Assisted Multiuser Detection for Synchronous
CDMA Systems
KaiYen and Lajos Hanzo
Dept. of ECS, Univ. of Southampton, SO17 1BJ, UK.
Tel: +44-703-593 125; Fax: +44-703-594 508
Email: [email protected]; http://www-mobile.ecs.sot on.a c.uk
Abstract|Aspatialdiversityreceptionassistedmultiuser
CDMAdetectorbasedongenetic algorithms(GAs)is
pro-posed. Two dierentdiversityselectionstrategiesare
con-sidered. In our rst approach the so-called individuals of
the GA are chosen for selection based purely on thesum
of theircorresponding gures ofmerit associated withthe
individualantennas,anapproachwhichisanalogousto
em-ploying the conventional log-likelihood function (LLF) for
diversityreception. According toour second strategy, the
individualsarechosenforselectionbasedon theconceptof
theso-calledParetooptimality,whichusestheinformation
fromtheindividualantennasindependently. Computer
sim-ulationsshowedthattheGAsemployingthelatterstrategy
achievealowerBERascomparedtotheformerapproach.
I. Introduction
It is well known that the multiple access interference
(MAI)presentincodedivisionmultipleaccess(CDMA)[1,
2] systems can seriously deteriorate the quality of
recep-tion. An attractive solution for eliminating or reducing
theeectsofMAIismultiuserdetection[3]. Theoptimum
multiuser detector [4] { which is based on the maximum
likelihood(ML)ruleinthecontextofsynchronousCDMA
systems { searches exhaustively for the specic sequence
oftheusers'transmittedbitsthatmaximisestheso-called
log-likelihoodfunction(LLF).Sincethenumberofpossible
bitsequencecombinationsisexponentiallyproportionalto
the numberof users,the optimummultiuser detector has
acomputational complexitythat is exponentially
increas-ing with the numberof users. Hence it is impractical to
implement. Thislimitationledto numerousso-called
sub-optimal multiuser detection proposals, highlighted in [5]
and in thereferences therein,which sacriceperformance
forthesakeofareducedcomplexity.
Geneticalgorithms (GAs) [6,7]havebeenemployedfor
solving many complex optimization problems in
numer-ous elds. While GAsare by nomeans perfect, i.e. they
do not always nd the optimal point in the optimization
space,theyareveryeÆcientinattainingnear-optimal
solu-This work has been performed in the framework of the
Pan-Euroepan IST project IST-1999-12070 (TRUST), which is partly
fundedbytheEuropeanUnion. Theauthorswouldliketo
acknowl-edge the contributions of their colleagues, although the views
ex-pressedarethoseoftheauthors.
ThenancialsupportoftheEPSRC,Swindon,UKisalsogratefully
acknowledged.
tionssignicantlyfaster,thanconventional point-by-point
exhaustive search techniques, especially in large solution
spaces. GA-basedmultiuser detection has been rst
pro-posedin[8,9],wheretheanalysiswasbasedontheAWGN
channel without invoking diversity techniques. In [10],
Rayleighfadingchannelswereconsidered,again,inthe
ab-senceofdiversitytechniques.
Inthiscontribution,wepresentanovelapproachto the
problem of multiuser detection in DS/CDMA over
at-fading channels assisted by antenna diversity [11] based
on a genetic algorithm innovation. The antennas are
as-sumed to be suÆcientlyseparated such that the received
signalsat theantennas are faded independently, resulting
in anindependent LLF foreach antenna. This imposes a
problem onthe optimization process due to the fact that
whileaspecic bitsequencemayoptimizetheLLF ofone
antenna, the samebit sequence may notnecessarily
opti-mizetheLLFoftheotherantenna. Inordertoresolvethis
dilemmatwodierentGA-baseddiversityselection
strate-gies areconsidered. Inourrst approach, theindividuals
of the GA are chosen for selection based purely on the
sum of their correspondinggures of merit from the two
antennas,anapproachwhichis analogousto invoking the
conventional LLF for diversity reception [12]. According
tooursecondstrategy,theindividualsassociatedwiththe
GA are chosen for selection based on the concept of the
so-calledPareto optimality[6],whichusestheinformation
fromtheantennasindependently.
