14. Stochastic Processes · 2021. 1. 9. · process X(t). (see Fig 14.1). For example where is a...
Transcript of 14. Stochastic Processes · 2021. 1. 9. · process X(t). (see Fig 14.1). For example where is a...
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 1
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. The
set of and the time
index t can be continuous
or discrete (countably
infinite or finite) as well.
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
),( tX
}{ k
Si
),( 11 tXX
),( tX
14. Stochastic Processes
t
1t
2t
),(n
tX
),(k
tX
),(2
tX
),(1
tX
Fig. 14.1
),( tX
0
),( tX
Introduction
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 2
process X(t). (see Fig 14.1). For example
where is a uniformly distributed random variable in
represents a stochastic process. Stochastic processes are everywhere:
Brownian motion, stock market fluctuations, various queuing systems
all represent stochastic phenomena.
If X(t) is a stochastic process, then for fixed t, X(t) represents
a random variable. Its distribution function is given by
Notice that depends on t, since for a different t, we obtain
a different random variable. Further
represents the first-order probability density function of the
process X(t).
),cos()( 0 tatX
})({),( xtXPtxFX
),( txFX
(14-1)
(14-2)
(0,2 ),
dx
txdFtxf X
X
),(),(
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 3
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 task
in reality).
})(,)({),,,( 22112121 xtXxtXPttxxFX
(14-3)
(14-4)
),, ,,,( 2121 nn tttxxxfX
),, ,,,( 2121 nn tttxxxfX
niti , ,2 ,1 ,
2
1 2 1 21 2 1 2
1 2
( , , , )( , , , )
X
X
F x x t tf x x t t
x x
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 4
Mean of a Stochastic Process:
represents the mean value of a process X(t). In general, the mean of
a 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.
(14-5)
(14-6)
*
1
*
212
*
21 )}]()({[),(),( tXtXEttRttRXXXX
(14-7)
.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 X
R t t E X t X t x x f x x t t dx dx
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 5
3. represents a nonnegative definite function, i.e., for any
set of constants
Eq. (14-8) follows by noticing that
The function
represents the autocovariance function of the process X(t).
Example 14.1
Let
Then
.)(for 0}|{|1
2
n
iii tXaYYE
)()(),(),( 2
*
12121 ttttRttCXXXXXX
(14-9)
.)(
T
TdttXz
T
T
T
T
T
T
T
T
dtdtttR
dtdttXtXEzE
XX
2121
212
*
1
2
),(
)}()({]|[|
(14-10)
n
iia 1}{
),( 21 ttRXX
n
i
n
jjiji ttRaa
XX
1 1
* .0),( (14-8)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 6
Similarly
,0}{sinsin}{coscos
)}{cos()}({)(
0 0
0
EtaEta
taEtXEtX
).(cos2
)}2)(cos()({cos2
)}cos(){cos(),(
210
2
210210
2
2010
2
21
tta
ttttEa
ttEattRXX
(14-12)
(14-13)
Example 14.2
).2,0(~ ),cos()( 0 UtatX (14-11)
This gives
2
0 }.{sin0cos}{cos since
21 EdE
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 7
Stationary Stochastic ProcessesStationary 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
of X(ti) and X(ti+c) are the same for any c.
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 and
the right side corresponds to the joint density function of the random
variables
A process X(t) is said to be strict-sense stationary if (14-14) is
true for all
),, ,,,(),, ,,,( 21212121 ctctctxxxftttxxxf nnnn XX
(14-14)
)( , ),( ),( 2211 nn tXXtXXtXX
).( , ),( ),( 2211 ctXXctXXctXX nn
. and ,2 ,1 , , ,2 ,1 , canynniti
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 8
For a first-order strict sense stationary process,
from (14-14) we have
for any c. In particular c = – t gives
i.e., the first-order density of X(t) is independent of t. In that case
Similarly, for a second-order strict-sense stationary process
we have from (14-14)
for any c. For c = – t2 we get
),(),( ctxftxfXX
(14-16)
(14-15)
(14-17)
)(),( xftxfXX
[ ( )] ( ) , E X t x f x dx a constant.
