stochastic processes -...
TRANSCRIPT
Stochastic ProcessesAdopted FrompChapter 9
Probability, Random Variables and Stochastic Processes, 4th EditionA. Papoulis and S. Pillai
9. Stochastic Processes Introduction Let denote the random outcome of an experiment. To every such outcome suppose a waveform
is assigned.
),( tX
),( tXis assigned.
The collection of such waveforms form a t h ti Th
),(
),(n
tX
),(k
tX
stochastic process. The set of and the time index t can be continuous
}{ k ),(2
tX
)( tX
or discrete (countablyinfinite or finite) as well.For fixed (the set ofSi
t1
t2
t
),(1
tX
Fig. 9.1
0
For fixed (the set of all experimental outcomes), is a specific time function.For fixed t,
Si
),( 11 itXX
),( tX
2is a random variable. The ensemble of all such realizationsover time represents the stochastic
),( 11 i
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process X(t). (see Fig 9.1). For example
)()(
where is a uniformly distributed random variable in t t h ti St h ti h
),cos()( 0 tatX
(0,2 ),represents a stochastic process. Stochastic processes are everywhere:Brownian motion, stock market fluctuations, various queuing systemsall represent stochastic phenomena.
If X(t) is a stochastic process, then for fixed t, X(t) representsa random variable. Its distribution function is given by
Notice that depends on t, since for a different t, we obtaina different random ariable F rther
})({),( xtXPtxFX
),( txFX
(9-1)
a different random variable. Further (9-2)
dxtxdFtxf X
X
),(),(
3
represents the first-order probability density function of the process X(t).
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For t = t1 and t = t2, X(t) represents two different random variablesX X( ) d X X( ) i l Th i j i di ib i iX1 = X(t1) and X2 = X(t2) respectively. Their joint distribution is given by
and
})(,)({),,,( 22112121 xtXxtXPttxxFX (9-3)
and
(9-4)2
1 2 1 21 2 1 2
1 2
( , , , )( , , , ) XX
F x x t tf x x t tx x
represents the second-order density function of the process X(t).Similarly represents the nth order density),, ,,,( 2121 nn tttxxxf X
1 2
function of the process X(t). Complete specification of the stochasticprocess X(t) requires the knowledge of for all and for all n (an almost impossible task
),, ,,,( 2121 nn tttxxxf X niti 21
4
for all and for all n. (an almost impossible taskin reality).
niti ,,2 ,1 ,
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Mean of a Stochastic Process:
(9 5) ( ) { ( )} ( )t E X t f t d
represents the mean value of a process X(t). In general, the mean of a process can depend on the time index t
(9-5)( ) { ( )} ( , )Xt E X t x f x t dx
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
(9-6)* *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
p pX1 = X(t1) and X2 = X(t2) generated from the process X(t).
Properties:Properties:
1. *1
*212
*21 )}]()({[),(),( tXtXEttRttR XXXX (9-7)
52.
( )
.0}|)({|),( 2 tXEttRXX
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(Average instantaneous power)
3. represents a nonnegative definite function, i.e., for anyset of constants na }{
),( 21 ttRXX
set of constants
2n
iia 1}{
n
i
n
jjiji ttRaa XX
1 1
* .0),( (9-8)
Eq. (9-8) follows by noticing that The function
.)(for 0}|{|1
2
i
ii tXaYYE
)()()()( * ttttRttC (9-9)
represents the autocovariance function of the process X(t).Example 9 1
)()(),(),( 212121 ttttRttC XXXXXX (9-9)
Example 9.1Let
.)(
T
TdttXz
Then
T
T T
dtdttXtXEzE
212*
12 )}()({]|[|
6
T
T
T
T
T T
dtdtttR
dtdttXtXEzE
XX
2121
2121
),(
)}()({]|[|
(9-10)PILLAI/Cha
Example 9.2
)20()()( (9 11)).2,0(~ ),cos()( 0 UtatX (9-11)
This gives
,0}{sinsin}{coscos)}{cos()}({)(
0 0
0
EtaEtataEtXEtX
(9-12)
Similarly
2
0 }.{sin0cos}{cos since 2
1 EdE
Similarly
)}cos(){cos(),(2
20102
21
a
ttEattRXX
)(
)}2)(cos()({cos2
2
210210
tta
ttttEa
(9 13)7
).(cos2 210 tt (9-13)
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Stationary Stochastic ProcessesStationary processes exhibit statistical properties that are
invariant to shift in the time index. Thus, for example, second-orderstationarity 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. { ( 1) , ( 2) } { ( 1 ) , ( 2 )} ySimilarly 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 theIn strict terms, the statistical properties are governed by thejoint probability density function. Hence a process is nth-orderStrict-Sense Stationary (S.S.S) if
for any c, where the left side represents the joint density function of
),, ,,,(),, ,,,( 21212121 ctctctxxxftttxxxf nnnn XX (9-14)
y , p j ythe random variables andthe right side corresponds to the joint density function of the randomvariables
)( , ),( ),( 2211 nn tXXtXXtXX
)()()( 2211 ctXXctXXctXX
8
variables A process X(t) is said to be strict-sense stationary if (9-14) is true for all
).( , ),( ),( 2211 ctXXctXXctXX nn
. and ,2,1 ,,,2,1 , canynniti PILLAI/Cha
For a first-order strict sense stationary process,from (9-14) we have( )
for any c In particular c = – t gives
),(),( ctxftxf XX (9-15)
for any c. In particular c = – t gives
i h fi d d i f X( ) i i d d f I h
(9-16))(),( xftxf XX
i.e., the first-order density of X(t) is independent of t. In that case
(9-17) [ ( )] ( ) , E X t x f x dx a constant.
