NON-ITERATIVE JOINT CHANNEL EQUALISATION AND CHANNEL
DECODING
A. Knickenberg, B.-L. Yeap, J. Hamorsky,M. Breiling, L. Hanzo
Dept. ofElectr. andComp. Sc.,Univ. ofSouthampton,SO17 1BJ, UK.
Tel: +44-703-593125,Fax: +44-703-593045
Email: [email protected]
http://www-mobile.ecs.sot on.a c.u k
ABSTRACT
Anon-iterativeturboequaliserschemeisproposed,
whichoutperforms theiterative turbo equaliserby
about0.7dBataBERof10
3
overasymbol-spaced
two-pathchannelandbyabout 3.4dBataBERof
10 3
overave-pathGaussian channel.
1. ITERATIVETURBOEQUALISATION
In the context of iterative turbo equalisation [1 , 4] both
theconvolutionalencoderandthechannelcanbedescribed
withtheaidofatrellisoranitestatemachine,whichare
eectivelyconcatenatedandthemaximumlikelihoodtrellis
pathcanbeidentiedineach,forexampleusingtheViterbi
algorithmorBahl's MAPalgorithm[3 ]. Thisiterative
al-gorithmhasalsobeenemployedforthedecodingofserially
concatenatedcodes[2 ]. Itisanticipated, however,thatan
amalgamatedschemeachieves abetterperformance, than
twoseparatealgorithms.
The joint detection scheme, namely the turbo
equali-sation scheme of Reference [4] is portrayed in Figure 1.
The scheme is based on a Maximum Aposteriori (MAP)
equaliser, followed byaMAPdecoder, whichexchange in
eachiterationa-prioriinformationabouttheencodedbits.
BeforeweelaboratefurtheronthestructureofFigure1,
wedenethelog-likelihoodratio(LLR),asthenatural
log-arithmofthequotientoftheprobabilitythatthebitdkwas
abinary'1'andtheprobabilitythatitwasa'0':
LLR (dk)=ln
P(d
k
=1)
P(d
k
=0)
(1)
Thisrepresentationallowstheharddecisionconcerningthe
estimatedbitstobeconvenientlycarriedoutattheendof
theiterationswithrespecttothethresholdofzero.
The MAP equaliser and decoder blocks are so-called
soft-in soft-out blocks, which allow them to readily
ex-change'soft'-informationduringeachiteration. TheMAP
equaliser of Figure 1 is fed with the channel output
val-ues~yandwiththea-priori informationL
D e
(^c 0
)concerning
TheauthorsarewithDepartmentofElectronicsand
Com-puterSciences,UniversityofSouthampton,SO171BJ,U.K.
E-mail:[email protected],http://www-mobile.ecs.soton.ac.uk.
Thenancialsupportofthefollowingorganisationsis
grate-fully acknowledged: MotorolaECID, Swindon, UK; European
Community,Brussels, Belgium; Engineering and Physical
Sci-encesResearchCouncil,Swindon,UK;MobileVirtualCentreof
Excellence,UK.;TheRoyalSociety(NATOFellowship).
theencodedbits^c
0
. Asexpected, intherst iterationwe
haveL
D e
(^c 0
)=0. Upon invokingtheMAPalgorithm,the
equaliser of Figure 1outputs thea-posteriori LLR values
L E
(^c 0
)related totheencodedbits, where thea-posteriori
LLR iscomposed ofthe channel information, thea-priori
informationplustheextrinsicinformation. However,in
or-der togenerate thedecodedbits, we want to passto the
deinterleaveranddecoderblockofFigure1onlythe
chan-nel information and the extrinsic information, where the
latter is dened as the information concerning a certain
bit c
k 0,
whichiscalculatedbyconsideringthereceived
se-quenceuponexplicitlyexcludingthevaluesduetothebit
c k
0 -hencethetermextrinsic.
Therefore,inordertoremovetheinuenceofthea-priori
information,itissubtractedfromL
E (^c
0
),yieldingthesum
of channel information and extrinsic information, namely
L E
(^c 0
),whichisformulatedas:
L E
(^c 0
)=L
E (^c
0
) L
D e
(^c 0 ):
These valuesL
E
(^c 0
)arethendeinterleavedin Figure1,in
ordertobeintherightorderforfeedingthemintotheMAP
decoder,asa-prioriinformation.
