Digital Communication - Quick Guide - Tutorialspoint

This unit has dealt with the introduction, the digitization of signals, the advantages and the elements of digital communications. In the coming chapters, ... Home CodingGround Jobs Whiteboard Tools Business Teachwithus DigitalCommunicationTutorial DigitalCommunication-Home AnalogtoDigital PulseCodeModulation Sampling Quantization DifferentialPCM DeltaModulation Techniques LineCodes DataEncodingTechniques PulseShaping DigitalModulationTechniques AmplitudeShiftKeying FrequencyShiftKeying PhaseShiftKeying QuadraturePhaseShiftKeying DifferentialPhaseShiftKeying M-aryEncoding InformationTheory SourceCodingTheorem ChannelCodingTheorem ErrorControlCoding SpreadSpectrumModulation DigitalCommunicationResources QuickGuide DigitalCommunication-Resources DigitalCommunication-Discussion SelectedReading UPSCIASExamsNotes Developer'sBestPractices QuestionsandAnswers EffectiveResumeWriting HRInterviewQuestions ComputerGlossary WhoisWho DigitalCommunication-QuickGuide Advertisements PreviousPage NextPage  DigitalCommunication-AnalogtoDigital Thecommunicationthatoccursinourday-to-daylifeisintheformofsignals.Thesesignals,suchassoundsignals,generally,areanaloginnature.Whenthecommunicationneedstobeestablishedoveradistance,thentheanalogsignalsaresentthroughwire,usingdifferenttechniquesforeffectivetransmission. TheNecessityofDigitization Theconventionalmethodsofcommunicationusedanalogsignalsforlongdistancecommunications,whichsufferfrommanylossessuchasdistortion,interference,andotherlossesincludingsecuritybreach. Inordertoovercometheseproblems,thesignalsaredigitizedusingdifferenttechniques.Thedigitizedsignalsallowthecommunicationtobemoreclearandaccuratewithoutlosses. Thefollowingfigureindicatesthedifferencebetweenanaloganddigitalsignals.Thedigitalsignalsconsistof1sand0swhichindicateHighandLowvaluesrespectively. AdvantagesofDigitalCommunication Asthesignalsaredigitized,therearemanyadvantagesofdigitalcommunicationoveranalogcommunication,suchas− Theeffectofdistortion,noise,andinterferenceismuchlessindigitalsignalsastheyarelessaffected. Digitalcircuitsaremorereliable. Digitalcircuitsareeasytodesignandcheaperthananalogcircuits. Thehardwareimplementationindigitalcircuits,ismoreflexiblethananalog. Theoccurrenceofcross-talkisveryrareindigitalcommunication. Thesignalisun-alteredasthepulseneedsahighdisturbancetoalteritsproperties,whichisverydifficult. Signalprocessingfunctionssuchasencryptionandcompressionareemployedindigitalcircuitstomaintainthesecrecyoftheinformation. Theprobabilityoferroroccurrenceisreducedbyemployingerrordetectinganderrorcorrectingcodes. Spreadspectrumtechniqueisusedtoavoidsignaljamming. CombiningdigitalsignalsusingTimeDivisionMultiplexing(TDM)iseasierthancombininganalogsignalsusingFrequencyDivisionMultiplexing(FDM). Theconfiguringprocessofdigitalsignalsiseasierthananalogsignals. Digitalsignalscanbesavedandretrievedmoreconvenientlythananalogsignals. Manyofthedigitalcircuitshavealmostcommonencodingtechniquesandhencesimilardevicescanbeusedforanumberofpurposes. Thecapacityofthechanneliseffectivelyutilizedbydigitalsignals. ElementsofDigitalCommunication Theelementswhichformadigitalcommunicationsystemisrepresentedbythefollowingblockdiagramfortheeaseofunderstanding. Followingarethesectionsofthedigitalcommunicationsystem. Source Thesourcecanbeananalogsignal.Example:ASoundsignal InputTransducer Thisisatransducerwhichtakesaphysicalinputandconvertsittoanelectricalsignal(Example:microphone).Thisblockalsoconsistsofananalogtodigitalconverterwhereadigitalsignalisneededforfurtherprocesses. Adigitalsignalisgenerallyrepresentedbyabinarysequence. SourceEncoder Thesourceencodercompressesthedataintominimumnumberofbits.Thisprocesshelpsineffectiveutilizationofthebandwidth.Itremovestheredundantbits(unnecessaryexcessbits,i.e.,zeroes). ChannelEncoder Thechannelencoder,doesthecodingforerrorcorrection.Duringthetransmissionofthesignal,duetothenoiseinthechannel,thesignalmaygetalteredandhencetoavoidthis,thechannelencoderaddssomeredundantbitstothetransmitteddata.Thesearetheerrorcorrectingbits. DigitalModulator Thesignaltobetransmittedismodulatedherebyacarrier.Thesignalisalsoconvertedtoanalogfromthedigitalsequence,inordertomakeittravelthroughthechannelormedium. Channel Thechanneloramedium,allowstheanalogsignaltotransmitfromthetransmitterendtothereceiverend. DigitalDemodulator Thisisthefirststepatthereceiverend.Thereceivedsignalisdemodulatedaswellasconvertedagainfromanalogtodigital.Thesignalgetsreconstructedhere. ChannelDecoder Thechanneldecoder,afterdetectingthesequence,doessomeerrorcorrections.Thedistortionswhichmightoccurduringthetransmission,arecorrectedbyaddingsomeredundantbits.Thisadditionofbitshelpsinthecompleterecoveryoftheoriginalsignal. SourceDecoder Theresultantsignalisonceagaindigitizedbysamplingandquantizingsothatthepuredigitaloutputisobtainedwithoutthelossofinformation.Thesourcedecoderrecreatesthesourceoutput. OutputTransducer Thisisthelastblockwhichconvertsthesignalintotheoriginalphysicalform,whichwasattheinputofthetransmitter.Itconvertstheelectricalsignalintophysicaloutput(Example:loudspeaker). OutputSignal Thisistheoutputwhichisproducedafterthewholeprocess.Example−Thesoundsignalreceived. Thisunithasdealtwiththeintroduction,thedigitizationofsignals,theadvantagesandtheelementsofdigitalcommunications.Inthecomingchapters,wewilllearnabouttheconceptsofDigitalcommunications,indetail. PulseCodeModulation Modulationistheprocessofvaryingoneormoreparametersofacarriersignalinaccordancewiththeinstantaneousvaluesofthemessagesignal. Themessagesignalisthesignalwhichisbeingtransmittedforcommunicationandthecarriersignalisahighfrequencysignalwhichhasnodata,butisusedforlongdistancetransmission. Therearemanymodulationtechniques,whichareclassifiedaccordingtothetypeofmodulationemployed.Ofthemall,thedigitalmodulationtechniqueusedisPulseCodeModulation(PCM). Asignalispulsecodemodulatedtoconvertitsanaloginformationintoabinarysequence,i.e.,1sand0s.TheoutputofaPCMwillresembleabinarysequence.ThefollowingfigureshowsanexampleofPCMoutputwithrespecttoinstantaneousvaluesofagivensinewave. Insteadofapulsetrain,PCMproducesaseriesofnumbersordigits,andhencethisprocessiscalledasdigital.Eachoneofthesedigits,thoughinbinarycode,representtheapproximateamplitudeofthesignalsampleatthatinstant. InPulseCodeModulation,themessagesignalisrepresentedbyasequenceofcodedpulses.Thismessagesignalisachievedbyrepresentingthesignalindiscreteforminbothtimeandamplitude. BasicElementsofPCM ThetransmittersectionofaPulseCodeModulatorcircuitconsistsofSampling,QuantizingandEncoding,whichareperformedintheanalog-to-digitalconvertersection.Thelowpassfilterpriortosamplingpreventsaliasingofthemessagesignal. Thebasicoperationsinthereceiversectionareregenerationofimpairedsignals,decoding,andreconstructionofthequantizedpulsetrain.FollowingistheblockdiagramofPCMwhichrepresentsthebasicelementsofboththetransmitterandthereceiversections. LowPassFilter Thisfiltereliminatesthehighfrequencycomponentspresentintheinputanalogsignalwhichisgreaterthanthehighestfrequencyofthemessagesignal,toavoidaliasingofthemessagesignal. Sampler Thisisthetechniquewhichhelpstocollectthesampledataatinstantaneousvaluesofmessagesignal,soastoreconstructtheoriginalsignal.ThesamplingratemustbegreaterthantwicethehighestfrequencycomponentWofthemessagesignal,inaccordancewiththesamplingtheorem. Quantizer Quantizingisaprocessofreducingtheexcessivebitsandconfiningthedata.ThesampledoutputwhengiventoQuantizer,reducestheredundantbitsandcompressesthevalue. Encoder Thedigitizationofanalogsignalisdonebytheencoder.Itdesignateseachquantizedlevelbyabinarycode.Thesamplingdonehereisthesample-and-holdprocess.Thesethreesections(LPF,Sampler,andQuantizer)willactasananalogtodigitalconverter.Encodingminimizesthebandwidthused. RegenerativeRepeater Thissectionincreasesthesignalstrength.Theoutputofthechannelalsohasoneregenerativerepeatercircuit,tocompensatethesignallossandreconstructthesignal,andalsotoincreaseitsstrength. Decoder Thedecodercircuitdecodesthepulsecodedwaveformtoreproducetheoriginalsignal.Thiscircuitactsasthedemodulator. ReconstructionFilter Afterthedigital-to-analogconversionisdonebytheregenerativecircuitandthedecoder,alow-passfilterisemployed,calledasthereconstructionfiltertogetbacktheoriginalsignal. Hence,thePulseCodeModulatorcircuitdigitizesthegivenanalogsignal,codesitandsamplesit,andthentransmitsitinananalogform.Thiswholeprocessisrepeatedinareversepatterntoobtaintheoriginalsignal. DigitalCommunication-Sampling Samplingisdefinedas,“Theprocessofmeasuringtheinstantaneousvaluesofcontinuous-timesignalinadiscreteform.” Sampleisapieceofdatatakenfromthewholedatawhichiscontinuousinthetimedomain. Whenasourcegeneratesananalogsignalandifthathastobedigitized,having1sand0si.e.,HighorLow,thesignalhastobediscretizedintime.ThisdiscretizationofanalogsignaliscalledasSampling. Thefollowingfigureindicatesacontinuous-timesignalx(t)andasampledsignalxs(t).Whenx(t)ismultipliedbyaperiodicimpulsetrain,thesampledsignalxs(t)isobtained. SamplingRate Todiscretizethesignals,thegapbetweenthesamplesshouldbefixed.ThatgapcanbetermedasasamplingperiodTs. $$Sampling\:Frequency=\frac{1}{T_{s}}=f_s$$ Where, $T_s$isthesamplingtime $f_s$isthesamplingfrequencyorthesamplingrate Samplingfrequencyisthereciprocalofthesamplingperiod.Thissamplingfrequency,canbesimplycalledasSamplingrate.Thesamplingratedenotesthenumberofsamplestakenpersecond,orforafinitesetofvalues. Forananalogsignaltobereconstructedfromthedigitizedsignal,thesamplingrateshouldbehighlyconsidered.Therateofsamplingshouldbesuchthatthedatainthemessagesignalshouldneitherbelostnoritshouldgetover-lapped.Hence,aratewasfixedforthis,calledasNyquistrate. NyquistRate Supposethatasignalisband-limitedwithnofrequencycomponentshigherthanWHertz.Thatmeans,Wisthehighestfrequency.Forsuchasignal,foreffectivereproductionoftheoriginalsignal,thesamplingrateshouldbetwicethehighestfrequency. Whichmeans, $$f_S=2W$$ Where, $f_S$isthesamplingrate Wisthehighestfrequency ThisrateofsamplingiscalledasNyquistrate. Atheoremcalled,SamplingTheorem,wasstatedonthetheoryofthisNyquistrate. SamplingTheorem Thesamplingtheorem,whichisalsocalledasNyquisttheorem,deliversthetheoryofsufficientsamplerateintermsofbandwidthfortheclassoffunctionsthatarebandlimited. Thesamplingtheoremstatesthat,“asignalcanbeexactlyreproducedifitissampledattheratefswhichisgreaterthantwicethemaximumfrequencyW.” Tounderstandthissamplingtheorem,letusconsideraband-limitedsignal,i.e.,asignalwhosevalueisnon-zerobetweensome–WandWHertz. Suchasignalisrepresentedas$x(f)=0\:for\:\midf\mid>W$ Forthecontinuous-timesignalx(t),theband-limitedsignalinfrequencydomain,canberepresentedasshowninthefollowingfigure. Weneedasamplingfrequency,afrequencyatwhichthereshouldbenolossofinformation,evenaftersampling.Forthis,wehavetheNyquistratethatthesamplingfrequencyshouldbetwotimesthemaximumfrequency.Itisthecriticalrateofsampling. Ifthesignalx(t)issampledabovetheNyquistrate,theoriginalsignalcanberecovered,andifitissampledbelowtheNyquistrate,thesignalcannotberecovered. Thefollowingfigureexplainsasignal,ifsampledatahigherratethan2winthefrequencydomain. TheabovefigureshowstheFouriertransformofasignalxs(t).Here,theinformationisreproducedwithoutanyloss.Thereisnomixingupandhencerecoveryispossible. TheFourierTransformofthesignalxs(t)is $$X_s(w)=\frac{1}{T_{s}}\sum_{n=-\infty}^\inftyX(w-nw_0)$$ Where$T_s$=SamplingPeriodand$w_0=\frac{2\pi}{T_s}$ Letusseewhathappensifthesamplingrateisequaltotwicethehighestfrequency(2W) Thatmeans, $$f_s=2W$$ Where, $f_s$isthesamplingfrequency Wisthehighestfrequency Theresultwillbeasshownintheabovefigure.Theinformationisreplacedwithoutanyloss.Hence,thisisalsoagoodsamplingrate. Now,letuslookatthecondition, $$f_s<2W$$ Theresultantpatternwilllooklikethefollowingfigure. Wecanobservefromtheabovepatternthattheover-lappingofinformationisdone,whichleadstomixingupandlossofinformation.Thisunwantedphenomenonofover-lappingiscalledasAliasing. Aliasing Aliasingcanbereferredtoas“thephenomenonofahigh-frequencycomponentinthespectrumofasignal,takingontheidentityofalow-frequencycomponentinthespectrumofitssampledversion.” ThecorrectivemeasurestakentoreducetheeffectofAliasingare− InthetransmittersectionofPCM,alowpassanti-aliasingfilterisemployed,beforethesampler,toeliminatethehighfrequencycomponents,whichareunwanted. Thesignalwhichissampledafterfiltering,issampledatarateslightlyhigherthantheNyquistrate. ThischoiceofhavingthesamplingratehigherthanNyquistrate,alsohelpsintheeasierdesignofthereconstructionfilteratthereceiver. ScopeofFourierTransform Itisgenerallyobservedthat,weseekthehelpofFourierseriesandFouriertransformsinanalyzingthesignalsandalsoinprovingtheorems.Itisbecause− TheFourierTransformistheextensionofFourierseriesfornon-periodicsignals. Fouriertransformisapowerfulmathematicaltoolwhichhelpstoviewthesignalsindifferentdomainsandhelpstoanalyzethesignalseasily. AnysignalcanbedecomposedintermsofsumofsinesandcosinesusingthisFouriertransform. Inthenextchapter,letusdiscussabouttheconceptofQuantization. DigitalCommunication-Quantization Thedigitizationofanalogsignalsinvolvestheroundingoffofthevalueswhichareapproximatelyequaltotheanalogvalues.Themethodofsamplingchoosesafewpointsontheanalogsignalandthenthesepointsarejoinedtoroundoffthevaluetoanearstabilizedvalue.SuchaprocessiscalledasQuantization. QuantizinganAnalogSignal Theanalog-to-digitalconvertersperformthistypeoffunctiontocreateaseriesofdigitalvaluesoutofthegivenanalogsignal.Thefollowingfigurerepresentsananalogsignal.Thissignaltogetconvertedintodigital,hastoundergosamplingandquantizing. Thequantizingofananalogsignalisdonebydiscretizingthesignalwithanumberofquantizationlevels.Quantizationisrepresentingthesampledvaluesoftheamplitudebyafinitesetoflevels,whichmeansconvertingacontinuous-amplitudesampleintoadiscrete-timesignal. Thefollowingfigureshowshowananalogsignalgetsquantized.Thebluelinerepresentsanalogsignalwhilethebrownonerepresentsthequantizedsignal. Bothsamplingandquantizationresultinthelossofinformation.ThequalityofaQuantizeroutputdependsuponthenumberofquantizationlevelsused.Thediscreteamplitudesofthequantizedoutputarecalledasrepresentationlevelsorreconstructionlevels.Thespacingbetweenthetwoadjacentrepresentationlevelsiscalledaquantumorstep-size. Thefollowingfigureshowstheresultantquantizedsignalwhichisthedigitalformforthegivenanalogsignal. ThisisalsocalledasStair-casewaveform,inaccordancewithitsshape. TypesofQuantization TherearetwotypesofQuantization-UniformQuantizationandNon-uniformQuantization. ThetypeofquantizationinwhichthequantizationlevelsareuniformlyspacedistermedasaUniformQuantization.Thetypeofquantizationinwhichthequantizationlevelsareunequalandmostlytherelationbetweenthemislogarithmic,istermedasaNon-uniformQuantization. Therearetwotypesofuniformquantization.TheyareMid-RisetypeandMid-Treadtype.Thefollowingfiguresrepresentthetwotypesofuniformquantization. Figure1showsthemid-risetypeandfigure2showsthemid-treadtypeofuniformquantization. TheMid-Risetypeissocalledbecausetheoriginliesinthemiddleofaraisingpartofthestair-caselikegraph.Thequantizationlevelsinthistypeareeveninnumber. TheMid-treadtypeissocalledbecausetheoriginliesinthemiddleofatreadofthestair-caselikegraph.Thequantizationlevelsinthistypeareoddinnumber. Boththemid-riseandmid-treadtypeofuniformquantizersaresymmetricabouttheorigin. QuantizationError Foranysystem,duringitsfunctioning,thereisalwaysadifferenceinthevaluesofitsinputandoutput.Theprocessingofthesystemresultsinanerror,whichisthedifferenceofthosevalues. ThedifferencebetweenaninputvalueanditsquantizedvalueiscalledaQuantizationError.AQuantizerisalogarithmicfunctionthatperformsQuantization(roundingoffthevalue).Ananalog-to-digitalconverter(ADC)worksasaquantizer. Thefollowingfigureillustratesanexampleforaquantizationerror,indicatingthedifferencebetweentheoriginalsignalandthequantizedsignal. QuantizationNoise Itisatypeofquantizationerror,whichusuallyoccursinanalogaudiosignal,whilequantizingittodigital.Forexample,inmusic,thesignalskeepchangingcontinuously,wherearegularityisnotfoundinerrors.SucherrorscreateawidebandnoisecalledasQuantizationNoise. CompandinginPCM ThewordCompandingisacombinationofCompressingandExpanding,whichmeansthatitdoesboth.Thisisanon-lineartechniqueusedinPCMwhichcompressesthedataatthetransmitterandexpandsthesamedataatthereceiver.Theeffectsofnoiseandcrosstalkarereducedbyusingthistechnique. TherearetwotypesofCompandingtechniques.Theyare− A-lawCompandingTechnique UniformquantizationisachievedatA=1,wherethecharacteristiccurveislinearandnocompressionisdone. A-lawhasmid-riseattheorigin.Hence,itcontainsanon-zerovalue. A-lawcompandingisusedforPCMtelephonesystems. µ-lawCompandingTechnique Uniformquantizationisachievedatµ=0,wherethecharacteristiccurveislinearandnocompressionisdone. µ-lawhasmid-treadattheorigin.Hence,itcontainsazerovalue. µ-lawcompandingisusedforspeechandmusicsignals. µ-lawisusedinNorthAmericaandJapan. DigitalCommunication-DifferentialPCM Forthesamplesthatarehighlycorrelated,whenencodedbyPCMtechnique,leaveredundantinformationbehind.Toprocessthisredundantinformationandtohaveabetteroutput,itisawisedecisiontotakeapredictedsampledvalue,assumedfromitspreviousoutputandsummarizethemwiththequantizedvalues.SuchaprocessiscalledasDifferentialPCM(DPCM)technique. DPCMTransmitter TheDPCMTransmitterconsistsofQuantizerandPredictorwithtwosummercircuits.FollowingistheblockdiagramofDPCMtransmitter. Thesignalsateachpointarenamedas− $x(nT_s)$isthesampledinput $\widehat{x}(nT_s)$isthepredictedsample $e(nT_s)$isthedifferenceofsampledinputandpredictedoutput,oftencalledaspredictionerror $v(nT_s)$isthequantizedoutput $u(nT_s)$isthepredictorinputwhichisactuallythesummeroutputofthepredictoroutputandthequantizeroutput Thepredictorproducestheassumedsamplesfromthepreviousoutputsofthetransmittercircuit.Theinputtothispredictoristhequantizedversionsoftheinputsignal$x(nT_s)$. QuantizerOutputisrepresentedas− $$v(nT_s)=Q[e(nT_s)]$$ $=e(nT_s)+q(nT_s)$ Whereq(nTs)isthequantizationerror Predictorinputisthesumofquantizeroutputandpredictoroutput, $$u(nT_s)=\widehat{x}(nT_s)+v(nT_s)$$ $u(nT_s)=\widehat{x}(nT_s)+e(nT_s)+q(nT_s)$ $$u(nT_s)=x(nT_s)+q(nT_s)$$ Thesamepredictorcircuitisusedinthedecodertoreconstructtheoriginalinput. DPCMReceiver