CONVOLUTIONAL CODES AND SYSTEMATIC SPACE TIME TRELLIS
CODES
B. L. Yeap, T. H. Liew and L. Hanzo
Dept. of Electr. and Comp. Sc., Univ. of Southampton, SO17 1BJ, UK.
Tel: +44-703-593 125, Fax: +44-703-593045
Email: [email protected]
http://www-mobile.ecs.sot on.a c.u k
Abstract - A novel turbo equalization
ar-rangement isproposed,whichis constituted by
aseriallyconcatenatedsystematicconvolutional
codec and a systematic space time trellis code
(CC-SSTTC)assistedscheme. TheBER
perfor-manceof thisscheme usingvarious interleaving
sizeswascharacterisedoveranequal-power
two-pathRayleighfadingchannel. Aftereightturbo
equalization iterations, an interleaving gain of
approximately 5.5 dB was recorded at BER =
10 4
for the CC-SSTTC system using a
6400-bitSpace Time Trellis (STT) interleaver,when
comparedto the sameCC-SSTTC scheme
util-isinga 100-bit STT interleaver. It was also
ob-servedthattheCC-SSTTCschemeemployinga
6400-bitSTTinterleaveroutperformedtheSTT
codedsystemutilisingthesameSTTinterleaver
but without channel coding by approximately
5.7dB at BER=10 4
.
1. INTRODUCTION
TheIMT2000systemaimstosupportfullcoverageand
mobilityforabitrateofat least144kbps,but
prefer-ably384atkbpsandtoprovidealimitedcoverageand
mobilityforabitrateof2Mbps[1]. The3Gproposals
also aim to provide an improved spectrum eÆciency
comparedtoexistingsystems.
However, there are problems associated with high
dataratetransmissions,sincesystemstransmittingat
highbitrates,suchas2Mbps,experienceahighgrade
ofchannel-induceddispersionandsuerfromInter
Sym-bolInterference(ISI).Hence,theequalizerdesignmust
becapableofmitigating theeectsofISI. In
conjunc-tion with equalization, channel decoding can also be
employedinorderto furtherimprovetheperformance
ofthesystem. Powerfulerrorcorrectionschemes,such
as turbo codes [2], have been shown to yield
perfor-discoveryofturbocodesandturbodecodingledtothe
development of turbo equalization [3]. Turbo
equal-izationis atechniquethat performs channel
equaliza-tionandchanneldecodingjointlyanditeratively. This
schemehasbeenshowntosuccessfullymitigatethe
ef-fectsofchannel ISI.
Recentlyarangeoftransmissiondiversitytechniques
knownasSpaceTimeTrellis(STT)coding[4]havealso
been introduced in order to providediversity gainfor
MobileStations(MSs)byupgradingtheBaseStations
(BSs). STTcoding[4]jointlydesignsthechannel
cod-ing, modulation, transmit diversity and the optional
receiver diversity. Subsequently, Bauch et al. [5]
pro-posed a jointequalizationand STT decoding scheme,
whichyieldedanimprovedperformancedueto
exploit-ingthesoft-decision basedfeedbackfromtheSTT
de-codertotheequalizer'sinput.
Motivated by these trends, in this contribution we
aim to improvethe performance of the STT encoded
systembyemployingadditionalchannel encoding and
turbo equalization. Specically, we propose a turbo
equalizationscheme for a system, which serially
con-catenatesasystematicConvolutionalCodeanda
Sys-tematicSTTCode (CC-SSTTC).The performance of
thissystemisinvestigatedforvariousSTTinterleaver
sizes
s
, namely 100, 400, 800, 1600, 3200 and 6400
bits.
The outline of this paper is as follows. Section 2
presentsanoverviewoftheinvestigatedsystem. Thisis
followedbyadescriptionoftheturboequalization
prin-ciplein thecontext of ourserial concatenatedscheme
inSection3. Performanceresultsarepresentedin
Sec-tion4andnally,themainconclusionsaresummarised
inSection5.