), ,,(), ,,( 21212121 ctctxxfttxxfXX
) ,,(), ,,( 21212121 ttxxfttxxfXX
(14-18)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 9
i.e., the second order density function of a strict sense stationary
process depends only on the difference of the time indices
In that case the autocorrelation function is given by
i.e., the autocorrelation function of a second order strict-sense
stationary process depends only on the difference of the time
indices
Notice that (14-17) and (14-19) are consequences of the stochastic
process being first and second-order strict sense stationary.
On the other hand, the basic conditions for the first and second order
stationarity – Eqs. (14-16) and (14-18) – are usually difficult to verify.
In that case, we often resort to a looser definition of stationarity,
known as Wide-Sense Stationarity (W.S.S), by making use of
.21 tt
.21 tt
(14-19)
*
1 2 1 2
*
1 2 1 2 1 2 1 2
*
1 2
( , ) { ( ) ( )}
( , , )
( ) ( ) ( ),
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
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 10
(14-17) and (14-19) as the necessary conditions. Thus, a process X(t)
is said to be Wide-Sense Stationary if
(i)
and
(ii)
i.e., for wide-sense stationary processes, the mean is a constant and
the autocorrelation function depends only on the difference between
the time indices. Notice that (14-20)-(14-21) does not say anything
about the nature of the probability density functions, and instead deal
with the average behavior of the process. Since (14-20)-(14-21)
follow from (14-16) and (14-18), strict-sense stationarity always
implies wide-sense stationarity. However, the converse is not true in
general, the only exception being the Gaussian process.
This follows, since if X(t) is a Gaussian process, then by definition
are jointly Gaussian random
variables for any whose joint characteristic function
is given by
)}({ tXE
(14-21)
(14-20)
),()}()({ 212
*
1 ttRtXtXEXX
)( , ),( ),( 2211 nn tXXtXXtXX
nttt ,, 21
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 11
where is as defined on (14-9). If X(t) is wide-sense
stationary, then using (14-20)-(14-21) in (14-22) we get
and hence if the set of time indices are shifted by a constant c to
generate a new set of jointly Gaussian random variables
then their joint characteristic
function is identical to (14-23). Thus the set of random variables
and have the same joint probability distribution for all n and
all c, establishing the strict sense stationarity of Gaussian processes
from its wide-sense stationarity.
To summarize if X(t) is a Gaussian process, then
wide-sense stationarity (w.s.s) strict-sense stationarity (s.s.s).
Notice that since the joint p.d.f of Gaussian random variables depends
only on their second order statistics, which is also the basis
),( ki ttCXX
1 ,
( ) ( , ) / 2
1 2( , , , )XX
n n
k k i k i k
k l k
X
j t C t t
n e
(14-22)
12
1 1 1 1
( )
1 2( , , , )XX
n n n
k i k i k
k k
X
j C t t
n e
(14-23)
n
iiX 1}{ n
iiX 1}{
),( 11 ctXX
)(,),( 22 ctXXctXX nn
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 12
for wide sense stationarity, we obtain strict sense stationarity as well.
From (14-12)-(14-13), (refer to Example 14.2), the process
in (14-11) is wide-sense stationary, but
not strict-sense stationary.
Similarly if X(t) is a zero mean wide
sense stationary process in Example 14.1,
then in (14-10) reduces to
As t1, t2 varies from –T to +T, varies
from –2T to + 2T. Moreover is a constant
over the shaded region in Fig 14.2, whose area is given by
and hence the above integral reduces to
),cos()( 0 tatX
2
z
.)(}|{|
2121
22
T
T
T
Tz dtdtttRzEXX
21 tt
)(XX
R
)0(
dTdTT )2()2(2
1)2(
2
1 22
.)1)((2|)|2)((2
2 2
||2
2
2
T
TT
T
Tz dRTdTR
XXXX
(14-24)
T T
T
T2
2t
1t
Fig. 14.2
21tt
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 13
Systems with Stochastic InputsA deterministic system1 transforms each input waveform into
an output waveform by operating only on the
time variable t. Thus a set of realizations at the input corresponding
to a process X(t) generates a new set of realizations at the
output associated with a new process Y(t).
