PrinciplesofCommunications2Chapter4Digitizationofanalogsignal4.1IntroductionTwocategoriesofinformationsources:analogsignal,digitalsignalThreestepsofA/Dconversion: sampling,quantization,codingMostpopularA/Dconversionmethod: PulseCodeModulation(PCM)DigitalCommunicationSystemTopicstobecovered54.2Samplingofanalogsignal

4.2.1Samplingoflow-passanalogsignalUsuallysamplingatequaltimeintervalTTheoretically,samplingprocess＝periodicalunitimpulseanalogsignalPractically,samplingprocess

＝periodicalnarrowpulseanalogsignalSamplingtheorem:Ifthehighestfrequencyofacontinuousanalogsignals(t)islessthanfH,andifitissampledbyperiodicimpulseswithintervaltimeT1/2fH,thens(t)canbecompletelydecidedbythesesamples.6Proofofsamplingtheorem

Let:

s(t)－

signalwithhighestfrequencylessthanfH

T(t)－periodicalunitimpulsewithrepetitionperiodT

andrepetitionfrequencyfs=1/T,thenthesampledsignalis: LettheFouriertransformofsk(t)isSk(f),thenwhere,Sk(f)－spectrumofsk(t)S(f)－spectrumofs(t)(f)－spectrumofT(t)(f)isthefrequencyspectrumofperiodicalunitimpulse,itcanbefoundasAfastvaryingsignalshouldbesampledmorefrequently!TheoreticallygovernedbytheNyquistsamplingtheoremTheideallowpassfilterisunabletobeachieved.Thecut-offedgeofapracticalfiltercannotbesosharp.Therefore,thepracticalsamplingratefs

mustbemuchlargerthan

2fH.Forexample,thehighestfrequencyoftypicaltelephonesignalislimitedto3,400Hz,andthesamplingfrequencyusedis8,000Hz.8

4.2.2Samplingofband-passanalogsignalThefrequencybandofaband-passsignalislimitedtobetweenfLandfH,i.e.,thecut-offfrequencyofitsfrequencyspectrumatthelowerendisobviouslylargerthan0.Therequiredsamplingfrequencyfs:whereB－signalbandwidth,n－largestintegerlessthanfH/B, 0<k<1。9

4.2.3AnalogpulsemodulationPAM

PDM

PPM

（a）Basebandsignal (b)PAMsignal(c)PDMsignal (d)PPM signalFigure4.2.6Analogpulsemodulation104.3Quantizationofsampledsignal

4.3.1PrinciplesofquantizationPurposeofquantization:Todigitizethesampledsignal.Quantizationisaroundingprocess,eachsampledsignalpointisroundedtothe“nearest”valuefromafinitesetofpossiblequantizationlevels.圖4.3.1抽樣信號的量化Methodofquantization:Lets(kT)bethesampledvalue.IfNbitsofbinarysymbolsareusedtoexpressit,thenonlyM=2^Ndifferentsampledvaluescanbeexpressed.TherearetotallyMdiscretelevels.Theyarecalledquantizationlevel.ThemethodofrepresentingcontinuoussamplesbyMquantitionlevelsiscalledquantization.Example:inthefigure

Itisuniformquantizationinthefigure.

PerformanceMeasureofQuantizationQuantizationnoise:Average(meansquareerror)Noise:Signal-to-quantizationnoiseratio(SQNR):

13

4.3.2UniformquantizationAssume:themagnitudeoftheanalogsamplesiswithina～b, andthenumberofquantizationlevels＝M,

thenthequantizationintervalis: theboundarypointsare:Ifquantizedoutputlevelqi

isthemidpointofthequantization

interval,then14FindaveragepowerofquantizationnoiseNq

：

where,skissampleofsignal,i.e.,s(kT)

sqismagnitudeofquantizedsignal,i.e.,sq(kT)

f(sk)isprobabilitydensityofsignalsamplesk

Eexpressesoperationofstatisticalmean.