TheblockdiagramofDPCMReceiverconsistsofadecoder,apredictor,andasummercircuit.FollowingisthediagramofDPCMReceiver. Thenotationofthesignalsisthesameasthepreviousones.Intheabsenceofnoise,theencodedreceiverinputwillbethesameastheencodedtransmitteroutput. Asmentionedbefore,thepredictorassumesavalue,basedonthepreviousoutputs.Theinputgiventothedecoderisprocessedandthatoutputissummedupwiththeoutputofthepredictor,toobtainabetteroutput. DigitalCommunication-DeltaModulation ThesamplingrateofasignalshouldbehigherthantheNyquistrate,toachievebettersampling.IfthissamplingintervalinDifferentialPCMisreducedconsiderably,thesampleto-sampleamplitudedifferenceisverysmall,asifthedifferenceis1-bitquantization,thenthestep-sizewillbeverysmalli.e.,Δ(delta). DeltaModulation Thetypeofmodulation,wherethesamplingrateismuchhigherandinwhichthestepsizeafterquantizationisofasmallervalueΔ,suchamodulationistermedasdeltamodulation. FeaturesofDeltaModulation Followingaresomeofthefeaturesofdeltamodulation. Anover-sampledinputistakentomakefulluseofthesignalcorrelation. Thequantizationdesignissimple. TheinputsequenceismuchhigherthantheNyquistrate. Thequalityismoderate. Thedesignofthemodulatorandthedemodulatorissimple. Thestair-caseapproximationofoutputwaveform. Thestep-sizeisverysmall,i.e.,Δ(delta). Thebitratecanbedecidedbytheuser. Thisinvolvessimplerimplementation. DeltaModulationisasimplifiedformofDPCMtechnique,alsoviewedas1-bitDPCMscheme.Asthesamplingintervalisreduced,thesignalcorrelationwillbehigher. DeltaModulator TheDeltaModulatorcomprisesofa1-bitquantizerandadelaycircuitalongwithtwosummercircuits.Followingistheblockdiagramofadeltamodulator. ThepredictorcircuitinDPCMisreplacedbyasimpledelaycircuitinDM. Fromtheabovediagram,wehavethenotationsas− $x(nT_s)$=oversampledinput $e_p(nT_s)$=summeroutputandquantizerinput $e_q(nT_s)$=quantizeroutput=$v(nT_s)$ $\widehat{x}(nT_s)$=outputofdelaycircuit $u(nT_s)$=inputofdelaycircuit Usingthesenotations,nowweshalltrytofigureouttheprocessofdeltamodulation. $e_p(nT_s)=x(nT_s)-\widehat{x}(nT_s)$ ---------equation1 $=x(nT_s)-u([n-1]T_s)$ $=x(nT_s)-[\widehat{x}[[n-1]T_s]+v[[n-1]T_s]]$ ---------equation2 Further, $v(nT_s)=e_q(nT_s)=S.sig.[e_p(nT_s)]$ ---------equation3 $u(nT_s)=\widehat{x}(nT_s)+e_q(nT_s)$ Where, $\widehat{x}(nT_s)$=thepreviousvalueofthedelaycircuit $e_q(nT_s)$=quantizeroutput=$v(nT_s)$ Hence, $u(nT_s)=u([n-1]T_s)+v(nT_s)$ ---------equation4 Whichmeans, Thepresentinputofthedelayunit =(Thepreviousoutputofthedelayunit)+(thepresentquantizeroutput) AssumingzeroconditionofAccumulation, $u(nT_s)=S\displaystyle\sum\limits_{j=1}^nsig[e_p(jT_s)]$ AccumulatedversionofDMoutput=$\displaystyle\sum\limits_{j=1}^nv(jT_s)$ ---------equation5 Now,notethat $\widehat{x}(nT_s)=u([n-1]T_s)$ $=\displaystyle\sum\limits_{j=1}^{n-1}v(jT_s)$ ---------equation6 DelayunitoutputisanAccumulatoroutputlaggingbyonesample. Fromequations5&6,wegetapossiblestructureforthedemodulator. AStair-caseapproximatedwaveformwillbetheoutputofthedeltamodulatorwiththestep-sizeasdelta(Δ).Theoutputqualityofthewaveformismoderate. DeltaDemodulator Thedeltademodulatorcomprisesofalowpassfilter,asummer,andadelaycircuit.Thepredictorcircuitiseliminatedhereandhencenoassumedinputisgiventothedemodulator. Followingisthediagramfordeltademodulator. Fromtheabovediagram,wehavethenotationsas− $\widehat{v}(nT_s)$istheinputsample $\widehat{u}(nT_s)$isthesummeroutput $\bar{x}(nT_s)$isthedelayedoutput Abinarysequencewillbegivenasaninputtothedemodulator.Thestair-caseapproximatedoutputisgiventotheLPF. Lowpassfilterisusedformanyreasons,buttheprominentreasonisnoiseeliminationforout-of-bandsignals.Thestep-sizeerrorthatmayoccuratthetransmitteriscalledgranularnoise,whichiseliminatedhere.Ifthereisnonoisepresent,thenthemodulatoroutputequalsthedemodulatorinput. AdvantagesofDMOverDPCM 1-bitquantizer Veryeasydesignofthemodulatorandthedemodulator However,thereexistssomenoiseinDM. SlopeOverloaddistortion(whenΔissmall) Granularnoise(whenΔislarge) AdaptiveDeltaModulation(ADM) Indigitalmodulation,wehavecomeacrosscertainproblemofdeterminingthestep-size,whichinfluencesthequalityoftheoutputwave. Alargerstep-sizeisneededinthesteepslopeofmodulatingsignalandasmallerstepsizeisneededwherethemessagehasasmallslope.Theminutedetailsgetmissedintheprocess.So,itwouldbebetterifwecancontroltheadjustmentofstep-size,accordingtoourrequirementinordertoobtainthesamplinginadesiredfashion.ThisistheconceptofAdaptiveDeltaModulation. FollowingistheblockdiagramofAdaptivedeltamodulator. Thegainofthevoltagecontrolledamplifierisadjustedbytheoutputsignalfromthesampler.Theamplifiergaindeterminesthestep-sizeandbothareproportional. ADMquantizesthedifferencebetweenthevalueofthecurrentsampleandthepredictedvalueofthenextsample.Itusesavariablestepheighttopredictthenextvalues,forthefaithfulreproductionofthefastvaryingvalues. DigitalCommunication-Techniques Thereareafewtechniqueswhichhavepavedthebasicpathtodigitalcommunicationprocesses.Forthesignalstogetdigitized,wehavethesamplingandquantizingtechniques. Forthemtoberepresentedmathematically,wehaveLPCanddigitalmultiplexingtechniques.Thesedigitalmodulationtechniquesarefurtherdiscussed. LinearPredictiveCoding LinearPredictiveCoding(LPC)isatoolwhichrepresentsdigitalspeechsignalsinlinearpredictivemodel.Thisismostlyusedinaudiosignalprocessing,speechsynthesis,speechrecognition,etc. Linearpredictionisbasedontheideathatthecurrentsampleisbasedonthelinearcombinationofpastsamples.Theanalysisestimatesthevaluesofadiscrete-timesignalasalinearfunctionoftheprevioussamples. Thespectralenvelopeisrepresentedinacompressedform,usingtheinformationofthelinearpredictivemodel.Thiscanbemathematicallyrepresentedas− $s(n)=\displaystyle\sum\limits_{k=1}^p\alpha_ks(n-k)$forsomevalueofpandαk Where s(n)isthecurrentspeechsample kisaparticularsample pisthemostrecentvalue αkisthepredictorco-efficient s(n-k)isthepreviousspeechsample ForLPC,thepredictorco-efficientvaluesaredeterminedbyminimizingthesumofsquareddifferences(overafiniteinterval)betweentheactualspeechsamplesandthelinearlypredictedones. Thisisaveryusefulmethodforencodingspeechatalowbitrate.TheLPCmethodisveryclosetotheFastFourierTransform(FFT)method. Multiplexing Multiplexingistheprocessofcombiningmultiplesignalsintoonesignal,overasharedmedium.Thesesignals,ifanaloginnature,theprocessiscalledasanalogmultiplexing.Ifdigitalsignalsaremultiplexed,itiscalledasdigitalmultiplexing. Multiplexingwasfirstdevelopedintelephony.Anumberofsignalswerecombinedtosendthroughasinglecable.Theprocessofmultiplexingdividesacommunicationchannelintoseveralnumberoflogicalchannels,allottingeachoneforadifferentmessagesignaloradatastreamtobetransferred.Thedevicethatdoesmultiplexing,canbecalledasaMUX.Thereverseprocess,i.e.,extractingthenumberofchannelsfromone,whichisdoneatthereceiveriscalledasde-multiplexing.Thedevicewhichdoesde-multiplexingiscalledasDEMUX. ThefollowingfiguresrepresentMUXandDEMUX.Theirprimaryuseisinthefieldofcommunications. TypesofMultiplexers Therearemainlytwotypesofmultiplexers,namelyanaloganddigital.TheyarefurtherdividedintoFDM,WDM,andTDM.Thefollowingfiguregivesadetailedideaonthisclassification. Actually,therearemanytypesofmultiplexingtechniques.Ofthemall,wehavethemaintypeswithgeneralclassification,mentionedintheabovefigure. AnalogMultiplexing Theanalogmultiplexingtechniquesinvolvesignalswhichareanaloginnature.Theanalogsignalsaremultiplexedaccordingtotheirfrequency(FDM)orwavelength(WDM). FrequencyDivisionMultiplexing(FDM) Inanalogmultiplexing,themostusedtechniqueisFrequencyDivisionMultiplexing(FDM).Thistechniqueusesvariousfrequenciestocombinestreamsofdata,forsendingthemonacommunicationmedium,asasinglesignal. Example−Atraditionaltelevisiontransmitter,whichsendsanumberofchannelsthroughasinglecable,usesFDM. WavelengthDivisionMultiplexing(WDM) WavelengthDivisionmultiplexingisananalogtechnique,inwhichmanydatastreamsofdifferentwavelengthsaretransmittedinthelightspectrum.Ifthewavelengthincreases,thefrequencyofthesignaldecreases.Aprismwhichcanturndifferentwavelengthsintoasingleline,canbeusedattheoutputofMUXandinputofDEMUX. Example−OpticalfibercommunicationsuseWDMtechniquetomergedifferentwavelengthsintoasinglelightforcommunication. DigitalMultiplexing Thetermdigitalrepresentsthediscretebitsofinformation.Hence,theavailabledataisintheformofframesorpackets,whicharediscrete. TimeDivisionMultiplexing(TDM) InTDM,thetimeframeisdividedintoslots.Thistechniqueisusedtotransmitasignaloverasinglecommunicationchannel,byallottingoneslotforeachmessage. OfallthetypesofTDM,themainonesareSynchronousandAsynchronousTDM. SynchronousTDM InSynchronousTDM,theinputisconnectedtoaframe.Ifthereare‘n’numberofconnections,thentheframeisdividedinto‘n’timeslots.Oneslotisallocatedforeachinputline. Inthistechnique,thesamplingrateiscommontoallsignalsandhencethesameclockinputisgiven.TheMUXallocatesthesameslottoeachdeviceatalltimes. AsynchronousTDM InAsynchronousTDM,thesamplingrateisdifferentforeachofthesignalsandacommonclockisnotrequired.Iftheallotteddevice,foratime-slot,transmitsnothingandsitsidle,thenthatslotisallottedtoanotherdevice,unlikesynchronous.ThistypeofTDMisusedinAsynchronoustransfermodenetworks. RegenerativeRepeater