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ADAPTIVE MULTIUSER DETECTION Sergio Verdh Department of Electncal Engineering Pnnceton University Princeton, NJ 08544, USA The foregoing multiuser detectors depend on various parameters 1. Introduction such as received ampliNdes and crosscorrelations which are usually not fixed and known beforehand. Therefore. another important thrust in research in multiuser detection is the design of adaptive detectors, which self-tune the detector p a r a m e m from the observation of the received Spurred by its applications in Code Dlvision Multiple Access. mul- waveform. The very recent, and already considerable. literature on flus uuser detection has grown from its origins more than ten years ago to a subject is surveyed in the present tutorial paper. vibrant research and development acuvity in industry and acadenua As the needs to increase capacity in muluuser radio channels become more Sections 2 and 3 contain background material used throughout the pressing. i t is safe to expect that the interest in the s u b s t will grow in paper on the multiaccess channel model, optimum multiuser detection the near future and the decorrelating detector. A comprehensive Ntorial exposition of these and othex topics can be found in [171. Section 4 deals with the The development of multiuser detection has proceeded along a path MMSE linear multiuser detector and its adaptive implementations. Sec- which is typical of other areas in communicaiions. Initially, optimum tion 5 gives an overview of adaptive tentativedecision based detectors solutions were obtained along with the best possible performance achtev- such as those that use successive cancellation and decision-feedback. able in Gaussian noise channels [I]. Those results showed a huge gap Section 6 deals with blind multiuser detection. and in particular, it between the optimum performance and the performance of the conven- presents a multiuser detector which is optimally near-far reistaut and tional single-user detector (which neglects the presence of multiaccess requires no more knowledge than the conventional single-user detector. interference). In particular. they showed that the near-far problem is not Section 7 is devoted to multiuser detection using learning neural- a flaw of CDMA. as widely beheved, but of the inability of the conven- networks tional receiver to exploit the suucmre of the multiaccess interference. This feature of multiuser detection sidesteps the need for sophisticated high-precision power conuol in mobile communication systems. Thus, an increase in the complexity of the base station enables a considerable reduction in the complexity of the mobile transmitters. Equally impor- 2. Optimum Detection tant to the near-far resistant property of optimum multiuser detection, is the performance gain that it promises even in situations of exact power control (equal-power reception). This performance gain results in lower The asynchronous CDMA white Gaussian noise channel considered in power consumpuon and processing gain requirements, which translate thls paper is into increased battery lifes and lower bandwidth in order to support the same information rates. The second stage in the development of multiuser detection was devoted to the analysis and design of multiuser detectors that could aclueve significant performance gains over the convenoonal receiver where without incurring in the exponential (in the number of users) complexity of the optimum multiuser detector. Notable among those efforts were the decorrelating detector of Lupas and Verdu [Z]. [31; the multistage * K is the number of users. detector of Varanasi and Aazhang [41, [51; the decision-feedback mul- tiuser detectors of Duel-Haen [61. [71; and the suboptimum detectors of Xie, Rushforth and Short [SI, [91. - A, is the received amplitude of the kth user. Motivated by the channel environments encountered in many CDMA applications. the design of multiuser detectors for channels with - ua bk[i] E {-l,+l] is the data s e m modulated by the kth user. fading, multipath. or noncoherent modulation has amacted considerable attention, as exemplified by the works of Varanasi [IO]: Vasudevan and st is the unit-energy signature waveform of the kth user. Varanasi [ I l l , [121: Zvonar and Brady [131, [141. [ I f : Fawer and Aazhang [161. - h T is t e inverse of the data rate (duration of st) - 7, E [O.T) is the kth user's offset "(1) is normalized whte Gaussian noise. 