Cooper Extracts

8/11/2019 Cooper Extracts

1/47

2 0 R RRL IONS

he pevously dned coaon as spy a nub snce the ado vables ee ecessaly dened s beng assocatd t e nctons . n the llog ase, hoeve, evea f andom vaabs can b ad by t te spaaton betee the, ad the oelall be a cton of ths sepaaton us t bcoms appopat to dee a coeationfction hh th agument s t spaaton of the to andom vaables If the to a

vaables coe om te sam ando pocss ths ncto ll be ko as the autocoeationnction If the coe o deent and posses t l be called the cosscoeationction e ll cosde autocoelaton ctos st

If X(t) s a sapl ncton om a ado poces nd the do vales e deeto be

X X(t

= (t

he the autocoelaton ncton s dened to be

1

hs dento s vald both statonay and nonstatonay ando poesses Hoeve, teest s pmaly statonay pocesses hch futhe splato of (1 s le a b ecaled o the pevous chpt that a desese statoay poess l s eeble aveages ae ndependent of the te o Accodgly, a desese sat e

R(t t R t + T t + T)

E [t )X(t + T)

Se ths epesso s depedet of the choce of e og, e a se T -t

R(t t R( t t E [X () (t - t )

It s seenthat ths psso depends only o the te dee t - t Sett s deece equa to r t t and suppessg the zeo the aguet of R O t - t

can ete (1) as

R(r) E[t t + r)]

hs s the epesson the autocoelato to of a statoay oess ad ee o r ad ot o e value of t Because of ths lack of depedece o the l e tat hch the nsle avages ae taen t s coo pate t te 2) t

subscpt thusR (r E[ t)t + r)


2/47

INTRODUCTION 1 1

Wenever coelation nctions relate to nonstatonary processes, since theyaeenent on theparticuar time at which the ensemble average is taken as well as on the time ierece betweensamples, they must be written as Rx C ) or Rx ( ) In all cases in this an subseentchapters, uness specically state otherwise, it s assume that al correlation nctions reateto widesense staionary ranom rocesse

t is also ossible to ene a i aooaio nction r a partcular sampe nctio as 1 1 T() = lim

2x (t)x (t+r) dt= x (t )x (t + r)

o T&3

For the special case of an oi po ( () ( + ) is he same r every () an euato Rx ) That is,

r an ergoic process

The assumption of ergodiciy, where it is not obviously invalid, oen simplies the compuationof coeation nctions.

From (2) it is seen reaily thatr = O, since x(O) = E[X ( )X () ] ; the autocoelationnction s eual to the mean-suare value of the rocess . For values of other than 0 theautocoelation nction Rx ( ) can be thought o as a measure of the similarity of the wavermX () and the waverm ( + ). To illusate this oint rther, let () be a sampe ctionom a zero-mean stationary random rocess an rm the new nction

Y() = X () X ( + )

By etermining the value of tht nizes the mean-square value of Y() we will havea measure of how much of the waverm ( + ) is containe in the waverm X () . Thedetermination of is mae by computing the vaance of Y(), setting the erivative of thevaance with respect o eual to zero, an solving r The oerations are as llows:

E{ [Y ()} = E{[X () ( + )] }

= E{X() 2 X ()X ( + ) + X ( + ) }

; = i 2x() + i_Y = 2Rx() + 2i = 0

x()= -

.-

0

&5

It is seen om &5) that is iectly relate to x () an is actly the oaio oidene in Secon Te coecient can be thought of as the action o te waveshae

e symbol ( ) is sed to denote time aveaging.


3/47

22 CH APTER CORREATO FCOS

of X(t) eaiig afte r seonds has elapsed It ust be eebed that p as alao a statistial basis ad that i is the aveage etetio of aveshape ove the eseb aot this popety in ay patua saple tio that is ipotat As sho pevo hoelatio oeet p an vay o 1 to 1 Fo a value of p 1 he aveshape be idetialthat is opletely oelated Fo p 0 the aves ould be opuoelated; that i , o pat of the ave X(t+r) ol be otaied i X(t Fo p = 1 the aveshapes ould be idetial, eept oppost i hat h av (t + r)ould be he egative of X(t)

Fo a egodi poess o f oado sials the eoi iteptaio a b a tes of aveage poe istead of vaiae ad i tes of the tie oelatio io aof the esebe oelatio ntio

Sie R ( r) is depedet both o the aout of oelatio p ad the va of th poai, it i s ot possible t estiat the sigiae of soe patiul value of (r) ithokoig oe o the othe of these quatities Fo eaple, if the ado poe ha a z

ea ad the autooelatio futio hs a positive value, th ost that a b a s hathe ado vaiables X (t a X (t + r) pobably have te sae s2 If the atooelaotio as a negative value it is likely that the ado vaiables have opposite sis I eay zeo te andom vaiables ae about as likey to have oppsite ss as they ae to havthe sae sig

Exercise 6- .

A rdom pro h mpl fuio of h fom

X (t) A

0

0 : t : 1

elsehee

wher A i rdom ribl h i uiforml diib u from 0 o 1 0 gh b i dfii io of h uoorrlio fuio gi b Equao (6

1 ) fid h uoorrlio fu io of hi p ro.

Awer

Rx (t , t) 33 3

= 0 elsehee

2hs s strct on i f (x1 ) s smmetrcal about the axs


4/47

6- EXAMP LE : ATOCORRELAT ON NT N A B NA RY PRE 23

xecise 6 .2

Defie rdom vrible Z )

Zt = t t r1)wee i mle fuio from ioy dom poe woeaoorrelio fuio i

Rx (r = exp r

Wrie expreio for he uoorrelio fio o rdom roe .

Awe

62 xampe: Atoorreatio Ftio of a Bia roess

e above eas ay be ae soewhat lae by oeg, as a speal eaple, a aooess havg a vey sple autooelato to Fgue shows a typal sapl too a sete, statoay; zeo-mea ao poess whh oly two values, A aeossbe e sae to ethe a hage o oe value to the othe evey t seoso e the se, equal pobablty he te t s a ao vaable wh espet to theeee o ossble t tos a s uy sbute ove a teal of legth ts eas , as fa as the eseble s oee, that hages value a ou at ay teth eqa obablty It s also assume that the value of X t ay oe teval s statstallyeeet o ts value ay othe teval

thoh the a oess esibe the abve paagaph ay seem otve t auallyeesets a vey pata stuato I oe gtal omuato systes, the essagesto e oveye e ovete to bay sybols s s oe by st saplg the essage ateo te stats a the quatzg the saples to a te ube of apltue lvelsas ssse Seto 27 oeto wth the u pobabty esty to ahte level s the epesete by a blok of bay syols eaple, 25 ampltueees a eah be uqey epesete by a blok of 8 bay sybols he bay sybolsa be epesete by a voltage level of A o _ Thus, a sequee obay sybolsbeoes a ave o the type show Fgue1 Slaly, ths wave s typa o those tal outes o ouato ls oetg oputes ogethe Hee,


5/47

214 CH APTER 6 CORRELAO FCTOS

A

a I

a:1Xt1

= 1>

-

T'

gre c1 A dscete statonay saple fncton.

igre c2 Autocoelaton fncton o the pocessn Fure 61.

