Overview Random Processes
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Overview of Random Processes
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Let denote the random outcome of an experiment.
To every such outcome, suppose a waveform
is assigned.The collection of such
waveforms form a
stochastic process.
For fixed (the set of
all experimental outcomes),
is a specific time function.
For fixed t,is a random variable.
The ensemble of all such realizations
over time represents the stochastic process X(t).
),(tX
Si
),( 11 itXX =
),( tX
t
1t
2t
),(n
tX
),( ktX
),(2
tX
),(1
tX
M
M
M
),( tX
0
),( tX
Random (stochastic) Processes
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Random (stochastic) Processes
For example
where is a uniformly distributed random variable in
represents a stochastic process.
Stochastic processes are everywhere:
Noise, detection and classification problems, pattern recognition,
stock market fluctuations, various queuing systems
all represent stochastic phenomena.
),cos()( 0 += tatX
(0,2 ),
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Random (stochastic) Processes
IfX(t) is a stochastic process, then for fixed t, X(t) representsa random variable. Its distribution function (cdf) is given by
Notice that depends on t, since for a different t, we obtaina different random variable. Further
represents the first-order probability density function (pdf) of the
process X(t).
})({),( xtXPtxFX
=
),( txFX
dx
txdFtxf X
X
),(),( =
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Random (stochastic) Processes
For t= t1 and t= t2, X(t) represents two different random variables
X1 = X(t1) and X2 = X(t2) respectively. Their joint distribution is
given by
and
represents the second-order density function of the process X(t).
Similarly represents the nth order density
function of the process X(t). Complete specification of the stochastic
process X(t) requires the knowledge of
for all and for all n. (an almost impossible taskin reality).
})(,)({),,,( 22112121 xtXxtXPttxxFX =
),,,,,( 2121 nn tttxxxfX LL
),,,,,( 2121 nn tttxxxfX LL
niti ,,2,1, L=
21 2 1 2
1 2 1 2
1 2
( , , , )( , , , )
X
X
F x x t t f x x t t
x x
=
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Random (stochastic) Processes
Mean of a Stochastic Process:
represents the mean value of a process X(t). In general, the mean ofa process can depend on the time index t.
Autocorrelation function of a process X(t) is defined as
and it represents the interrelationship between the random variables
X1 = X(t1) and X2 = X(t2) generated from the process X(t).
Properties:
1.
2.
*
1 2 2 1( , ) ( , )
XX XXR t t R t t=
.0}|)({|),(2
>= tXEttRXX (Average instantaneous power)
( ) { ( )} ( , )X
t E X t x f x t dx+
= =
* *
1 2 1 2 1 2 1 2 1 2 1 2( , ) { ( ) ( )} ( , , , )XX XR t t E X t X t x x f x x t t dx dx= =
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Random (stochastic) Processes
The function
represents the autocovariance function of the process X(t).
)()(),(),( 2*
12121 ttttRttC XXXXXX =
Similarly
0
2
00
( ) { ( )} {cos( )}1 cos( ) 0.
2
X t E X t aE t
t d
= = +
= + =
).(cos
2
)}2)(cos()({cos2
)}cos(){cos(),(
210
2
210210
22010
2
21
tta
ttttEa
ttEattRXX
=
+++=
++=
Example ).2,0(~),cos()( 0 UtatX +=
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Stationary Random Processes
Stationary processes exhibit statistical properties that are
invariant to shift in the time index.
Thus, for example, second-order stationarity implies that
the statistical properties of the pairs
{X(t1) , X(t2) } and {X(t1+c) , X(t2+c)} are the same for any c.
Similarly first-order stationarity implies that the statistical
properties ofX(ti) and X(ti+c) are the same for any c.
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Stationary Random Processes
In strict terms, the statistical properties are governed by the
joint probability density function. Hence a process is nth-order
Strict-Sense Stationary (S.S.S) if
for any c, where the left side represents the joint density function of
the random variables andthe right side corresponds to the joint density function of the random
variables
A process X(t) is said to be strict-sense stationary if above eqn. is
true for all
),,,,,(),,,,,( 21212121 ctctctxxxftttxxxf nnnn XX +++ LLLL
)(,),(),( 2211 nn tXXtXXtXX === L
).(,),(),( 2211 ctXXctXXctXX nn +=+=+= L
.and,2,1,,,2,1, canynniti LL ==
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Stationary Random Processes
For a first-order strict sense stationary process,
for any c.
