Low Complexity Joint Iterative Equalization and Multiuser Detection in Dispersive DS-CDMA Channels

0.23 MB·English

by  Husheng Li

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Reduced Complexity Joint Iterative Equalization
and Multiuser Detection in Dispersive DS-CDMA
Channels
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Husheng Li and H. Vincent Poor
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Abstract

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Communications in dispersive direct-sequence code-divisionmultiple-access channels suffer from intersymbol
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c andmultiple-accessinterference,whichcansignificantlyimpairperformance.Jointmaximumaposterioriprobability
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equalization and multiuser detection with error control decoding can be used to mitigate this interference and to
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v achievetheoptimalbiterrorrate. Unfortunately,suchoptimaldetectiontypicallyrequiresprohibitivecomputational
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complexity.Thisproblemisaddressedinthispaperthroughthedevelopmentofareducedstatetrellissearchdetection
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algorithm,basedondecisionfeedbackfromchanneldecoders. Theperformanceofthisalgorithmisanalyzedinthe
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large-systemlimit.Thisanalysisandsimulationsshowthatthisreducedcomplexityalgorithmcanexhibitnear-optimal
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s performanceunder moderate signal-to-noise ratio and attains larger system load capacity than parallel interference
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v: cancellation.
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I. INTRODUCTION
Overthepasttwodecadestherehasbeenconsiderableresearchondirectsequencecodedivisionmultiple
access(DS-CDMA)communications,whichofferstheadvantagesofsoft-capacitylimit,inherentfrequency
diversity and high data rate [18], and which is the fundamental signaling technique of third-generation
(3G) cellular communications and other emerging applications.. However, DS-CDMA suffers from inter-
ference, particularly multiple-access interference (MAI), which is due to the non-orthogonality of different
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (email: {hushengl, poor}@princeton.edu). This
researchwassupportedinpartbytheArmyResearchLaboratoryunderContractDAAD19-01-2-0011andinpartbytheNewJerseyCenterfor
WirelessTelecommunications.
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users’spreadingcodes, andintersymbolinterference(ISI), whichiscaused bymultipathfadinginhigh-rate
systems. Itiswellknownthatmitigationofthesetypesofinterferencecansubstantiallyimprovesystemper-
formance. In recent years, muchprogresshasbeen achievedtowardsuchmitigationthroughtheapplication
ofequalization(EQ)and multiuserdetection(MUD).
Equalization can be roughly categorized into three classes. One class is based on maximum likelihood
(ML) detection [15], which can be implemented efficiently with the Viterbi algorithm (VA). A second
class is linear equalization based on some criterion, such as minimum peak distortion or minimum mean
squareerror(MMSE).Thethirdclassisdecisionfeedbackequalization(DFE),inwhichpreviouslydetected
symbols are used to cancel the intersymbol interference [15]. In multiuser detection, we can find the
counterparts for these three kinds of equalizers. In particular, ML based multiuser detection and MMSE
multiuser detection are both well known [24]. As a combination of both techniques, the problem of joint
ISI an MAImitigationis discussedin [3].
In recent years, the turbo principle, namely the iterative exchange of soft information among different
blocks in a communication system to improve the system performance, has been applied to equalization
and multiuser detection in channel coded systems, thus resulting in turbo equalization [11] and turbo mul-
tiuser detection [23]. In these algorithms, soft decisions from channel decoding are fed back to be used
a priori probabilities by a maximum a posteriori probability (MAP) based equalizer or multiuser detector
and enhance the performance iteratively. However, the computational cost of MAP based detection is pro-
hibitive for large numbers of users or long delay spread. Thus, it is necessary to reduce the complexity
of such iterative algorithms for practical applications. For memoryless synchronous multiaccess channels,
the MAP turbo multiuser detection can be simplified to parallel interference cancellation (PIC) [1], whose
performance can be enhanced with an MMSE filter [23] ora decorrelating interference canceller [10]. For
systems with memory, an alternative way to simplify these detectors is to reduce the number of states by
either truncatingthe channel memory to a fixed order or eliminatingthe states with reliable decisionsat the
channel decoder. The former strategy has been used in the equalization of ISI channels [6] [7] [13] and
in asynchronous multiuser detection of channel coded CDMA systems [16], while the latter scheme is of
dynamic complexity and is widely used in iterative decoding algorithms. In speech recognition [25], this
schemeisappliedtotrimtheacousticorgrammarnodeswithlowmetrics. Inearlydetectionbaseddecoding
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of parallel turbo codes [9], the trellis is simplified by splicing the state with reliable a priori probabilities.
