Chapter 3. Baseband Pulse and Digital Signalingyadwang/ECE375_Lec6.pdf · 3.1. Introduc:on The...

Chapter3.BasebandPulseandDigitalSignaling

ChapterObjec:ves

Ø  Analog-to-digitalsignaling

Ø  Binaryandmul:leveldigitalsignals

Ø  Spectraandbandwidthsofdigitalsignals

Ø  Preven:onofintersymbolinterference

Ø  Timedivisionmul:plexing

Ø  Packettransmission

3.1.Introduc:on

Thefollowingarethefourmaingoalsofthischapter

²  Tostudyhowanalogwaveformscanbeconvertedtodigitalwaveform.Themostpopulartechniqueiscalledpulsecodemodula,on(PCM).

²  Tostudyhowtocomputethespectrumfordigitalsignals.

²  Toexaminehowthefilteringofpulsesignalsaffectsourabilitytorecoverthedigitalinforma?onatthereceiver.Thisfilteringcanproducewhatiscalledintersymbolinterference(ISI)intherecovereddatasignal.²  Tostudyhowwecanmul?plex(combine)datafromseveraldigitalbitstreamsintoonehigh-speeddigitalstreamfortransmissionoveradigitalsystem.Onesuchtechnique,called,me-divisionmul,plexing(TDM),willbestudiedinthischapter.

3.2.PulseAmplitudeModula:on

²  Pulseamplitudemodula,on(PAM)isanengineeringtermthatisusedtodescribetheconversionoftheanalogsignaltoapulse-typesignalinwhichtheamplitudeofthepulsedenotestheanaloginforma?on.

²  ThepurposeofPAMsignalingistoprovideanotherwaveformthatlookslikepulses,yetcontainstheinforma?onthatwaspresentintheanalogwaveform.

²  TherearetwoclassesofPAMsignals:ü  PAMthatusesNaturalSampling(Ga:ng)ü  PAMthatusesInstantaneousSampling(Flat-TopPAM)

3.2.PulseAmplitudeModula:onNaturalSampling(Ga:ng)

DEFINIATION:Ifw(t)isananalogwaveformbandlimitedtoBhertz,thePAMsignalthatusesnaturalsamplingis:whereisarectangularwaveswitchingwaveformand

ws(t) = w(t)s(t)

s(t) = Π(t − kTsτ

)k=−∞

∞

∑fs =1/Ts ≥ 2B


ws(t)=w(t)s(t)


THEOREM:ThespectrumforanaturallysampledPAMsignaliswhere,,thedutycycleofs(t)is,andisthespectrumoftheoriginalunsampledwaveform.

Ws( f ) =ℑ[ws(t)]= dsinπndπnd

W ( f − nfs )n=−∞

∞

∑

fs =1/Ts ws = 2π fs

d = τ /Ts W ( f ) =ℑ w(t)[ ]

NaturalSampling(Ga:ng)


• Thedutycycleoftheswitchingwaveformisd=τ/Ts=1/3.

• Thesamplingrateisfs=4B.

NaturalSampling(Ga:ng)

3.2.PulseAmplitudeModula:onGenera:onofPAMwithnaturalsampling(ga:ng)

ThePAMwaveformwithnaturalsamplingcanbegeneratedusingaCMOS(complementarymetal-oxide-semiconductor)circuitconsis?ngofAclockandanalogswitchasshown.

3.2.PulseAmplitudeModula:onRecoveringNaturallySampledPAM

²  Atthereceiver,theoriginalanalogwaveform,w(t)canberecoveredfromthePAMsignal,ws(t),bypassingthePAMsignalthroughalowpassfilter,wherethecutofffrequencyis:B<fcutoff<fs-B.

