channel lecture important 2

8/6/2019 Channel Lecture Important 2

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EE 400: Communication Networks (091) Dr. Wajih A. Abu-Al-Saud

Ref: A. Leon Garcia and I. Widjaja, Communication Networks, 2nd Ed. McGraw Hill, 2006Latestupdateofthislecturewason17 12 2009

Lecture13:DigitalTransmissionFundamentals

1

ChannelAmplitudeResponseH(f)

Achannelischaracterizedbyeitherits:

1. ImpulseResponseh(t):Thisisthetimedomainrepresentationofthechannel.Itrepresentstheoutputofthechannelwhentheinputofthechannelisadeltafunction(t),whichisanimpulse

at t = 0. The impulse response indicates how a channel spreads the input signal. An ideal

channeldoesnotspreadthesignalatallwhileanonidealchannelspreadstheinputsignal.The

spreadingoftheinputsignalisequaltothewidthoftheimpulseresponse.

2. Transfer function H(f): This is the frequency domain representation of the channel. Itrepresentstheamplitudeof theoutputof thechannelwhenasinusoidwithamplitude1V is

andfrequencyf isinputtothechannel.

Intimedomain,anidealchannelwithinfinitebandwidthisonethathasan impulseresponsethat isa

delta function at time 0 or later. If it has an impulse response of delta at 0, the channel does not

introduceanydelayinsignal(thereiszerodelaybetweentransmittingthesignalandreceivingitatthe

outputofthechannel).Iftheimpulseresponseisadelayeddeltafunction,thechannelisstillidealbut

there is a nonzero delay between transmitting the signal and receiving it at the other side of the

channel.Infrequencydomain,anidealchannelwillhaveatransferfunctionwithaflatmagnitudeforall

frequencies.Thephaseofthetransferfunctioniszeroifthechannelhaszerodelayandwillbeastraight

line(linearcurve)ifthechannelhasanonzerodelay.

Inreality,nochannelhasinfinitebandwidthbutallchannelshavesomebandwidthWthatisfinite(less

than infinity). A channelwith finite bandwidth can still be considered ideal if the response of that

channelisflat(constant)oversomefrequencyrange(forexamplefrom0toWforalowpasschannels)

asseeninpart(a)ofthefigurebelow.Asignaltransmittedoversuchachannelwillexperienceanideal

channelifthebandwidthofthesignalislessthanW.IfthebandwidthofthesignalisgreaterthanW,the

channelwillnotactasanidealchannelforthissignal,butsomedistortionwilloccurinthesignalasitis

transmittedthroughthechannel.

Furthermore, in reality,channelsarenever ideal (somechannelsmayonlybeapproximated tobeing

idealforspecificapplications).Atypical lowpasschannel isshown inpart(b)ofthefigurebelow.The

bandwidthofthechannelinthiscasemaybeoneofseveralvaluesdependingontheapplication,but

themostwidelyusedisthe3dBbandwidth.


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(a) (b)

Now consider thatwe have a digital signal thatwewould like to transmit over a specific channel.

Assumethat:

BandwidthofDigitalSignal=s

W

BandwidthofTransmissionChannel= cW

Wecaneasilyverifythefollowingtwocases(assumingnonoiseexists):

Forthecaseof sW W :The input signalwill completelypass through the channel andwillnotexperienceanydistortion

giving an output signal that is similar to the input signal with the exception of

amplification/attenuationanddelay.

Forthecaseofs

W W> :

The input signalwillonly partially pass through the channel and thereforewill experience some

distortion that results from filtering out high frequencies. The signal may also experience

amplification/attenuationandpossiblysomedelay.


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NyquistSignalingRate

Nyquistcameupwithaformulathatgivesthemaximumnumberofpulsesyoucantransmitthrougha

channelthathasbandwidthc

W .TheNyquistSignalingRatetheoremstatesthatthemaximumrateat

whichpulsescanbetransmittedoverachannelwithBandwidthc

W is

Maximumpulsetransmissionrate maxR = 2 cW pulses/s

Note:theNyquestSignalingRatetheoremgivesthemaximumPULSERATEnotthemaximumBITRATE.