),( itX )],([),( ii tXTtY
)},({ tY
Our goal is to study the output process statistics in terms of the input
process statistics and the system function.
1A stochastic system on the other hand operates on both the variables t and .
][T )(tX )(tY
t t
),(i
tX ),(
itY
Fig. 14.3
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 14
Deterministic Systems
Systems with Memory
Time-Invariant
systems
Linear systems
Linear-Time Invariant
(LTI) systems
Memoryless Systems
)]([)( tXgtY
)]([)( tXLtY Time-varying
systemsFig. 14.3
.)()(
)()()(
dtXh
dXthtY( )h t( )X t
LTI system
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 15
Memoryless Systems:The output Y(t) in this case depends only on the present value of the
input X(t). i.e., (14-25))}({)( tXgtY
Memoryless
system
Memoryless
system
Memoryless
system
Strict-sense
stationary input
Wide-sense
stationary input
X(t) stationary
Gaussian with)(
XXR
Strict-sense
stationary output.
Need not be
stationary in
any sense.
Y(t) stationary,but
not Gaussian with
(see (14-26)).
).()( XXXY
RR
(see (9-76), Text for a proof.)
Fig. 14.4
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 16
Theorem: If X(t) is a zero mean stationary Gaussian process, and
Y(t) = g[X(t)], where represents a nonlinear memoryless device,
then
Proof:
where are jointly Gaussian random
variables, and hence
)(g
)}.({ ),()( XgERRXXXY
(14-26)
212121 ),()(
)}]({)([)}()({)(
21dxdxxxfxgx
tXgtXEtYtXER
XX
XY
(14-27)
)( ),( 21 tXXtXX
* 1
1 2
/ 2
1 2
1 2 1 2
* *
12 | |
(0) ( )
( ) (0)
( , )
( , ) , ( , )
{ } XX XX
XX XX
X X
x A x
T T
A
R R
R R
f x x e
X X X x x x
A E X X LL
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 17
where L is an upper triangular factor matrix with positive diagonal
entries. i.e.,
Consider the transformation
so that
and hence Z1, Z2 are zero mean independent Gaussian random
variables. Also
and hence
The Jacobaian of the transformation is given by
. 0
22
1211
l
llL
IALLLXXELZZE 11 *1**1*
}{}{
* * *1 * 1 2 2
1 2 .x A x z L A Lz z z z z
22222121111 , zlxzlzlxzLx
1 1 1 2 1 2( , ) , ( , )T TZ L X Z Z z L x z z
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 18
Hence substituting these into (14-27), we obtain
where This gives
.|||||| 2/11 ALJ
2 21 2
1/ 211 1 12 2 22 2
11 1 22 2 1 21 2
12 2 22 2 1 21 2
/ 2 / 21 1| | 2 | |
1 2
1 2
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
XY J A
z z
z z
z zR l z l z g l z e e
l z g l z f z f z dz dz
l z g l z f z f z dz dz
22
212 22
22
11 1 1 22 2 21 2
12 2 22 2 22
/ 2
2
2
1 2
2
1
2
/ 21
2
( ) ( ) ( )
( ) ( )
( ) ,
z z
z
z
u lll
l z f z dz g l z f z dz
l z g l z f z dz
e
ug u e du
0
22 2 .u l z
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 19
2
22
222 22
2
2
( )
/ 2112 22 2
( )( )
( ) ( )
( ) ( ) ( ) ,
u
XY
uu
XX
f u
u
df uf u
du
u
l
l
ul
R l l g u e du
R g u f u du
Hence).( gives since 2212
* XX
RllLLA
the desired result, where Thus if the input to
a memoryless device is stationary Gaussian, the cross correlation
function between the input and the output is proportional to the
input autocorrelation function.
),()}({)(
})()(|)()(){()(
XXXX
XXXY
RXgER
duufugufugRR uu
0
)].([ XgE
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 20
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 (14-28) and (14-30), then it corresponds to
a linear time-invariant (LTI) system.
