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  • 3/31/2013

    1

    RandomProcesses

    Dr.AliMuqaibel

    Dr.AliHusseinMuqaibel 1

    Introduction Reallife:time(space)waveform(desired+undesired) Ourprogressanddevelopmentrelaysonourabilitytodealwithsuchwave

    forms. Thesetofallthefunctionsthatareavailable(orthemenu)iscallthe

    ensembleoftherandomprocess.

    Dr.AliHusseinMuqaibel

    Thegraphofthefunction , ,versus tfor fixed,iscalledarealization,Samplepath,orsamplefunction oftherandomprocess.

    Foreachfixedfromtheindexedset, , isarandomvariable

    2

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    2

    FormalDefinition Considerarandomexperimentspecifiedbytheoutcomes fromsome

    samplespace ,andbytheprobabilitiesontheseevents. Supposethattoeveryoutcome ,weassignafunctionoftime

    accordingto somerule: , , . Wehavecreatedanindexedfamilyofrandomvariables, , , . Thisfamilyiscalledarandomprocess (stochasticprocesses). Weusuallysuppressthe anduse todenote arandomprocess. Astochasticprocessissaidtobediscretetime iftheindexset isa

    countable set(i.e.,thesetofintegersorthesetofnonnegativeintegers).

    Acontinuoustime stochasticprocessisonewhich is continuous(thermalnoise)

    Dr.AliHusseinMuqaibel 3

    DeterministicandnondeterministicProcesses

    Nondeterministic:futurevaluescannot bepredictedfromcurrentones. mostoftherandomprocessesarenondeterministic.

    Deterministic: like:

    ( , ) cos(2 ) X t s s t t

    ( , ) cos(2 )Y t s t s

    Sinusoidwithrandomamplitude[1,1]

    Sinusoidwithrandomphase ,Dr.AliHusseinMuqaibel 4

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    3

    DistributionandDensityFunctions Ar.v.isfullycharacterizedbyapdf orCDF.Howdowecharacterize

    randomprocesses? Tofullydefinearandomprocesses,weneed dimensionaljoint

    densityfunction. DistributionandDensityFunctions Firstorder:

    ; Secondorder jointdistributionfunction

    , ; , , Nth orderjointdistributionfunction , , ; , , , . , , , ; , , , , ; , ,

    Dr.AliHusseinMuqaibel 5

    StationaryandIndependence StatisticalIndependence , , , , , ; , , , , , =, , , ; , , , , ; , , Stationary

    Ifallstatisticalpropertiesdonotchangewithtime FirstorderStationaryProcess ; ; ,stationarytoorderone => Proof

    , ; ; Let

    Dr.AliHusseinMuqaibel 6

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    4

    Cyclostationary

    Dr.AliHusseinMuqaibel

    ).,...,,(

    ),...,,(

    21)(),...,(),(

    21)(),..,(),(

    21

    21

    kmTtXmTtXmTtX

    ktXtXtX

    xxxF

    xxxF

    k

    k

    Wesaythat iswidesensecyclostationary ifthemeanandautocovariancefunctionsareinvariantwithrespecttoshiftsinthetimeoriginbyintegermultiplesof ,thatis,foreveryinteger .

    ).,(),()()(

    2121 ttCmTtmTtCtmmTtm

    XX

    XX

    Notethatif iscyclostationary,theniffollowsthat isalsowidesensecyclostationary.

    Adiscretetimeorcontinuoustimerandomprocess issaidtobecyclostationary ifthejointcumulativedistributionfunctionofanysetofsamplesisinvariantwithrespecttoshiftsoftheoriginbyintegermultiplesofsomeperiod

    Forall , andallchoicesofsamplingtimes , ,

    7

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    1

    CrossCorrelationFunctionanditsproperties

    , IfXandYarejointlyw.s.s.wemaywrite

    . Orthogonalprocesses , 0 IfXandYarestatisticallyindependent =

    Ifinadditiontobeingindependenttheyareatleastw.s.s.