Similar strategies have been applied to the iterative decoding of LDPC codes [8] and concatenated codes
[4].
InthispaperwestudychannelcodedDS-CDMAsystemsoperatingoverfrequencyselectivefadingchan-
nels. Althoughjointdetection and decodingcan achievehigherchannel capacity, we consideronlysystems
in which detection and decoding are separate due to the increased feasibility of such systems in practical
applications. In particular, we consider joint equalization and multiuser detection based on MAP detection
with decision feedback (MAP EQ-MUD) from the decoder. Similarly to MAP turbo equalization or mul-
tiuserdetection, theMAPEQ-MUD alsosuffers from prohibitivecomputationalcost. In thispaper, we first
decompose the multiuser trellis into single-user trellises, thus linearizing the number of states in terms of
the number of users. Then we apply both the state-reducing techniques in channels with memory, namely
shortening the channel memory to a fixed order and partitioning the decision feedback of channel decoders
into unreliable and reliable sets of symbols using a simple confidence metric. The states are constructed
with the unreliable set and the interference from the reliable set is cancelled using the soft decisions from
the decoder. We call this reduced complexity algorithm reduced state equalization and multiuser detection
(RS EQ-MUD).
Thispaperisorganizedasfollows. SectionIIintroducesthesystemmodel,inwhichthesignalmodeland
decoder are explained. The optimalMAP EQ-MUD algorithmis described in Section III, while Section IV
is focused ondevelopingtheRS EQ-MUD algorithm. Anasymptoticanalysisofthesystemperformanceis
carried outin Section V, and correspondingnumerical resultsare givenin Section VI. Final conclusionsare
drawnin Section VII.
II. SYSTEM MODEL
A. Signalmodel
For a channel-coded DS-CDMA system, let K denote the number of active users, N the spreading gain
andM thecodedsymbolblocklength. DenotethesymbolperiodandchipperiodbyT andT respectively,
s c
andnotethatT = NT . Foruserk,thebinaryphase-shiftkeying(BPSK)modulatedcontinuous-timesignal
s c
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at thetransmitterisgivenby
M
r˜ (t) = b (i)s (t iT )
k k k s
−
i=1
X
M N
= b (i) s(i)(n)Ψ(t iT nT ), (1)
k k − s − c
i=1 n=1
X X
whereb (i) isthei-th(binary)channel coded symbolsent by userk, Ψ(t) isthechip waveform,and s(i)(n)
k k
isthenormalizedbinaryspreadingcodeofuserk,whichsatisfies s(i)(n) = 1 . Thesuperscriptiins(i)(n)
k √N k
(cid:12) (cid:12)
implies that the spreading code varies with the symbol period, s(cid:12)ince lon(cid:12)g (aperiodic) codes are considered
(cid:12) (cid:12)
inthispaper.
The signal (1) passes through a frequency selectivefading channel whose impulseresponse is h˜ (t), and
k
thechanneloutputistheconvolutionoftheinputsignaland thechannel response:
r (t) = r˜ (t)⋆h˜ (t);
k k k
M N
= b (i) s(i)(n)g (t iT nT ), (2)
k k k − s − c
i=1 n=1
X X
where
g (t iT nT ) = Ψ(t iT nT )⋆h˜ (t). (3)
k s c s c k
− − − −
Assumingsynchronoustransmissionamongusers1,thesignalat thereceiveris givenby
K
r(t) = r (t)
k
k=1
X
K M N
= b (i) s(i)(n)g (t iT nT ). (4)
k k k − s − c
k=1 i=1 n=1
XX X
The received signal is sampled at the chip rate 1 , and the corresponding discrete output of the sampler at
Tc
chipperiod l is
K M N
y(l) = r(lT ) = b (i) s(i)(n)g ((l n iN)T )
c k k k − − c
k=1 i=1 n=1
XX X
K M
= b (i)h(i)(l iN), (5)
k k −
k=1 i=1
XX
1Notethatthisassumptionofsynchronoustransmissiondoesnotlimitthegeneralityofthismodel,sincedelayoffsetsbetweenuserscanbe
incorporatedintothechannelimpulseresponse˜h (l),...,h˜ (l).
