Iterative Multiuser Detection and Decoding for DS-CDMA System WithSpace-Time Linear Dispersion

Xiao, P., Wu, J., Sellathurai, M., Ratnarajah, T., & Strom, E. G. (2009). Iterative Multiuser Detection andDecoding for DS-CDMA System With Space-Time Linear Dispersion. IEEE Transactions on VehicularTechnology, 58(5), 2343-2353. https://doi.org/10.1109/TVT.2008.2008652

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009 2343

Iterative Multiuser Detection and Decoding forDS-CDMA System With Space-Time

Linear DispersionPei Xiao, Member, IEEE, Jinsong Wu, Member, IEEE, Mathini Sellathurai, Senior Member, IEEE,

T. Ratnarajah, Senior Member, IEEE, and Erik G. Ström, Senior Member, IEEE

Abstract—This paper considers a Q-ary orthogonal direct-sequence code-division multiple-access (DS-CDMA) system withhigh-rate space-time linear dispersion codes (LDCs) in time-varying Rayleigh fading multiple-input–multiple-output (MIMO)channels. We propose a joint multiuser detection, LDC decoding,Q-ary demodulation, and channel-decoding algorithm and applythe turbo processing principle to improve system performance inan iterative fashion. The proposed iterative scheme demonstratesfaster convergence and superior performance compared with theV-BLAST-based DS-CDMA system and is shown to approach thesingle-user performance bound. We also show that the CDMAsystem is able to exploit the time diversity offered by the LDCsin rapid-fading channels.

Index Terms—Code-division multiple access (CDMA), itera-tive detection and decoding, linear dispersion codes (LDCs),multiple-input–multiple-output (MIMO), multiuser detection(MUD).

I. INTRODUCTION

IN this paper, we study a Q-ary orthogonally modulatedorthogonal direct-sequence code-division multiple-access

(DS-CDMA) system that employs high-rate space-time lineardispersion codes (LDCs) and an iterative detection and de-coding receiver. The orthogonal modulation of the DS-CDMAsystem is accomplished by Walsh codes, which combines theadvantages of spreading and coding to achieve improved per-formance for spread spectrum [code-division multiple-access(CDMA)] systems. Multiuser detection (MUD) [1] is an ef-fective tool to increase the capacity of interference-limitedCDMA systems while reducing some technical requirementssuch as power control. Several iterative MUD schemes wereproposed, e.g., in [2], for uncoded Q-ary orthogonal systems

Manuscript received January 16, 2008; revised July 25, 2008 and September 9,2008. First published October 31, 2008; current version published May 11,2009. This work was supported by the U.K. Engineering and Physical SciencesResearch Council under Grant EP/D07827X/1. The review of this paper wascoordinated by Dr. D. W. Matolak.

P. Xiao, M. Sellathurai, and T. Ratnarajah are with the Institute ofElectronics, Communications, and Information Technology, School of Elec-tronics, Electrical Engineering, and Computer Science, Queen’s Univer-sity Belfast, BT3 9DT Belfast, U.K. (e-mail: pei.xiao@ecit.qub.ac.uk;m.sellathurai@ecit.qub.ac.uk; t.ratnarajah@ecit.qub.ac.uk).

J. Wu is with the Department of Electrical and Computer Engineer-ing, Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail: jwu@ee.queensu.ca).

E. G. Ström is with the Division for Communications Systems and Infor-mation Theory, Department of Signals and Systems, Chalmers University ofTechnology, 412 96 Göteborg, Sweden (e-mail: erik.strom@chalmers.se).

Digital Object Identifier 10.1109/TVT.2008.2008652

with affordable complexity, which is significantly less than thatof an optimum receiver. It has been shown that the performanceof an iterative MUD receiver is far better than that of theconventional receiver, particularly in high-load networks inwhich the interference from other users is severe [2].

Channel coding is usually employed in practical systemsto improve the error detection and correcting capability andpower efficiency. The CDMA systems exhibit their full poten-tial when combined with forward error-correction coding [3].In this context, the joint MUD and decoding, including softinterference cancellation, linear minimum mean square error(MMSE) filtering, trellis-based Log-MAP multiuser detector,and blind Bayesian MUD are some of the schemes studied in[4]–[7] to reduce the deteriorative effect of interference. In [8],we provide a thorough treatment of joint MUD, channel estima-tion, demodulation, and decoding for the serially concatenatedCDMA system (orthogonal modulation concatenated with outerconvolutional code) over multipath fading channels. Decision-directed interference cancellation/suppression and channel es-timation were proposed to combat the effect of multiple-accessinterference (MAI) and to improve the reliability of the demod-ulation process. For a general reference to iterative methods forjoint detection, see [9].

In recent years, multiple-transmit–multiple-receive[multiple-input–multiple-output (MIMO)] antenna systemshave attracted extensive interest and research. In particular,space-time block coding (STBC) has emerged as one of themost promising technologies for meeting the high data rate andhigh service quality requirements for wireless communications[10]–[12]. A critical issue for high-data-rate transmission is theSTBC designs for large MIMO arrays. In particular, full-rateSTBCs cannot be found for complex constellations when thenumber of transmit antennas is greater than two. Hassibi andHochwald proposed a high-rate space-time coding framework,called LDCs [13]. The principle is to transmit substreamsof data in linear combinations over space and time. LDCsare designed to optimize the mutual information betweenthe transmitted and received signals and, at the same time,retain decoding simplicity due to their linear structure. Theyprovide a powerful means to combat fading by dispersing thetransmitted signals over time and space, which is equivalentto creating better effective channels and improving effectivesignal-to-interference-plus-noise ratio, leading to an improvedsystem performance. It is shown in [13] that LDC may achieve

0018-9545/$25.00 © 2009 IEEE

2344 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009

Fig. 1. Block diagram of the transmitter for the LDC-coded DS-CDMA system for the kth user.

a coding rate of up to one and outperform the well-knownfull-rate uncoded V-BLAST [14] scheme.