    Dr.AliHusseinMuqaibel 8

    Somepropertiesfor

    0 0 0 0 Thegeometricmeanistighterthanthearithmeticmean

    0 0 0 0

    Dr.AliHusseinMuqaibel 9

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    2

    MeasurementofCorrelationFunction

    Inreallife,wecannevermeasurethetruecorrelation. Weassumeergodicity anduseportionoftheavailabletime. Assumeergodicity,noneedtoprovemathematicallyphysicalsense Assumejointlyergodic =>stationary Let 0, 2 Similarly,wemayfind &

    Delay

    Delay

    Product12 . 2

    2 12

    Dr.AliHusseinMuqaibel 10

    Example Usetheabovesystemtomeasurethe for . 2 cos cos cos cos 2 2 where cos cos 2 Ifwerequirethe tobeatleast20timeslessthanthelargestvalueof

    thetrueautocorrelation 0.05 0 0.05 Waitenoughtime!Dependingonthefrequency

    Dr.AliHusseinMuqaibel 11

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    3

    Matlab:Measuringthecorrelation

    %Dr.AliMuqaibel %MeasurementofCorrelationfunction clearall closeall clc T=100; A=1; omeg=0.2; t=T:T; thet=2*pi*rand(1,1); X=A*cos(omeg*t+thet); [R,tau]=xcorr(X,'unbiased'); %R=R/(2*T); True_R=A^2/2*cos(omeg*tau); Err=A^2/2*cos(omeg*tau+2*thet)*sin(2*omeg*T)/(2*omeg*T); subplot(3,1,1) plot(tau,True_R,tau,R+Err,':') title('TrueVariance') subplot(3,1,2) plot(tau,R,':') title('Measured') %error subplot(3,1,3) plot(tau,Err) title('Error') %error

    -100 -80 -60 -40 -20 0 20 40 60 80 100-0.5

    0

    0.5True Variance

    -100 -80 -60 -40 -20 0 20 40 60 80 100-0.5

    0

    0.5Measured

    -100 -80 -60 -40 -20 0 20 40 60 80 100-0.05

    0

    0.05Error

    -40 -30 -20 -10 0 10 20 30 40-1

    0

    1True Variance

    -40 -30 -20 -10 0 10 20 30 40-1

    0

    1Measured

    -40 -30 -20 -10 0 10 20 30 40-0.1

    0

    0.1Error

    T=50,OMEGA=0.2

    T=20,OMEGA=0.2

    Notetheerrorislessthan5%

    Dr.AliHusseinMuqaibel 12

    Arandomprocess isaGaussianrandomprocess ifthesamples, ,

    arejointlyGaussian randomvariablesforall ,andallchoicesof , , .Thisdefinitionappliesfordiscretetimeandcontinuoustimeprocesses.Thejointpdf ofjointlyGaussianrandomvariablesisdeterminedbythevectorofmeansandbythe covariancematrix:

    212/

    )()(21

    1,.....,,)2(

    ),....,(1

    21C

    exxfk

    mXCmX

    kXXX

    T

    k

    where

    )(...

    )(

    m

    1

    kX

    X

    tm

    tm

    )(...)(

    ....

    ...),(...),(),()(...),(),(

    ,1,

    22212

    ,12111

    kkXkX

    kXXX

    kXXX

    ttCttC

    ttCttCttCttCttCttC

    C

    Gaussian Random Processes

    Dr.AliHusseinMuqaibel 13

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    4

    Letthediscretetimerandomprocess beasequenceofindependentGaussianrandomvariableswithmean andvarianceThecovariancematrixforthetimes , , is

    ,}{)},({ 221 IttC ijjX

    where 1 when and0otherwise,and istheidentitymatrix.Thusthecorrespondingjointpdf is

