Digital-Communications--Fundamentals-And-Applicati

ConciseDSPTutorialInformationsourceMessagesymbolsChannelsymbolsFromothersourcesDigitalinputDigitaloutputBitstreamDigitalbasebandwaveformDigitalbandpasswaveformChannelimpulseresponseChannelhc(t)si(t)r(t)z(T)gi(t)ûiuimiˆmiChannelsymbolsTootherdestinationsMessagesymbolsInformationsinkSynch-ronizationFormatSourceencodeEncryptChannelencodeMultiplexPulsemodulateBandpassmodulateFrequencyspreadMultipleaccessXMTFormatSourcedecodeDecryptChanneldecodeDemultiplexDetectDemodulate&SampleFrequencydespreadMultipleaccessRCVSECONDEDITIONFundamentalsandApplicationsSECONDEDITIONCDCompaniontheROBERTW.STEWARTDANIELGARCÍA-ALÍSBERNARDSKLARCDBook3of3ThispageisblankiiiConciseDSPTutorialThisdocumentisaprintableversionoftheConciseDSPTutorialfrom:DigitalCommunications:FundamentalsandApplications(2ndEdition),theCompanionCDBERNARDSKLAR,PublishedbyPrenticeHall,2001.ThisDocumentCopyright©BlueBoxMultimedia2001-(FFT)303GeneralDigitalFilters344FiniteImpulseResponse(FIR)Filter364.1FIRFilterz-domainZeroes414.2FIRLinearPhase424.3Minimum/MaximumPhase454.4OrderReversedFIRFilter454.5MovingAverage(MA)FIRFilter484.6CombFilter504.7Differentiator554.8All-passFIRFilter554.8.1PhaseCompensation594.8.2FractionalDelayImplementation605InfiniteImpulseResponse(IIR)Filter635.1IIRFilterStability635.2Integrator646FilterDesignTechniques666.1RealTimeImplementation706.2FIRFilterWordlengthConsiderations706.3FilteringBitErrors717AdaptiveFilters737.1GenericAdaptiveSignalProcessor737.2LeastSquares767.3LeastSquaresResidual797.4Wiener-HopfSolution797.5LeastMeanSquares(LMS)Algorithm837.5.1Convergence877.5.2Misadjustment887.5.3TimeConstant897.5.4Variants897.6RecursiveLeastSquares(RLS)917.6.1ExponentiallyWeighted9751IntroductiontheCompanionCDCDCDCDDIGITALCOMMUNICATIONSBERNARDSKLAR2ndEdition1IntroductionInthisdocumentwereviewsomestandardmathematicsandkeyanalysispointsofdigitalsignalprocessing.Specificallythisreviewwillcoverfrequencydomainanalysis,digitalfilteringandadaptivedigitalfiltering.TheaimofthethisdocumentistoprovideadditionalmathematicalinformationtotheGettingStartedandBookExercisebooksonthecompanionCD.6ConciseDSPTutorial-RobertW.Stewart,DanielGarcía-AlísDSPedia2FrequencyDomainAnalysis2.1FourierSeriesThereexistsmathematicaltheorycalledtheFourierseriesthatallowsanyperiodicwaveformintimetobedecomposedintoasumofharmonicallyrelatedsineandcosinewaveforms.ThefirstrequirementinrealisingtheFourierseriesistocalculatethefundamentalperiod,,whichistheshortesttimeoverwhichthesignalrepeats,i.e.forasignal,then:(1)Foraperiodicsignalwithfundamentalperiodseconds,theFourierseriesrepresentsthissignalasasumofsineandcosinecomponentsthatareharmonicsofthefundamentalfrequency,Hz.TheFourierseriescanbewritteninanumberofdifferentways:(2)Txt()xt()xtT+()xt2T+()…xtkT+()====timeT1f0----=xt()The(fundamental)periodofasignalidentifiedas.Thefundamentalfrequency,,iscalculatedas.Clearly.xt()Tf0f01T⁄=xt0()xt0T+()xt02T+()==t0t0T+t02T+Tf01T⁄=xt()An2πntT------------cosn0=∞∑Bn2πntT------------sinn1=∞∑+=A0An2πntT------------cosBn2πntT------------sin+n1=∞∑+=72FrequencyDomainAnalysistheCompanionCDCDCDCDDIGITALCOMMUNICATIONSBERNARDSKLAR2ndEditionwhereandaretheamplitudesofthevariouscosineandsinewaveforms,andangularfrequencyisdenotedbyradians/second.Dependingontheactualproblembeingsolvedwecanchoosetospecifythefundamentalperiodicityofthewaveformintermsoftheperiod(),frequency(),orangularfrequency()asshowninEq.2.Notethatthereisactuallynorequirementtospecificallyincludeatermsince,althoughthereisanterm,since,whichrepresentsanyDCcomponentthatmaybepresentinthesignal.InmoredescriptivelanguagetheaboveFourierseriessaysthatanyperiodicsignalcanbereproducedbyaddinga(possiblyinfinite)seriesofharmonicallyrelatedsinusoidalwaveformsofamplitudesor.Therefore,ifaperiodicsignalwithafundamentalperiodofsay0.01secondsisidentified,thentheFourierserieswillallowthiswaveformtoberepresentedasasumofvariouscosineandsinewavesatfrequenciesof100Hz(thefundamentalfrequency,),200Hz,300Hz(theharmonicfrequencies,)andsoon.Theamplitudesofthesecosineandsinewavesaregivenby.....andsoon.Sohowarethevaluesofandcalculated?Theanswercanbederivedbysomebasictrigonometry.TakingthelastlineofEq.2,ifwemultiplybothsidesby,whereisana