`Errors-and-Erasures' Decoding in RS coded
DS-CDMA Using Noncoherent M-ary
Orthogonal Modulation
Lie-LiangYang,LajosHanzo
Dept. ofECS,Univ. ofSouthampton,SO171BJ,UK.
Tel: +442380593125,Fax: +442380594508
Email:[email protected],http://www-mobile.ecs.soton.ac.uk
Abstract|In this contribution we investigate the
per-formance of Reed-Solomon (RS) coded DS-CDMA using
noncoherentM-aryorthogonalmodulationovermultipath
Rayleighfadingchannels. `Errors-and-erasures'RS
decod-ingisconsidered,whereerasuresaredeclaredbasedon
var-iouserasureinsertionschemes,suchasthemaximum
Out-put Threshold Test (OTT), RatioThresholdTest(RTT)
andthejointMaximumOutputandRatioThresholdTest
(MOR-TT).Furthermore,theperformanceof
`errors-and-erasures' decodingemploying the jointMOR-TT is
com-pared to that of `error-correction-only' decoding and to
that of `errors-and-erasures' decoding using the OTT as
wellasthe RTT. The numericalresults showthat, when
using`errors-and-erasures'decoding,RScodesof agiven
code rate can achieve a signicantly higher coding gain
thanwithouterasureinformation,andthattheMOR-TT
techniqueoutperformsboththeRTTandtheOTT
tech-niques.
I. Introduction
InDirectSequenceCode-DivisionMultiple-Access
(DS-CDMA)cellularsystemsForwardErrorCorrection(FEC)
coding is often used for mitigating the eects of
fad-ing and interference. For thewell-knownReed-Solomon
(RS) codes, `errors-and-erasures' decoding is preferable
to `error-correction-only' decoding, since
`errors-and-erasures' decoding can typically achieve a signicantly
lower codeword decoding error probability, than
`error-correction-only' decoding, if a reliable erasure insertion
scheme is invoked. Hence, inthe context of RS
decod-ing it is benecial to determinethe reliability ofthe
re-ceived symbols and to erase the low-reliability symbols
prior to thedecoding process. Inthe context of
`errors-and-erasures'decoding,theerasureinsertionschemesare
expected to exhibitlow-complexity and low-delay, while
achievingaslowacodeworddecodingerrorprobabilityas
possible.
On the up-link of a DS-CDMA cellular system,
due to the high complexity of coherent
modula-tion/demodulation, which would require a pilot signal
foreachuser,noncoherentM-aryorthogonalmodulation
ThisworkhasbeencarriedoutintheframeworkoftheISTproject
IST-1999-12070TRUST,whichispartlyfundedbytheEuropeanUnion.
Theauthorswouldliketoacknowledge thecontributionsoftheir
col-leagues from Siemens AG, France Telecom - CNET, Centre Suisse
d'ElectroniqueetdeMicrotechniqueS.A.,King'sCollegeLondon,
Mo-torola Ltd, Panasonic European Laboratories GmbH, Robert Bosch
GmbH,TelefonicaInvestigationYDesarrolloS.A.Unipersonal,Toshiba
ResearchEuropeLtd.,TTINorteS.L.,UniversityofBristol,University
ofSouthampton.
using M = 64, i.e. 6-bit symbols has been proposed
for example for the reverse link of IS-95 [1]. Analysis
ofDS-CDMAsystemsusingM-aryorthogonal signaling
- both with and without RS FEC - has been provided
forexamplein[2]-[7] overmultipathfading channels. In
thecontextof`errors-and-erasures'decodingofRScoded
DS-CDMAusingM-aryorthogonalmodulation,the
era-sure insertion schemes based on the Output Threshold
Test(OTT) [3],[6], [8], RatioThresholdTest (RTT)[7]
as wellas joint Maximum Outputand Ratio Threshold
Test(MOR-TT)[9]constituteattractive,low-complexity
andlow-delay erasureinsertion schemes. Inthistreatise
theabovementionederasureinsertionschemesaswellas
theerasureinsertion schemebasedontheBayesianTest
(BT)[10] arerstreviewed. Then,the performanceofa
RScodedDS-CDMAsystemisinvestigated,whenM-ary
orthogonal signaling is employed in conjunction with a
OTT-,RTT-orMOR-TT-basederasureinsertionscheme
over dispersive Rayleigh-fading channels. Furthermore,
theperformanceoftheassociatedM-aryDS-CDMA
sys-teminconjunctionwithRScodesassistedby
`errors-and-erasures'decodingemployingtheOTT,RTTand
MOR-TT are estimatedand compared. Wealso compare the
codeword decoding error performance of the MOR-TT
based erasure insertion scheme to that of using
`error-correction-only'decodingwithoutsideinformation. Note
that, since the complexity of the BT-based erasure
in-sertion scheme is signicantly higher than that of the
OTT, RTTor MOR-TT, the performance investigation
for the RS coded M-ary DS-CDMA system using
BT-based`errors-and-erasures'decodingisnotconsideredin
thiscontribution.
