The joint analytical optimisation of the spreading gain and coding gain of nonbinary BCH coded CDMA communication systems is considered in both single-cell and multi-cell scenarios. Furthermore, two types of detectors were employed, namely the minimum mean square error multiuser detector and the classic single-user matched filter detector. It is shown that the optimum coding rate varied over a wide range

Hanzo, L.

Wei, H.

English

Southampton (e-Prints Soton)

Coding against spreading gain optimisationof nonbinary BCH coded CDMA systemH. Wei and L. HanzoThe joint analytical optimisation of the spreading gain and coding gainof nonbinary BCH coded CDMA communication systems is consid-ered in both single-cell and multi-cell scenarios. Furthermore, twotypes of detectors were employed, namely the minimum mean squareerror multiuser detector and the classic single-user matched ﬁlterdetector. It is shown that the optimum coding rate varied over awide range.Introduction: When considering the CDMA system of Fig. 1,h a v i n gaﬁxed bandwidth, the ratio of G=Rc has to be a constant, where G is thespreading gain and Rc is the channel coding rate. Therefore, it ispossible to jointly optimise the spreading gain G a n dc o d i n gg a i nRc.Recently, Yue and Wang [1] investigated the coding against spreadinggain trade-offs in the context of a binary LDPC-coded CDMA systemusing BPSK modulation and simulations. By contrast, in this Letter weinvestigate the joint optimisation of the coding gain and spreading gainusing nonbinary BCH codes in the context of a QPSK-modulatedCDMA communication system by ﬁnding an analytical solution. Inpractice, the coding rate Rc assumes values of 1=2o r1 =3, while thespreading gain G may be in the range of 64 to 128. Hence in this Letterwe considera general systemhavinga ﬁxed ratioofG=Rc¼256, whereQPSK modulation was invoked for communicating over a Rayleighfading channel in both single-cell and multi-cell scenarios.Fig. 1 Schematic of transmitterAsymptotic performance analysis of CDMA: Tse and Hanly [2]investigated the asymptotic performance of CDMA systems, wheret h en u m b e ro fu s e r sK a n dt h es p r e a d i n gg a i nG tend to inﬁnity, andthe ratio of a¼K=G is ﬁxed. Accordingly, the output signal-to-interference-plus-noise ratio (SINR) of the linear minimum meansquare error (MMSE) multiuser detector (MUD) can be expressedas [2]:g1 ¼P1s2 þ aEp½IðP;P1;g1Þ ð1ÞwhereEp½IðP;P1;g1Þ  ¼PP1P1 þ Pg1ð2Þand P1, g1 and P denote the power of the desired user, i.e user 1, theoutput SINR of the linear MMSE receiver for user 1 and the total powerof the interfering users. Furthermore, Tse and Hanly [2] also indicatedthat the output SINR of the MF can be expressed as:g1 ¼P1s2 þ 1=NPKi¼2 Pið3ÞIn [3],Y uet al. extended this result to spectrally efﬁcient M-aryquadrature amplitude modulation (QAM) communicating over theuncorrelated non-dispersive Rayleigh fading channel:PM ¼ 1  1M    2r 1  1M   1ﬃﬃﬃﬃﬃMp þ2p1  1ﬃﬃﬃﬃﬃMp tan 1 r     ð4Þwherer ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃg12=3ðM   1Þþg1rPerformance of nonbinary BCH codes: Nonbinary BCH codes [4]constitute an efﬁcient class of linear codes and have the capability ofcorrecting=detecting symbol errors. A BCH(N, Q) code has a total of Nencoded symbols and Q original information symbols is capable ofcorrecting up to t¼b(N Q)=2c random symbol errors and the corre-sponding coding rate is Rc¼Q=N. In this Letter, we considered error-correction-only decoding [4]. According to [4], the decoded symbolerror probability (SEP) after ‘error-correction-only’ decoding can beexpressed as [4]:Ps ¼nNP Nn¼bðN QÞ=2cþ1Nn  ðPMÞnð1   PMÞN n ð5Þwhere the Rayleigh channel’s symbol error probability PM beforechannel decoding is given by (4).Numerical results: We considered both single-cell and multi-cellscenarios using pseudo-noise (PN) spreading codes. For the multi-cell scenario, we assume that each cell is surrounded by six adjacentcells having the same interference load. More explicitly, we consid-ered a system supporting a total of 7K users having the powerdistribution of: P1¼P2   ¼PK¼P, PKþ1¼   ¼P7K¼P¼12.The classic Gaussian approximation of the interference was used.Again, we had a constant bandwidth associated with G=Rc¼256,while the channel was a single-path Rayleigh fading channel [4] andQPSK modulation was invoked in the system.Fig. 2 Symbol error probability (SEP) against coding gain at Eb=N0 ¼20 dB for K¼60, 70, 80 and 90 users and a constant bandwidthassociated with G=Rc¼256, when communicating over uncorrelatednon-dispersive Rayleigh channels in a multi-cell scenarioN¼256, Q¼RcNFig. 