1 K
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where
h(i)(l) = s(i)(l)⋆g (lT ). (6)
k k k c
Suppose the support of h(i)(l) is (0,(L 1)N), where L is the dispersion length. We can simplify (5) to a
k −
vector form. (For notational convenience, we henceforth use t to designate discrete time advancing at the
symbolrate.) Thisvectorisgivenby2
y(t) = (y(tN),y(tN +1),...,y((t+1)N 1))T ,
−
and
T
h(i)(j) = h(i)(jN),h(i)(jN +1),...,h(i)((j +1)N 1)) .
k k k k −
(cid:16) (cid:17)
Usingthevectorformand consideringthethermalnoiseat thereceiver,wewrite
K t
y(t) = b (i)h(i)(t i)+n(t), t = 1,2,...,M (7)
k k −
k=1i=t L+1
X X−
where n(t) is additive white Gaussian noise (AWGN) that satisfies E n(t)n(t)H = σ2I . The devel-
n N N
×
opmentintheremainingofthispaper willbebased on thisdiscretemo(cid:8)del(7). (cid:9)
B. Equivalentspreadingcode
Wetermh(i)(t i)theequivalentspreadingcodeofthei-thsymbolofuserkinthet-th(i t)symbolpe-
k − ≤
riod. Inordertoexplorethepropertiesoftheequivalentspreadingcode,weneedtoplacesomeassumptions
onthediscretechannel responseg (lT )in (6):
k c
Causality. g (lT ) = 0when l < 0.
k c
•
Normality. Weassumethat g (lT ) isacomplex-valuedGaussianrandom sequencewithzero mean
• { k c }l
and exponentiallydecayingvariance
σ2(l) = λke−λkNl , (8)
k N
where λk isthedecay factor andsatisfies e−λkL ≈ 0 and limN→∞ ∞l=1σk2(l) = 1.
Independence. We assumethatthescatteringcaused byfading isuPncorrelated, whichmeans
•
E g (lT )g (jT ) = 0, ifl = j ork = n. (9)
k c n c ∗
{ } 6 6
2T denotestransposition,Hdenotesconjugatetranspositionand*denotesconjugation.
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Thusg (lT ) and g (jT ) areindependentsincetheyarejointlyGaussian.
k c n c
In non-dispersive multiple-access channels, the cross correlation of the spreading codes plays a key role
in the system performance. Thus we need to discuss thecross correlation between the equivalent spreading
codes. Define the equivalent cross correlation between the equivalent spreading codes of user k and user n
tobe
H
ρtτ (i,j) = h(t)(i) h(τ)(j). (10)
kn k n
(cid:16) (cid:17)
Notethatρtτ (i,j)is arandomvariablesinceht(i) and hτ(j) are random.
kn k n
C. Channel decoder
The diagram of the system discussed in this paper is given in Fig. 1. At the transmitter, the information
symbolsare encoded, interleavedwith an infinitelengthinterleaverand spread. TheBCJR algorithm [2]is
used in the channel decoder, which followsa deinterleaver, to obtain the a posterioriprobability P(b (t) =
k
b yM), whereyM = y(t) t = 1,...,M . Thisprobabilitycan bedecomposedintotwoparts,
| 1 1 { | }
P(b (t) = b yM) P(b (t) = b y(t))ζt(b),
k | 1 ∝ k | k
where ζt(b) is the extrinsic information about b (t) from the other coded symbols. We use the extrinsic
k k
informationto constructthesoft decisionfeedback:
ˆb (t) = 2ζt(1) 1, (11)
k k −
whichisusedtocancelboththeISIandMAI.Theestimationerrorisdenotedby∆b (t) = b (t) ˆb (t);its
k k k
−
expectationiszero dueto symmetryand itsvariancecan beobtainedbysimulation.
III. OPTIMAL MAP TURBO EQ-MUD
In this section we develop the BCJR algorithm [2] for MAP EQ-MUD in a way similar to that used for
turbomultiuserdetection in [23].
First we define the state at symbol period t as the set of all symbols of all users from symbol period
t L+2 to t:
−
S = b (l) l = t L+2,...,t, k = 1,...,K . (12)
t k
{ | − }
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Thus for each symbol period we have 2(L 1)K states with which to construct a trellis. We call state m and
−
m compatiblewhenthestatecan transitfromm to m and denotethisconditionby m m.