II. System Overview
A. Transmitted Signal andChannel Model
ThetransmitterschematicofthecodedDS-CDMA
sys-temusingM-aryorthogonalmodulationisshowninFig.1.
Inthetransmitterof Fig.1the information bitsare rst
groupedinto b-bitsymbols, whereb=log
2
M. Then,K
information symbols are encoded into an N-symbolRS
codeword. Following RSencodingthe codewordsare
in-terleaved,inordertorandomizetheburstysymbolerrors.
Finally,eachRScodedsymbolis M-arymodulated, DS
spread and carrier-modulated using theapproach of [2],
inorder to formthe transmitted signal. Let us assume
Information
bits
RSEncoder
M-ary
orthogonal
modulation Interleaver
a i
I
(t) cos!ct
a i
Q
(t) sin!ct
Delay
T0
si(t)
+
Fig.1. Transmitterblock diagramofthecodedDS-CDMAsystemusing M-ary orthogonalmodulation,RSencodingandsymbol
interleaving.
i=1;2;:::;K,canbeexpressedas[2]
si(t) = p
PRcW j
(t)a i
I
(t)cos!ct (1)
+ p
PR
c W
j
(t T
0 )a
i
Q (t T
0 )sin!
c
t; 0tT
s ;
where P isthetransmittedsymbolpowerwithoutFEC,
Rc is the coding rate, Ts is the symbol duration, !c is
the carrier's angular frequency andT0 is anoset time.
Furthermore,W j
(t)isthejthWalsh-Hadamard
orthogo-nalfunction,whichrepresentsthejthorthogonalsignalof
theithuser'ssymbols,whilea i
I
(t)anda i
Q
(t)representthe
spreadingwaveformsoftheIn-phase(I)and Quadrature
(Q)phasechannels,respectively. Thesequadrature
com-ponentsare expressedas a i
# (t)=
P
1
h= 1 a
#;i
h
p(t hTc)
where # 2 fI;Qg, a #;i
h
represents independent
identi-cally distributed (i.i.d) random variables assuming
val-ues of +1 and -1 with equal probability of 1/2.
Fur-thermore, Tc represents the chip duration, and p(t) is
assumed to be the rectangular chip waveform, which is
dened over the interval (0;Tc]. Moreover, we assume
that Ns = Ts=Tc =bTb=Tc = bN, where Tb is the
bit-durationandN =Tb=Tc.
Thedispersivefading channelsare assumedtobe
dis-crete and time-invariant, having a channel impulse
re-sponse of h
i (t) =
P
L
l=1
il Æ(t
il
)exp( j
il
)
experi-encedbyuseri, where
il ;
il and
il
represent the
at-tenuationfactor,delay andphase-shiftforthe l th
multi-path component ofthe channel, respectively, while L is
thetotalnumberofdiversitypathsandÆ(t)isthe
Delta-function. Weassumethattheithuser'smultipath
atten-uations f
il
; l=1;2;:::;Lg are independent
Rayleigh-distributedrandomvariableshavingaPDFgivenby
f
il (R ) =
2R
exp
R 2
; (2)
where =E
(
il )
2
. The phasesf
il
; l=1;2;:::;Lg
of the dierent paths are assumed to be uniformly
dis-tributed randomvariablesin[0;2), whilethe ithuser's
pathdelaysof n
il
; l=1;2;:::;L o
are modeledas
ran-anduniformlydistributedin[0;T
s
). Wealsoassumethat
ideal powercontrol is employed, andhence the received
signalpowersare thesameforall Kusers. Thenthe
re-ceivedsignalatthebasestationgeneratedbytheKusers
canbeexpressedas
r(t)= K
X
i=1 y
i
(t)+n(t); (3)
where
yi(t)= L
X
l=1 p
P
il
W j
(t
il )a
i
I (t
il
)cos(!c(t
il
)
il )
+W j
(t T
0
il )a
i
Q (t T
0
il )sin(!