Similarly, for a second-order strict-sense stationary processwe have from (9-14)
)()( ff
for any c. For c = – t2 we get
), ,,(), ,,( 21212121 ctctxxfttxxf XX
9) ,,(), ,,( 21212121 ttxxfttxxf XX (9-18)
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i.e., the second order density function of a strict sense stationary process depends only on the difference of the time indices .21 ttp p yIn that case the autocorrelation function is given by
21
*1 2 1 2( , ) { ( ) ( )}XXR t t E X t X t
(9-19)
*1 2 1 2 1 2 1 2
*1 2
( , , )
( ) ( ) ( ),X
XX XX XX
x x f x x t t dx dx
R t t R R
i.e., the autocorrelation function of a second order strict-sensestationary process depends only on the difference of the time
( )1 2( ) ( ) ( ),XX XX XX
y p p yindices Notice that (9-17) and (9-19) are consequences of the stochastic process being first and second-order strict sense stationary
.21 tt
process being first and second-order strict sense stationary. On the other hand, the basic conditions for the first and second order stationarity – Eqs. (9-16) and (9-18) – are usually difficult to verify.I th t ft t t l d fi iti f t ti it
10
In that case, we often resort to a looser definition of stationarity,known as Wide-Sense Stationarity (W.S.S), by making use of
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(9-17) and (9-19) as the necessary conditions. Thus, a process X(t)is said to be Wide-Sense Stationary ify(i)and(ii)
)}({ tXE
(9-21)
(9-20)
)()}()({ * ttRtXtXE(ii)
i.e., for wide-sense stationary processes, the mean is a constant and h l i f i d d l h diff b
(9-21)),()}()({ 2121 ttRtXtXE XX
the autocorrelation function depends only on the difference between the time indices. Notice that (9-20)-(9-21) does not say anything about the nature of the probability density functions, and instead deal p y ywith the average behavior of the process. Since (9-20)-(9-21) follow from (9-16) and (9-18), strict-sense stationarity always implies wide-sense stationarity However the converse is not true inimplies 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
j i tl G i d)()()( XXXXXX11
are jointly Gaussian randomvariables for any whose joint characteristic function is given by
)( , ),( ),( 2211 nn tXXtXXtXX
PILLAI/Chanttt ,, 21
h i d fi d (9 9) If X(t) i id)(C
1 ,
( ) ( , ) / 2
1 2( , , , )XX
n n
k k i k i kk l k
X
j t C t t
n e
(9-22)
where is as defined on (9-9). If X(t) is wide-sense stationary, then using (9-20)-(9-21) in (9-22) we get
),( ki ttCXX
1n n n
and hence if the set of time indices are shifted by a constant c to
12
1 1 1 1
( )
1 2( , , , )XXk i k i k
k kX
j C t t
n e
(9-23)
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 f ti i id ti l t (9 23) Th th t f d i bl nX }{
),( 11 ctXX )(,),( 22 ctXXctXX nn
function is identical to (9-23). 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
iiX 1}{ niiX 1}{
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).
12
wide sense stationarity (w.s.s) strict sense stationarity (s.s.s).Notice that since the joint p.d.f of Gaussian random variables dependsonly on their second order statistics, which is also the basis
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for wide sense stationarity, we obtain strict sense stationarity as well.From (9-12)-(9-13), (refer to Example 9.2), the process
i (9 11) i id t ti b t)cos()( tatX in (9-11) is wide-sense stationary, butnot strict-sense stationary.
