IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 769

Effect of Overlapping Between Successive Carriers of Multicarrier CDMA onthe Performance in a Multipath Fading Channel

Said M. Elnoubi and Ahmed El-Beheiry

Abstract—The effect of the overlapping between successivecarriers of a multicarrier code-division multiple-access system onthe performance in a multipath fading channel is investigated.Transmitted data bits are serial to parallel converted to a numberof parallel branches. On each branch each bit is direct-sequencespread-spectrum modulated and transmitted with certain numberof carriers. The separation between successive carriers is allowedto vary to achieve various overlapping percentages. The systemis analyzed with conventional matched filter receiver. It is shownthat for each number of carriers there exists an optimum over-lapping-percentage at which the performance of the system isoptimized. The results show that a considerable improvement canbe achieved.

Index Terms—Code-division multiple access, fading channels,multicarrier CDMA.

I. INTRODUCTION

M ULTICARRIER (MC) modulation techniques have beenproposed for high bit rate services in fading channels to

reduce intersymbol interference (ISI) and interchip interference(ICI). The performance of orthogonal MC code-division mul-tiple access (MC-CDMA) in nonfading and nonselective fadingchannels was presented in [1] and [2]. The performance in amultipath fading channel was presented in [3]. It was shownthere that as the number of carriers increases, the bandwidth(BW) on each carrier is reduced and it is subject to less resolv-able multipath. With sufficient number of carriers, the condi-tion of single-path fading for each carrier is achieved. It wasshown in [3] that MC system can outperform the RAKE receiverand has the advantage that channel parameters estimation is notneeded. The performance of nonorthogonal MC-CDMA systemwas presented in [4]. For the analysis in [3], successive carrierswere allowed to overlap with an overlapping percentage of 50%,while in [4] successive carriers were not allowed to overlap.Using higher overlapping percentage, the spread-spectrum (SS)processing gain is increased but multiple-access interference ishigher.

Without multipath fading, and for 50% overlapping, allcarriers transmitted by the same user are orthogonal and do notcross-interfere. However, in multipath fading, matched filter(MF) tuning to any carrier and path will get interference

Paper approved by B. Aazhang, the Editor for Spread-Spectrum Networks ofthe IEEE Communications Society. Manuscript received April 15, 1998; revisedMay 9, 2000. This paper was presented in part at the IEEE International Con-ference on Universal Personal Communications, San Diego, CA, October 1997.

S. M. Elnoubi is with the Electrical Engineering Department, Faculty ofEngineering, Alexandria University, Alexandria, Egypt (e-mail: selnoubi@hot-mail.com).

A. El-Beheiry was with the Electrical Engineering Department, Facultyof Engineering, Alexandria University, Alexandria, Egypt. He is now withMobinil, Cairo, Egypt.

Publisher Item Identifier S 0090-6778(01)04077-6.

from other paths of other carriers . Hence,carrier orthogonality is lost in multipath fading and 50%overlapping becomes suboptimum. In this letter, the analysis in[3] is generalized to incorporate variable overlapping betweensuccessive carriers, and the system performance is evaluatedfor different overlapping percentages (from 0% to 95%) anddifferent number of carriers to achieve a single-path fadingper carrier. It is shown that for each number of carriers, thereexists an optimum overlapping percentage at which the systemperformance is optimized. The performance at the obtainedoptimum overlapping percentages is compared to that at 50%overlapping presented in [3] assuming the same total transmis-sion BW and same number of users. The results show that aconsiderable improvement can be achieved in some cases.

This letter is organized as follows. The proposed nonorthog-onal multicarrier system is explained in Section II. The channelmodel is presented in Section III. In Section IV, the interferenceis analyzed based on the standard Gaussian approximation andthe probability of error is derived. In Section V, numerical re-sults are presented and discussed. Finally, conclusions are drawnin Section VI.

II. NONORTHOGONALMC SYSTEM

The system analyzed in this letter was first described in [3].At the transmitting side, the bit stream with bit duration isserial-to-parallel converted into parallel streams. The newbit duration on each stream becomes . Each streamfeeds parallel streams such that the same data exists on thebranches. All data streams are spread by the same code of length

and chip duration such that . One of car-riers is used for binary phase-shift keying (BPSK) modulationof each stream. In [3], the spectra of successive carriers wereallowed to overlap by 50%. In this letter, we consider variableoverlapping such that the separation between successive carriersis , where is allowed to vary between 0.1–2, which re-sults in overlapping percentage of 95% to 0%. Note thatmay be considered as the special case that was analyzed in [3].

