3.5.LineCodesandSpectraLineCodeinPCM

TheoutputofanADCcanbetransmi5edoverabasebandchannel.

• Thedigitalinforma<onmustfirstbe

convertedintoaphysicalsignal.

• Thephysicalsignaliscalledalinecode.Linecodersusetheterminology

marktomeanbinaryoneandspacetomeanbinaryzero.

ADC

PCMsignal

Sample

Quan?ze

AnalogInputSignal

Encode

LineCode

X

XQ

Xk

x(t)

3.5.LineCodesandSpectra

DEFINIATION:Binary1’sand0’s,suchasinPCMsignaling,mayberepresentedinvariousserial-bitsignalingformatscalledlinecodes.

BinaryLineCoding

Therearetwomajorcategories:return-to-zero(RZ)andnonreturn-to-zero(NRZ).WithRZcoding,thewaveformreturnstoazero-voltlevelforapor<on(usuallyon-half)ofthebitinterval.

3.5.LineCodesandSpectra

Unipolarsignaling:“1”--+A(highlevel)“0”--0(zerolevel)Alsocalled:on-offkeying

polarsignaling:“1”--+A“0”---A

Bipolarsignaling:“1”--alternatelyposi<veornega<vevalues“0”--0

Manchestersignaling:“1”--posi<vehalf-bitfollowedbyanega<vehalf-bitperiodpulse.“0”–nega<vehalf-bitfollowedbyaposi<vehalf-bitperiodpulse

3.5.LineCodesandSpectra

Desirableproper?esofalinecode:²  Self-synchroniza5on:thereisenough<meinforma<onbuiltin.²  Lowprobabilityoferror:receiverscanrecoverthebinarydatawhen

theinputdatasignalsiscorrupted.²  Aspectrumthatissuitableforthechannel:thesignalbandwidth

needstobesufficientlysmallcomparedtothechannelbandwidth²  Transmissionbandwidth:thisshouldbeassmallaspossible²  Errordetec5oncapability:thisfeatureiseasilyimplementedbythe

addi<onofchannelencodersanddecoders.²  Transparency:everypossiblesequenceofdataisfaithfullyand

transparentlyreceived.

BinaryLineCoding

3.5.LineCodesandSpectra

BinaryLineCoding

Line Coder an

DigitalData PhysicalWaveform

s t( ) = an f (t − nTs )n=−∞

∞

∑

²  Theinputtothelinecoderisasequenceofvalues,thatisafunc<onofadatabitoranADCoutputbit.

²  Theoutputofthelinecoderisawaveform s t( ) = an f (t − nTs )n=−∞

∞

∑

an

Wheref(t)isthesymbolpulseshapeandTsisthedura<onofonesymbol.Forbinarysignaling,Ts=Tb,whereTbisthe<methatittakestosend1bit.Formul<levelsignaling,Ts=lTb.{an}istheamplitude,“A”or“0”forNRZ,forexample.

3.5.LineCodesandSpectraTypesofLineCodes

²  Eachlinecodeisdescribedbyasymbolmappingfunc5on,andapulseshapep(t)through

²  Categoriesoflinecodes:

s t( ) = an f (t − nTs )n=−∞

∞

∑

an

anØ  Symbolmappingfunc<ons():UnipolarPolarBipolar

Ø  Pulseshape(p(t)):NRZ(Nonreturn-to-zero)RZ(ReturntoZero)Manchester(splitphase)

3.5.LineCodesandSpectra

Unipolarsignaling:“1”--+A(highlevel)“0”--0(zerolevel)Alsocalled:on-offkeying

polarsignaling:“1”--+A“0”---A

Bipolarsignaling:“1”--alternatelyposi<veornega<vevalues“0”--0

Manchestersignaling:“1”--posi<vehalf-bitfollowedbyanega<vehalf-bitperiodpulse.“0”–nega<vehalf-bitfollowedbyaposi<vehalf-bitperiodpulse