43 the reciprocal of the Y drmem of the invuse of the ucWscorreMon - d is the background noise power s p a l density matnx: The model in (I) CM k gencnliwd to take into acwunt a number of fearurcs that are relevant in prrctiee such as q u a d " and nonbinary modularion. signature waveforms spanning mme than one bit epoch. In practical terms, this mum that is a huge p u f c " c e gap c intasymbol i m a f u a w k a . For wmaptual clarity it is best not to gen- l benveen the c o n v d o d singkusea m d Em and tbc optimum k cralizc tbc modcl in rbom dircctim; insrud. it is usually concepmaUy achievable p a f " ~ . . For example, while tbc =-far resisancc of tbc advanagoous to consider the special CBSC of (1) where the users are conventional m i v a is mo. (he cxpcacd opinnsm m-fv mistance symbol-synchronous: using direct-scqucofc spread-spar" s i g " wiveforms wim N chips p symbol is I o w a bounded by I201 a In most CIUCS. the analysis of multiwu detectors for the synchro- nous channel (2) contains all the key Ingredients necessary for the These notable p u f o r m gains ~ obtained a the expense of: ~ ~ are t analysis of the more gcnual channd (I). In multiuser detection, it is frequently useful to examine perfor- the signstun wavdorms of all users must be hown mance in the situation of low baelrground noise. 0 + 0. To that end, the asympfon'c multiuser efficiency of thc kth us4 (whwe bit Mor rate is * the received ampli!xde of all users m s k known ut denoted by Pt(o) is defined as - the timing of all uwrs mustk acquired (3) * exponential complexity m the number of was. which is simply the degradation (measured in SNR) suffered by a u w * a centralized smcture which demodulats all t r ~ ~ m i m s due to the presence of Dther users in the channel. The worst-case asymptotic multiuser efficiency over all received intfafering amplitudes IS called the nearfar resistance, denoted by &. For mwt multiuser detec- tors. near-far resistan= is n an overly conservalve performance a Remarkably, as we will set in Section 6,recent progress in adap- measure because the worst-case usually doeS not wcur at large inmfer- tive multiuser daecton has resulted in a receiva which achieves ing ratios. Therefore, it is an attractive performance measure even for optimally near-far resistant multiuser detection with none of the above receivers that employ some power control. shortcomings. The opumum receiver for (2) processes the received waveform with a bank of matched filters. whlch produce a vector of observables: 3. Dewrrelating Detector y = R A b + n (4) - where A diagiA,. - A K ) , b [b,, bKIr, n is a zero mean Gaus- sian vector with c o v m m e mamx R and R is the crosscorrelanon matnx The decorrelating dewtor outputs the signs of the matched filta outputs in (4) multiplied by the inverse crasscorrelation matnx R", i.e., it takes the sign of the the v e c m whose 11 coefficient is P,, -J r J,O) J,O) (5) Ply - A b + R% 0 Thus, in the hypahetical absence of hdrground noise, tbc decorrclating lna transformation r e w v a s tbc rranunitred bits without multiuser ier The optimum detector that selects the mmt Likely data v e m r based intuferewe. In the asyncbronoua ure, tbc d "' daoaor gen- c q on the obsavption of y must solve UI NP-complete combinatorial optim- aalizes to an infinite impllsanaponse film [3]. The decarelating ization algorithm [181. Thus, no Qorithm polynomial in K is known detector is the maximum likelihood solution in tbc absence of any for optimal multiusa M o n . Ill tbc asynchronous case,the receiver knowledge about the &'fed rmplitudcs. A major mult of h p a s and consists of a matched film f r o n t 4 followed by a Vitabi algcdthm Vudb [21, I31 is tbat (he Occmdab dctcaor achieves o i IIUT- 'ng p " [l]. The number of stptcs of the Vitcrbi algaithm is c x p o m t i d in K far resismce. Thc bit aror rate of the defcmluiog dncna is mdcpcn- with murics that are v u y simple to compuk in tsrrm of the matched dent of thc i - n unplituds. This is because the dccomlating filter wtputs and crcWscare1ations. The optimum I t h USCT --far resis- linear traosformation projects tbc received waveform on a subspace t n e [191. I31 is equal to the m i n i " mcqy of my multiuser signal ac which is uthogolul to tbc space spanned by the intufaing signature modulated by (-l,O.