Performance of RS coded DS-CDMA
Using M-ary Orthogonal Signalling
Lie-Liang Yang and Lajos Hanzo
Dept. of ECS, University of Southampton, SO17 1BJ, UK.
Tel: +44-703-593 125, Fax: +44-703-594 508
Email: [email protected] and [email protected]
http://www-mobile.ecs.soto n.ac .uk
Abstract|Inthiscontributionweinvestigatethe
perfor-manceofRScodedDS-CDMAusingM-aryorthogonal
sig-nalling over multipath Rayleigh fading channels.
`Errors-And-Erasures(EAE)'RSdecodingisconsidered,where
era-sures are declared based on the joint Maximum Output
and Ratio Threshold Test (MOR-TT). Two classes of
di-versityreceptiontechniquessuitablefornoncoherentM-ary
detection are invoked, namely either Equal Gain
Combin-ing (EGC) or Selection Combining (SC) and their related
performanceisevaluated forboth uncoded and coded
DS-CDMA systems. The performanceof theproposed
MOR-TT technique is compared to that of the Ratio Threshold
Test(RTT)andtheOutputThresholdTest(OTT),aswell
as that of `Error-Correction-Only (ECO)' decoding. The
numericalresultsshowthatwhen usingEAE decoding,RS
codesofagivencoderatearecapableofachievinga
signi-cantlyhighercodinggainthanwithouterasureinformation.
Furthermore,theMOR-TTtechniqueoutperformsboththe
RTTand theOTTtechniques.
I. Introduction
In DS-CDMA cellular systems Forward Error
Correc-tion(FEC)isoftenusedformitigatingtheeectsoffading
and interference. For the well-known RS codes,
`Errors-And-Erasures (EAE)' decoding is preferable to
`Error-Correction-Only (ECO)' decoding, since EAE decoding
typicallyachievesasignicantlylowercodeworddecoding
errorprobability,than ECOdecoding, providedareliable
erasureinsertionschemeisinvoked. InthecontextofM
-ary orthogonal modulation using RS coding, anumberof
methods for generating reliability-based side information
andtheirperformancehasbeenanalyzedforexamplein[1]
- [5]. In[4], [5]ithas beenshownthat theRatio
Thresh-old Test (RTT) and thedemodulator'sOutput Threshold
Test(OTT)constituteapairofpromisingerasureinsertion
schemes exhibitinglow-complexityandlow-delay.
Speci-cally, in theRTT[1]anerasureis declared, wheneverthe
ratioofthemaximumto the`second'maximumofthe
in-putoftheMaximum-LikelihoodDetector(MLD)doesnot
exceed apre-set threshold. By contrast, in the OTT [5]
Thiswork has been funded inthe frameworkof the ISTproject
IST-1999-12070 TRUST, which is partly funded by the European
Union. Theauthorswouldliketoacknowledge thecontributionsof
theircolleagues.
the erasure insertion is based on the observation of only
the maximum of the MLD inputs, i.e. based on the
de-modulator'soutput. Anerasureisactivated,wheneverthe
amplitudeof themaximumoftheMLD's inputs doesnot
exceedapre-setthreshold.
In thiscontributiona novelerasure insertionscheme is
investigated, where an erasure is activated based on the
observation of not only the maximum, but also the
ra-tio of the maximum to the `second' maximum of the
in-putstotheMLDassociatedwiththeM-arydemodulator.
Equivalently, also the ratio of the `second' maximum to
themaximumcanbeused. Werefertothis erasure
inser-tionschemeasjointMaximumOutput andRatio
Thresh-old Test (MOR-TT)[6]. Furthermore,weinvestigate the
performance of a RS coded DS-CDMA system [4], [7],
whenM-aryorthogonal signalingis employedin
conjunc-tionwithaMOR-TT-basederasureinsertionschemeover
dispersive Rayleigh-fading channels. Two dierent
diver-sitycombiningschemes[4],[7]-namelyEqual-Gain
Com-bining (EGC) and Selection Combining (SC) - are
con-sideredand their performanceis evaluatedin thecontext
oftheproposed MOR-TTbasederasureinsertionscheme.