Foranycommunicationsystemtobereliable,itshouldtransmitandreceivethesignalseffectively,withoutanyloss.APCMwave,aftertransmittingthroughachannel,getsdistortedduetothenoiseintroducedbythechannel. Theregenerativepulsecomparedwiththeoriginalandreceivedpulse,willbeasshowninthefollowingfigure. Forabetterreproductionofthesignal,acircuitcalledasregenerativerepeaterisemployedinthepathbeforethereceiver.Thishelpsinrestoringthesignalsfromthelossesoccurred.Followingisthediagrammaticalrepresentation. Thisconsistsofanequalizeralongwithanamplifier,atimingcircuit,andadecisionmakingdevice.Theirworkingofeachofthecomponentsisdetailedasfollows. Equalizer Thechannelproducesamplitudeandphasedistortionstothesignals.Thisisduetothetransmissioncharacteristicsofthechannel.TheEqualizercircuitcompensatestheselossesbyshapingthereceivedpulses. TimingCircuit Toobtainaqualityoutput,thesamplingofthepulsesshouldbedonewherethesignaltonoiseratio(SNR)ismaximum.Toachievethisperfectsampling,aperiodicpulsetrainhastobederivedfromthereceivedpulses,whichisdonebythetimingcircuit. Hence,thetimingcircuit,allotsthetimingintervalforsamplingathighSNR,throughthereceivedpulses. DecisionDevice Thetimingcircuitdeterminesthesamplingtimes.Thedecisiondeviceisenabledatthesesamplingtimes.Thedecisiondevicedecidesitsoutputbasedonwhethertheamplitudeofthequantizedpulseandthenoise,exceedsapre-determinedvalueornot. Thesearefewofthetechniquesusedindigitalcommunications.Thereareotherimportanttechniquestobelearned,calledasdataencodingtechniques.Letuslearnabouttheminthesubsequentchapters,aftertakingalookatthelinecodes. DigitalCommunication-LineCodes Alinecodeisthecodeusedfordatatransmissionofadigitalsignaloveratransmissionline.Thisprocessofcodingischosensoastoavoidoverlapanddistortionofsignalsuchasinter-symbolinterference. PropertiesofLineCoding Followingarethepropertiesoflinecoding− Asthecodingisdonetomakemorebitstransmitonasinglesignal,thebandwidthusedismuchreduced. Foragivenbandwidth,thepowerisefficientlyused. Theprobabilityoferrorismuchreduced. Errordetectionisdoneandthebipolartoohasacorrectioncapability. Powerdensityismuchfavorable. Thetimingcontentisadequate. Longstringsof1sand0sisavoidedtomaintaintransparency. TypesofLineCoding Thereare3typesofLineCoding Unipolar Polar Bi-polar UnipolarSignaling UnipolarsignalingisalsocalledasOn-OffKeyingorsimplyOOK. Thepresenceofpulserepresentsa1andtheabsenceofpulserepresentsa0. TherearetwovariationsinUnipolarsignaling− NonReturntoZero(NRZ) ReturntoZero(RZ) UnipolarNon-ReturntoZero(NRZ) Inthistypeofunipolarsignaling,aHighindataisrepresentedbyapositivepulsecalledasMark,whichhasadurationT0equaltothesymbolbitduration.ALowindatainputhasnopulse. Thefollowingfigureclearlydepictsthis. Advantages TheadvantagesofUnipolarNRZare− Itissimple. Alesserbandwidthisrequired. Disadvantages ThedisadvantagesofUnipolarNRZare− Noerrorcorrectiondone. Presenceoflowfrequencycomponentsmaycausethesignaldroop. Noclockispresent. Lossofsynchronizationislikelytooccur(especiallyforlongstringsof1sand0s). UnipolarReturntoZero(RZ) Inthistypeofunipolarsignaling,aHighindata,thoughrepresentedbyaMarkpulse,itsdurationT0islessthanthesymbolbitduration.Halfofthebitdurationremainshighbutitimmediatelyreturnstozeroandshowstheabsenceofpulseduringtheremaininghalfofthebitduration. Itisclearlyunderstoodwiththehelpofthefollowingfigure. Advantages TheadvantagesofUnipolarRZare− Itissimple. Thespectrallinepresentatthesymbolratecanbeusedasaclock. Disadvantages ThedisadvantagesofUnipolarRZare− Noerrorcorrection. OccupiestwicethebandwidthasunipolarNRZ. Thesignaldroopiscausedattheplaceswheresignalisnon-zeroat0Hz. PolarSignaling TherearetwomethodsofPolarSignaling.Theyare− PolarNRZ PolarRZ PolarNRZ InthistypeofPolarsignaling,aHighindataisrepresentedbyapositivepulse,whileaLowindataisrepresentedbyanegativepulse.Thefollowingfiguredepictsthiswell. Advantages TheadvantagesofPolarNRZare− Itissimple. Nolow-frequencycomponentsarepresent. Disadvantages ThedisadvantagesofPolarNRZare− Noerrorcorrection. Noclockispresent. Thesignaldroopiscausedattheplaceswherethesignalisnon-zeroat0Hz. PolarRZ InthistypeofPolarsignaling,aHighindata,thoughrepresentedbyaMarkpulse,itsdurationT0islessthanthesymbolbitduration.Halfofthebitdurationremainshighbutitimmediatelyreturnstozeroandshowstheabsenceofpulseduringtheremaininghalfofthebitduration. However,foraLowinput,anegativepulserepresentsthedata,andthezerolevelremainssamefortheotherhalfofthebitduration.Thefollowingfiguredepictsthisclearly. Advantages TheadvantagesofPolarRZare− Itissimple. Nolow-frequencycomponentsarepresent. Disadvantages ThedisadvantagesofPolarRZare− Noerrorcorrection. Noclockispresent. OccupiestwicethebandwidthofPolarNRZ. Thesignaldroopiscausedatplaceswherethesignalisnon-zeroat0Hz. BipolarSignaling Thisisanencodingtechniquewhichhasthreevoltagelevelsnamely&plus;,-and0.Suchasignaliscalledasduo-binarysignal. AnexampleofthistypeisAlternateMarkInversion(AMI).Fora1,thevoltagelevelgetsatransitionfrom&plus;to–orfrom–to&plus;,havingalternate1stobeofequalpolarity.A0willhaveazerovoltagelevel. Eveninthismethod,wehavetwotypes. BipolarNRZ BipolarRZ Fromthemodelssofardiscussed,wehavelearntthedifferencebetweenNRZandRZ.Itjustgoesinthesamewayheretoo.Thefollowingfigureclearlydepictsthis. TheabovefigurehasboththeBipolarNRZandRZwaveforms.ThepulsedurationandsymbolbitdurationareequalinNRZtype,whilethepulsedurationishalfofthesymbolbitdurationinRZtype. Advantages Followingaretheadvantages− Itissimple. Nolow-frequencycomponentsarepresent. OccupieslowbandwidththanunipolarandpolarNRZschemes. ThistechniqueissuitablefortransmissionoverACcoupledlines,assignaldroopingdoesn’toccurhere. Asingleerrordetectioncapabilityispresentinthis. Disadvantages Followingarethedisadvantages− Noclockispresent. Longstringsofdatacauseslossofsynchronization. PowerSpectralDensity Thefunctionwhichdescribeshowthepowerofasignalgotdistributedatvariousfrequencies,inthefrequencydomainiscalledasPowerSpectralDensity(PSD). PSDistheFourierTransformofAuto-Correlation(Similaritybetweenobservations).Itisintheformofarectangularpulse. PSDDerivation AccordingtotheEinstein-Wiener-Khintchinetheorem,iftheautocorrelationfunctionorpowerspectraldensityofarandomprocessisknown,theothercanbefoundexactly. Hence,toderivethepowerspectraldensity,weshallusethetimeauto-correlation$(R_x(\tau))$ofapowersignal$x(t)$asshownbelow. $R_x(\tau)=\lim_{T_p\rightarrow\infty}\frac{1}{T_p}\int_{\frac{{-T_p}}{2}}^{\frac{T_p}{2}}x(t)x(t+\tau)dt$ Since$x(t)$consistsofimpulses,$R_x(\tau)$canbewrittenas $R_x(\tau)=\frac{1}{T}\displaystyle\sum\limits_{n=-\infty}^\inftyR_n\delta(\tau-nT)$ Where$R_n=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_ka_ka_{k+n}$ Gettingtoknowthat$R_n=R_{-n}$forrealsignals,wehave $S_x(w)=\frac{1}{T}(R_0+2\displaystyle\sum\limits_{n=1}^\inftyR_n\cosnwT)$ Sincethepulsefilterhasthespectrumof$(w)\leftrightarrowf(t)$,wehave $s_y(w)=\midF(w)\mid^2S_x(w)$ $=\frac{\midF(w)\mid^2}{T}(\displaystyle\sum\limits_{n=-\infty}^\inftyR_ne^{-jnwT_{b}})$ $=\frac{\midF(w)\mid^2}{T}(R_0+2\displaystyle\sum\limits_{n=1}^\inftyR_n\cosnwT)$ Hence,wegettheequationforPowerSpectralDensity.Usingthis,wecanfindthePSDofvariouslinecodes. DataEncodingTechniques Encodingistheprocessofconvertingthedataoragivensequenceofcharacters,symbols,alphabetsetc.,intoaspecifiedformat,forthesecuredtransmissionofdata.Decodingisthereverseprocessofencodingwhichistoextracttheinformationfromtheconvertedformat. DataEncoding Encodingistheprocessofusingvariouspatternsofvoltageorcurrentlevelstorepresent1sand0softhedigitalsignalsonthetransmissionlink. ThecommontypesoflineencodingareUnipolar,Polar,Bipolar,andManchester. EncodingTechniques Thedataencodingtechniqueisdividedintothefollowingtypes,dependinguponthetypeofdataconversion. AnalogdatatoAnalogsignals−ThemodulationtechniquessuchasAmplitudeModulation,FrequencyModulationandPhaseModulationofanalogsignals,fallunderthiscategory. AnalogdatatoDigitalsignals−Thisprocesscanbetermedasdigitization,whichisdonebyPulseCodeModulation(PCM).Hence,itisnothingbutdigitalmodulation.Aswehavealreadydiscussed,samplingandquantizationaretheimportantfactorsinthis.DeltaModulationgivesabetteroutputthanPCM. DigitaldatatoAnalogsignals−ThemodulationtechniquessuchasAmplitudeShiftKeying(ASK),FrequencyShiftKeying(FSK),PhaseShiftKeying(PSK),etc.,fallunderthiscategory.Thesewillbediscussedinsubsequentchapters. DigitaldatatoDigitalsignals−Theseareinthissection.Thereareseveralwaystomapdigitaldatatodigitalsignals.Someofthemare− NonReturntoZero(NRZ) NRZCodeshas1forHighvoltageleveland0forLowvoltagelevel.ThemainbehaviorofNRZcodesisthatthevoltagelevelremainsconstantduringbitinterval.Theendorstartofabitwillnotbeindicatedanditwillmaintainthesamevoltagestate,ifthevalueofthepreviousbitandthevalueofthepresentbitaresame. ThefollowingfigureexplainstheconceptofNRZcoding. Iftheaboveexampleisconsidered,asthereisalongsequenceofconstantvoltagelevelandtheclocksynchronizationmaybelostduetotheabsenceofbitinterval,itbecomesdifficultforthereceivertodifferentiatebetween0and1. TherearetwovariationsinNRZnamely− NRZ-L(NRZ–LEVEL) Thereisachangeinthepolarityofthesignal,onlywhentheincomingsignalchangesfrom1to0orfrom0to1.ItisthesameasNRZ,however,thefirstbitoftheinputsignalshouldhaveachangeofpolarity. NRZ-I(NRZ–INVERTED) Ifa1occursattheincomingsignal,thenthereoccursatransitionatthebeginningofthebitinterval.Fora0attheincomingsignal,thereisnotransitionatthebeginningofthebitinterval. NRZcodeshasadisadvantagethatthesynchronizationofthetransmitterclockwiththereceiverclockgetscompletelydisturbed,whenthereisastringof1sand0s.Hence,aseparateclocklineneedstobeprovided. Bi-phaseEncoding Thesignallevelischeckedtwiceforeverybittime,bothinitiallyandinthemiddle.Hence,theclockrateisdoublethedatatransferrateandthusthemodulationrateisalsodoubled.Theclockistakenfromthesignalitself.Thebandwidthrequiredforthiscodingisgreater. TherearetwotypesofBi-phaseEncoding. Bi-phaseManchester DifferentialManchester Bi-phaseManchester Inthistypeofcoding,thetransitionisdoneatthemiddleofthebit-interval.ThetransitionfortheresultantpulseisfromHightoLowinthemiddleoftheinterval,fortheinputbit1.WhilethetransitionisfromLowtoHighfortheinputbit0. DifferentialManchester Inthistypeofcoding,therealwaysoccursatransitioninthemiddleofthebitinterval.Ifthereoccursatransitionatthebeginningofthebitinterval,thentheinputbitis0.Ifnotransitionoccursatthebeginningofthebitinterval,thentheinputbitis1. ThefollowingfigureillustratesthewaveformsofNRZ-L,NRZ-I,Bi-phaseManchesterandDifferentialManchestercodingfordifferentdigitalinputs. BlockCoding Amongthetypesofblockcoding,thefamousonesare4B/5Bencodingand8B/6Tencoding.Thenumberofbitsareprocessedindifferentmanners,inbothoftheseprocesses. 