3.5.LineCodesandSpectraPowerSpectraforBinaryLineCode

²  Adigitalsignalisrepresentedby:

1.)thepulseshape2.)sta<s<calproper<esofdataexpressedbytheautocorrela<onfunc<on

²  PSDcanbecalculatedusingtheautocorrela<onfunc<on,anditdependson:

s t( ) = an f (t − nTs )n=−∞

∞

∑

whereanisthedigitalnumber(e.g.“0”or“1”),andf(t)isthesymbolpulseshape(e.g.ForunipolarNRZ)f (t) =Π t

Ts

⎛

⎝⎜⎜

⎞

⎠⎟⎟

3.5.LineCodesandSpectraPowerSpectraforBinaryLineCode

ThegeneralexpressionforthePSDofadigitalsignalis:

ps ( f ) =F ( f )

2

TsR(k)e j2πkfTs

k=−∞

∞

∑

WhereF(f)istheFouriertransformofthepulesshapef(t)andR(k)istheautocorrela<onofthedata.

R(k) = (anan+k )i Pii=1

I

∑

Whereanandan+karethe(voltage)levelsofthedatapulsesatthenthand(n+k)thsymbolposi<ons,respec<vely.Piistheprobabilityofhavingtheithanan+kproduct.

3.5.LineCodesandSpectraPSDforUnipolarNRZsignaling

LineCodeofUnipolarNRZ

PSDofUnipolarNRZ

punipolarNRZ ( f ) =A2Tb4

sinπ fTbπ fTb

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

1+ 1Tbδ( f )

⎡

⎣⎢

⎤

⎦⎥

A= 2 1/Tb = R

3.5.LineCodesandSpectraPSDforPolarNRZsignaling

LineCodeofPolarNRZ

PSDofPolarNRZ

pPolarNRZ ( f ) = A2Tb

sinπ fTbπ fTb

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

3.5.LineCodesandSpectraDifferen?alCoding

Poten<alissuesisunipolarNRZ,polarNRZ,andManchesterNRZ:²  Thewaveformisoeenuninten5onallyinverted(happensina

twisted-pairtransmissionlinechannelbyswitchingthetwoleads)²  Results:1->0,0->1

²  Thisisnotaissueforbipolarsignal

3.5.LineCodesandSpectraDifferen?alCoding

Thetechnologycalleddifferen<alcodingcanbeusedtoamelioratethisproblem.

en = dn ⊕ en−1!dn = !en ⊕ !en−1

Whereisamodulo2adderoranexclusive-ORgate(XOR)opera<on.

⊕

3.5.LineCodesandSpectraDifferen?alCoding

Encoding

Input sequence dn 1 1 0 1 0 0 1

Encoded sequence en 1 0 1 1 0 0 0 1

Reference digit

Decoding (with correct channel polarity)

Receiver sequence 1 0 1 1 0 0 0 1

(Correct polarity)

Decoded sequence 1 1 0 1 0 0 1

Decoding (with inverted channel polarity)

Received sequence 0 1 0 0 1 1 1 0

(Inverted polarity)

Decoded sequence 1 1 0 1 0 0 1

1 −⊕= nnn ede

1~~d~ −⊕= nnn ee

3.5.LineCodesandSpectra

Regenera?veRepeaters

Whenasignal(waveform)istransmi5edoverahardwirechannel,itisfiltered,a5enuated,andcorruptedbynoise.Consequently,forlongdistance,thedatacannotberecoveredatthereceivingendunlessrepeatersareu<lized.

²  Foranalogsignal(suchasPAM),onlylinearamplifierswithappropriatefiltercouldbeused.àonlyincreasetheamplitude,thein-banddistor<on(suchasrandomnoise)couldaccumulate.

²  Fordigitalsignal(suchasPCM),nonlinearprocessingcanbeusedtoregeneratea“noise-free”digitalsignal.Thistypeofnonlinearprocessingiscalledaregenera5verepeater.