),cos()( 0 tatX2
t
Similarly if X(t) is a zero mean wide sense stationary process in Example 9.1, then in (9-10) reduces to2
T T
T
1
t
21tt
then in (9 10) reduces to
A t t i f T t +T i
z
.)(}|{|
2121
22
T
T
T
Tz dtdtttRzE XX
tt
T2
As t1, t2 varies from –T to +T, variesfrom –2T to + 2T. Moreover is a constantover the shaded region in Fig 9.2, whose area is given by
21 tt )(XXR
)0(
Fig. 9.2
and hence the above integral reduces to
dTdTT )2()2(21)2(
21 22
13
and hence the above integral reduces to
PILLAI/Cha.)1)((|)|2)((
2
2 2||
212
2 2
T
t TTT
tz dRdTR XXXX
(9-24)
Systems with Stochastic InputsA deterministic system1 transforms each input waveform into),( itX y pan 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
),( i)],([),( ii tXTtY
)}({ tYto a process X(t) generates a new set of realizations at the output associated with a new process Y(t).
)},({ tY
)(tY
][T )(tX )(tY
),(i
tX ),(
itY
][t t
Fig. 9.3
Our goal is to study the output process statistics in terms of the inputprocess statistics and the system function.
141A stochastic system on the other hand operates on both the variables t and .
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Deterministic Systems
Systems with MemoryMemoryless Systems
)]([)( tXgtY
Time-Invariantsystems
Linear systems)]([)( tXLtY
Time-varyingsystems systems
Linear Time Invariant
)]([)(systemsFig. 9.3
Linear-Time Invariant(LTI) systems
)()()(
dXthtY( )h t( )X t
15
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)()()(
dtXh
dXthtY( )h t( )X t
LTI system
Memoryless Systems:The output Y(t) in this case depends only on the present value of the input X(t). i.e., (9-25))}({)( tXgtY
i Memorylesssystem
Strict-sense stationary input
Strict-sense stationary output.
MemorylessWide-sense Need not be
(see (9-76), Text for a proof.)
Memorylesssystem
Wide sense stationary input stationary in
any sense.
Memorylesssystem
X(t) stationary Gaussian with
)(R
Y(t) stationary,butnot Gaussian with
)()( RR
16
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)(XXR(see (9-26)).
).()( XXXY RRFig. 9.4
Theorem: If X(t) is a zero mean stationary Gaussian process, andY(t) = g[X(t)], where represents a nonlinear memoryless device, )(gthen
)}.({ ),()( XgERR XXXY (9-26)
Proof:
)}]({)([)}()({)( tXgtXEtYtXERXY
212121 ),()(21
dxdxxxfxgx XX (9-27)
where are jointly Gaussian random variables, and hence
)( ),( 21 tXXtXX
* 1 / 21( ) x A xf x x e
1 2 1 2
1 2 1 2
2 | |( , )
( , ) , ( , )
X X
T T
Af x x e
X X X x x x
17
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* *
(0) ( )
( ) (0){ } XX XX
XX XX
R R
R RA E X X LL
where L is an upper triangular factor matrix with positive diagonal entries. i.e.,
ll
Consider the transformation
. 0
22
1211
lll
L
so that
1 1 1 2 1 2( , ) , ( , )T TZ L X Z Z z L x z z
so that
d h Z Z i d d t G i d
IALLLXXELZZE 11 *1**1* }{}{
and hence Z1, Z2 are zero mean independent Gaussian random variables. Also
zlxzlzlxzLx
and hence* * *1 * 1 2 2A L A L
22222121111 , zlxzlzlxzLx
18The Jacobaian of the transformation is given by
1 1 2 21 2 .x A x z L A Lz z z z z
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Hence substituting these into (9 27) we obtain.|||||| 2/11 ALJ
Hence substituting these into (9-27), we obtain2 21 2
1/ 211 1 12 2 22 2
/ 2 / 21 1| | 2 | |
( ) ( ) ( )XY J Az zR l z l z g l z e e
11 1 22 2 1 21 2
1 2
( ) ( ) ( )
( ) ( ) ( )
z zl z g l z f z f z dz dz
l z g l z f z f z dz dz
12 2 22 2 1 21 2 1 2 ( ) ( ) ( )z zl z g l z f z f z dz dz
( ) ( ) ( )l f d l f d
0
11 1 1 22 2 21 2
12 2 22 2 22
1 2
2
( ) ( ) ( )
( ) ( )
z z
z
l z f z dz g l z f z dz
l z g l z f z dz
22
212 22
/ 2
2
12
/ 21( )
z
u ll
e
ug u e du
19where This gives222 2
( ) ,l ug u e du
PILLAI/Cha22 2 .u l z
222
2
2
( )
/ 2112 22( ) ( )
uf u
u lulR l l g u e du
222 22212 22 2
( )( )
( ) ( ) XY
uu
df uf u
du
llR l l g u e du
( ) ( ) ( ) ,XX uR g u f u du
Hence)(givessince * RllLLA Hence).( gives since 2212 XXRllLLA
})()(|)()(){()(
XXXY duufugufugRR uu
0
),()}({)(
})()(|)()(){()(
XXXX
XXXY
RXgER
fgfg uu
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
)].([ XgE
20
u c o be wee e pu d e ou pu s p opo o o einput autocorrelation function.