4B/5BEncoding InManchesterencoding,tosendthedata,theclockswithdoublespeedisrequiredratherthanNRZcoding.Here,asthenameimplies,4bitsofcodeismappedwith5bits,withaminimumnumberof1bitsinthegroup. TheclocksynchronizationprobleminNRZ-Iencodingisavoidedbyassigninganequivalentwordof5bitsintheplaceofeachblockof4consecutivebits.These5-bitwordsarepredeterminedinadictionary. Thebasicideaofselectinga5-bitcodeisthat,itshouldhaveoneleading0anditshouldhavenomorethantwotrailing0s.Hence,thesewordsarechosensuchthattwotransactionstakeplaceperblockofbits. 8B/6TEncoding Wehaveusedtwovoltagelevelstosendasinglebitoverasinglesignal.Butifweusemorethan3voltagelevels,wecansendmorebitspersignal. Forexample,if6voltagelevelsareusedtorepresent8bitsonasinglesignal,thensuchencodingistermedas8B/6Tencoding.Henceinthismethod,wehaveasmanyas729(3^6)combinationsforsignaland256(2^8)combinationsforbits. Thesearethetechniquesmostlyusedforconvertingdigitaldataintodigitalsignalsbycompressingorcodingthemforreliabletransmissionofdata. DigitalCommunication-PulseShaping Aftergoingthroughdifferenttypesofcodingtechniques,wehaveanideaonhowthedataispronetodistortionandhowthemeasuresaretakentopreventitfromgettingaffectedsoastoestablishareliablecommunication. Thereisanotherimportantdistortionwhichismostlikelytooccur,calledasInter-SymbolInterference(ISI). InterSymbolInterference Thisisaformofdistortionofasignal,inwhichoneormoresymbolsinterferewithsubsequentsignals,causingnoiseordeliveringapooroutput. CausesofISI ThemaincausesofISIare− Multi-pathPropagation Non-linearfrequencyinchannels TheISIisunwantedandshouldbecompletelyeliminatedtogetacleanoutput.ThecausesofISIshouldalsoberesolvedinordertolessenitseffect. ToviewISIinamathematicalformpresentinthereceiveroutput,wecanconsiderthereceiveroutput. Thereceivingfilteroutput$y(t)$issampledattime$t_i=iT_b$(withitakingonintegervalues),yielding− $y(t_i)=\mu\displaystyle\sum\limits_{k=-\infty}^{\infty}a_kp(iT_b-kT_b)$ $=\mua_i+\mu\displaystyle\sum\limits_{k=-\infty\\k\neqi}^{\infty}a_kp(iT_b-kT_b)$ Intheaboveequation,thefirstterm$\mua_i$isproducedbytheithtransmittedbit. Thesecondtermrepresentstheresidualeffectofallothertransmittedbitsonthedecodingoftheithbit.ThisresidualeffectiscalledasInterSymbolInterference. IntheabsenceofISI,theoutputwillbe− $$y(t_i)=\mua_i$$ Thisequationshowsthattheithbittransmittediscorrectlyreproduced.However,thepresenceofISIintroducesbiterrorsanddistortionsintheoutput. Whiledesigningthetransmitterorareceiver,itisimportantthatyouminimizetheeffectsofISI,soastoreceivetheoutputwiththeleastpossibleerrorrate. CorrelativeCoding Sofar,we’vediscussedthatISIisanunwantedphenomenonanddegradesthesignal.ButthesameISIifusedinacontrolledmanner,ispossibletoachieveabitrateof2WbitspersecondinachannelofbandwidthWHertz.SuchaschemeiscalledasCorrelativeCodingorPartialresponsesignalingschemes. SincetheamountofISIisknown,itiseasytodesignthereceiveraccordingtotherequirementsoastoavoidtheeffectofISIonthesignal.ThebasicideaofcorrelativecodingisachievedbyconsideringanexampleofDuo-binarySignaling. Duo-binarySignaling Thenameduo-binarymeansdoublingthebinarysystem’stransmissioncapability.Tounderstandthis,letusconsiderabinaryinputsequence{ak}consistingofuncorrelatedbinarydigitseachhavingadurationTaseconds.Inthis,thesignal1isrepresentedbya&plus;1voltandthesymbol0bya-1volt. Therefore,theduo-binarycoderoutputckisgivenasthesumofpresentbinarydigitakandthepreviousvalueak-1asshowninthefollowingequation. $$c_k=a_k+a_{k-1}$$ Theaboveequationstatesthattheinputsequenceofuncorrelatedbinarysequence{ak}ischangedintoasequenceofcorrelatedthreelevelpulses{ck}.ThiscorrelationbetweenthepulsesmaybeunderstoodasintroducingISIinthetransmittedsignalinanartificialmanner. EyePattern AneffectivewaytostudytheeffectsofISIistheEyePattern.ThenameEyePatternwasgivenfromitsresemblancetothehumaneyeforbinarywaves.Theinteriorregionoftheeyepatterniscalledtheeyeopening.Thefollowingfigureshowstheimageofaneye-pattern. Jitteristheshort-termvariationoftheinstantofdigitalsignal,fromitsidealposition,whichmayleadtodataerrors. WhentheeffectofISIincreases,tracesfromtheupperportiontothelowerportionoftheeyeopeningincreasesandtheeyegetscompletelyclosed,ifISIisveryhigh. Aneyepatternprovidesthefollowinginformationaboutaparticularsystem. Actualeyepatternsareusedtoestimatethebiterrorrateandthesignal-to-noiseratio. ThewidthoftheeyeopeningdefinesthetimeintervaloverwhichthereceivedwavecanbesampledwithouterrorfromISI. Theinstantoftimewhentheeyeopeningiswide,willbethepreferredtimeforsampling. Therateoftheclosureoftheeye,accordingtothesamplingtime,determineshowsensitivethesystemistothetimingerror. Theheightoftheeyeopening,ataspecifiedsamplingtime,definesthemarginovernoise. Hence,theinterpretationofeyepatternisanimportantconsideration. Equalization Forreliablecommunicationtobeestablished,weneedtohaveaqualityoutput.Thetransmissionlossesofthechannelandotherfactorsaffectingthequalityofthesignal,havetobetreated.Themostoccurringloss,aswehavediscussed,istheISI. TomakethesignalfreefromISI,andtoensureamaximumsignaltonoiseratio,weneedtoimplementamethodcalledEqualization.Thefollowingfigureshowsanequalizerinthereceiverportionofthecommunicationsystem. Thenoiseandinterferenceswhicharedenotedinthefigure,arelikelytooccur,duringtransmission.Theregenerativerepeaterhasanequalizercircuit,whichcompensatesthetransmissionlossesbyshapingthecircuit.TheEqualizerisfeasibletogetimplemented. ErrorProbabilityandFigure-of-merit Therateatwhichdatacanbecommunicatediscalledthedatarate.Therateatwhicherroroccursinthebits,whiletransmittingdataiscalledtheBitErrorRate(BER). TheprobabilityoftheoccurrenceofBERistheErrorProbability.TheincreaseinSignaltoNoiseRatio(SNR)decreasestheBER,hencetheErrorProbabilityalsogetsdecreased. InanAnalogreceiver,thefigureofmeritatthedetectionprocesscanbetermedastheratioofoutputSNRtotheinputSNR.Agreatervalueoffigure-of-meritwillbeanadvantage. DigitalModulationTechniques Digital-to-Analogsignalsisthenextconversionwewilldiscussinthischapter.ThesetechniquesarealsocalledasDigitalModulationtechniques. DigitalModulationprovidesmoreinformationcapacity,highdatasecurity,quickersystemavailabilitywithgreatqualitycommunication.Hence,digitalmodulationtechniqueshaveagreaterdemand,fortheircapacitytoconveylargeramountsofdatathananalogmodulationtechniques. Therearemanytypesofdigitalmodulationtechniquesandalsotheircombinations,dependingupontheneed.Ofthemall,wewilldiscusstheprominentones. ASK–AmplitudeShiftKeying Theamplitudeoftheresultantoutputdependsupontheinputdatawhetheritshouldbeazeroleveloravariationofpositiveandnegative,dependinguponthecarrierfrequency. FSK–FrequencyShiftKeying Thefrequencyoftheoutputsignalwillbeeitherhighorlow,dependingupontheinputdataapplied. PSK–PhaseShiftKeying Thephaseoftheoutputsignalgetsshifteddependingupontheinput.Thesearemainlyoftwotypes,namelyBinaryPhaseShiftKeying(BPSK)andQuadraturePhaseShiftKeying(QPSK),accordingtothenumberofphaseshifts.TheotheroneisDifferentialPhaseShiftKeying(DPSK)whichchangesthephaseaccordingtothepreviousvalue. M-aryEncoding M-aryEncodingtechniquesarethemethodswheremorethantwobitsaremadetotransmitsimultaneouslyonasinglesignal.Thishelpsinthereductionofbandwidth. ThetypesofM-arytechniquesare− M-aryASK M-aryFSK M-aryPSK Allofthesearediscussedinsubsequentchapters. AmplitudeShiftKeying AmplitudeShiftKeying(ASK)isatypeofAmplitudeModulationwhichrepresentsthebinarydataintheformofvariationsintheamplitudeofasignal. Anymodulatedsignalhasahighfrequencycarrier.ThebinarysignalwhenASKmodulated,givesazerovalueforLowinputwhileitgivesthecarrieroutputforHighinput. ThefollowingfigurerepresentsASKmodulatedwaveformalongwithitsinput. TofindtheprocessofobtainingthisASKmodulatedwave,letuslearnabouttheworkingoftheASKmodulator. ASKModulator