²  Iftheanalogsignalisundersampledfs<2B,theeffectofspectraloverlappingiscalledAliasing.Thisresultsinarecoveredanalogsignalthatisdistortedcomparedtotheoriginalwaveform

3.2.PulseAmplitudeModula:onRecoveringNaturallySampledPAM

3.2.PulseAmplitudeModula:onInstantaneousSampling(Flat-TopPAM)


DEFINIATION:Ifw(t)isananalogwaveformbandlimitedtoBhertz,thePAMsignalthatusesinstantaneoussamplingis:whereissamplingpulseshape,wereand

ws(t) = w(kTs )k=−∞

∞

∑ h(t − kTs )

h(t) =∏ tτ

"

#$%

&'=

1, | t |< τ / 2

0, | t |> τ / 2

(

)*

+*

τ ≤ Ts =1/ fs fs ≥ 2B


THEOREM:Thespectrumforaflat-topPAMsignalis

Ws( f ) =1TsH ( f ) W ( f − kfs )

k=−∞

∞

∑

H ( f ) =ℑ[h(t)]= τ sinπτ fπτ f

"

#$

%

&'

²  ThistypeofPAMsignalconsistsofinstantaneoussamples² w(t)issampledatt=kTs²  Thesamplevaluesw(kTs)determinetheamplitudeoftheflat-toprectangularpulses


Limita:onsofPAM

²  Thetransmissionofeithernaturallyofinstantaneouslysampled

PAMoverachannelrequiresaverywidefrequencyresponse.

²  Thebandwidthrequiredismuchlargerthanthatoftheoriginal

analogsignal.

²  ThenoiseperformanceofthePAMsystemcanneverbebeSerthan

achievedbytransmiTngtheanalogsignaldirectly.

²  PAMisnotverygoodforlong-distancetransmission.

3.3.PulseCodeModula:onDEFINITION:Pulsecodemodula5on(PCM)isessen?allyanalog-to-digitalconversionofaspecialtypewheretheinforma?oncontainedintheinstantaneoussamplesofananalogsignalisrepresentedbydigitalwordsinaserialbitstream.

ADVANTAGES:²  Rela?velyinexpensivedigitalcircuitrymaybeusedextensively.²  PCMsignalsderivedfromalltypesofanalogsourcesmaybemergedwithdatasignalsandtransmiSedoveracommonhigh-speeddigitalcommunica?onsystem.²  Inlong-distancedigitaltelephonesystemsrequiringrepeaters,aclean

PCMwaveformcanberegeneratedattheoutputofeachrepeater,wheretheinputconsistsofanoisyPCMwaveform.

²  Thenoiseperformanceofadigitalsystemcanbesuperiortothatofananalogsystem.

²  Theprobabilityoferrorforthesystemoutputcanbereducedevenfurtherbytheuseofappropriatecodingtechniques.

3.3.PulseCodeModula:onPCMsignalisgeneratedbycarryingoutthreebasicopera5ons:

sampling

2 31

quan:zing encoding

3.3.PulseCodeModula:on1.Sampling(Flat-topPAMSignal)

3.3.PulseCodeModula:on

2.Quan:zing

M=8(M=2n)levelsareusedtoapproximatetheanalogsamplevalues(infinitenumber):-7,-5,-3,-1,1,3,5,7

ExampleofM=8=23uniformquan:zer


errorsareintroducedbecauseofthefinitenumberoflevels(M=8)isusedinthequan?zing(quan,zingnoise)

5 7 7 775

ThePAMsignalis“roundingoff”togetquan?zedPAMsignal

2.Quan:zing

3.3.PulseCodeModula:on3.Encoding

ThePCMsignalisobtainedfromthequan?zedPAMsignalbyencodingeachquan?zedsamplevalueintoadigitalword.

