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Performance evaluation of space-time-frequencyspreading for MIMO OFDM-CDMA systemsHaysam Dahman* and Yousef Shayan

Abstract

In this article, we propose a multiple-input-multiple-output, orthogonal frequency division multiplexing, code-division multiple-access (MIMO OFDM-CDMA) scheme. The main objective is to provide extra flexibility in usermultiplexing and data rate adaptation, that offer higher system throughput and better diversity gains. This is doneby spreading on all the signal domains; i.e, space-time frequency spreading is employed to transmit users’ signals.The flexibility to spread on all three domains allows us to independently spread users’ data, to maintain increasedsystem throughput and to have higher diversity gains. We derive new accurate approximations for the probabilityof symbol error and signal-to-interference noise ratio (SINR) for zero forcing (ZF) receiver. This study and simulationresults show that MIMO OFDM-CDMA is capable of achieving diversity gains significantly larger than that of theconventional 2-D CDMA OFDM and MIMO MC CDMA schemes.

Keywords: code-division multiple-access (CDMA), diversity, space-time-frequency spreading, multiple-input multi-ple-output (MIMO) systems, orthogonal frequency-division multiplexing (OFDM), 4th generation (4G)

1. IntroductionModern broadband wireless systems must support mul-timedia services of a wide range of data rates with rea-sonable complexity, flexible multi-rate adaptation, andefficient multi-user multiplexing and detection. Broad-band access has been evolving through the years, start-ing from 3G and High-Speed Downlink Packet Access(HSDPA) to Evolved High Speed Packet Access (HSPA+) [1] and Long Term Evolution (LTE). These are exam-ples of next generation systems that provide higher per-formance data transmission, and improve end-userexperience for web access, file download/upload, voiceover IP and streaming services. HSPA+ and LTE arebased on shared-channel transmission, so the key fea-tures for an efficient communication system are to max-imize throughput, improve coverage, decrease latencyand enhance user experience by sharing channelresources between users, providing flexible link adapta-tion, better coverage, increased throughput and easymulti-user multiplexing.An efficient technique to be used in next generation

wireless systems is OFDM-CDMA. OFDM is the main

air interface for LTE system, and on the other hand,CDMA is the air interface for HSPA+, so by combiningboth we can implement a system that benefits fromboth interfaces and is backward compatible to 3G and4G systems. Various OFDM-CDMA schemes have beenproposed and can be mainly categorized into two groupsaccording to code spreading direction [2-5]. One is tospread the original data stream in the frequency domain;and the other is to spread in the time domain.The key issue in designing an efficient system is to

combine the benefits of both spreading in time and fre-quency domains to develop a scheme that has thepotential of maximizing the achievable diversity in amulti-rate, multiple-access environment. In [6], it hasbeen proposed a novel joint time-frequency 2-dimen-sional (2D) spreading method for OFDM-CDMA sys-tems, which can offer not only time diversity, but alsofrequency diversity at the receiver efficiently. Each userwill be allocated with one orthogonal code and spreadits information data over the frequency and timedomain uniformly. In this study, it was not mentionedhow this approach will perform in a MIMO environ-ment, specially in a downlink transmission. On theother hand, in [7], it was proposed a technique, calledspace-time spreading (STS), that improves the downlink

* Correspondence: [email protected] of Electrical Engineering, Concordia University, Montreal, QC,Canada

Dahman and Shayan EURASIP Journal on Advances in Signal Processing 2011, 2011:139http://asp.eurasipjournals.com/content/2011/1/139

© 2011 Dahman and Shayan; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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performance, however they do not consider the multi-user interference problem at all. It was assumed thatorthogonality between users can somehow be achieved,but in this article, this is a condition that is not triviallyrealized. Also, in [8], multicarrier direct-sequence code-division multiple-access (MC DS-CDMA) using STSwas proposed. This scheme shows good BER perfor-mance with small number of users and however, theperformance of the system with larger MUI was not dis-cussed. Recently, in [9], they adopted Hanzo’s scheme[8], which shows a better result for larger number ofusers, but both transmitter and receiver designs arecomplicated.In this article, we propose an open-loop MIMO

