Precise bit-error-ratio (BER) analysis of an asynchronous QPSK-modulated direct-sequence code-division multiple-access system using random quaternary spreading sequences for transmission over Rayleigh channels is performed based on the characteristic-function approach. Its accuracy is verified by our numerical simulation results and also compared with those of the Gaussian approximation. Index Terms—Asynchronous direct-sequence code-division multiple-access (DS-CDMA), bit-error-ratio (BER), precise, QPSK, quarternary spreading, Rayleigh

Hanzo, L

Liu, X

English

Southampton (e-Prints Soton)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008 239Precise BER Formulas for AsynchronousQPSK-Modulated DS-CDMA SystemsUsing Random Quaternary SpreadingOver Rayleigh ChannelsXiang Liu and Lajos HanzoAbstract—Precise bit-error-ratio (BER) analysis of anasynchronous QPSK-modulated direct-sequence code-divisionmultiple-access system using random quaternary spreadingsequences for transmission over Rayleigh channels is performedbased on the characteristic-function approach. Its accuracy isveriﬁed by our numerical simulation results and also comparedwith those of the Gaussian approximation.Index Terms—Asynchronous direct-sequence code-divisionmultiple-access (DS-CDMA), bit-error-ratio (BER), precise,QPSK, quarternary spreading, Rayleigh.I. INTRODUCTIONTRADITIONALLY, the in-phase and quadrature-phasecomponents of QPSK-modulated direct-sequence code-division multiple-access (DS-CDMA) systems are separatelyspread [1]. Quaternary spreading sequences have been appliedin the wideband CDMA systems characterized in [2]. As a ben-eﬁt of their relative simplicity compared with other polyphasesequences [3], a number of quaternary spreading sequences,which are synonymously also referred to as complex-valuedspreading sequences, have been proposed in [4]–[6]. Someof them were claimed to outperform gold and m-sequencesin asynchronous DS-CDMA systems [5], [6]. A receiver ar-chitecture using quaternary spreading sequences and QPSKmodulation was proposed in [7].The bit-error-ratio (BER) performance of the family ofDS-CDMA systems using quaternary spreading and QPSKmodulation has been investigated using a variety of tech-niques, including precise analysis [6], [7], approximations[6], [8], [9], bounding techniques [9], or simulations [5], [8].The achievable BER performance over both additive whiteGaussian noise (AWGN) [5], [7]–[9] and Rayleigh [6] channelswas also documented. The BER performance attainable usingManuscript received October 4, 2005; revised October 29, 2006 and May 8,2007. This work was supported in part by the Engineering and PhysicalSciences Research Council (EPSRC), U.K., and in part by the EU under theauspices of the NEWCOM and Phoenix Projects. This paper was presentedat the IEEE Wireless Communications and Networking Conference 2006, LasVegas, NV. The review of this paper was coordinated by Prof. H. Leib.X. Liu iswith theUniversity of Southampton,SO17 1BJSouthampton,U.K.,and also with ArrayComm LLC, GU2 7YD Guildford, U.K.L. Hanzo is with the University of Southampton, SO17 1BJSouthampton, U.K.Digital Object Identiﬁer 10.1109/TVT.2007.905257both deterministic quaternary spreading sequences [5]–[7],[9] and random quaternary spreading sequences [8] was alsostudied.In contrast to the performance analysis of [6] and [7], we pro-vide the precise BER analysis of the asynchronous DS-CDMAsystems using the random quaternary spreading sequencesand the QPSK modulation for transmission over the Rayleighchannels rather than employing the deterministic quaternaryspreading sequences [6], [7] for transmission over the