EQUALISATION
S. Vlahoyiannatos, S. X. Ng, L. Hanzo
Dept. of Electronics and Computer Science,
University of Southampton, SO17 1BJ, UK.
Tel: +44-23-8059 3125, Fax: +44-23-8059 4508
Emails: fsv97r, sxn99r, [email protected]
http://www-mobile.ecs.soton.ac.uk
ABSTRACT
In this contribution, the Per-Survivor Processing
(PSP) based blind turbo equalization is combined
withvariouscodedmodulationschemes,which
em-ploycombinedmodulationandchannelcoding
tech-niques.Theproposedmethodexploitstheenhanced
dataprotectionoeredbyeitherTrellis{Coded
Mod-ulation(TCM),Bit{InterleavedCodedModulation
(BICM)orTurbo{TCM(TTCM)andexhibitsgood
performance in termsof itsoutput Bit ErrorRate
(BER). Explicitly, a BER comparableto that of a
trainedturbo equaliser is attained at the cost of a
modestcomplexity increase.
1. INTRODUCTION
Blindequalizationhas attractedsignicant research
inter-ests duringrecent years. The blindequalizer proposed in
this contribution belongsto the classof sequence
estima-tiontechniques. Itincorporates aPer-SurvivorProcessing
(PSP)basedequalizer[1],modiedfor producingsoft
out-puts,asin[2 ,3 ]anditinvolveschannelcodingnotonlyto
protectthetransmitteddatafromthechannel'seects,but
alsoto assistthe PSP equalizerduring itsconvergence by
utilizingafeedbackloop.
Thepaperisorganisedasfollows. Thecommunications
system is described in Section 2, while in Section 3 the
proposedcodedmodulationassistedturbo-PSPequaliseris
described. Specically,inSection4theencoderanddecoder
usedinthecodedmodulationschemesaredetailed. Finally,
inSection5performanceresultsareprovidedforbothstatic
andfadingdispersivechannels.
2. SYSTEMDESCRIPTION
Thecommunicationssystem underconsideration isshown
inFigure1.TheinformationbitsaremappedtoQAM
sym-bolsbyachannelencoder,whichisatrelliscoded
modula-tionencoder[4], aturbotrelliscodedmodulationencoder
[5 ]oraconvolutionalencoderinthecaseofBICM[6 ]. The
QAMsymbolsaretheninterleaved,inordertodispersethe
channel's bursty errors as well as to enhance the turbo{
equaliser'sperformance, generating thetransmitted QAM
symbolsa(n). Thesesymbolsareconvolvedwiththe
Chan-nel'sImpulseResponse(CIR)h
i
andthenthechannelnoise
Information
Bits
Interleaver
Channel
Noise
Decoded
Bits
Turbo-PSP
Equalizer
Encoder
Channel
Figure1:Equalisedcommunicationssystem
LLR
eq
a-priori
Soft
Output
PSP
Equalizer
Received
y(n)
Symbols
-+
eq
LLR
e
eq
Π
Π
−1
+
-
+
LLR
e
LLR
LLR
eq
in
+
Bit-Symbol
Converter
LLR
dec
Symbol-Bit
Converter
LLR
cod
Decoder
Channel
Figure2: Theturbo{PSPequaliser
e(n)isadded,yieldingthereceivedsymbolsy(n)as:
y(n)= L
2
X
i= L
1
hia(n i)+e(n): (1)
Therestorationoftheoriginalinformationbitsisperformed
bytheturbo{PSPequaliser. Inthenextsectionweprovide
furtherdetailsconcerning theoperationof theturbo{PSP
equaliserusingTCM,TTCMandBICM.