MisnumberofquantizationlevelsFindaveragepowerofsignalsk:Averagesignaltoquantizationnoisepowerratiocanbefoundfromtheabovetwoequations.OriginalfileQuantizedat4bitsQuantizedat6bits17【Example4.1】AssumethenumberofquantizationlevelsforauniformquantizerisM,andthesampleoftheinputsignalhasuniformprobabilitydensityintheinterval[-a,a].Findtheaveragesignaltoquantizationnoiseratioforthequantizer.

Solution:fromeq.(4.3-5),wehave ∵ ∴

或

18

4.3.2NonuniformquantizationDisadvantageofuniformquantization:

1.thequantizationnoiseNqisconstant.2.realaudiosignals(speechandmusic)aremoreconcentratednearzero.

3.Humanearismoresensitivetoquantizationerrorsatsmallvalues.WhyweneedNonuniformquantization?

Nonuniformquantizationcanimprovethesignaltoquantizationnoiseratiowhenthesignalissmallandquantizationintervalsissmallernearzero.

PrinciplesofnonuniformquantizationCompressXbeforequantization,g(x)=y,thenexpandindecoder,that’swhywecalltheoverallsystemascompandor.

y=g(x)Whentherearemanyquantizationintervals,thecurveinthefigureineachquantizationintervalmayberegardedapproximatelyasasegmentofastraightline.Hence,theslopeofthelinesegmentcanbewrittenas20

Assumerangesoftheinputandtheoutputvoltagesarelimitedtobetween0and1,andtheordinateyisuniformlydividedintoNquantizationintervals,thentheintervalofeachquantizationintervalwouldbe

∴

21Sowehave,Thesolutionoftheabovelineardifferentialequationis

Forfindingc,substitutingtheboundarycondition(whenx=1,y=1)intotheaboveequation,weobtaink+c=0，i.e.,c=-k.Substitutingthevalueofc

intotheaboveequation,weobtainTheoreticallythecharacteristicofcompressionisrequiredtobelogarithmcharacteristic.

Fortelephonesignal,ITUconstituted2recommendations,i.e.,A-lawand-law,andcorrespondingapproximatealgorithms---13brokenlinemethodand15brokenlinemethod.22A-lawwhere,xisthenormalizedinputvoltageofthecompressor

y

isthenormalizedoutputvoltageofthecompressor

A

isaconstantdecidingthecompression

extent.Inpractice,Aischosentobeequalto87.6.Thenitcouldbeapproximatelyrealizedby13brokenlinecompression2313brokenlinecompressioncharacteristicThecharacteristicof13brokenlineisclosetothatofA-law.Inthefigure,xininterval0~1isnonuniformlydividedinto8segments.Segmentbetween?~1iscalledthe8thsegment;segmentbetween?~?iscalledthe7thsegment;segmentbetween1/8~1/4iscalledthe6thsegment;andsoon,untilsegmentbetween0~1/128iscalledthe1stsegment.Ordinateyisuniformlydividedinto8segments.2413brokenlinecompressioncharacteristicThecorrespondingcoordinates(x,y)ofthe8segmentsareconnectedtoformabrokenline.Exceptthe1standthe2ndsegments,theslopesofothersegmentsofthebrokenlinearedifferent.ForA.C.signal,thereare13segmentsintotal.

SegmentNo.-8-7-6-5-4-3-2-112345678Slope??1248161616168421??25-lawand15brokenlinecompressioncharacteristics

-lawcanbeobtained:15brokenline:closeto-law26Thefirstandsecondsegmentscan’tbecombinedintoonestraightlinebecausetheirslopesaredifferent.Whenpositiveandnegativevoltagesofsignalareconsidered,15segmentsareobtainedintotal.27Comparisonof13brokenlinemethodand15brokenlinemethod

SNRforsmallsignalgivenby15brokenlineisabouttwicegivenby13brokenline

SNRforlargersignalgivenby15brokenlineisalittlelowerthanthe13broken284.4Pulsecodemodulation(PCM)

4.4.1

BasicprinciplesofPCMSamplingquantizationcodingBlockdiagram:29Example:seethefigure3.153011 3.96410030

