1

14. Stochastic Processes

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 continuousor 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 realizationsover time represents the stochastic

ξ

),( ξtX

}{ kξ

Si ∈ξ

),( 11 itXX ξ=

),( ξtX

PILLAI/Cha

t1

t2

t

),(n

tX ξ

),(k

tX ξ

),(2

ξtX

),(1

ξtX

Fig. 14.1

),( ξtX

0

),( ξtX

Introduction

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 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 variable. Further

represents the first-order probability density function of the process X(t).

),cos()( 0 ϕω += tatX

ϕ

})({),( xtXPtxFX ≤=

),( txFX

(14-1)

(14-2)

PILLAI/Cha

(0,2 ),π

dxtxdFtxf X

X

),(),( =∆

3

For t = t1 and t = t2, X(t) represents two different random variablesX1 = 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 densityfunction 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 taskin reality).

})(,)({),,,( 22112121 xtXxtXPttxxFX ≤≤= (14-3)

(14-4)

),, ,,,( 2121 nn tttxxxf X

),, ,,,( 2121 nn tttxxxf X

niti , ,2 ,1 , =

PILLAI/Cha

21 2 1 2

1 2 1 21 2

( , , , )( , , , )

XX

F x x t tf x x t tx x

∂=

∂ ∂∆

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 variablesX1 = X(t1) and X2 = X(t2) generated from the process X(t).

Properties:

1.

2.

(14-5)

(14-6)

*1

*212

*21 )}]()({[),(),( tXtXEttRttR XXXX == (14-7)

.0}|)({|),( 2 >= tXEttRXX

PILLAI/Cha

(Average instantaneous power)

( ) { ( )} ( , )Xt E X t x f x t dxµ

+∞

−∞= = ∫∆

* *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= = ∫ ∫

∆

5

3. represents a nonnegative definite function, i.e., for anyset of constants

Eq. (14-8) follows by noticing that The function

represents the autocovariance function of the process X(t).Example 14.1Let

Then

.)(for 0}|{|1

2 ∑=

=≥n

iii tXaYYE

)()(),(),( 2*

12121 ttttRttC XXXXXX µµ−= (14-9)

.)(

∫−=T

TdttXz

∫ ∫

∫ ∫

− −

− −

=

=T

T

T

T

T

T

T

T

dtdtttR

dtdttXtXEzE

XX

2121

212*

12

),(

)}()({]|[|

(14-10)

niia 1}{ =

),( 21 ttRXX

∑∑= =

≥n

i

n

jjiji ttRaa XX

1 1

* .0),( (14-8)

PILLAI/Cha

6

Similarly

,0}{sinsin}{coscos)}{cos()}({)(

0 0

0

=−=+==

ϕωϕωϕωµ

EtaEtataEtXEtX

).(cos2

)}2)(cos()({cos2

)}cos(){cos(),(

210

2

210210

22010

221

tta

ttttEa

ttEattRXX

−=

+++−=

++=

ω

ϕωω

ϕωϕω

(14-12)

(14-13)

Example 14.2

).2,0(~ ),cos()( 0 πϕϕω UtatX += (14-11)

This gives

PILLAI/Cha

∫ ===π

ϕϕϕϕ π2

0 }.{sin0cos}{cos since 2

1 EdE

7

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. 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 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 the random variables andthe right side corresponds to the joint density function of the randomvariables 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 == PILLAI/Cha

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 processwe have from (14-14)

for any c. For c = – t2 we get

),(),( ctxftxf XX +≡

(14-16)

(14-15)

(14-17)

)(),( xftxf XX =

[ ( )] ( ) , E X t x f x dx a constant.µ

+∞

−∞= =∫

), ,,(), ,,( 21212121 ctctxxfttxxf XX ++≡

) ,,(), ,,( 21212121 ttxxfttxxf XX −≡ (14-18)

PILLAI/Cha

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-sensestationary 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 τ=− tt

.21 tt −=τ

(14-19)

