PAGE

2

un

z-1

AdaptiveLinearPrediction

un-1

z-1

…

z-1

un-m

w1*

X

w2*

X

wm

*

X

u?(n|un1,un2,...,unm)

Theideaistousealinearfiltertopredictthecurrentinput,withthedifferencebetweentheestimateandtheactualvaluehavingthenewinformation.Theestimateischosenasminimizingthevarianceofthisdifferencevariable.Foralongenoughfilter,thisresultsinthedifferencevariablebeinganuncorrelatedsequence.

EE230BProfGregPottie 3

LinearPrediction

Definethedatavectoras

u(un1,...,unm)

Wethensolvetheleastsquaresproblem

wr,where

E[uuH]

r(0)

r*(1)

r(1)

r(0)

...

r(m2)

r(m1)

r*(m1)

r*(m2) ...

r(0)

r E[uu];noteuisascalar.

r*(1)

*

*

n n

r*(m)

EE230BProfGregPottie

4

LinearPredictionII

Wecaninpracticeusethe

algorithm,usingthedifference

variableastheerrorterm.Thisisknownastheforwardpredictionerror,givenby

m

f(n)u wu

*

m

n

knk

k1

Thereissimilarlyastructureknownasabackwardpredictor,whichusesreceivedvaluestopredict(estimate)anearlierreceivedinput.

EE230BProfGregPottie

5

PredictiveDFE

Theideahereistousealinearequalizer(whichproducescorrelatednoise)andthenusealinearpredictortoremovethecorrelationinthedistortion.Sincetheforwardfiltercanbeadaptedindependently,convergenceisfaster,butusuallyattheexpenseofmoreadaptivecoefficientsintotalthantheconventionalDFE.

LEQ

y+

-

-

+

e2

e1

-

Noisepredictionfilter

+

EE230BProfGregPottie

6

PredictiveDFEII

Inthelimitofinfinitefilterlength,thepredictiveDFEhasthesameperformanceastheconventionalDFE

BoththeconventionalandpredictiveDFE’sleadtorelatedMLSEandprecodingstructures.

EE230BProfGregPottie

7

Precoding

TheDFEhastwopotentialdrawbackscomparedtotheLEQ:

AtlowerSNR,itcanbesubjecttoerrorpropagation;onewrongdecisiontriggersalongstringoferrors

Channelcodingrequiresdecisiondelay;thisisfataltoaDFE

Precodingavoidsbothofthispitfallsbymovingthefeedbacksectiontothetransmitter.Inthissectionwewillcover:

theD-transform,convenientnotationfordescribinghowthedeviceswork

modulo-reductionandwhyitisnecessary

costsandbenefitsofprecoding(nothingcomesfor )

EE230BProfGregPottie

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D-Transform

Thisisreallyjustthez-transform,whereD=one-symboldelay=z-1.Itisconvenientsincesequencescanbeexpressedaspolynomials.

Thereceivedsequenceis

r(D)x(D)h(D)n(D)

wherex(D)xxDxD2...

0 1

2

n(D)w(D),q(D)1qD...,

q(D)

1

theMMSEpredictionerrorfilterforn(D)

h(D)h(D1)Kp(D)p(D1)

p(D)1pDpD2...,

1

2

causal,minphase,spectrallyequivalenttoh(D)

EE230BProfGregPottie

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Infini

engthConventionalDFE

Usingthisnotation,theforwardfilterisdefinedby

c(D)q(D)[p(D)/h(D)]noisepredictor+allpasstokillprecursors

b(D)q(D)p(D);theFBFisthenb(D)1

ThefullpicturefromchannelinputtoDFEoutputisthenasbelow.

x(D)

+

n(D)

r(D)

p(D)/h(D)

decision

-

1/q(D)

c(D)

b(D)-1

w(D)

EE230BProfGregPottie

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h(D)

q(D)

Precoder

ConsideranL-pointPAMsignalsetwhereL2,levels1,3,...

i(D)inputdatasequence(Llevels)

x(D)precoderoutput;precoderhasresponseb(D)1

Thenxkikxkjbj2Lzk,k0,

j1

z=integerchosentominimizethevalueofx2.

k

k

Withthisprocedure,xkmustliewithin[-L,L).E.g.,forL=4

2Lz(D)

+

i(D)

-

-

+

b(D)-1

x(D)

EE230BProfGregPottie

11

Precoder

ternativepointofviewistofirstformthesignal

fkikxkjbj,k0,

j1

Thenreduceto[-L,L)usingamodulo2Loperation

i(D)

+

x(D)

-

b(D)-1

EE230BProfGregPottie

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mod

Receiver

Inthereceiver,aftertheforwardfilter(thesameasfortheDFE),get

v(D)v0v1(D),...

x(D)h(D)c(D)n(D)c(D)

y(D)w(D)

Recallc(D)q(D)[p(D)/h(D)]b(D)/h(D)

Thisiswhythenoiseiswhitefollowingtheforwardfilter

Also,givenx(D)[i(D)2Lz(D)]/b(D)

y(D)[i(D)2Lz(D)]h(D)b(D)

b(D)

i(D)2Lz(D)

h(D)

EE230BProfGregPottie

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ReceiverII

Theelementsofy(D)lieonananexpandedintegergrid,witharangethatisthesameasaconventionalDFE.Tothisisaddedthenoise.

Tomakeadecision,firstreducemod2Lto[-L,L)toobtainthefoldedsamples

v'(D)i(D)w(D)

ThisisthesamesequenceaswouldbeseenbythedecisiondeviceoftheDFE(assumingnoerrors),becausei(D)livesonlyon(-L,L)and|zk|=0orisgreaterthanorequalto1.Theoverallsystemis

i(D)

x(D)

r(D)

v(D)

mod h(D) c(D) mod decision

-

b(D)-1

n(D)

EE230BProfGregPottie

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BenefitsofModuloDevices

Withoutnoiseormodulodeviceswecouldhaveobtained

x(D)i(D)/b(D)

v(D)i(D)h(D)b(D)i(D)

b(D) h(D)

However,x(D)haslargepeakvaluesandthesequenceiscorrelated.

Themodulooperationinthetransmit

mountstosubtractionby

therandomsequence2Lz(D).Itreducesthepeakpoweranddecorrelatesthesequence(reducingaveragepoweralso).Withmoduloreduction,x(D)isagainapproxima ywhitewiththesamplesbeingroughlyuniformover(-L,L)(perfectlyasLgoestoinfinity,anexcellentapproximationalreadyforL=4).

EE230BProfGregPottie 15

CostsandBenefitsofPrecoding

Foruniformlydistributedxk,L=4,theaveragepoweris5.33,vs.5for4-PAM.ThissmallpowerpenaltyvanishesasLgrowslarge.

Noiseisunaffectedbythemodulooperations--sothisistheonlyperformancepenaltyforastaticchannel.Inexchangeweavoiderrorpropagationandcaneasilyusechannelcodes,whilegettingtheperformanceofaDFE

Adaptationismoredifficult:mustfirstlearnthefeedbackfilterbyadaptingaconventionalDFE,andthentransmitFBFcoefficientstothetransmitter.Thiscanbedoneonlyperiodically,ands