LTI systems can be uniquely represented in terms of their output to
a delta function
][L
)}({)( tXLtY
)}.({)}({)}()({ 22112211 tXLatXLatXatXaL (14-28)
][L
)()}({)}({)( 00 ttYttXLtXLtY
(14-29)
(14-30)
][L
LTI)(t )(th
Impulse
Impulse
response of
the system
t
)(th
Impulse
responseFig. 14.5
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 21
Eq. (14-31) follows by expressing X(t) as
and applying (14-28) and (14-30) to Thus)}.({)( tXLtY
)()()( dtXtX
(14-31)
(14-32)
(14-33).)()()()(
)}({)(
})()({
})()({)}({)(
dtXhdthX
dtLX
dtXL
dtXLtXLtY
By Linearity
By Time-invariance
then
LTI
)()(
)()()(
dtXh
dXthtYarbitrary
input
t
)(tX
t
)(tY
Fig. 14.6
)(tX )(tY
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 22
Output Statistics: Using (14-33), the mean of the output process
is given by
Similarly the cross-correlation function between the input and output
processes is given by
Finally the output autocorrelation function is given by
).()()()(
})()({)}({)(
thtdth
dthXEtYEt
XX
Y
(14-34)
).(),(
)(),(
)()}()({
})()()({
)}()({),(
2
*
21
*
21
*
21
*
21
2
*
121
thttR
dhttR
dhtXtXE
dhtXtXE
tYtXEttR
XX
XX
XY
*
*
(14-35)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 23
or
),(),(
)(),(
)()}()({
})( )()({
)}()({),(
121
21
21
2
*
1
2
*
121
thttR
dhttR
dhtYtXE
tYdhtXE
tYtYEttR
XY
XY
YY
*
).()(),(),( 12
*
2121 ththttRttRXXYY
(14-36)
(14-37)
h(t))(tX
)(tY
h*(t2) h(t1) ),( 21 ttRXY ),( 21 ttR
YY),( 21 ttR
XX
(a)
(b)
Fig. 14.7
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 24
In particular if X(t) is wide-sense stationary, then we have
so that from (14-34)
Also so that (14-35) reduces to
Thus X(t) and Y(t) are jointly w.s.s. Further, from (14-36), the output
autocorrelation simplifies to
From (14-37), we obtain
XXt )(
constant.a cdhtXXY
,)()(
(14-38)
)(),( 2121 ttRttRXXXX
(14-39)
).()()(
,)()(),( 21
2121
YYXY
XYYY
RhR
ttdhttRttR
(14-40)
).()()()( * hhRRXXYY
(14-41)
. ),()()(
)()(),(
21
*
*
2121
ttRhR
dhttRttR
XYXX
XXXY
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 25
From (14-38)-(14-40), the output process is also wide-sense stationary.
This gives rise to the following representation
LTI system
h(t)
Linear system
wide-sense
stationary process
strict-sense
stationary process
Gaussian
process (also
stationary)
wide-sense
stationary process.
strict-sense
stationary process
(see Text for proof )
Gaussian process
(also stationary)
)(tX )(tY
LTI system
h(t)
)(tX
)(tX
)(tY
)(tY
(a)
(b)
(c)
Fig. 14.8
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 26
White Noise Process:W(t) is said to be a white noise process if
i.e., E[W(t1) W*(t2)] = 0 unless t1 = t2.
W(t) is said to be wide-sense stationary (w.s.s) white noise
if E[W(t)] = constant, and
If W(t) is also a Gaussian process (white Gaussian process), then all of
its samples are independent random variables (why?).
For w.s.s. white noise input W(t), we have
),()(),( 21121 tttqttRWW
(14-42)
).()(),( 2121 qttqttRWW
(14-43)
White noise
W(t)
LTI
h(t)
Colored noise
( ) ( ) ( )N t h t W t
Fig. 14.9
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 27
and
where
Thus the output of a white noise process through an LTI system
represents a (colored) noise process.
Note: White noise need not be Gaussian.
“White” and “Gaussian” are two different concepts!