OFDM-CDMA system using space, time, and frequency(STF) spreading [10]. The main goal is to achieve higherdiversity gains and increased throughput by indepen-dently spreading data in STF with reasonable complex-ity. In addition, the system allows flexible data rates andefficient user multiplexing which are required for nextgeneration wireless communications systems. An impor-tant advantage of using STF-domain spreading inMIMO OFDM-CDMA is that the maximum number ofusers supported is linearly proportional to the productof the S-domain, T-domain and the F-domain spreadingfactors. Therefore, the MIMO OFDM-CDMA systemusing STF-domain spreading is capable of supporting asignificantly higher number of users than other schemesusing solely T-domain spreading. We will show throughthis article, that STF-domain spreading has significantthroughput gains compared to conventional schemes.Furthermore, spreading on all the signal domains pro-vides extra flexibility in user multiplexing and schedul-ing. In addition, it offers better diversity/multiplexingtrade-off. The performance of MIMO OFDM-CDMAscheme using STF-domain spreading is investigated withzero-forcing (ZF) receiver. It is also shown that largerdiversity gains can be achieved for a given number ofusers compared to other schemes. Moreover, highernumber of users are able to share same channelresources, thus providing higher data rates than conven-tional techniques used in current HSPA+/LTE systems.

2. System modelIn this section, joint space-time-frequency spreading isproposed for the downlink of an open-loop multi-usersystem employing single-user MIMO (SU-MIMO) sys-tem based on OFDM¬CDMA system.

A. MIMO-OFDM channel modelConsider a wireless OFDM link with Nf subcarriers ortones. The number of transmit and receive antennas areNt and Nr, respectively. We assume that the channel hasL’ taps and the frequency-domain channel matrix of the

qth subcarrier is related to the channel impulse responseas [11]

Hq =L′−1∑l=0

H(l)e−j2π lqNf , 0 ≤ q < Nf − 1, (1)

where the Nr × Nt complex-valued random matrixH(l) represents the lth tap. The channel is assumed tobe Rayleigh fading, i.e., the elements of the matricesH(l)(l = 0, 1, . . . , L′ − 1) are independent circularlysymmetric complex Gaussian random variables with zeromean and variance σ 2

l , i.e., [H(l)]ij ∼ CN(0,σ 2l ) .

Furthermore, channel taps are assumed to be mutuallyindependent, i.e., E[H(l)H(k)∗] = 0 , the path gains σ 2

l

are determined by the power delay profile of the channel.Collecting the transmitted symbols into vectors

xq = [x(0)q x(1)q . . . x(Nt−1)q ]T (q = 0, 1, . . . ,Nf − 1) with

x(i)q denoting the data symbol transmitted from the ith

antenna on the qth subcarrier, the reconstructed datavector after FFT at the receiver for the qth subcarrier isgiven by [12,13]

yq =√EsHqxq + nq, k = 0, 1, . . . , Nf − 1, (2)

where

yq = [y(0)q y(1)q . . . y(Nr−1)q ]T (q = 0, 1, . . . , Nf − 1) with

y(i)q denoting the data symbol received from the jth

antenna on the qth subcarrier, nq is complex-valuedadditive white Gaussian noise satisfying

E{nqnHl } = σ 2

n INrδ[q − l] . The data symbols x(i)q are

taken from a finite complex alphabet and having unitaverage energy (Es = 1).

B. MIMO OFDM-CDMA systemWe will now focus on the downlink of a multi-accesssystem that employs multiple antennas for MIMOOFDM-CDMA system. As shown in Figure 1a, the sys-tem consists of three different stages. The first stageemploys the Joint Spatial, Time, and Frequency (STF)spreading which is illustrated in details in Figure 1b.The second stage is multi-user multiplexing (MUX)where all users are added together, and finally the thirdstage is IFFT to form the OFDM symbols. Then cyclicshifting is applied on each transmission stream. Specifi-cally as shown in Figure 1, the IFFT outputs associatedwith the ith transmit antenna are cyclicly shifted to theright by (i - 1)L where L is a predefined value equal orgreater to the channel length.Now, we will describe in details the Joint STF spread-

ing block shown in Figure 1b, where the signal is firstspread in space, followed by time spreading and then


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time-frequency mapping is applied to ensure signalindependency when transmitted and hence maximizingachievable diversity [14] on the receiver side.