The performance of the proposed MOR-TT technique is
compared to that ofthe RTT, OTT, andto that of ECO
decoding. We will show that when using EAEdecoding,
RS codes of a given code rate are capable of achieving a
signicantly higher coding gain than without erasure
in-formation, and that the MOR-TT techniqueoutperforms
boththeRTTas wellastheOTTtechniques.
II. System andChannel Model
TheDS-CDMAsystemusingM-aryorthogonal
modula-tionconsideredin thispaperhasbeendescribedin[5],[6],
[7]. Hence, readersinterestedin the details ofthe system
are referred to [7], [5], [6]. We assume that there are K
activeuserstransmittingtheirsignalsasynchronously
acti-vating theircorrespondingorthogonalsignalsand
spread-ing sequences. The channel betweenthe transmitter and
its corresponding receiver is a multipath fading channel,
Rayleigh-The receiver schematic of the studied DS-CDMA
sys-tem including noncoherent M-ary demodulation,
multi-path diversity combining, maximum-likelihood detection
(MLD), MOR-TT-based erasure insertion, deinterleaving
andRSEAEdecodingisshowninFig.2of[6]. The
square-law-basednoncoherentM-arydemodulatorhasastandard
form,and henceit isthesameasthat used in[7]. Hence,
theinterestedreadersarereferredto[7]forfurtherdetails.
Inthereceiver,multipathdiversitycombininginvokingthe
EGCortheSCprinciplewasimplementedinthe`EGCor
SC'block. TheMLDblockselectsthelargestfromitsinput
variables and computes thevalue of - namely the ratio
of the`second'largestto thelargestof theMLD's inputs.
Followingthe MLD stage,the next blockmay output an
M-ary RScodesymbolorinsertanerasure. Finally,after
symbol-baseddeinterleaving,theRSdecoderinvokesEAE
decodingandthenoutputsthereceivedinformationbits.
III. Erasure Insertion Schemes
In thissection the three erasureinsertionschemes
con-sidered in this paper are dened. Let U i 1 ;U i 2 ;:::;U
i
M
representthedecisionvariablesaftercombiningthe
multi-pathcomponentsusingEGCorSC,whichformtheinputs
oftheMLDblockinFig.2of[4]. Weexpressthemaximum
andthe`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 ; (1) Y 2 =max 2 U i 1 ;U i 2 ;:::;U
i
M
: (2)
Furthermore, let H
1
and H
0
represent the hypotheses
of correct decision and erroneous decision, respectively,
concerning an M-ary symbol using the MLD algorithm.
Hence,wehaveP(H
0
)=1 P(H
1 ).
In the context of the OTT, the decision variable
sub-jectedtoanerasureinsertionisY
1
,i.e.,theactual
demod-ulatoroutputisobserved. LetY
T
beathresholdassociated
withmakinganerasuredecisionbasedontheOTT.Then,
ifY
1 Y
T
,theassociateddemodulatedsymbolisreplaced
by an erasure. Otherwise, if Y
1 > Y
T
, the demodulator
outputsaRScodesymbol.
TheratioinvolvedinViterbi'sRTTisdenedastheratio
of the maximumto the `second'maximum [1], or
equiva-lently, it canbedened as theratio ofthe `second'
maxi-mumtothemaximum,which canbeexpressedas
= Y
2
Y
1
; 01: (3)
Itwasshownin[1],[4]thatthedemodulatedsymbols
hav-ingarelativelylowratioofaremorereliable,thanthose
having relatively high values of . Consequently, a
pre-set threshold
T
can be invoked, in order to erase these
low-reliabilitysymbolsassociatedwitharatioof
T .
In the context of the joint MOR-TT [6], the erasure
insertion is based on the observation of both the
max-imum Y
1
of Eq.(1) and the ratio of Eq.(3).