+ = ' l

0+ 8: 0+ III'

0a

the adom poess beg osdeed hee s not oly oe of the smplest oes to aale, also oe of the most patal ones the eal old

The autooelaton to of ths poess ll be detemed by heust age rathrtha by goous devato I e st plae he s lage tha t the t ad t + r = aot le the same nteval, d X ad X ae statstally depedet Se X and Xhave zeo mea, the expeted value of the podut must be eo, as sho ( that ,

> t

se X = X = he s less ta t the t ad t + a o a o e same teval, depedg upo the value of t Se t a be ahee, th eqa proailitythe pobablt tht hey do le the same teval s popotoal to he deee twnt d I patula, t s see that t t < t + t h edt + < t t Hee,

P (t1 ad t + ae the same teval)

= P [(t + - t < t t ) 1 t - r= [t - (t + t) = t t


6/47

6 XP : UTOCORRELAON UNCON O A B NA RY PROCE 2

e e oalty desty tion i s us 1 / he r < i s ee tat : t r :i < to + t yelds - t < + r . Thu

ee eea

P t ad t ae in he same iea

= P - < :

r ]

1 t + r= t + r - t - ] = t t

P ( d - t a t r ae same ea = t

e e e te sae teval, te podu of X1 and alays he they ae o ,

e eeted odt zeo Hee

Rx (r)=Azr ta- l r l ]=Az[ 1- J' t =

o keted e2 teet to ode the physal itepeain of his uooelao to lh

of e evo dso Note tat e l r l is smal (less han

he a iesed

aly tat Xt) ad X r ll ave he same vaue ad he autooelaton to otve e l r l s eate tha t t s equally pbable ha X ad X r ll havee e vae ta tey ll ave opposte value, and he auoeaio o s zeoFor 0 te atooelato to yeds the meanquae vale of

Exerise 6- 1

A seech waeform is samled 4000 imes seod d eh smles quazed o 6 amliude lels The es me lel eeresee b biar olae hai les of 5 Assm ha esse ba smbols are saisll ideede wie he ooeliocto o he ba roess.

Aswer

R (

25[ 32l r l ]

=

1 b

63 roperties of Autooreation FuntonsIf autooelato to ae to pa a uel oe epeet a ee aal of yte th ado put t ee to e ale to eate ete autooelatio to to the opetes of the ado poess t epeset ue of the popeties that ae poseed all autooelato to t egod _ado poeses ae suazed he tudet should p atula tet epopete beaue they ll oe up a e ue uo

Rx (0) = Hee te eauae vae e ao poe la e sp b ettg r = 0.

It should be ephaszed that Rx (0) gve the eaae vae ethe e e s ozo ea value o ot f the poes s zeo ea the the e v e the vae of the poe Rx( = Rx(-r) . he autooeto to een r

ost ea

yee pehap he eve

hh the ae a the eebleaveae ae poess t ae he te aveae ake ve ea e


8/47

6- POP ETES OF ATOCOEL ATO F CTOS 17

of h deto oe of te te tos s shfted hs set popet s eteeuse devg te autooelato to of a ado poess beause t ps tat edevato eeds be aed out ol postve values of T ad the esult egatve Tdeteed b set hus , te devato sho the exaple Seto 2 t oudhave ee eessa to osde ol the ase T 2 0. o a ostatoa poess, teset oet does ot eessa appl I Rx(r) I Rx(O) e lagest value of the autooelat to alas ous atT ee a be ote vaues of T hh t s ust as b ample, see the peodase eo but t aot be lage hs s o easl osdg

E[ X X)] E[X f + Xi 2X X] 2 0E[X f Xi 2RxO) 2 I E2XX) I 2Rx) I

ad ts Rx(O) 2 I Rx) I

f Xt) as a d opoet o ea value, te Rx) ll have a ostat opoeto eale f Xt) , the

Rx(r) [(t )X(t + )] = E[] = oe geea f

X(t)as a ea vaue an a zeo ea opoet

N(t)so that

te

e

Xt) = X + Nt)

Rx(r E{ [ + Nt ] [X + N(t + ) ] } E[(X) + XN( ) + Xt + ) + N(t )N(t + r) ] + R()

E[N(t ) E[N t + )] 0us eve s ase Rx(T) otas a osat opoet

8

9

o egod poees te agude of the ea value of the poess a be deteedy oog at the autooeato to as T appoaes t povded that a peodoet e autooeat to ae goed te t e ol the squa' of e oted o auato t s ot possle to detee te sg of e vaue e oe tatoa ut ot egod e vaue o Rx T a ot eld a


9/47

8 CHAPTER 6 C O R R E L A TI O N F U N C TI O N S

information regarding he mean value. For example, a random process hav smple ncsof e

X(t) = A

were A i a random variabe wit zero mea and variance al , has an autocoelaon cto oRx() = al

r a T e auooein nio oe not anish at = even tough te procesa zer ea Ti rage ret i a onsequene o te rocess beng nonergodi and ouldo our r a ergoi roe

X() a a peiodc omponent, then Rx() l also have a perodc component, the ame peiod. For example e

X(t) = A os ( w t +0)

were A ad w ae nan ad 0 i a radom varae unirmly dstrut over a range o2 Ta

Then

f(O) =

= elsewhere

Rx() = E[ o (wt1 + O)A os (wt1 + w + O)

A A = E z os (wt1 + w + 20) + z cos A

{2 1=

o

2[os (wt1 + w + 20) + co w dO

A= os (

2

I e more genera ae in w

X(t) = A cos (wt + 0) N(t)

(610

where 0 and (t1) are statstcall ndependent r all t1 , the method sd n tn (5-9),t s eas to show tat


10/47

6- PROPERT ES OF ATOCORRELATON FN!ONS 29

AzRr) T os r + R r) (611 )

Hee, the autooelatio ton still ontains a peodi oponente above popety a be eteed to onside ndo poesses that otai ay ube

o eod !poets If the ado vaiables assoiated ith the peiodi opoets aesttstll depedet the the autooeatio nt of the sm of the peiodi opoetss spy the su of the peiodi atooeation ntions of h mpoet This stateets e egadless of hethe the peiodi oponents ae hnialy eated o ot

evey saple tio of the ado poess is peiodi and an be epesented by a Fouesees, the esultg atooelaio is also peodi and a also be epeseted by a Foiesees Hoeve, this Fouie seies inlude oe than st the sm f the atooelatiotos o the dvdual tes if the a vaiables ssoiated ith the vaios opoetso the sape tio ae ot statistially inepende A omon sitation in hih the ado

bes e ot depedet is the as i hih thee is oly one ado vaiable theoess , aey a ado delay on eah saple tion that is nily distbuted ove thedaetal peod

I {Xt ) } s egod ad zeo ea and has o iodi omponents thenJi Rr) =

I T l-o(62

ge vaes o r se the eet of pas vaues teds to die ot as time pogesses thedo vaables ted t beoe tatistially independet

utooelatio tios aot have an abitay shape One ay of speifyig sapest ae pessble is tes of the Fouie tans of the atooeatio futio Thats,

[Rr) ] 1: Rr)ej dre the estti is

[R r ) ]

0 all (613)

e easo ths esttio ill beoe appaet afte the disussio of spetal desity pte 7 Aog ote thigs this estitio peles the eistene of autooeltotos th at tops vtial sides o ay disotiity i aplitde

ee s oe uthe poit tha shuld be ephasized in oetio ith autooelatiotos Although a o lege of the oit obablity density ts of the ado poesss set to obtai a uiqe atooelatio ti the ovese is ot tue Thee ay beay deet ado poesses that a yield the sae atooelatio futio Fuheoe,as ll be sho late, the eet of liea systes o the atooelatio tio o the ut e oputed ithout oig aythng abot the pobability desi tios Hee, te


11/47

220 CH APTE 6 CORRELATO FCTOS

ii i i i qiv t ition bbii i i ib fii

Execise 6-3

A rodi rdom pro h uoorrlio fuio of he form

R () = 9e-4IT + 1 1 + 1

Fid he mequ lu me lue and rae of is roess.b A rodi rdom pro h uoorrlio fuio of e form

4r2 + R () =

r2 + 1

Fid he me-qur lu m lue d ari of hi ros.