PILLAI/Cha

*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

τ

τ τ

=

= = −

= − = = −∫ ∫

∆

∆

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 randomvariables for any whose joint characteristic function is given by

µ=)}({ tXE

(14-21)

(14-20)

),()}()({ 212*

1 ttRtXtXE XX −=

)( , ),( ),( 2211 nn tXXtXXtXX ===

PILLAI/Chanttt ,, 21

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, thenwide-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

),( ki ttCXX

1 ,

( ) ( , ) / 2

1 2( , , , )XX

n n

k k i k i kk l k

X

j t C t t

n eµ ω ω ω

φ ω ω ω =

−∑ ∑∑= (14-22)

12

1 1 1 1

( )

1 2( , , , )XX

n n n

k i k i kk k

X

j C t t

n eµω ω ω

φ ω ω ω = = =

− −∑ ∑∑= (14-23)

niiX 1}{ =

niiX 1}{ =′

⇒

PILLAI/Cha

),( 11 ctXX +=′)(,),( 22 ctXXctXX nn +=′+=′

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, butnot 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, variesfrom –2T to + 2T. Moreover is a constantover the shaded region in Fig 14.2, whose area is given by

and hence the above integral reduces to

),cos()( 0 ϕω += tatX

PILLAI/Cha

2zσ

.)(}|{|

212122 ∫ ∫− −

−==T

T

T

Tz dtdtttRzE XXσ

21 tt −=τ)(τXXR

)0( >τ

τττττ dTdTT )2()2(21)2(

21 22 −=−−−−

.)1)((|)|2)((2

2 2||

212

2 2 ∫∫ −−

−=−=T

t TT

T

tz dRdTR XXXX τττττσ τ

(14-24)

T− T

T−τ

τ

τ−T2

2t

1t

Fig. 14.2

21tt −=τ

13

Systems with Stochastic InputsA deterministic system1 transforms each input waveform intoan 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 ξ)],([),( ii tXTtY ξξ =

)},({ ξtY

Our goal is to study the output process statistics in terms of the inputprocess statistics and the system function.

1A stochastic system on the other hand operates on both the variables t and .ξ

PILLAI/Cha

][⋅T→ )(tX → )(tY

t t

),(i

tX ξ),(

itY ξ

Fig. 14.3

14

Deterministic Systems

Systems with Memory

Time-Invariantsystems

Linear systems

Linear-Time Invariant(LTI) systems

Memoryless Systems

)]([)( tXgtY =

)]([)( tXLtY =

PILLAI/Cha

Time-varyingsystems

Fig. 14.3

.)()(

)()()(

∫

∫∞+

∞−

∞+

∞−

−=

−=

τττ

τττ

dtXh

dXthtY( )h t( )X t

LTI system

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)

PILLAI/Cha

)}({)( tXgtY =

Memorylesssystem

Memorylesssystem

Memorylesssystem

Strict-sense stationary input

Wide-sense stationary input

X(t) stationary Gaussian with

)(τXXR

Strict-sense stationary output.

Need not bestationary in any sense.

Y(t) stationary,butnot Gaussian with

(see (14-26)).).()( τητ XXXY RR =

(see (9-76), Text for a proof.)

Fig. 14.4

16

Theorem: If X(t) is a zero mean stationary Gaussian process, andY(t) = g[X(t)], where represents a nonlinear memoryless device,

then

Proof:

where are jointly Gaussian random variables, and hence

)(⋅g

)}.({ ),()( XgERR XXXY′== ητητ (14-26)

212121 ),()(

)}]({)([)}()({)(

21dxdxxxfxgx

tXgtXEtYtXER

XX

XY

∫ ∫=

−=−= τττ

(14-27)