This paperis organizedasfollows. Section II describes
our synchronous CDMA system communicating over
un-correlated non-frequency-selective fading channels using
twoantennas. SectionIII describes theGAsused for
im-plementing ourproposed detector in conjunction with
di-versityreception. Oursimulationresults arepresentedin
SectionIV,whileSectionV concludesthepaper.
II. System Description
We consider a symbol-synchronous CDMA system,
where K users are simultaneously transmitting binary
phaseshiftkeying(BPSK)modulatedbitshavingaperiod
T
b
. Eachbitisspreadbyauser-specicsignaturesequence
having achip period T
c
, hencethe processing gainof the
systembecomesN
c =T
b =T
Estimation
Channel
Estimation
Channel
LLF
Computation
LLF
Computation
Diversity antenna 1
Diversity antenna 2
Matched
filter
User 1
Matched
filter
User K
Matched
filter
User 1
Matched
filter
User K
GA
Processing
r1(t) r2(t) z1;1 zK;1 z1;2 zK;2 C1 C2 1 ~ b (y) p 2 ~ b (y) p ~ b (y) p ~ b (y) p ^ b1 ^ bKFig.1. Blockdiagramofthereceivermodel
separatedspatially, such that thesignalsreceivedare
sta-tistically independent. Asumming ideal lowpass ltering,
the baseband received signal at the ith antenna is given
by:
r
i (t)=a
T
(t)C
i b+n
i
(t); 0t<T
b
(1)
where a(t) = [ a
1 (t)a
2 (t):::a
K (t)]
T
is the
sig-nature sequence vector for the K users, =
diag p 1 p 2 ::: p K
is the received bit energy matrix
of the K users during the bit-interval considered, C
i = diag 1;i e j 1;i 2;i e j 2;i ::: K;i e j K;i
is the CIR matrix
describing a frequency-nonselective slowly Rayleigh
fad-ing channel of each of the K users, b = [ b
1 b
2 :::b
K ]
T
is
thetransmittedbit vectorofthe K usersin thebit
inter-valconcernedandn
i
(t)isthezero-meancomplexAdditive
WhiteGaussianNoise(AWGN)withindependentrealand
imaginarycomponents,each having adouble-sidedpower
spectraldensityofN
0
=2. Theaveragereceivedbit
energy-to-noiseratioofthekthuserisgivenby:
k = E 2 k ;1 +E 2 k ;2 k ; (2) where k = k =N 0
isthe received bit signal-to-noiseratio
(SNR) in theabsenceof fading. Thepath amplitudes are
normalized, such that E h 2 k ;1 i +E h 2 k ;2 i
= 1 for k =
1;2;:::;K.
Ateachantenna,abankofltersmatchedtothe
corre-sponding set of the users' signaturesequences is sampled
attheendofeachbitinterval. Hence,theoutputz
i ofthe
matchedlterbankattheithdiversityantennaisgivenby
thevector:
z i =[z 1;i z 2;i :::z
K;i ]
T
=R C
i b+n
i
; (3)
where R = R
Tb
0 a(t)a
T
(t) is a KK user signature
se-Based on theobservation vector z
i
givenin Eq. (3), it
can be shown that the LLF for the ith antenna is given
by[13]:
i
(b)=2< n b T C i z o b T C i R C i
b ; fori=1;2: (4)
The decisionrule for the optimum multiuser detector
as-sociated with the ith antenna is to choose the bit vector
^
b, which maximisesthe LLF givenin Eq. (4). Hence, the
estimated transmitted bit vector of the K users is given
by:
^
b=arg max b [ i (b )] : (5)
Sincethechannel characteristicsforeachantennaare
sta-tisticallyindependent,wehavetypically
1
( b)6=
2 (b)for
theLLFsofthetwoantennae. Incertainscenariossuchas
duringdeepfades,theaboveinequalityimpliesthat :
arg max b [ 1 (b )] = ^
b6=arg max b [ 2 (b) ] (6)
orviceversa. Thiscreatesaso-calledmulti-objective
opti-mizationproblem,sincetheoptimizationofbothLLFscan
sometimesleadtotwopossiblesolutions. Notethatin the
conventionaloptimumdetector[12],theLLFs
correspond-ing tothe twodiversityantennas are summedin order to
produceascalargureofmerit,asgivenby[12]:
(b) = 2 X i=1 i (b ) = 2< n ~ b T ~ C ~ ~z o ~ b T ~ C ~ ~ R ~ ~ C ~
b ; (7)
where ~
b=[b
1 b
1 :::b
K b
K ]
T
and~z=[z
1;1 z
1;2 :::z
K;1 z
K;2 ]
T
are vectors of dimension 2K 1, while ~ C = diag 1;1 e j1;1 1;2 e j1;2 ::: K;1 e jK;1 K;2 e jK;2 and ~ = diag p 1 p 1 ::: p K p K
are diagonal matrices of
di-mension 2K 2K. The decision rule is then to nd the
estimatedtransmitted bitvector ^
bthatmaximizes(b ).
