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doi:10.1016/j.si

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Signal Processing 86 (2006) 3934–3940

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Fast communication

Effect of carrier-frequency offset on the performance of group-orthogonal multicarrier CDMA systems

Wei Yanga,b,�, Jun-Ying Liua, Shi-Xin Chengb

aBeijing Jiaotong University, Shangyuncun 3], Xizhimenwai, Haidian District, Beijing 100044, PR ChinabSoutheast University, PR China

Received 24 September 2005; received in revised form 7 July 2006; accepted 2 August 2006

Available online 31 August 2006

Abstract

In this letter, the downlink performance of a group-orthogonal multicarrier code-division multiple-access (GO-MC-

CDMA) system with binary-phase shift keying (BPSK) modulation is investigated, when communicating over Rayleigh

fading channels in the presence of carrier-frequency offset (CFO). The bit error rate (BER) performance of the GO-MC-

CDMA is compared with that of a conventional MC-CDMA for given system resources and for a given value of the CFO.

Our study and results show that the GO-MC-CDMA system is capable of achieving a better BER performance than that

of a conventional MC-CDMA scheme in the presence of CFO, when the number of users per group is relatively small. By

contrast, when the number of users per group is high, two MC-CDMA systems considered achieve similar BER

performance.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Group-orthogonal MC-CDMA; Carrier-frequency offset; Bit error rate; Fading channel

1. Introduction

Multicarrier code-division multiple-access (MC-CDMA) system has been proposed as one of thecandidates for the future generations of wirelesscommunications because of its attractive featuressuch as high data rate, high capacity, low complex-ity of implementation due to using fast Fouriertransform (FFT)-based carrier modulation, etc. [1].In order to further improve the performance of theconventional MC-CDMA systems a novel group-orthogonal MC-CDMA has recently been proposed

e front matter r 2006 Elsevier B.V. All rights reserved

gpro.2006.08.002

ing author. Beijing Jiaotong University, Shang-

himenwai, Haidian District, Beijing 100044, PR

86 10 51682162.

ess: wyang@center.njtu.edu.cn (W. Yang).

[2]. Specifically, in group-orthogonal (GO)-MC-CDMA, the data of a group of users are transmittedon a set of carefully selected subcarriers [2].

Multicarrier systems are usually sensitive to thecarrier-frequency offset (CFO), which generallygives rise to significant performance degradationespecially for the conventional multicarrier systems[3–7]. In this letter, we investigate the effect of CFOon the downlink performance of the GO-MC-CDMA system. By contrast, in [2] the uplinkperformance of the GO-MC-CDMA has beeninvestigated without considering the CFO. In thisletter, the closed-form bit error rate (BER) expres-sion for the downlink GO-MC-CDMA systemusing binary-phase shift keying (BPSK) modulationis derived, when assuming that the sub-carriers conveying the same data bits experience

.

ARTICLE IN PRESSW. Yang et al. / Signal Processing 86 (2006) 3934–3940 3935

independent Rayleigh fading in addition to theCFO. However, the derived closed-form bit errorrate (BER) expression can be employed for evaluat-ing the BER of the GO-MC-CDMA, even when thesubcarriers conveying the same data bits exhibitslight correlation. Specifically, as shown by ournumerical and simulation results, the theoreticalevaluated BER performance provides a goodapproximation to that obtained by simulation,provided that the channel correlation coefficientbetween two adjacent subcarriers is sufficiently low,for example, lower than 0.3.

Let us first describe the GO-MC-CDMA schemein the context of the downlink transmission, as wellas the channel model.

2. Downlink GO-MC-CDMA system and channel

model

The block diagram of a downlink GO-MC-CDMA system is shown in Fig. 1. In the consideredGO-MC-CDMA the entire available bandwidth isutilized associated with N subcarriers, and thespacing between two adjacent subcarriers is 1/T,where T ¼ NTc with T and Tc denoting both thesymbol duration and chip duration, respectively. InGO-MC-CDMA the N su-bcarriers are partitionedinto G groups, each group hence has M ¼ N/Gsubcarriers, if N ¼MG. Let us assume that theBPSK signal ag,k(i) represents the ith transmittedsymbol of the kth user in the gth group. Let ck ¼

ck;1; ck;2; . . . ; ck;M

� �be the spreading code for the kth

user. Then, as shown in Fig. 1, ag,k(i) is first spread

ag,k (i) Subcarrier

SelectionP

Group

Demux⊗

⊗⊗

⊗

⊗ ⊗

•••

•••

•••

•••

*h1,1

*h1,2

*h1,M

∑

v1,1

ck,1

ck,2

ck,M

1,1c

1,2c

c1,M

⊗

⊗

⊗

Fig. 1. Simplified transmitter and receiver diagram of a downlink GO

in the frequency domain using ck. After thefrequency domain spreading, a set of subcarriers isselected, in order to transmit the users’ signals in thegth group.

