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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
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3.6.IntersymbolInterference
Page 3
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∑
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3.6.IntersymbolInterference
Example3-13IntersymbolInterferenceCausedbyRCFiltering
PlottheoutputwaveformwhenachannelfiltersaunipolarNRZsignal.AssumethattheoverallfilteringeffectofthetransmiIer,channel,andthereceiveristhatofanRC-lowpassfilterwherethe3dBbandwidthis1HZ.AssumethattheunipolarNRZinputsignalhasabitrateofRb=1HZandthatthedataontheunipolarNRZsignalis[1001011010].Plotthewaveformatthereceiveroutputandobservetheintersymbolinterference.
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3.6.IntersymbolInterference
Example3-13IntersymbolInterferenceCausedbyRCFiltering
Page 6
• 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
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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
⎧
⎨⎪
⎩⎪
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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
=
=
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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.
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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
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( )
( )
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
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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
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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
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
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3.6.IntersymbolInterference
Amoregeneralclassoffilter:RaisedCosine-RolloffFilter
FrequencyresponseandimpulseresponsesofRaisedCosinepulsesforvariousvaluesoftherolloffparameter.
r Br ISI↑→ ↑
↑→ ↓
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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.
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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
⎧
⎨⎪⎪
⎩⎪⎪
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3.6.IntersymbolInterference
Problem3.50
AnanalogsignalistobeconvertedintoaPCMsignalthatisabinarypolarNRZlinecode.ThesignalistransmiIedoverachannelthatisabsolutelybandlimitedto4kHz.AssumethatthePCMquanWzerhas16stepsandthattheoverallequivalentsystemtransferfuncWonisoftheraisedcosine-rollofftypewithr=0.5.a.)FindthemaximumPCMbitratethatcanbesupportedbythissystemwithoutintroducingISI.b.)FindthemaximumbandwidththatcanbepermiIedfortheanalogsignal.