Applications of low-rate orthogonal space-time block codesto CDMA systems have been studied, e.g., in [15]–[17]. Tosupport future high-data-rate CDMA systems, the use of high-rate space-time block codes, e.g., LDCs, may be desirable.However, very limited efforts have been made in investigat-ing the application of LDCs to CDMA systems. In [18], anLDC decoder combined with a blind subspace-based multiuserdetector is studied for the downlink of a DS-CDMA system,and a subspace-based sphere decoding algorithm is proposedto further improve the performance. The iterative decoding ofLDCs in a frequency-selective channel is considered in [19],where only a single-user approach is studied, and multiuserscenarios are not investigated. To the best of our knowledge,the issue of iterative MUD and LDC decoding has not beenaddressed for the orthogonally modulated DS-CDMA systemsin the existing literature.

In this paper, we propose a joint MUD, LDC decoding, andQ-ary demodulation algorithm for the system under investiga-tion. The turbo processing principle is employed to improve thesystem performance in an iterative manner, while maintaininga reasonable computational complexity. The role of LDC inthis work is to exploit space and time diversity for multipleusers in DS-CDMA systems. The performance of the LDC-coded DS-CDMA system is compared to that of the V-BLAST-based system, and time diversity gains obtained by the LDCsare investigated for the case when the channel gains changewithin one LDC codeword.

The remainder of this paper is organized as follows. The sys-tem model is introduced in Section II. Different LDCs suitablefor the orthogonally modulated CDMA system are discussedin Section III. Iterative MUD, demodulation, and decodingschemes are introduced in Section IV. Different algorithms areexamined and compared numerically in Section V. Conclusionsare drawn in Section VI.

II. SYSTEM MODEL

Fig. 1 illustrates a block diagram of the baseband receivedsignal due to the kth user. The kth user’s lth information bit

is denoted as bkl ∈ {+1,−1} (k = 1, . . . , K, l = 1, . . . , Lb,

where Lb is the block length). The information bits are con-volutionally encoded into code bits {uk

n,l} ∈ {+1,−1}, whereuk

n,l denotes the nth code bit due to bkl . For example, in the

case of a rate-1/3 code, bkl is encoded into uk

0,l, uk1,l, and

uk2,l. Code bits are subsequently interleaved, and each block

of log2 Q coded and interleaved bits {u′kn,l} ∈ {+1,−1} is

mapped into wk(j) ∈ {w0, . . . ,wQ−1}, which is one of theQ Walsh symbols (j is the symbol index). The interleaverand deinterleaver are denoted as Π and Π−1, respectively, inFigs. 1 and 2. The Walsh codeword wk(j) ∈ {+1,−1}Q is thenencoded with an LDC as follows:

Wk(j) =Q∑

q=1

wqk(j)Aq (1)

where the wqk(j) denotes the qth bit of the Walsh codeword

wk(j), and Wk(j) ∈ CT×Nt is the LDC-encoded matrix (T

is the number of time slots or channel uses needed to transmitQ LDC symbols, Nt is the number of transmit antennas, andthe symbol C denotes the complex field). The matrices Aq ∈C

T×Nt , q = 1, . . . , Q, are called dispersion matrices, whichtransform data symbols (Walsh codeword in this context) into aspace–time matrix.

In a more general case, when the data sequence is modulatedusing complex-valued symbols γq = αq + iβq , chosen from anarbitrary constellation (e.g., r-PSK or r-QAM), an LDC SLDC

is defined as [13]

SLDC =Q∑

q=1

(αqAq + iβqBq) (2)

where i =√−1. Note that wq

k in (1) is real valued; therefore,Bq becomes irrelevant for the system under investigation.

To facilitate LDC decoding in the receiver, it is desir-able to reorder Wk(j) column by column in a vector form.Define the vec(·) operation of the m × n matrix K as

XIAO et al.: ITERATIVE MULTIUSER DETECTION AND DECODING FOR DS-CDMA SYSTEM 2345

vec(K) = [KT.1 KT

.2 . . . KT.n]T , where T denotes the

transpose operation, and K.i is the ith column of K. Denoting

Gvec =[vec

(AT

1

)vec

(AT

2

)· · · vec

(AT

Q

)]wk(j) =

[w1

k(j) w2k(j) · · · wQ

k (j)]T

we can now express the jth LDC codeword of the kth user invector form as

xk(j) = vec(Wk(j)T

)= Gvecwk(j) (3)

where the vectors xk(j) and wk(j) and the LDC generator ma-trix Gvec are of sizes TNt × 1, Q × 1, and TNt × Q, respec-tively. The LDC codeword xk(j) is then repetition encoded intosymbol sequence sk(j) = rep{xk(j), Nc}, where rep{·, ·} de-notes the repetition encoding operation, its first argument is theinput bits, and the second one is the repetition factor. Therefore,each LDC symbol per channel use is spread (repetition coded)into Nc symbols, which causes bandwidth expansion (signalspreading in the frequency domain). The spread sequencesk(j) is then scrambled (randomized) by a scrambling codeunique to each user to form the transmitted symbol sequenceak(j) = Ck(j)sk(j), where Ck(j) ∈ {−1,+1}NcTNt×NcTNt

is a diagonal matrix whose diagonal elements correspond to thescrambling code for the kth user’s jth symbol. The purposeof scrambling is to separate users. In this paper, we focus onthe use of long codes, e.g., the scrambling code differs fromsymbol to symbol. The scrambled sequence ak(j) is trans-mitted over a MIMO channel via multiple transmit antennas.For simplicity, we only consider a flat-fading channel here.The received signal is the sum of K users’ signals plus theadditive white complex Gaussian noise. After descramblingand despreading, the received signal due to the kth user’s jthtransmitted sequence can be written in a vector form as yk(j) =NcHk(j)xk(j) = NcHk(j)Gvecwk(j). Denoting the channelgain of the path between the mth transmit antenna and the nthreceive antenna for the tth channel use of Wk(j) as hk,t

m,n,the MIMO channel matrix corresponding to the kth user’s jthsymbol can be expressed as

Hk(j) =

⎡⎢⎣H(1)

k (j) · · · 0...

. . ....

0 · · · H(T )k (j)

⎤⎥⎦ (4)

where

H(t)k (j) =

⎡⎢⎣

hk,t1,1(j) · · · hk,t

Nt,1(j)

.... . .