We assume that the mth subcarrier of the gthgroup in the GO-MC-CDMA system experiencesindependent flat fading. Let the channel gain in thefrequency domain be expressed as

hg;m ¼ bg;mejfg;m ; g ¼ 1; . . . ;G; m ¼ 1; . . . ;M,

(1)

where bg,m and fg,m are, respectively, the amplitudeand phase responses of the fading channel asso-ciated with the mth subcarrier of the gth group.In our analysis, bg,m is assumed to be theindependent, identically distributed (i.i.d.) Rayleigh

random variable (RV) with E b2g;mh i

¼ s2, where

E [ � ] denotes the mathematical expectation. Bycontrast, fg,m is assumed to be the uniformlydistributed in [0,2p). Furthermore, we assume thatthe symbol duration T is significantly longer thanthe delay spread of the channel, resulting in thateach subcarrier signal experiences flat fading.Consequently, when the K number of GO-MC-CDMA signals is transmitted over the above-described fading channels in the presence of CFO,the received baseband equivalent signal after the CPremoval [2,8] can be represented as

rðtÞ ¼Xþ1

i¼�1

XG

g¼1

XK

k¼1

XMm¼1

ffiffiffiffiSp

bg;mag;kðiÞck;mu t� iTð Þ

� ej 2pDf gþ m�1ð ÞG��½ �ðt�iTÞþfg;mf g þ n tð Þ ð2Þ

AWGN

/S

CP

RemoveFFTr(t)

•••

Other users

s(t)

channel

S/P

CPInsertion

⊕

⊕

-MC-CDMA system in the context of user 1 in the first group.

ARTICLE IN PRESSW. Yang et al. / Signal Processing 86 (2006) 3934–39403936

where u(t) represents the rectangular pulse wave-form defined over [0,T], S is the transmitted powerof a user with respect to one subcarrier, n(t) is theAWGN with single-sided power spectrum density ofN0. Furthermore, in (2) e represents the CFOnormalized by the spacing Df ¼ 1/T between twoadjacent subcarriers.

3. Bit error rate analysis

The receiver schematic diagram for the GO-MC-CDMA is also shown in Fig. 1. As shown in Fig. 1,after removing the CP, S/P conversion and FFT-based subcarrier demodulation, the received sam-ples belonging to different groups of subcarriers aredemultiplexed. Following the demultiplexing, max-imum ratio combining (MRC) is employed in orderto form the decision variables for detecting thedesired signals. Consequently, when assuming thatthe channel coefficients are perfectly estimated, theith decision variable v1,1(i) for signals of user 1 in thefirst group is given by

v1;1ðiÞ ¼1

T

Z T

0

rðiÞXMm0¼1

c1;m0h�1;m0e

�j2pDf m0�1ð ÞGþ1½ �t dt

¼ Dþ Zþ ICIþMAI0 þMAI1 þMAI2,

ð3Þ

where Z is a Gaussian RV with zero mean and a

variance of s2Z ¼ N0

PMm0¼1b

21;m0

� �.4T , while D

represents the desired output given asffiffiffiffiSp

a1;1ðiÞsin ðp�Þ

p�

PMm0¼1b

21;m0 . Furthermore, in (3) the

different types of interference are analyzed asfollows.

A.

Self-intercarrier interference from the othersubcarriers of the same group: ICIIn (3), ICI represents the self-intercarrier inter-ference imposed by the subcarriers within thesame group of the desired subcarrier. The ICIcan be obtained from (3) by letting k ¼ 1, g ¼ 1and m 6¼m0, yielding

ICI ¼ffiffiffiffiSp

a1;1ð0Þsin p�ð Þ

p�

XMm0¼1

XMm¼1

mam0

b1;m0 b1;m c1;m0 c1;m

� cos f1;m0 � f1;m � p�� � �

m0 �mð ÞG þ �

� .