Awr 2 2 33

Execise 6-3

For eh of followi fuios of r, dermi e lre lue of theo A for whih h uio ould be lid uoorrelio uio 4 1 T I _ A21 T I

b e- I T+ I

(2r) A r)Awer 0 2 O

64 easureet of Autorreato FutosSine t ooti ntin n int in t anli of in tmwit rnom input, n imn i b i tat of tning t ntion


12/47

6 MEEMET F TCRRELT FCT 221

eimenta od andom oee n gna t annot b aatd om t jointdnit ntion in te denit ntion a dom nown No an an nmb aagb mad beau the i uua on on ame ntion m t nmb aaiab Und iumta th on aaiab odu i t aat a tim atooatio ntion a nite tim inta nd t amtion tat t o odi

To itate thi aum that a atiua otag o nt wam x t) a b obeoe a tim intea om 0 to T on t i tn oib to dn an estimted oationnction a thi atiu wam a

) = t)t + ) dt1 Tr . )Oe te enmb o am ntion t tiat i a andom aiab dnotd b ) Note tat te aagin tim i T ate tan T ba ti i t on oion o tobeved data in whi bot () and t + ) a aaiabIn mot atia a it i not oib to a ot t itgation ad in (14)becaue a mathmatia ion () i not aaiab An atat od i toapoximate te inta b aming t ontino tim nCtion at dit intant o timeand eong th dit qiant t (14) T am o a atia amnction a taken at time intant o 0 t , 2t , . , N t and i t oreonding aue of ae x , x, N , t dit qiant to (14) i

n= 0, 1

2, , M 5)M N

Th etimate i ao a andom aiae o t n and a u i dentd b nt) ince N i qute ag (on the ode o ea toand) ti oatio i bt od ! adigital comute

To eauate the quait o etimate it necea to dtemine the mean and e vaiGeo

t)ine it i a adom ab woe i au dnd on t atiua

ample ntion bing ued and the atiua t o am taken Te an i ea toobtn ince [ 1 n ]E) =

1 L XkXk+n N n k=O

l Nn l Nn= E[XkX+n] = Rnt)

- l .

N -

.

k=O k

= Rn)


13/47

C 6 CI FUCI

Thus, the epecte vaue ofthe estimate is the true value of the autocorrelation nction an tss biasd estimate of autocorrelation nction

Although the estimate escbed by (15) is unbiased, it is not necessrly the best estimatein the mean-square error sense an is not the rm that is most commonly use Instea t scustomary to use

= 0 1 2 . M 6-16

This s a biase estimate, s cn be seen reaily om the e"aluation ofD) given abover the estimte of (6-15). since only the ctor by which the sum is ivie is ierent n tepresent case, the epecte value of is new esimate is simply

)=

1

]

D) +

Note that f the bs is smll Athough ths estimate is biase, in most cases, the totameansquare eor is slightly less thn r the estimate of (6-5) Furthermore, (6-16 is slightlyeasier to calculate

It is much more cult to etrmine the varance of the estime, an te etails of ths ebeyon the scope of te presnt iscussion It is possible to show, however, tat the vnce of estmate must e mler thn

2 MVr D) L D)kM1

Ts epression r e vance assumes that the 2M + 1 estmate values of the autocoeonncion spn e regon n whch the utocoeton nction s a sgncnt mple If vle of (M + oo sma, te vince gven y 7 ay e too sm If memtical of te utocoelation ncton is on, or can be euce om emesurements that a me, a ore accurate masure of the vnce of the estimate s

2 10

V ) o ) d 8

T s e eng of the observe speAs ston of t hs result means in tes of e number of spes reqre r

a vn egree of accrcy, suppose that it is esre to estimate a coelaton ncton of e sow n Figure 6 wt ur points on eiter se of center (M = ) If n rms error of5%3 or less s requre, hn 7 mples that snce = D)

3 This imlies ha he sadd deaio of he es should b o rer 5% of he e malue of h rado arible


14/47

6 UT F UTCTIN FU CT 223

(0.05A2)2tA4 [ 1- lk lLt 2N 4 4Lt

hs can be solve r N to obtan

N

t s clear hat long samples of ata an extensve calculatons are necessary f accurat estimateso coelaton nctions a to be mae

The Stent' s Eton of MATLAB oes not have a nction r computng the autocoelatonncton of a vector f ata samples However there are sval ways to reaily accomplsh theclclaton The one cosered here makes se of the convoluton nction n a methoescrbe Chapter 7 maes use of the st Fourier transrm The raw convolution of two

vectos an b, of ata leas to a new vector of ata whose elements a of the rmck = )b )

j

hee he smmaton s taken over all values of r whch ) an ) are vali elementsof e vectos of ata The mot wdely se estmate of the autocelaion ncton e thebae esmate has elements of the rm

I

k) = ) )N = 1 . . . N - I

s e atocoelaton ncton can be compted by convoluton of he ata vector wth aevese copy of tself an weghtng the reslt with the ctor I / N I ) . The llowng specalM ncton caes ot ths calclaton

uncton [nt R] cor(a,,} cor ase correlation fuction a are equal lngth sample time ctins s the sapl ig frequecy nt s the lag value for time e laysenth(aRconv(a,lpl r(/(N+1 cal of crrelation fuctint((1 N1 *1 /f cal of lag values

s ncton calclates vales of nLt) r N :

n:

(N r a total of 2N I

elemens Te maxmm ale occrs a N) coesponng to ) an the atocoelationncton s symmetrcal abot ths pont As an example of the use of ts nction t will be


15/47

4 CHA PTER 6 CORREATO FCTOS

ued to etimate t autooato tio o a a o a aia aom roe TheMAAB ogam i taigta a o

%corxm1 m exam le of auocorrelaio calcu laio

ra(see 1 000}; use ee o make reeaable

x1O*ra(1 1 001 ; % geerae raom samlst 1 .1 : 1 ; % samlg i extcor(xx 1 000; % autocorrelao

sot( 1 1 ; o(1 x; xlabe(TIM ;l(X

sot( 1 ; lo(t ;xabel (LAG;label( x

The reig a tio ad atooratio tio a o "i igur 3 It is seeha he autoorratio ftio i etia zo aa o t oigi wre it i oetrated.

Thi aratiti of iga woe am ar oatd a te are i thi ae. romte rogram it i tat the taar dviatio o t aom iga i 0 and, erre evae i 100 orreondig to a ag of zo o te ga o te autoorreaio ion.