c (t
il
)
il )
:
Furthermore,n(t)representstheAdditiveWhite
Gaus-sianNoise(AWGN),whichis modeledasarandom
vari-ablewithzeromeananddouble-sidedpowerspectral
den-sity of N
0
=2. After the receiver's bandpass lter, n(t)
becomes anarrow-bandnoise process, whichcanbe
ex-pressed as n(t) = n
c (t)cos!
c t+n
s (t)sin!
c
t [2], where
n
c
(t)and n
s
(t) represent lowpass-ltered Gaussian
pro-cesses.
B. ReceiverModel
Thereceiverschematicconsideredinthispaperforthe
RScodedDS-CDMAsystemisshowninFig.2,which
in-cludesnoncoherentM-arydemodulation,equalgain
com-bining(EGC),maximumlikelihooddetection(MLD),
era-sureinsertionbasedontheOTT,RTTorMOR-TT,
dein-terleaving and RS `errors-and-erasures' decoding.
Fur-thermore, the square-law-based noncoherent M-ary
de-modulationblockisthesameasthatusedin[2]andhence
theinterestedreadersarereferredto[2]forfurtherdetails.
Multipathdiversity combininginvokingtheEGC
princi-plewasimplementedinthe`EGC'block. TheMLDblock
selectsthe largestfromitsinputvariablesandcomputes
thevalueof-theratioofthe`second'maximumtothe
r(t)
Square-law
noncoherent
M-ary
demodulation U
i
(1;1)
U i
(1;L)
U i
(m;1)
U i
(m;L)
U i
(M;1)
U i
(M;L)
EGC EGC EGC
U i
1
U i
m
U i
M
Y1
Select
the
largest
Output
or
erase
anM-ary
RScode
symbol
RSdecoder
Information
bits Deinterleaver
(MLD)
Fig.2. ReceivermodeloftheRScodedDS-CDMAsysteminvestigatedusingnoncoherentM-arydemodulation,multipathdiversity
combining,Maximum-LikelihoodDetection(MLD),MOR-TTbasederasureinsertion,deinterleavingandRS`errors-and-erasures'
decoding.
insertanerasure. Finally,aftersymbol-based
deinterleav-ing,theRSdecoderinvokes`errors-and-erasures'decoding
andthenoutputsthereceivedinformationbits.
III. Overview of Erasure Insertion
Schemes
In this section four erasure insertion schemes
asso-ciated with the noncoherent M-ary orthogonal
demod-ulation are reviewed. Let
U i
1 ;U
i
2 ;:::;U
i
M
represent
the decision variables input to the MLD block of Fig.2.
We expressthe maximumand the`second'maximumof
U i
1 ;U
i
2 ;:::;U
i
M as
Y
1 =max
1
U i
1 ;U
i
2 ;:::;U
i
M
; (4)
Y2=max
2
U i
1 ;U
i
2 ;:::;U
i
M
: (5)
Furthermore,letH1 andH0 representthehypothesesof
correctdecisionanderroneousdecisionconcerninganM
-arysymbolin theMLD block of Fig.2. Hence, we have
P(H0)=1 P(H1).
A. OTT
Inthe context of theOTT, the decisionvariable
sub-jectedtoanerasureinsertionisY1,i.e.,theactual
demod-ulator outputisobserved. Let fY
1
(yjH1)and fY
1 (yjH0)
be the conditional probability density functions (PDF),
giventhattheassociated demodulated symboliscorrect
(H1)andincorrect(H0),respectively.Asanexample,the
distributions offY
1
(yjH1) andfY
1
(yjH0) overmultipath
Rayleigh fading channels are shownin Fig.3, where the
diversity orderisL=3,theaveragesignal-to-noiseratio
(SNR)perbitis
b
=5dBandM =16;64;256. Fromthe
gureweobservethat foragivenvalueofM,fY
1 (yjH1)
ismainlydistributedoveranormalizeddemodulator
out-put range associated with relatively high values of Y1,
while fY
1
(yjH0) is spread overa rangehaving relatively
low values of Y1. Consequently, we can argue that the
demodulated symbols havingrelatively low valuesof Y1
are lessreliable thanthose havingrelatively high values
of Y1. Let YT beathresholdassociated withmaking an
erasuredecisionbasedontheOTT.Then,ifY1YT,the
Fig.3. ThePDFsofY1=maxfgunderthehypothesesofH1
andH0usingM=16;64;256,L=3;b=5dBovermultipath
Rayleigh-fadingchannels.
associated demodulated symbolshould be erased.
