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PART 4
Classification of Random Processes
Huseyin Bilgekul
Eeng571 Probability and astochastic Processes
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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1.7 Statistics of Stochastic Processes
•
•
•
•
•
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n-th Order Distribution (Density)
Expected Value
Autocorrelation
Cross-correlation
Autocovariance
Cross-covariance
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First-Order and 2nd-Order Distribution
• First-Order Dustribution
For a specific t, X(t) is a random variable with
first-order distribution function
FX(t)(x) = P{X(t)  x},
The first-order density of X(t) is defined as
f X ( t ) ( x) 
FX (t ) ( x)
x
• 2nd-Order Distribution
FX(t1)X(t2) (x1, x2 ) = P{X(t1)  x1, X(t2)  x2}
 2 F ( x1 , x2 )
f X (t1 ) X (t2 ) ( x1 , x2 ) 
x1x2
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nth-Order Distribution
• Definition
The nth-order distribution of X(t) is the joint
distribution of the random variables X(t1),
X(t2), …, X(tn), i.e.,
FX(t1)…X(tn)(x1, …, xn ) = P{X(t1)  x1,…, X(tn) 
xn}
• Properties
FX ( t1 ) ( x1 )  FX ( t1 ) X ( t2 ) ( x1 , )
f X ( t1 ) ( x1 )  


f X ( t1 ) X ( t 2 ) ( x1 , x2 )dx2
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Expected Value
• Definition：
The expected value of a stochastic process X(t)
is the deterministic function

X(t) = E[X(t)]   x(t ) f X (t ) ( x)dx
• Example 1.10：
If R is a nonnegative random variable, find the
expected value of X(t)=R|cos2ft| .
Sol：
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Autocorrelation
Definition
The autocorrelation function of the stochastic
process X(t) is cf : RX (t1, t2 )  E[ X (t1 ) X (t2 )]
RX(t,) = E[X(t)X(t+ )] ,
The autocorrelation function of the random
sequence Xn is cf : RX [m, n]  E[ X m X n ]
RX[m,k] = E[XmXm+k].
P.S. If X(t) and Xn are complex, then
RX(t,) = E[X(t)X*(t+ )] , RX[m,k] =
E[XmX*m+k],
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Complex Process and Vector Processes
Definitions
The complex process Z(t) = X(t) + jY(t) is
specified in terms of the joint statistics of the
real processes X(t) and Y(t).
The vector process (n-dimensional process) is
a family of n stochastic processes.
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Example 1.11
Find the autocorrelation RX(t,) of the process
X(t) = rcos(t+),
where the R.V. r and  are independent and  is
uniform in the interval (, ).
Sol：
1
RX (t , )  E[r 2 ] cos 
2
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Cross-correlation
Definition
The cross-correlation function of two stochastic
processes X(t) and Y(t) is
RXY(t,) = E[X(t)Y*(t+ )] ,
The cross-correlation function of two random
sequences Xn and Yn is
RXY[m,k] = E[XmY*m+k]
If , RXY(t,) = 0 then X(t) and Y(t) are called
orthogonal.
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Autocovariance
Definition
The autocovariance function of the stochastic
processes X(t) is
CX(t, ) = Cov[X(t), X(t+)]
= E{[X(t) X(t)][X*(t+) *X (t
+)]}
The autocovariance function of the random
sequence Xn is
CX[m, k] = Cov[Xm, Xm+k]
=
E{[XmX(m)][X*m+k*X(m+k)]}
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Example 1.12
The input to a digital filter is an iid random sequence …, X-1,
X0, X1, X2,…with E[Xi] = 0 and Var[Xi] = 1. The output is a
random sequence…, Y-1, Y0, Y1, Y2,…related to the input
sequence by the formula
Yn = Xn + Xn-1 for all integers n.
Find the expected value E[Yn] and autocovariance CY[m, k]
.
Sol：
2  k , k  1,0,1
CY [m, k ]  
otherwise.
0,
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Example 1.13
Suppose that X(t) is a process with
E[X(t)] = 3, RX(t,) = 9 + 4e0.2|| ,
Find the mean, variance, and the covariance of
the R.V. Z = X(5) and W = X(8).
Sol：
CZW (t , )  C X (5) X (8) (5,3)  e
0.6
 2.195
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Cross-covariance
Definition
The cross-covariance function of two stochastic
process X(t) and Y(t) is
CXY(t,) = Cov[X(t), Y(t+)]
= E{[X(t) X(t)][Y*(t+) *Y(t+)]}
The cross-covariance function of two random
sequences Xn and Yn is
CXY[m, k] = Cov[Xm, Ym+k]
= E{[XmX(m)][Y*m+k*Y(m+k)]}
Uncorrelated： CXY(t,) = 0, CXY[m,k] = 0.
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Correlation Coefficient, a-Dependent
Definitions
The correlation coefficient of the process X(t) is
rX (t , ) 
C X (t , )
C X (t ,0)C X (t   ,0)
The process X(t) is called a-dependent if X(t) for
t < t0 and for t > t0+a are mutually independent.
Then we have,
CX(t,) = 0, for | |>a.
The process X(t) is called correlation adependent if CX(t,) satisfies
CX(t,) = 0, for | |>a.
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Example 1.14
(a) Find the autocorrelation RX(t,) if X(t) = aejt.
(b) Find the autocorrelation RX(t,) and