2. SYSTEM OVERVIEW
Systematic
encoder
Systematic
space time
trellis encoder
Source
bits
TX1
TX2
c
s
π
s
π
π
Figure1: Transmitteroftheseriallyconcatenated
sys-tematicconvolutionalcodedandsystematicSTTcoded
system.
andsystematicSTT code, asshownin Figure 1. The
transmittedsourcebitsareconvolutionallyencodedand
directedtoarandomchannelinterleaver
c
. The
con-volutionalencoderdenotedasCC(2,1,5)isa 1
2
-rate
Re-cursiveSystematicConvolutional(RSC)codehavinga
constraint lengthof K = 5and octal generator
poly-nomialsofG
0
=35and G
1
=23. TheRSCcodeword
consists of a systematic information bit and aparity
bit. Subsequently, the encoded bits are passed to a
0 01 02 03
0 21 22 23
0 31 32 33
10 11 12 13
Figure2: Trellisdiagram ofthesystematic4-state,
4-PSKSTTcode[4].
systematicSTTencoderusingtwotransmitantennas,
which is illustratedin Figure 2. Thesystematic STT
encoderisreferredtoasSSTTC(n=4,m=4),sinceit
isan=4-state, m=4-PSKSTTcode. Upon
receiv-ing an input symbol, the SSTTC produces a symbol
ineachtransmitter arm. These symbolsaredisplayed
atthe sideof each trellisstatein Figure 2,whichcan
be treated asthe STT codeword, consisting of a
sys-tematicinformationsymbolandaparitysymbol. Note
thatwehaveemployedthesimpleSSTTC(4,4)scheme
insteadof morecomplexsystematic STTcodes, since
ouraim wasto highlight theturbo equalization
prin-ciple. An exampleof amorecomplexsystematicSTT
code is the SSTTC(8,8) scheme of reference [4]. The
STTencodedsymbolsareinterleavedbyarandomSTT
interleaverrepresentedas
s .
tureconsistsof100datasymbols. Atwo-path,
symbol-spacedfadingChannelImpulseResponse(CIR)ofequal
weights was used, where the Rayleigh fading
statis-ticsobeyedanormalisedDopplerfrequencyof3:3615
10 5
. The fadingmagnitudeand phasewaskept
con-stantfor the duration ofa transmissionburst, a
con-dition which we referto asemploying burst-invariant
fading. Furthermore,in order toinvestigatethe lower
boundperformanceofthesesystems,wehaveassumed
that the CIR was perfectly estimated at the receiver.
TheLog-MAPalgorithm[6]wasemployedinthe
chan-nelequalizer,inthesystematicSTTdecoderandinthe
RSCconvolutionaldecoder.
3. PRINCIPLE OFTURBO
EQUALIZATION
Letusnowdescribetheturboequalizationprinciplein
thecontextoftheCC-SSTTCsystem. Avitalrulefor
turboequalizersisthattheinputinformationofa
par-ticularstage in thecurrentiterationmustnotinclude
the information contributed by that particular stage
fromthepreviousiteration,sincethentheinformation
used in consecutive iterationswould be dependent on
each other [7]. Thechoice of convolutional codes and
STT codes wasconstrained to the systematic variant
of bothcodes. This is essential when calculating the
extrinsic information of theSTT decoderand that of
theconvolutionaldecoder.
Wewillelaborate onthisissuebyusingthechannel
equalizerof Figure 3as our rstexample. Note that
theLogLikelihoodRatios(LLRs)areexpressedinthe
vectorial form of [7], but using dierent actual
nota-tions. Here,thenotationsGandF areused to
repre-senttheLLRsofthesystematicandparitysymbolsof
theSTTcodeword,respectively, whileB and Arefers
to the LLRsof the systematic and parity bits of the
convolutional codeword. The superscript denotes the
natureoftheLLR,namely`a'denotestheapriori
in-formation,`c'isusedforthecompositeaposteriori[8]
information,`i'forthecombinedchannelandextrinsic
[8]informationand`e'fortheextrinsic[8]information.
Thesedenitionswillbefurtheraugmentedduringour
forthcomingdiscourse.Furthermore,thesubscriptsare
used to represent theiteration index, while the
argu-mentwithin thebrackets()indicatestheindexofthe
receiverstage,wheretheequalizersaredenotedasstage
0,theSTTdecoderasstage1andthechanneldecoder
as stage 2. In the gures, we have also marked the
connectedpointsusingcircled boldnotations.