Inthenextsectionwewillhighlightthephilosophyofour
GA-assisted diversity-aided multiuser detector with
em-phasisonthediversityselectionstrategy,inordertodetect
theusers'transmitted bits.
III. GeneticAlgorithmbased Multiuser
Detectionwith DiversityReception
GAs [6,7] can be invoked in robust global search and
optimization procedures that are well suited for solving
complex optimization problems. In this contribution, we
willemployGAsinordertodetecttheestimated
transmit-ted bit vector ^
b, where the so-called objectivefunction is
denedbytheLLFofEq. (4)forthetwoantennas.
GAscommencetheirsearchfortheoptimumsolutionat
theso-called0thgenerationbyrandomlycreatingP
num-berofsequences,eachconsistingofKantipodalbits,where
Pisknownasthepopulationsize,whichmayassumevalues
between1and2 K
represents a possiblesolution of ^
b. We shall express the
pthindividualhereas ~
b (y)
p =
h
~
b (y)
p;1 ;
~
b (y)
p;2 ;:::;
~
b (y)
p;K i
,where y
denotestheythgeneration.
Associated with each individual wehave twoguresof
merit, which are associated with the two antennas and
are referred to as the so-called tness values of the GA.
These antenna-specic tness values are derived by
eval-uating Eq. (4) with b as dened by the individual. The
diversity-basedtnessvalueofthepthindividualisdenoted
asf
~
b (y)
p
= h
1
~
b (y)
p
2
~
b (y)
p i
, which is afunction
of the LLFs associatedwith the twoantennas. Based on
the evaluated diversity-specic tness, the population of
P individuals at theyth generationof theGAevolves,in
order to create anew populationof P individuals for the
(y+1)th generation. The above-mentioned evolution of
apopulationinvolvesseveralprocesses,whicharereferred
to in GA parlance as selection, crossover, mutation and
elitism[6].
As suggested by the terminology, the selection process
selectstwoso-calledparents fromamatingpoolconsisting
ofT individuals{where2T <P {in orderto produce
twoso-calledospringforthenextgenerationpopulation.
Weshalldenotetheindividualsinthematingpoolas
b (y)
q
forq=1;:::;T. TwodierentGA-assisteddiversity
com-biningstrategiesareevaluatedhere,inordertodetermine,
whichT outofthePindividualsinapopulationareplaced
inthematingpool. Astraightforwardstrategyistosimply
sumthe twoantenna-specictnessvaluesofeach
individ-ual in order toproduce adiversity-combined tness value,
denoted here as
~
b (y)
p
=
1
~
b (y)
p
+
2
~
b (y)
p
, which
was stated explicitly also inEq. (7), for p=1;:::;P.
In-dividuals having the T highest diversity-combined tness
valuesinthepopulationarethenplacedinthematingpool.
Hence ourGA-aidedoptimizationemployingthis strategy
is basedontheconventional LLF-assisteddiversity
recep-tion [12]. Intuitively, we can see that this strategy relies
moreontheexploitationratherthanontheexplorationof
the solutionspace, since thealgorithm always favours
in-dividualshavingthehighesttness,particularlyifT P.
