InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter channelchannel

( ( X(tX(t( ( XXTT(t(t

TimingTiming

ReceiverReceiverfilterfilter

ClockClock recoveryrecovery networknetwork

A/DA/D

++ Channel noiseChannel noise

n(t(n(t(

OutputOutput

Block diagram of an Binary/M-ary signalingBlock diagram of an Binary/M-ary signaling schemescheme

++

HHTT(f((f(

HHRR(f((f(

Y(t)Y(t)

HHcc(f((f(

InformationInformationsourcesource

TransmitterTransmitter ChannelChannel ReceiverReceiver DecisionDecision

Communication system

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

( ( X(tX(t ((XXTT(t(tTimingTiming

Block diagram DescriptionBlock diagram Description

HHTT(f((f(

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

TTbb)t(pg

TTbb

)t(pg

TTbb

For bFor bkk=1=1

For bFor bkk=0=0

;"0"a;"1"a

a kk

k

d if d if

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

( ( X(tX(t ((XXTT(t(tTimingTiming

Block diagram Description Block diagram Description (Continue - 1((Continue - 1(

HHTT(f((f(

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

TTbb

)t(pg

TTbb

For bFor bkk=1=1

For bFor bkk=0=0TransmitterTransmitter

filterfilter

TTbb

)t(pg

TTbb

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

( ( X(tX(t ((XXTT(t(tTimingTiming

Block diagram Description Block diagram Description (Continue - 2((Continue - 2(

HHTT(f((f(

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

k

bgk kTtpatX )()(

TTbb

100110100110

2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

( ( X(tX(t ((XXTT(t(tTimingTiming

Block diagram Description Block diagram Description (Continue - 3((Continue - 3(

HHTT(f((f(

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

k

bgk kTtpatX )()(

TTbb

100110100110

2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

( ( X(tX(tTimingTiming

Block diagram Description Block diagram Description (Continue - 4((Continue - 4(

HHTT(f((f(

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

Channel noise n(t(Channel noise n(t(

++

tt

ReceiverReceiverfilterfilter

HHRR(f((f(

Block diagram Description Block diagram Description (Continue - 5((Continue - 5(

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

tt

k

bdrk tnkTttpAtY )()()( 0

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter channelchannel

( ( X(TX(T

((XXtt(T(T

TimingTiming

ReceiverReceiverfilterfilter

ClockClock recoveryrecovery networknetwork

A/DA/D

++ Channel noiseChannel noise

n(t(n(t(

OutputOutput

Block diagram of an Binary/M-ary signalingBlock diagram of an Binary/M-ary signaling schemescheme

++

HHTT(f((f(

HHRR(f((f(

Y(t)Y(t)

HHcc(f((f(

k

bdrk tnkTttpAtY )()()( 0

Block diagram DescriptionBlock diagram Description

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

tt

1 0 0 1 0 0 00 1 0 1 0

1 0 0 1 1 01 0 0 1 1 0

tt

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter channelchannel

( ( X(tX(t( ( XXTT(t(t

TimingTiming

ReceiverReceiverfilterfilter

ClockClock recoveryrecovery networknetwork

A/DA/D

++ Channel noiseChannel noise

n(t(n(t(

OutputOutput

Block diagram of an Binary/M-ary signalingBlock diagram of an Binary/M-ary signaling schemescheme

++

HHTT(f((f(

HHRR(f((f(

Y(t)Y(t)

HHcc(f((f(

TransTransfilterfilter channelchannel

PPgg(t((t(ReceiverReceiver

filterfilter

Explanation of PExplanation of Prr(t((t(

HHTT(f((f( HHRR(f((f(HHcc(f((f(

k

bdrk tnkTttpAtY )()()( 0

PPrr(t((t(

HHTT(f( H(f( Hcc(f( H(f( HRR(f((f( PPgg(f((f( PPrr(f((f(1)0(pr

The output of the pulse generator X(t(,is given byThe output of the pulse generator X(t(,is given by

Tpa bgk

kkttX

PPgg(t( is the basic pulse whose amplitude a(t( is the basic pulse whose amplitude akk depends depends on .the kon .the kthth input bit input bit

For For ttm m =mT=mTbb+t+tdd and t and tdd is the total time delay in the is the total time delay in the system, we get. system, we get.