In therstiterationat p=1,thechannel equalizer
ofFigure3onlyevaluatesthereceivedsignalr(t),since
thereisnoapriori feedbackinformation,i.e. F a
1 (0)=
0andG a
1
+
Equalizer
Stage 0
F
p (0)=F
p 1 (1)
G a
p (0)=G
e
p 1 (1)+G
e p 1 (2) G i p (0) F i p (0) r(t) F c p (0) G c p (0) H J K L I
Figure 3: Schematic of the channel equalizer showing
thereceivedLLRsandtheLLRsgenerated.
ori LLR values G c
1
(0) of the systematic STT coded
symbolandtheaposteriori LLRsF c
1
(0)oftheparity
symbols. Subsequently, these LLR valuesare
deinter-leavedbytheSTTdeinterleaver 1
s
-whichisnot
ex-plicitlyshowninthegures-andpassedtotheSSTTC(4,4)
decoder of Figure 4. However, in the subsequent
it-erations, the channel equalizer will receive the
addi-tionalapriori information G a
p
(0)andF a
p
(0)
concern-ing the STT codeword | consisting of the
system-atic and parity symbols | from the other decoding
stages. In order to avoid passing the a priori
infor-mationcontributed by theother concatenated
decod-ingstagesbackto these stages,wesubtractthe a
pri-ori LLRsF a
p
(0) and G a
p
(0) from thecorrespondinga
posteriori LLRs F c
p
(0) and G c
p
(0) in order to derive
thecombinedchannel andextrinsic informationF i p (0) andG i p
(0), asshownin Figure3.
MUX
STT Decoder
Stage 1
G i p (0) F i p (0) G a p (1)=Ge p 1 (2) G e p (1) F e p (1) A e p (1) B e p (1) F c p (1) G c p (1) H J K L M N I
Figure 4: Schematic of the systematic STT decoder
showingtheinputandoutputLLRs.
Let us now consider the operation of the STT
de-coderin Figure4. Thedecoderreceivesthecombined
channel and theextrinsic LLRsF i
p
(0) andG i
p
(0)
pro-ducedbythechannel equalizerofFigure3in the
cur-rentpthiterationand theapriori LLRsG a
p
(1) ofthe
systematicsymbol,whichistheextrinsicLLRG e
p 1 (2)
generated by the channel decoder of Figure 5 during
thepreviousturboequalizationiteration,namely
iter-ationp 1. UsingtheseLLRvalues,theSTTdecoder
computesthe a posteriori LLRs G c
(1) and F c
(1) of
theSTTcodedsystematicandparitysymbols,
respec-tively. As depicted in Figure 4, the a priori LLRs
G a
p
(1) and the combined channel and extrinsic LLRs
G i
p
(0)correspondingtothesystematicsymbolare
sub-tractedfromthecompositeaposteriori LLRsG c
p (1),in
orderto obtaintheextrinsic informationG e
p
(1)of the
systematic symbol. By contrast, the extrinsic LLRs
F e
p
(1)ofthespacetimecoded`parity'symbolsare
ob-tainedbysubtractingthecombinedchanneland
extrin-sicLLRsF i
p
(0)fromtheaposteriori LLRsF c
p
(1)
cor-respondingtothespace-diversityrelated`parity'
sym-bols.
Notethatifanon-systematicSTTencoderwere
em-ployed,theSTTdecoderwouldbeunabletodetermine
the extrinsic LLRs by subtracting the apriori LLRs,
which constitute the LLRs of the systematic
compo-nentoftheSTTcodeword,fromtheaposteriori LLRs.
ThisisbecausetheaposterioriLLRsoftheSTTcode
donotcorrespond to thesystematic STTcoded
sym-bol. Specically,theycorrespondtothediversity-related
STTcoded`parity'symbol. Therefore,unlesstheSTT
codeweresystematic,theaprioriinformationG a p (1)= G e p 1
(2)produced bythe convolutional decoder could
notberemoved, potentiallyallowing this information
tobefedbackintotheconvolutionaldecoder. Hence,it
isimportantthattheSTTencoderusedissystematic.
Subsequently, these extrinsic LLRsare then passed
to the channel deinterleaver, which is not explicitly
shown in the gures. The extrinsic LLRs F e p (1) and G e p
(1)oftheSTTcodewordseeninFigure4becomea
prioriLLRsofthechannelequalizer,whilethe
extrin-sicLLRsB e
p
(1) correspondingto thechannel encoded
systematicbitsareextractedfromG e
p
(1)tobecomethe
apriori LLRsoftheconvolutionaldecoder. Notethat
theextrinsic LLRsA e
p
(1) referring to thechannel
en-codedparitybitsarenotusedinthechanneldecoding
operation, sincethea priori information
correspond-ingtothedecodertrellistransitionprobabilityisonly
inuencedbythesystematicbit,butnotbytheparity
bit.