40

20

0-20

400

1 50

1 00

500

50

0.2

0

.

04

T IM E

'

0

LG

06

J

08

05

Fgr 63 Sple ncon uocoeon ncon of uncoele nose

1


16/47

4 MAS RMNT O ATOORRLT ON N ONS 225

Conside now an exampe in whic the sapes ae t uceated The data vecto wi eobtaind om that used n the pevious exampe cang out a unning aeage the datawith the aveage etending ove ponts The pogam that caies out this cacuatin i s asows

%orxmp2m exmpe 2 of oorreio irad'eed' 1 )

x1 =1 Ord1 1 )

h= 1 /1 ) oe1 51 )

x=ox1 h) %eh o or i 1 +5

x=x22525+ 1 ) eep eo h

1 = : 1 1 mpi i

[ R]=orbxx 1 ) oorei

ulo2 1 ) po x)xbe T ') y e'X)supo2 1 2) po R) xbe' ') y 'R ')

igue shows the esutng sampe nct ad the autcoeati ction It is seenthat the is consideab moe oeation awa o the igin and the meansquae vaue iseduced The eduction i meansquae vaue ccus ecause the covoution wth the ectangulanction is a tpe of owpass teng peati that eiates ee the hihequenccomponents in the wavem as can be seen the uppe pat Figue 6.

The standad deviation of the autocoeatn etmate i the eampe igue 6 can be

und using (17 The MATLA pogam this s as w

%orxmp3 m of drd de iio f ri im

M = ehR)

V = 2/M)u m R. "2)

S = qV)

The esut is S 7 It is evident that a much nge sampe wud e equied i a highdege of accuac was desied

Eercse 61

An ergodic randomprocessh n tocrreltio functio ofhe form

= e

a) er wh re of e m he oorreio fio of hproe e emaed i o rde o i ud e e of ) reer h1 % of he maximu m


17/47

22

2

0

2

CHA PTER 6 CORRELATON FCTO

4

02 04 06 08 1E

0.

0

-o.s . -

0 0LGFigr utocoelaton fncton o ptaly coelated nose

05

b If 23 eie M = 22 of he uoorrelio fu io ae o be ai he i el eif ied i , h hould he l ig i el be

How my mle lue of he rdom roe e equi re o ah rm e ror of he ime i le h of he rue i u aof he uoorreio fuio?

Awer 0 1 23 403

Execise 6.

Ui g he ie oud gie by he i egrl of (61 8, f i e ole oi requ i red for he uorrelio fuio eiae o Es1

Awer 2000


18/47

EXAMPLES O A TOORRELAT ON N T ONS 227

5 Examles of Atocoelatio cios

gog o to osde ossoelaton funtons t s othhl t look at som tyalooelato tos sugest th cumstancs und hch thy mght ase ad lstle plaos hs dsusso s not ntnded to b hastve but s tdd pmaly ode some deas

agl oelaton to shon n Fgu 2 s typal of andom bna gals shg must ocu at unmly spaed tm ntvals Suh a sga ases may of omucato ad otol systms n hh th ontnuous sgnals ae sampled atod stats of tme ad the esultg sampl ampltudes convtd to bnay numbes helto futo sho n Fgue 2 assums that the andom poess has a mea lue ofo ut hs s ot alays the ase I ampl th andom sgnal ould assume valueso A ad (athe tha -A) the the poess has a men valu o A/2 and a measquev of A/2 he sultg autooelato ton shon n Fgue 5 llos om alto of (69).

ot ll a tm tos have tagula autooelaton ntons hoeve Fo eampleohe ommo type of ba sgal s on n hch th sthng ocus at andomly spaedts of tme If all tmes ae equally pobabe thn th pobablty dnsty ncton ssoated e duato of eah teal s epoental as sho Sto 2. he esultgaoelt to s also epoetal as sho Fgue 66 he sul mathemataletto o sh a auoelato to s

9)

e a s e aveage umbe of teals pe seond ls ad oelat ntos of th type sho Fgue quy ase

eto t doatve motog deves he andomly oug pulses at te ouof le deteto e used to tgge a pop ut that eates the bay sl ype of sgal s a ovet oe measung the the avag tm teval beteeles o the aveage ate of ouen It s usually eed to n the lteat as theRandomeeaph Wae

igr Autocolaton fncton o ana pocss wth a nonzo man vau


19/47

22 H APTR ORRLATON NTONS

A -

_

A(a) b

igr (a A bary ga rady pad g ad (b pdigaurra

Noby g o e epoe oeon o Fo exampe vewdebd oe (g mo y pobby dey on) paed ouh a op C e e g ppeg e oupu o e e w e eay exponenauooeo o eu ow de Cpe

o e gu uooeo o d e epoe uooeaon noe oe ue wo og bo e e auooeao non ha douou dee e og Rdom poee whoe auooeaon none popey e d o be nndeenable A odeeabe poe one hoedee n e e Fo empe do oae avn a xponenaauooeo o apped o po e eug ue popoona o hedee o e oge d ue woud e e ne Sne h doe omke ee o py b e mpo dom poee havn uy auo uy epoe uooe o ao e e ea wod In pe o honuo w deed ue bo e ngu d exponena auooeao opode ue mode my uo e mu be e owee no o ue hee moden y uo w e dee o e dom poe needed beaue he eunuo mo eta o be wog

o e oeo o dued o e bee pove aue o r . Th o eey owee d o ommo ype o uooeao uno hae eaveeo ae ge by

ad

RA2 Jr

x (

)

=

-Jr

0

nd ae uaed Fgue 7 Te auooean non o 2) a he op oe now bd bdpa e whoe npu ey deband noe, whe ha o 21)


20/47


21/47

230

b

d}

e}

I e 1e

! + 4 + 8

CH PTER 6 CORRELATO CTOS

Ae: b c e e o id mode

Crossorelatio Fos

It o pobe o onde te oeton between two ando vabe o deendom poee tuton e wen tee mo han one ando n ben appeto ytem o wen one we to ompe ndo voae o uent oun at deepont n te yem I te ndom poee e on on n the wde ene a fape nton o tee poee e dened a (t) ad t) e o avbe

= t Y = t pobe to dene the crosscorreltion fnction

(622)

Te ode o ubpt nant; the eond bp e o e o e ent t +

Tee o note ooeton nton tt an be deed e e o enn T et

Y = (t) = (t +

nd dee

4 s s n rbtry conventon, c s by no ens unvers uthos e denitions soulbe cecked n every cse


22/47

66 CSSCLTN FUN CTNS 231

(623)

t t cas oth ranom procsss ar assmd to ointly statonary thse crosorrlin nctins dpnd only pon th tim dirnc r

I t is imprtant that th procsss jointly statonar and not jst indvidaly statonaryIt is qit possil to hav two individay staton random procsss that ar not jontystionay In sch a cas th crosscorration nction dpnds pon tim as we as the tminc r .

me csscorrelaonnctions may dnd a er r a part par of apenctins as

1

1T

ly (r = lim - x t)yt + ) dtT-o 2T T

1 1 Tly(r = lim - yt)xt + r dtT-o 2 T

4)

65)

e anm pcsss ar jointly rgodi thn (624) and (625) yild th sam va reve p f spl nctins Hnc r rgodic procsss

ly

(r) = Rx (r

ly(r) = RYX (r

66)

67)

I eel, physcal intrprtation of crosscoation nctions s no more onrte t f tclaon nctions It is simply masr of how mch thse two randomls n pn n anor In th latr stdy of systm analyss howvr th pssltion nction twn systm inpt and ot will tak on a vry dnt andimpt physical sigcanc

xercise .1

o jont statona rano prcesses have samp fucis f he fm

) = 2 cs S + )

adY) = 10 sin 5t +


23/47

232 CH PTER CORRELTON FNCTONS

where is a random variable that is u niformly distributed from O to 2 Findthe crosscorrelation funcUon Y for these two p rocesses.

