t); Y(t) = X(t) + N(t)
– Find the cross-correlation between Y(t) and X(t)
assuming that X(t) and N(t) are independent r.p.
 R X ,Y (t1 , t 2 )  E[ X (t1 )Y (t 2 )]
 E[ X (t1 ){ X (t 2 )  N (t 2 )}]
 E[ X (t1 ) X (t 2 )]  E[ X (t1 ) N (t 2 )]
 R X (t1 , t 2 )  E[ X (t1 )] E[ N (t 2 )]
 R X (t1 , t 2 )  m X (t1 )m N (t 2 )