1) Spatial spreadingLets denote xk as the transmitted symbol from user k. Itwill be first spread in space domain using orthogonal

code such as Walsh codes or columns of an FFT matrixof size Nt, as they are efficient short orthogonal codes.Let’s denote x’k as the spread signal in space for user k

x’k = skxk= [x′

k,1, x′k,2, . . . , x′

k,Nt ], k = 1, 2, . . . , M(3)

L

IFFT

IFFT

IFFT

1

2

Nt

User# 1

x1

xM

User# M

x2

User# 2

Joint STF Spreading

Joint STF Spreading

Joint STF Spreading

CS by

OFDM + Cyclic ShiftJoint Space-Time-Frequency SpreadingMUX

(Nt − 1)×CS by

L

(a) MIMO OFDM-CDMA system

ck

sk

ck

ck MappingT-Fsk,2xk

User# k

ck,1sk,Ntxk

ck,Ncsk,Nt

xk

ck,1sk,2xk

ck,Ncsk,2xkxk

ck,Ncsk,1xk

sk,Ntxk

x′′

kx′

k

Joint STF Spreading

sk,1xk

ck,1sk,1xk

(b) Joint STF Spreading block diagram

Figure 1 MIMO OFDM-CDMA system block diagram.


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where M is the number of users in the system, andsk = [sk,1, sk,2, . . . , sk,Nt ]

T is orthogonal code with sizeNt for user k.

2) Time SpreadingThen each signal in x’k is spread in time domain withck orthogonal code for user k with size Nc. Let’s denotex”k as spread signal in time,

x”k,i = ckx′k,i,

= [x′′k,i,1, x′′

k,i,2, . . . , x′′k,i,Nc ]

T , i = 1, 2, . . . , Nt(4)

where x′′k,i,n is the transmitted signal for user k from

antenna i at time n.

3) Time-Frequency mappingThe output of the space-time spreading is then mappedin time and frequency before IFFT. Figure 2 describes theTime-Frequency mapping method used in this system foruser 1 at a particular transmit antenna. Without loss ofgenerality all users will use the same mapping method ateach antenna. Let’s consider the mapping for x”k,1 and

assume x′′k,1,1 occupies OFDM symbol 1 at subcarrier

K1, x′′k,1,2 occupies OFDM symbol 2 at subcarrier K2,...,

and x′′k,1,Nc

occupies OFDM symbol Nc at subcarrier KNc .

The next transmitted symbol x′′k,1,1 occupies OFDM sym-

bol 1 at subcarrier K1 + 1, x′′k,1,2 occupies OFDM symbol

2 at subcarrier K2 + 1,..., and x′′k,1,Nc

occupies OFDM

symbol Nc at subcarrier KNc + 1. Next symbols x′′k,i are

spread in the same manner as symbols 1 and 2.The assignment for each OFDM subcarrier is calcu-

lated from the fact that the IFFT matrix for our OFDMtransmitted data for symbol 1 isF = [fK1 , fK2 , . . . , fKNc

]H with size Nc × Nf, where FH ⊂FFT matrix with size Nf. F matrix in this paper is aWIDE matrix Nc × Nf where the rows are picked froman FFT matrix and complex transposed (Hermitian). Forthis matrix to satisfy the orthogonality condition and tomaintain independence, those rows needs to be pickedas every Nf /Nc column, so then and ONLY then, eachcolumn and row are orthogonal. The max rank cannot

be more than Nc. The frequency spacing or jump intro-duced, made it possible to achieve the max rank, whereeach row and column is orthogonal within the rank. Inorder to achieve independent fading for each signal andhence maximizing frequency diversity, we need to haveFHF = I. FHF = I is only possible if FH is constructedfrom every Nf/Nc columns of the FFT matrix,F = [f1, fNf

/Nc, f2Nf

/Nc, . . . , f(Nc−1)Nf

/Nc]H . Therefore, if

K1 = 1, then K2 = Nf/Nc,..., and KNc = (Nc − 1)Nf/Nc .

3. ReceiverA. Received signal of SU-MIMO systemOn the receiver side, let us consider the detection of

symbol xk at receive antenna j. Let y(j)Knbe the received

signal of the Kn-th subcarrier at the j-th receive antenna.Note that Kn is the K-th subcarrier at time n (n = 1, 2,...,Nc).

y(j)Kn= fHKn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h1,j 0L . . . 0L

0L−L′... . . .