tt

tt

tt

tmY

tt11

tt22 tt33

ttmm

The input to the A/D converter isThe input to the A/D converter is

k

bdrk tnkTttpAtY )()()( 0

tnTpAAt m0brmK

kmmkmY

tt

tmY

tt11

tt22 tt33

ttmm

The output of the A/D converter at the sampling timeThe output of the A/D converter at the sampling time

ttm m =mT=mTbb+t+tdd

k

bdrk tnkTttpAtY )()()( 0

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

tnTpAAt m0brmK

kmmkmY

tt

tmY

tt11

tt22 tt33

ttmm

ISI - ISI - IInter nter SSymbolymbolIInterferencenterference

TransTransfilterfilter channelchannel

PPgg(t((t(ReceiverReceiver

filterfilter

Explanation of ISIExplanation of ISIHHTT(f((f( HHRR(f((f(HHcc(f((f(

PPrr(t((t(

PPgg(f((f( PPrr(f((f(TransTransfilterfilter channelchannel ReceiverReceiver

filterfilter

HHTT(f((f( HHRR(f((f(HHcc(f((f(

tt

ff

FourierFourier TransformTransform

BandPassBandPassFilterFilter

ff

FourierFourier TransformTransform

tt

Explanation of ISI - ContinueExplanation of ISI - Continue

tt

ff

FourierFourier TransformTransform

BandPassBandPassFilterFilter

ff

FourierFourier TransformTransform

tt

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

-The pulse generator output is a pulse waveform

k

bgk kTtpatX )()(

aa

a

p

k

g 1)0(

If kth input bit is 1

if kth input bit is 0

k

bdrk tnkTttpAtY )()()( 0

-The A/D converter input Y(t)

-The pulse generator output is a pulse waveform

k

bgk kTtpatX )()(

aa

a

p

k

g 1)0(

If kth input bit is 1

if kth input bit is 0

k

bdrk tnkTttpAtY )()()( 0

-The A/D converter input Y(t)

5.2 BASEBAND BINARY PAM SYSTEMS

Pulse shapesp g(t)

p r(t) H R (f) H T(t)

D esign o f a basebandb inary PAM system

- minimize the combined effects of inter symbol interference and noise in order to achieve minimum probability of error for given data rate.

0nfor00nfor1

)nT(p br

5.2.1 Baseband pulse shaping

The ISI can be eliminated by proper choice of received pulse shape pr (t(.

Doe’s not Uniquely Specify Pr(t( for all values of t.

0nfor00nfor1

)nT(pThen

T2/1fforT)Tkf(Pif

br

kbb

br

k

T2/)1k2(

T2/)1k2(rr

rr

b

b

df)ft2jexp()f(p)t(p

df)ft2jexp()f(p)t(p

TheoremTheorem

ProofProof

To meet the constraint, Fourier Transform Pr(f) of Pr(t), shouldTo meet the constraint, Fourier Transform Pr(f) of Pr(t), shouldsatisfy a simple condition given by the following theoremsatisfy a simple condition given by the following theorem

b

b

b

b

T2/1

T2/1k b

brbr

k

T2/1

T2/1b

brbr

df)fnT2jexp())Tkf(p()nT(p

'df)nT'f2jexp()Tk'f(p)nT(p

k

Tk

Tkbrbr

b

b

dftfnTjfpnTp2/)12(

2/)12(

)2exp()()(

b

b

T2/1

T2/1bbbr n

)nsin(df)fnT2jexp(T)nT(p

Which verify that the Pr(t) with a transform Pr(f) Which verify that the Pr(t) with a transform Pr(f) Satisfy Satisfy ZERO ISIZERO ISI

The condition for removal of ISI given in the theorem is calledThe condition for removal of ISI given in the theorem is calledNyquist (Pulse Shaping) CriterionNyquist (Pulse Shaping) Criterion

mk

m0brkmm )t(n)T)km((PAA)t(Y

0nfor00nfor1

)nT(p br

TTbb 2T2Tbb-T-Tbb-2T-2Tbb

11

n

)nsin()nT(p br

The Theorem gives a condition for the removal of ISI using a Pr(f) withThe Theorem gives a condition for the removal of ISI using a Pr(f) witha bandwidth larger then rb/2/. a bandwidth larger then rb/2/.

ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.

HHTT(f( H(f( Hcc(f( H(f( HRR(f((f( PPgg(f((f( PPrr(f((f(TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

R ate o f decayof p r(t)

Shap ing filte rs

p r(t)

Particular choice of Pr(t) for a given application

The smallest values near Tb, 2Tb, …The smallest values near Tb, 2Tb, …In such that timing error (Jitter)In such that timing error (Jitter)

will not Cause large ISIwill not Cause large ISI

Shape of Pr(f) determines the easeShape of Pr(f) determines the easewith which shaping filters can be with which shaping filters can be

realized.realized.

A Pr(f) with a smooth roll - off characteristics is preferable A Pr(f) with a smooth roll - off characteristics is preferable over one with arbitrarily sharp cut off characteristics.over one with arbitrarily sharp cut off characteristics.

Pr(f)Pr(f) Pr(f)Pr(f)