Infact,differentpulsesmaycarrydifferentnumberofbitsperpulse(1bitforpulsesthattakeoneof

twovalues5Vand+5V,or2bitsforpulsesthattakeoneof4values0V,1V,2V,and3V,or3bitsfor

pulsesthattakeoneofthe8values0V,1V,2V,3V,4V,5V,6V,and7V).

Ifeachpulserepresents1bit,thenbitrateinthiscaseis 2 cW bits/s Usingmultileveltransmissionwithnumberof levelsof 2mM = (where m isnumberofbits),we

getamaximumtransmissionbitrateof

max

pulses bits bits2 2

second pulse secondc c

R W m W m= =

SignaltoNoiseRatio(SNR)andMulti LevelTransmissions

Onemeasureof thequalityofasignal is theSignal toNoiseRatio (SNR).Thesignaltonoiseration is

often defined as shown in the following equation. Sometimes the SNR becomes useful if it is

representedindBformasshownbelow.


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( ) ( )10

Average Signal Power(Linear)Average Noise Power

dB 10log (Linear)

SNR

SNR SNR

=

=

ShannonChannelCapacity

Shannon(anAmericancommunicationengineerandmathematician)provedthatforaspecificchannel

withaspecificsignaltonoiseratioSNR,youcantransmitamaximumnumberofbitspersecondequal

toCthatisgivenbytheformula(thesignaltonoiseratioisthelinearSNR)

( )2bits

log 1+SNR(Linear)second

cC W=

Where Wc is the channel bandwidth in (Hz), and SNR(Linear) is the linear signal to noise ratio. The

ShannonChannelCapacitystatesthatforanoisychannel,youcantransmitreliablyamaximumnumber

ofbitspersecondequalto C . Ifyou trytoexceedthechannelcapacity foraparticularchannel,the

noisewillcauseuncontrollablehighratesoferrors.

BandwidthofAnalogSignals

One characteristic of an analog signal is its bandwidth. The bandwidth of analog signals is usually

computedusingoneofthefollowingtwomethods:

Bandwidth isequal to thehighest frequency component in the signal: theproblemwith thismethod of computation of the bandwidth is that there is a possibility that the frequency

componentsofthesignalaboveaspecificfrequencyareverysmallthattheycontainnegligible

power,yet,thebandwidthbecomessolargeduetotheseverylowpowercomponents

Bandwidthisequaltothefrequencyatwhichthepowerofthesignalcomponentsdropbelowthe power of the signal components at zero frequency: this bandwidth is less than the

bandwidthaboveand itconsidersthesignalcomponentsabovethatbandwidthnottobepart

ofthesignal(theycanbecompletelyremovedwithoutmuchdistortiontothesignal.

SamplingofAnalogSignals(NyquistSamplingTheorem)

Beforeananalogsignalcanbetransmittedoveradigitalcommunicationsystem,theanalogsignalmust

besampledandthenquantized,whichconvertstheanalogsignalintoasequenceofbits.Accordingto


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NyquistSamplingTheorem,topreservethequalityofananalogsignal,itmustbesampledatarateATLEASTdouble thebandwidthof thesignal (called theNyquistsamplingrate). Ifsampling isdoneatalowerrate,aliasingoccurswhichcausesthereconstructedsignaltobedistorted.Sincesignalsmayhave

componentswith frequency above the bandwidth, the proper thing todo before sampling to avoid

aliasing from occurring is to pass the analog signal through a lowpass filter that removes any

componentsaboveonehalftheNyquistsamplingrate.Thisisillustratedinthefollowingfigure:


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Quantization

Onceananalogsignalissampled,thenextstepistoquantizeit,meaningthatafiniteprecessionvalueis

used todescribe thevalueofeachsample. Inquantization,ananalogsamplewithanamplitude that

maytakeanyvalueinaspecificrangeisconvertedtoadigitalsamplewithanamplitudethattakesone

ofaspecificpredefinedsetofquantizationvalues(withfinitenumberofdigitsneededtodescribethe

value

of

each

sample).