...... h2,j

. . . 0... 0L−L′

. . . 0...

...... 0

......

. . . hNt ,j

0 0 . . . 0Nf−(Nt−1)L−L′

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

ck,nskxk + n(j)Kn

(5)

Stacking y(j)Knin one column, we have

⎡⎢⎢⎢⎢⎢⎢⎢⎣

y(j)K1...

y(j)Kn...

y(j)KNc

⎤⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸y(j)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

fHK1ck,1...

fHKtck,n...

fHKNcck,Nc

⎤⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸Fc

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h1,jsk,10L−L′

h2,jsk,20L−L′

...hNt ,jsk,Nt

0Nf−(Nt−1)L−L′

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸hsj

xk + nj

(6)

K2

K1 Nf

KNc

Symbol 1

Symbol 2

Symbol Nc

Figure 2 (T-F) Time-frequency mapping.


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Here, fKn stands for the Kn-th column of the (Nf × Nf)FFT matrix, L is the cyclic shift on each antenna whereL > L’ (L’ is the channel length), and hi,j is the impulseresponse from the i-th transmit antenna to the j-threceive antenna. Here, cyclic shifting in time has trans-formed the effective channel response j-th receive

antenna to hsj as shown in Equation (6) instead of the

addition of all channel responses. This will maximizethe number of degrees of freedom from 1 to Nt.In our scheme, we assumed that all users transmit on

same time and frequency slots. As shown in Figure 1,we have the ability to achieve flexible scheduling in bothtime and frequency. This will contribute in more flexiblesystem design for next-generation wireless systems ascompared to other schemes.

B. Achievable Diversity in SU-MIMOLet us assume that x, and x’ are two distinct transmittedsymbols from user k, and y(j), y’(j) are the correspondingreceived signals at receive antenna j, respectively. Tocalculate diversity, we first calculate the expectation ofthe Euclidian distance between the two received signalsE[||y’(j) - y(j)||2], where y(j) is defined by Equation (6),

E[||�y(j)||2] = E[||Fchsj ||2|�x|2]

= E[hsHj FHc Fch

sj |�x|2]

= E[hsHj Fch

sj |�x|2]

(7)

In Equation (7), Fc is a toeplitz matrix (Nf × Nf) whereit is all zero matrix except for the r where

r =∑Nc

t=1

∣∣ck,n∣∣2 , and all non-zero values are spaced Nc

entries apart, where

Fc =

⎡⎢⎣1 . . . 1...

. . ....

1 . . . 1

⎤⎥⎦ ⊗

⎡⎢⎣

r 0. . .

0 r

⎤⎥⎦

= 1Nf/Nc

⊗ rINc

(8)

The rank of the Fc matrix is found as,

rank(Fc) = Nc (9)

Since the maximum achievable degrees of freedom forthe transmitter is equal to NtL’, diversity can be foundas d = min(Nc, NtL’) [15]. For this reason, in order toachieve maximum spatial diversity, we need to choosetime spreading length Nc ≥ NtL’.

C. Receiver DesignNow, let’s assume all the users send data simultaneouslywhere each user is assigned different spatial spreadingcode sk and time spreading code ck generated from a

Walsh-Hadamard function.

yKn=

M∑k=1

(HKnck,nsk)xk + nKn , 1 ≤ Kn ≤ Nf (10)

where k stands for user index and Kn is the K-th sub-carrier at time n (n = 1, 2,..., Nc).Stacking yKn in one column, we have⎡⎢⎢⎢⎣

yK1

yK2...

yKNc

⎤⎥⎥⎥⎦

︸︷︷︸y

= Hs1x1 + Hs2x2 + . . . + HsMxM + n

=[H1 H2 . . . HM

]︸︷︷︸G

x + n

(11)

where H is the modified channel matrix for the Nc

subcarriers, Hk is the effective channel (NcNr × 1) foruser k, and sk = ck ⊗ sk is the combined spatial-timespreading code, where

H = diag{HK1 ,HK2 , . . . , HKNc

}(12)

sk =

⎡⎢⎢⎢⎣

ck,1skck,2sk...

ck,Ncsk

⎤⎥⎥⎥⎦ (13)

At the receiver, the despreading and combining proce-dure with the time-frequency spreading grid patterncorresponding to the transmitter can not be processeduntil all the symbols within one super-frame arereceived. Then by using a MMSE or ZF receiver, datasymbols could be recovered for all users [16,17]

x = (GHG + σ 2I)−1GHy (MMSE) (14)

x = (GHG)−1GHy (ZF) (15)

where x =[x1, x2, . . . , xM

], and M is the number of

users.