    )()...()(

    2/)(exp)2(

    1),.....,,(

    21

    1

    22

    2221,.....1

    kXXX

    k

    iikkXX

    xfxfxf

    mxxxxfk

    Example iid GaussianSequence

    Dr.AliHusseinMuqaibel 14

    ExampleofaGaussianRandomProcess

    AGaussianRandomProcesswhichisW.S.S. 4 and25 | | 16

    Specifythejointdensityfunctionforthreer.v. , 1,2,3 ,,

    , 1,2,3, 25 16 25 16 4

    251

    11

    Dr.AliHusseinMuqaibel 15

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    5

    ComplexRandomProcesses

    Acomplexrandomprocess isgivenby , ,

    Notetheconjugate

    Therecouldbeafactorof insomebooks SeeexampleinPeebles

    Dr.AliHusseinMuqaibel 16

    Supposeweobserveaprocess ,whichconsistsofadesiredsignalplusnoise .

    Findthecrosscorrelationbetweentheobservedsignalandthedesiredsignal assumingthat and areindependentrandomprocesses.

    , 1 2 1

    1

    1 1

    1 2 1 2

    1 2

    2

    2 2

    2 2

    1 2

    ( , ) [ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ]

    ( ){

    ( , ) [ ( )] [ ( )] ( , ) ( ) ( )

    ( ) ( )}( ) ( )

    X Y

    XX

    XX X N

    R t t E X t Y tX t N t

    X tE X tE X t E X tR t t E X t E N tR t t m

    N t

    t m t

    wherethethirdequalityfollowedfromthefactthat andareindependent.

    Example SignalPlusNoise

    Dr.AliHusseinMuqaibel 17

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    6

    Dr.AliHusseinMuqaibel 18

    See LeonGarciaProbability,Statistics,andRandomProcessesforElectricalEngineers,3rd Edition

    9.5GAUSSIANRANDOMPROCESSES,WIENERPROCESS,ANDBROWNIANMOTION

    EXAMPLES OF DISCRETE_TIME & Continuous-Time RANDOM PROCESSES

    iid Random Processes

    Let beadiscretetimerandomprocessconsistingofasequenceofindependent,identicallydistributed(iid)randomvariableswithcommoncdfmean andvariance . Thesequence iscalledtheiid randomprocess.Thejointcdf foranytimeinstants , . , isgivenby

    ),()...()(

    ),...,,[),....,,(

    21

    221121,...1

    kXXX

    kkkXX

    xFxFxFxXxXxXPxxxF

    K

    whereforsimplicity denotes .Theequationaboveimpliesthatif isdiscretevalues,thejointpmf factorsintotheproductofindividualpmfs,andif iscontinuousvalued,thejointpdf factorsintotheproductoftheindividualpdfs.

    Themeanofaniid process isobtained

    Thus,themeanisconstant.Theautocovariance functionisobtainedfromasfollows.If ,then

    nmXEnm nX allfor ][)(

    EXAMPLES OF DISCRETE_TIME RANDOM PROCESSES

    Dr.AliHusseinMuqaibel 19

  • 3/31/2013

    7

    1 2

    1 2

    1 2( , ) [( )( )]

    [( )] [( )] 0X n n

    n n

    C n n E X m X m

    E X m E X m

    since and areindependentrandomvariables.If then

    ])[(),( 2221 mXEnnC nX

    Wecanexpresstheautocovariance oftheiid processincompactformasfollows:

    ,),(21

    221 nnX nnC

    whereif and0otherwiseTheautocorrelationfunctionoftheiid processis:

    121nn

    22121 ),(),( mnnCnnR XX

    Dr.AliHusseinMuqaibel 20

    Let beasequenceofindependentBernoullirandomvariables. isthenaniid randomprocesstakingonvaluesfromtheset{0,1}.ArealizationofsuchaprocessisshowninFigure.Forexample, couldbeanindicatorfunctionfortheeventalightbulbfailsandisreplacedondayn.Since isaBernoullirandomvariable,ithasmeanandvariance

    1Theindependenceofthe makesprobabilitieseasytocompute.Forexample,theprobabilitythatthefirst4bitsinthesequenceare1001is

    1, 0, 0, 1 1 0 0 11

    Similarly,theprobabilitythatthesecondbitis0andtheseventhis1is0, 1 0 1 1

    Realization of