Three–bitM=8Graycode

-7-5-3-11357

010011001000100101111110

5 7 7 7 7 5

3.3.PulseCodeModula:onPrac:calPCMCircuits

²  Threepopulartechniquesareusedtoimplementtheanalog-to-digitalconverter(ADC)encodingopera?on:

1.  Thecoun?ngorramp,(MaximICL7126ADC)2.  Serialorsuccessiveapproxima?on,(AD570)3.  Parallelorflashencoders.(CA3318)

²  Theobjec?veofthesecircuitsistogeneratethePCMword.²  Paralleldigitaloutputobtained(fromoneoftheabovetechniques)

needstobeserializedbeforesendingovera2-wirechannel

²  Thisisaccomplishedbyparallel-to-serialconverters[SerialInput-Output(SIO)chip]

²  UART,USRTandUSARTareexamplesforSIO’s

3.3.PulseCodeModula:onBandwidthofPCMSignals

²  ThebandwidthofPCMsignalisnotdirectlyrelatedtothespectrumofinputanalogysignal.(Nonlinearfunc?on:3steps)

²  Thebandwidthof(serial)binaryPCMwaveformsdependsonthebitrateRandthewaveformpulseshapeusedtorepresentthedata.

²  ThebitrateRis:R=nfswherenisthenumberofbitsinthePCMword(M=2n)andfsisthesamplingrate(PAM).

²  ThebandwidthofPAMsignalcanbeobtainedasafunc?onofthespectrumoftheinputanalogsignal(Linearfunc?on).

3.3.PulseCodeModula:onBandwidthofPCMSignals

²  ForPCMwaveformgeneratedbyrectangularpulses,theFirst-nullbandwidthis:BPCM=R=nfs

²  Fornoaliasingcase(fs>=2B,Bisthebandwidthoftheanalogsignal,thatistobeconvertedtoPCMsignal),theMINIMUMbandwidthofPCMis:

BPCM=R/2=nfs/2TheMinimumbandwidthofnfs/2isobtainedonlywhensin(x)/xpulseisusedtogeneratethePCMwaveform.

²  Forareasonablevalueofn,thebandwidthoftheserialPCMsignalwillbesignificantlylargerthanthebandwidthofthecorrespondinganalogsignalthatitrepresents.

²  IfthebandwidthofthePCMsignalisreducedbyimproperfilteringofpoorfrequencyresponse,onebitwillsmearintoadjacentbitslot.Thisiscalledintersymbolinterference(ISI).

3.3.PulseCodeModula:onEffectsofNoise

TheanalogsignalthatisrecoveredatthePCMsystemoutputiscorruptedbynoise.Twomaineffectsproducethisnoiseordistor?on:

²  Quan?zingnoisethatiscausedbytheM-stepquan?zeratthePCMtransmiSer.

²  BiterrorsintherecoveredPCMsignal.Thebiterrorsarecausedbychannelnoise,aswellasimproperchannelfiltering,whichcausesISI(intersymbolinterference).

Theinputanalogsignalneedstobesufficientlybandlimited(withalow-passan?aliasingfilter)andsampledfastenoughsothatthealiasingnoiseontherecoveredanalogsignalisnegligible.

3.3.PulseCodeModula:onEffectsofNoise

Thera:ooftherecoveredanalogPEAKsignalpowertothetotalaveragenoisepowerisgivenby:

Thera:ooftheAVERAGEsignalpowertotheaveragenoisepoweris:

SN!

"#

$

%&pkout

=3M 2

1+ 4(M 2 −1)Pe

SN!

"#

$

%&pkout

=M 2

1+ 4(M 2 −1)Pe

WhereMisthenumberofquan?zedlevelsusedinthePCMsystemandPeistheprobabilityofbiterrorintherecoveredbinaryPCMsignalatthereceiverDACbeforeitisconvertedbackintoananalogsignal

3.3.PulseCodeModula:onExample3-3.Averagesignal-to-noisera:oforarecoveredanalogsignal

CalculatetheaverageSNRdBoftheanalogsignalthatisrecoveredfromaPCMsignalthathaserrorbitswithaprobabilityoferrorofPe.PlottheSNRdBforPeoverarangefrom10-7to10-1.

SN!

"#

$

%&pkout

=M 2

1+ 4(M 2 −1)Pe

M=2n

n=6


IfPeisnegligible,therearenobiterrorsresul?ngfromchannelnoiseandnoISI,thepeakSNRresul?ngfromonlyquan?zingerroris

Thera:ooftheAVERAGEsignalpowertotheaveragenoisepoweris:

SN

!