...hk,t

1,Nr(j) · · · hk,t

Nt,Nr(j)

⎤⎥⎦ .

In this paper, we focus on asynchronous transmission. With-out loss of generality, we assume that τ1 < τ2 < · · · < τk · · · <τK , where τk is the propagation delay for user k and is assumedto be a multiple of chip intervals. The maximum delay spread isassumed to be less than or equal to a symbol interval (Nc chipintervals). After descrambling and despreading, the received

signal corresponding to the kth user’s jth transmitted sequencecan now be expressed as

r(j) =K∑

k=1

yk(j) + n(j) = NcHk(j)Gvecwk(j)︸ ︷︷ ︸desired signal

+n(j)︸︷︷︸noise

+k−1∑s=1

(Nc − τk + τs)Hs(j)Gvecws(j)︸ ︷︷ ︸MAI

+k−1∑s=1

(τk − τs)Hs(j + 1)Gvecws(j + 1)

︸ ︷︷ ︸MAI

+K∑

s=k+1

(τs − τk)Hs(j − 1)Gvecws(j − 1)

︸ ︷︷ ︸MAI

+K∑

s=k+1

(Nc − τs + τk)Hs(j)Gvecws(j)

︸ ︷︷ ︸MAI

(5)

where n(j) ∈ CTNt is a vector of independent identically

distributed complex Gaussian noise samples with zero meanand variance matrix N0, i.e., n(j) ∼ CN (0, N0I).

III. LDCS

In this section, we discuss two classes of LDCs that arewell suited for the DS-CDMA system with orthogonal signalingunder study.

A. Rectangular U-LDC

As an extension of rate-one square LDC matrices of sizeNt × Nt defined in [13, eq. (31)], a class of algebraicallydesigned rate-one rectangular LDCs of arbitrary size T × Nt,called uniform LDCs (U-LDCs), have been reported in [22].They are particularly suited to the DS-CDMA applicationunder investigation due to their flexible choice of space-timedimensions.

1) Case of T ≤ Nt: The T × Nt LDC dispersion matricesare defined as

ANt(k−1)+l =1√T

Dk−1ΓΠl−1 (6)

where k = 1, . . . , T , l = 1, . . . , Nt, D = diag(1, ei(2π/T ), . . . ,

ei(2π(T−1)/T )), Π =[01×(Nt−1) 1

INt−1 0(Nt−1)×1

], and Γ =

[IT 0T×(Nt−T )], where diag(a) denotes a diagonal matrixwith vector a on the main diagonal, i =

√−1, D is of size

T × T , Π is of size Nt × Nt, Γ is of size T × Nt, IX denotesan identity matrix of size X × X , and 0X×Y denotes anall-zero matrix of size X × Y .

2346 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009

2) Case of T > Nt: The T × Nt LDC dispersion matricesare defined as

ANt(k−1)+l =1√N t

Πk−1ΓDl−1 (7)

where k = 1, . . . , T , l = 1, . . . , Nt, D = diag(1, ei(2π/Nt),

. . . , ei(2π(Nt−1)/Nt)), Π =[01×(T−1) 1

IT−1 0(T−1)×1

], and Γ =[

INt

0(T−Nt)×Nt

], where D is of size Nt × Nt, Π, as previously

defined, is of size T × T , and Γ is of size T × Nt.

B. FDFR LDC

In general, space-time LDCs [13] do not necessarily reachany guaranteed space diversity order. LDC was applied tospace–time CDMA systems in [18]; however, the selected LDC(see [13, eq. (31)]) does not offer full diversity. Damen et al.proposed a class of high-rate full-transmit diversity-basedspace-time CDMA systems in [23]. However, they actually usespace codes instead of space-time codes. Note that in [23],Nt source data symbols are transformed using a signal spacediversity rotation to obtain Nt coded symbols, and each ofthe coded symbols is spread through a signature sequence.Although these spread symbols are over time, the signatures arenot spread over multiple data symbols. The approach in [23]actually limits the potential to exploit time diversity in signallevels.

To obtain full diversity over multiple-symbol time channeluses, we adopt the full-diversity full-rate (FDFR) complex-fieldcode proposed in [24] and [25] in this work. It is a class of full-space-diversity LDCs with equal real and imaginary dispersionmatrices in each source data symbol. The generator matrix ofFDFR codes is calculated as

Gvec =[(P1Dβ) ⊗ [θ1]T . . . (PNt

Dβ) ⊗ [θJNt]T

]T(8)

where [θk]T is the kth row vector of Θ, and

Θ =1

JNt[FJNt

]Hdiag(1, α, . . . , αJNt−1)

= [θ1, . . . , θJNt]T

Dβ = diag(1, β, . . . , βNt−1)

Pm =[0(Nt−m+1)×(m−1) Im−1

INt−m+1 0(m−1)×(Nt−m+1)

](9)

where the superscript operator ()H denotes the conjugate trans-pose operation, and FQ denotes a discrete Fourier transformmatrix of dimension Q × Q. The constants α and β for Nt = 2k

(k is a positive integer) are specified as

α ={

eiπ/(2Nt), for Design Aeiπ/(Nt)

3, for Design B

β ={

eiπ/(4(Nt)3), for Design A

α1/Nt , for Design B.(10)

In [24], FDFR codes of size Nt × Nt were proposed asspace-time codes; FDFR codes of size JNt × Nt (J is aninteger number) were proposed as space-frequency codes andspace-Doppler codes. In this paper, we consider FDFR codesof size JNt × Nt as a class of full-space-diversity rectangularspace-time LDCs.

Defining the LDC coding rate as RsymLDC = Q/(NtT ) [20],

one can see that both U-LDC and FDFR codes mentionedabove have a rate of one (Q = NtT ). This is in contrast to allorthogonal space-time codes, for which the rate is less than one(even the so-called full-rate Alamouti code [11] has a rate of1/2 by this definition).