AWGNchannels [7]. Furthermore, we do not assume the independencebetween the real and imaginary parts of the received signal,which was implicitly assumed in [7] when calculating symbolerror ratio. In contrast to the performance analysis of [9], weuse quadriphase data rather than binary data.II. SYSTEM MODELWe consider a general asynchronous QPSK-modulated DS-CDMA system using the random quaternary spreading and arectangular chip waveform for transmission over a Rayleigh-fading channel. We assume that there are K simultaneouslytransmitting users, and the zeroth user’s signal is the desiredone. Since exactly the same system model has been described in[7], we commence our discussions from the decision statistics  Z of the received signal at the output of the correlation receivermatched to the zeroth user’s signal  Z = h0L  b0,0 +K−1  k=1  Xkhkej∆k +   η (1)where L is the length of the spreading sequences. The sig-nal amplitudes {hk}K−1k=0 of the kth user are assumed to beindependently Rayleigh-distributed with a second uncenteredmoment of {2σ2k}, whereas the quaternary data sequences{  bk,m}∞m=−∞ are assumed to be mutually independent andare uniformly distributed with a probability of P{  bk,m =±1 ± j} =1 /4. Finally,   η is the noise component, which isa zero-mean complex-valued Gaussian variable. The phase-shift difference {∆k}K−1k=1 between the kth and zeroth usersis deﬁned as ∆k = −ωc(τk − τ0)+( θk − θ0), where ωc isthe common carrier frequency, Tc is the chip duration, andthe phase shifts {θk}K−1k=0 and the time delays {τk}K−1k=0 are0018-9545/$25.00 © 2008 IEEE240 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008independentlyanduniformlydistributedin[0,2π)and[0,LT c),respectively. The random variable   Xk may be further expressedas [10], [11]  Xk =L−2  m=0  Yk,m (1 − νk)+12  a0,m  a∗0,m+1νk +   Yk,L−1νk +   Yk,L(1 − νk) (2)where the (L +1 ) random variables {  Yk,m}Lm=0 are mutu-ally independent and uniformly distributed, which are condi-tioned on the zeroth user’s spreading sequence {  a0,m}L−1m=0.Furthermore, the relative chip shifts {νk}K−1k=1 between thekth and zeroth users, which are normalized by the chip du-ration, are mutually independent and uniformly distributed in[0, 1) [10], [11].III. BER ANALYSISThe (L − 1) possible chip combinations of {(1/2)  a0,m  a∗0,m+1}L−2m=0 can be categorized into four sets according tothe relative change of the chip value. Let A, B, C, and Ddenote the number of the relative phase changes of 0◦, 180◦,270◦, and 90◦ between the adjacent two chips’ values withinthe zeroth user’s spreading sequence, respectively. Then, wehave A + B + C + D = L − 1. Upon deﬁning the cochannelinterference (CCI)   Ik =   Xkhkej∆k incurred by the kth user,the characteristic function of  {  Ik} conditioned on A, B, andC can be expressed asΦ {  Ik}|A,B,C(ω)=2 −2(L+1)    d1∈A   d2∈B   d3∈C   d4∈D× A {  d1}+A2  A {  d1}+A2  B {  d2}+B2  B {  d2}+B2 × C {  d3}+C2  C {  d3}+C2  D {  d4}+D2  D {  d4}+D2 ×   Yk,L−1,  Yk,LΦ {  Ik}|λ0,λ1,λ2(σkω) (3)where the sets A, B, C, and D are deﬁned asA =   d1         d1 ,    d1 ∈{ − A,−(A − 2),...,A} B =   d2         d2 ,    d2 ∈{ − B,−(B − 2),...,B} C =   d3         d3 ,    d3 ∈{ − C,−(C − 2),...,C} D =   d4         d4 ,    d4 ∈{ − D,−(D − 2),...,D} .