Answer: 20 5

Execse 6-6.

Tw samp le functins from tw randm processes have the form

= 2 co Stand

=10

Find the time c rosscorrelation function for ( ) and ( + r

Answer: 0 5

7 Poeties o f Cosscoeatio FctiosThe geea popete o a cocoeato cto ae qute deent om thoe autocoeaton cto . They may be uazed a ow :

1 . The quatte () and () have no patcua phyca gcance and do otepeet maquae vaue It tue, howeve, that () = ()

2 Cocoeato ncto ae ot geeay even cto o r Thee a pe oymmety, howeve, a dcated by the eato

8)T eut ow om the ct that a ht o one decton (n me) euvaeo a o te ote decto .

. cocoeto cto doe ot eceay have t maxmm vae at r ca be how, howeve, that

() 0)] 9)

wth a ma eatohp

( ) The mxmm o the cocoeao co coccu aywee, but t caot exceed th above vae Futhemoe o ceet vaue anywee.


24/47

PROPERT ES OF CROSSCORRELATO CTOS

. I the two dom poee e ttty depedet, the

Rxy (r)= E[X1 , Y2] =E[Xi ]E [Yz] = X

R ( )

3)

If, ddto, ete poe eo me, the the ooton nton vanhe r The oee of t ot eey tue, howe he t that the ooeatonto zeo d tt o poe h zeo me oe t mpy tht the adompoee e tty depedet, eept oty u dom be

5 If X(t) ttoy dom poe d (t) t dete wth epet to tme, teooeto to of X(t) d (t) ge by

dR(r)R (r) _

31)

whh th ght de of 3 1 the dete of the autooeto nton thepet to T ey how by empoyg the fudmet deto o devte

Hene,

(t) mX(t X(t)

R (r) E[X(t)(t r) ]= E

{lim

X(t )X (t+r+e)- X (t)X (t+]

e (r e) R(r) d(r) m e d(r)

The tehge of the mt opeto d the epetto pembe wheneve (t)ext I the boe poe epeted, t o pobe to how th the utooeto

nto o (t)

32)

whee the ght de th eod dete of the b utooeto nton thepet to r

It wot otg tht te equemet the extee o ooeton toe moe eed th toe the etee o utooeato ton Cooetion

uto e geney ot ee nton o r the Foue tanf do ot hve to bepotve a aue ow d t ot ee neey tht the Foue tfm be eaThee atte to pot e dued moe deta the next hpte


25/47

3 C 6 C UC

Exercise 7

Prve he ieqaiy shw Equai (29 This i s s easil one y

evaluaig he expeced value f he quaiy

Exercise 7

Xi f2 2Rx ) JRy )

Tw radm prcesses have saple fucis he fr

X cos (wt + an Y(t) B sn (Wt + where e is a radm variabe ha s uifrly diriued eeen an ad ad B are csasa Find he crsscrrelai funcis an

Wha is he sigficace f he values f hese crsscoeaon funconsa O?

Anser sin w

Eamples and Applictions of Crossrraton Funnt s nte previously that one of the applcatons of crosscoelaton ctons s n concoth systes th to or or rano nputs. To eplore ts n ore eta conse a o

prcess hose saple nctons e of the

Z) ) Y) ch ) an Y )e also sale nctons ofrano pocsses . Then e e ovarables as

Z ) Y)

Z2 2 Y2

+ ) Y +

te utocoelaton ncton of Z) s


26/47

6 - XAMPES A APP CATOS

Rz(' = E[Z1 Z E[(X ) X ) ]

= E [X1 X + 1 X1 1X= Rx( + R( RXY ( RYX (

235

33

t asy xtn to th sum of any numbr of random variabls. n gnral, thooaton ncton o sch a su wll b h sum of a h acaion nctons pl o t cocoaton ncons.

I two _anom rocs bng consrd sasticaly ndpndn and n of hm has an, tn both o th crosscorrlaion ncions in (633)vanish and h aucoelatoncton o th sm s jst th sm of th auocolaon ncions. An xampl of h mporanco t as n conncton wh h xtracion ofriodic signals om rdom nos. L b gna amp nction of th rm

X() = A cos (w + ) 3

w s a andom varabl urmly dsbud ovr ( 2) . is shown prvousy tat thocoaton ncion of this procss s

Rx( = A cos w 2

t, t Y() b a samp nction of zroman ranom nos that s statstcay npnnt

o t ga an spcfy tha t has an autocoration ncton of th rm "

Th obv quany s Z() which om (633) has an autocorraon ncon of

Rz( = Rx( + R (

=-A

2cos wr+B2e-a 1 T I

2

35

Th ncton s skch in Fgur 68 r a cas in which h avrag nos powr, s mchg tan th avag sgna owr A t s clar om th skch that r larg va o t atocoaton ncton pns mosy upon th sgn, snce nos autocotonnction tn to zo as tn nnity. Thus, shou b possb to xact tny amontso nsoal sgna om arg amouns of nois by usng an appropra mtho masng tcoaton ncton of th rcv sgna plus nos.

noth mtho of xtrctng a sma knwn sgnal om a combnaon of sgna an no o o a cocoation opaton. A typca xamp f ths mght b a ra tm tat nttng a sgna X (t) Th sgna that s rtu om any targt s a vy mc mao o X an ha bn ay n m by th proagaton tim to tagt an back


27/47

236 CHTE 6 CETN FUNCTNS

+

ftgre Autocoelaton fncton o snusodal snal plus nose.

Snce nose s awas esen a e inu e aa eceie e a eie sina ()ma be eesene as

() ( ) + () (3)

we a num mu sma an 1 is un-i a i f sinaan () is ec ns n a ia siuain aae ow of e eu sinaX ( -1 ) , s e muc sma an aae owe f noise ()

Te ossorelaon ncon of ansi sina an e eci inu is

Y () = () ( + ) ]= () ( + ) + ( )( + )

= ( ) + ( )

(37)

Snce he sina an nose ae saisa ineenn an ae ze mean (ecause e are banass sns) e ossceaon ncin ween X () an () is zeo a ausf . Tus (637 bcomes

Y ( ) = ( ) (38)

Rememben ta autocoeaion ncions ae e maimum aues a e on s aa f is ajuse so a e meas au of Y (} s a amum n = an saue ncaes e san e &

In sm siuaons inoin wo anm sss i is ssi os o te suman e ience f e wo osses u n ac n iniiua n s ase one mabe nteese n e ossoeain wen sum n ien a eans f einsme aou em us eam a we a aaia w esss escie

( = X( ) + ()

() = () ()

(39)