In particular c= tgives
i.e., the first-order density of X(t) is independent of t.
In that case
),(),( ctxftxfXX
+
)(),( xftxfXX
=
[ ( )] ( ) ,E X t x f x dx a constant.+
= =
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Stationary Random Processes
i.e., the second order density function of a SSS process depends only
on the difference of the time indices
In that case, the autocorrelation function is given by
i.e., it depends only on the difference of the time indices.
2 1 .t t =
*
1 2 1 2
*
1 2 1 2 2 1 1 2
*
2 1
( , ) { ( ) ( )}
( , , )
( ) ( ) ( ),
XX
X
XX XX XX
R t t E X t X t
x x f x x t t dx dx
R t t R R
=
= =
= = =
Similarly, for a second-order strict-sense stationary process
for any c. For c= t1 we get
),,,(),,,( 21212121 ctctxxfttxxf XX ++
1 2 1 2 1 2 2 1( , , , ) ( , , )X Xf x x t t f x x t t
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Stationary Random Processes
The basic conditions for the first and second order stationarity areusually difficult to verify.
In that case, we often resort to a looser definition of stationarity,known as Wide-Sense Stationarity (W.S.S).
A process X(t) is said to be Wide-Sense Stationary if
(i)and(ii)
i.e., the mean is a constant and the autocorrelation functiondepends only on the difference between the time indices.
=)}({ tXE
*
1 2 2 1{ ( ) ( )} ( ),XXE X t X t R t t=
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Stationary Random Processes
Remarks:
1. Notice that these conditions do not say anything about the
nature of the probability density functions, and instead dealwith the average behavior of the process.
2. Strict-sense stationarity always implies wide-sense
stationarity. SSSWSS
However, the converse is not truein general, the only exceptionbeing the Gaussian process.
If X(t) is a Gaussian process, then wide-sense stationarity (w.s.s) strict-sense stationarity (s.s.s).
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Ergodic Random Processes
If almost every member of the ensemble shows the same statisticalbehavior as the whole ensemble, then it is possible to determine thestatistical behavior by examining only one typical sample function.
Ergodic process
For ergodic process, the mean values and autocorrelation functioncan be determined by time averages as well as by ensembleaverages, that is,
Ergodic in the mean process:
Ergodic in the autocorrelation process:
These conditions can exist if the process is stationary.
ergodicstationary(not vice versa)
{ }1
( ) ( )2
limT
TT
E X t X t dtT
=
( ) { }* *1
( ) ( ) ( ) ( )2
limT
XXT
T
R E X t X t X t X t dtT
= + = +
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Power Spectral Density
Power spectrum of X(t)
Autocorrelation of X(t)
Power spectrum and Autocorrelation function
are Fourier transform pair
Total average power=
2( ) [ ( )] ( ) j fx x x
S f F R R e d
= =
1 2
( ) [ ( )] ( )
j f
x x xR F S f S f e df
= =
(0) ( )x x
R S f df
=
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Random Processes as Inputs/Outputs to LTI Sytems
A deterministic system transforms each input waveform intoan output waveform by operating only on thetime variable t.
Thus a set of realizations at the input corresponding to a processX(t)generates a new set of realizations at theoutput associated with a new process Y(t).
),( itX )],([),( ii tXTtY =
)},({ tY
Our goal is to study the output process statistics in terms of the inputprocess statistics and the system function.
][T)(tX
)(tY
t t
),(i
tX ),(
itY
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Random Processes as Inputs/Outputs to LTI Sytems
Linear Systems: represents a linear system if
Let represent the output of a linear system.
Time-Invariant System: represents a time-invariant system if
i.e., shift in the input results in the same shift in the output also.
If satisfies both, then it corresponds to a linear time-invariant (LTI)system.