On the other hand, choosing a higherT value may allow
certain low-tness individuals to beplaced in the mating
poolandhencemayreducetherateofconvergence.The
ef-fectofthematingpoolsizeT isinvestigatedinSectionIV.
Oursecond GA-assisted diversitycombiningstrategyis
basedontheconceptoftheso-calledPareto optimality[6].
Thisstrategyfavourstheso-callednon-dominated
individ-uals andignorestheso-calleddominated individuals. The
pth individual is considered to be dominated by the qth
individual i the latterhas ahigher gure of merit, such
astheLLF ofEq.(4),whichisformulatedas[14]:
8i2f1;2g:
i
~
b (y)
q
i
~
b (y)
p
^
9i2f1;2g:
j
~
b (y)
q
>
j
~
b (y)
p
: (8)
IfanindividualisnotdominatedinthesenseofEq.(8)by
crossover mask :
1
-1
1
1
1
1
1
1
1
-1 -1 -1 -1 -1 -1 -1
1
1
1
0
0
1
0
0
-1 -1 -1
1
1
-1
1
1
1
1
1 -1 -1
1 -1 -1
Offspring
Parents
==>
==>
~
b (y)
i :
~
b (y)
j :
Fig.2. Anexampleof uniformcrossoverbetween two parent bits
strings
considered tobenon-dominated. According to oursecond
diversity combining strategy, all the non-dominated
indi-viduals are placed in the mating pool. Hence the valueof
T in this case is not xed, since it depends on the
num-berofnon-dominated individuals. Observethat thelatter
strategyusestheinformationfrombothantennas
indepen-dently, in order to decide which individuals areplaced in
the matingpool. Bycontrast, our former strategy based
itsdecisionsonasinglemetricbycombiningthe
antenna-specicguresofmeritaccordingtoEq.(7).
Theindividuals in the matingpool areselected as
par-ents according to a probabilistic function based on their
correspondinggureofmerit
b (y)
q
. Inordertoprevent
premature convergence to a `local optimum' without
ex-ploringtheglobalsolutionspace, theso-calledsigma
scal-ing [7] is employed. Under sigma scaling, the selection
probabilityp
b (y)
q
ofanindividualisafunctionofitsown
tnessaswellasthatof thematingpool'smeantness
and its associated standard deviation
, as formulated
below[7]:
p
b (y)
q
= 8
<
: 1:0+
b (y )
q
2
if
6=0
1:0 if
=0;
(9)
where
= 1
T T
X
q=1
b (y)
q
;
=
v
u
u
t P
T
q=1 h
b (y)
q
i
2
T 1
:
(10)
Theantipodalbitsoftheparentvectorsarethenexchanged
using the so-called uniform crossover [15] process with a
probability p
c
in order to produce twoospring.
Speci-cally,uniformcrossoverinvokesaso-calledcrossovermask,
whichisasequenceconsistingofKrandomlygenerated1s
and0s. Antipodal bitsareexchangedbetweenthepairof
parentsat locations correspondingto a1in thecrossover
mask. An illustrationof theuniform crossoverprocess is
shownin Fig.2. Theselectionof parentsfrom themating
poolis repeated until anew population of P ospring is
producedin ordertoperformthecrossoverprocess.
Themutationprocessreferstothealterationofthevalue
of an antipodal bit in the ospring from 1 to -1 or vice
versa,withaprobabilityp
m
. Finally,under elitism[7],we
identify the lowest-merit ospring in the population and
replaceitwiththehighest-meritindividualfrom the
mat-ingpool. Thiswillensurethatthehighest-meritindividual
ispropagatedthroughouttheevolutionprocess.