The total transmission BW is assumed to be the passbandnull-to-null BW , where is the pseudonoise (PN)code ship duration for single-carrier case . Thetotal system BW in case of overlapping carriers is

(1)

To keep the BW fixed for any selections of and , the PNcode chip duration must follow

(2)

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770 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001

Consequently with , and , the periodof the PN sequence must follow

(3)

Assuming CDMA users, all employing the proposed MCsystem with equal selection of and and the same powerfor all carriers, the transmitted signal of useris given by

(4)

where such that is the bitstream on the identical-bit branches fed from the pth branch inthe transmitter proposed in [3], areindependent and identically distributed (i.i.d.) for all , and. is the transmitted power per carrier, is the th carrier

frequency, is a random phase for each carrier, uniformlydistributed and i.i.d. for all and , and consistsof a periodic train of chips , that takes the values

1. The bit and chip waveforms are rectangular. The carrierfrequencies are related by

(5)

where is the absolute carrier number in the system. Note thatif the relative carrier number within groupis , where

, then the absolute carrier number in the system is

(6)

We will use the receiver proposed in [3, Fig. 2], where the re-ceiver of user employs MF detectors, each tuned andsynchronized to one of the carriers. The MF outputs of the iden-tical-bit carriers are deinterleaved and the decision statistics ofthe same bit are added prior to the threshold device.

III. CHANNEL MODEL

We will use the multipath model proposed in [3]. The com-plex low-pass impulse response of the channel for carrieranduser is given by

(7)

where is the number of resolvable paths,is a complex Gaussian random variable with

zero mean and variance , and isthe delay of theth path of the th user, assumed equal for allcarriers of the same user. The unit energy on the fading processimplies

(8)

where is the number of resolvable paths. We will use uniformmultipath profiles

(9)

When the maximum delay spread of the channel is thenumber of resolvable pathsis given in [5] by

(10)

Assume for some integer and by applying (2)is given by

(11)

where is the number of resolvable paths for the single-car-rier case . Obviously from (11), the number ofresolvable paths depends on both the number of carriersand the separation between two successive carriers. If

(12)

then and the channel is a single-path fading channel foreach carrier.

IV. I NTERFERENCE ANDPROBABILITY OF ERRORANALYSIS

We followed the same approach given in [3] to analyze inter-ference and derive the probability of error in terms of correla-tion functions defined in [6]. Some of the equations of [3] weremodified to take variable overlapping into consideration. For anasynchronous CDMA system with users, and with applyingpower control, the derived probability of error is given by

(13)

where is the sum of Rayleigh random variables with cor-relation coefficient and is given by the following equationusing the identities given in [7] for random codes:

(14)

where is given by

(15)

and is related to the average signal-to-noise ratio (SNR)bythe following equation for equal-gain diversity combining, withthe assumption that only the Rayleigh envelopes of successiveidentical-bit carriers are correlated with correlation coefficient:

(16)For the results presented in the following section, we usedrandom codes because it has been shown in [3] that the averageperformance of 46 gold codes is very similar to the results ofrandom codes.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 771

On evaluating (13), a Monte Carlo integration is performed[8]. The random variable is computer generated for 100 000times, substituted in the , and the sum of the trials is di-vided by 100 000.

V. NUMERICAL RESULTS AND DISCUSSION

Consider a system with ten users and or the closestsmaller integer that produces an integer in (3) when , andchange. The number of userswas taken as 10 and tocompare the results with those in [3]. Using uniform multipathprofile of (9) in (14) gives

(17)

Since in (13) is the sum of Rayleigh random variables,each of variance , the best performance is always achievedwhen . Therefore, the bit-error rate (BER) curves increaseabruptly when changes from 1 to 2 or from 2 to 3.

For a given , the BER is minimum when given by(17) is minimum. For a given and , this occurs whenis maximum. From (3), is increased by decreasingor in-creasing the overlapping percentage. However, when the over-lapping percentage exceeds 50%,increases more rapidly than

, and consequently and the BER increase again. The sim-ulation results show a local BER minimum at 50% overlapping.However, for some cases, there exists a global minimum lessthan this local minimum.

It has been shown in [3] that the performance of an MCsystem in a multipath fading channel is improved by increasingthe number of carriers due to improved frequency diversity,with an abrupt improvement when a reduction in the number ofpaths occurs, this reduction was only achieved by increasing

in [3, eq. (11a)].In the new proposed system, the reduction incan be

achieved by increasing both the number of carriers andthe separation between them , where is allowed tovary between 0.1–2, which results in variable overlappingpercentage of 95% to 0%. In [3], a 50% overlapping was used,i.e., , which results in two paths/carrier forand one path/carrier for . For the new system, we canachieve one path/carrier for with variable overlapping.