Speci-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. 1. The 2D joint PDFsof f
Y
1 ;
(y;rjH
1 ) and f
Y 1 ; (y;rjH 0 )
forthe EGCschemeusing the parametersof M = 64, L=3,
b
=5dBandK=1overdispersiveRayleigh-fadingchannels.
cally,letf
Y1; (y;rjH
1 )andf
Y1; (y;rjH
0
)bethejoint
two-dimensional(2D) PDFs ofthemaximumY
1
and theratio
associatedwith the MOR-TT, conditionedonboth the
correct detection and erroneousdetection of M-ary
sym-bols. It canbe shown that ifa demodulated symbol has
amaximumoutputvalueof Y
1
and aratioof,that had
fallenin themainrangeoff
Y1; (y;rjH
0
),thesymbol
con-cernedmustbealow-reliabilitysymbolandhenceitmust
bereplacedby anerasure. Bycontrast, ifademodulated
symbolhasvaluesof Y
1
and that hadfallen outsidethe
main rangeof f
Y
1 ;
(y;rjH
0
), then this symbolis likelyto
be correct and hence the corresponding RS code symbol
canbeforwardedto theRSdecoder. Consequently,in
or-der to erase the low-reliability RS coded symbols in the
context of MOR-TT, we assumethat Y
T and
T
are two
thresholds, which activate an erasure insertion, whenever
Y 1 Y T and T .
IV. Statistics Properties of MOT-TT
Thepropertiesof theproposed MOR-TTbasederasure
insertion schemes can be studied with the aid of the 2D
PDFs of f
Y
1 ;
(y;rjH
1 ) and f
Y
1 ;
(y;rjH
0
) shown in Fig.1
andFig.2associatedwithEGCandSC,respectively. The
rangeofparametersusedis shownin thecaptions. Inthe
gures,M isthedynamicrangeofM-arysymbols,L
rep-resentsthe numberof diversity components combined, K
isthenumberofuserstransmittingsimultaneouslyand
-nally,
b
istheaverageSignaltoNoiseRatio(SNR)perbit,
whichwasobtainedbycomputing
b =L c =log 2 M,where log 2
M isthenumberofbitspersymbol. Fromthe2D
sur-faces ofFig.1 andFig.2 we observethat for thegiven
pa-rametersthepeakofthedistributionoff
Y1; (y;rjH
1 )is
lo-catedatarelativelyhighvalueofyorarelativelylowvalue
of r, while the peak of the distribution of f
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.2. The2DjointPDFsoff
Y
1 ;
(y;rjH
1 )andf
Y 1 ; (y;rjH 0 )for
theSCschemeusingtheparametersofM=64,L=3,
b =5dB
andK=1overdispersiveRayleigh-fadingchannels.
is located at a relatively low value of y and a relatively
high value of r. The above observations in turn imply
thatf
Y1; (y;rjH
1
)ismainlydistributedoverthearea
hav-ing relatively high values of y or relatively low values of
r, while f
Y
1 ;
(y;rjH
0
)is mainly distributed overthearea
havingrelativelylowvaluesofy andrelativelyhighvalues
ofr. Therefore, ifademodulated symbolhasamaximum
output valueof Y
1
and aratioof that had fallen in the
mainareaoff
Y
1 ;
(y;rjH
0
),thesymbolconcernedmustbe
alow-reliabilitysymbolandhencemustbereplacedbyan
erasure. By contrast, if a demodulated symbol has
val-ues of Y
1
and that had fallen outside themain areaof
f
Y1; (y;rjH
0
), thenthissymbolislikelyto becorrectand
hencethecorrespondingRScodesymbolcanbeforwarded
totheRSdecoder. Consequently,inordertoerasethe
low-reliabilityRS coded symbols, wecanassumethat Y
T and
T
aretwothresholds,whichactivateanerasureinsertion,
whenever Y 1 Y T and T
. Based onthe above
era-sure decision scheme, thesymbol erasure probability, P
e ,
andtherandomsymbolerrorprobability,P
t
,aftererasure
insertioncanbeformulatedas
P
e
= P(H
1 ) Z YT 0 Z 1 T f Y1; (y;rjH 1 )dydr
+ P(H
0 ) Z YT 0 Z 1 T f Y 1 ; (y;rjH 0 )dydr; (4) P t
= P(H
0 ) " 1 Z YT 0 Z 1 T f Y1; (y;rjH 0 )dydr # ;(5) wheref Y1; (y;rjH 1 )andf
Y1; (y;rjH
0
)arethejointPDFs
of Y
1
and corresponding to the EGC or SC diversity
combiningscheme,respectively.