3. TURBO{PSPEQUALISER DESCRIPTION
Theturbo{PSP equaliserperforms joint channel
equalisa-tionand coded modulation decoding using the schematic
shown in Figure 2. The operation of this system is the
sameasthatoftheturbo{PSPequaliserusingconvolutional
coding[7 ],wheretheLogLikelihoodRatio(LLR)valuesare
denedas:
LLR (Bit)=l n
Prob(Bit=1)
Prob(Bit=0)
decoderbya TCMdecoder. Hence, abit{to{symbol
con-verterisplacedbeforetheTCMdecoderonordertoconvert
theLLRvaluestosymbolprobabilities,whicharenecessary
forfacilitating TCM decoding. Similarly,asymbol{to{bit
converterisemployedattheoutputofthedecoder. The
cal-culationof thesymbolprobabilities fromthe probabilities
oftheinputbitsisbasedonthefollowing formula:
Prob(Symbol=A
i
) =
j
Prob(Bit
j
(Symbol )=Bit
j (A
i ));
i=0;;Q, j=0;;K; (3)
whereQisthe numberofsymbolsinthemodulation
con-stellationused,Kisthenumberofbitspersymbol,A
i
rep-resentstheconstellationsymbolsandthefunctionBitj(Ai)
returnsthevalueofthej{thbitofsymbolAi. The
assump-tion implicitly stipulated here is that the bits of a
sym-bolare independentofeachother. This,however, isnota
validassumption,sincechannelcodinghasdeliberately
im-posedacertainamountofcorrelationorinterdependenceon
thebits,inorder toexploitthisredundancyfor correcting
transmission errors. Hence the symbol probability
calcu-lation is somewhat inaccurate. Fortunately the eects of
thisinaccuracyaremitigatedbytheiterativeturbo
equal-izer. Similarlytothebit{to{symbolconversionofEquation
3,the symbol{to{bit conversion is based onthe following
formula:
Prob(Bit
j
=Bit)= X i Prob(Symbol i ;Bit j (Symbol i
)=Bit):
(4)
Thisformula is accurate inthe sensethat it does not
re-quire any assumptions concerning the correlation of the
input symbols. However, the coding-imposed correlation
amongstthebitsofasymbolwouldhavealreadybeenlost
atthe bit{to{symbolconverter ofFigure 2and hencethe
inputofthesymbol{to{bitconverterisuncorrelated.
Havingdescribedthebasicfunctionsofthecomponent
modulesoftheturbo{PSPequaliserinFigure2,wewillnow
proceedtohighlightingtheoperationofthesecomponents.
ThePSP equaliseris the samemodule asthe oneusedin
[7 ]. Therefore, we will not elaborate onit further inthis
paper. Inthenextsectionwedescribetheoperationofthe
codedmodulationencoderanddecoder.
4. CODED MODULATIONSCHEMES
Trellis Coded Modulation (TCM) [4] was proposed
origi-nallyforGaussianchannelsand itwas laterfurther
devel-opedforapplicationsinmobilecommunications[8 ]. Turbo
TrellisCodedModulation(TTCM)[5]isamorerecentjoint
coding and modulationscheme that hasa structure
simi-lar to that of the family of power eÆcient binary turbo
codes[2],butemploysTCM schemesascomponentcodes.
BothTCM andTTCMuse symbol{basedinterleaversand
Set{Partitioning(SP)basedsignallabeling. Anothercoded
modulation scheme distinguishing itself by utilising bit{
basedinterleavinginconjunctionwithGraysignal
constel-lationlabelingisreferredtoasBit{InterleavedCoded
Mod-ulation(BICM)[6 ]. Thenumberofparallelbit{interleavers
equalsthenumberofcodedbitsinasymbolfortheBICM
schemeproposedin[6 ].
TheTCM decoderoperatessimilarlytothecorresponding
moduleofatrainedturboequaliser[3 ]. Anon{binaryMAP
decoderisinvokedfor thechanneldecodermodule. Given
the LLRvalueLLR (Bit)ofabinarybit,wecancalculate
the probability of Bit=+1or Bit= 1as follows.