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Linear Systems: represents a linear system if][L
)}({)}({)}()({ tXLatXLatXatXaL (9-28)
Let )}({)( tXLtY
)}.({)}({)}()({ 22112211 tXLatXLatXatXaL (9 28)
(9-29)
represent the output of a linear system.Time-Invariant System: represents a time-invariant system if][L
i.e., shift in the input results in the same shift in the output also.
)()}({)}({)( 00 ttYttXLtXLtY (9-30)
If satisfies both (9-28) and (9-30), then it corresponds to a linear time-invariant (LTI) system.LTI systems can be uniquely represented in terms of their output to
][L
y q y p pa delta function
LTI)(t )(th
Impulseresponse ofthe system
)(th
21
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LTI)(t )(th
Impulse
t
Impulseresponse
Fig. 9.5
then)(tX
)(tY
LTI
)()()( dXthtY
bit
t
t
Fig 9 6
)(tX )(tY
Eq. (9-31) follows by expressing X(t) as(9-31)
)()( dtXh
arbitraryinput
Fig. 9.6
q ( ) y p g ( )
and applying (9-28) and (9-30) to Thus)}({)( tXLtY
)()()( dtXtX (9-32)
and applying (9-28) and (9-30) to Thus)}.({)( tXLtY
})()({)}({)(
dtXLtXLtY
)}({)(
})()({
dtLX
dtXL By Linearity
By Time-invariance
22(9-33)PILLAI/Cha
.)()()()(
dtXhdthX
Output Statistics: Using (9-33), the mean of the output processis given byg y
)()()()(
})()({)}({)(
hdh
dthXEtYEtY
(9 34)
Similarly the cross-correlation function between the input and outputi i b
).()()()( thtdth XX (9-34)
processes is given by)}()({),(
*
2*
121 tYtXEttRXY
*
)()}()({
})()()({ *
21
*
21
dhtXtXE
dhtXtXE
*
*
)()(
)(),(*
*
21
thttR
dhttRXX
(9 35)23Finally the output autocorrelation function is given by
).(),( 221 thttRXX (9-35)
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})()()({
)}()({),( *
2*
121
tYdhtXE
tYtYEttRYY
)()}()({
})( )()({
21
21
dhtYtXE
tYdhtXE
*
),(),(
)(),(
121
21
thttR
dhttR
XY
XY
(9-36)
or ),(),( 121XY
)()()()( * ththttRttR (9 37)).()(),(),( 122121 ththttRttR XXYY (9-37)
h(t))(tX )(tY
h*(t2) h(t1) ),( 21 ttRXY )( 21 ttR)( 21 ttR
(a)
24
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h (t2) h(t1) ),( 21 ttRYY),( 21 ttRXX(b)
Fig. 9.7
In particular if X(t) is wide-sense stationary, then we haveso that from (9-34)
XX t )(so that from (9 34)
Al th t (9 35) d t
constant.a cdht XXY ,)()(
(9-38)
)()( ttRttRAlso so that (9-35) reduces to)(),( 2121 ttRttR XXXX
)()(),( *
2121 dhttRttR XXXY
Thus X(t) and Y(t) are jointly w.s.s. Further, from (9-36), the output
(9-39). ),()()( 21*
ttRhR XYXX
Thus X(t) and Y(t) are jointly w.s.s. Further, from (9 36), the output autocorrelation simplifies to
,)()(),( 21
2121 XYYY ttdhttRttR
From (9-37), we obtain).()()(
YYXY RhR
(9-40)
25).()()()( * hhRR XXYY (9-41)
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From (9-38)-(9-40), the output process is also wide-sense stationary.This gives rise to the following representationg g p
LTI systemwide-sense wide-sense)(tX )(tY
LTI systemh(t)stationary process
wide-sense stationary process.
(a)
strict-sense t ti
strict-senseLTI systemh(t)
)(tX )(tYstationary process stationary process
(see Text for proof )
h(t)(b)
Linear systemGaussianprocess (also
Gaussian process(also stationary)
)(tX )(tY
26
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p (stationary) (c)
Fig. 9.8
White Noise Process:W(t) is said to be a white noise process if ( ) p
i e E[W(t1) W*(t2)] = 0 unless t1 = t2
),()(),( 21121 tttqttRWW (9-42)
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
).()(),( 2121 qttqttRWW (9-43)
its samples are independent random variables (why?).