ð4Þ

With the aid of the central limit theorem, whenthe value M is high the ICI can be approximated

by a Gaussian RV with mean zero. The varianceof the ICI can be obtained by considering thestatistics of f1;m0 ;f1;m and b1;m, respectively. Itcan be shown that, conditioned on {b1,m0}, thevariance of the ICI can be expressed as

s2ICI ¼Ss2

2p2XMm0¼1

XMm¼1

mam0

sin2 p�ð Þ

m�mð ÞG þ �½ �2b21;m0 . (5)

B.

Type-0 multiple-access interference: MAI0In (3) MAI0 represents the multiple-accessinterference imposed by the interfering users inthe same group of the desired user associatedwith g ¼ 1 and m ¼ m0 for k 6¼1. The MAI0 canbe expressed as

MAI0 ¼ffiffiffiffiSp sin ðp�Þ

p�

XMm0¼1

b21;m0XK

k¼2

a1;kðiÞc1;m0ck;m0 .

(6)

Since the data transmitted by the (K�1) inter-fering users in the same group are independentand mutually uncorrelated, it can be shown that,when KKb1, MAI0 can be approximated by aGaussian RV with mean zero and a varianceconditioned on {b1,m0} given by

s2MAI0¼ SðK � 1Þs2

sin2 ðp�Þ

ðp�Þ2XMm0¼1

b21;m0 . (7)

C.

Type I multiple-access interference: MAI1The multiple-access interference MAI1 in (3) isalso imposed by the interfering users in the samegroup as the desired user, but associated withthe subcarriers different from the considered.MAI1 can be obtained from (3) with k 6¼1, g ¼ 1and m 6¼m0, yielding

MAI1 ¼ffiffiffiffiSp sinðp�Þ

p�

XK

k¼2

XMm0¼1

XMm¼1m0¼1

a1;kðiÞc1;m0ck;mb1;m0b1;m

� cos f1;m0 � f1;m � p�� � �

ðm0 �mÞG þ �

� .

ð8Þ

Again, when Kb1, MAI1 can be approximatedas a Gaussian RV with mean zero, and avariance conditioned on {b1,m0} given by

s2MAI1¼

SðK � 1Þs2

2p2XMm0¼1

XMm¼1

mam0

sin2 ðp�Þ

ðm0 �mÞG þ �½ �2b21;m0 .

(9)

ARTICLE IN PRESSW. Yang et al. / Signal Processing 86 (2006) 3934–3940 3937

Type-II multiple-access interference: MAI2

D. It can be shown that in GO-MC-CDMA, whenpresence of CFO, the orthogonality amongdifferent groups may be destroyed, which resultsin interference from the other groups. Therefore,in (3) MAI2 represents the inter-group inter-ference, which can be obtained by letting g 6¼1,and can be expressed as

MAI2 ¼ffiffiffiffiSp sin ðp�Þ

p�

XG

g¼2

XK

k¼1

XMm0¼1

XMm¼1

ag;kðiÞ

�ck;mck;m0bg;mb1;m0�

gþ ðm0 �mÞG � �

�

� cos p gþ m0 �mð ÞG � �½ � þ fg;m0 � f1;m

� � �,

ð10Þ

which, when Gaussian approximation is em-ployed, can be viewed as a zero-mean GaussianRV with the variance conditioned on {b1,m0}given by

s2MAI2¼

Ss2

2p2XG

g¼1

XMm0¼1

XMm¼1

K sin2 ðp�Þ

gþ ðm0 �mÞG þ �½ �2b21;m0 .

(11)

Consequently, when the above-mentioned Gaus-sian approximation is employed, v1,1 in (3) can beapproximated as a Gaussian RV with mean givenby D, and a variance conditioned on {b1,m0} given by

s2I ¼ s2ICI þ s2MAI0þ s2MAI1

þ s2MAI2þ s2Z

¼XMm0¼1

Am0b21;m0 , ð12Þ

where by definition, Am0 is given by

Am0 ¼Ss2

2p2XMm¼1

mam0

sin2 ðp�Þ

ðm0 �mÞG þ �½ �2

þ SðK � 1Þs2sin2 ðp�Þ

ðp�Þ2þ

N0

4T

þSðK � 1Þs2

2p2XMm¼1

mam0

sin2 ðp�Þ

ðm0 �mÞG þ �½ �2

þSs2

2p2XG

g¼1

XMm¼1

K sin2 ðp�Þ

gþ ðm0 �mÞG þ �½ �2; ð13Þ

which represents the total interference coefficientassociated with the m0th subcarrier. Let us approx-imate Am0 by a positive i.i.d. RV with mean mA and avariance of s2A. Then, it can be shown that, when