X
(
t
)

a
e

autocovariance CX(t,), if
,
where the random variables ai are
uncorrelated with zero mean and variance i2.
Sol：
j
it
i
i
(b) RX (t , )    i2 e ji ,
i
(a) RX (t , )  E[ a ]e
2
j
C X (t , )  RX (t , )
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Theorem 1.2
The autocorrelation and autocovariance functions of a
stochastic process X(t) satisfy
CX(t,) = RX(t,) X(t) *X(t+)
The autocorrelation and autocovariance functions of the
random sequence Xn satisfy
CX[m,k] = RX[m,k] X(m) *X(m+k)
The cross-correlation and cross-covariance functions of a
stochastic process X(t) and Y(t) satisfy
CXY(t,) = RXY(t,) X(t) *Y(t+)
The cross-correlation and cross-covariance functions of
two random sequences Xn and Yn satisfy
CXY[m,k] = RXY[m,k] X(m) *Y(m+k)
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1.8 Stationary Processes
Definition
A stochastic process X(t) is stationary if and only
if for all sets of time instants t1, t2, …, tm, and any
time difference ,
f X (t1 ),, X (tm ) ( x1 ,, xm )  f X (t1  ),, X (tm  ) ( x1 ,, xm )
A random sequence Xn is stationary if and only if
for any set of integer time instants n1, n2, …, nm,
and integer time difference k,
f X n ,, X n ( x1 ,, xm )  f X n k ,, X n
1
m
1
m k
( x1 ,, xm )
Also called as Strict Sense Stationary (SSS).
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Theorem 1.3
Let X(t) be a stationary random process. For
constants a > 0 and b, Y(t) = aX(t)+b is also a
stationary process.
1
y b
y b
Pf： fY (t ),,Y (t ) ( y1 ,, yn )  a n f X (t ),, X (t ) ( a ,, a )
n
1
1
n
X(t) is stationary
1
n
1
 n f X (t1  ),, X (tn  ) ( y1ab ,, ynab )
a
 fY (t1  ),,Y (t1  ) ( y1 ,, yn )
 Y(t) is stationary.
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Theorem 1.4(a)
For a stationary process X(t), the expected value,
the autocorrelation, and the autocovariance have
the following properties for all t：
(a) X(t) = X
(b) RX(t,) = RX(0,) = RX()
(c) CX(t,) = RX()  X2 = CX()
Pf：
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Theorem 1.4(b)
For a stationary random sequence Xn, the
expected value, the autocorrelation, and the
autocovariance satisfy for all n：
(a) E[Xn] = X
(b) RX[n,k] = RX[0,k] = RX[k]
(c) CX[n,k] = RX[k]  X2 = CX[k]
Pf：D.I.Y.
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Example 1.15
At the receiver of an AM radio, the received signal
contains a cosine carrier signal at the carrier frequency fc
with a random phase  that is a sample value of the
uniform (0,2) random variable. The received carrier
signal is
X(t) = A cos(2 fc t + ).
What are the expected value and autocorrelation of the
process X(t) ?
Sol：
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Wide Sense Stationary Processes
Definition
X(t) is a wide sense stationary (WSS) stochastic
process if and only if for all t,
E[X(t)] = X , and RX(t,) = RX(0,) = RX().
(ref: Example 1.15)
Xn is a wide sense stationary random sequence
if and only if for all n,
E[Xn] = X , and RX[n,k] = RX[0,k] = RX[k].
Q：SSS implies WSS?
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Theorem 1.5
For a wide sense stationary stochastic process X(t),
the autocorrelation function RX() has the following
properties：
RX(0)  0, RX() = RX() and RX(0)  | RX()| .
If Xn is a wide sense stationary random sequence：
RX[0]  0 , RX[n] = RX[n] and RX[0]  |RX[0]|.
Pf：RX (0)  E[ X 2 (t )]  Var[ X (t )]   X2  Var[ X (t   )]   X2
RX ( )  E[ X (t ) X (t   )]  C X ( )   X2
C X ( )
 1,
Var[ X (t )] Var[ X (t   )]
RX ( )   X2  Var[ X (t )]  RX (0)   X2
 RX (0)  RX ( )
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Average Power
Definition
The average power of a wide sense stationary
process X(t) is RX(0) = E[X2(t) ].
The average power of a wide sense stationary
random sequence Xnis RX[0] = E[Xn2].
Example：use x(t) = v2(t)/R=i2(t)R to model the
instantaneous power.
1
X (T ) 
2T