Stage 2
Conv Decoder
MUX
DEMUX
G i p (0) B a p (2)=Be p (1) A i p (0) B i p (0) B e p (2) A e p (2) G e p (2) A c p (2) B c p (2) L M J
Figure 5: Schematic of the systematic convolutional
decoderdepictingtheinputandoutputLLRs.
As for the convolutional decoder depicted in
B
p
(2) concerning the systematic bit of the
convolu-tionalcodewordandthecombinedchanneland
extrin-sicLLRs G i
p
(0) of the systematic STT coded symbol
generatedby thechannel equalizerof Figure3, which
consistsoftheconvolutionalcodedsystematicand
par-itybits. UsingtheseLLRvalues,thedecodercomputes
the aposteriori LLRsA c
p
(2) and B c
p
(2) of the
code-word, as seen in Figure 5. Inorder to determine the
extrinsicLLRsB e
p
(2)ofthesystematicbits,thea
pri-oriLLRsB a
p
(2)andthecombinedchanneland
extrin-sicLLRsB i
p
(0)correspondingtothesystematicbitare
subtractedfrom thea posteriori LLRsB c
p
(2) related
tothe systematicbits, asshown in Figure5. By
con-trast, the extrinsic LLRs A e
p
(2) of the parity bits are
extractedbysubtractingthecombinedchanneland
ex-trinsicLLRs A i
p
(0) from the aposteriori LLRsA c
p (2)
correspondingtotheparitybits. Forthesamereasons
asmentionedabove,uponusinganon-systematic
con-volutionaldecoder,theapriori informationcouldnot
be removedfrom the aposteriori information.
Con-sequently,theapriori LLRsoriginatingfromtheSTT
decoder would beinevitably passed back to the STT
decoderin thefollowingiteration. Therefore,it is
im-portantthat systematic convolutional codesare
em-ployedintheturbo-equalizedCC-SSTTCsystem. The
extrinsicLLRsB e
p
(2)andA e
p
(2)ofthesystematicand
paritybits,respectively,arecombined,inordertoform
G e
p
(2) and subsequently passed to the channel
inter-leaver. Thechannel-deinterleavedextrinsicLLRsofthe
channelencodedbits areused as theapriori LLRsof
theSTT decoderin thenext iteration. These
extrin-sicLLRs are also added to the extrinsic LLRs G e
p (1)
of the STT decoder and subsequentlycombined with
F e
p
(1)generatedby theSTTdecoderin order toform
theapriori LLRsofthechannelequalizerforthe
forth-coming(p+1)thiteration.
4. RESULTS AND DISCUSSION
Letus nowcharacterisetheperformanceof the
turbo-equalizedCC-SSTTCsystem. Figure 6showsthe
per-formanceoftheinvestigatedsystemusingvariousSTT
interleavingsizes,namely100,400,800,1600,3200and
6400 bits after eightturbo equalization iterations. It
wasobservedinFigure6thatbyincreasingtheSTT
in-terleavingsize from100to 6400, theinterleavinggain
attained was approximately 5.5 dB at BER = 10 4
.
Therefore,asexpected,higherinterleavinggainscanbe
achievedbyusinglongerSTTinterleavers.Asournext
performancemetric,wedened theinterleavergainas
theachievableE
b =N
o
reductionat BER=10 4
upon
increasing theinterleaver memory relative to a
mem-ory of 100 bits. The values of this metric were then
extracted from Figure 6 and plotted in Figure 7. As
0
2
4
6
8
10
12
E
b
/N
o
[dB]
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
BER
Uncoded SSTTC
CC-SSTTC
6400
3200
1600
800
400
100
STT interleaver size
Figure6: BERperformanceoftheturbo-equalized
seri-allyconcatenatedconvolutionalcodedandSTTcoded
system using various STT interleaver sizes, namely
100, 400, 800, 1600, 3200 and 6400 bits, after eight
turboequalizationiterations. Theperformanceof the
uncoded SSTTC system is also shown as a
bench-mark. Thethroughputof theCC-SSTTCsystem is1
bits/symbol(BPS),whiletheuncodedSSTTCsystem
hasaBPSthroughputof2.