The MAPdecoder then delivers thea-posteriori values
of the encoded bits L
D (^c
0
). The LLRvalues of thedata
bitsarethen subjectedtoharddecision following thelast
iteration. Now again, we haveto subtractfrom the LLR
valuesoftheencodedbitsthequantityL
E
(^c),in orderto
acquiretheextrinsicinformationL
D e
(^c)concerningthe
en-codedbits,whichisgivenby:
L D
e
(^c )=L
D
(^c) L
E
(^c):
After passingthroughtheinterleaverof Figure1, the
ex-trinsic information ofthe decoder is readyto be used as
a-prioriinformationbytheequaliserforthenextiteration.
By feeding back the extrinsic information iteratively, we
willgetanimprovedBERfromiteration toiteration. Let
us now consider a novel technique, which we refer to as
non-iterativejointequalisation/decoding.
In summary,in iterative turboequalisation and
decod-ingtherearetwoconsecutivetrellises,whichexchange
soft-decision information concerning theencodedbits, passing
theassociatedlog-likelihoodvaluesbackandfortha
num-beroftimes,inordertoiterativelyimprovethesystem'sbit
error rate (BER) performance. The maximum likelihood
+
+
-L E
(^c 0
) L
E
(^c 0 )
L D
(^c) L
D e
(^c) L
D e
(^c 0 )
L E
(^c)
L D
( ^ d) ~
y
Equaliser MAP
Interleaver
Deinterleaver
Decoder MAP
Figure1: TheprincipleofiterativeturboequalisationofReference[4].
thismethoduponincreasingthenumberofiterations. Let
usnowconsideranon-iterativetechniqueinthenext
sec-tion.
2. NON-ITERATIVE TURBOEQUALISATION
InReference[5]theoptimumnon-iterativeturbo-decoding
algorithmwasintroducedforthedecodingofparallel
con-catenatedcodes, whicharealsoreferredtoasturbocodes.
Bycontrast,inthiscontributionweproposea
non-iterative joint channel decoding and equalisation
scheme forthe serially concatenated convolutional
encoderplusdispersivechannelarrangementof
Fig-ure2,whichoutperformsiterativeturboequalisers.
Incontrasttoiterativeturboequalisationalgorithms[1 ,4 ],
wherewe needacertainnumber ofiterations,in orderto
achieveagood performance,accordingtoourproposed
so-lution we carry out equalisation anddecoding in a joint,
non-iterativestep. Wenoteherethatconceptuallya
mul-tipathchannelcan be seenas aspecial convolutional
en-coder,havingacodingrateofR=1andamemorylength
K 1, whichisequaltothemaximumdelayofthe
chan-nelexpressedintermsofthetimeunitsk
0
characteristicof
theoutputofthedecoder. Wethenhaveasystemoftwo
serialencoders,namelytheconvolutionalencoderandthe
channel, separated by an interleaver, namelythe channel
interleaver.
For the proposed non-iterative joint
equalisa-tion/decoding process we have to construct a nite-state
machine,whichmodelsthewholesystem. Theinputofthis
nite-statemachineisconstitutedbythetheoriginaldata
bitsanditsoutputisthechanneloutput. Inanite-state
machinethe outputy~
k
andthe nextstate S
k +1
are fully
determinedby thecurrent stateS
k
andthecurrentinput
databitd
k
. However,theinterleaverrendersthereceiver's
taskcomplex.
LetusconsidertheexampleofFigure2,wherethe
mul-tipathchannelhasL=2symbol-spacedpaths,
correspond-ingtoaninnerencodermemorylengthofI=L 1=1.
TheconvolutionalouterencoderofFigure2hasaconstraint
lengthofK =3andamemorylength of
o
=K 1=2.
Each encoded bit is determinedby thecurrent incoming
databit d
k
andthe
o
previousones. These encodedbits
passthroughtheinterleaverandenterthemultipath
chan-nelintheinterleavedorder. Wedenotetheinterleaver
func-tionbyl (k
0
),whichassignstheinputtimeinstantk
0
tothe
outputtimeinstantl (k
0
). Hencetheoutputofthechannel
atthetimeinstantk
0
dependsonc
l(k 0
) ;c
l(k 0
1) ;:::;c
l(k 0
I
) .
Each convolutionally encoded bit in turn depends on the
last
o
+1 data bits. Finally, we get the output of the
channelasafunctionofthedatabitsasfollows:
y k
0 = g0[d
l(k )
d
l(k ) 1
d
l(k ) o
]+
+ g
1 [d
l(k 1)
d
l(k 1) 1
d
l(k 1)
o
]+
. . .