)()()(
)()()()(
*
*
qhqh
hhqRnn
(14-45)
.)()()()()(
**
dhhhh (14-46)
(14-44)
[ ( )] ( ) ,
WE N t h d
a constant
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 28
Upcrossings and Downcrossings of a stationary Gaussian process:
Consider a zero mean stationary Gaussian process X(t) with
autocorrelation function An upcrossing over the mean value
occurs whenever the realization X(t)
passes through zero with
positive slope. Let
represent the probability
of such an upcrossing in
the interval
We wish to determine
Since X(t) is a stationary Gaussian process, its derivative process
is also zero mean stationary Gaussian with autocorrelation function
(see (9-101)-(9-106), Text). Further X(t) and
are jointly Gaussian stationary processes, and since (see (9-106), Text)
).(XX
R
t
). ,( ttt .
Fig. 14.10
)(tX
)()( XXXX
RR
)(tX
,)(
)(
d
dRR XX
XX
Upcrossings
t
)(tX
Downcrossing
The derivative of an even function is an odd function and vice versa
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 29
we have
which for gives
i.e., the jointly Gaussian zero mean random variables
are uncorrelated and hence independent with variances
respectively. Thus
To determine the probability of upcrossing rate,
0
)()(
)(
)()(
XX
XXXX
XXR
d
dR
d
dRR
(14-48)
(14-47)
(0) 0 [ ( ) ( )] 0XX
R E X t X t
)( and )( 21 tXXtXX (14-49)
,
0 )0( )0( and )0( 2
2
2
1 XXXXXX
RRR (14-50)
2 21 1
2 21 2
1 2 1 2 1 2
1 2
2 21( , ) ( ) ( ) .
2X X X X
x x
f x x f x f x e
(14-51)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 30
we argue as follows: In an interval the realization moves
from X(t) = X1 to
and hence the realization intersects with the zero level somewhere
in that interval if
i.e.,
Hence the probability of upcrossing
in is given by
Differentiating both sides of (14-53) with respect to we get
and letting Eq. (14-54) reduce to
), ,( ttt
,)()()( 21 tXXttXtXttX
1 2 .X X t
(14-52)
) ,( ttt
(14-53)
t
)(tX
)(tX
)( ttX
ttt
Fig. 14.11
.)()(
),(
21
0
1
0 2
0 21
0
21
212
2 2121
dxxdxfxf
dxxdxxft
tx
x txx
XX
XX
,t
(14-54)2 1
2 2 2 2 0( ) ( )
X Xf x x f x t dx
,0t
1 2 1 20, 0, and ( ) 0 X X X t t X X t
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 31
[where we have made use of (5-78), Text]. There is an equal
probability for downcrossings, and hence the total probability for
crossing the zero line in an interval equals where
It follows that in a long interval T, there will be approximately
crossings of the mean value. If is large, then the
autocorrelation function decays more rapidly as moves
away from zero, implying a large random variation around the origin
(mean value) for X(t), and the likelihood of zero crossings should
increase with increase in agreeing with (14-56).
)0(
)0(
2
1)/2(
2
1
)0(2
1
)()0(2
1)0()(
2
0 222
0 222 12
XX
XX
XX
X
XX
XX
R
R
R
dxxfxR
dxfxfx
(14-55)
) ,( ttt ,0
t
.0 )0(/)0(1
0
XXXXRR
(14-56)
T0
)0( XX
R )(
XXR
(0),XX
R
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 32
Discrete Time Stochastic Processes:
A discrete time stochastic process Xn = X(nT) is a sequence of
random variables. The mean, autocorrelation and auto-covariance
functions of a discrete-time process are gives by
and
respectively. As before strict sense stationarity and wide-sense
stationarity definitions apply here also.
For example, X(nT) is wide sense stationary if
and
)}()({),(
)}({
2
*
121 TnXTnXEnnR
nTXEn
*
2121 21),(),( nnnnRnnC
(14-57)
(14-58)
(14-59)
constanta nTXE ,)}({ (14-60)
(14-61)* *[ {( ) } {( ) }] ( ) n nE X k n T X k T R n r r
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 33
i.e., R(n1, n2) = R(n1 – n2) = R*(n2 – n1). The positive-definite
property of the autocorrelation sequence in (14-8) can be expressed
in terms of certain Hermitian-Toeplitz matrices as follows:
Theorem: A sequence forms an autocorrelation sequence of
a wide sense stationary stochastic process if and only if every
Hermitian-Toeplitz matrix Tn given by
is non-negative (positive) definite for
Proof: Let represent an arbitrary constant vector.
