8. Couch,L. W., II, DigitalandAnalog CommunicationSystems,MacmillanPublishingCompany,NewYork, 1982.

9. Lender,A., "CorrelativeDigitalCommunicationTechniques,"IEEE Trans.Commun.Technol.,Dec.1964,pp.128-135.

101

xo(t)

H(f)

~r-~ ~

2 2

Figure P2.1

cos (2rrmfst)

Localoscillator

Hw

NaturallysampledPAM

Problems

PROBLEMS

Tt 2.1. You wantto transmittheword"HOW" usingan8-arysystem.'\ (a) Encodethe word "HOW" into a sequenceof bits,using7-bitASCII coding,followedby an eighthbit for error detection,per character.The eighthbit is

chosenso thatthenumberof onesin the8 bitsis an evennumber.How manytotalbitsaretherein themessage?

(b) Partitionthe bit streaminto k =3 bit segments.Representeachof the 3-bitsegmentsasanoctalnumber(symbol).How manyoctalsymbolsaretherein themessage?

(c) If thesystemweredesignedwith 16-arymodulation,howmanysymbolswouldbeusedtorepresenttheword"HOW"?

(d) If thesystemweredesignedwith256-arymodulation,howmanysymbolswouldbeusedtorepresenttheword"HOW"?

-"\ 2.2. Wewanttotransmit800characters/s,whereeachcharacterisrepresentedbyits7-bitASCII codeword,followedby aneighthbit for errordetection,percharacter,asinProblem2.1.A multilevelPAM waveformwithM =16levelsisused.(a) Whatistheeffectivetransmittedbitrate?(b) Whatisthesymbolrate?

-"'\ 2.3. We wishto transmita lOO-characteralphanumericmessagein2 s,using7-bitASCIIcoding,followedby an eighthbit for errordetection,percharacter,asin Problem2.1.A multilevelPAM waveformwithM =32levelsis used.(a) Calculatetheeffectivetransmittedbitrateandthesymbolrate.(b) Repeatpart (a) for 16-levelPAM, eight-levelPAM, four-levelPAM, andPCM

(binary)waveforms.

~ 2.4. GivenananalogwaveformthathasbeensampledatitsNyquistrate,f,.,usingnaturalsampling,provethata waveform(proportionalto the originalwaveform)canberecoveredfrom thesamples,usingthe recoverytechniquesshownin FigureP2.1.Theparametermfsisthefrequencyof thelocaloscillator,wherem isaninteger.

2.5. An analogsignal is sampledat its Nyquist rate 1fT" and quantizedusingL quantiza-tion levels.The derived digital signalis then transmittedon somechannel.(a) Show that the time duration. T, of one bit of the transmitted binary encoded

signal mustsatisfyT

QUESTIONS

'{What are the similaritiesand differencesbetweenthe terms"formatting"and"sourcecoding"?(SeeChapter2,introduction.)

2.2 In theprocessof formattinginformation,whyis it oftendesirableto performover-sampling?(SeeSection2.4.3.)

2.3 In usingpulsecodemodulation(PCM) for digitizinganaloginformation,explainhowtheparametersfidelity,bandwidth,andtimedelaycanbetradedoff. (SeeSec-tion2.6.)

2.4 Why is it often preferredto use unitsof normalizedbandwidth,WT (or time-bandwidthproduct),comparedwithbandwidthalone?(SeeSection2.8.3.)

103

Using the Companion CD, run the exercisesassociatedwith Chapter 2.

Questions

EXERCISES

"\ 2.15. A signalin thefrequencyrange300to 3300Hz is limitedto apeak-to-peakswingof10V. It is sampledat 8000samples/sandthe samplesarequantizedto 64 evenlyspacedlevels.Calculateandcomparethebandwidthsandratioof peaksignalpowerto rmsquantizationnoiseif thequantizedsamplesaretransmittedeitherasbinarypulsesor asfour-levelpulses.Assumethatthesystembandwidthis definedby themainspectrallobeof thesignal.

2.16. In thecompactdisc(CD) digitalaudiosystem,ananalogsignalis digitizedso that'"" theratioof thepeak-signalpowertothepeak-quantizationnoisepow~is atleast96

dB.Thesamplingrateis 44.1kilosamples/s.(a) How manyquantizationlevelsof theanalogsignalareneededfor (S/N,)peak=

96dB?(b) Howmanybitspersampleareneededforthenumberof levelsfoundinpart(a)?(c) Whatisthedataratein bits/s?

2.17. Calculatethe differencein requiredsignalpowerbetweentwo PCM waveforms,NRZ andRZ, assumingthateachsignalingschemehasthesamerequirementsfordata-rateandbit-errorprobability.Also assumeequallylikelysignaling,andthatthedifferencebetweenthehigh-voltageandlow-voltagelevelsis thesamefor boththeNRZ andRZ schemes.If thereis a poweradvantagein usingoneof thesignalingschemes,what,if any,is thedisadvantagein usingit?

2.18. In theyear1962,AT &T firstoffereddigitaltelephonetransmissionreferredto asT1

,~, - service.Withthisservice,eachT1 frameispartitionedinto24channelsor timeslots.l.'-i ,.. Eachtimeslotcontains8 bits(onespeechsample),andthereis oneadditionalbit~ , \ perframefor alignment.The frameis sampledattheNyquistrateof8000samples/s,

I.; 1,1"coC:nd thebandwidthusedfor transmittingthecompositesignalis 386kHz. Find the",' n bandwidthefficiency(bits/s/Hz)for thissignalingscheme.~

\ 2.19. (a) Considerthatyoudesireadigitaltransmissionsystem,suchthatthequantizationdistortionof anyaudiosourcedoesnotexceed2% of thepeak-to-peakanalogsignalvoltage.If the audiosignalbandwidthand the allowabletransmissionbandwidthareeach4000Hz, andsamplingtakesplaceattheNyquistrate,whatvalueofbandwidthefficiency(bits/s/Hz)isrequired?

(b) Repeatpart(a)exceptthattheaudiosignalbandwidthis 20kHz (highfidelity),yettheavailabletransmissionbandwidthisstill4000Hz.