(4)The function Φ {  Ik}|λ0,λ1,λ2(ω) is deﬁned as (5), shown at thebottom of the page, where the coefﬁcients λ0, λ1, and λ2 aregiven byλ0=      d1 +   d2 +   d3 +   d4 +   Yk,L    2λ1=     d1+   d2+   d3+   d4+  Yk,L × −2  d2−(1−j)  d3−(1+j)   d4+  Yk,L−1−  Yk,L ∗ λ2=     −2  d2−(1−j)  d3−(1+j)   d4+  Yk,L−1−  Yk,L     2. (6)Since the CCI {  Ik}K−1k=1 conditioned on A, B, and C is inde-pendent [10], [11], the BER of the zeroth user’s real componentconditioned on A, B, and C can be shown as [10]Pre|A,B,C =12−σ0L√2π∞  0exp −12ω2 σ20L2 + σ2 {  η}  ×K−1  k=1Φ {  Ik}|A,B,C(ω)dω. (7)Then, the BER of the zeroth user’s real component is obtainedby averaging Pre|A,B,C over all spreading sequences [10]. Fol-lowing the same approach, we may conclude that the averageBER of the zeroth user’s imaginary component Pie has the samevalue as Pre. Finally, we arrive at the overall BER Pe averagedover both the real and imaginary components of the zeroth useryieldingPe = Pre =4 −(L−1)L−1  A=0L−1−A  B=0L−1−A−B  C=0 L − 1A × L − 1 − AB  L − 1 − A − BC Pre|A,B,C. (8)Φ {  Ik}|λ0,λ1,λ2(ω)=      exp −12λ0ω2 , if λ2 =01ω π2λ2 exp 12 λ21λ2 − λ0 ω2 × erfc λ1ω √2λ2 − erfc  λ22 1+λ1λ2 ω  , if λ2  =0(5)LIU AND HANZO: PRECISE BER FORMULAS FOR ASYNCHRONOUS QPSK-MODULATED DS-CDMA SYSTEMS 241Fig. 1. BER versus the number of users K in the asynchronous QPSK-modulated DS-CDMA system using the random quaternary spreading se-quences. The lengths of the random quaternary spreading sequences are L =7and 31, respectively, the average power of all users at the receiver is equal, andthe background noise is ignored, i.e., when γSNR = ∞.IV. NUMERICAL RESULTSWe will compare the BER results acquired by our pre-cise analysis derived in Section III with that of the standardGaussian approximation (SGA) [10], [11], with that of theBPSK system of [10], and with our simulations described inthis section. Similar to the discussions in [10] and [11], thecomplexity of our analysis, the SGA, and the simulation is onthe order of Θ(L3), Θ(1), and Θ(4L), respectively, where Θ isthe big-Theta notation.1 Fig. 1 shows that the results obtainedby our precise analysis exactly match those obtained by sim-ulations for two different-length random spreading sequenceswhen using L =7and 31. However, the SGA overestimates theBER,particularlyinthescenariowherethereisalownumberofinterfering users and when short spreading sequences are used.The BER of the QPSK system is higher than that of the BPSKsystems due to the crosstalk between the real and imaginarycomponents. Fig. 2 also conﬁrms that the BER results obtainedby our precise analysis match those obtained by simulation forboth of the different-length random spreading sequences, i.e.,for L =7and 31. By contrast, the SGA slightly overestimatesthe BER, particularly when the SNR is high and short spreadingsequences are used. The BER of the QPSK systems is higherthan that of the BPSK systems due to the crosstalk between thereal and imaginary components.V. C ONCLUSIONIn this paper, we provided exact BER formulas for theasynchronous QPSK-modulated DS-CDMA system using therandom quaternary spreading sequences for transmission over1A function is in Θ(f(n)) if it does not signiﬁcantly increase faster or slowerthan f(n), i.e., there exist two positive real constants c1 and c2 and a positiveinteger n0 so that c1n ≤ Θ(f(n)) ≤ c2n for ∀n ≥ n0.Fig. 2. BER versus per-bit SNR in the asynchronous QPSK-modulatedDS-CDMA system using the random quaternary spreading sequences. Thelengths of the random quaternary spreading sequences are L =7and 31,respectively, and the average power of all users at the receiver is equal. Thenumber of users is K =4 .the Rayleigh channels. Its accuracy was veriﬁed by our nu-merical simulation results and also compared with those of theGaussian approximation.ACKNOWLEDGMENTThe authors would like to thank F.-C. Kuo for her valuablesuggestions that help improve this paper. The authors wouldalso like to thank the anonymous reviewers for the helpfulcritique.REFERENCES[1] F. D. Garber and M. B. Pursley, “Performance of offset quad-riphase spread-spectrum multiple-access communications,” IEEE Trans.Commun., vol. COM-29, no. 3, pp. 305–314, Mar. 1981.