+ g
I
[d
l(k
I )
d
l(k
I
) 1
d
l(k
I
)
o ];
where gi represents thesymbol-spaced dispersivechannel
impulseresponsetaps. In conventionaltrellises, which are
used todescribe convolutional codes, theencoderstate is
determinedby the bitsin theencoder'sshift-register. In
ourseriallyconcatenatedconvolutionalencoderplus
chan-nelwedenea super-state S
k
,whichisdetermined
by allthe bitsthatare used tocalculatethe
chan-neloutput,excepttheactualinputbit,whichisoften
referredtoasthestatetransitionbit.
In ourforthcomingdiscourseweproposeanon-iterative
equalisation /decoding algorithmandhighlight its
opera-tionusingthesimpleexampleofa2N blockinterleaver
andahalf-rateconvolutionalencoder.
2.1. Non-iterative joint equalisation/decoding
us-ing a2N interleaver
Again,Figure2showsasimplemanifestationoftheserially
concatenatedsystemusinganon-recursive,half-rate
convo-lutional encoderwith aconstraint length ofK = 3, octal
generatorpolynomialsG[0]=5andG[1]=7,a2Nblock
interleaverandatwo-pathchannel. Itisreadilyseenthat
thistwo-columninterleaverwillsimplyseparateforeach
en-codedbitblockoflength2Ntheevenandtheoddencoded
bits. Firstall theoddindexedencodedbitsgeneratedby
therst generator polynomial willenter the channel,
Interleaver c 1 k d k c 2 k + + z 1 z 1 g 1 g 0 ~ y k = y 1 k y 2 k c l (k 0 1) c l (k 0 ) + z 1 n k 0 P MUX Modulator c k 0
Figure2: Modelofthetransmitterconsistingoftheconvolutionalencoder,interleaver,modulatorandcomplexchannel.
generator polynomial. The super state S
k
is constituted
byallthepreviousbits,whichinuencethechanneloutput
sequence,namelyy
1 k
andy
2 k
inFigure2. Letusdenotethe
encodedbitsemanatingfromtherstgeneratorpolynomial
atthetimekbyc
1
k
andthoseduetothesecondonebyc
2
k .
The channeloutputsy
1 k
andy
2 k
inFigure2arecalculated
accordingtothefollowingequations:
y 1 k = g 0 c 1 k +g 1 c 1 k 1 (2)
= g0(dkdk 2)+g1(dk
1 dk 3) y 2 k = g 0 c 2 k +g 1 c 2 k 1 (3) = g 0 (d k d k 1 d k 2
)+g
1 (d k 1 d k 2 d k 3 ); where y 1 k
is transmitted during the rst half of the
en-coded output block of length 2N andy
2 k
during the
sec-ondhalf. ThesuperstateS
k
isdeterminedbythedatabits
(dk 1;:::;dk 3), while dk isthecurrentencoderinputbit.
The next superstate, namely S
k +1
will be determinedby
(d k
;:::;d
k 2
).
In our example the outer encoder's and the inner
en-coder's memorylength are given by
o
= 2 and
I
= 1,
respectively,hencethenumberofsuperstatebitsiso+I
andthenumberofsuperstatesis2
o
+ I
. Witheach
addi-tionalmemory-elementinthechannel,wehavetoconsider
oneadditionalpreviousencodedbit,implyingthatwehave
toconsideranadditionalbitinthesuperstate.
Letus now considertheinner-working ofthe proposed
non-iterative scheme at the commencement of operation
andalsonearthemiddleofthetransmittedsequence,where
wechangefromtransmittingc
1 k
toc
2 k
,andlastly,attheend
ofeachencodedblockoflength2N. Letusassumeforthe
sakeofillustrationthatwehavea210interleaver. Then
wehave20 encodedbits andhence10 input data bitsat
the encoder. In order tobe in apre-dened state at the
end ofeachencoded block we willuse aterminated code
with3zero-valuedterminationbits. Thelengthofthis
ter-minationsequence isequaltothenumberofconcatenated
encoderstatebitsi.e. thenumberofbitsinthesuperstate.
Thisimpliesthatinthe10-bitencodedsequencewehave7
data bitsand3terminationbits, whichare,again,logical
'0's, since we are usinga non-recursiveencoder. Table 1
showsthesituationattheabove-mentioned'critical'
opera-tionalpointsoftheencoder. Observeintheexamplethatif
theinputbitsd8;:::;d10 arezero,thenc
1 10 andc 2 10 arealso zero.