2
3
4
5
6
7
8
9
10
T
10
-5
2
5
10
-4
2
5
10
-3
2
5
10
-2
2
5
10
-1
2
5
10
0
BER
P = 30
P = 20
-k
=10dB for k=1,...,K
k
- =20dB for k=1,...,K
Y = 10
K = 10
Selection strategy: S1
Fig.3. TheBERperformanceofthe GA-basedmultiuserdetector
usingselectionstrategyS1asafunctionofthematingpoolsizeT
withpopulationsizesP =20;30usingbinaryrandomsignature
sequencesoflength31at
k
=10;20dBfork=1;2;:::;K.The
GA parametersused are the probability of crossovergivenby
p
c
=1, the probabilityofmutation specied by p
m
=0:1and
theevolutionwasterminatedafterY =10generations
is the detectedK numberof users' bit sequenceassociated
with the bit-interval considered. For a given population
size P, thenumberof LLF evaluationsrequired to detect
^
bisequalto PY 2foratwin-antennabaseddiversity
scheme. Hence thecomplexityoftheproposeddetectoris
not directly related to the number of users. The values
of P and Y canbe adaptivelyselected, in order to nda
trade-o betweencomputationalcomplexityandoptimum
performance.
IV. Simulation Results
Inthissectionoursimulationresultsarepresented,in
or-dertocharacterizetheBitErrorRate(BER)performance
of the GA-basedmultiuser detector employing the above
twoGA-baseddiversity selectionstrategies highlightedin
theprevioussection. Thestrategybasedonthesumofthe
guresofmeritfrombothantennasisdenotedasS1,while
thestrategy basedontheParetooptimalityisdenoted as
S2. Alltheresultsinthissectionwerebasedonevaluating
theBERperformanceofabit-synchronous10-userCDMA
system employing twin-antenna based diversity reception
over uncorrelated slowly Rayleigh fading channels. The
processing gainwas N
c
=31and thesignature sequences
wererandomlygenerated.
Fig.3showstheeectsofthematingpoolsizeT onthe
BER performance using the selectionstrategy S1. While
the eects of the pool size were negligible at an SNR of
k
=10dB,thedeteriorationoftheBERperformanceupon
increasing T is signicant at
k
= 20dB and P = 20.
Furthermore, we can see that the value of T required
for achievingthe best BERperformancewas dierent for
diÆerent valuesof P, although thedependence onT was
notpronounced. AccordingtoFig.3,T =3andT =5gave
thebest resultsfor P =20and P =30, respectively. We
willusethisresultforoursubsequentperformancestudies.
Fig.4showstheBERperformanceagainst theSNR
0
5
10
15
20
25
30
35
40
k
for k = 1,...,K
-10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
BER
single-user bound
P = 30
P = 20
K = 10
Y = 10
T = 3 for S1, P = 20
T = 5 for S1, P = 30
S1
S2
S1
S2
Fig. 4. The BER performance of the GA-basedmultiuser
detec-toremployingtheselectionstrategiesofS1andS2with
popula-tionsizesofP=20;30usingbinaryrandomsignaturesequences
oflength 31and supporting K =10users. The average
re-ceivedenergyattheantennaswasassumedtobeequal,
i.e.E
2
k ;i
=0:5fori=1;2andk=1;:::;K.TheGA
param-etersusedaretheprobabilityofcrossovergivenbyp
c
=1,the
probabilityofmutationspeciedbypm=0:1andthe evolution
wasterminatedafterY =10generations
fortheGA-basedK=10-userdetectoremployingselection
strategy S1 and S2 with equal average received energy at
the twoantennas,i.e. forE h
2
k ;1 i
=E h
2
k ;2 i
=0:5.
Per-fectpowercontroland CIRestimation was assumed. The
single user bound, which assumed equalaverage received
energyatbothantennas,wascomputedusingthefollowing
equation[13]:
P
2 =
1
2 (1 )
2
(2+); (11)
where= q
k
1+
k
. Anerroroorisobservedfortheresults
shown in the gure. As mentioned in Section I, GAs do
not always nd the optimal solution. In this multiuser
scenarioitwastheGAthatcausedtheerroroorandnot
the multiuser interference. It is seen in Fig. 4 that the
BERperformanceimproved,whenthepopulationsizewas
increased from P = 20 to P = 30. However, this also
increasedthecomputationalcomplexity. Hencethevalues
of P can be selected,in order to nda trade-o between
computational complexity and performance. We also see
from Fig. 4 that the GA employing S2 performs better,
exhibiting a lower error oor, as compared to employing
S1. Nevertheless,bothstrategieswerecapableofmatching
the single-userbound performance up to
k
=16dB and
k
=24dBforP=20andP =30,respectively.