Oth-erwise,if Y1 >YT,the demodulatoroutputs aRS code
symbol.
B. RTT
Theratio involved inViterbi's RTT is dened as the
ratio of the maximum to the `second' maximum [8], or
equivalently,itcanbedenedastheratioofthe`second'
maximumtothemaximum,whichcanbeexpressedas
= Y
2
Y
1
; 01: (6)
Letf(rjH1)and f(rjH0) bethe conditionalPDFs
un-der the hypotheses of H1 associated with correct
deci-sions and H0 of erroneousdecisionof the MLD,
respec-tively. The distributions of f
(rjH1) and f
(rjH0)
us-ing the sameparameters as thoseinFig.3 are shownin
Fig.4,whenmultipathRayleighfadingchannelswere
Fig.4. ThePDFsof= Y
2
Y
1
underthehypothesesofH
1 and
H0using M =16;64;256, L =3; b =5dBovermultipath
Rayleigh-fadingchannels.
mainly spread over the range havingrelatively low
val-ues of , while f
(rjH
0
) is mainly distributed over the
range havingrelatively high values of. Therefore, the
demodulatedsymbolshavingarelativelylowratioofare
morereliable,thanthosehavingrelativelyhighvaluesof
. Consequently,apre-setthresholdR
T
canbeinvoked,
inordertoerasetheselow-reliability symbolsassociated
witharatioofR
T .
C. Joint MOR-TT
In the context of the joint MOR-TT, the erasure
in-sertion is based on the observation of both the
maxi-mum Y
1
of Eq.(4) and the ratio of Eq.(6).
Specif-ically, let f
Y1;
(y;rjH1) and f
Y1;
(y;rjH0) be the joint
two-dimensional(2D)PDFsofthemaximumY1 andthe
ratioassociatedwiththeMOR-TT,conditionedonboth
the correct detection and erroneous detection of M-ary
symbols. The properties of the MOR-TT can be
stud-ied with the aid of the 2D PDFs of f
Y
1 ;
(y;rjH
1 ) and
f
Y1;
(y;rjH0),for example,whichareshowninFig.5for
theparametersofM =64; L=3and
b
=5dB.Fromthe
2DsurfacesofFig.5weobservethatforthegiven
param-etersthepeakofthedistributionf
Y1;
(y;rjH1)islocated
atarelativelyhighvalueofY1orarelativelylowvalueof
,while thepeakof thedistributionf
Y1;
(y;rjH0) is
lo-catedatarelativelylowvalueofY1 andarelativelyhigh
value of . Theabove observations inturn imply that
f
Y1;
(y;rjH1)ismainlyspreadovertherangehaving
rel-ativelyhighvaluesofY1orrelativelylowvaluesof,while
f
Y1;
(y;rjH0)ismainlydistributedovertherangehaving
relatively low values of Y1 and relatively high values of
. Therefore, ifa demodulatedsymbolhas a maximum
outputvalueofY1 andaratioof,thathadfalleninthe
mainrangeoffY
1
;(y;rjH0),thesymbolconcernedmust
be a low-reliability symboland mustbe replacedby an
erasure. By contrast, if a demodulated symbolhas
val-uesofY1andthathadfallenoutsidethemainrangeof
fY
1
;(y;rjH0),thenthissymbolislikelytobecorrectand
0
0.2
0.4
0.6
0.8
1 0
5
10
15
20
25
30
0
0.3
0.6
0.9
1.2
1.5
H
1
H
0
M=64, L=3,
b
=5dB
Variable
y
Variable
r
Probability
density
conditioned
on
H
1
and
H
0
Fig.5. The2DjointPDFsoff
Y
1 ;
(y;rjH1)andf
Y
1 ;
(y;rjH0)
usingtheparametersofM=64,L=3,b=5dBover
disper-siveRayleigh-fadingchannels.
theRSdecoder. Consequently,inordertoerasethe
low-reliabilityRScodedsymbols,weassumethatYT andRT
are two thresholds, which activate an erasure insertion,
wheneverY1YT andRT.