0


28/47

-8 EXAMPLES AN APPL CAT ONS 237

X() ad () ae no necessai zeo mean no saisiall nependen eossoelaon nion been U ) ad ) is

v r) = () ( + r)]

= X() + ()] [X( + r) ( + r) = X()X( + r) + () ( + r) () ( + r) - () ( + r)

Ea o e expeed vales in (1) ma be idenied as an auocoelaion o o aossoelaio nion Ts,

v r) = Rr) + R r) R r) Rr) 2)

a sla a e eade ma veif easil a e oe cossoelaion io s

v r) = Rr) - Rr ) + R r) - R r) (63)

bo X and ae eo ean and saisicall inependen, bo ossoelain nosede o te same cion, namel

Rv r) = v r) = r) - r) (6)

e aa easeme o ossoelaion nios an be caied o in m e sae

a as ta sesed measin aooelaion ncions in Secion Tis pe ofeaseen is sill nbased en cossoelaion ncions ae bein onsideed, te

esl iven in (17) he vaace of te esimae is no lone stctl epatilal ifoe o e sinals onins addiive noelated noise, as in e ada example js disssedeeal spea e mb o samples eied o obain a iven vaiace i e esaeo a ossoelao on is mc eae a a eied an aooelao o

o llsae ossoelaon ompaions usin e compe onside e llo exale A sinal () = sin + 8) is mesed in the pesence of assian nise av aaddh of 50 Hz and a sandad deviaion of 5 Tis oesponds o a sinalonoise (poe)

aio o0 5 /5

=0 08 o- 1 1 dB is sinal is sampled a a ae of 1000 samples pe seod 05 seod ivin 501 samples. These samples ae pocessed in o as : b ompi

te aooelao no o e sial and b ompin e ossoelao ion o esal ad aoe dees sial s ( ) o pposes of s example i bessed a e ao vable akes on e vale of /4 e lo AA poaeeaes e sals es e poessin, and plos e esls

% comp.m ceocorrelaio exampleT = 0. f = 000 = 1 / fo = 0 N = T/

t 1 =0 : .001 : x =*si(*fo*p i*t + .*p i*oe(ie(1 ra(ee, 1000


29/47

238 CH APTER 6 CORRELATION FUNCTIONS

y=rand(1 ,N+1 ) ;

[b, a]=butter(2,50500) ; %2nd order 50Hz LP fter

y= lter(b,a,y1 ); %fi ler noise

y=*y/td(y);

Z=X

+ y;[ u = cob, ,);

x = i n(2*o*i* ) ;

[3] = corb(x,x1 ,) ;

ubpo(3 1 1 ) ; lot(t ) ;xlabel( 'TIME' ) y lael( 'z(t) ' )

ubot(3 ) ; lot(t2 u) ;xlabel( 'LAG') ylabel( 'Rzz') ;

uot(3, 3) ; lot(2,v);xlabel( 'LA') ;yabel( 'Rx') ;

e results are sown in Figure 69. Te autocoeation funtion of te signal idctes thpossblty of a sinusoidal signal being present but not distinctly. However te cosscorrelto

20

z{t)0

-200

5

Rz0

-50

0.1

xz

05

0.1 .2

0

0

LAG

0.3 0.4

ftgre6 Signal, autocoelation fnction,ad crosscoelation fctio CORP4.

0.5

05

5


30/47

6 9 CAT MATC FR AMP F CTO 39

ay sos psn h sign . I ould b possib dmin pasOf 'sinusi by masuring h im ag of h pe of h crosoelaion nion m tn and uiping by 2 / T whr T is h pod of he sinusoid.

Exercse .1A rano process has sape functios of the for in which a rano variale that has a ea valu of 5 and a variance of 10Sapl functions fro this procs ca osed ly i the prsenceof nepeent no se having an autocorreation fucto o

RN() = 10 xp (-2 1

a Fin the atocorelatio function of the sum o these two processes

I the autocorrelato function of th sum is osd, find th vaue of at which this auocorrelaton fuctio is wit 1 % of its valu t = Answers: 1 .8 , 35 + 1 e

Exercise .2A rano na process suc as that decried i ection 2 has sapleunctos with aplitudes of 1 ad t = .1 . is appi ed to the half-wave

rectfer circuit shown eow

d d

()

r i

+

Y

a Fn te autocorrelaon unction of he output, .

Fn the crosscorrelaton functon Y

c Fn the crosscorrelatin functon y

Ansers 9 + 9(1 o ) 3[ - ]


31/47

Z CHAPTER CORRELATO FCTOS

69 Correla Maties for Sampled Ftios

Te dscsson of coelaon s fa as concenaed on only o andom vaables Ts, saonay pocesses e coelaon ncons can be expessed as a ncion o e sinevaable

r Tee ae many pa

cal saons oeve, n c ee may be many

dom vaiables and s necessay o develop some convenn meod epeseni emany aocoelaons and cosscoelaions a ase Te se o veco noaion povdesa convenen ay of epesenn a se of andom vaables, and e podc o veos as necessay o oban coelaons esls n a max I s moan, eee, o dissssome saons n c e veco epesenaon s sel and o descibe some o epopees of e sln coelaon maces. A saon n c veco noaion is sel n epesenin a snal aes n e case of a snle me ncon a is samped apeiodic me nsans If ony a ne nmbe of sc samples e o be consideed, say

N en eac sample vale can become a componen o an (N 1) veco Ts, i sampln mes ae t , t2 , . . . , tN , e veco epesenn e me non X(t may be expessed as X (t

X = X t2- (tN

If (t s a sample ncon om a andom pocess en eac of e componens o e vecoX s a andom vaable

I s no possble o dene a coelaon max a s (N N) and ves e coeaonbeeen evey pa of anom vaables Ts,

[X(t1)(ti )

T t2( t R = E[XX ] = E : tN(t

X(t i X(t2 (t2( t2

(t X(tN

tNX(tNee XT s e anspose X Wen e expeced vale o eac elemen ofe max s aken,a elemen becomes a pacla vale of e aocoelaion fncon o e andom pocessom c t came. Ts.

Rx t t

R =Rx

2 t

Rx tN t

R t t2

Rx (t2 t2

(5)


32/47

6 - 9 CT TC P D UCT 1

When the rndom process om hich X t cme is idesense sttionry, then ll thcomponents of R become nctions of time dierence only f the intervl beteen smlvlues s t, then

nd

t1 t + t

t3 = t + 2 t

tN = ti+ t

Rx[O] Rx[t]R Rx:[t] RxO]Rx t]

Rx[ /t ] RxO]

)

here use hs been mde of the syetry of the utocoeltion nction; tht is , [ it ] =Rx[- t] . Note tht s consequence of the symmey, R is symmetric mti (v ne nonsttonry cse), n tht s consequence of sttionrity, the mjor dionl (nd llonls pllel to t) hve entcl elements

ouh e

Rjust ene s ogicl consequence of previous denitions, it is not th

ost customry y of designtin the coelton mtri of rnom vector consistin ofsple vues A more common procedure is to dene covaance mt, hic contins thevrances n covrince of he rndom vrbles The enerl covrince :eten two domvribes is dened s

E{Xt; ) Xt; ) tj ) Xtj ) } = UUjPj

here Xt; ) = men vlue of X t; )

