4 randomsignal.docx

4 randomsignal.docx

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4randomsignal

4.RandomSignals

4.1Introduction

Sofarwehavedealtwithcontinuousandsampledsignalshavingdefinedwaveshapes.Suchsignalsaredescribedas‘deterministic’,andthefrequencyspectrawecalculateforthemspecifythemagnitudesandrelativephasesofthesinusoidalwaveswitch,ifaddedtogether,wouldexactlyresynthesisetheoriginalwaveforms.Bycontrast,thevalueofarandomwaveformorsignalisnotspecifiedateveryinstantoftime,norisitpossibletopredictitsfuturewithcertaintyonethebasisofitspasthistory.Thereasonforthisisgenerallythatwehaveinsufficientunderstandingofthephysicalprocessproducingtherandomsignals;onotheroccasions,theunderstandingispresent,buttheeffortinvolvedinpredictingthesignalistoogreattobeworthwhile.Insuchcasesitisusualtoevaluatesomeaveragepropertiesofthesignalwhichdescribeitadequatelyforthetaskinhand.

Acommoninitialreactionisthatarandomsignalwithill-definedpropertiescanhavelittleplaceinanyscientifictheoryofsignalanalysis,buttheoppositeisinfactnearerthetruth.Suppose,forexample,itisdesiredtosendamessagealongatelegraphlink.Itisalmostvaluelesstosendaknown,deterministic,messagesincethepersonatthereceivingendlearnssolittlebyitsreception.Asasimple,itwouldbepointlesstotransmitacontinuoussinusoidaltone,sinceonceitsamplitude,frequencyandphasecharacteristicshavebeendeterminedbythereceiver,nofurtherinformationisconveyed.ButifthetoneisswitchedonandoffasinMorse-codemessage,thereceiverdoesnotknowwhethera‘dot’or‘dash’istobesentnext,anditisthisveryrandomnessoruncertaintyaboutthefutureofthesignalwhichallowsusefulinformationtobeconveyed.OntheotherhanditisquitepossibletosaysomethingabouttheaveragepropertiesofMorse-codemessages,andsuchknowledgemightbeveryusefultothedesignerofatelegraphsystem,eventhoughhedoesnotknowwhatdot-dashsequenceistobesentonanyparticularoccasion.Viewedinthislightitisunsurprisingthatrandomsignaltheoryhasplayedamajorroleinthedevelopmentofmoderncommunicationssystems.

Asignalmayberandominavarietyofways,perhapsthemostcommontypeofrandomnessisintheamplitudeofasignalwaveform,illustratedinfigure4.1（a）.Inthiscasefuturevaluesofthesignalcannotbepredictedwithcertainty,evenwhenitspasthistoryistakenintoaccount.Figure4.1（b）showsanothercommonformofrandomness,ofthegeneraltypedisplayedbytheMorse-codemessagejustmentioned.Herethesignalisalwaysatoneorotheroftwodefinitevaluesbutthetransitionsbetweenthesetwolevelsoccuratrandominstants.Signalsinwhichthetimesofoccurrenceofsomedefiniteeventortransitionbetweenstatesarerandomariseinmanydiversefieldsofstudy,suchasqueuingtheory,nuclearparticlephysicsandneurophysiology,aswellasinelectroniccommunications.Figure4.1（c）showasignalpossessingboththecommontypesofrandomnesssofarmentioned,andrepresentsanelectrocardiogram,orrecordingoftheelectricalactivityoftheheart.Theheartbeatissomewhatirregular,sothatthetimingofsuccessiveECGcomplexeshasarandomcomponent;furthermore,thewaveshapeitselfisnotexactlyrepetitiveinform.

Inpracticeasignalquiteoftencontainsbothrandomanddeterministiccomponents.Forexample,theECGcomplexesoffigure4.1（c）mightusefullybeconsideredtoconsistofastrictlyrepetitivesignalplussmallrandomamplitudeandtimingcomponents.Veryoftensucharandomcomponentistheresultofrecordingormeasurementerrors,orarisesatsomepointinthesystemwhichisoutsidethecontroloftheexperimenter.Randomdisturbancesarewidelyencounteredinelectroniccircuits,wheretheyarereferredtoaselectrical‘noise’.However,themethodsofanalysisofrandomwaveformsdescribedinthislessonapplywhetherawaveformrepresentsausefulsignaloranunwanted‘noise’.Itisimportanttobeabletodescribeanoisewaveformquantitatively,notleastsothattheeffectsonitofsignalprocessingdevicesmaybeassessed.Andweshallseelaterthatamostimportanttypeofsignalprocessingoperationisoneinwhichattemptismadetoextractorenhanceasignalwaveforminthepresenceofunwanteddisturbances.

Figure4.1

Figure4.1Signaldisplaying（a）randomamplitude,（b）randomtimingoftransitionsbetweenfixedlevel,and（c）randomamplitudeandtimingcomponentsinanessentiallyrepetitivewaveform.