D. Performance Evaluation for Zero Forcing ReceiverIn this section, we will calculate probability of bit errorfor Zero-Forcing receiver (ZF) [18,19] to examine theperformance of our space-time-frequency spreading. ZFis considered in our paper, because of its simpler design.ZF is more affordable in terms of computational com-plexity and lower cost. As well, the impact of noiseenhancement from ZF is reduced due to the inherent


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property of avoiding poor channel quality using space,time and frequency spreading. Without the loss of gen-erality, the signal from first user is regarded as thedesired user and the signals from all other users asinterfering signals. With coherent demodulation, thedecision statistics of user 1 symbol is given as,

x1 = (HH1 H1)−1H

H1 y

=(sH1 H

HHs1

)−1sH1 H

H(Hs1x1 + Hs2x2 + . . . + HsMxM + n

) (16)

Then, the desired signal, multiple access interference(MAI) and the noise are S, I, h, respectively.

S = x1 (17)

I =(sH1 H

HHs1

)−1 M∑k=2

(sH1 H

HHsk

)xk (18)

η =(sH1 H

HHs1

)−1sH1 H

Hn (19)

To compute signal-to-interference noise ratio (SINR),which is defined as Γ, we will assume S, I, h are uncor-related,

�H =E|S2|

E[|η|2] + E[|I|2]=

E[S2]

σ 2I + σ 2

η

(20)

where, xk(MAI) are assumed to be mutually indepen-

dent, therefore input symbols {xk}Mk=1 are assumed Gaus-sian with unit variance. The expectation is taken overthe user symbols xk, k = 1,..., M and noise k.Since the effective channel is denoted as Hn = Hsk ,

then

HHk Hl = sHk H

HHsl (21)

Desired signal average power is defined as,

E[S2] = 1 (22)

Multiple access interference (MAI) is defined as,

σ 2I = E

[(sH1 H

HHs1

)−2 M∑k=2

∣∣∣sH1 HHHsk

∣∣∣2]

= E

[∣∣∣HH1 H1

∣∣∣−2 M∑k=2

∣∣∣HH1 Hk

∣∣∣2] (23)

where HH1 Hk is the projection of H1 on Hk . Without

loss of generality, let’s assume in Equation (23) that

H1 =√H

H1 HkPe1 , where P is any permutation matrix,

and e1 is the 1-st column of the I identity matrix,

σ 2I = E

[∣∣∣HH1 H1

∣∣∣−2 M∑k=2

∣∣∣∣√H

H1 H1 eH1 (P

HHk)

∣∣∣∣2]

= E

[∣∣∣HH1 H1

∣∣∣−1 M∑k=2

∣∣∣eH1 PHHk

∣∣∣2]

=(

1M − 1

) M∑k=2

∣∣zk∣∣2/(1

NtNc

)NcNt∑m=1

∣∣xm∣∣2(24)

where∣∣zk∣∣2 and

∣∣xm∣∣2 are chi-squared random vari-

ables, as Equation (21) shows that Hk is gaussian ran-dom variable ~ CN(0, 1)Noise average power is defined as,

σ 2η = E

{(sH1 H

HHs1

)−2sH1 H

HnnHHs1

}

= E[∣∣∣HH

1 H1

∣∣∣−2sH1 H

HHs1

]σ 2

= E[∣∣∣HH

1 H1

∣∣∣−1]

σ 2

= σ 2/(

1NtNc

)NcNt∑m=1

∣∣xm∣∣2(25)

Therefore, the probability of error can be simply givenby

P(e) = Q(√

�) (26)

From Equations (22), (24), and (25), we can obtainSINR

� =E[S2]

σ 2I + σ 2

η

=1(

1M − 1

)M∑k=2

∣∣zk∣∣2(1

NtNc

)NcNt∑m=1

∣∣xm∣∣2+

σ 2(1

NtNc

)NcNt∑m=1

∣∣xm∣∣2=

1(1/Fa,b

)+(σ 2

/χ2

)

(27)

where Fa,b is F-distribution random variable (ratiobetween two chi-squared random variables) where a =NtNc and b = M - 1 degrees of freedom, and c2 is chi-squared random variable with NtNc degrees of freedom.It is clear that when interference is small enough, themost dominant part will be the c2 which agrees withRaleigh fading channel where no MUI exists. When the


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MUI dominates channel noise, Equation (27) can beapproximated as Γ = Fa,bNow, by assuming all users are scheduled to transmit

at similar symbol rates Rs at a time instance, we couldcalculate BER using Equation (26) by statistically aver-aging over the probability density function of Fa,b (seeAppendix), i.e., by substituting Equation (27) in Equa-tion (26).