Re-memberingthat Prob(Bit= 1) =1 Prob(Bit=+1),
andtakingtheexponentofbothsidesinEquation2wecan
write:
e LLR(Bit)
=
Prob(Bit=+1)
1 Prob(Bit=+1)
: (5)
Hencewehave:
Prob(Bit=1)= e
LLR(Bit)
1+e LLR(Bit)
; (6)
and,
Prob(Bit= 1)= 1
1+e LLR(Bit)
: (7)
Then,assumingthatthebitsofasymbolareindependentof
eachother,theprobabilityofaSymbolwhichisrepresented
bythebitsBit 1
;:::;Bit n
,canbecalculatedasinEquation
3,wherewehaveSymbol2(0;:::;2 n
1)forthe2 n
{array
modulationschemeused. Theprobability ofthetransition
fromstatess 0
tostates,commonlydenedas(s 0
;s),used
inthenon{binaryMAPdecoder[2 ],canbecalculatedas:
(s 0
;s)=
Symbol i=n Y i=1 Prob(Bit i ); (8)
where Symbol is the trellis transition branch label from
states 0
tostatesand
Symbol
isthe\apriori"information
ofthe Symbol . Thenthe and valuescanbeobtained
from: k (s)= X s 0 k (s 0 ;s) k 1 (s 0 ) (9) k 1(s 0 )= X s k(s 0
;s)k(s): (10)
Thenumberoftransitionsemergingfromaspecic
trel-lisstateisequalto2 k
,wherekisthenumberofinformation
bitspern{bitmodulationsymbol. Thecodingrateusedis
R= k
n
,wherek=n 1. Thereforethelog{MAPdecoder
usedisnon{binary,whenk>1. Bycontrast,ifk=1,then
the numberof transitions emerging from atrellis state is
equalto2 1
=2,i.e. abinaryMAP decoder is used. The
a{posteriori probability (APP) of the information symbol
ut,ut 2(0;:::;2 k
1)at timeinstant tcanbecomputed
from[2 ]as:
APP(u
t )= X s 0 !s;u t k 1 (s 0 ) k (s 0 ;s) k (s): (11)
The nal decoded information symbolat instant t is the
harddecisionbasedsymbolgeneratedfromtheseAPP
val-ues. However, wehave tofeed back theLLRvalues ofall
thencodedbits,ratherthanjustthoseofthekinformation
bits,tothePSP equaliser,afterimprovingtheirreliability
bythechanneldecoder. TheAPPofthecodedsymbolxt,
xt2(0;:::;2 n
1)attimeinstanttcanbecomputedfrom:
Trellis
Encoder 1
Trellis
Encoder 2
Signal
Mapper
Signal
Mapper
Selector
Π
Π
-1
Figure3: Turbotrellis{codedmodulationencoder.
Symbol
by
Symbol
MAP
Symbol
by
Symbol
MAP
Metric
Metric
LLR
eq
in
Π
-1
LLR
Π
-1
dec
Π
-1
"0"
"0"
LLR
cod
Π
[e&s]
[e&s]
a-priori
a-priori
Π
Figure4:TTCM decoder.
while Equation (11) formulated the APP of the original
encodedinformationsymbol. Theprobabilityofbititobe
1inacodedsymbolxiscalculatedfrom:
Prob(Bit i
=1)= x=2
n
1
X
x=0
APP(x i
=1); (13)
wherex i
denotes the binaryvalue at bitposition iofthe
symbolx, x i
2 (0;1) and inverbal termsthe probability
ofBit i
=1is givenbythe sumof theprobabilities of all
symbolsfrom the setof 2 n
1numberofphasors, which
have a binary1 at bit position i. A similar procedure is
invokedfordeterminingProb(Bit i
=0)andnallytheLLR
ofthebitscanbecomputedfromEquation2.
4.2. TTCMTurbo{PSP
Anextensionoftheturbo{PSPequaliseremployingTurbo
Trellis{CodedModulation (TTCM) has also been
consid-ered. TTCM was proposed in[5 ], where the information
bitsare sent onlyonce,while theparity bits areprovided
alternatively by the two constituent TCM encoders. The
TTCMencoderisshowninFigure3,whichcomprisestwo
identical TCM encoders linked by the symbolinterleaver
. TheTTCMdecoderstructureshowninFigure4is
sim-ilartothatofbinaryturbocodes,exceptforthedierence
inthenature of theinformation passed from onedecoder
totheother. Eachdecoderalternatelyprocessesits
corre-sponding encoder's channel impaired outputsymbol, and
thentheotherencoder'schannelimpairedoutputsymbol.