LTI Colored noiseWhite noise
W(t)
LTIh(t)
Co o ed o se( ) ( ) ( )N t h t W t
Fig 9 927For w.s.s. white noise input W(t), we have
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Fig. 9.9
(9-44)
[ ( )] ( ) ,WE N t h d
a constant
and
(9 44)[ ( )] ( ) , WE N t h d a constant
)()()(
)()()()(*
*
qhqh
hhqRnn
(9-45)
where
Thus the output of a white noise process through an LTI system
.)()()()()( **
dhhhh (9-46)
Thus the output of a white noise process through an LTI system represents a (colored) noise process.Note: White noise need not be Gaussian.
“Whit ” d “G i ” t diff t t !28
“White” and “Gaussian” are two different concepts!
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Upcrossings and Downcrossings of a stationary Gaussian process:Consider a zero mean stationary Gaussian process X(t) withConsider a zero mean stationary Gaussian process X(t) with
autocorrelation function An upcrossing over the mean value occurs whenever the realization X(t)
h h i h
).(XXR
passes through zero withpositive slope. Let represent the probability
tUpcrossings
)(tX
p p yof such an upcrossing inthe interval We wish to determine
). ,( ttt .
t
We wish to determine
Since X(t) is a stationary Gaussian process, its derivative process i l t ti G i ith t l ti f ti
Fig. 9.10
)(tX
Downcrossing
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))()( XXXX RR
)(tX
29,)()(
ddRR XX
XX
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we have
)()()()( XXXX RdRdRR
(9-47)
which for gives0
)()(
)(
XXXX Rdd
R
(9 47)
i e the jointly Gaussian zero mean random variables
(9-48)(0) 0 [ ( ) ( )] 0XXR E X t X t
i.e., the jointly Gaussian zero mean random variables
l t d d h i d d t ith i
)( and )( 21 tXXtXX (9-49)
are uncorrelated and hence independent with variances
0 )0( )0( and )0( 22
21 XXXXXX RRR (9-50)
respectively. Thus 2 21 1
2 21 2
1 2 1 22 21( , ) ( ) ( ) .X X X X
x x
f x x f x f x e
(9-51)
30To determine the probability of upcrossing rate, ,
1 2 1 2 1 21 2
( , ) ( ) ( )2X X X Xf f f
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we argue as follows: In an interval the realization moves from X(t) = X1 to
),,( ttt )()()( tXXttXtXttX from X(t) X1 to
and hence the realization intersects with the zero level somewherein that interval if
,)()()( 21 tXXttXtXttX
i.e., 1 2 .X X t
(9-52)
)(tX
1 2 1 20, 0, and ( ) 0 X X X t t X X t
Hence the probability of upcrossingin is given by ) ,( ttt t
)(tX)( ttX
ttt
0
(9-53)
)(tX Fig. 9.11.)()(
),(
1
12
0 2
0 210
21
12
2 21 21
xdxfxdxf
dxxdxxftx txx
XX
XX
Differentiating both sides of (9-53) with respect to we get(9 53))()( 1120 2
2 12ff
tx XX
,t
(9 54)
( ) ( )f x x f x t dx
31
PILLAI/Chaand letting Eq. (9-54) reduce to
(9-54)2 12 2 2 20( ) ( )X Xf x x f x t dx
,0t
)()0(2
1)0()(
0 222
0 222 X
XX
XX dxxfxR
dxfxfx
)0()0(
21)/2(
21
)0(21
2XX
XX
XX
XX
RR
R
(9-55)
[where we have made use of (5-78), Text]. There is an equal probability for downcrossings, and hence the total probability for
i h li i i l l h)( crossing the zero line in an interval equals where),( ttt ,0 t
.0 )0(/)0(10 XXXX RR (9-56)
It follows that in a long interval T, there will be approximately crossings of the mean value If is large then the
)()(0 XXXX (9 56)
T0)0(R crossings of the mean value. If is large, then the
autocorrelation function decays more rapidly as movesaway from zero, implying a large random variation around the origin ( l ) f X( ) d th lik lih d f i h ld
)0( XXR)(XXR
32
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(mean value) for X(t), and the likelihood of zero crossings should increase with increase in agreeing with (9-56). (0),XXR