′ ′ ′
⇒
Next wedefinesomeintermediatevariables[2]:
forward probability: α (m) = P(S = m,yt);
• t t 1
backward probability: β (m) = P(yM S = m);
• t t+1| t
transitionprobability: γ (m,m) = P(y(t),S = m S = m).
t ′ t t 1 ′
• | −
Theforward and backward probabilitiescan becomputedrecursivelyviatheequations
α (m) = α (m)γ (m,m), (13)
t t 1 ′ t ′
−
m′
X
and
β (m) = β (m)γ (m,m). (14)
t t+1 ′ t+1 ′
m′
X
Invoking Bayes’ formula, we can rewrite the transition probability as γ (m,m) = P(y(t) S = m,S =
t ′ t t 1
| −
m)P(S = m S = m). Sinceweassumethattheinterleaverhas infinitelength,thesymbolsofdifferent
′ t t 1 ′
| −
users anddifferent symbolperiodsare independent. Thuswehave
P(S = m S = m) = P(b (t) =˜b ,...,b (t) =˜b ˜b ,...,˜b m)
t t 1 ′ 1 1 K K 1 K
| − | ∈
K
= P b (t) =˜b ˜b m
k k k
| ∈
kY=1 (cid:16) (cid:17)
= ζt ˜b
k k
˜bkY∈m (cid:16) (cid:17)
, Λ (m), (15)
t
which can be regarded as the a priori probabilities of the symbols at symbol period t. When no apriori
informationisavailable,weassumethesymbolstobeuniformlydistributed. Sincethechanneldecoder can
providetheextrinsicinformationfromtheothercoded symbols,wecan feed thesoftoutputsofthedecoder
back as thea prioriprobability. Hence
Λ (m) 1 exp y(t)−yˆm,m′(t) 2 ifm m;
γt(m′,m) =  t √2πσn2 −(cid:13)(cid:13)(cid:13) 2σn2 (cid:13)(cid:13)(cid:13) ! ′ ⇒ ,

 0 ifm ; m.
′



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whereyˆm,m′(t) is theestimatedreceivedsignalduetothestatesm and m′:
K t
yˆm,m′(t) = ˜bk(i)h(kt)(t−i), ˜bk(i) ∈ m′ ∪m. (16)
k=1i=t L+1
X X−
When theforward and backward probabilitiesare available,wecan computethejointprobability
P S = m,yM = α (m)β (m), (17)
t 1 t t
(cid:0) (cid:1)
and thencomputethea posterioriprobability:
P b (t) =˜b yM P S = m,yM , (18)
k k| 1 ∝ t 1
(cid:16) (cid:17) ˜bXk∈m (cid:0) (cid:1)
where denotesproportionality. Theappropriatenormalizationis
∝
P b (t) = 1 yM +P b (t) = 1 yM = 1.
k | 1 k − | 1
(cid:0) (cid:1) (cid:0) (cid:1)
Toavoidthereuseoftheextrinsicinformation,weneedtocancelthea prioriprobability. Thereforethesoft
inputtothedecoder, denotedby (t), forsymbolt ofuserk is normalizedbythesoftdecisionfeedback:
k
L
P(bk(t)=1|yM1 )
(t) = (cid:20)P(bk(t)=−1|yM1 )(cid:21). (19)
k
L ζ(bk(t)=1)
ζ(bk(t)=−1)
h i
IV. REDUCED STATE EQ-MUD
A. Independenceassumptionand trellisdecomposition
Facingthesameproblemaseitherequalizationormultiuserdetection,theoptimalMAPEQ-MUDsuffers
from prohibitive computational complexity because the number of states increases exponentially with the
numberofusersK andthedispersionlengthL. Thus,theoptimalMAPEQ-MUDisprimarilyoftheoretical
valueand cannotbeimplementedformanypractical applications.