2 portrays the symbol error probability (SEP) characterising thejoint optimum of the coding gain and spreading gain, when commu-nicating over an uncorrelated Rayleigh channel in an multi-cellscenario, where an MMSE MUD was invoked at the receiver. Thenumber of users supported was varied from K¼60 to K¼90 and theEb¼N0 value in the reference cell was 20 dB. From this Figure we canobserve that the optimum coding rate in this scenario was Rc¼1=2.To further characterise the system, Fig. 3 illustrates the optimumcoding rate Rc against the number of users supported, when invokingdifferent receivers in both single- and multi-cell environments. Theseresults were computed from (5). Note however that at a higher user loadthe SEP is inevitably higher, despite using the optimum G and Rcvalues. Observe in Fig. 3 that the optimum coding rate for the matchedﬁlter (MF) detector varied between 0.3 and 0.2 in both the single-celland multi-cell environments. However, for the MMSE MUD commu-nicating in a single-cell environment we can observe that the optimumcoding rate Rc is increased from 0.22 to 0.7, as the number of userssupported is increased from 10 to 200. By contrast, for the MMSEMUD operating in a multi-cell environment, we can observe that theoptimum coding rate Rc varied from 0.3 to 0.5 when the number of usersupported was between 10 and 200.Finally, Fig. 4 illustrates the optimum coding rate Rc against Eb¼N0,when invoking an MF and an MMSE MUD operating in both thesingle-cell and multi-cell environments, while the number of userssupported was K¼40. From this Figure we may conclude that for theMF detector, the optimum code rate is approximately 1=3, while for theMMSE detector, optimum range is between 0.3 and 0.5.ELECTRONICS LETTERS 7th July 2005 Vol. 41 No. 14Fig. 3 Optimum coding rate Rc evaluated from (5) against number of userssupported, when invoking the single user MF and MMSE MUD receiversin both single- and multi-cell environments at Eb=N0¼20 dBAn uncorrelated Rayleigh channel was used. N¼256, Q¼RcNFig. 4 Optimum coding rate Rc evaluated from (5) against Eb=N0,w h e ninvoking single user MF and MMSE MUD receivers in both single- andmulti-cell environments at K¼40An uncorrelated Rayleigh channel was used. N¼256, Q¼RcNConclusions: In this Letter, we investigated the coding against spread-ing gain trade-offs in a nonbinary BCH coded CDMA system. Theoptimum coding rate Rc of the MF receiver is about 1=3, while that ofthe MMSE MUD ranges from 0.4 to 0.7, depending on both the userload K and the single against multi-cell scenario encountered.# IEE 2005 9 May 2005Electronics Letters online no: 20051702doi: 10.1049/el:20051702H. Wei and L. Hanzo (School of ECS, University of Southampton,SO17 1BJ, United Kingdom)E-mail: wh01r@ecs.soton.ac.ukReferences1 Yue, G., and Wang, X.: ‘Coding-spreading tradeoff in LDPC-codedCDMA with turbo multiuser detection’, IEEE Trans. Wirel. Commun.,2004, 3, pp. 1734–17442 Tse, D.N.C., and Hanly, S.: ‘Linear multiuser receivers: effectiveinterference, effective bandwidth and user capacity’, IEEE Trans. Inf.Theory, 1999, 45, pp. 614–6573 Yu, K., Evans, J.S., and Collings, I.B.: ‘Performance analysis of LMMSEreceivers for M-ary QAM in Rayleigh faded CDMA channels’, IEEETrans. Veh. Technol., 2003, 52, pp. 1242–12534 Proakis, J.G.: ‘Digital communications’ (McGraw-Hill InternationalEditions, 1995, 3rd edn.)ELECTRONICS LETTERS 7th July 2005 Vol. 41 No. 14

Coding against Spreading Gain Optimization of Nonbinary BCH Coded CDMA System

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