Discrete Time Stochastic Processes:
A discrete time stochastic process Xn = X(nT) is a sequence of p n ( ) qrandom variables. The mean, autocorrelation and auto-covariance functions of a discrete-time process are gives by
)}({ ( )
d)}()({),(
)}({
2*
121 TnXTnXEnnR
nTXEn
(9-57)
(9-58)and
*2121 21),(),( nnnnRnnC (9-59)
respectively. As before strict sense stationarity and wide-sense stationarity definitions apply here also.For example X(nT) is wide sense stationary ifFor example, X(nT) is wide sense stationary if
dconstanta nTXE ,)}({ (9-60)
33
and
PILLAI/Cha(9-61)* *[ {( ) } {( ) }] ( ) n nE X k n T X k T R n r r
i.e., R(n1, n2) = R(n1 – n2) = R*(n2 – n1). The positive-definite property of the autocorrelation sequence in (9-8) can be expressed in terms of certain Hermitian Toeplitz matrices as follows:in terms of certain Hermitian-Toeplitz matrices as follows: Theorem: A sequence forms an autocorrelation sequence ofa wide sense stationary stochastic process if and only if every
i i li i i b
}{ nr
Hermitian-Toeplitz matrix Tn given by
*210 nrrrr
*
***
110*
1
nn
n Trrrr
T
(9-62)
is non-negative (positive) definite for P f L bi
0, 1, 2, , .n
011 nn rrrr
TProof: Let represent an arbitrary constant vector.Then from (9-62),
Tnaaaa ],,,[ 10
n n
ikkin raaaTa ** (9-63)
34since the Toeplitz character gives Using (9-61),Eq. (9-63) reduces to PILLAI/Cha
i k
ikkin0 0
(9 63)
.)( , ikkin rT
(9-64)2
* * * *{ ( ) ( )} ( ) 0.n n n
n i k ka T a a a E X kT X iT E a X kT
From (9-64), if X(nT) is a wide sense stationary stochastic processh i i d fi i i f
( )0 0 0
{ ( ) ( )} ( )n i k ki k k
then Tn is a non-negative definite matrix for everySimilarly the converse also follows from (9-64). (see section 9.4, Text)
.,,2,1,0 n
If X(nT) represents a wide-sense stationary input to a discrete-timesystem {h(nT)}, and Y(nT) the system output, then as before the crosscorrelation function satisfiescorrelation function satisfies
and the output autocorrelation function is given by)()()( * nhnRnR XXXY (9-65)
or)()()( nhnRnR XYYY
).()()()( * nhnhnRnR XXYY
(9-66)
(9-67)
35Thus wide-sense stationarity from input to output is preserved for discrete-time systems also.
PILLAI/Cha
Auto Regressive Moving Average (ARMA) Processes
Consider an input – output representation
)()()( qp
knWbknXanX (9-68)
where X(n) may be considered as the output of a system {h(n)}
,)()()(01
k
kk
k knWbknXanX ( )
driven by the input W(n). Z – transform of (9-68) gives
h(n)W(n) X(n)
Fi 9 12(9 68) gives
(9-69)00 0
( ) ( ) , 1p q
k kk k
k kX z a z W z b z a
Fig.9.12
or0 0k k
1 2
0 1 2( ) ( )( ) ( )q
qk b b z b z b zX z B zH h k
36
1 20 1 2
( ) ( )( ) ( )( ) ( )1
qkp
k p
H z h k zW z A za z a z a z
(9-70) PILLAI/Cha
represents the transfer function of the associated system response {h(n)}in Fig 9.12 so that
Notice that the transfer function H(z) in (9-70) is rational with p poles
(9-71).)()()(0
k
kWknhnX
and q zeros that determine the model order of the underlying system.From (9-68), the output undergoes regression over p of its previous values and at the same time a moving average based on )1()( nWnWvalues 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))
X( ) G ll th i t {W( )} t f
),1(),( nWnW)( , qnW
process X(n). Generally the input {W(n)} represents a sequence of uncorrelated random variables of zero mean and constant variance so that (9 72)
2W
)()( 2 nnR
If in addition, {W(n)} is normally distributed then the output {X(n)} also represents a strict-sense stationary normal process.