Wethen investigated theBERperformance ofthe
GA-based multiuser detector employing selection strategy S1
andS2withunequalaveragereceivedenergyatthe two
an-tennas,settingE h
2
k ;1 i
=0:8andE h
2
k ;2 i
=0:2. Perfect
powercontrolandCIRestimationwasassumedagain. The
results were shown in Fig. 5in comparison to the
single-user bound given by Eq. (11). Again, we can see that
0
5
10
15
20
25
30
35
40
k
for k=1,...,K
-10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
BER
single-user bound
P = 30
P = 20
K = 10
Y = 10
T = 3 for S1, P = 20
T = 5 for S1, P = 30
S1
S2
S1
S2
Fig.5. TheBERperformanceofthe GA-basedmultiuserdetector
employingselectionstrategiesS1andS2withpopulationsizesof
P =20;30usingbinary random signaturesequences of length
31andsupportingK=10users. Theaveragereceived
en-ergy at the antennas was assumed to be unequal with
E
2
k ;1
=0:8 and E
2
k ;2
=0:2. TheGAparametersused
aretheprobabilityofcrossovergivenbyp
c
=1,theprobabilityof
mutationspeciedbypm=0:1andtheevolutionwasterminated
afterY =10generations
0
2
4
6
8
10
12
k
/
1
in dB for k=2,...,K
10
-5
2
5
10
-4
2
5
10
-3
2
5
10
-2
2
5
10
-1
2
5
10
0
BER
P = 30
P = 20
K = 10
Y = 10
T = 3 for S1, P = 20
T = 5 for S1, P = 30
1
= 16 dB
-S1
S2
S1 & S2
P = 20
P = 30
Fig.6. Relativenear-farresistenceof the GA-baseddetector
em-ploying selection strategies S1 and S2 with population sizes
P =20;30usingbinary random signaturesequences of length
31at
1
= 16dBand supporting K = 10 users. Theaverage
receivedenergyattheantennaswereassumedtobeequal. The
GA parametersused are the probability of crossovergivenby
pc=1, the probabilityofmutation specied by pm =0:1and
theevolutionwasterminatedafterY =10generations
to strategyS1.
Finally,thenear-farresistenceoftheGA-basedmultiuser
detectorisdemonstratedinFig.6forthedesireduser. The
averageenergy-to-noiseratioofthedesireduser
1
issetto
16 dB,while theenergiesofall otheruserswerevariedin
therange of0-12dBhigher,thanthat ofthe desireduser.
Perfect CIR estimation was assumed and the average
re-ceived energy at both antennas were similar. Wecansee
that atpopulationsizeP =20,theBERperformance
de-terioratesslightly,astheinterferingusers'energybecomes
higherrelativetothedesireduser. Ontheotherhand,the
BERperformanceassociatedwith P =30remainsalmost
thesame,evenwhentheinterferingusers'energyis10dB
V. Conclusions
Inconclusion,wedevelopedasuboptimalmultiuser
de-tector based on GAs in order to circumvent the
compu-tationalcomplexityproblem presentin theoptimum
mul-tiuserdetector [4]. Tomitigate theeects of fading, dual
antennadiversitytechniques were used. Twodiversity
se-lectionstrategieswerehighlightedfortheGAs. Inourrst
solutionthematingpoolwasformed basedon the
combi-nationofthestatisticsderivedfromthediversityantennas
andwehadaxedmatingpoolsize. Accordingtoour
sec-ond strategy,thestatisticsweretreatedindependently, in
orderto selectthenon-dominated individualsto form the
mating pool. Hence, the matingpool size wasnot xed.
WehaveshownthatGAsemployingthelatterstrategy
al-ways exhibit a lower BER compared to those employing
the former strategy. We have also shown that the BER
performancecanbeimprovedbyincreasingthepopulation
size. TheGA-baseddetectorisalsonear-farresistent. Our
futureworkwillattempttoextendtheseadvancesto
adap-tivebeam-steeringassisted asynchronousCDMA systems,
aswellastoinvokingspace-timecoding.
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