Time State c
l(k 0 ) c l(k 0 1) Output k 0 y i k
1 (0;0;0) c
1 1
=d
1
0 0 y
1 1
2 (d
1
;0;0) c
1 2 =d 2 0 c 1 1 =d 1 0 y 1 2 . . . . . . . . . . . . . . . 10 (d 9 ;d 8 ;d 7 ) c 1 10 =d 10 d 8
| {z }
=0 c 1 9 =d 9 d 7 y 1 10
11 (0;0;0) c
2 1
=d
1
00 c
1 10 =d 10 d 8
| {z }
=0 y 2 1 12 (d 1
;0;0) c
2 2 =d 2 d 1 0 c 2 1 =d 1
00 y
2 2 . . . . . . . . . . . . . . . 20 (d 9 ;d 8 ;d 7 ) c 2 10 =d 10 d 9 d 8
| {z }
=0 c 2 9 =d 9 d 8 d 7 y 2 10
Table1: Thecontentsoftheconcatenatedencoderseenin
Figure2versustimek
0 .
The proposed decoder operates on the basis of
constructing theencoder'ssuper-trellisconstituted
bythepreviouslydenedsuper-statesS
k
described
bythedatabits(dk 1;:::;dk 3)andinvokingthe
Log-MAP algorithm for joint channelequalisation and
decoding. AlthoughinthiscontextalsotheViterbi
algorithm could have been utilized, since in this
particular application there is no need for
produc-ing the soft outputs provided by the MAP
algo-rithm. Moreexplicitly, the proposed non-iterative
schemerefrainsfrompassingsoft-decision
informa-tion iterativelybetweentheseparatetrellisesofthe
equaliser andchanneldecoder andoperates on the
basisofthesuper-trellis,ratherthanonthetwo
in-dependenttrellises oftheequaliser andchannel
de-coder. Invoking the supertrellis in a non-iterative
stepimprovestheperformanceoftheamalgamated
3. SYSTEMPERFORMANCE
We have conducted simulations using the proposed
non-iterativealgorithmdescribedaboveandcomparedits
per-formance to that ofthe iterative turbo equalisation
algo-rithmusingthesameparameters. Inbothsystemswe
em-ployedBPSKmodulation.
0
1
2
3
4
5
6
7
8
E
b
/N
o
(dB)
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
BER
BER Against Eb/N0
2 path iterative, 8th iteration
2 path non-iterative
Figure3: Comparisonofiterativeversusnon-iterativejoint
equalisation/decodingoveratwo-pathdispersiveGaussian
channel,whenusingaconvolutionalcodehavingthe
follow-ingparameters: R=
1
2
,K=5,G[0]=37,G[1]=21,channel
impulseresponsetaps ofg0 =g1 =
1 p 2
andaninterleaver
of22052bits
In Figure 3 we have compared the iterative and
non-iterative algorithms, when transmittingover a dispersive
two-pathGaussianchannelusingeightiterations. Weused
anon-recursiverateR=
1
2
,constraintlengthK=5
convo-lutionalcodewithageneratorpolynomialofG[0]=37and
G[1]=21,whichareexpressedinoctal format. The
chan-nelimpulseresponsetapsweregivenbyg
0
=g
1 =
1 p 2
and
theinterleaversizewas22052=4104. Asseeninthe
g-ure, thenon-iterativealgorithmout-performstheiterative
schemebyabout0.7dBataBERof10
3 .
Inordertofurtherquantifytheproposedscheme's
perfor-mance,inFigure4wecomparedthetwoalgorithmsovera
moredispersiveve-pathchannel,employingthefollowing
parameters: R=
1
2
;constraintlengthofK =5;G[0]=37
andG[1]=21;channelcoeÆcientsofg0=g4=0:227,g1=
g3 = 0:46, g2 = 0:688; interleaversize of22052=4104;
eightiterations. Over thismoredispersivechannel
appar-entlythenon-iterativescheme'sperformancebecomeseven
more attractive, resultingin an SNR-gainof 3:4 dBat
aBERof10
3
. Wealsonotethat theperformanceofthe
Max-Log-MAPandtheLog-MAPalgorithmswasalso
com-paredandwasfoundtobevirtuallyidentical. These
per-formanceimprovementswereachievedinthecontextofthe
2N interleaver at thecost of areasonable complexity.
Forexample,inthecaseofK=5andatwo-pathchannel
wehave
o
+
I
=4+1statebits,hencethereare2
5
=32
states. Even forK =5andave-pathchannelwehavea
moderatenumberof2
8
=256states.