Theinformation bits, i.e. the systematicbits, are
consti-tutedbythecorrespondingsystematicTCMencoder's
out-putbits receivedoverthe channelinbothcases. The
sys-tematicinformation and theparity informationare
trans-mittedtogetherinthesamesymbol. Hence,thesystematic
information component cannot be separated fromthe
ex-trinsicinformation, since the noise that aectsthe parity
component ofaTTCM symbolalsoaects thesystematic
information component. Theoutputofeachsymbol{based
MAPdecoderofFigure4canbesplitintotwocomponents
[2 ]:
1. theaprioricomponent,and
2. theamalgamated(extrinsicandsystematic)[e&s]
com-ponent.
EachdecoderofFigure4hastopassonlythe[e&s]
compo-nenttothe otherdecoder,whichiswritteninparentheses
inordertoemphasizetheinseparabilityoftheextrinsicand
systematiccomponents.Thereasonthea{prioricomponent
issubtractedfromtheoutputofthesymbol{basedMAP
de-coder of Figure4is that this information component was
generated by the otherdecoder and henceit mustnotbe
fed back to it. Otherwisethe probability estimates ofthe
twodecodersbecomedependentoneachother,preventing
theiterativeenhancementofthedecoder'sdecision
reliabil-ity. TheLLR in
eq
outputoftheequaliserisforwardedtothe
\metric" calculation block of Figure 4, inorder to
gener-ate asetof 2 n
symbolreliabilities. Theselectors, infront
of theSymbolbySymbolMAPdecoderofFigure4,select
the current symbol'sreliabilities from the\metric"
calcu-lation block,ifthecurrentreceivedsymbolcorrespondsto
that component decoder, otherwise depuncturing will be
applied, wherethe reliabilitiesof thesymbolsare setto a
\0" corresponding to noa{priori information. The
\met-ric" calculation block provides the decoder with the
par-ity and systematic [p&s] information, and the second
in-put tothe symbolbysymbolMAP decoder ofFigure4is
the a prioriinformation acquiredfromthe otherdecoder.
The MAPdecoder thenprovides theaposteriori
informa-tionconstitutedbythe(apriori+[e&s])componentsasthe
output. Thenthea prioriinformation issubtracted from
the a posterioriinformation, again so that informationis
notusedmorethanonceintheotherdecoder. The
result-ing [e&s] information is interleaved (orde{interleaved) in
ordertocreatetheaprioriinputoftheotherdecoder. This
decodingprocesswillcontinueiteratively,inorderto
gener-ateanimprovedversionofthesetofsymbolreliabilitiesfor
theotherdecoder. Oneiterationcomprisesthedecodingof
the receivedsymbolsby both of thecomponent decoders.
Finally,theaposterioriinformationofthelowercomponent
decoderof Figure4will bede{interleaved, inorder to
ex-tract(n 1)decodedinformationbitspersymbol. Onthe
otherhand,theaposterioriinformationofthencodedbits
is de{interleaved, inorderto convey the LLR cod
signalof
Figure2totheequaliser'sinput.
4.3. BICMTurbo{PSP
The major dierence between BICM [6 ]incomparison to
TCM [4] is that bit-based interleaving is used and non{
systematic convolutional codes are utilised for protecting
the information. ThedecodingofBICM issimilartothat
ofTCM,asitwashighlightedinSection4.1.
Thesetwocodedmodulationschemesexhibitasimilar
cod-ingrate,dependingonthemodulationmodeused.
Speci-cally,arate{1/2codeisusedforQPSKandarate{2/3code
is employedfor 8{PSK. Inour investigations, Gray signal
constellation labelling is invoked,since we foundthat the
performanceofthecombinedsystemwasbetter,whenusing
Gray-codedconstellationlabellinginsteadofset-partitioning.