)( ),( 21 τ−== tXXtXX

PILLAI/Cha

* 1

1 2

/ 21 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

π

τ

τ

−−=

= =

= = =

∆

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

=

lll

L

IALLLXXELZZE ===−− −− 11 *1**1* }{}{

* * *1 * 1 2 21 2 .x A x z L A Lz z z z z− −= = = +

22222121111 , zlxzlzlxzLx =+=⇒=

PILLAI/Cha

1 1 1 2 1 2( , ) , ( , )T TZ L X Z Z z L x z z− −= = = =∆ ∆

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

πτ

+∞ +∞ − −

−∞ −∞

+∞ +∞

−∞ −∞

+∞ +∞

−∞ −∞

= + ⋅

=

+

=

∫ ∫

∫ ∫

∫ ∫

22

212 22

22

11 1 1 22 2 21 2

12 2 22 2 22

/ 2

2

2

1 2

2

12

/ 212

( ) ( ) ( )

( ) ( )

( ) ,

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

π

π

−

+∞ +∞

−∞ −∞

+∞

−∞

+∞ −

−∞

+

=

∫ ∫

∫

∫

0

PILLAI/Cha22 2 .u l z=

19

222

222 22

2

2

( )

/ 2112 22 2

( )( )

( ) ( )

( ) ( ) ( ) ,

u

XY

uu

XX

f u

u

df uf u

du

u

ll

ulR l l g u e du

R g u f u du

πτ

τ

+∞ −

−∞

′− =−

+∞

−∞

=

′= −

∫

∫

Hence).( gives since 2212* τXXRllLLA ==

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 theinput autocorrelation function.

PILLAI/Cha

),()}({)(

})()(|)()(){()(

τητ

ττ

XXXX

XXXY

RXgER

duufugufugRR uu

=′=

′+−= ∫∞+

∞−

∞+∞−

0

)].([ XgE ′=η

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

PILLAI/Cha

LTI)(tδ )(th

Impulse

Impulseresponse ofthe system

t

)(th

Impulseresponse

Fig. 14.5

21

Eq. (14-31) follows by expressing X(t) as

and applying (14-28) and (14-30) to Thus)}.({)( tXLtY =

∫∞+

∞−−=

)()()( ττδτ dtXtX

(14-31)

(14-32)

(14-33)PILLAI/Cha

.)()()()(

)}({)(

})()({

})()({)}({)(

∫∫

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

−=−=

−=

−=

−==

ττττττ

ττδτ

ττδτ

ττδτ

dtXhdthX

dtLX

dtXL

dtXLtXLtY

By Linearity

By Time-invariance

then

LTI

∫

∫∞+

∞−

∞+

∞−

−=

−=

)()(

)()()(

τττ

τττ

dtXh

dXthtYarbitrary

input

t

)(tXt

)(tY

Fig. 14.6

)(tX )(tY

22

Output Statistics: Using (14-33), the mean of the output processis given by

Similarly the cross-correlation function between the input and outputprocesses is given by

Finally the output autocorrelation function is given by

).()()()(

})()({)}({)(

thtdth

dthXEtYEt

XX

Y

∗=−=

−==

∫

∫∞+

∞−

∞+

∞−

µτττµ

τττµ

(14-34)

).(),(

)(),(

)()}()({

})()()({

)}()({),(

2*

21

*

21

*

21

*

21

2*

121

thttR

dhttR

dhtXtXE

dhtXtXE

tYtXEttR

XX

XX

XY

∗=

−=

−=

−=

=

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

ααα

ααα

ααα

*

*

(14-35)

PILLAI/Cha

23

or ),(),(

)(),(

)()}()({

})( )()({

)}()({),(

121

21

21

2*

1

2*

121

thttR

dhttR

dhtYtXE

tYdhtXE

tYtYEttR

XY

XY

YY

∗=

−=

−=

−=

=

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

βββ

βββ

βββ

*

).()(),(),( 12*

2121 ththttRttR XXYY ∗∗=

(14-36)

(14-37)

PILLAI/Cha

h(t))(tXµ )(tYµ

h*(t2) h(t1) → ),( 21 ttRXY→ → ),( 21 ttRYY),( 21 ttRXX

(a)

(b)

Fig. 14.7

24

In particular if X(t) is wide-sense stationary, then we haveso 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