X t = men vlue of X t? = vrince of X t = vrice of Xtj )P = nolized covrince coecient of X t nd X t

= 1 , hen = e covaance is ene s

TAx = E[X - X)(X - X ]here X i the men vlue of X Usin the covrnce enitions leds immeditely to

(6-7)

(8)


33/47

242 HA PTE 6 CORRELAION FUNCTIONS

[ u2UU1 P21

A =

UNUI NI

(69)

since p;; = 1 i = 1 , 2, N By expanding (69) it is easy to sow tha Ax is Rx by

-TAx = x - X X

If te andompocess as a zeo mean ten Ax = Rx

(650)

Te above epesentation te covaiance matix is vaid bot satio o

tionay pocesses. In te case of a ide-sense stationay pocess, oweve, a c

te same and te coeation coefcients in a given diagonal ae te sae Tus,

Pij =Pli-j l

and

P

P1 1

P2 P

Ax = u2

PN- 1

Suc a matix is sai to beToepitz.

i , j = l , 2 , . . . N

i , j = 2, N

P2

P1

1 P1

N

N

1 1

(651)

Asan llustrationof someoftheabove concepts, supposewehave a staordoprocess

whose autocorrelation fnctionis given by

R () = lOe ' + 9 (62)

To keep te example simple assume tat tee andom viables sepa by 1 second

be consideed. Tus N = 3 and ft = 1. Evaluating (652) r = 0 1 2 yis h vauestat are needed te coelation matix. Tus te coelation max becomes

[19 12.68

Rx = 12 .68 1910 .35 2 .68

10 35

]12 .6819Sce te aiance of tis pocess is I0 an its ean value i 3, e coc is


34/47

6 CI D U C 3

[ 1 36 . 1 35 J = 36 0368.35 .36 1Anoter sitation in which the se of vcto notaion is convnint aises whn h rado

variables coe o dierent rando processes n this cas, the vctor repesenting all erando varables iht be written as

The coelation ati is now dened as

in whch

= E[ (t) (t + i ( )=

z( 1 ( )

; ) = E[X; (t)X; (t + ( ) = E[X; (t X (t +

(653)

Note that in this case, eleents of the coelation ai nctions of rather than ubersas they were in the case of e coelation ati associated wi saples taken o a single rando process Sitations in whch sch a coelation ai ght occr arise in connection withantnna aays or aays of sisc detectors. sch syste8, the nois signals at each antennaelent, or each seisc detector, a b o dirnt, bt correlated, ano pocsss

Bee we leave the sbject of covariance atrices, it is woh noing the impont olat these atrices play in connection with the joint probabiliy dnsit nction r N randovariables o a Gassian process t was noted earlier at th assian process was on ohe w r whch it is possible to wt a oint probability density nction r any nbof rando variables Th derivation of ths joint density nction is beyond the scope of hisiscussion, but it can be shown tha it becoes

f x = f[ (t1) (t) (t

e - x - x x -x1

[1

] (2) l p 2 ( where is e deterinant o and 1 is its inverse

(54)


35/47

2 CHA PTER CORRELATO FC

e conce of coelaion maices can also be exended to rprsnt crosscolaionncions Sppose e ave o andom vecos t) and t) r ach vctor conains anom vaiables . s le

t ) X t)t) X t)

Y t) Y t)t) = Y t)

B analo o 53 e cosscoelaion maix can be ened as

e no

xy r = E[t ) t + r) ]

RI r)R (r

R r )

R! r)Rr)

R;; r) = E[X; t)Y t r) ]

Rj r) = E [X; t)Yj t + r) ]

In man siaions e veco of anom pocesses t) is e sm of th vctor X() an saisicall inepenen noise veco t) a as eo mean In s cas (55) rucs o e aocoelaion maix of 53 becase e osscoelaion btn X() an N() isidenicall zeo ee ae oe sitaions in wic e elemens of h vctor Y() ar idelaed vesions of a sinle andom pocess Xt) . Also nlike th autocolation ai of53, i is no necessa a t) and t) ave e same nmbr of insions I X() isa colmn veco of sie an t) is a colm veco o sie th crosscolaion aiill be an x maix insea of a sae maix is pe of marix may aris if () ise sinle ieban anom inp o a ssem and e veco t) is coposd of rsponss a

vaios poins in e ssem As discsse e in a sbseent cptr, h crosscolaionmaix ic is no a 1 x N o veco, can be intepeed as e se of impuls rsponss aese vaios poins

Exercise 69.1

A random proe ha an autooeatn fc f he fr

Rr) = lOe- 1' 1 cos


36/47

PROBLES

We he oelation matix assoiated wih four random variables deinedo ime instants sepaaed by 0.5 second .

nwe: Elements in the fi rst row inc ude 3 2228, 1 0.0 , 6064

Exercise 9

oaane maix for a stationay random process has he form[ 1 06 10.4 060

n he lank spaes in this matrix.

nswers: 0 6 0 2 , 0 .

04 ]0.6

06 1

5

61. 1 A saona andm poess avn sampe nons of t) as an aooelaonnon o

R r) = 5e

Anoe andom poess as sample nons o

Yt) = t) + bt 0 1 )

a) nd e vale of b a mnmzes e meansqae vale o Y)

b) nd e val o e mnmm meansqae vale of Yt) .

) If lb l 1 , nd e maxmm meansqae vale of Yt ) .

61.2 o ea of e aocoelaon ons ven belo, sae ee e poess epesens be desense saona o anno be desense saona


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PROBLEMS

3. t o te ntons son belo cannot be vald autocoelaton nctons? oea ase explan y t s not an autocoelaton ncton

2

1 0

g

0

2

2 2

3. A ando poe as saple nctons of te mX (t Y cos (wt + 8)

0

0

Y w ad e ae statstally ndependent andom vaables Assume te Y asa ean vale o and a vaane of 9 tat e s ny dsibuted om to ,and tat wa s n dstbuted om to

a) s ts poess statoay? Is t eodc?

b) nd e ean and meansae value of te pocess

) nd te atooelato functon of te pocess

3.3 A atoay andom pocess as an autocoelaton ncton of te m2

( lOOeT cos + 10 os

a) nd te ean vale, meansuae value, and te vaance ofs pocess .

b) at dete eency omponents ae pesent?

) d te allet vale of te andom vaables X (t and X (t + aeoelatd


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cPT 6 CTI FU CI

63A Conside a ntion o r o te m l I JVr 1 l r T 0

l r l>

Te te oie ansm o tis ntion and so tat it is a valid atooelatontion only T .

.1 A stationa andom poess is sampled at time instants sepaated b 0.01 seodse sample vales ae

k Xk k Xk k Xk0 0.19 7 1. 1 1.51 09 8 1 .88 15 0.8 1. 9 0.31 1 6 053 0.83 10 1 . 1 8 1 7 0.3 0.01 11 1 . 70 18 0.915 1 3 1 0.57 19 0 . 196 1. 1 3 0.95 0 0.

a) ind te sample mean

b) ind the estimated atooelation ntion 001 n) n 0 1 , 3 sineation 615 .

) Repeat b) sin eation 616.