0
1
2
3
4
5
6
7
8
9
10
11
E
b
/N
o
(dB)
10
-1
10
-2
10
-3
10
-4
BER
BER Against Eb/N0
5 path iterative, 8th iteration
5 path non-iterative
Figure4: Comparisonofiterativeversusnon-iterativeturbo
equalisationoverave-pathGaussianchannelusinga
con-volutional code havingthefollowing parameters: R =
1
2 ,
K=5, G[0] =37, G[1] =21, g0 =g4 = 0:227,g1 = g3 =
0:46,g2=0:688,22052interleaver.
3.1. Eectof Interleaver Depth
Let usnowconsider theeect ofinterleaver-depth inthis
section. Ifwedonotusean interleaver,thentheencoded
bitsenterthechannelinthesameorder astheywere
cal-culatedbytheencoder. Wethenhavetwochanneloutput
bits, namelyy
1 k
andy
2 k
,whichwereduetotheinputdata
bit d
k
, when thesuper-state was S
k
. If we consider the
exampleofFigure2,wehavethefollowingequations:
y 1 k
= g0c
1 k
+g1c
2
k 1
(4)
= g
0 (d
k
d
k 2
)+g
1 (d
k 1
d
k 2
d
k 3
)
y 2
k
= g0c
2
k
+g1c
1
k
(5)
= g0(d
k
d
k 1
d
k 2
)+g1(d
k
d
k 2
):
Hence we have three state bits in the superstate S
k
=
(dk 1;dk 2;dk 3). Duetothelackofinterleavingwehave
nowo+bI=2cstatebits,whichimpliesthatweneedless
statebits,thanthepreviouso+Ivalue,whichwas
asso-ciatedwiththe2Ninterleaver.
Whenusing a4N interleaver, we have tocope with
ahigherdetectioncomplexity. Specically,thisinterleaver
producesfromthesequence:
:::;c 2
4 ;c
1
4 ;c
2
3 ;c
1
3 ;c
2
2 ;c
1
2 ;c
2
1 ;c
1
1
=)
thefollowing outputsequence:
:::;c 2 6
;c 2 4
;c 2 2
;:::;c 1 6
;c 1 4
;c 1 2
;:::;c 2 5
;c 2 3
;c 2 1
;:::;c 1 5
;c 1 3
;c 1 1 :
Intheshift-registerstagesofFigure2modellingthechannel
wenowhavethefollowingencodedbits:c
i k
;c i
k 2
;:::. Hence,
incomparisontothe2Ninterleaverwenowalwayshave
a displacement oftwo clock intervals between theindices
ofthecodedbits,whichconsecutivelyenterthechannel. If
outputvaluesarecalculatedasfollows:
y 1 k
= g0c
1 k
+g1c
1
k 2
(6)
= g0(dkdk 2)+g1(dk
2
dk 4)
y 2 k
= g
0
c
2 k
+g
1
c
2
k 2
(7)
= g0(dkdk
1
dk 2)+g1(dk
2
dk
3
dk 4):
Thisleadstothefollowing superstate:
S k
=(d
k 1
;d
k 2
;d
k 3
;d
k 4
):
Thisexample showsthatwe nowneed
o
+2
I state
bits,sincewehavetostoreallthepreviousbits,whichare
either used to calculate the channel output or whichare
usedtodeterminethenextstate. Inthiscasethenumber
of statesincreases rapidly with thememoryordispersion
ofthechannel. Forexample,forave-pathchannelanda
convolutionalencoderofconstraintlengthK =5thiswould
resultin2
4+24
=2
12
=4096states,whichisirrealistically
high.
0
1
2
3
4
5
6
7
8
E
b
/N
o
(dB)
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
BER
BER Against Eb/N0
2 path non-iterative with 4xN interleaver
2 path non-iterative with 2xN interleaver
2 path non-iterative without interleaver
Figure5: Comparisonofdierentinterleavers, using
non-iterativejointequalisation/decodingovera2pathchannel.
R=
1
2
,K=5,G[0]=37,G[1]=21,g0=g1=
1 p 2
.
Wehaveconducted simulations fora two-path channel
using the same parameters, as in Section 2.1, except for
the interleaver. Figure 5 shows the results for the
two-pathchannelwithoutinterleaver,witha2N ora4N
interleaver.
4. CONCLUSIONS
The proposed non-iterative joint equaliser/decoding
out-performedtheiterativeturboequaliserbyabout0.7dBat
aBERof10
3
overthetwo-pathchannelandbyabout3.4
dBataBERof10
3
overtheve-pathGaussianchannel.
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