4.4.2NaturalbinarycodeandfoldbinarycodeCharacteristicoffoldbinarycodeImagerelationship:Codeworderrorhaslessinfluenceonsmallvoltage.0000000100100011010001010110011101110110010101000011001000010000Negativepolarity

765432101111111011011100101110101001100011111110110111001011101010011000Positivepolarity

15141312111098Foldbinarycodeword

Naturalbinarycodeword

PolarityofquantizedvoltageNo.ofquantizedvalue

31TheFoldbinarycodeusedin13brokenlinemethodinChinathereare8bitsintotal:c1toc8

c1:polarityc2

～c4:segmentbits

－8segmentslopesc5

～c8：inner-segmentbits

－16quantizationlevelsNo.ofsegmentsegmentbitsc2c3c481117110610151004011301020011000QuantizationintervalInner-segmentbitsc5c6c7c815111114111014110112110011101110101091001810007011160110501014010030011200101000100000

Sothequantizationlevelfor1stsegmentis1/2048.Togainthesameperformancebyuniformquantizer,weneed11bits.324.4.3QuantizationnoiseinPCMsystemSignaltoquantizationnoiseratioofuniformquantizer,whereMisthenumberofquantizationlevels:

S/Nq=M2ForPCMsystem,,

S/Nq=22N

,whereNisbitsofcodewords.

ForPCMsystem,bysamplingtheorem,thesamplerateisnotlessthan2NfHb/s.Hence,therequiredsystembandwidthisatleastB=NfH

accordingtoNyquistcriterion.So

TheaboveequationshowsthatoutputsignaltoquantizationnoiseofPCMsystemincreaseswiththebandwidthBofthesystemaccordingtoexponentialrule.

334.5Differentialpulsecodemodulation(DPCM)4.5.1

BasicprinciplesoflinearpredictionPredictionofthecurrentsamplebyusinglinearcombinationofprevioussamplesincalledlinearprediction.Differencebetweenthecurrentsampleandthepredictedvalueiscalledpredictionerror.Thepredictedvalueisclosetothesamplebecausethereisstrongcorrelationbetweenadjacentsamples,i.e.,thepossiblerangeofthepredictederrorissmall.Thenumberofbitsmaybelessforencodingpredictederror,andthebitrateisconsequentlyreduced.34BlockdiagramoflinearpredictionEncoder:

s(t)－inputsignal

sk

＝s(kT)－sampleofs(t)

sk

－predictedvalue

ek

－predictionerror

rk

－quantizedpredictionerror

s*k

－inputofpredictor

35BlockdiagramoflinearpredictionEncoder:

Meaningofs*k

:

s*kistheskwithquantizationerror

Input~outputrelationshipofpredictor: where,pisthepredictionorder

ai

isthepredictioncoefficient36BlockdiagramoflinearpredictionDecoder:

whenthereisnotransmissionerror,i.e.,whentheoutputoftheencoderisjusttheinputofthedecoder,inputsignalsofthetwoaddersareidentical,i.e.,rk=rk.Therefore,

sk*=sk*＋

(a)Decoder

(b)EncoderBasicprincipleofDPCM:whenp=1，a1=1,sk

=s*k-1,thepredictorisreducedtodelaycircuit,thedelaytimeisT.Now,linearpredictionbecomesDPCM.38

4.5.2QuantizationnoiseandsignaltoquantizationnoiseratioinDPCMsystemQuantizationnoise:i.e.,quantizationerrorqk

Assume:(+,-)－rangeofpredictionerrorek

M－numberofquantizationlevelsofthequantizer v－quantizationinterval thewehave

+-vv0vM1M2M3M4圖4.5.2,和M之間關(guān)系39distributedin(-v,+v),Assume:fs

－

samplingfrequency

N=log2

M

－numberofsymbolsofthecodewordforeachsample

Nfs

－symbolrateoftheDPCMencoderoutput

E(qk2)isuniformlydistributedinfrequencyrange(0,Nfs)thenthepowerspectraldensityofE(qk2)is:

Thequantizationnoisepowerafterpassingalow-passfilterwithcut-offfrequencyfLequals

－outputquantizationnoiseoftheDPCMsystem40Signal