3.6.IntersymbolInterference

Intersymbolinterference(ISI)occurswhenapulsespreadsoutinsuchawaythatitinterfereswithadjacentatthesampleinstant.

Example:assumepolarNRZlinecode.Thechanneloutputsareshownas“smeared”(widthTbbecomes2Tb)pulses(Spreadingduetobandlimitedchannel)

Data 1

bT− 0 bT0bT bT−

Data 0

bT− 0 bT0bT bT−

Channel Input

Pulse width Tb

Channel Output

Pulse width Tb

3.6.IntersymbolInterference

3.6.IntersymbolInterference

He(t)

win (t) = π anh(t − nTs )n∑ wout (t) = anδ(t − nTs )

n∑⎡

⎣⎢

⎤

⎦⎥*he (t)

he (t) = h(t)*hT (t)*hC (t)*hR(t)

He ( f ) = H ( f )HT ( f )HC ( f )HR( f )

ℑ

H ( f ) =ℑ ΠtTs

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥=Ts

sinπTs fπTs f

⎛

⎝⎜⎜

⎞

⎠⎟⎟where

wout (t) = anhe (t − nTs )n∑

3.6.IntersymbolInterference

Example3-13IntersymbolInterferenceCausedbyRCFiltering

PlottheoutputwaveformwhenachannelfiltersaunipolarNRZsignal.AssumethattheoverallfilteringeffectofthetransmiIer,channel,andthereceiveristhatofanRC-lowpassfilterwherethe3dBbandwidthis1HZ.AssumethattheunipolarNRZinputsignalhasabitrateofRb=1HZandthatthedataontheunipolarNRZsignalis[1001011010].Plotthewaveformatthereceiveroutputandobservetheintersymbolinterference.

3.6.IntersymbolInterference

Example3-13IntersymbolInterferenceCausedbyRCFiltering

•  Nyquist three criteria –  Pulse amplitudes can be

detected correctly despite pulse spreading or overlapping, if there is no ISI at the decision-making instants

•  1: At sampling points, no ISI

•  2: At threshold, no ISI •  3: Areas within symbol

period is zero, then no ISI –  At least 14 points in the finals

•  4 point for questions •  10 point like the homework

3.6.IntersymbolInterference

3.6.IntersymbolInterferenceNyquist’sFirstMethod(ZeroISI)

Nyquist’sfirstmethodforeliminaWngISIistouseanequivalenttransferfuncWonHe(f):

He ( f ) =1fsΠffs

⎛

⎝⎜⎜

⎞

⎠⎟⎟

he (t) =sinπ fstπ fst

where fs =1Ts

0f

He(f)1/fs

fs/2-fs/2

he (kTs +τ ) =C, k = 0

0, k ≠ 0

⎧

⎨⎪

⎩⎪

3.6.IntersymbolInterferenceNyquist’sFirstMethod(ZeroISI)

ThistypeofpulsewillallowsignalingatabaudrateofD=1/Ts=2B(forBinaryR=D)

s

MINIMUM BANDAbsolute bandwidth is: 2

Signaling Rate is: =1 2 Pulses/se

ID

c

W THsfB

D T B

=

=

3.6.IntersymbolInterferenceNyquist’sFirstMethod(ZeroISI)

ThistypeofpulsewillallowsignalingatabaudrateofD=1/Ts=2B(forBinaryR=D)

he(t)

0f

He(f)1/fs

fs/2-fs/2

²  Sincepulsesarenotpossibletocreatedueto:InfiniteWmeduraWon.SharptransiWonbandinthefrequencydomain.

²  TheSincpulseshapecancausesignificantISIinthepresenceofWmingerrors.Ifthereceivedsignalisnotsampledatexactlythebitinstant(SynchronizaWonErrors),thenISIwilloccur.

3.6.IntersymbolInterferenceNyquist’sFirstMethod(ZeroISI)

THEOREM:AfilterissaidtobeaNyquistfilteriftheeffecWvetransferfuncWonis:

He ( f ) =Π

f2 f0

⎛

⎝⎜⎜

⎞

⎠⎟⎟+Y ( f ), f < 2 f0

0, elsewhere

⎧

⎨⎪⎪

⎩⎪⎪

Y(f)isarealfuncWonandevensymmetricaboutf=0;Y(f)=Y(-f),|f|<2f0Yisoddsymmetricaboutf=f0;Y(-f+f0)=Y(-f+f0),|f|<2f0

Therewillbenointersymbolinterferenceatthesystemoutputifthesymbolrateis:

D=fs=2f0

( )

( )

0

0

0 0 0

( ) is a real function and even symmetric about = 0:

( ), 2

Y is odd symmetric about :

( ),

Y f fY f Y f f f

f fY f f Y f f f f

− = <

=

− + = − + <

3.6.IntersymbolInterference

3.6.IntersymbolInterference

Amoregeneralclassoffilter:RaisedCosine-RolloffFilter

He ( f ) =

1, f < f1

1/ 2 1+ cosπ f − f1( )2 f

Δ

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪, f1 < f < B

0, f > B

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

WhereBistheabsolutebandwidthandtheparameters

fΔ= B− f0

f1 = f0 − fΔand

f0isthe6-dBbandwidthofthefilter.Therollofffactorisdefinedtobe

r = fΔf0

3.6.IntersymbolInterference

Amoregeneralclassoffilter:RaisedCosine-RolloffFilter

he (t) =ℑ−1 He ( f )⎡⎣ ⎤⎦= 2 f0

sin 2π f0t2π f0t

⎛

⎝⎜⎜

⎞

⎠⎟⎟cos2π f

Δt

1− 4 fΔt( )2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3.6.IntersymbolInterference

Amoregeneralclassoffilter:RaisedCosine-RolloffFilter

FrequencyresponseandimpulseresponsesofRaisedCosinepulsesforvariousvaluesoftherolloffparameter.

r Br ISI↑→ ↑

↑→ ↓

3.6.IntersymbolInterference

Amoregeneralclassoffilter:RaisedCosine-RolloffFilter

He ( f ) =

1, f < f1

1/ 2 1+ cosπ f − f1( )2 f

Δ

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪, f1 < f < B

0, f > B

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

D = 2B1+ rBaudrate:

WhereBistheabsolutebandwidthofthesystemandristhesystemrollofffactor.

3.6.IntersymbolInterference

Problem3.49

Assumethatapulsetransmissionsystemhastheoverallraisedcosine-rolloffNyquistfiltercharacterisWcdescribedby

He ( f ) =

1, f < f1

1/ 2 1+ cosπ f − f1( )2 f

Δ

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪, f1 < f < B

0, f > B

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

FindtheY(f)NyquistfuncWonofHe ( f ) =

Πf2 f0

⎛

⎝⎜⎜

⎞

⎠⎟⎟+Y ( f ), f < 2 f0

0, elsewhere

⎧

⎨⎪⎪

⎩⎪⎪

3.6.IntersymbolInterference

Problem3.50

AnanalogsignalistobeconvertedintoaPCMsignalthatisabinarypolarNRZlinecode.ThesignalistransmiIedoverachannelthatisabsolutelybandlimitedto4kHz.AssumethatthePCMquanWzerhas16stepsandthattheoverallequivalentsystemtransferfuncWonisoftheraisedcosine-rollofftypewithr=0.5.a.)FindthemaximumPCMbitratethatcanbesupportedbythissystemwithoutintroducingISI.b.)FindthemaximumbandwidththatcanbepermiIedfortheanalogsignal.