(9-72)).()( nnR WWW
37
also represents a strict sense stationary normal process.If q = 0, then (9-68) represents an AR(p) process (all-pole
process), and if p = 0, then (9-68) represents an MA(q) PILLAI/Cha
process (all-zero process). Next, we shall discuss AR(1) and AR(2)processes through explicit calculations.AR(1) A AR(1) h th f ( (9 68))AR(1) process: An AR(1) process has the form (see (9-68))
)()1()( nWnaXnX (9-73)and from (9-70) the corresponding system transfer
(9-74)
111)( nn zazH
provided | a | < 1. Thus
( )
011
)(naz
represents the impulse response of an AR(1) stable system. Using
1|| ,)( aanh n (9-75)
(9-67) together with (9-72) and (9-75), we get the output autocorrelation sequence of an AR(1) process to be
|||| a n
kk
38
PILLAI/Cha
22
0
||22
1}{}{)()(
aaaaaannR
k
kknnnWWWXX
(9-76)
where we have made use of the discrete version of (9-46). The normalized (in terms of RXX (0)) output autocorrelation sequence isi bgiven by
.0 || ,)0()()( || na
RnRn n
XX
XXX (9-77)
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) Thusterm associated with observing a first order AR process X(n). Thus
h X( ) AR(1) i (9 73) d V( ) i l t d d
)()()( nVnXnY (9-78)
where X(n) ~ AR(1) as in (9-73), and V(n) is an uncorrelated randomsequence with zero mean and variance that is also uncorrelated with {W(n)}. From (9-73), (9-78) we obtain the output
2V
autocorrelation of the observed process Y(n) to be )()()()()( 2 nnRnRnRnR VXXVVXXYY
39
PILLAI/Cha
)(1
22
||2 n
aa
VW
n
(9-79)
so that its normalized version is given by
1 0( )( ) YYnR n
where(9-80)
2
| |
( )( ) (0) 1, 2,
YYY
YYnn
R c a n
Eqs. (9-77) and (9-80) demonstrate the effect of superimposing
.1)1( 222
ac
VW
W
(9-81)
q ( ) ( ) p p gan error sequence on an AR(1) model. For non-zero lags, the autocorrelation of the observed sequence {Y(n)}is reduced by a constantfactor compared to the original process {X(n)}factor compared to the original process {X(n)}.From (9-78), the superimposederror sequence V(n) only affectsh di i Y( )
1)0()0( YX
the corresponding term in Y(n)(term by term). However,a particular term in the “input sequence” n
)()( kkYX
0
40
p p qW(n) affects X(n) and Y(n) as well asall subsequent observations.
PILLAI/ChaFig. 9.13
k0
AR(2) Process: An AR(2) process has the form
)()2()1()( 21 nWnXanXanX (9-82)
and from (9-70) the corresponding transfer function is given by
)()()()( 21 ( )
1 bb
so that
(9-83)12
21
1
1
02
21
1 1111)()(
z
bz
bzaza
znhzHn
n
so that
d i t f th l f th t f f ti
(9-84)2 ),2()1()( ,)1( ,1)0( 211 nnhanhanhahh
d and in term of the poles of the transfer function, from (9-83) we have
(9-85)0)( nbbnh nn
21 and
that represents the impulse response of the system. From (9-84)-(9-85), we also have
(9 85)0 ,)( 2211 nbbnh
1 1221121 abbbb
41
From (9 84) (9 85), we also have From (9-83),
PILLAI/Cha
. ,1 1221121 abbbb
, , 221121 aa (9-86)
and H(z) stable impliesFurther, using (9-82) the output autocorrelations satisfy the recursion
.1|| ,1|| 21
)}()({)( *XXER
)}()({
)}()]2()1({[
)}()({)(
*
*21
XWE
mXmnXamnXaE
mXmnXEnRXX
0
and hence their normalized version is given by(9-87))2()1(
)}()({
21
*
nRanRamXmnWE
XXXX
0
and hence their normalized version is given by
B di t l l ti i (9 67) th t t t l ti
(9-88)1 2( )( ) ( 1) ( 2).(0)
XXX X X
XX
R nn a n a nR
By direct calculation using (9-67), the output autocorrelations are given by
*2* )()()()()()( nhnhnhnhnRnR WWWXX
0
*2 )()( k
khknhW
(9-89)
42
PILLAI/Cha
22
*2
22
*21
*2
*21
2*1
*12
*1
21
*1
212
||1)(||
1)(
1)(
||1)(||
nnnn bbbbbbW
(9 89)
where we have made use of (9-85). From (9-89), the normalizedoutput autocorrelations may be expressed as
)(
where c1 and c2 are appropriate constants.
(9-90)nn
XX
XXX cc
RnRn *
22*11)0(
)()(
w e e c1 d c2 e pp op e co s s.Damped Exponentials: When the second order system in (9-83)-(9-85) is real and corresponds to a damped exponential response the poles are complex conjugate which gives 2 4 0a a response, the poles are complex conjugate which gives in (9-83). Thus
1 24 0a a
*1 2 1 1jr e r (9-91)
In that case in (9-90) so that the normalized correlations there reduce to
* 1 2
jc c c e 1 2 1, , 1.r e r (9 91)
But from (9 86)
(9-92)).cos(2}Re{2)( *11 ncrcn nn
X
43
But from (9-86)
PILLAI/Cha,1 ,cos2 2
2121 arar (9-93)
and hence which gives21 22 sin ( 4 ) 0r a a
)4( 221 aa (9 94)
Also from (9-88)
.)4(tan1
21
aaa
(9-94)
so o (9 88)
so that
)1()1()0()1( 2121 XXXX aaaa
so that
(9-95))cos(21
)1(2
1
cra
aX
where the later form is obtained from (9-92) with n = 1. But in (9-92) gives
1)0( X2
( ) g
Substituting (9 96) into (9 92) and (9 95) we obtain the normalized
(9-96).cos2/1or ,1cos2 cc
44
Substituting (9-96) into (9-92) and (9-95) we obtain the normalizedoutput autocorrelations to be
PILLAI/Cha
1 ,cos
)cos()()( 22/
2
anan nX
(9-97)
where satisfies
1)cos( a
cos
Thus the normalized autocorrelations of a damped second order
.11cos
)cos(22
1
aaa
(9-98)
Thus the normalized autocorrelations of a damped second order system with real coefficients subject to random uncorrelated impulses satisfy (9-97).