3.5.LineCodesandSpectra

Regenera?veRepeaters

Increasestheamplitude

Minimizetheeffectofchannelnoise&ISI

Producesasamplevalue

Generatesaclockingsignal

Producesahighlevelo/pifsamplevalue>VT

3.5.LineCodesandSpectra

Regenera?veRepeaters

(a)Transmi5edsignal.(b)Receiveddistortedsignal(withoutnoise).(c)Receiveddistortedsignal(withnoise).(d)Regeneratedsignal(delayed).

Sampleandhold

3.5.LineCodesandSpectra

Regenera?veRepeaters

²  Inlong-distancedigitalcommunica<onsystems,manyrepeatersmaybeusedincascade,andthedistancebetweentherepeatersisgovernedbythepathlossofthetransmissionmediumandtheamountofnoisethatisadded.

²  ArepeaterisrequiredwhentheSNRatapointalongthechannelbecomeslowerthanthevaluethatisneededtomaintaintheoverallprobability-of-bit-errorspecifica<on.

²  Formrepeatersincascade,theoverallprobabilityofbiterrorsPmeis

Pme=mPewherePeistheprobabilityofbiterrorforasinglerepeater

3.5.LineCodesandSpectra

BitSynchroniza?on

²  Synchroniza<onsignalareclock-typesignalsthatarenecessarywithinareceiver(orrepeater)fordetec<on(orregenera<on)ofthedatafromthecorruptedinputsignal.

²  Theseclocksignalshaveprecisefrequencyandphaserela<onshipwithrespecttothereceivedinputsignal,andtheyaredelayedcomparedtotheclocksignalsatthetransmi5er.

²  Atleasethreetypesofsynchroniza<onsignalsareneeded:Ø  bitsyncØ  framesyncØ  carriersync

3.5.LineCodesandSpectra

clock-typesignal

BitSynchroniza?on

Whysyncsignalisnecessary??

3.5.LineCodesandSpectraBitSynchroniza?on

Square-lawbitsynchronizerforNRZsignal

3.5.LineCodesandSpectraPowerSpectraforMul?levelPolarNRZsignals

Whydoweneedmul<levelsignal?

Forbinarysignal:N(dimensionnumber)=n(bitnumber)

w(t) = wkϕk (t)k=1

N

∑ 0<t<T0

wkisthedigitalnumber(e.g.“0”and“1”),T0isthe<meperiodtosendoutamessage

Formul?levelsignal:N(dimensionnumber)=n/l(bitnumber)

3.5.LineCodesandSpectraPowerSpectraforMul?levelPolarNRZsignals

Binarytomul<levelconversionisusedtoreducethebandwidthrequiredbythebinarysignaling.

² Mul<plebits(lnumberofbits)areconvertedintowordshavingSYMBOLdura<onsTs=lTbwheretheSymbolRateortheBAUDrateD=1/Ts=1/lTb

²  ThesymbolsareconvertedtoaLlevel(L=2l)mul<levelsignalusinganl-bitDAC.

²  NotethatnowtheBaudrateisreduced(D=R/l),thusthebandwidthisalsoreduced.

3.5.LineCodesandSpectraPowerSpectraforMul?levelPolarNRZsignals

Forbinarysignal:

Bandwidth(minimum)ofthewaveformrepresen<ngthedigitalsignal:

B = N2T0

=12D Hz

Formul<levelsignal:

B = N2T0

=12D

B = N2T0L

=12LD

3.5.LineCodesandSpectraPowerSpectraforMul?levelPolarNRZsignals

(c)L=8=23LevelPolarNRZWaveformOut

3.5.LineCodesandSpectraSpectralEfficiency

DEFINATION:Thespectralefficiencyofadigitalsignalisgivenbythenumberofbitspersecondofdatathatcanbesupportedbyeachhertzofbandwidth.

( )Bit s

HzRB

η =

max 2log 1C SB N

η ⎛ ⎞= = +⎜ ⎟⎝ ⎠