IV. JOINT LDC DECODING AND Q-ARY

SYMBOL DEMODULATION

The proposed iterative detection and decoding scheme isillustrated in Fig. 2. The soft metric λ(u′k

n,l;O) from theinner soft-input–soft-output (SISO) block is deinterleaved toλ(uk

n,l; I). The kth user’s Log-MAP decoder computes an aposteriori log-likelihood ratio (LLR) of each information bitλ(bk

l ;O) and each code bit λ(ukn,l;O) based on the soft input

λ(ukn,l; I) and the trellis structure of the convolutional code.

The former is used to make a decision on the transmittedinformation bit at the final iteration, while the latter is usedfor MUD in the inner SISO block in the next iteration. We usethe notations λ(·; I) and λ(·;O) to denote the input and outputports of a SISO device.

The algorithms discussed above require the design of aninner SISO block that can produce soft reliability values foreach bit u′k

n,l from the received signal to enable soft-inputchannel decoding. To this end, we propose an integrated MUD,LDC decoding, and symbol demodulation scheme, which willbe described next. To simplify the notation, we sometimessuppress the index j from xk(j), wk(j), Hk(j), etc., wheneverno ambiguity arises.

A. Single-User Approach

Let rk denote the delay-aligned version of the received vectordue to the transmission of the kth user’s jth symbol. The single-user detection (SUD) approach is based on the assumptionthat different users’ scrambling codes are (nearly) orthogonalto each other (their cross correlations are approximately zero)and their autocorrelation approximates the delta function. Thereceived vector corresponding to the kth user’s jth symbol afterdescrambling and despreading can be expressed as

zk = NcHkxk + vk = NcHkGvecwk + vk (11)

where zk is the descrambled and despread version of rk, andvk denotes the combined MAI and noise, which is a complexGaussian random vector, i.e., vk ∼ CN (0, NvI) [26], whereNv is the combined MAI and noise variance.

The Walsh codeword wk (or equivalently the jth Walshsymbol for user k) can be estimated by a linear MMSE algo-rithm, i.e.,

wk =ΦHzk =ΦH(NcHkGvecwk + vk)=Uwk + ξk (12)

XIAO et al.: ITERATIVE MULTIUSER DETECTION AND DECODING FOR DS-CDMA SYSTEM 2347

Fig. 2. Iterative receiver structure.

where the matrix Φ is designed to minimize E[‖wk − wk‖2],leading to the solution Φ = R−1P, where

R = E[zkzHk

]= E

[(NcHkGvecwk + vk)

(NcwH

k GHvecH

Hk + vH

k

)]= N2

c HkHHk + NvI (13)

P = E[zkwH

k

]= E

[(NcHkGvecwk + vk)wH

k

]= NcHkGvec

U = ΦHNcHkGvec = ΦHP. (14)

Equations (13) and (14) are derived utilizing the fact thatfor Walsh codewords, E[wkwH

k ] = IQ, and Gvec is a uni-tary matrix for both U-LDC and FDFR codes (the proof forU-LDC is given in [22], and a proof can be similarly de-rived for FDFR). Therefore, GvecGH

vec = IQ. The noise termξk = ΦHvk is Gaussian, since it is a linear transformation ofa Gaussian random vector, with zero mean and covarianceE[ξkξHk ] = NvΦHΦ. The probability density function of theMMSE filter output wk, conditioned on that the mth Walshsymbol is transmitted, can be expressed as

f(wk|wm)

=exp

[−(wk − Uwm)H(NvΦHΦ)−1(wk − Uwm)

]πQ det(NvΦHΦ)

=1

πQ det(NvΦHΦ)exp

[−‖Φ−Hwk − Pwm‖2

Nv

].

The soft metric for the bit u′kn,l can thus be computed in terms

of LLR as

λ(u′k

n,l;O)

= ln

∑m:u′k

n,l=+1 f(wk|wm)∑

m:u′kn,l

=−1 f(wk|wm)

≈ lnmaxm:u′k

n,l=+1 f(wk|wm)

maxm:u′kn,l

=−1 f(wk|wm)

= lnmaxm:u′k

n,l=+1 exp

(−‖Φ−Hwk − Pwm‖2/Nv

)maxm:u′k

n,l=−1 exp (−‖Φ−Hwk − Pwm‖2/Nv)

=1

NvRe

{[2(Φ−1Pw+)Hwk − ‖Pw+‖2

]−

[2(Φ−1Pw−)Hwk − ‖Pw−‖2

]}(15)

TABLE IMAPPING BETWEEN INPUT BITS, SYMBOL INDICES, AND

WALSH CODEWORDS

where m : u′kn,l = ±1 denotes the set of Walsh code-

words {wm} that correspond to the code bit u′kn,l = ±1,

w+ denotes the Walsh codeword wm that corresponds tomaxm:u′k

n,l=+1 f(wk|wm), and w− denotes the Walsh code-

word wm that corresponds to maxm:u′kn,l

=−1 f(wk|wm).In the case where Q = 8, the kth user’s jth Walsh codeword

wk(j) corresponds to three coded and interleaved bits: u′k0,l,

u′k1,l, and u′k

2,l. We know from Table I that u′k0,l = +1 holds for

m = 0, 1, 2, 3 and u′k0,l = −1 holds for m = 4, 5, 6, and 7.

According to (15)

λ(u′k

0,l;O)≈ max {zk(0), zk(1), zk(2), zk(3)}

−max {zk(4), zk(5), zk(6), zk(7)}

where zk(m) = (1/Nv)Re{2(Φ−1Pwm)Hwk − ‖Pwm‖2}.Similarly

λ(u′k

1,l;O)≈ max {zk(0), zk(1), zk(4), zk(5)}

− max {zk(2), zk(3), zk(6), zk(7)}

λ(u′k

2,l;O)≈ max {zk(0), zk(2), zk(4), zk(6)}

− max {zk(1), zk(3), zk(5), zk(7)} .

From (15), we can see that LDC decoding and symboldemodulation, as well as symbol-to-LLR mapping, are allintegrated into one step; the complexity of the receiver is thusgreatly reduced.