XX t µµ =)(

constant.a cdht XXY ,)()(

µττµµ == ∫

∞+

∞−(14-38)

)(),( 2121 ttRttR XXXX −=

(14-39)

).()()(

,)()(),( 21

2121

τττ

τβββ

YYXY

XYYY

RhR

ttdhttRttR

=∗=

−=−−= ∫∞+

∞−

(14-40)

).()()()( * ττττ hhRR XXYY ∗−∗= (14-41)PILLAI/Cha

. ),()()(

)()(),(

21*

*

2121

ttRhR

dhttRttR

XYXX

XXXY

−==−∗=

+−= ∫∞+

∞−

ττττ

ααα∆

25

From (14-38)-(14-40), the output process is also wide-sense stationary.This gives rise to the following representation

PILLAI/Cha

LTI systemh(t)

Linear system

wide-sense stationary process

strict-sense stationary process

Gaussianprocess (alsostationary)

wide-sense stationary process.

strict-sensestationary process(see Text for proof )

Gaussian process(also stationary)

)(tX )(tY

LTI systemh(t)

)(tX

)(tX

)(tY

)(tY

(a)

(b)

(c)Fig. 14.8

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 −= δ (14-42)

).()(),( 2121 τδδ qttqttRWW =−= (14-43)

White noiseW(t)

LTIh(t)

Colored noise( ) ( ) ( )N t h t W t= ∗

PILLAI/Cha

Fig. 14.9

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!

)()()(

)()()()(*

*

τρττ

τττδτ

qhqh

hhqRnn

=∗−=

∗−∗=

(14-45)

.)()()()()(

** ∫

∞+

∞−+=−∗= αταατττρ dhhhh (14-46)

PILLAI/Cha

(14-44)

[ ( )] ( ) , WE N t h dµ τ τ

+∞

−∞= ∫ a constant

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 withpositive slope. Let represent the probabilityof such an upcrossing inthe 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)

).(τXXR

t∆ρ

). ,( ttt ∆+.ρ

Fig. 14.10

)(tX ′

)()( ττ XXXX RR ′′−=′′)(tX ′

,)()(ττ

τd

dRR XXXX −=′

PILLAI/Cha

Upcrossings

t

)(tX

Downcrossing

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=τ

)()()(

)()( τττ

ττ

τ XXXXXX

XX Rd

dRd

dRR ′′ −==−−

−=−

(14-48)

(14-47)

(0) 0 [ ( ) ( )] 0XXR E X t X t′′= ⇒ =

)( and )( 21 tXXtXX ′== (14-49)

,ρ

0 )0( )0( and )0( 22

21 >′′−=== ′′ XXXXXX RRR σσ (14-50)

2 21 1

2 21 2

1 2 1 2 1 21 2

2 21( , ) ( ) ( ) .2X X X X

x x

f x x f x f x e σ σ

πσ σ

−

+= = (14-51)

PILLAI/Cha

30

PILLAI/Cha

we argue as follows: In an interval the realization moves from X(t) = X1 to and hence the realization intersects with the zero level somewherein 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.)()(

),(

1

12

0 2

0 210

21

2 12

2 21 21

xdxfxdxf

dxxdxxft

tx

x txx

XX

XX

∫∫

∫ ∫∞

∆−

∞

∞

= ∆−=

=

=∆ρ

,t∆

(14-54)2 1

2 2 2 2 0

( ) ( )X Xf x x f x t dxρ∞

= − ∆∫,0→∆t

1 2 1 20, 0, and ( ) 0 X X X t t X X t< > + ∆ = + ∆ >

31

PILLAI/Cha

[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 movesaway 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(

21)/2(

21

)0(21

)()0(2

1)0()(

2

0 222

0 222

XX

XX

XX

X

XX

XX

RR

R

dxxfxR

dxfxfx

′′−==

== ∫∫∞∞

ππσ

π

πρ

(14-55)

) ,( ttt ∆+ ,0 t∆ρ

.0 )0(/)0(10 >′′−= XXXX RR

πρ (14-56)