Asalreadynoted,themethodusedtodescriberandomsignalsistoassesssomeaveragepropertiesofinterest.Thebranchofmathematicsinvolvedisstatistics,whichconcernsitselfwiththequantitativepropertiesofapopulationasawhole,ratherthanofitsindividualelements.Inthepresentcontext,wemaythinkofthispopulationasbeingmadeupfromaverylargenumberofsuccessivevaluesofarandomsignalwaveform.Theotherbranchofmathematicsofdirectinterestforrandomsignalanalysisisthatofprobability,whichiscloselyrelatedtostatistics.Probabilitytheoryconcernsitselfwiththelikelihoodofvariouspossibleoutcomesofrandomprocessorphenomenon,whereasstatisticsseektosummarisetheactualoutcomesusingaveragemeasures.Inordertoallowustodevelopusefulaveragemeasuresforrandomsignals,wefirstexaminesomeofthebasicnotionsofprobabilitytheory.

4.2Elementsofprobabilitytheory

4.2.1Theprobabilityofanevent

Supposeadiewithsixfacesisthrownrepeatedly.Asthenumberoftrailsincreases,andprovidedthedieisfair,weexpectthatthenumberoftimesitlandsonanyonefacetobeclosetoonesixthofthetotalnumberofthrows.Ifaskedaboutthechanceofthedielandingonaparticularfaceinthenexttrial,wewouldthereforeassessitasoneinsix.Formally,ifthetrialisrepeatedNtimesandtheeventAoccursntimes,theprobabilityofeventAisdefinedas

（4-1）

Thedefinitionofprobabilityastherelativefrequencyoftheeventisgreatintuitiveappeal,althoughitgivesrisetoconsiderabledifficultyinmorerigorousmathematicaltreatments.Themainreasonforthisisthatanyactualexperimentnecessarilyinvolvesafinitenumberoftrialsandmayfailtoapproachthelimitinaconvincingway.Eveninasimplecointossingexperiment,avastnumberoftossesmayfailtoyieldtheprobabilitiesof0.5expectedforboth‘heads’and‘tails’.Forthisreasonanalternativeapproachtothedefinitionofprobability,baseduponasetofaxioms,issometimesadopted.Whicheverdefinitionisused,however,itisclearthataprobabilityof1denotescertaintyandaprobabilityof0impliesthattheeventneveroccurs.

Nextsupposethatwedefinetheevent（AorB）asoccurringwhenevereitherAorBoccurs.InaverylargenumberNoftrials,supposeAoccursntimesandBoccursmtimes.IfAandBaremutuallyexclusive（sothattheyneveroccurtogether）theevent（AorB）occurs（n+m）times.Hence

（4-2）

Thisisthebasicadditivelawofprobability,whichmaybeextendedtocoverthecaseofanexperimentortrialwithanynumberofmutuallyexclusiveoutcomes.

4.2.2Jointandconditionalprobabilities

Supposewenowconductanexperimentwithtwosetsofpossibleoutcomes.Thenthejointprobabilityp（AandB）istheprobabilitythattheoutcomeAfromonesetoccurstogetherwithoutcomeBfromtheotherset.Forexample,theexperimentmightconsistoftossingtwodicesimultaneously,orofdrawingtwocardsfromapack.SupposethatofNexperimentsperformednproduceoutcomeA;ofthese,supposethatmalsoproduceoutcomeB.ThenmisthenumberofexperimentswhichgiverisetobothAandB,sothat

（4-3）

Notethatthelimitof（n/N）asN→∞isp（A）,andthatthelimitof（m/n）istheprobabilitythatoutcomeBoccursgiventhatoutcomeAhasalreadyoccurred.Thislatterprobabilityiscalledthe'conditionalprobabilityofBgivenA'andwillbedenotedbysymbolp（B/A）.Thus

（4-4）

Bysimilararguments,itmaybeshownthat

（4-5）

andhence

（4-6）

or

（4-7）

Thisresult,knownasBayes'rule,relatestheconditionalprobabilityofAgivenBtothatofBgivenA.

ItisobviousthatifoutcomeBiscompletelyunaffectedbyoutcomeA,thentheconditionalprobabilityofBgivenAwillbejustthesameastheprobabilityofAalone,hence

（4-8）

And

（4-9）

InthiscasetheoutcomesAandBaresaidtobestatisticallyindependent.

Toillustratetheseresults,supposewehaveaboxcontaining3redballsand5blueballs.Thefirstexperimentconsistsoftakingoneballfromtheboxatrandom,replacingitandthentakinganother,andweareinterestedintheprobabilitythatthefirstballwithdrawnisred（outcomeA）andthesecondoneblue（outcomeB）.Inthiscase,theresultofthefirstpartoftheexperimentinnowayaffectsthatofthesecond,sincethefirstballisreplaced.Hencetheprobabilityofthejointevent（redfollowedbyblue）is

（4-10）

Wenowrepeattheexperimentwithoutreplacingthefirstballbeforetakingoutthesecond,sothatthetwopartsoftheexperimentarenolongerstatisticallyindependent.Ifaredballiswithdrawnfirst,then5blueand2redballsremain,givingtheconditionalprobability

（4-11）

Thebriefdiscussionofjointandconditionalprobabilities,whichhasbeenappliedtothecaseofanexperimentwithtwosetsofpossibleoutcomes,maybeextendedtocoveranynumberofsuchsets.Itsrelevancetosignalsandsignalanalysismaybeillustratedbyreferencetofigure4.2（a）,whichshowsaportionofarandomsampled-datasignalinwhicheachsamplevaluetakesononeofsixpossiblelevels.Thisisanalogoustothediew