D. BT
For M-ary orthogonal modulation systems using the
Bayesiantest,allthedecisionvariablesinputtotheMLD
mustbeconsidered,inordertomakeanerasuredecision.
Specically, let si be the ith code symbolof the
alpha-bet, and U = U i
1 ;U
i
2 ;:::;U
i
M
be the vectorof M-ary
decisionvariables input to the MLD unitof Fig.2. Let
f(ujsi) be the conditionalPDF of U given that si was
sent,andibetheaprioriprobabilityofsendingsi. The
BTdecidesthatareceivedsymbolshouldbeerasedif
max
j
j f(ujs
j )
M
X
i=1
if(ujsi)
(1 ); (7)
where 0< < 1. Since inthe BT-based erasure
inser-tionschemethePDFsoff(ujsi)fori=1;2;:::;M have
tobeestimatedformakinganerasureinsertiondecision,
thecomplexityoftheBTissignicantlyhigherthanthat
ofthe OTT,RTTorMOR-TT,whichcarryoutan
era-suredecisionbasedontheestimationofeitherY1or,or
both. Furthermore,thenumericalresultsin[10]indicate
thattheBT-basederasureinsertionschemeonlyslightly
outperforms the RTT-based erasure insertion scheme, if
theSignaltoInterferenceplusnoiseratio(SINR)is
suÆ-cientlyhigh.
E. Performance
In this section the performance of RS coded M-ary
4
6
8
10
12
14
16
18
20 0
0.2
0.4
0.6
0.8
1
-11
-10
-9
-8
-7
Log
of
codeword
decoding
error
probability
Ratio
threshold
(R
T
)
Amplitude
threshold
(Y
T
)
Fig.6. Codeworddecodingerrorprobabilityversustheamplitude
threshold,YT andtheratiothreshold,RT fortheRS(32,20)
FECcodeusing `errors-and-erasures'decodingbased onthe
MOR-TTerasure insertionscheme over dispersive Rayleigh
fadingchannelsusingtheparametersofM=32,L=3,K=1,
b
=12dBandRS(32,20).
schemes is estimated and compared for a given set of
parameters. Fig.6 shows the codeword decoding error
probabilityofthe M-ary DS-CDMAsystemover
disper-sive Rayleigh fading channels usingthe joint MOR-TT.
The RS(32,20) code constructed over the Galois Field
GF(32)=GF(2 5
)correspondingto5-bitsymbolswasused
and`errors-and-erasures'decodingbasedontheMOR-TT
insertionschemewasemployed. Fromtheresults we
ob-servethatthereexistsanoptimumthresholdvalueofR
T
or Y
T
, for which`errors-and-erasures' decoding achieves
theminimumcodeworddecodingerrorprobability. This
observationinturnimpliesthatforgivenvaluesofM,
di-versityorderofL,numberofusersK,SNRperbitof
b as
wellasforagivenRScode,thereexistoptimum
thresh-olds Y
T and R
T
, for which the `errors-and-erasures'
de-codingusingthejointMOR-TTerasureinsertionscheme
achievestheminimumcodeworddecodingerror
probabil-ity. Thisminimum codeword decoding errorprobability
islower,thanthatassociatedwithusingtheRTTaloneor
theOTTalone. NotethatinFig.6thepoint
correspond-ingtoY
T
=0and
T
=1representsthecodeword
decod-ing error probability using`error-correction-only'
decod-ing. Therefore,wecanobservethat`errors-and-erasures'
decoding outperforms`error-correction-only'decoding,if
the appropriate thresholdsare invoked. However, ifthe
threshold Y
T
is too highand simultaneouslythe
thresh-old T is too low, too many erasures will be activated,
potentially erasing correct demodulated symbols.
Con-sequently, the codeword decoding error probability
us-ing `errors-and-erasures' decoding might be signicantly
higher,thanthatusing`error-correction-only'decoding.
Finally,inFig.7wecomparedthe`errors-and-erasures'
RSdecodingperformanceofoursystemsusingtheOTT,
Fig.7. Codeword decodingerror probability versus SNR per
bitforRS(32,20)`erasures-and-erasures'decodingusingOTT,
RTTandjointMOR-TTerasureinsertionschemes.
nels. The parameters used was M = 32, the number
of users was K = 10, the numberof chips per bit was
N =128and the diversity order was L =2;3;4. From
the results we observethat the proposed MOR-TT
era-sureschemeoutperformsboththeRTTandOTTerasure
insertionschemes.
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