The above scheme is suboptimal because the orthogonal con-dition is hardly satisfied in uplink transmission where each usertransmits asynchronously. However, this kind of single-userapproach has been adopted in practical CDMA systems, e.g.,IS-95, due to its low computational complexity. Next, we

2348 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009

introduce an MUD technique to increase the capacity ofinterference-limited CDMA systems. Among different MUDtechniques, the multistage parallel interference cancellation(PIC) scheme [26] is known to be simple and effective formitigation of MAI in long-code DS-CDMA systems. In whatfollows, we shall explain how this PIC-based MUD tech-nique can be incorporated into the demodulation and decodingprocess to mitigate the effect of MAI.

B. Multiuser Approach

Once the transmitted signals are estimated for all the usersat the previous iteration, interference can be removed by sub-tracting the estimated signals of the interfering users from thereceived signal r to form a new signal vector r′k for demodu-lating the signal transmitted from user k. The descrambled anddespread signal corresponding to the kth user’s jth symbol afterinterference cancellation can now be expressed as

r′k(j) = r(j) −k−1∑s=1

(Nc − τk + τs)Hs(j)Gvecws(j)

−k−1∑s=1

(τk − τs)Hs(j + 1)Gvecws(j + 1)

−K∑

s=k+1

(τs − τk)Hs(j − 1)Gvecws(j − 1)

−K∑

s=k+1

(Nc − τs + τk)Hs(j)Gvecws(j)

= NcHk(j)Gvecwk(j) + v′k(j) (16)

where ws(j) is an estimate of ws(j) using hard decisions, andv′

k(j) denotes the combined cancellation residual and noise.The vector r′k is the interference-canceled version of r aftersubtracting the contributions from all the other users usingdecision feedback. The symbol index j is sometimes omittedfor simplicity. With the interference cancellation technique, v′

k

contains much less MAI compared with vk in (11), leading toa significant performance improvement. In case of perfect in-terference cancellation, the cancellation residual vanishes, andv′

k ∼ CN (0, NcN0I). As will become apparent in Section V,the assumption of perfect cancellation can be approached byproper design of the iterative receiver and by proper choice offull-diversity LDCs. The rest of LDC decoding and derivationof LLR values for u′k

n,l is the same as described in the previoussection, except that zk is replaced by r′k, and Nv is replaced byNcN0 in (12)–(15).

The conventional interference cancellation is subject to per-formance degradation due to incorrect decisions on interfer-ence that are subtracted from the received signal. To preventerror propagation from the decision feedback, soft interferencecancellation was proposed, e.g., in [27], for uncoded Q-aryDS-CDMA systems. The rationale is that the cancellation with

hard decisions tends to propagate errors and increase the inter-ference with incorrect decision feedback, while with soft can-cellation, an erroneously estimated symbol usually has a smallLLR and does not make much contribution to the feedback, andtherefore, the error propagation problem is alleviated. In ourcase, the interference cancellation scheme using soft symbolestimates can be reformed as (16) with ws(j) replaced byws(j) (the soft estimate of ws(j)).

This MUD-based iterative scheme is shown in Fig. 2. It usesthe soft information of xk(j), denoted as xk(j), for interferencecancellation to reduce the error propagation. To this end, wecompute wk(j) = [w0

k(j) w1k(j) · · · wQ−1

k (j)]T , the softestimate of the codeword wk(j), from its LLR λ(wk(j)),which is derived by feeding λ(u′k

n,l) = Π{λ(ukn,l;O)} into a

soft modulator. The soft estimate wk(j) is then LDC encodedto produce the LDC codeword xk(j).

When soft information is to be used for interference cancel-lation, a serially concatenated system would be rather differentfrom the uncoded or nonconcatenated systems in that the softvalues are not directly available for all the inner code bitsfrom the outer decoder. In particular, in our case, only thesoft information can be extracted for the systematic bits of theWalsh codewords from a SISO channel decoder. To tackle thisproblem, we design a soft modulator to derive the soft estimatesfor parity bits, which will be explained next.

In the case where Q = 8, the code bits to the Walsh codeword(Walsh symbol) mapping rule is given in Table I. The threesystematic bits are w1

k(j), w2k(j), and w4

k(j), where wqk(j)

denotes the qth bit of the codeword. The columns correspondingto the systematic bits are highlighted in the table. For easeof understanding, we use Q = 8 as an example. However, theextension of the proposed algorithms to other values of Q isstraightforward. In Table I, we can see that the parity bits areformed by systematic bits w1

k(j), w2k(j), and w4

k(j) as

w0k(j)= + 1

w3k(j)= w1

k(j) ⊕ w2k(j)

w5k(j)= w1

k(j) ⊕ w4k(j)

w6k(j)= w2

k(j) ⊕ w4k(j)

w7k(j)= w1

k(j) ⊕ w2k(j) ⊕ w4

k(j).

The LLRs for systematic bits are

λ(w1

k(j))

=λ(u′k

0,l

)λ

(w2

k(j))

=λ(u′k

1,l

)λ

(w4

k(j))

=λ(u′k

2,l

).

Considering the fact that the interleaver breaks the mem-ory of the convolutional encoding process, the bits u′k

0,l, u′k1,l,

and u′k2,l can be modeled as statistically independent random

variables. Additionally, assuming that they are independently

XIAO et al.: ITERATIVE MULTIUSER DETECTION AND DECODING FOR DS-CDMA SYSTEM 2349

conditioned on the received signal, the LLRs for parity bits canthus be computed according to [28] by

λ(w3

k(j))

= λ(w1

k(j) ⊕ w2k(j)

)= 2arc tanh

{tanh

(λ

(u′k

0,l

)/2

)· tanh

(λ

(u′k

1,l

)/2

)}λ

(w5

k(j))

= λ(w1

k(j) ⊕ w4k(j)

)= 2arc tanh

{tanh

(λ

(u′k

0,l

)/2

)· tanh

(λ

(u′k

1,l

)/2

)}λ

(w6

k(j))

= λ(w2

k(j) ⊕ w4k(j)

)= 2arc tanh

{tanh

(λ

(u′k

1,l

)/2

)· tanh

(λ

(u′k

2,l

)/2

)}λ

(w7

k(j))

= λ(w1

k(j) ⊕ w2k(j) ⊕ w4

k(j))

= 2arc tanh

{2∏

n=0

tanh(λ

(u′k

n,l

)/2

)}. (17)

Finally, the soft estimate (expected value given the receivedobservation) for each bit of the Walsh codeword wp

k(j) iscomputed based on λ(wp

k(j)) as

wpk(j) = E [wp

k(j)|z′k] = (+1) × P {wpk(j) = +1|z′k}

+ (−1) × P {wpk(j) = −1|z′k}

= (+1)eλ(wp

k(j))

1 + eλ(wpk(j))

+ (−1)e−λ(wp

k(j))

1 + e−λ(wpk(j))

= tanh {λ (wpk(j)) /2} .

