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

PILLAI/Cha),( tX

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

PILLAI/Cha.)()(

)()()(

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

PILLAI/Cha

)(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

PILLAI/Cha

* *

(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

PILLAI/Cha

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.

PILLAI/Cha

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

PILLAI/Cha

(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