][L
)}({)( tXLtY =
)}.({)}({)}()({ 22112211 tXLatXLatXatXaL +=+
][L
)()}({)}({)( 00 ttYttXLtXLtY ==
][L
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Random Processes as Inputs/Outputs to LTI Sytems
LTI
+
+
=
=
)()(
)()()(
dtXh
dXthtYarbitrary
input
t
)(tX
t
)(tY
)(tX )(tY
LTI systems can be uniquely represented in terms of their outputto a delta function
LTI)(t )(th
Impulse
Impulse
response ofthe system
t
)(th
Impulse
response
then
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Random Processes as Inputs/Outputs to LTI Sytems
Output Statistics: The mean of the output process is given by
).()()()(
})()({)}({)(
thtdth
dthXEtYEt
XX
Y
==
==
+
+
h(t))(tX )(tY
In particular if X(t) is wide-sense stationary, then we haveso that
XXt =)(
constant.acdhtXXY
,)()( == +
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Random Processes as Inputs/Outputs to LTI Sytems
Output Statistics: The autocorrelation function of the output is given by
*( ) ( ) ( ) ( ).YY XX
R R h h =
h() h*(-)( )XYR ( )
YYR ( )
XXR
h*(-) h()
( )YXR ( )YYR
( )XX
R
Thus, Y(t) is w.s.s process.X(t) and Y(t) are jointly w.s.s.
LTI systemh(t)
wide-sensestationary process
wide-sensestationary process.
)(tX )(tY
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Random Processes as Inputs/Outputs to LTI Sytems
Output Statistics: The power spectral density function of the output is
2 *( ) ( ) ( ) ( ) ( ) ( )
YY XX XX S f S f H f S f H f H f = =
H(f) H*(f)( )XYS f ( )
YYS f( )
XXS f
H*(f) H(f)( )YXS f ( )
YYS f( )XXS f
{ } { }*( ) ( ) ( ) ( )XY XYS f F R F E X t Y t = = +
{ } { *( ) ( ) ( ) ( )YX YX S f F R F E Y t X t = = +
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White Noise
A random process X(t) is called a white process if it
has a flat power spectrum.
If Sx(f) is constant for all f
It closely represents thermal noise
f
Sx(f)
The area is infinite(Infinite power !)
0( )
2n
NS f =
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White Gaussian Noise
The sampled random variables will be statisticallyindependent Gaussian random variables
Sn(f)
N0/2
f
N0/2
Rx()
=0 0
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Poisson Random Process
Poisson random variable:
L,2,1,0,!"durationofinterval
aninoccurarrivals" ==
kk
ekPk
=== T
Tnp
0 T
43421arrivalsk
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Poisson Random Process
Definition: X(t) = n(0, t) represents a Poisson process if
the number of arrivals n(t1, t2) in an interval (t1, t2) of length
t= t2t1 is a Poisson random variable with parameter
Thus
.t
1221 ,,2,1,0,
!
)(}),({ tttk
k
tekttnP
kt
==== L
ttnEtXE == )],0([)]([
).,min(),( 2121221 ttttttRXX +=
X(t) : not WSS
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Poisson Impulse Process
Although X(t) does not represent a wide sense stationaryprocess, its derivative doesrepresent a wide sensestationary process (called Poisson Impulse Process).
)(tX
)(tX )(tXdt
d )(
constantadt
td
dt
tdt X
X,
)()(
===
2
1 2 1 2( ) ( ).X XR t , t t t = +
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Gaussian Random Process
A random process X(t) is a Gaussian process if for all
n and for all , the random variableshas a jointly Gaussian density function, which may be
expressed as
where
1 2( , , , )nt t tK
2{ ( ), ( ), , ( )}i nX t X t X tK
1
/ 2 1/ 21 1( ) exp[ ( ) ( )]
2(2 ) [det( )]
T
nf x x m C x m
C
=
2[ ( ), ( ), , ( )]
T
i nx X t X t X t=
K( )m E X=
{ } (( )( ))ij i i j jC c E x m x m= =
: n random variables: mean value vector: nxn covariance matrix
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