TheASKmodulatorblockdiagramcomprisesofthecarriersignalgenerator,thebinarysequencefromthemessagesignalandtheband-limitedfilter.FollowingistheblockdiagramoftheASKModulator. Thecarriergenerator,sendsacontinuoushigh-frequencycarrier.ThebinarysequencefromthemessagesignalmakestheunipolarinputtobeeitherHighorLow.Thehighsignalclosestheswitch,allowingacarrierwave.Hence,theoutputwillbethecarriersignalathighinput.Whenthereislowinput,theswitchopens,allowingnovoltagetoappear.Hence,theoutputwillbelow. Theband-limitingfilter,shapesthepulsedependingupontheamplitudeandphasecharacteristicsoftheband-limitingfilterorthepulse-shapingfilter. ASKDemodulator TherearetwotypesofASKDemodulationtechniques.Theyare− AsynchronousASKDemodulation/detection SynchronousASKDemodulation/detection Theclockfrequencyatthetransmitterwhenmatcheswiththeclockfrequencyatthereceiver,itisknownasaSynchronousmethod,asthefrequencygetssynchronized.Otherwise,itisknownasAsynchronous. AsynchronousASKDemodulator TheAsynchronousASKdetectorconsistsofahalf-waverectifier,alowpassfilter,andacomparator.Followingistheblockdiagramforthesame. ThemodulatedASKsignalisgiventothehalf-waverectifier,whichdeliversapositivehalfoutput.Thelowpassfiltersuppressesthehigherfrequenciesandgivesanenvelopedetectedoutputfromwhichthecomparatordeliversadigitaloutput. SynchronousASKDemodulator SynchronousASKdetectorconsistsofaSquarelawdetector,lowpassfilter,acomparator,andavoltagelimiter.Followingistheblockdiagramforthesame. TheASKmodulatedinputsignalisgiventotheSquarelawdetector.Asquarelawdetectorisonewhoseoutputvoltageisproportionaltothesquareoftheamplitudemodulatedinputvoltage.Thelowpassfilterminimizesthehigherfrequencies.Thecomparatorandthevoltagelimiterhelptogetacleandigitaloutput. FrequencyShiftKeying FrequencyShiftKeying(FSK)isthedigitalmodulationtechniqueinwhichthefrequencyofthecarriersignalvariesaccordingtothedigitalsignalchanges.FSKisaschemeoffrequencymodulation. TheoutputofaFSKmodulatedwaveishighinfrequencyforabinaryHighinputandislowinfrequencyforabinaryLowinput.Thebinary1sand0sarecalledMarkandSpacefrequencies. ThefollowingimageisthediagrammaticrepresentationofFSKmodulatedwaveformalongwithitsinput. TofindtheprocessofobtainingthisFSKmodulatedwave,letusknowabouttheworkingofaFSKmodulator. FSKModulator TheFSKmodulatorblockdiagramcomprisesoftwooscillatorswithaclockandtheinputbinarysequence.Followingisitsblockdiagram. Thetwooscillators,producingahigherandalowerfrequencysignals,areconnectedtoaswitchalongwithaninternalclock.Toavoidtheabruptphasediscontinuitiesoftheoutputwaveformduringthetransmissionofthemessage,aclockisappliedtoboththeoscillators,internally.Thebinaryinputsequenceisappliedtothetransmittersoastochoosethefrequenciesaccordingtothebinaryinput. FSKDemodulator TherearedifferentmethodsfordemodulatingaFSKwave.ThemainmethodsofFSKdetectionareasynchronousdetectorandsynchronousdetector.Thesynchronousdetectorisacoherentone,whileasynchronousdetectorisanon-coherentone. AsynchronousFSKDetector TheblockdiagramofAsynchronousFSKdetectorconsistsoftwobandpassfilters,twoenvelopedetectors,andadecisioncircuit.Followingisthediagrammaticrepresentation. TheFSKsignalispassedthroughthetwoBandPassFilters(BPFs),tunedtoSpaceandMarkfrequencies.TheoutputfromthesetwoBPFslooklikeASKsignal,whichisgiventotheenvelopedetector.Thesignalineachenvelopedetectorismodulatedasynchronously. Thedecisioncircuitchooseswhichoutputismorelikelyandselectsitfromanyoneoftheenvelopedetectors.Italsore-shapesthewaveformtoarectangularone. SynchronousFSKDetector TheblockdiagramofSynchronousFSKdetectorconsistsoftwomixerswithlocaloscillatorcircuits,twobandpassfiltersandadecisioncircuit.Followingisthediagrammaticrepresentation. TheFSKsignalinputisgiventothetwomixerswithlocaloscillatorcircuits.Thesetwoareconnectedtotwobandpassfilters.Thesecombinationsactasdemodulatorsandthedecisioncircuitchooseswhichoutputismorelikelyandselectsitfromanyoneofthedetectors.Thetwosignalshaveaminimumfrequencyseparation. Forbothofthedemodulators,thebandwidthofeachofthemdependsontheirbitrate.Thissynchronousdemodulatorisabitcomplexthanasynchronoustypedemodulators. DigitalCommunication-PhaseShiftKeying PhaseShiftKeying(PSK)isthedigitalmodulationtechniqueinwhichthephaseofthecarriersignalischangedbyvaryingthesineandcosineinputsataparticulartime.PSKtechniqueiswidelyusedforwirelessLANs,bio-metric,contactlessoperations,alongwithRFIDandBluetoothcommunications. PSKisoftwotypes,dependinguponthephasesthesignalgetsshifted.Theyare− BinaryPhaseShiftKeying(BPSK) Thisisalsocalledas2-phasePSKorPhaseReversalKeying.Inthistechnique,thesinewavecarriertakestwophasereversalssuchas0°and180°. BPSKisbasicallyaDoubleSideBandSuppressedCarrier(DSBSC)modulationscheme,formessagebeingthedigitalinformation. QuadraturePhaseShiftKeying(QPSK) Thisisthephaseshiftkeyingtechnique,inwhichthesinewavecarriertakesfourphasereversalssuchas0°,90°,180°,and270°. Ifthiskindoftechniquesarefurtherextended,PSKcanbedonebyeightorsixteenvaluesalso,dependingupontherequirement. BPSKModulator TheblockdiagramofBinaryPhaseShiftKeyingconsistsofthebalancemodulatorwhichhasthecarriersinewaveasoneinputandthebinarysequenceastheotherinput.Followingisthediagrammaticrepresentation. ThemodulationofBPSKisdoneusingabalancemodulator,whichmultipliesthetwosignalsappliedattheinput.Forazerobinaryinput,thephasewillbe0°andforahighinput,thephasereversalisof180°. FollowingisthediagrammaticrepresentationofBPSKModulatedoutputwavealongwithitsgiveninput. Theoutputsinewaveofthemodulatorwillbethedirectinputcarrierortheinverted(180°phaseshifted)inputcarrier,whichisafunctionofthedatasignal. BPSKDemodulator TheblockdiagramofBPSKdemodulatorconsistsofamixerwithlocaloscillatorcircuit,abandpassfilter,atwo-inputdetectorcircuit.Thediagramisasfollows. Byrecoveringtheband-limitedmessagesignal,withthehelpofthemixercircuitandthebandpassfilter,thefirststageofdemodulationgetscompleted.Thebasebandsignalwhichisbandlimitedisobtainedandthissignalisusedtoregeneratethebinarymessagebitstream. Inthenextstageofdemodulation,thebitclockrateisneededatthedetectorcircuittoproducetheoriginalbinarymessagesignal.Ifthebitrateisasub-multipleofthecarrierfrequency,thenthebitclockregenerationissimplified.Tomakethecircuiteasilyunderstandable,adecision-makingcircuitmayalsobeinsertedatthe2ndstageofdetection. QuadraturePhaseShiftKeying TheQuadraturePhaseShiftKeying(QPSK)isavariationofBPSK,anditisalsoaDoubleSideBandSuppressedCarrier(DSBSC)modulationscheme,whichsendstwobitsofdigitalinformationatatime,calledasbigits. Insteadoftheconversionofdigitalbitsintoaseriesofdigitalstream,itconvertsthemintobitpairs.Thisdecreasesthedatabitratetohalf,whichallowsspacefortheotherusers. QPSKModulator TheQPSKModulatorusesabit-splitter,twomultiplierswithlocaloscillator,a2-bitserialtoparallelconverter,andasummercircuit.Followingistheblockdiagramforthesame. Atthemodulator’sinput,themessagesignal’sevenbits(i.e.,2ndbit,4thbit,6thbit,etc.)andoddbits(i.e.,1stbit,3rdbit,5thbit,etc.)areseparatedbythebitssplitterandaremultipliedwiththesamecarriertogenerateoddBPSK(calledasPSKI)andevenBPSK(calledasPSKQ).ThePSKQsignalisanyhowphaseshiftedby90°beforebeingmodulated. TheQPSKwaveformfortwo-bitsinputisasfollows,whichshowsthemodulatedresultfordifferentinstancesofbinaryinputs. QPSKDemodulator TheQPSKDemodulatorusestwoproductdemodulatorcircuitswithlocaloscillator,twobandpassfilters,twointegratorcircuits,anda2-bitparalleltoserialconverter.Followingisthediagramforthesame. ThetwoproductdetectorsattheinputofdemodulatorsimultaneouslydemodulatethetwoBPSKsignals.Thepairofbitsarerecoveredherefromtheoriginaldata.Thesesignalsafterprocessing,arepassedtotheparalleltoserialconverter. DifferentialPhaseShiftKeying InDifferentialPhaseShiftKeying(DPSK)thephaseofthemodulatedsignalisshiftedrelativetotheprevioussignalelement.Noreferencesignalisconsideredhere.Thesignalphasefollowsthehighorlowstateofthepreviouselement.ThisDPSKtechniquedoesn’tneedareferenceoscillator. ThefollowingfigurerepresentsthemodelwaveformofDPSK. Itisseenfromtheabovefigurethat,ifthedatabitisLowi.e.,0,thenthephaseofthesignalisnotreversed,butcontinuedasitwas.IfthedataisaHighi.e.,1,thenthephaseofthesignalisreversed,aswithNRZI,inverton1(aformofdifferentialencoding). Ifweobservetheabovewaveform,wecansaythattheHighstaterepresentsanMinthemodulatingsignalandtheLowstaterepresentsaWinthemodulatingsignal. DPSKModulator DPSKisatechniqueofBPSK,inwhichthereisnoreferencephasesignal.Here,thetransmittedsignalitselfcanbeusedasareferencesignal.FollowingisthediagramofDPSKModulator. DPSKencodestwodistinctsignals,i.e.,thecarrierandthemodulatingsignalwith180°phaseshifteach.TheserialdatainputisgiventotheXNORgateandtheoutputisagainfedbacktotheotherinputthrough1-bitdelay.TheoutputoftheXNORgatealongwiththecarriersignalisgiventothebalancemodulator,toproducetheDPSKmodulatedsignal. DPSKDemodulator InDPSKdemodulator,thephaseofthereversedbitiscomparedwiththephaseofthepreviousbit.FollowingistheblockdiagramofDPSKdemodulator. Fromtheabovefigure,itisevidentthatthebalancemodulatorisgiventheDPSKsignalalongwith1-bitdelayinput.ThatsignalismadetoconfinetolowerfrequencieswiththehelpofLPF.Thenitispassedtoashapercircuit,whichisacomparatororaSchmitttriggercircuit,torecovertheoriginalbinarydataastheoutput. DigitalCommunication-M-aryEncoding Thewordbinaryrepresentstwobits.Mrepresentsadigitthatcorrespondstothenumberofconditions,levels,orcombinationspossibleforagivennumberofbinaryvariables. Thisisthetypeofdigitalmodulationtechniqueusedfordatatransmissioninwhichinsteadofonebit,twoormorebitsaretransmittedatatime.Asasinglesignalisusedformultiplebittransmission,thechannelbandwidthisreduced. M-aryEquation Ifadigitalsignalisgivenunderfourconditions,suchasvoltagelevels,frequencies,phases,andamplitude,thenM=4. Thenumberofbitsnecessarytoproduceagivennumberofconditionsisexpressedmathematicallyas $$N=\log_{2}{M}$$ Where Nisthenumberofbitsnecessary Misthenumberofconditions,levels,orcombinationspossiblewithNbits. Theaboveequationcanbere-arrangedas $$2^N=M$$ Forexample,withtwobits,22=4conditionsarepossible. TypesofM-aryTechniques Ingeneral,Multi-level(M-ary)modulationtechniquesareusedindigitalcommunicationsasthedigitalinputswithmorethantwomodulationlevelsareallowedonthetransmitter’sinput.Hence,thesetechniquesarebandwidthefficient. TherearemanyM-arymodulationtechniques.Someofthesetechniques,modulateoneparameterofthecarriersignal,suchasamplitude,phase,andfrequency. M-aryASK ThisiscalledM-aryAmplitudeShiftKeying(M-ASK)orM-aryPulseAmplitudeModulation(PAM). Theamplitudeofthecarriersignal,takesonMdifferentlevels. RepresentationofM-aryASK $S_m(t)=A_mcos(2\pif_ct)\quadA_m\epsilon{(2m-1-M)\Delta,m=1,2...