T0ρ)0( XXR ′′−

τ)(τXXR

(0),XXR′′−

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

=

=µ

*2121 21),(),( nnnnRnnC µµ−=

(14-57)

(14-58)

(14-59)

constanta nTXE ,)}({ µ= (14-60)

PILLAI/Cha(14-61)* *[ {( ) } {( ) }] ( ) n nE X k n T X k T R n r r−+ = = =∆

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 ofa wide sense stationary stochastic process if and only if everyHermitian-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 =

=

−

−

Tnaaaa ] , , ,[ 10=

(14-62)

PILLAI/Cha

∑∑= =

−=n

i

n

kikkin raaaTa

0 0

** (14-63)

.)( , ikkin rT −=

34

From (14-64), if X(nT) is a wide sense stationary stochastic processthen Tn is a non-negative definite matrix for everySimilarly the converse also follows from (14-64). (see section 9.4, Text)

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 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= = =

= = ≥

∑∑ ∑

PILLAI/Cha

)()()( * nhnRnR XXXY −∗=

)()()( nhnRnR XYYY ∗=

).()()()( * nhnhnRnR XXYY ∗−∗=

(14-65)

(14-66)

(14-67)

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 kk k

k kX z a z W z b z a− −

= == ≡∑ ∑

∆1 2

0 1 21 2

0 1 2

( ) ( )( ) ( )( ) ( )1

qqk

pk p

b b z b z b zX z B zH z h k zW z A za z a z a z

− − −∞−

− − −=

+ + + += = = =

+ + + +∑(14-70) PILLAI/Cha

Fig.14.12

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 varianceso 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) PILLAI/Cha

(14-72)

(14-71).)()()(0∑∞

=−=

kkWknhnX

),1( ),( −nWnW

2Wσ

).()( 2 nnR WWW δσ=

)( , qnW −

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

PILLAI/Cha

)()1()( nWnaXnX +−= (14-73)

1|| ,)( <= aanh n (14-75)

(14-74)∑∞

=

−− =

−=

011

1)(n

nn zaaz

zH

2

||2

0

||22

1}{}{)()(

aaaaaannR

n

k

kknnnWWWXX −

==∗∗= ∑∞

=

+− σσδσ

(14-76)

38

where we have made use of the discrete version of (14-46). The normalized (in terms of RXX (0)) output autocorrelation sequence isgiven 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 randomsequence 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

PILLAI/Cha

)()()( nVnXnY +=

.0 || ,)0()()( || ≥== na

RnRn n

XX

XXXρ

(14-78)

(14-77)

2Vσ

)(1

)()()()()(

22

||2

2

na

a

nnRnRnRnR

VW

VXXVVXXYY

n

δσσ

δσ

+−

=

+=+=

(14-79)

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, the autocorrelation of the observed sequence {Y(n)}is reduced by a constantfactor compared to the original process {X(n)}.From (14-78), the superimposederror sequence V(n) only affectsthe 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 asall subsequent observations.

PILLAI/Cha

(14-80)

.1)1( 222

2

<−+

=a

cVW

W

σσσ

(14-81)

Fig. 14.13

nk

)()( kkYX

ρρ >

1)0()0( ==YX

ρρ

0

| |

1 0( )( ) (0) 1, 2,

YYY

YYn

nR nnR c a n

ρ=

= = = ± ±

∆

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),

PILLAI/Cha

)()2()1()( 21 nWnXanXanX +−+−= (14-82)

(14-83)

(14-84)

(14-85)

12

21

1

1

02

21

1 1111)()( −−

∞

=−−

−

−+

−=

−−== ∑ z

bz

bzaza

znhzHn

n

λλ

2 ),2()1()( ,)1( ,1)0( 211 ≥−+−=== nnhanhanhahh

0 ,)( 2211 ≥+= nbbnh nn λλ

. ,1 1221121 abbbb =+=+ λλ

, , 221121 aa −==+ λλλλ (14-86)

21 and λλ

41

and H(z) stable impliesFurther, 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

PILLAI/Cha

(14-88)