The simulation results of Fig. 1 show how the BER perfor-mance is affected by variable overlapping with steps of 5% for

. From this figure, it is clear that for , the optimumoverlapping percentage is 50%, but for there exists an-other optimum overlapping percentage. In the following, theseresults are justified and an analytical method to predict the op-timum overlapping percentage is given.

For , the inequality of (12) gives to make. If , then (3) gives . The next largest value

of making is 19. Substituting in (3) givesand overlapping percentage of 28%. For , the

inequality of (12) gives to make . Substitutingin (3) gives . To make for the maximum

value of , substitute in (3), thus givingcorresponding to 46% overlapping percentage. For , theinequality of (12) gives for . Similarly, by

Fig. 1. BER versus overlapping percentage forM = 1 (SNR = 30 dB).

substituting in (3), corresponding to 57%overlapping percentage. Therefore, it is concluded that 50% isthe optimum overlapping percentage for , while46% and 28% are the optimum overlapping percentages for

and , respectively.The simulation results of Fig. 2 show how the BER perfor-

mance is affected by variable overlapping with 5% steps for. For and , the inequality of (12) gives

to make . If , then (3) gives .The next largest value of making is 39. Substituting

in (3) gives and overlapping percentageof 30%. For and , the inequality of (12) gives

to make . If , then (3) gives .The next largest value of making is 39. Substituting

in (3) gives and overlapping percentage of58%. Similarly, by substituting in (3), corre-sponding to 70% overlapping percentage for and .Therefore, it is concluded that 30% is the optimum overlappingpercentage for and , while 50% is the optimumoverlapping percentage for and . Following sim-ilar calculations for , we found that the largest value ofgiving is 59. The corresponding overlapping percentagesare 59% and 74% for and , respectively. Thus, weconclude that 50% is the optimum overlapping percentage forthese cases.

772 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001

Fig. 2. BER versus overlapping percentage forM > 1 (SNR = 30 dB).

TABLE IRESULTS FORM = 1 AND

S = 2

For the case and , to make(no overlapping). Substituting in (3) gives . Thenext largest value of making and is 19. Sub-stituting in (3), yields . Simulation gives

for this value of . It is not possible to showthis in Fig. 1 because the first point to the left in Fig. 1 corre-sponds to . At this value of and simulationgives , which is higher than previous value. Bydecreasing (or increasing overlapping percentage), the BERdecreases again to a local minimum of at ,then increases abruptly to at . The last localminimum at corresponds to 46% overlapping per-centage, , and . corresponds to 50%overlapping percentage, , and . These results aresummarized in Table I.

From these results, we conclude that the best overlapping is0% corresponding to .

In all the above results, the SNR was taken as 30 dB. Theeffect of the SNR on the BER performance is shown in Figs. 3and 4 for .

Fig. 3. BER versus SNR for optimum overlapping and 50% overlapping.

Fig. 4. BER versus SNR for optimum overlapping and 50% overlapping.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 773

Fig. 3 shows the BER performance of MC system with op-timum overlapping for different values of and , whichis achieved by sufficient interleavingbetween the same bit onidentical-bit carriers. The performance for 50% overlapping ob-tained in [3] is also shown on this figure. For and ,a significant improvement in performance is obtained by usingthe optimum overlapping obtained from previous figures. For

, 50% overlapping is optimum and hence, no further im-provement can be obtained.

Fig. 4 shows the BER performance ifis increased to 0.25.The effect of changing is very slight.

VI. CONCLUSION

The effect of variable overlapping between successive car-riers of a MC-CDMA system in multipath fading was investi-gated in this letter. It is shown that the condition of single-pathfading for each carrier can be achieved by increasing both thenumber of carriers and the separation between successive car-riers. It is shown that for each number of carriers, there existsan optimum overlapping percentage at which the system per-formance is optimized. System complexity may be reduced by

reducing the number of the carriers. Therefore, when takingor for to reduce complexity, the optimum

overlapping can be determined from the presented results.

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[3] , “Performance of orthogonal multicarrier CDMA in a multipathfading channel,”IEEE Trans. Commun., vol. 44, pp. 356–367, Mar.1996.

[4] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMAsystems,”IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996.

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