Then from (14-62),
since the Toeplitz character gives Using (14-61),
Eq. (14-63) reduces to
}{ nr
0, 1, 2, , .n
*
0
*
1
*
1
*
110
*
1
210
n
nn
n
n
n T
rrrr
rrrr
rrrr
T
T
naaaa ] , , ,[ 10
(14-62)
n
i
n
kikkin raaaTa
0 0
**(14-63)
.)( , ikkin rT
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 34
From (14-64), if X(nT) is a wide sense stationary stochastic process
then Tn is a non-negative definite matrix for every
Similarly the converse also follows from (14-64). (see section 9.4, Text)
If X(nT) represents a wide-sense stationary input to a discrete-time
system {h(nT)}, and Y(nT) the system output, then as before the cross
correlation function satisfies
and the output autocorrelation function is given by
or
Thus wide-sense stationarity from input to output is preserved
for discrete-time systems also.
.,,2 ,1 ,0 n
(14-64)
2
* * * *
0 0 0
{ ( ) ( )} ( ) 0.n n n
n i k ki k k
a T a a a E X kT X iT E a X kT
)()()( * nhnRnRXXXY
)()()( nhnRnRXYYY
).()()()( * nhnhnRnRXXYY
(14-65)
(14-66)
(14-67)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 35
Auto Regressive Moving Average (ARMA) Processes
Consider an input – output representation
where X(n) may be considered as the output of a system {h(n)}
driven by the input W(n).
Z – transform of
(14-68) gives
or
,)()()(01
q
kk
p
kk knWbknXanX (14-68)
(14-69)
h(n)W(n) X(n)
00 0
( ) ( ) , 1p q
k k
k kk k
X z a z W z b z a
1 2
0 1 2
1 20 1 2
( ) ( )( ) ( )
( ) ( )1
q
qk
pk p
b b z b z b zX z B zH z h k z
W z A za z a z a z
(14-70)
Fig.14.12
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 36
represents the transfer function of the associated system response {h(n)}
in Fig 14.12 so that
Notice that the transfer function H(z) in (14-70) is rational with p poles
and q zeros that determine the model order of the underlying system.
From (14-68), the output undergoes regression over p of its previous
values and at the same time a moving average based on
of the input over (q + 1) values is added to it, thus
generating an Auto Regressive Moving Average (ARMA (p, q))
process X(n). Generally the input {W(n)} represents a sequence of
uncorrelated random variables of zero mean and constant variance
so that
If in addition, {W(n)} is normally distributed then the output {X(n)}
also represents a strict-sense stationary normal process.
If q = 0, then (14-68) represents an AR(p) process (all-pole
process), and if p = 0, then (14-68) represents an MA(q)
(14-72)
(14-71).)()()(0
k
kWknhnX
),1( ),( nWnW
2
W
).()( 2 nnRWWW
)( , qnW
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 37
process (all-zero process). Next, we shall discuss AR(1) and AR(2)
processes through explicit calculations.