[2] 3GPP TS 25.213 V6.3.0. (2005, Jun.). Spreading and Modulation (FDD).3rd Generation Partnership Project. [Online]. Available: http://www.3gpp.org/[3] P. Fan and M. Darnell, Sequence Design for CommunicationsApplications. Hoboken, NJ: Wiley, 1996.[4] S. R. Park, I. Song, S. Yoon, and J. Lee, “A new polyphase sequencewith perfect even and good odd cross-correlation functions for DS/CDMAsystems,” IEEE Trans. Veh. Technol., vol. 51, no. 5, pp. 855–866,Sep. 2002.[5] S. Rahardja, W. Ser, and Z. Lin, “UCHT-based complex sequences forasynchronous CDMA system,” IEEE Trans. Commun., vol. 51, no. 4,pp. 618–626, Apr. 2003.[6] S. Xie and S. Rahardja, “Performance evaluation for quaternaryDS-SSMA communications with complex signature sequences overRayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 1,pp. 266–277, Jan. 2005.[7] T. G. Macdonald and M. B. Pursley, “The performance of direct-sequencespread spectrum with complex processing and quaternary data modu-lation,” IEEE J. Sel. Areas Commun., vol. 18, no. 8, pp. 1408–1417,Aug. 2000.[8] J. H. Cho, Y. K. Jeong, and J. S. Lehnert, “Average bit-error-rate per-formance of band-limited DS/SSMA communications,” IEEE Trans.Commun., vol. 50, no. 7, pp. 1150–1159, Jul. 2002.[9] A. Lam and F. Ozluturk, “Performance bounds for DS/SSMA commu-nications with complex signature sequences,” IEEE Trans. Commun.,vol. 40, no. 10, pp. 1607–1614, Oct. 1992.242 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008[10] J. Cheng and N. Beaulieu, “Accurate DS-CDMA bit-error probabilitycalculation in Rayleigh fading,” IEEE Trans. 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COM-35, no. 1, pp. 87–98,Jan. 1987.Xiang Liu is currently working toward the Ph.D. de-gree in electronics at the University of Southampton,Southampton, U.K.He is currently a Software Engineer with Array-Comm LLC, Guildford, U.K.Lajos Hanzo received the degree in electronics in1976 and the Ph.D. degree in 1983.During his 31-year career in telecommunications,he has held various research and academic posts inHungary, Germany, and the U.K. Since 1986, he hasbeen with the School of Electronics and ComputerScience, University of Southampton, Southampton,U.K., where he is the Chair in telecommunications.Currently, he is directing an academic research team,working on a range of research projects in the ﬁeld ofwireless multimedia communications sponsored byindustry, the Engineering and Physical Sciences Research Council, U.K., theEuropean IST Programme, and the Mobile Virtual Centre of Excellence, U.K.He is a coauthor of 15 books on mobile radio communications, totaling morethan 10000 pages, and has published more than 700 research papers. He isan enthusiastic supporter of industry and academia, and he offers a range ofindustrial courses.Dr. Hanzo is an IEEE Distinguished Lecturer of both the CommunicationsSociety (ComSoc) and the IEEE Vehicular Technology Society (VTS). He isalso the Governor of both ComSoc and the VTS. He was the Technical ProgramCommittee Chair of several IEEE conferences, has presented keynote lectures,and is the recipient of a number of distinctions. He received the ComSoc WTVAchievement Award in 2007.

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Precise BER Formulas for Asynchronous QPSK-Modulated DS-CDMA Systems Using Random Quaternary Spreading Over Rayleigh Channels

Precise BER Formulas for Asynchronous QPSK-Modulated DS-CDMA Systems Using Random Quaternary Spreading Over Rayleigh Channels

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