This

is

performed

by

dividing

the

range

of

possible

values

of

the

analog

samples

intoL different levels, and assigning the center value of each level to any sample that falls in thatquantization interval. The problemwith this process is that it approximates the value of an analog

samplewiththenearestof thequantizationvalues.So, foralmostallsamples, thequantizedsamples

willdifferfromtheoriginalsamplesbyasmallamount.Thisamountiscalledthequantizationerror.To

getsomeideaontheeffectofthisquantizationerror,quantizingaudiosignalsresultsinahissingnoise

similartowhatyouwouldhearwhenyouplayarandomsignal.

Assume thatasignalwithpowerPsis tobequantizedusingaquantizerwithL = 2n levelsranging in

voltagefrommp to mpasshowninthefigurebelow.

Quantizer Output Samplesq

x

Quantizer Output Samples x

Wecandefinethevariable v tobetheheightoftheeachoftheLlevelsofthequantizerasshown

above.Thisgivesavalueof v equalto


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2 pmv

L = .

Therefore,forasetofquantizerswiththesamemp,thelargerthenumberoflevelsofaquantizer,the

smaller the size of eachquantization interval, and for a setofquantizerswith the samenumber of

quantizationintervals,thelargermpisthelargerthequantizationintervallengthtoaccommodateallthe

quantizationranges.

Aftersimplemanipulation,wecanwritetheSignaltoNoise(orQuantizationNoise)Ratio(SNRorSQNR)

as

2

2

Signal Power

Noise Power

3.

s

q

s

p

PSNR

P

LP

m

= =

=

IngeneralthevaluesoftheSNRaremuchgreaterthan1,andamoreusefulrepresentationoftheSNR

canbeobtainedbyusing logarithmicscaleordB.WeknowthatLofaquantizer isalwaysapowerof

twoorL = 2n.Therefore,

( )

( )

22

10 10 102 2

10 102

3 310 log 10 log 10 log 2

3

10 log 20 log 2

6 dB.

n

dB s s

p p

sp

LSNR P P

m m

P nm

n

= = +

= +

= +

Note that shown in the above representation of the SNR is a constant for a specific signalwhen

differentquantizerswiththesamempareused.

ItisclearthattheSNRofaquantizerindBincreaseslinearlyby6dBasweincreasethenumberofbits

that thequantizerusesby1bit.Thecost for increasing theSNRofaquantizer is thatmorebitsare


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generatedand thereforeeitherahigherbandwidthor longer timeperiod isrequired to transmit the

outputdigitaldataofthequantizer.

DigitalTransmissionofAnalogSignals

Now,oncethesignalhasbeenquantizedbythequantizer,thequantizerconvertsittobits(1sand0s)

andoutputsthesebits.Lookingattherepresentationofthequantizershownabove,weseethateachof

the levelsof thequantizer isassignedacode from 000000 for the lowestquantization interval to

111111 forthehighestquantization intervalasshown inthecolumn totherightofthefigure.The

outputbitsarecalledaPulseCodeModulation.ThePCMsignalisobtainedbyoutputtingthebitsofthe

differentsamplesonebitaftertheotherandonesampleaftertheother.Thiswouldbethecodethat

would be transmitted in a digital communication system. So, it is clear that either increasing the

samplingrateorthenumberofquantizationlevelsofthequantizerwouldincreasetheinformationthat

wewouldhavetotransmit.

t4T

sT

s3T

s5T

s2T

s0

mp

mp

L = 2n

L levels

n bits

0.000

0.001

0.010

1.111

.

.

.

.

PCM Code

n bits/sample

0

v

Quantizer Output Samplesq

x

Quantizer Output Samples x

A quantization interval Corresponding quantization value

channel lecture important 2

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