AR(1) process: An AR(1) process has the form (see (14-68))
and from (14-70) the corresponding system transfer
provided | a | < 1. Thus
represents the impulse response of an AR(1) stable system. Using
(14-67) together with (14-72) and (14-75), we get the output
autocorrelation sequence of an AR(1) process to be
)()1()( nWnaXnX (14-73)
1|| ,)( aanh n (14-75)
(14-74)
011
1)(
n
nn zaaz
zH
2
||2
0
||22
1}{}{)()(
a
aaaaannR
n
k
kknnn
WWWXX
(14-76)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 38
where we have made use of the discrete version of (14-46). The
normalized (in terms of RXX (0)) output autocorrelation sequence is
given by
It is instructive to compare an AR(1) model discussed above by
superimposing a random component to it, which may be an error
term associated with observing a first order AR process X(n). Thus
where X(n) ~ AR(1) as in (14-73), and V(n) is an uncorrelated random
sequence with zero mean and variance that is also uncorrelated
with {W(n)}. From (14-73), (14-78) we obtain the output
autocorrelation of the observed process Y(n) to be
)()()( nVnXnY
.0 || ,)0(
)()( || na
R
nRn n
XX
XX
X
(14-78)
(14-77)
2
V
)(1
)()()()()(
2
2
||2
2
na
a
nnRnRnRnR
VW
VXXVVXXYY
n
(14-79)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 39
so that its normalized version is given by
where
Eqs. (14-77) and (14-80) demonstrate the effect of superimposing
an error sequence on an AR(1) model. For non-zero lags, n ≥1, the
autocorrelation of the observed sequence {Y(n)}is reduced by a constant
factor compared to the original process {X(n)}.
From (14-78), the superimposed
error sequence V(n) only affects
the corresponding term in Y(n)
(term by term). However,
a particular term in the “input sequence”
W(n) affects X(n) and Y(n) as well as
all subsequent observations.
(14-80)
.1)1( 222
2
a
cVW
W
(14-81)
Fig. 14.13
nk
)()( kkYX
1)0()0( YX
0
| |
1 0( )( )
(0) 1, 2,
YY
Y
YY
n
nR nn
R c a n
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 40
AR(2) Process: An AR(2) process has the form
and from (14-70) the corresponding transfer function is given by
so that
and in term of the poles of the transfer function,
from (14-83) we have
that represents the impulse response of the system.
From (14-84)-(14-85), we also have
From (14-83),
)()2()1()( 21 nWnXanXanX (14-82)
(14-83)
(14-84)
(14-85)
1
2
2
1
1
1
02
2
1
1 111
1)()(
z
b
z
b
zazaznhzH
n
n
2 ),2()1()( ,)1( ,1)0( 211 nnhanhanhahh
0 ,)( 2211 nbbnh nn
. ,1 1221121 abbbb
, , 221121 aa (14-86)
21 and
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 41
and H(z) stable implies
Further, using (14-82) the output autocorrelations satisfy the recursion
and hence their normalized version is given by
By direct calculation using (14-67), the output autocorrelations are
given by
(14-88)
(14-87)
.1|| ,1|| 21
)2()1(
)}()({
)}()]2()1({[
)}()({)(
21
*
*
21
*
nRanRa
mXmnWE
mXmnXamnXaE
mXmnXEnR
XXXX
XX
0
2
2
*
2
2
2
*
21
*
2
*
21
2
*
1
*
12
*
1
2
1
*
1
2
12
0
*2
*2*
||1
)(||
1
)(
1
)(
||1
)(||
)()(
)()()()()()(
nnnn
k
bbbbbb
khknh
nhnhnhnhnRnR
W
W
WWWXX
(14-89)
1 2
( )( ) ( 1) ( 2).
(0)
XX
X X X
XX
R nn a n a n
R
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 42
where we have made use of (14-85). From (14-89), the normalized
output autocorrelations may be expressed as
where c1 and c2 are appropriate constants.
Damped Exponentials: When the second order system in
(14-83)-(14-85) is real and corresponds to a damped exponential
response, the poles are complex conjugate which gives
in (14-83). Thus
In that case in (14-90) so that the normalized
correlations there reduce to
But from (14-86)
(14-90)nn
XX
XX
Xcc
R
nRn *
22
*
11)0(
)()(
2
1 24 0a a
* 1 2
jc c c e
* 1 2 1, , 1.jr e r
(14-91)
(14-92)).cos(2}Re{2)( *
11 ncrcn nn
X
,1 ,cos2 2
2
121 arar (14-93)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 43
and hence which gives
Also from (14-88)
so that
where the later form is obtained from (14-92) with n = 1. But
in (14-92) gives
Substituting (14-96) into (14-92) and (14-95) we obtain the normalized
output autocorrelations to be
2
1 22 sin ( 4 ) 0r a a
1)0( X
.)4(
tan1
2
2
1
a
aa (14-94)
(14-95)
(14-96)
)1()1()0()1( 2121 XXXXaaaa
)cos(21
)1(2
1
cra
aX
.cos2/1or ,1cos2 cc
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 44
where satisfies
Thus the normalized autocorrelations of a damped second order
system with real coefficients subject to random uncorrelated
impulses satisfy (14-97).