0
2000
4000
6000
STT Interleaver size
0
1
2
3
4
5
6
Interleaver
gain
[dB]
Figure7: InterleavergainachievedbytheCC-SSTTC
systematBER=10 4
forSTTinterleaversizesof100,
1600bits. When theBER performance of the
turbo-0
2000
4000
6000
STT Interleaver size
0
1
2
3
4
5
6
E
b
/N
o
[dB]
iterations at BER=10
-4
Coding gain after 8 TEQ
Figure8: CodinggainachievedbytheCC-SSTTC
sys-temoverthe uncoded SSTTC(4,4) systemat BER=
10 4
for STT interleaversizes of 100,400, 800,1600,
3200and6400bits. ThethroughputoftheCC-SSTTC
system is 1 bits/symbol (BPS), while the uncoded
SSTTCsystemhasaBPSthroughputof2.
equalized CC-SSTTC system is compared to that of
the STT coded scheme using no channel coding, as
expected, thechannelcoded systemoutperformedthe
uncoded scheme. It was observedfrom Figure 8that
a coding gain of approximately 5.7 dB was achieved
bytheCC-SSTTCsystemusinga6400-bitSTT
inter-leaver, whencompared to anunprotected STT coded
systematBER=10 4
.
5. CONCLUSIONS
Aturboequalizationschemewasproposedforaserially
concatenatedsystematicconvolutional codedand
sys-tematicSTTcodedsystem. ItwasobservedinFigure6
thathigherinterleavinggainswere achievedforlonger
STTinterleavers.Byusinga6400-bitSTTinterleaver,
aninterleavinggain of approximately5.5 dB was
ob-tained,whencompared to theCC-SSTTCscheme
us-inga100-bitSTTinterleaver. Figure7showedthatthe
interleavinggainsaturatedforSTTinterleaversizesin
excess of 1600 bits. Finally, the performance of the
CC-SSTTCsystemwascomparedtotheuncodedSTT
coded system, in order to determine the
correspond-ing coding gain. It was observed from Figure 8 that
a coding gain of approximately 5.7 dB was achieved
by the CC-SSTTC system using a 6400-bit STT
in-terleaver, when compared to the STT coded system
4
be achievedin jointly detected channel coded,
turbo-equalizedSTTcodedsystems.
Acknowledgement
ThisworkhasbeenfundedintheframeworkoftheISTproject
IST-1999-12070TRUST,whichispartlyfundedbytheEuropean
Union.Theauthorswouldliketoacknowledgethecontributions
oftheircolleagues.
6. REFERENCES
[1] T. Ojanpera and R. Prasad, Wideband CDMA for Third
GenerationMobileCommunications. ArtechHouse,1998.
[2] C.Berrou,A.Glavieux,andP.Thitimajshima,\Near
Shan-non Limit Error-Correcting Coding and Decoding: Turbo
Codes," inProceedings of theInternational Conference on
Communications,(Geneva,Switzerland),pp.1064{1070,
23-26May1993.
[3] C.Douillard,A.Picart,M. Jezequel, P.Didier,C.Berrou,
and A. Glavieux, \Iterative correction of intersymbol
in-terference: Turbo-equalization," European Transactionson
Communications, vol. 6, pp. 507{511, September-October
1995.
[4] V.Tarokh,N.Seshadri,andA.R.Calderbank,\Space-Time
Codes for High Data Rate Wireless Communication:
Per-formanceCriterionandCodeConstruction,"IEEE
Transac-tions onInformation Theory, vol.44, pp.744{765, March
1998.
[5] G.Bauch,A.F.Naguib,andN.Seshadri,\MAP
Equaliza-tion of Space-timeCodedSignals overFrequency Selective
Channels,"inProceedingsof theWirelessCommunications
andNetworkingConfeerence,(NewOrleans,USA),
Septem-ber1999.
[6] P. Robertson, E. Villebrun, and P. Hoeher, \A
Compari-sonofOptimalandSub-OptimalMAPDecodingAlgorithms
OperatingintheLogDomain,"inProceedingsof the
Inter-national Conference on Communications, (Seattle, United
States),pp.1009{1013,18-22June1995.
[7] M.J.GertsmanandJ.L.Lodge,\Symbol-by-symbolMAP
demodulation of CPM and PSK signals on Rayleigh
at-fading channels," IEEE Transactions onCommunications,
vol.45,pp.788{799,July1997.
[8] L.R.Bahl,J.Cocke,F.Jelinek,andJ.Raviv,\Optimal
De-codingofLinearCodesforMinimisingSymbolErrorRate,"
IEEE Transactions on Information Theory, pp. 284{287,