1
X (T ) 
2T
T
2
T
T

T
x(t )dt
x 2 (t )dt
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Ergodic Processes
Definition
For a stationary random process X(t), if the
ensemble average equals the time average, it is
called ergodic.
1 T
 X  lim X (T )  lim
T 
T 
2T 
T
x(t )dt
For a stationary process X(t), we can view the
time average X(T), as an estimate of X .
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Theorem 1.6
Let X(t) be a stationary process with expected
value X and autocovariance CX() . If


X (T ), X (2T ),isan unbiased,
 C X ( ) d ,then
consistent sequence of estimates of X .
Pf： E[ X (T )]  1 E[ T X (t )dt ]  1 T E[ X (t )]dt
2T

T
2T
1

2T

T

T
 X dt  X
unbiased.
consistent sequence: we must show that
T
lim Var[ X (T )]  0.
T 
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Jointly Wide Sense Stationary Processes
Continuous-time random process X(t) and Y(t)
are jointly wide sense stationary if X(t) and Y(t)
are both wide sense stationary, and the crosscorrelation depends only on the time difference
between the two random variables： RXY(t,) =
RXY().
Random Sequences Xn and Yn are jointly wide
sense stationary if Xn and Yn are both wide
sense stationary, and the cross-correlation
depends only on the index difference between
the two random variables ：RXY[m,k] = RXY[k].
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Theorem 1.7
If X(t) and Y(t) are jointly wide sense stationary
continuous-time processes, then
RXY() = RYX().
If Xn and Yn are jointly wide sense stationary
random sequences, then
RXY[k] = RYX[k].
Pf：
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Example 1.16
Suppose we are interested in X(t) but we can observe
only and Y(t) = X(t) + N(t), where N(t) is a noise
process. Assume X(t) and N(t) are independent WSS
with E[X(t)] = X and E[N(t)] = N = 0. Is Y(t) WSS?
Are X(t) and Y(t) jointly WSS ? Are Y(t) and N(t)
jointly WSS ?
Sol：
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Example 1.17
Xn is a WSS random sequence with RX[k]. The random
sequence Yn is obtained from Xn by reversing the sign
of every other random variable in Xn：Yn = (1)n Xn.
(a) Find RY[n,k] in terms of RX[k].
(b) Find RXY[n,k] in terms of RX[k].
(c) Is Yn WSS?
(d) Are Xn and Yn jointly WSS?
Sol：
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