InthispaperwemitigatethecomplexityoftheMAPEQ-MUD,bydecomposingitstrellisintosingle-user
sub-trellises,thuslinearizingthenumberofstateswithrespecttothenumberofusers. Foruserk,wedefine
thesub-stateatsymbolperiodtas
Sk = b (t L+2),...,b (t) .
t { k − k }
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Thesub-statesbelongingtooneuserconstructasingle-usersub-trellis. Then anystatedefined in(12)isthe
union of the corresponding sub-states of all users. In order to distinguish sub-state and state, we use non-
bold fonts to designate the sub-states in the rest of this paper. Similar to Section III, we define the forward
and backward probabilitiesforthesub-stateofuserk:
αk(m) = P Sk = m,yt , (20)
t t 1
(cid:0) (cid:1)
and
βk(m) = P Sk = m,yM . (21)
t t t+1
(cid:0) (cid:1)
Compared with the analogous definition of Section III, the definition of the backward probability βk(m) is
t
slightlydifferenthere;thiswillfacilitatetheapplicationofAssumptionIV.1,whichfollowsimmediately,on
thebackwardprobability. IftheBCJRalgorithmcanbeconfinedtoeachsub-trellis,thenumberofsub-states
will be reduced to K2L 1. However, the forward and backward probabilities defined in Section III involve
−
joint distributions of different users’ symbols; thus the sub-trellis searches for different users are coupled,
whichprohibitstheexact decompositionintosingle-usertrellises.
However, we can approximate the joint distribution with the product of marginal distributions, which
resultsinthefollowingassumption.
AssumptionIV.1: For any state S defined in (12) which is the union of the correspongding sub-states,
t
S = K Sk, wehave
t k=1 t
S K
P S yt = P Sk yt , (22)
t| 1 t| 1
k=1
(cid:0) (cid:1) Y (cid:0) (cid:1)
and
K
P S yM = P Sk yM . (23)
t| t+1 t| t+1
k=1
Thisassumptionisbasedonthefactt(cid:0)hatP S(cid:1)k yt iYsconc(cid:0)entrated(cid:1)around0and1,providedthatthenoise
t| 1
power is small enough. It can be validated b(cid:0)y the s(cid:1)imulation results in Section VI, which state that, with
Assumption IV.1, the reduced complexity algorithm in this paper can achieve near optimal performance
in moderate energy region. Under this assumption, which is only an approximation, we can decompose
the forward and backward probabilities of a state into the product of the probabilities of the corresponding
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sub-states:
K αk(Sk)
α (S ) = k=1 t t , (24)
t t Q(P(yt1))K−1
and
K βk(Sk)
β (S ) = k=1 t t . (25)
t t QP(yMt+1) K−1
Wecan neglectthecommonfactors (P(yt))K−1 and(cid:0) P(yM )(cid:1) K−1, whichdo notaffect thefinal result.
1 t+1
With this assumption, we can develop recursive f(cid:0)ormulas(cid:1)similar to (13) and (14) with respect to the
sub-states. In particular,fortheforward probability,wehave
αk(m) = P Sk = m,yt
t t 1
(cid:0) (cid:1)
= P S = m ,S = m,yt
t ′′ t 1 ′ 1
−
m′ m m′′
X X∈ (cid:0) (cid:1)
= P yt S = m ,S = m P (S = m S = m)P (S = m)
1| t ′′ t−1 ′ t ′′| t−1 ′ t−1 ′
m′ m m′′
X X∈ (cid:0) (cid:1)
= P yt 1,S = m P (y(t) S = m ,S = m)P (S = m S = m)
1− t−1 ′ | t ′′ t−1 ′ t ′′| t−1 ′
m′ m m′′
X X∈ (cid:0) (cid:1)
K
= αi (m ) λk(m,m) , (26)
 t 1 i  t ′ 
−
Xm′ i=1,Ymi∈m′
  
whereλk(m,m) istheprobabilityoftransitionfrom statem tosub-statem.
t ′ ′
λk(m,m) , P (y(t),S = m S = m)
t ′ t ′′| t−1 ′
m m′′
X∈
= P (y(t) S = m ,S = m)Λ (m )
t ′′ t 1 ′ t ′′
| −
m m′′
X∈
= Λ (m ) 1 exp ky(t)−wˆm′′,m′(t)k2 , (27)
t ′′
2πσ2 − 2σ2
m m′′ n (cid:18) n (cid:19)
X∈
p
wherewˆm′′,m′(t) istheestimatedreceivedsignalduetothestatesm′ and m′′:
K t
wˆm′′,m′(t) = ˜bn(i)h(ni)(t−i), (28)
n=1i=t L+1
X X−
where˜b (i) m m .
n ′ ′′
∈ ∪