.2 a) o the data o Poblem 6.1 nd an ppe bond on te vaiane o te estimatedatooelation ntion sin te estimated vales o pat b)

b) Repeat a) sin te estimated vales o pat )

3 An eodi andom poess has an atooelation ntion o te m r 0sin )

a) ve hat ane o vles mst he atooelation ntion o tis poess beestimated in ode to inlde the st to zeos o te atooelation ntion?

b If 2 estimtes M f the utcetin ae to be made in te inteval speiedn a), ht shold the samplin inteval be?


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9

c) Ho many sampl valus of th random procss ar rquird so tat ths rror ofth smat is ss han 5 prcnt of h tru maximum valu of th autocolatonnction?

.4 Assum that h r autocorration nction of th ndom procss om hich h

data of Poblm . 1 cos has th m

and is z lshr.

a) Find th valus ofA and T that provid th bst t to stimatd autocolationnction valus of Problm (b) in th lastman-squar sns (S Sc

b) Using h rsults of part (a) and quation 18, nd anothr uppr bound on thvarianc of th stimat of th autocorrlation nction. Compar ith th rsult ofProblm .2(a).

A random procss has an autocoation ncion of h rmRx() 10- l cos 20

f his procss is sampld v 001 scond, nd th numbr of sapls rquird tostimat h autocorrlation nction ih a standard dviationthat is no mor 1 %of h varianc f h procss .

Th lloing ATAB program gnrats 1000 sampls of a banditd noisprocss. Mak a plot of h sampl nction and th tim autocoation ncion ofth procss Mak an pandd plot of th autocolation nction r lag vus of01 scond around th origin Th sampling rat is 1 H

x = ad12000b ,a] = buer420/500y = flteraxy = y/std(y

7 Us th computr t mak plots of th ti autolation nctions of llowindtministic sinals

a) rct (4t

sin 2t r (4t).


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20 CHAPTER CORRELATO FCTOS

c cs 2 ect 4 65 nsd a and pcss avn saple nctins of the fm shon in iue

(a and assue tat te te intevals beteen sithin times e independent, exnental distibuted and vaiables (See Se. 27 . Sho that the atooelationctin ts pcess s a tsided exponential as shon i iue (b.

65 Suppse that eac saple nctin f the andm pocess in Poblem5 is sithinbetwen and instead f beteen ind the autocolation ntion of tecess n

65 tene te ean valu and the vaiance f each f te andm pcsses avinte win autcoelatn ntins:

a oe-r2

b) cs

c 8

654 Cns ide a and pcess avn an autelatio nction of

a Fnd te an and vaiance f ts pocess

b Is ti s pcess dieentiable W?

6 w ndpndent statina and pcesses avin sample ntions of t andY ave autcelatin nctns f

andR = 5e-J

Olr cs

sn 5Ry (r )= 5

a Fnd te autceatn nctn X Y b Fnd te autceatn nctin

Xt - Y .c Fnd bt cssceatn nctins te t poesses dene by (a) and (b.


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PROBLEMS 25

d) Fd e uooeo fuo of X( t )Y ( t ) .7.2 Fo e wo poee o obe 67. (c) u ue e oo

eo uo e ug bou of uo (6-29) Cope boudw e u u ue ee ooo fuco e

7.3 A oy do poe uocoeo c o

) Fd Rx (r .

b) Fd R- ( .

rRx (r =

r

7.4 wo oy dom poee e ooe o uo of

Rxy (r ) = 16e


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252 C ATE 6 CLATN FUNCTNS

(t) = 10 cos (OOt +


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EMS 253

bb lld s T h l Y1 d Y p

dnRy= Rn (O)E

{ dg1 X1 ) . dn g2 X2 dp dX' X

v el el s Cd d

gX) = gX) = + 1

1

n = 1 ' e s

> 0

X <

t t) = s (p) p = [ t1 t)] t1 t) b ued hu r u ul b us

lsv " u Ths du s alled pli d elato

dl be e u sa d ps d h es s T b ed b ddv s ls a eo e s ss ad t) b e il es ad Yt) = X tr )+t) be h v ,_ de r , d se t) Te ll MATAB Mle s s X t) ad Yt ) be sd ls d d a M ae

P_8

clear w clear y

rann('see', )rou(200*sq(pi));=nn( 1 1 0 + );=sq(01)*z(g 10+g) + ran(1 , 1 0000) ; -1 0B SN=Z(1:10000)

d r us the pek o he elao o ud ul d elao Hnt use he sign o ad he = = peao l dee olao)

E v ve h v Xt) s


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2 CHAPTE 6 COELAO FCTOS

Vbao ssos a mou o h o ad a axs o a mov vhc o pckup h adom vbaos du o h ouhnss o he oad suac Th sa om ho sso ma b modld as

t) = t) + n t)

h h sa t) ad he ose n t) a om depd adom pocsss Thesa om h a sso s modd as

t) = t - 1 ) + nz t)

h n t s os ha s dpd o boh t) ad n t ) . A pocsss havo ma The da r dpds upo h spac oh ssos ad h spd o hvehc

a h ssos a pacd 5 m apa, dv a aoshp b r ad h vhcspd v

b kch a bock daam o a ssm ha ca b usd o masu vehc spd ova a o 5 m p scod o 50 m p cod pc h maxmum ad mmumda vaus ha a ued a aao coao s usd

c h s hee a mmum spd ha ca b masud hs a?

d a da coao s usd, ad sas a ach sampd a a a o1 2

sampspe scod, ha s h maxmum vhc sp ha ca b masud

Th a to dsa sas ca b masud b cosscoa h oupus o o dspaad aas ad masu h da ud o axm h cosscoaoco Th omt o b cosdd s sho bo hs ssm, h dsacb aas s oma 500 , bu has a sadad dvao o 0.0 1 m sdsd o asu a h a sadad dvao o no mo ha I mada a 0 ad 1 4 adas Fd a upp bound o he sadad dvao oh dla asum od o accomplsh hs . Hint Us h oa da olaz ao

I /I / , "If

. I /

s t T1 ) + n t ) s t ) + , t )


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PROBEMS 55

. A aia adm pce hav a auocoelao ncon ofRr) = 36eZl r co J H

i amped a peiodic me a epaaed 05 ecod We he \ae

max u cecuive ample ake om th poc.

. A Gaia am vec

a a vaace maix o

[001 .s .5 .5 0.5 1

i h expeced value E[TA- 1 ]

.3 A ava e i a apped dela le ih he oupu om he vaou ap ehe a e a h belo

Y

If la bee ap i t he oupu om he ap can e epee a eo [ t) ]t X -

X

:

keie, he ehn fcto o he ou p can e te a a veco


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6 I FUTI

[I ]

a) Wri an xprssion r th oupu of h ransvrsal r, (t) , in s of thvcors X(t) and .

) I f X(t) is om a saiona random procss ih an atocoation nctin ofRx(), ri an xprssion r h auocolaion nction R (

L h inpu o h ransvrsal ltr of Prolm 9.3 hav an autocorration ncionof

and zro lshr.

I IRx ( ) =

!t

a) If h ansvrsal r has aps (i.., N = 3) and h ihing cor r ach tapis ; = 1 r all i dn and skch h auocorraon ncin of h outpu.

) Rpa part (a) if h ghting cors ar a; i i 0 3

RfSee the fences for Chapter Ofparticular interest for the material ofthis chapter are the book by

Davenport and Root, Helstm, and Papoulis.

Documents

Cooper Extracts