"#

$

%&pkout

=3M 2

1+ 4(M 2 −1)Pe

SN!

"#

$

%&out

=M 2

1+ 4(M 2 −1)Pe

SN!

"#

$

%&pkout

= 3M 2

SN!

"#

$

%&out

=M 2

Aboveequa?onscanbeexpressedindecibelsas:

SN!

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$

%&dB= 6.02n+α Where,M=2n

α=4.77forpeakSNRα=0foraverageSNR

3.3.PulseCodeModula:onExample3-4.DesignofaPCMsignalfortelephonesystems

Assumethatananalogaudiovoice-frequency(VF)telephonesignaloccupiesabandfrom300to3,400Hz.ThesignalistobeconvertedtoaPCMsignalfortransmissionoveradigitaltelephonesystem.Theminimumsamplingfrequencyis2x3.4=6.8ksample/sec.Tobeabletouseofalow-costlow-passan?aliasingfilter,theVFsignalisoversampledwithasamplingfrequencyof8ksamples/sec.ThisisthestandardadoptedbytheUnitesStatestelephoneindustry.Assumethateachsamplevaluesisrepresentedby8bits;thenthebitrateofthebinaryPCMsignalis:

R=(fssamples/s)(nbits/sample)=(8ksamples/s)(8bits/sample)=64kbits/s

This64kbits/siscalledaDS-0signal(digitalsignal,typezero)

TheminimumabsolutebandwidthofthebinaryPCMsignalis

BPCM ≥R2=nfs2= 32kHZ

Thisbandwidthisrealizedwhena(sinx)/xpulseshapeisused.

3.3.PulseCodeModula:onExample3-4.DesignofaPCMsignalfortelephonesystems

Werequireabandwidthof64kHZtotransmitthisdigitalvoicePCMsignal,whereasthebandwidthoftheoriginalanalogvoicesignalwas,atmost,4kHz.

Weobservethatthepeaksignal-to-quan?zingnoisepowerra?ois:

SN

!

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$

%&pkout

= 3(28)2 = 52.9dB

Ifarectangularpulseforsampling,theabsolutebandwidthisinfinity,andthefirstnullbandwidthis:

BPCM = R = 64kHZ

3.3.PulseCodeModula:onNonuniformQuan:za:on

Manysignals(e.g.voiceanalog)haveanon-uniformdistribu?on--Theamplitudeismorelikelytobeclosetozerothantobeathigherlevels--Ex.Ifthepeakvalueallowedis1V,weakpassagesmyhavevoltagelevelsontheorderof01.V(20dBdown)

2 4 6 8

2

4

6

-2

-4

-6

InputsampleX

OutputsampleXQ

-2-4-6-8

Example: Nonuniform 3 bit quantizer


w1(t)Compression(nonlinear)

Analogsignal PCMcircuit(uniformquan?zer)

w2(t) =ln 1+µ w1(t)( )ln(1+µ)

μ-lawtype:

IntheUnitedStates,Canada,andJapan,thetelephonecompaniesuseaμ=255compressioncharacteris?cintheirPCMsystems.


M=8Quan?zerCharacteris?c μ-lawcharacteris?c


w1(t)Compression(nonlinear)

Analogsignal PCMcircuit(uniformquan?zer)

w2(t) =

A w1(t)1+ lnA

0 ≤ w1(t) ≤1A

1+ ln(A w1(t) )1+ lnA

1A≤ w1(t) ≤1

"

#

$$

%

$$

A-lawtype:

InEurope,theA-lawisgenerallyused.


SN!

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%&dB= 6.02n+α

TheOutputSNRfollowsthe6-dBlaw

Whereα=4.77-10log(V/xrms)(uniformquan?zing)

α≈4.77-10log[ln(1+μ)](μ-lawquan?zing)

α≈4.77-10log[1+lnA](A-lawquan?zing)

Chapter 3. Baseband Pulse and Digital Signalingyadwang/ECE375_Lec6.pdf · 3.1. Introduc:on The...

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