V. NUMERICAL RESULTS

In the simulations, we employ a rate Rc = 1/3 convolutionalcode with constraint length Lc = 5 and generator polynomials(25, 33, 37) in octal form for all the users, unless otherwisestated. Each block of three interleaved bits from each user isthen converted into one of Q = 8 Walsh symbols, which issubsequently encoded with an LDC. In this paper, we use theU-LDC expressed by (7) and the FDFR code (design A) ex-pressed by (8)–(10). The parameter setting is chosen to be Nt =2, Nr = 2 and 4, and T = 4. In this case, an 8-bit Walsh code-word is dispersed into two transmit antennas, each of whichaccommodates four LDC symbols (T = 4 channel uses beforespreading). Each LDC symbol is spread (repetition encoded) toNc = 8 symbols. The effective spreading factor of the systemis given as the reciprocal of the overall code rate divided byNt, i.e., NcQ/(NtRc log2 Q) = 32 LDC symbols/informationbit/transmit antenna. Hence, the spectral efficiency of a singlelink is equal to 1/32 times the spectral efficiency of an uncodedunspread system. The number of users is chosen to be K = 18,which represents a fairly heavily loaded system, as a K = 32-

Fig. 3. Comparison of different MUD schemes. LDC versus V-BLAST.Curves with the same marker represent the performance of the same schemeat different iterations (including the first SUD stage).

user system would be a fully loaded system. The long scram-bling codes Ck are randomly generated, and all the transmitantennas of a specific user are assigned the same scramblingcode. The noise variance N0 and Ck, as well as path delaysτ1, τ2, . . . , τK , are assumed to be known to the receiver, and thedifferent paths of the MIMO channels for the same user havethe same delay. We assume uplink asynchronous transmission,and the delays are therefore different for different users.

We compare the performance of the proposed LDC iterativescheme with that of the soft demodulation and decoding al-gorithm [8] for the Q-ary orthogonally modulated DS-CDMAsystem without LDCs in a 2 × 2 or 2 × 4 V-BLAST system.In this case, information-bearing signals are divided into mul-tiple substreams, each encoded and modulated independently.Unlike in the LDC systems, each user’s two transmit antennasare assigned with different scrambling codes and simultane-ously transmit the data. The employed convolution code andspreading factor are the same as in the LDC system. To transmitone Walsh symbol, the V-BLAST system needs Q = 8 channeluses (time slots), whereas the LDC system only needs T = 4channel uses at both transmit antennas. However, due to si-multaneous transmission from two transmit antennas in theV-BLAST system, the two systems have the same data rate.

The channel gain hk,tm,n is a complex circular Gaussian

process with autocorrelation function E[hk,t∗m,nhk,t+τ

m,n ] =P k

m,nJ0(2πfDτ), where fD is the maximum Doppler fre-quency, J0(x) is the zeroth-order Bessel function of the firstkind, and P k

m,n is the average power of hk,tm,n. The amplitude

of hk,tm,n follows a Rayleigh distribution. The Doppler shifts

are due to the relative motion between the base station andmobile units. Perfect slow power control is assumed in thesense that Pk =

∑m,n E[|hk,t

m,n|2], the average receivedpower, is equal for all users, and it is normalized such thatPk = 1 for all k. However, the instantaneous power |hk,t

m,n|2may vary from one user to another. Perfect knowledgeof the channel state information (CSI) is assumed in oursimulations. In Figs. 3–7, we assume that channel gains remain

2350 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009

Fig. 4. Convergence property of the iterative schemes. → represents thedecoding process, and ↑ represents the MUD process.

Fig. 5. Performance comparison with (5, 7) convolutional code in 2Tx-2Rxsystems. Curves with the same marker represent the performance of the samescheme at different iterations.

constant during the transmission of one LDC codeword, i.e.,hk,t

m,n(j) = hk,t+1m,n (j) = hk,t+2

m,n (j) = hk,t+3m,n (j), and vary from

one codeword to another. The normalized Doppler frequencyis assumed to be fDTs = 0.01, where Ts is the symbol (LDCcodeword) duration. During each Monte Carlo run, the blocksize is set to 1526 information bits followed by 4 tail bitsto terminate the trellis. The coded bits are passed through arandom interleaver. The simulation results are averaged overrandom fading, noise, delays, and scrambling codes with aminimum of 50 blocks of data transmitted and at least 100 biterrors generated.

In Fig. 3, we compare the performance of the iterativeMUD algorithms at different iterations. The PIC-based softdemodulation scheme [8] is used for the V-BLAST system.Here, Eb is defined as the received bit energy and is normalizedwith the number of receive antennas. The single-user scheme isused in the beginning of the iterative process to obtain an initial

Fig. 6. Performance comparison with (575, 623, 727) convolutional code in2Tx-2Rx systems.

Fig. 7. Performance comparison with (5, 7) convolutional code in 2Tx-4Rxsystems.

estimate of the data, which is needed for MUD at subsequentiterations. We notice that the V-BLAST system has to iteratethree times before reaching convergence (excluding the SUDstage), while the LDC system only needs to iterate two times toconverge. The use of LDC leads to faster convergence for theiterative receiver, as well as superior performance, comparedwith the V-BLAST system. By comparison, the performance ofthe U-LDC is not as good as that of the FDFR codes, especiallyat high SNRs (the curves for the U-LDC are not included toconserve space). To fully exploit the space diversity, FDFRcodes should be used.