If we assume that the symbol errors and symbol
era-sureswithinacodewordareindependentduetousing
suf-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
(
T
)
Amplitude
threshold
(Y
T
)
Fig.3. EGC:Codeworddecodingerrorprobabilityversusthe
ampli-tudethreshold,Y
T
andtheratiothreshold,
T
fortheRS(32,20)
FECcodeusing`errors-and-erasures(EAE)'decodingbasedon
theMOR-TTerasureinsertionschemeoverdispersiveRayleigh
fadingchannels.
cient channel interleaving, then the codeword decoding
errorprobabilityafterEAERSdecodingcanbeexpressed
intheformof [2]:
P W = N X i=0 N i X j=j 0 (i) N i N i j P i t P j e (1 P t P e )
N i j
; (6)
where j
0
(i) = maxf0;N K+1 2ig, while P
e and P
t
representthesymbolerasureprobabilityandrandom
sym-bol errorprobability before decoding, respectively, which
aregivenbyEq.(4)andEq.(5).
V. Examples of DecodingPerformance
In this section the performance of RS coded M-ary
CDMAusingEAEdecoding associatedwiththeproposed
MOR-TT erasure insertion scheme is estimated with the
aid of the equations from [6] and compared to that of
using `error-correction-only' decoding, and that of using
`erasures-and-erasures'decodingassociatedwitheitherthe
RTTorOTTerasureinsertionschemes[1],[5],[4].
As an example,Fig.3 and 4 show thecodeword
decod-ingerrorprobabilityof(6)overdispersiveRayleighfading
channels in the context of EGC (Fig.3) and SC (Fig.4)
using the proposed joint MOR-TT. The RS(32,20) code
denedovertheGaloisFieldGF(32)=GF(2 5
)
correspond-ingto5-bitsymbolswasusedandEAEdecodingbasedon
theproposedMOR-TTinsertionschemewasemployed. In
additionto theparameterswedened previously,inFig.3
and4thevariableN representsthenumberofchipsofthe
spreadingsequenceswithinabitinterval,andN isrelated
to the spreadingfactor, N
s , by N
s
= Nlog
2
0
2
4
6
8
10
12 0
0.2
0.4
0.6
0.8
1
-8
-7
-6
-5
Log
of
codeword
decoding
error
probability
Ratio
threshold
(
T
)
Amplitude
threshold
(Y
T
)
Fig.4. SC:Codeworddecodingerrorprobabilityversusthe
ampli-tudethreshold,Y
T
andtheratiothreshold,
T
fortheRS(32,20)
FECcodeusing EAEdecoding basedon theMOR-TTerasure
insertionschemeoverdispersiveRayleighfadingchannels.
the resultsof Fig.3and 4weobservethat there exists an
optimumthresholdvalueof
T or Y
T
, forwhich theEAE
decoding achieves the minimum codeworddecoding error
probability. Thisobservationinturnimpliesthatforgiven
values of M, for adiversity order of L, for K numberof
users, for a SNR per bit of
b
as well asfor a given RS
code,thereexistoptimumthresholdsY
T and
T
,forwhich
the EAE decoding using the joint MOR-TT erasure
in-sertion scheme achievesthe minimumcodeword decoding
errorprobability. This minimumcodeworddecodingerror
probability is lower, than that associated with using the
RTT alone or OTT alone. Note that for both the EGC
scheme ofFig.3 and fortheSC arrangementof Fig.4, the
point correspondingto Y
T
=0and
T
=1representsthe
codeworddecodingerrorprobabilityusingECOdecoding.
Therefore,wecanobservethatforbothEGCandSC,the
EAE decoding outperforms theECO decoding, ifthe
ap-propriatethresholdsareinvoked. However,ifthethreshold
Y
T
is toohighandsimultaneouslythethreshold
T istoo
low,toomanyerasureswillbeactivated,potentially
eras-ingcorrectdemodulatedsymbols. Consequently,the
code-worddecodingerrorprobabilityusingEAEdecodingmight
besignicantlyhigher,thanthat usingECOdecoding.