Pe =∫

p(Fa,b)Q(√

�Fa,b)dFa,b

≤ (P/σ 2)baabb

β(b, a)

∫ ∞

0

ya−1((P/σ 2)b + ay

)a+b(16e−y +

12e−

43 y)dy

(28)

In Equation (28) y is SINR defined in Equation (27),P/s2 is the signal-to-noise ratio (SNR), a is equal toNtNc, and b = M - 1.In Figure 3, we compare the SINR PDFs for our pro-

posed scheme defined by Equation (27) and 2D OFDM-CDMA [6]. It is clear that the probability of SINR hashigher values in our proposed OFDM-CDMA systemcompared to 2D OFDM-CDMA system, which meansthat the average SINR for our proposed system will bemore likely to be higher than that of the 2D OFDM-CDMA system. This is confirmed by numerically evalu-ating P(SINR <20 dB) for our proposed system and 2DOFDM-CDMA system, which are 0.6479 and 0.5468respectively. This improvement will lead to better multi-user diversity gains. In Figure 4, the PDF curves of theproposed scheme with various number of users are

provided. From Figures 3 and 4, it can be seen that theSINR PDF curve of the proposed scheme with 32 usersis close to that of the 2D scheme with 16 users. Thisshows that the proposed scheme supports twice thenumber of users in a system with 4 transmit and 4receive antennas. It is also interesting to note that thesimulated results match well with our analytical resultsprovided by Equation (27). Figure 4 shows that the aver-age SINR is 20 dB for all users, and the most probableSINR decreases as the number of users increases.

E. ComplexityThe process of spreading each bit on space, time andfrequency in a parallel manner was considered to be acomplicated issue [20]. However, the proposed OFDM-CDMA has efficient mapping in bit allocation in space,time and frequency without degrading overall systemperformance, and therefore it is less complex. In otherOFDM-CDMA systems, RAKE receiver is widely usedto take advantage of the entire frequency spread of aparticular bit, that adds to overall system hardware com-plexity. In our proposed open-loop MIMO OFDM-CDMA, RAKE receiver is not needed as each bit isspread in time and frequency, occupying different timeand frequency slots, where each bit is spread to ensurefrequency independence as shown in Figure 2. Also,other systems that use space-time-frequency (STF) cod-ing as in [16], has more complexity than our proposedsystem. Their spreading technique uses space-time block

SINR (dB.)

%

Proposed OFDM CDMA (sim.)

2D OFDM CDMA (sim.)

Proposed OFDM CDMA (theo.)

2D OFDM CDMA (theo.)

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

20

Figure 3 Probability density function for SINR for Es/s2 = 20 dB for our proposed scheme (solid) and 2D OFDM-CDMA (dotted), forboth simulated and calculated (Nt, Nr = 4, Nc = 16, and M = 16).


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codes or space-time trellis codes and then uses subcar-rier selectors to map signals to different OFDM fre-quency subcarriers. Our proposed STF spreadingmethod does not involve coding or precoding, just bitspreading to maintain signal orthogonality and maxi-mize diversity at receiver side. Figure 5 shows that ourproposed system has better performance than [16], byimproving both diversity and coding gains.

4. Simulation resultsComputer simulations were carried out to investigatethe performance gain of the proposed open-loop MIMOOFDM-CDMA system with joint space-frequency-timespreading. The channel is a multipath channel modelledas a finite tapped delay line with L = 4 Rayleigh fadingpaths. Walsh-Hadamard (WH) codes are utilized forboth space and time spreading. Different codes are

SINR (dB.)

%

64 Users32 Users16 Users8 Users1 User

1 User

64 Users

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

Figure 4 Probability density function for SINR for Es/s2 = 20 dB for our proposed scheme with different number of users.