More on ARMA processes
From (9-70) an ARMA (p, q) system has only p + q + 1 independentcoefficients, and hence its impulse response sequence {hk} also must exhibit a similar dependence among
( , 1 , , 0 ),k ia k p b i q
45
response sequence {hk} also must exhibit a similar dependence amongthem. In fact according to P. Dienes (The Taylor series, 1931),
PILLAI/Cha
an old result due to Kronecker1 (1881) states that the necessary and sufficient condition for to represent a rational
t (ARMA) i th t0( ) k
kkH z h z
system (ARMA) is that
det 0, (for all sufficiently large ),nH n N n (9-99)
where0 1 2 nh h h h
h h h h
(9-100)1 2 3 1
1 2 2
.nn
h h h hH
h h h h
i.e., In the case of rational systems for all sufficiently large n, theHankel matrices Hn in (9-100) all have the same rank.
1 2 2n n n nh h h h
The necessary part easily follows from (9-70) by cross multiplyingand equating coefficients of like powers of 0 1 2kz k
46
and equating coefficients of like powers of
1Among other things “God created the integers and the rest is the work of man.” (Leopold Kronecker)PILLAI/Cha
, 0, 1, 2, .z k
This gives
b h0 0
1 0 1 1
b hb h a h
(9-101)
0 1 1
0 1 1 1 10 1q q q mb h a h a h
h a h a h a h i
(9-102)
For systems with
0 1 1 1 10 , 1.q i q i q i q ih a h a h a h i (9-102)
1, letting , 1, , 2q p i p q p q p q in (9-102) we get
0 1 1 1 1 0p p p ph a h a h a h
1 1 2 1 1 2 0p p p p p ph a h a h a h
(9-103)
47
PILLAI/Cha
which gives det Hp = 0. Similarly gives 1,i p q
0 1 1 1
1 1 2 2
0
0p p p
p p p
h a h a h
h a h a h
1 1 2 2
1 1 2 2 2 0,
p p p
p p p p ph a h a h
(9-104)
and that gives det Hp+1 = 0 etc. (Notice that )(For sufficiency proof, see Dienes.)I i ibl b i i il d i i l di i f ARMA
0, 1, 2,p ka k 1 1 2 2 2p p p p p
It is possible to obtain similar determinantial conditions for ARMA systems in terms of Hankel matrices generated from its outputautocorrelation sequence.q
Referring back to the ARMA (p, q) model in (9-68), the input white noise process w(n) there is uncorrelated with its ownpast sample values as well as the past values of the system outputpast sample values as well as the past values of the system output.This gives
*{ ( ) ( )} 0, 1E w n w n k k (9-105)
48
PILLAI/Cha
{ ( ) ( )} ,
*{ ( ) ( )} 0, 1.E w n x n k k
( )
(9-106)
Together with (9-68), we obtain*{ ( ) ( )}E i
* *
{ ( ) ( )}
{ ( ) ( )} { ( ) ( )}
ip q
k k
r E x n x n i
a x n k x n i b w n k w n i
1 0
*
1 0{ ( ) ( )}
k kp q
k i k kk k
a r b w n k x n i
(9-107)
and hence in general
1 0k k
p
and1
0,p
k i k ik
a r r i q
(9-108)
and
10, 1.
p
k i k ik
a r r i q
(9-109)
49
PILLAI/ChaNotice that (9-109) is the same as (9-102) with {hk} replaced
by {rk} and hence the Kronecker conditions for rational systems canbe 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} satisfiess s es
where
1rank rank , 0,p p kD D p k (9-110)
where
0 1 2 kr r r r
(9-111)1 2 3 1kk
r r r rD
r r r r
represents the Hankel matrix generated from It follows that for ARMA (p q) systems we have
( 1) ( 1)k k 0 1 2k kr r r r
1 2 2k k k kr r r r
50
PILLAI/Cha
It follows that for ARMA (p, q) systems, we have
det 0, for all sufficiently large .nD n (9-112)
0 1 2, , , , , .k kr r r r