Havingdescribedthemostimportantcomponentsofthe
0.707 0.707
0
0
1
1
0.577
0.577
0.577
2
(a) Channel A
Time (in symbols)
Time (in symbols)
(b) Channel B
Figure5: TheCIRsusedinoursimulations
Framelength(bits) 174
Interleaverblocklength 5x174
Carrierfrequency 1900MHz
Symbolrate 2:6MBaud
Dopplerspeed 48Km/h
Equaliserstep{size 5x10 3
Convergencecontrol D2[7 ]
TCMmemory 6
TTCMmemory 3
BICMmemory 6
TTCMnumberofiterations 4
Table 1: Theturbo{PSP equaliserparametersusedinthe
simulations. Observein the tablethat for the shake of a
faircomparisonbetweenTCM,TTCMandBICM,thecode
memorywas6,3and6respectively,sinceTTCMusedfour
iterations, which resulted in a similar complexity for all
threeschemes.
willnowprovideperformanceresultscharacterisingthe
var-iousschemes, whichwereobtained by meansof computer
simulations.
5. PERFORMANCE RESULTS
Inthissectionwepresentperformanceresultsfortheturbo
PSP{equaliserdescribedinSection3.Theresultsarebased
oncomputersimulations. Thechannelsassumedare static
orfading,havingtheCIRsgiveninFigure5. The
conver-gencecontrolis assumedto beD2,asit wasdened in[7 ]
where thevarianceof the dierence betweenthe previous
andcurrentequalizer output LLRsis measured and
com-pared against a xed threshold. The general turbo{PSP
equaliserparameters aregiveninTable 1. InFigure 6we
haveplottedthecodinggainofthevarioustechniquesata
BERof510 5
forQPSKfortransmissionsoverthestatic
2{path channelof Figure 5 versus their relative
complex-ity. It hasbeen assumedthat the complexityof the PSP
equaliserisapproximately the samefor all the algorithms
andthereforetheonlyfactorthataectsthetotal
complex-ity isthe complexity ofthechanneldecoder. InFigures 7
and8theBERversusBitSNRcurvesaregivenforQPSK
and8{PSK,overthefadingchannelsofFigure5. TheBit
SNRisdenedas:
BitSNR= Ea
IBPSEe
; (14)
whereE
a
istheaverageQAMsignalpower,E
e
isthe
aver-agenoisepowerandIBPSisthenumberofinformationbits
persymbol. This denitionassists inthe realistic
assess-mentofthebenetsofusingchannelcodingforprotecting
thedata.
WeobservefromFigure6thatthecodinggainisthehighest
forBICMatalowerdecodingcomplexityforstatic
disper-sivechannels. Whenthecomplexityisincreased, thenthe
codinggainofTTCMbecomesmarginallybetter. This
hap-pensat acodinggain ofapproximetaly 5.7dB.Inthe
fad-ing channels, BICM exhibits the worstperformance. The
performance of BICM is only marginally lower thanthat
of TCM or TTCM in conjunction with QPSK. However,
BICMperformssignicantlyworse,thanTCMandTTCM,
when 8{PSK modulation is employed. TCM is found to
performthebestfortheQPSKmodulationandmarginally
worsethanTTCM,when8{PSKisemployedoverthe
dis-persivefadingchannelshavingtheCIRofFigure5.
The important trend we inferred from our simulations is
thesuperiorityofBICMfortransmissionsoverstatic
chan-nels, butthistrend isreversedfor fadingchannels.
Essen-tially BICM is a binaryversion of TCM, whichis related
to the turbo{PSP algorithm of[7 ]. WhileTCM usesset{
partitioning for maximising the Euclidean distanceof the
constellation points represented by the uncodedbits of a
symbol, BICMis concernedonlywithbits,not with
sym-bols. However,thePSP equaliserexhibitsadegraded
per-formance,whenusingnoGraycoding,sincethenthe
neigh-bouring symbols of the constellation can have more than
onedierentbits. Withtheaidofsimulationswefoundthat
the PSP equaliser'sperformance degradation encountered
duetousingnoGray coding was higher, thanthe
perfor-mancedegradationincurredwhenusingGray{codedrather
thanSP-basedTCM. ThusinoursimulationsGray{coded
ratherthanSP-basedmodulationwas usedinconjunction
withTCM, TTCM andBICM. This isone ofthe reasons
for which TCM appears to performworse than BICM in
static channel scenarios. A further reason of the
perfor-mancedegradationofTCMincomparisonwithBICMover
static, but dispersive, channels is due to using bit LLRs
inthe turbo{PSP equaliser. In fact, a TCM turbo{PSP
equaliserwouldperformbetter,ifthebitprobabilities
rep-resentedbytheLLRswerereplacedbysymbolprobabilities.