\:.M}\quadand\quad0\leqt\leqT_s$ SomeprominentfeaturesofM-aryASKare− ThismethodisalsousedinPAM. Itsimplementationissimple. M-aryASKissusceptibletonoiseanddistortion. M-aryFSK ThisiscalledasM-aryFrequencyShiftKeying(M-aryFSK). Thefrequencyofthecarriersignal,takesonMdifferentlevels. RepresentationofM-aryFSK $S_i(t)=\sqrt{\frac{2E_s}{T_s}}\cos\left(\frac{\pi}{T_s}\left(n_c+i\right)t\right)$$0\leqt\leqT_s\quadand\quadi=1,2,3...\:..M$ Where$f_c=\frac{n_c}{2T_s}$forsomefixedintegern. SomeprominentfeaturesofM-aryFSKare− NotsusceptibletonoiseasmuchasASK. ThetransmittedMnumberofsignalsareequalinenergyandduration. Thesignalsareseparatedby$\frac{1}{2T_s}$ Hzmakingthesignalsorthogonaltoeachother. SinceMsignalsareorthogonal,thereisnocrowdinginthesignalspace. ThebandwidthefficiencyofM-aryFSKdecreasesandthepowerefficiencyincreaseswiththeincreaseinM. M-aryPSK ThisiscalledasM-aryPhaseShiftKeying(M-aryPSK). Thephaseofthecarriersignal,takesonMdifferentlevels. RepresentationofM-aryPSK $S_i(t)=\sqrt{\frac{2E}{T}}\cos\left(w_ot+\phi_it\right)$$0\leqt\leqT\quadand\quadi=1,2...M$ $$\phi_i\left(t\right)=\frac{2\pii}{M}\quadwhere\quadi=1,2,3...\:...M$$ SomeprominentfeaturesofM-aryPSKare− Theenvelopeisconstantwithmorephasepossibilities. Thismethodwasusedduringtheearlydaysofspacecommunication. BetterperformancethanASKandFSK. Minimalphaseestimationerroratthereceiver. ThebandwidthefficiencyofM-aryPSKdecreasesandthepowerefficiencyincreaseswiththeincreaseinM. Sofar,wehavediscusseddifferentmodulationtechniques.Theoutputofallthesetechniquesisabinarysequence,representedas1sand0s.Thisbinaryordigitalinformationhasmanytypesandforms,whicharediscussedfurther. DigitalCommunication-InformationTheory Informationisthesourceofacommunicationsystem,whetheritisanalogordigital.Informationtheoryisamathematicalapproachtothestudyofcodingofinformationalongwiththequantification,storage,andcommunicationofinformation. ConditionsofOccurrenceofEvents Ifweconsideranevent,therearethreeconditionsofoccurrence. Iftheeventhasnotoccurred,thereisaconditionofuncertainty. Iftheeventhasjustoccurred,thereisaconditionofsurprise. Iftheeventhasoccurred,atimeback,thereisaconditionofhavingsomeinformation. Thesethreeeventsoccuratdifferenttimes.Thedifferenceintheseconditionshelpusgainknowledgeontheprobabilitiesoftheoccurrenceofevents. Entropy Whenweobservethepossibilitiesoftheoccurrenceofanevent,howsurprisingoruncertainitwouldbe,itmeansthatwearetryingtohaveanideaontheaveragecontentoftheinformationfromthesourceoftheevent. Entropycanbedefinedasameasureoftheaverageinformationcontentpersourcesymbol.ClaudeShannon,the“fatheroftheInformationTheory”,providedaformulaforitas− $$H=-\sum_{i}p_i\log_{b}p_i$$ Wherepiistheprobabilityoftheoccurrenceofcharacternumberifromagivenstreamofcharactersandbisthebaseofthealgorithmused.Hence,thisisalsocalledasShannon’sEntropy. Theamountofuncertaintyremainingaboutthechannelinputafterobservingthechanneloutput,iscalledasConditionalEntropy.Itisdenotedby$H(x\midy)$ MutualInformation LetusconsiderachannelwhoseoutputisYandinputisX LettheentropyforprioruncertaintybeX=H(x) (Thisisassumedbeforetheinputisapplied) Toknowabouttheuncertaintyoftheoutput,aftertheinputisapplied,letusconsiderConditionalEntropy,giventhatY=yk $$H\left(x\midy_k\right)=\sum_{j=0}^{j-1}p\left(x_j\midy_k\right)\log_{2}\left[\frac{1}{p(x_j\midy_k)}\right]$$ Thisisarandomvariablefor$H(X\midy=y_0)\:...\:...\:...\:...\:...\:H(X\midy=y_k)$withprobabilities$p(y_0)\:...\:...\:...\:...\:p(y_{k-1)}$respectively. Themeanvalueof$H(X\midy=y_k)$foroutputalphabetyis− $H\left(X\midY\right)=\displaystyle\sum\limits_{k=0}^{k-1}H\left(X\midy=y_k\right)p\left(y_k\right)$ $=\displaystyle\sum\limits_{k=0}^{k-1}\displaystyle\sum\limits_{j=0}^{j-1}p\left(x_j\midy_k\right)p\left(y_k\right)\log_{2}\left[\frac{1}{p\left(x_j\midy_k\right)}\right]$ $=\displaystyle\sum\limits_{k=0}^{k-1}\displaystyle\sum\limits_{j=0}^{j-1}p\left(x_j,y_k\right)\log_{2}\left[\frac{1}{p\left(x_j\midy_k\right)}\right]$ Now,consideringboththeuncertaintyconditions(beforeandafterapplyingtheinputs),wecometoknowthatthedifference,i.e.$H(x)-H(x\midy)$mustrepresenttheuncertaintyaboutthechannelinputthatisresolvedbyobservingthechanneloutput. ThisiscalledastheMutualInformationofthechannel. DenotingtheMutualInformationas$I(x;y)$,wecanwritethewholethinginanequation,asfollows $$I(x;y)=H(x)-H(x\midy)$$ Hence,thisistheequationalrepresentationofMutualInformation. PropertiesofMutualinformation ThesearethepropertiesofMutualinformation. Mutualinformationofachannelissymmetric. $$I(x;y)=I(y;x)$$ Mutualinformationisnon-negative. $$I(x;y)\geq0$$ Mutualinformationcanbeexpressedintermsofentropyofthechanneloutput. $$I(x;y)=H(y)-H(y\midx)$$ Where$H(y\midx)$isaconditionalentropy Mutualinformationofachannelisrelatedtothejointentropyofthechannelinputandthechanneloutput. $$I(x;y)=H(x)+H(y)-H(x,y)$$ Wherethejointentropy$H(x,y)$isdefinedby $$H(x,y)=\displaystyle\sum\limits_{j=0}^{j-1}\displaystyle\sum\limits_{k=0}^{k-1}p(x_j,y_k)\log_{2}\left(\frac{1}{p\left(x_i,y_k\right)}\right)$$ ChannelCapacity Wehavesofardiscussedmutualinformation.Themaximumaveragemutualinformation,inaninstantofasignalinginterval,whentransmittedbyadiscretememorylesschannel,theprobabilitiesoftherateofmaximumreliabletransmissionofdata,canbeunderstoodasthechannelcapacity. ItisdenotedbyCandismeasuredinbitsperchanneluse. DiscreteMemorylessSource Asourcefromwhichthedataisbeingemittedatsuccessiveintervals,whichisindependentofpreviousvalues,canbetermedasdiscretememorylesssource. Thissourceisdiscreteasitisnotconsideredforacontinuoustimeinterval,butatdiscretetimeintervals.Thissourceismemorylessasitisfreshateachinstantoftime,withoutconsideringthepreviousvalues. SourceCodingTheorem TheCodeproducedbyadiscretememorylesssource,hastobeefficientlyrepresented,whichisanimportantproblemincommunications.Forthistohappen,therearecodewords,whichrepresentthesesourcecodes. Forexample,intelegraphy,weuseMorsecode,inwhichthealphabetsaredenotedbyMarksandSpaces.IftheletterEisconsidered,whichismostlyused,itisdenotedby“.”WhereastheletterQwhichisrarelyused,isdenotedby“--.-” Letustakealookattheblockdiagram. WhereSkistheoutputofthediscretememorylesssourceandbkistheoutputofthesourceencoderwhichisrepresentedby0sand1s. Theencodedsequenceissuchthatitisconvenientlydecodedatthereceiver. LetusassumethatthesourcehasanalphabetwithkdifferentsymbolsandthatthekthsymbolSkoccurswiththeprobabilityPk,wherek=0,1…k-1. LetthebinarycodewordassignedtosymbolSk,bytheencoderhavinglengthlk,measuredinbits. Hence,wedefinetheaveragecodewordlengthLofthesourceencoderas $$\overline{L}=\displaystyle\sum\limits_{k=0}^{k-1}p_kl_k$$ Lrepresentstheaveragenumberofbitspersourcesymbol If$L_{min}=\:minimum\:possible\:value\:of\:\overline{L}$ Thencodingefficiencycanbedefinedas $$\eta=\frac{L{min}}{\overline{L}}$$ With$\overline{L}\geqL_{min}$wewillhave$\eta\leq1$ However,thesourceencoderisconsideredefficientwhen$\eta=1$ Forthis,thevalue$L_{min}$hastobedetermined. Letusrefertothedefinition,“Givenadiscretememorylesssourceofentropy$H(\delta)$,theaveragecode-wordlengthLforanysourceencodingisboundedas$\overline{L}\geqH(\delta)$." Insimplerwords,thecodeword(example:MorsecodeforthewordQUEUEis-.-..-...