Their convergence behavior is further examined at Eb/N0 =7 dB in Fig. 4 using the extrinsic information transfer (EXIT)chart, which traces the evolution of the mutual informationIMi /IM

o ∈ [0, 1] between input/output LLR and u′kn,l for the

multiuser detector and the mutual information IDi /ID

o betweeninput/output LLR and uk

n,l for the Log-MAP decoder. See [29]for a detailed discussion of this analytical method and to [30]

XIAO et al.: ITERATIVE MULTIUSER DETECTION AND DECODING FOR DS-CDMA SYSTEM 2351

and [31] for its application to the analysis of iterative schemesin CDMA and MIMO systems. It should be noted that in ourcase, IM

i > 0 at the starting point. This is due to the SUDstage applied in the beginning of the iterative process to yieldan initial estimate of data; therefore, the a priori informationis not zero when the MUD process starts. The output LLRof the detector IM

o is forwarded to the decoder as input, i.e.,IDi = IM

o ; the output LLR of the decoder IDo is fed back to the

detector, i.e., IMi = ID

o , and so on. As indicated by the transfercurves in Fig. 4, the output LLR IM

o becomes more reliable (itsvalue increases) as the input LLR IM

i becomes more reliablein the detector. The iterative detection and decoding processis depicted by a staircase trace between the transfer curvesof the detector and decoder. The trace shows that two (three)iterations of detection/decoding are needed for the LDC (V-BLAST) system to converge (reach the maximum ID

o ). Thisis in close agreement with the results presented in Fig. 3. Theinitial value of IM

i is obtained through simulations for bothLDC and V-BLAST systems.

In Figs. 5–7, we compare the two systems with differentchannel codes and diversity orders. Two convolutional codesare tested: 1) a weak code with rate Rc = 1/2, constraint lengthLc = 3, and generator polynomials (5, 7) and 2) a strong codewith rate Rc = 1/3, constraint length Lc = 9, and generatorpolynomials (575, 623, 727), which is an optimum distancespectrum code [32]. Comparing Fig. 5 with Fig. 6, one can seethat the advantages of LDCs over V-BLAST become smallerwhen a stronger code is used. For example, with the (5, 7) code,the LDC system outperforms the V-BLAST system by 1.0 dBupon reaching convergence at a target bit error rate (BER) of10−4, whereas the difference is only 0.5 dB when the (575,623, 727) code is used. Obviously, it is more advantageous toapply LDC with a weak code. The single-user bounds for theFDFR-LDC-coded systems are shown in Figs. 3 and 5. They areobtained by the proposed scheme in a single-user environment,no interference mitigation is needed in this case, and they givelower bounds on the best achievable performance by applyingthe MUD technique. It can be seen that the performance of theproposed iterative MUD approach upon reaching convergenceis very close to the single-user bound, meaning that MAI canbe effectively eliminated by proper design of iterative MUDschemes.

Fig. 7 shows the performance comparison between the twosystems with the (5, 7) code in a 2 × 4 antenna setup. Com-paring with Fig. 5, it is obvious that the V-BLAST systemconverges faster, and the performance gap between the twosystems becomes smaller as the number of receive antennas(spatial diversity order) increases.

Note that the proposed system does not require a constantchannel during the transmission of one LDC codeword, channelgains can vary from one time slot to another, i.e., hk,t

m,n(j) �=hk,t+1

m,n (j) �= hk,t+2m,n (j) �= hk,t+3

m,n (j). This is in contrast withprevious work on LDC for CDMA systems, which were de-signed for static channel over the whole LDC codeword. InFig. 8, we examine the performance of the LDC-coded systemusing the proposed iterative MUD approach in faster fadingchannels when the channel gains keep changing at each timeslot within one LDC codeword. To this end, we redefine the

Fig. 8. BER performance versus normalized Doppler frequency for the LDC-coded MIMO CDMA system.

normalized Doppler frequency as fDT , where T is the durationof one time slot (channel use). The SNR is set to be Eb/N0 =5 dB. Slow power control is assumed so that the averagereceived power is equal for all users. The BER curve is plottedat the fourth iteration when the system reaches convergence.Fig. 8 shows that the system performance improves as theDoppler frequency increases, which clearly indicates the timediversity obtained by the LDCs, and the diversity gain is moreobvious by applying FDFR LDC rather than U-LDC. However,this time diversity is not exploited for slow-fading channels asin the previous cases, where the normalized Doppler frequencyis set to be fDTs = 0.01, and code-level channel stationarityis assumed. It was shown in [8] that for non-LDC systems,the performance degrades as the normalized Doppler frequencyincreases.

The complexity of the iterative MUD approach for theLDC-coded system and the V-BLAST system is compared inTable II, which shows the required number of complex mul-tiplications/divisions and additions/subtractions for one user’sone symbol estimate corresponding to the calculation of LLRsfor log2 Q code bits. One can see in the table that the proposedLDC system increases the complexity from O(Q2) to O(Q3)compared to the V-BLAST system, mainly due to the matrixinverse operation [see (15)] at symbol rate in the LDC decodingprocess. However, the complexity increase is partly compen-sated by the faster convergence achieved by the LDC system.The time diversity shown in Fig. 8 also justifies the use of LDCsin fast-fading channels.

VI. CONCLUSION

Iterative detection and decoding for an orthogonally modu-lated and LDC-coded MIMO DS-CDMA system has been in-vestigated in this paper. We have proposed an integrated designof MUD, LDC decoding, symbol demodulation, and symbol-to-LLR mapping to reduce the complexity of the iterative receiver.Numerical results show that the choice of LDCs is importantfor system performance, and the use of FDFR LDCs leadsto high transmission bandwidth efficiency and significantly

2352 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009

TABLE IICOMPLEXITY FOR ONE USER’S ONE SYMBOL ESTIMATE AT ONE ITERATION FOR THE ITERATIVE MUD ALGORITHMS CONSIDERED

improved BER performance. Under the assumption of perfectCSI, it enables the system to approach the single-user boundeven in heavily loaded systems. Compared with the V-BLASTsystem with the same transmission rate, the proposed LDCsystem shows superior performance and faster convergence.However, we have observed that the advantages of applyingLDC become smaller when a strong channel code is used orthe diversity order increases. Furthermore, by exploiting timediversity, LDCs also provide us with a powerful means tocombat impairment caused by rapid-fading channels. The ex-tension of the current work to frequency-selective channels byincorporating the orthogonal frequency-division multiplexingtechnique or using the turbo equalization method is a topic forfuture research.