(14-87)

.1|| ,1|| 21 << λλ

)2()1()}()({

)}()]2()1({[

)}()({)(

21

*

*21

*

−+−=++

−++−+=

+=

nRanRamXmnWE

mXmnXamnXaE

mXmnXEnR

XXXX

XX

0

−

+−

+−

+−

=

∗+=

∗−=∗−∗=

∑∞

=

22

*2

22

*21

*2

*21

2*1

*12

*1

21

*1

212

0

*2

*2*

||1)(||

1)(

1)(

||1)(||

)()(

)()()()()()(

λλ

λλλ

λλλ

λλ

σ

σ

σ

nnnn

k

bbbbbb

khknh

nhnhnhnhnRnR

W

W

WWWXX

(14-89)

1 2( )( ) ( 1) ( 2).(0)

XXX X X

XX

R nn a n a nR

ρ ρ ρ= = − + −∆

42

where we have made use of (14-85). From (14-89), the normalizedoutput 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)

PILLAI/Cha

(14-90)nn

XX

XXX cc

RnRn *

22*11)0(

)()( λλρ +==

21 24 0a a+ <

* 1 2

jc c c e ϕ= =

* 1 2 1, , 1.jr e rθλ λ λ−

= = < (14-91)

(14-92)).cos(2}Re{2)( *11 ϕθλρ +== ncrcn nn

X

,1 ,cos2 22

121 <−===+ arar θλλ (14-93)

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 normalizedoutput autocorrelations to be

PILLAI/Cha

21 22 sin ( 4 ) 0r a aθ = − + >

1)0( =Xρ

.)4(tan1

221

aaa +−

=θ (14-94)

(14-95)

(14-96)

)1()1()0()1( 2121 XXXX aaaa ρρρρ +=−+=

)cos(21

)1(2

1 ϕθρ +=−

= cra

aX

.cos2/1or ,1cos2 ϕϕ == cc

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 independentcoefficients, and hence its impulse response sequence {hk} also must exhibit a similar dependence amongthem. In fact according to P. Dienes (The Taylor series, 1931),

ϕ

.11cos

)cos(22

1

aaa

−−=

+θϕθ

(14-98)

1 ,cos

)cos()()( 22/

2 <−+

−= anan nX ϕ

ϕθρ (14-97)

( , 1 , , 0 ),k ia k p b i q= → = →

PILLAI/Cha

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, theHankel matrices Hn in (14-100) all have the same rank.

The necessary part easily follows from (14-70) by cross multiplyingand 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( ) kkkH 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

nn

n n n n

h h h hh h h h

H

h h h h

+

+ +

=

∆

46

This gives

For systems within (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 hb h a h

b h a h a h

h a h a h a h i−

+ + − + − +

== +

= + + +

= + + + + ≥

1, letting , 1, ,q p i 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

− −

+ − −

+ + + + =

+ + + + =

(14-102)

(14-101)

2 p q−

(14-103)

PILLAI/Cha

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 outputautocorrelation sequence.

Referring back to the ARMA (p, q) model in (14-68), the input white noise process w(n) there is uncorrelated with its ownpast sample values as well as the past values of the system 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)

PILLAI/Cha

*{ ( ) ( )} 0, 1E w n w n k k− = ≥

*{ ( ) ( )} 0, 1.E w n x n k k− = ≥

(14-105)

(14-106)

48

PILLAI/Cha

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

{ ( ) ( )}

{ ( ) ( )} { ( ) ( )}

{ ( ) ( )}

ip 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)

10,

p

k i k ik

a r r i q−=

+ ≠ ≤∑ (14-108)

10, 1.

p

k i k ik

a r r i q−=

+ = ≥ +∑ (14-109)

49

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} satisfies

where

represents the Hankel matrix generated from It follows that for ARMA (p, q) systems, we have

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

kk

k k k k

r r r rr r r r

D

r r r r

+

+ +

=

∆

PILLAI/Cha(14-112)det 0, for all sufficiently large .nD n=