-.)isalwaysgreaterthanorequaltothesourcecode(QUEUEinexample).Whichmeans,thesymbolsinthecodewordaregreaterthanorequaltothealphabetsinthesourcecode. Hencewith$L_{min}=H(\delta)$,theefficiencyofthesourceencoderintermsofEntropy$H(\delta)$maybewrittenas $$\eta=\frac{H(\delta)}{\overline{L}}$$ Thissourcecodingtheoremiscalledasnoiselesscodingtheoremasitestablishesanerror-freeencoding.ItisalsocalledasShannon’sfirsttheorem. ChannelCodingTheorem Thenoisepresentinachannelcreatesunwantederrorsbetweentheinputandtheoutputsequencesofadigitalcommunicationsystem.Theerrorprobabilityshouldbeverylow,nearly≤10-6forareliablecommunication. Thechannelcodinginacommunicationsystem,introducesredundancywithacontrol,soastoimprovethereliabilityofthesystem.Thesourcecodingreducesredundancytoimprovetheefficiencyofthesystem. Channelcodingconsistsoftwopartsofaction. Mappingincomingdatasequenceintoachannelinputsequence. InverseMappingthechanneloutputsequenceintoanoutputdatasequence. Thefinaltargetisthattheoveralleffectofthechannelnoiseshouldbeminimized. Themappingisdonebythetransmitter,withthehelpofanencoder,whereastheinversemappingisdonebythedecoderinthereceiver. ChannelCoding Letusconsideradiscretememorylesschannel(δ)withEntropyH(δ) Tsindicatesthesymbolsthatδgivespersecond ChannelcapacityisindicatedbyC ChannelcanbeusedforeveryTcsecs Hence,themaximumcapabilityofthechannelisC/Tc Thedatasent=$\frac{H(\delta)}{T_s}$ If$\frac{H(\delta)}{T_s}\leq\frac{C}{T_c}$itmeansthetransmissionisgoodandcanbereproducedwithasmallprobabilityoferror. Inthis,$\frac{C}{T_c}$isthecriticalrateofchannelcapacity. If$\frac{H(\delta)}{T_s}=\frac{C}{T_c}$thenthesystemissaidtobesignalingatacriticalrate. Conversely,if$\frac{H(\delta)}{T_s}>\frac{C}{T_c}$,thenthetransmissionisnotpossible. Hence,themaximumrateofthetransmissionisequaltothecriticalrateofthechannelcapacity,forreliableerror-freemessages,whichcantakeplace,overadiscretememorylesschannel.ThisiscalledasChannelcodingtheorem. DigitalCommunication-ErrorControlCoding NoiseorErroristhemainprobleminthesignal,whichdisturbsthereliabilityofthecommunicationsystem.Errorcontrolcodingisthecodingproceduredonetocontroltheoccurrencesoferrors.ThesetechniqueshelpinErrorDetectionandErrorCorrection. Therearemanydifferenterrorcorrectingcodesdependinguponthemathematicalprinciplesappliedtothem.But,historically,thesecodeshavebeenclassifiedintoLinearblockcodesandConvolutioncodes. LinearBlockCodes Inthelinearblockcodes,theparitybitsandmessagebitshavealinearcombination,whichmeansthattheresultantcodewordisthelinearcombinationofanytwocodewords. Letusconsidersomeblocksofdata,whichcontainskbitsineachblock.Thesebitsaremappedwiththeblockswhichhasnbitsineachblock.Herenisgreaterthank.Thetransmitteraddsredundantbitswhichare(n-k)bits.Theratiok/nisthecoderate.Itisdenotedbyrandthevalueofrisr<1. The(n-k)bitsaddedhere,areparitybits.Paritybitshelpinerrordetectionanderrorcorrection,andalsoinlocatingthedata.Inthedatabeingtransmitted,theleftmostbitsofthecodewordcorrespondtothemessagebits,andtherightmostbitsofthecodewordcorrespondtotheparitybits. SystematicCode Anylinearblockcodecanbeasystematiccode,untilitisaltered.Hence,anunalteredblockcodeiscalledasasystematiccode. Followingistherepresentationofthestructureofcodeword,accordingtotheirallocation. Ifthemessageisnotaltered,thenitiscalledassystematiccode.Itmeans,theencryptionofthedatashouldnotchangethedata. ConvolutionCodes Sofar,inthelinearcodes,wehavediscussedthatsystematicunalteredcodeispreferred.Here,thedataoftotalnbitsiftransmitted,kbitsaremessagebitsand(n-k)bitsareparitybits. Intheprocessofencoding,theparitybitsaresubtractedfromthewholedataandthemessagebitsareencoded.Now,theparitybitsareagainaddedandthewholedataisagainencoded. Thefollowingfigurequotesanexampleforblocksofdataandstreamofdata,usedfortransmissionofinformation. Thewholeprocess,statedaboveistediouswhichhasdrawbacks.Theallotmentofbufferisamainproblemhere,whenthesystemisbusy. Thisdrawbackisclearedinconvolutioncodes.Wherethewholestreamofdataisassignedsymbolsandthentransmitted.Asthedataisastreamofbits,thereisnoneedofbufferforstorage. HammingCodes Thelinearitypropertyofthecodewordisthatthesumoftwocodewordsisalsoacodeword.Hammingcodesarethetypeoflinearerrorcorrectingcodes,whichcandetectuptotwobiterrorsortheycancorrectonebiterrorswithoutthedetectionofuncorrectederrors. Whileusingthehammingcodes,extraparitybitsareusedtoidentifyasinglebiterror.Togetfromone-bitpatterntotheother,fewbitsaretobechangedinthedata.SuchnumberofbitscanbetermedasHammingdistance.Iftheparityhasadistanceof2,one-bitflipcanbedetected.Butthiscan'tbecorrected.Also,anytwobitflipscannotbedetected. However,Hammingcodeisabetterprocedurethanthepreviouslydiscussedonesinerrordetectionandcorrection. BCHCodes BCHcodesarenamedaftertheinventorsBose,ChaudariandHocquenghem.DuringtheBCHcodedesign,thereiscontrolonthenumberofsymbolstobecorrectedandhencemultiplebitcorrectionispossible.BCHcodesisapowerfultechniqueinerrorcorrectingcodes. Foranypositiveintegersm≥3andt<2m-1thereexistsaBCHbinarycode.Followingaretheparametersofsuchcode. Blocklengthn=2m-1 Numberofparity-checkdigitsn-k≤mt Minimumdistancedmin≥2t&plus;1 Thiscodecanbecalledast-error-correctingBCHcode. CyclicCodes Thecyclicpropertyofcodewordsisthatanycyclic-shiftofacodewordisalsoacodeword.Cycliccodesfollowthiscyclicproperty. ForalinearcodeC,ifeverycodewordi.e.,C=(C1,C2,......Cn)fromChasacyclicrightshiftofcomponents,itbecomesacodeword.Thisshiftofrightisequalton-1cyclicleftshifts.Hence,itisinvariantunderanyshift.So,thelinearcodeC,asitisinvariantunderanyshift,canbecalledasaCycliccode. Cycliccodesareusedforerrorcorrection.Theyaremainlyusedtocorrectdoubleerrorsandbursterrors. Hence,theseareafewerrorcorrectingcodes,whicharetobedetectedatthereceiver.Thesecodespreventtheerrorsfromgettingintroducedanddisturbthecommunication.Theyalsopreventthesignalfromgettingtappedbyunwantedreceivers.Thereisaclassofsignalingtechniquestoachievethis,whicharediscussedinthenextchapter. SpreadSpectrumModulation Acollectiveclassofsignalingtechniquesareemployedbeforetransmittingasignaltoprovideasecurecommunication,knownastheSpreadSpectrumModulation.Themainadvantageofspreadspectrumcommunicationtechniqueistoprevent“interference”whetheritisintentionalorunintentional. Thesignalsmodulatedwiththesetechniquesarehardtointerfereandcannotbejammed.Anintruderwithnoofficialaccessisneverallowedtocrackthem.Hence,thesetechniquesareusedformilitarypurposes.Thesespreadspectrumsignalstransmitatlowpowerdensityandhasawidespreadofsignals. Pseudo-NoiseSequence Acodedsequenceof1sand0swithcertainauto-correlationproperties,calledasPseudo-Noisecodingsequenceisusedinspreadspectrumtechniques.Itisamaximum-lengthsequence,whichisatypeofcycliccode. Narrow-bandandSpread-spectrumSignals BoththeNarrowbandandSpreadspectrumsignalscanbeunderstoodeasilybyobservingtheirfrequencyspectrumasshowninthefollowingfigures. Narrow-bandSignals TheNarrow-bandsignalshavethesignalstrengthconcentratedasshowninthefollowingfrequencyspectrumfigure. Followingaresomeofitsfeatures− Bandofsignalsoccupyanarrowrangeoffrequencies. Powerdensityishigh. Spreadofenergyislowandconcentrated. Thoughthefeaturesaregood,thesesignalsarepronetointerference. SpreadSpectrumSignals Thespreadspectrumsignalshavethesignalstrengthdistributedasshowninthefollowingfrequencyspectrumfigure. Followingaresomeofitsfeatures− Bandofsignalsoccupyawiderangeoffrequencies. Powerdensityisverylow. Energyiswidespread. Withthesefeatures,thespreadspectrumsignalsarehighlyresistanttointerferenceorjamming.Sincemultipleuserscansharethesamespreadspectrumbandwidthwithoutinterferingwithoneanother,thesecanbecalledasmultipleaccesstechniques. FHSSandDSSS/CDMA SpreadspectrummultipleaccesstechniquesusessignalswhichhaveatransmissionbandwidthofamagnitudegreaterthantheminimumrequiredRFbandwidth. Theseareoftwotypes. FrequencyHoppedSpreadSpectrum(FHSS) DirectSequenceSpreadSpectrum(DSSS) FrequencyHoppedSpreadSpectrum(FHSS) Thisisfrequencyhoppingtechnique,wheretheusersaremadetochangethefrequenciesofusage,fromonetoanotherinaspecifiedtimeinterval,hencecalledasfrequencyhopping.Forexample,afrequencywasallottedtosender1foraparticularperiodoftime.Now,afterawhile,sender1hopstotheotherfrequencyandsender2usesthefirstfrequency,whichwaspreviouslyusedbysender1.Thisiscalledasfrequencyreuse. Thefrequenciesofthedataarehoppedfromonetoanotherinordertoprovideasecuretransmission.TheamountoftimespentoneachfrequencyhopiscalledasDwelltime. DirectSequenceSpreadSpectrum(DSSS) WheneverauserwantstosenddatausingthisDSSStechnique,eachandeverybitoftheuserdataismultipliedbyasecretcode,calledaschippingcode.Thischippingcodeisnothingbutthespreadingcodewhichismultipliedwiththeoriginalmessageandtransmitted.Thereceiverusesthesamecodetoretrievetheoriginalmessage. ComparisonbetweenFHSSandDSSS/CDMA Boththespreadspectrumtechniquesarepopularfortheircharacteristics.Tohaveaclearunderstanding,letustakealookattheircomparisons. FHSS DSSS/CDMA Multiplefrequenciesareused Singlefrequencyisused Hardtofindtheuser’sfrequencyatanyinstantoftime Userfrequency,onceallottedisalwaysthesame Frequencyreuseisallowed Frequencyreuseisnotallowed Senderneednotwait Senderhastowaitifthespectrumisbusy Powerstrengthofthesignalishigh Powerstrengthofthesignalislow Strongerandpenetratesthroughtheobstacles ItisweakercomparedtoFHSS Itisneveraffectedbyinterference Itcanbeaffectedbyinterference Itischeaper Itisexpensive Thisisthecommonlyusedtechnique Thistechniqueisnotfrequentlyused AdvantagesofSpreadSpectrum Followingaretheadvantagesofspreadspectrum− Cross-talkelimination Betteroutputwithdataintegrity Reducedeffectofmultipathfading Bettersecurity Reductioninnoise Co-existencewithothersystems Longeroperativedistances Hardtodetect Noteasytodemodulate/decode Difficulttojamthesignals Althoughspreadspectrumtechniqueswereoriginallydesignedformilitaryuses,theyarenowbeingusedwidelyforcommercialpurpose. 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