SNR (dB)

SER

Proposed OFDM CDMASTF Block codes [16]

0 2 4 6 8 10 12 14 1610−4

10−3

10−2

10−1

100

Figure 5 SER vs SNR comparison of the proposed OFDM-CDMA scheme (dotted) and 2D STF block codes [16](solid) with 2Tx, 1Rx, Nf

= 64, L = 4 (multiray channels).


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assigned to different users. The OFDM super-framecontains 16 OFDM symbols, which is equal to thelength of the time spreading code Nc = 16, where eachOFDM symbol has 128 subcarriers. The channel estima-tion is assumed to be perfect, quadrature phase-shiftkeying (QPSK) constellation is used. We assume aMIMO channel with Nt = 4 transmit antennas and Nr =1, 2, 4 receive antennas. It is assumed that the meanpower of each interfering user is equal to the meanpower of the desired signal. The maximum number ofusers allowed by the system is Nc(min(Nt, Nr)).Figure 6 shows the Bit error rate (BER) performance

of OFDM-CDMA versus the average Es/N0 with differ-ent number of active users with slow fading channel for4 transmit and 4 receive antennas, where the solid linesstand for our proposed scheme, while the dotted linestands for the double-orthogonal coded (DOC)-STFS-CDMA scheme proposed in [9]. It is clear that ourscheme has better resiliency to the frequency selectivityof the channel due to the inherent property of avoidingpoor channel quality using the proposed space, time andfrequency spreading.Figure 7 shows the Block error rate (BLER) perfor-

mance of OFDM-CDMA versus the average Es/N0 withdifferent number of active users with slow fading chan-nel for 4 transmit and 4 receive antennas, where thesolid lines stand for our proposed scheme, while thedotted line stands for the 2D OFDM-CDMA. It isobvious that when we spread our signal on space, time,

and frequency, we had better performance as we wereable to maintain maximum achievable spatial diversityon the receiver side.Figures 8 and 9 show the BER performance of OFDM-

CDMA versus the average Eb/N0 for 1 and 2 receiveantennas, respectively. In our simulations, we compareour proposed scheme with 2D OFDM-CDMA describedin [6]. The maximum number of users allowed in Fig-ures 8 and 9 are 16, and 32 users, respectively. Simula-tion results show that our proposed system has betterperformance, but as the number of users increases tomax, diversity advantages are decreased due to the factof diversity/multiplexing trade-off. On the other hand,when we decrease the number of receive antennas toone, our proposed scheme is superior because we areable to maintain maximum possible spatial diversity onthe receiver side, but the other scheme is not able tocompensate when reducing the number of receiveantennas to one. Comparing both figures, our schemehas greater gains when reducing receive antennas from2 to 1, offering better diversity/multiplexing trade-off.Also, Figure 8 confirms that the results shown for SINRpdf in Figure 3 holds for 1 receive antenna, as BERcurves for the 2D OFDM-CDMA with 4 users coincideswith our proposed system but with 8 users. Therefore,our proposed scheme has twice the throughput with thesame BER performance.Figure 10 shows system user throughput. The pro-

posed system is able to have higher number of users

Eb/N0 (dB)

BER

1 User

16 User

32 User

48 User

1 User

16 User

32 User

48 User

0 2 4 6 8 10 12 14 16 18 2010−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure 6 BER comparison for OFDM-CDMA system with 4Tx, 4Rx with our proposed scheme (solid) and DOC-STFS-CDMA [9](dotted) ina slow fading frequency-selective environment.


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because we are able to fully exploit the spatial dimen-sion of the channel. This leads to lower BLER, andhigher diversity gains, that will contribute to increasednumber of users without degrading the system perfor-mance as shown in the SINR pdf graphs in Figure 3.The system is able to maintain reliable communicationwith reasonable super-frame drops up to 32 users, ascompared to 2D OFDM-CDMA. Also, we are able to

maintain double number of users with same BLER per-formance. At 32 users, the system is able to fully utilizethe channel at SNR = 10 dB.In Figure 11, we compare the upper-bound result in

Equation (28) with simulation result. It is clear that thetight bound we proposed matches our simulated resultsperfectly.