MX32 and sA

mAp0:05, according to (54a) and (54b)

of [8], s2I can be closely approximated as

s2I ¼XMm0¼1

Am0b21;m0 � mA

XMm0¼1

b21;m0 . (14)

Therefore, the BER of the down-link GO-MC-CDMA can be expressed as

PðeÞ ¼

Z 10

Qffiffiffiffiffiffiffi2gp

q� �f gp

gp

� �dgp (15)

where Q xð Þ ¼ 1=ffiffiffiffiffiffi2pp R1

xexp ð�t2=2Þdt, gp repre-

sents the conditional signal-to-interference-plus-noise ratio (SINR), which is defined as

gp ¼E2ðDÞ

2s2I

�S sin c2ðp�Þ

2mA

PMm0¼1b

21;m0 , while f gp

gp

� �is

the probability density function (pdf) for gp.

Since b1;mðm ¼ 1; . . . ;MÞ �

are the i.i.d. Rayleigh

RV’s, the summation ofPM

m¼1b21;m hence obeys the

chi-square distribution with 2M degree of freedom.Furthermore, according to [9], we can obtain thepdf of gp, which can be expressed as

f gpgp

� �¼

1

M � 1ð Þ!gM�1c

gM�1p e�gp=gc ; gpX0, (16)

where gc ¼ Ss2 sin c2ðp�Þ=mA.Finally, upon applying (16) to (15), the closed-

form BER expression for the GO-MC-CDMAusing BPSK modulation in the presence of CFOcan be expressed as [9]

PðeÞ ¼ð1� mÞ

2

� M XM�1j¼0

M � 1þ j

j

!ð1þ mÞ

2

� j

,

(17)

where by definition, m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigc

�1þ gc

� �q.

Let us now provide a range of numerical andsimulation results in order to characterize theperformance of the GO-MC-CDMA.

4. Simulation and numerical results

The effect of CFO on the BER performance ofthe GO-MC-CDMA system can be investigated byevaluating (17) associated with some given CFOs. Inthis section, the BER performance is also comparedwith that of the conventional MC-CDMA scheme.For a fair comparison, in our investigation weassume that BPSK was employed by both schemes.Furthermore, we assume that for both schemes thetotal system bandwidth, total number of subcarriersand the channel statistics were the same. Note thatin GO-MC-CDMA the same Walsh Hadamard

ARTICLE IN PRESSW. Yang et al. / Signal Processing 86 (2006) 3934–39403938

code matrix was chosen as the spreading codes forall the groups. The length of the spreading codes forGO-MC-CDMA was equal to M, which is thenumber of subcarriers in each group. By contrastthe length of the spreading code for MC-CDMAwas equal to N of the total number of subcarriers inthe MC-CDMA. As we mentioned previously inSection 2, the number of users per group in the GO-MC-CDMA system was K and the total number ofusers for both systems considered was K�G, where,again, G represents the number of groups in theGO-MC-CDMA system.

Fig. 2 shows the simulation and numerical BERperformance vs. the normalized CFO e for the GO-MC-CDMA system. In our simulations, the methodproposed in [10] was employed to generate thecorrelated Rayleigh fading channels. In Fig. 2 rdenotes the correlation coefficient between twoneighbor subcarriers which was assumed the valuesof r ¼ 0, 0.3 and 0.5. From the results of Fig. 2 itcan be observed that the computed BER resultsagree well with the simulation results when thesubcarriers experience independent fading corre-sponding to r ¼ 0. Furthermore, we can see that thetheoretical BER results can be a good approxima-tion to the practical channels when the correlationcoefficient r is lower than 0.3. However, whenincreasing the correlation coefficient, for example,

10-4 10-3 110-5

10-4

10-3

10-2

10-1

100

theoretical

simulation ρ=0

simulation ρ=0.3

simulation ρ=0.5

N=64,M=32,K=

N=128,M=64,K

BE

R

Fig. 2. BER vs. the CFO performance of the GO-MC-CDMA systems w

SNR per bit of Eb/N0 ¼ 20 dB.

r ¼ 0.5, the theoretical BER performance becomeslower than the actually achievable BER perfor-mance. This implies that, in the case of existinglarger correlation coefficients, the proposed analysisprovides a lower bound on the BER.