However, this would dramatically increase the associated
complexity,sinceitwouldmeanthatateachsymbolinstant
thesignalintheturbo{PSPdevicewouldberepresentedby
anincreasednumberofvalues,whichisequaltothe
num-berofsymbolsinthe QAMconstellation,ratherthanthe
numberofbits persymbol. Therefore, ifKis thenumber
ofbitspersymbol, we would haveafactorof2 K
=K more
LLRvalues to process. Inorderto avoid this complexity
problem,we haveusedbitLLRvalues,justas inthecase
oftheturbo{PSPof[7 ]. Aswehavementionedbefore,this
is notoptimum inconjunction with symbol{based coding
techniques,suchasTCMandTTCM,becauseweassumed
that the probability of a symbol is given by the product
of the probabilities of its constituent bits, which ignores
thecorrelation amongstthesebits,asitwashighlightedin
Section2.
6. SUMMARY
Inthispaperanovelblindequaliserwasproposed,
extend-ingtheturbo{PSPalgorithmof[7]byreplacingtheseparate
convolutional channel coding scheme withcoded
modula-tiontechniques. Theideaoriginatedfromthefact thatby
complexity-2p-4qam-static-2.gle
3
4
5
6
7
Relative Complexity
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
Coding
Gain
(dB)
TTCM
BICM
TCM
BER = 5x10
-5
Figure6: Thecoding gainof TCM,TTCM and BICMat
aBERof510 5
versuscomplexityfortransmissionsover
thestatic2{pathchannelofFigure5,employingQPSK.
$RSCfile: cm-fading-4qam.gle$
6
7
8
9
10
11
12
E
b
/N
o
(dB)
10
-4
2
5
10
-3
2
5
10
-2
2
BER
3-path
2-path
BICM
TTCM
TCM
Figure7: BERversusBitSNR curvesoverfading
chan-nels,exhibitingtheimpulseresponseshowninFigure5and
usingtheturbo{PSPequaliserparametersgiveninTable1
forQPSKmodulation.
$RSCfile: cm-fading-8psk.gle$
6
8
10
12
14
16
E
b
/N
o
(dB)
10
-3
2
5
10
-2
2
5
10
-1
BER
3-path
2-path
BICM
TTCM
TCM
Figure8: BERversusBitSNR curvesoverfading
chan-nels,exhibitingtheimpulseresponseshowninFigure5and
usingtheturbo{PSPequaliserparametersgiveninTable1
for8{PSKmodulation.
baseddetectiontechnique. However,threemaincalamities
exist inthe context of this technique. On the one hand,
whenthe symbolprobabilities replacethe bitLLRvalues,
thenthecomplexityandstoragerequirementsincrease
dra-matically, especially in the context of higher{order QAM
schemes. Thus the turbo equaliser was modied for
us-ingbit{basedLLRvaluesforcommunicatingamongstthe
detection blocks of Figure 2. This gave rise to the
sec-ond mainproblem. Specically, the conversion of the bit
LLRvalues to symbol probabilities was based onthe
as-sumptionthat the bits of asymbolare independent even
afterchannelencoding. However,thisassumptionimposes
a performance degradation on the TCM and TTCM
de-coders,sincethesymbolprobabilitiesbecamesomewhat
in-accurate, although this was partiallycompensated by the
iterativeTTCMscheme. Finally,theemploymentofGray{
coded modulation, which was preferable in terms of the
overall system'sperformance, gave rise to a performance
degradationoftheTCM andTTCM decoders, sincethese
schemeswereoptimisedforset{partitioningbasedmapping
ofthebitstothesymbols.
Nevertheless,inFigure6wefoundthattheTTCMscheme
slightly outperformed the BICM arrangement, which was
similarto the convolutional coding basedschemeused in
[7 ]inthe context of QPSKand staticchannels. By
con-trast, in fading environments TCM emerged as the best
performerfromthesetofschemesstudied.
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