SNR dB

FER

1 User8 Users16 Users32 Users64 Users1 User8 Users16 Users32 Users64 Users

0 2 4 6 8 10 12 14 16 18 20 2210−6

10−5

10−4

10−3

10−2

10−1

100

Figure 7 BLER comparison for OFDM-CDMA system with 4Tx, 4Rx with our proposed scheme (solid) and 2D OFDM-CDMA (dotted) ina slow fading frequency-selective environment.

Eb/N0

BER

1 User8 User16 User1 User8 User16 User

0 5 10 15 2010−6

10−5

10−4

10−3

10−2

10−1

100

Figure 8 BER comparison for OFDM-CDMA system with 4Tx, 1Rx of the proposed scheme (solid) and 2D OFDM-CDMA (dotted) in aslow fading frequency-selective environment.


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5. ConclusionIn this paper, we have proposed an open-loop MIMOOFDM-CDMA scheme using space-time-frequencyspreading (STFS), in the presence of frequency-selectiveRayleigh-fading channel. The BER and BLER perfor-mance of the OFDM-CDMA system using STFS hasbeen evaluated taking into consideration diversity/

multiplexing trade-off over frequency-selective Rayleigh-fading channels.We showed that our proposed system gives the advan-

tage of maintaining maximum achievable spatial diversityon the receiver side in the case of slow frequency-selectiveRayleigh-fading channels. Also, by appropriately selectingthe system parameters Nt, and Nc, the OFDM-CDMA

Eb/N0

BER

1 User12 User16 User32 User

1 User12 User16 User32 User

0 2 4 6 8 10 12 14 16 18 20 2210−6

10−5

10−4

10−3

10−2

10−1

100

Figure 9 BER comparison for OFDM-CDMA system with 4Tx, 2Rx of the proposed scheme (solid) and 2D OFDM-CDMA (dotted) in aslow fading frequency-selective environment.

SNR dB

NumberofUsers

Proposed OFDM-CDMA

2D OFDM-CDMA

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

Figure 10 System throughput comparison for OFDM-CDMA system with 4Tx, 4Rx of the proposed scheme (solid) and 2D OFDM-CDMA (dotted) in a slow fading frequency-selective environment.


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system using STFS is rendered capable of achieving highernumber of users than other schemes. System throughputhas increased as our proposed system was capable ofachieving higher SINR than other schemes at similarSNRs. Higher diversity gains than other systems wereshown, when number of receive antennas are reduced toone, as our system was able to maximize the number ofdegrees of freedom, by exploiting the spatial dimension ofthe channel. Our system showed great improvements, insystem performance and throughput compared to othersystems without sacrificing complexity.

AppendixUpper bound for PeIn this section, we will show the numerical evaluationthat led to Equation (28). When M is large enough, theinterference component will be the dominant compo-

nent,( 1M−1

)∑Mk=2

∣∣zk∣∣2 > σ 2 and Equation (27) can be

expressed as follows

SINR = y =P/σ 2

(1/x)

(29)

where x is fa,b-distribution with a = NtNc and b = M -1 degrees of freedom, the probability density function fa,b(x) is defined as

fa,b(x) =aabb

β(a, b)xa−1

(b + ax)a+b(30)

Substituting Equation (30) into Equation (29), weobtain the probability density function for SINR as,

f (y) =

((P/σ 2)

baabb

β(b, a)

)ya−1(

(P/σ 2)b + ay

)a+b (31)

As mentioned earlier, probability of error is defined as,

Pe =∫ ∞

0f (y)Q(

√y)dy (32)

In [21], it was shown that erfc(.) can be approximatedto a tighter bound than Chernoff-Rubin bound,

Q(√y) ≤ 1

6e−y +

12e−

43 y (33)

By substituting Equations (31) and (33) into Equation(32), we obtain the probability of error Pe,

Pe ≤ (P/σ 2)

baabb

β(b, a)

∫ ∞

0

ya−1((P/σ 2)b + ay

)a+b(16e−y +

12e−

43 y)dy (34)

Competing interestsThe authors declare that they have no competing interests.

Received: 12 February 2011 Accepted: 23 December 2011Published: 23 December 2011

Eb/N0(dB)

BER

Simulation

Theoretical

0 2 4 6 8 10 1210−6

10−5

10−4

10−3

10−2

10−1

Figure 11 Probability of error for analytical (solid) vs simulation (dotted).


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