Fig. 3 shows the BER vs. Eb/N0 performance withrespect to different e values for both the GO-MC-CDMA and MC-CDMA systems, when the corre-lation coefficient was r ¼ 0. It can be observed fromFig. 3 that the GO-MC-CDMA scheme outper-forms the MC-CDMA system when there exists noCFO or CFO is very small, for example, e ¼ 0.1.However, when the e value increases, for example,e ¼ 0.2 and 0:4, the BER performance of bothsystems decreases dramatically.

Fig. 4 shows the BER vs. e performance withrespect to the number of users K, when thecorrelation coefficient is r ¼ 0. It can be foundfrom Fig. 4 that GO-MC-CDMA scheme outper-forms MC-CDMA system in the presence of CFOwhen the number of K is small, for example, K ¼ 4.However, when the number of K increases, forexample, K ¼ 8 and 10, the BER performance ofboth systems is almost the same in the presence ofCFO. The reason is that the interference from thesame group is the main factor that causes theperformance degradation in the presence of CFOwhen the number of users per group is small, while

0-2 10-1 100

ε

4

=6

hen communicating over Rayleigh fading channels, at an average

ARTICLE IN PRESS

Fig. 4. BER vs. the CFO performance of both the GO-MC-

CDMA and the MC-CDMA systems, when communicating over

independent Rayleigh fading channels corresponding to r ¼ 0 at

an average SNR per bit of Eb/N0 ¼ 20 dB.

Fig. 5. BER vs. the number of total users of both the GO-MC-

CDMA and the MC-CDMA, when communicating over

independent Rayleigh fading channels corresponding to r ¼ 0

and using the parameters of Eb/N0 ¼ 20 dB, N ¼ 64, M ¼ 32.

Fig. 3. BER vs. the SNR per bit performance of both the GO-

MC-CDMA and the MC-CDMA, when communicating over

independent Rayleigh fading channels corresponding to r ¼ 0

using the parameters of N ¼ 64, K ¼ 4 and M ¼ 32.

W. Yang et al. / Signal Processing 86 (2006) 3934–3940 3939

the interference between different groups becomesthe main factor when the number of users per groupincreases. Therefore, the BER performance of thetwo systems tends to be the same in the presence ofCFO for large number of users. From Fig. 4, we canalso see that, when e is lower than 0.1, the BERperformance of both systems degrades very slightly.When e is larger than 0.1, the BER performance ofthe two systems starts to deteriorate.

Fig. 5 shows the BER performance vs. thenumber of total users for both the GO-MC-CDMA

and MC-CDMA systems, when the correlationcoefficient is r ¼ 0. We can see that the BERperformance of both system degrades gradually asthe number of total users increases when e is small.For large value of e, BER is large, even for a smallnumber of users. Again, it can be observed that fora small number of users the GO-MC-CDMAscheme outperforms MC-CDMA system, when thenumber of users increases the BER performance ofthe two systems is almost the same, both in thepresence of CFO.

5. Conclusions

In this letter, the BER performance of the down-link GO-MC-CDMA system using BPSK modula-tion has been investigated when communicating overRayleigh fading channels in the presence of CFO.A closed-form of BER expression has been obtainedfor the GO-MC-CDMA and it has been verifiedthrough simulations. From our numerical andsimulation results, we can conclude as follows.Firstly, in the presence of CFO, the BER perfor-mance of the downlink GO-MC-CDMA system isbetter than that of the conventional MC-CDMAscheme when the number of users per group is low.These two systems achieve similar BER performancewhen the number of users per group is relativelyhigh. Secondly, when the normalized CFO e is lessthan 0.1, the effect of the CFO on the BERperformance of the two systems is small. However,

ARTICLE IN PRESSW. Yang et al. / Signal Processing 86 (2006) 3934–39403940

when the normalized CFO e is larger than 0.1, theBER performance of both systems degrade signifi-cantly.

Acknowledgements

The authors would like to thank the reviewers fortheir constructive comments and suggestions. Thiswork was supported by the National NaturalScience Foundation of China (Grant no. 60572036).

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