More on ARMA processes
From (14-70) an ARMA (p, q) system has only p + q + 1 independent
coefficients, and hence its impulse
response sequence {hk} also must exhibit a similar dependence among
them. In fact according to P. Dienes (The Taylor series, 1931),
.1
1cos
)cos(
22
1
aa
a
(14-98)
1 ,cos
)cos()()( 2
2/
2
an
an n
X
(14-97)
( , 1 , , 0 ),k ia k p b i q
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 45
an old result due to Kronecker1 (1881) states that the necessary and
sufficient condition for to represent a rational
system (ARMA) is that
where
i.e., In the case of rational systems for all sufficiently large n, the
Hankel matrices Hn in (14-100) all have the same rank.
The necessary part easily follows from (14-70) by cross multiplying
and equating coefficients of like powers of
1Among other things “God created the integers and the rest is the work of man.” (Leopold Kronecker)
0( ) k
kkH z h z
det 0, (for all sufficiently large ),nH n N n (14-99)
(14-100)
, 0, 1, 2, .kz k
0 1 2
1 2 3 1
1 2 2
.
n
n
n
n n n n
h h h h
h h h hH
h h h h
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 46
This gives
For systems with
in (14-102) we get
which gives det Hp = 0. Similarly gives 1,i p q
0 0
1 0 1 1
0 1 1
0 1 1 1 10 , 1.
q q q m
q i q i q i q i
b h
b h a h
b h a h a h
h a h a h a h i
(14-102)
(14-101)
1, letting , 1, , 2q p i p q p q p q
0 1 1 1 1
1 1 2 1 1 2
0
0
p p p p
p p p p p p
h a h a h a h
h a h a h a h
(14-103)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 47
and that gives det Hp+1 = 0 etc. (Notice that )
(For sufficiency proof, see Dienes.)
It is possible to obtain similar determinantial conditions for ARMA
systems in terms of Hankel matrices generated from its output
autocorrelation sequence.
Referring back to the ARMA (p, q) model in (14-68),
the input white noise process w(n) there is uncorrelated with its own
past sample values as well as the past values of the system output.
This gives
0, 1, 2,p ka k
0 1 1 1
1 1 2 2
1 1 2 2 2
0
0
0,
p p p
p p p
p p p p p
h a h a h
h a h a h
h a h a h
(14-104)
*{ ( ) ( )} 0, 1E w n w n k k
*{ ( ) ( )} 0, 1.E w n x n k k
(14-105)
(14-106)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 48
Together with (14-68), we obtain
and hence in general
and
Notice that (14-109) is the same as (14-102) with {hk} replaced
*
* *
1 0
*
1 0
{ ( ) ( )}
{ ( ) ( )} { ( ) ( )}
{ ( ) ( )}
i
p q
k kk k
p q
k i k kk k
r E x n x n i
a x n k x n i b w n k w n i
a r b w n k x n i
(14-107)
1
0,p
k i k ik
a r r i q
(14-108)
1
0, 1.p
k i k ik
a r r i q
(14-109)
1/1/2021 Stochastic Processes - Ali Aghagolzadeh 49
by {rk} and hence the Kronecker conditions for rational systems can
be expressed in terms of its output autocorrelations as well.
Thus if X(n) ~ ARMA (p, q) represents a wide sense stationary
stochastic process, then its output autocorrelation sequence {rk}
satisfies
where
represents the Hankel matrix generated from
It follows that for ARMA (p, q) systems, we have
det 0, for all sufficiently large .nD n (14-112)
1rank rank , 0,p p kD D p k (14-110)
(14-111)
( 1) ( 1)k k
0 1 2, , , , , .k kr r r r
0 1 2
1 2 3 1
1 2 2
k
k
k
k k k k
r r r r
r r r rD
r r r r