InFig.5and6weevaluatedthecodeworddecodingerror
probability of a RS coded M-ary DS-CDMA system
em-ployingeitherECOor EAEdecodingfortheEGC(Fig.5)
andSC(Fig.6)schemesusingdiversityordersofL=1;2;3
and 4. Inthecomputations weassumed that theoptimal
thresholds,Y
T and
T
,wereemployedforanygivenvalue
oftheaveragereceivedSNRperbit,
b
. Theremaining
pa-rameterswerealsospeciedinthegures.Theresultsshow
thatunderdispersiveRayleighfadingconditions,while
as-suming a constant average SNR per bit and a constant
Fig.5. EGC: Codeworddecodingerrorprobabilityversusthe
av-erageSNRperbit,
b
fortheRS(32,20)FECcodeusing
`error-correctiononly(ECO)'decodingandEAEdecodingwith
param-etersofM=N =32,K=20,K=10,N=128,andadiversity
orderofL=1;2;3;4overdispersiveRayleighfadingchannels.
numberofmultipathdiversitycomponentscombined-i.e.,
aconstantcomplexity-thedecodingalgorithmsusingEAE
decoding based on the joint MOR-TT erasure insertion
scheme outperformECOdecoding. Asshownin Fig.5 for
EGCusingdiversityordersofL=3and4andatthe
code-worddecodingerrorprobabilityof10 6
,theEAEdecoding
scheme canachieveagain ofabout3.3dB and 2.3dB,
re-spectively, overthe ECO decoding scheme. Similarly, in
thecontextoftheSCschemeofFig.6usingagiven
diver-sity order, the EAE decoding scheme can also achieve a
signicant gainoverthe ECOdecoding scheme.
Further-more,theresultsshowthatforagivennumberofcombined
paths,LandforagivenSNRperbit,
b
theEGCscheme
hasalowercodeworddecodingerrorprobability, thanthe
SC scheme. This isbecauseEGCis theoptimal diversity
combining scheme for a noncoherent demodulation
tech-nique.
>From thepreviousresultsweconcludethat bothEGC
andSCexhibitsimilarcharacteristicsbothintermsoftheir
PDFsandintermsoftheassociatedcodeworddecoding
er-ror probabilities, although theEGC outperforms the SC.
Therefore,inFig.7 weusedEGCasanexampleand
com-paredtheEAERSdecodingperformanceusingthe
MOR-TT,RTTandOTT.TheresultsfortheproposedMOR-TT
were evaluated from Equationsin [6], while for the RTT
and OTTfrom theequationsderivedin [4]and[5]. From
the results we observe that the proposed MOR-TT
era-sureschemeoutperformsboththeRTTandOTT erasure
insertionschemes.
VI. Conclusions
In this contribution the performance of RS coded
SNRperbit,
b
fortheRS(32,20)FECcodeusingECOdecoding
and EAEdecoding withparametersof M =N =32,K=20,
K = 10, N = 128, and a diversity orderof L= 1;2;3;4 over
dispersiveRayleighfadingchannels.
RS decoding hasbeeninvestigatedand compared to that
using ECOdecoding aswellasto that usingEAERS
de-coding based on the low-complexity, low-delay RTT and
OTT erasureinsertionschemes,when noncoherentM-ary
orthogonal modulation and diversity reception using the
EGCorSCschemeswereconsidered. Weassumedthatthe
transmitted signalswere subjectedto dispersiveRayleigh
fading. The distributions of the joint PDFs associated
with the MOR-TT have been studied and their
proper-ties have been analyzed. The proposed MOR-TT
tech-nique was compared to the RTTand OTT erasure
inser-tion techniques. Specically, theperformance ofEAERS
decodingemployingtheMOR-TTwascomparedtothatof
ECORSdecodingwithoutusingside-information. The
nu-mericalresultssuggestedthat when using EAEdecoding,
RS codes of agiven code rate can achieve asignicantly
highercodinggainthanwithouterasureinformation,
pro-vided that appropriate thresholds were invoked.
There-fore, DS-CDMA systemsusing the proposed EAE
decod-ingcouldsupportmoresimultaneoususersatagivenerror
performance, than without erasureinformation.
Further-more,the numericalresultsdemonstratedthat the
MOR-TT technique outperforms both the RTT and the OTT
techniques.
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