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Pei Xiao (M’04) received the B.Eng. degree fromthe Huazhong University of Science and Technology,Wuhan, China, the M.Sc. and Licentiate of Technol-ogy degrees from the Tampere University of Tech-nology, Tampere, Finland, and the Licentiate ofEngineering and the Ph.D. degrees from theChalmers University of Technology, Göteborg,Sweden.

After receiving the M.Sc. degree, he was withNokia Networks, Tampere, Finland. After receivingthe Ph.D. degree, he was a Research Fellow with the

University of Wolverhampton, Wolverhampton, U.K., and the University ofNewcastle Upon Tyne, Newcastle, U.K. He is currently a Research Fellow withthe Institute of Electronics, Communications, and Information Technology,School of Electronics, Electrical Engineering, and Computer Science, Queen’sUniversity Belfast, Belfast, U.K.

Dr. Xiao is a member of the Institute of Electronics, Information, andCommunication Engineering.

XIAO et al.: ITERATIVE MULTIUSER DETECTION AND DECODING FOR DS-CDMA SYSTEM 2353

Jinsong Wu (M’07) received the B.Eng. degreein electrical engineering from Changsha RailwayUniversity (now part of Central South University),Changsha, China, in 1990, the M.A.Sc. degree inelectrical engineering from Dalhousie University,Halifax, NS, Canada, in 2001, and the Ph.D. degreein electrical engineering from Queen’s University,Kingston, ON, Canada, in 2006.

He held engineer positions of information tech-nology or telecommunications with Stone, China,Great-Wall Computer, China, Nortel Networks,

Canada, Philips Research North America, and Sprint Nextel. He is currentlywith the Department of Electrical and Computer Engineering, Queen’s Univer-sity. His current research interests lie in general communications theory andsignal processing, space-time-frequency processing and coding, cooperativecommunications, and network protocols with applications.

Dr. Wu was a Technical Program Committee member at the 2008 IEEEGlobal Communications Conference and the 2009 IEEE International Confer-ence on Communications.

Mathini Sellathurai (S’95–M’02–SM’06) receivedthe Technical Licentiate degree in electrical engi-neering from the Royal Institute of Technology,Stockholm, Sweden, in 1997 and the Ph.D. degreein electrical engineering from McMaster University,Hamilton, ON, Canada, in 2001.

From 2001 to 2006, she was with the Communi-cations Research Centre, Ottawa, ON, Canada, as aSenior Research Scientist in the Research Communi-cations Signal Processing Group. From August 2004to August 2006, she was with the Institute of Infor-

mation Systems and Integration Technology, Cardiff School of Engineering,Cardiff University, Cardiff, U.K., as a Senior Lecturer. She is currently a Readerin digital communications with the Institute of Electronics, Communications,and Information Technology, School of Electronics, Electrical Engineering,and Computer Science, Queen’s University Belfast, Belfast, U.K. Her currentresearch interests include adaptive signal processing, space-time and multime-dia wireless communications, sensor networks, information theory, and channelcoding.

Dr. Sellathurai was the recipient of the Natural Sciences and EngineeringResearch Council of Canada’s doctoral award for her Ph.D. dissertation and acorecipient of the IEEE Communication Society 2005 Fred W. Ellersick BestPaper Award.

T. Ratnarajah (S’94–A’96–M’05–SM’05) has re-ceived the B.Eng. (Hons), M.Sc., and Ph.D. degrees.

He is currently a Principal Research Engineer withthe Institute of Electronics, Communications, and In-formation Technology, School of Electronics, Elec-trical Engineering, and Computer Science, Queen’sUniversity Belfast, Belfast, U.K. Since 1993, he hasheld various research positions with the Universityof Ottawa, Ottawa, ON, Canada; Nortel Networks,Ottawa; McMaster University, Hamilton, ON; andImperial College London, London, U.K. He has

published more than 80 publications in these areas. He is the holder of fourU.S. patents. His research interests include random matrix theory, information-theoretic aspects of MIMO channels and ad hoc networks, wireless communi-cations, cognitive radio, signal processing for communications, statistical andarray signal processing, biomedical signal processing, and quantum informa-tion theory.

Dr. Ratnarajah is a member of the American Mathematical Society andInformation Theory Society.

Erik G. Ström (S’93–M’95–SM’01) received theM.S. degree in electrical engineering from the RoyalInstitute of Technology (KTH), Stockholm, Sweden,in 1990 and the Ph.D. degree in electrical engi-neering from the University of Florida, Gainesville,in 1994.

He accepted a postdoctoral position with the De-partment of Signals, Sensors, and Systems, KTH,in 1995 and was appointed Assistant Professor inFebruary 1996. In June 1996, he joined the ChalmersUniversity of Technology, Göteborg, Sweden, where

he has been a Professor of communication systems since June 2003 and is cur-rently the Head of the Division for Communications Systems and InformationTheory, Department of Signals and Systems. Since 1990, he has acted as aConsultant for the Educational Group for Individual Development, Stockholm,Sweden. His research interests include signal processing and communicationtheory in general and constellation labelings, channel estimation, synchro-nization, multiple access, medium access, multiuser detection, and wirelesspositioning in particular. He is a contributing author and an Associate Editorfor Roy Admiralty Publishers FesGas series.

Dr. Ström was a Co-Guest Editor for the IEEE JOURNAL ON SELECTED

AREAS IN COMMUNICATIONS Special Issue on Signal Synchronization in Dig-ital Transmission Systems (2001) and Special Issue on Multiuser Detection forAdvanced Communication Systems and Networks (2008). He was a member ofthe board of the IEEE Vehicular Technology/Communications Swedish Chapterfrom 2000 to 2006. He received the Chalmers Pedagogical Prize in 1998.