+Il. with fixed A t b i [ O 1 - l . In turns of the waveforms. In comparison with tbc obove requiruoents of the optimum crwscomlation matrix. the near-far mistpncc of the kth USCT is equal to detector. the dscrrelating deuctor has the following f c a m : 44 - the SignaNre waveforms of all users must be known energy dominates, then the MMSE detector approaches the conventional single user matched filter: on the other hand. as the background noise - the received amplitudes need not be known or esumated level vanishes U + 0 , the MMSE detector approaches t e decorrelating h detector. Therefore. the asymptotic multiuser efficiency and the near-far resistance of the MMSE detector are the same as those of the decorrelat- * the uming of all users must be acquired ing detector. In particular. it also achieves optimal near-far resistance. The linear MMSE multiuser detector was originally proposed by Xie. * the matrix inverse R-’ must be computed. Short and Rushforth [91 in the asynchronous case. and much earlier in single-user dual-polarization channels [ZSI, which can be viewed as - it lends itself to decenualized implementauon, demodulaung only the desired user. two-user synchronous channels. As long as the background noise is weak. there is little point in incurring in the additional complexity over the decorrelating detector required by the need w uack received amplitudes. However, the great The optimal near-far resistance property of the decorrelating detec- advantage of the linear MMSE detector is the ease with wluch it lends tor coupled with the fact that it does not require knowledge of the itself to adaptive implementation with uaining sequences. received amplitudes make the decorrelating detector atuacuve from the The conmbution of the kth user to the penalty function in (9) is standpoint of implementation. The main disadvantage is the computation equal to required to obtain the decorrelating coefficients from the crosscorrela- tions. In the case of synchronous direct-sequence spread-spectrum. Chen E Kbt - v , y>)’I. (10) and Roy [21] report a recursive least squares (RLS) computation of the decorrelating detector coefficients which requires knowledge of all signa- where the linear transformation has been denoted by c . The gradient of ture sequences but sidesteps the need to perform computations with the cost f u n d o n inside the expectation in (IO) is equal to crosscorrelations. In the asynchronous case, the processing window can be truncated to the bit of interest as suggested in [221. [231; or it can span a truncated sliding window as proposed in [241. In the lauer case. the dynamic updating of the decorrelating detector coefficients in response to variations in the crosscorrelations has been investigated in Because of the convexity of (IO) in c, the gradtent descent adaptive [25]. Mitra and Poor [261 advocate detecting the presence and identity algorithm is of a new transmitter by processing the residual signal that results by sub- tracung from the received signal the multiuser signal modulated by the decorrelating detector decisions. The optimabty of the near-far resistance of the decorrelating detec- tor with DPSK modulation has been established by Varanasi [IO]. The will converge (with infinitesimally s a l step size p) to the argument that ml decorrelating detector has also been used in the conjunction with DPSK minimizes the penalty function in (IO). The update of the impulse and individual rake matched fillers for each user (to combat multipath) i n response in (12) has the following features: [271. * the data stream (training sequence) of the desired usex must be known. 4. Linear MMSE Multiuser Detection * the received amplitudes need not be known or estimated The decorrelating detector may have worse bit error rate than the * the signature waveforms of the interferers need not be known conventtonal deteclor when all the interferers are very weak [3] Tlus means that it should indeed be possible to incorporate (exact or approxi- mate) knowledge of the received amphtudes in order to obmn a linear - the timing of the desired user must be acquired. muluuser detector that outperforms the decorrelating detector MiNmum mean-square error (MMSE) linear detection is one approach to tlus prob - the timing of interfering users need not be acquired lem According to tlus critenon. one chooses the K x K mamx M that aclueves - knowledge of the signature waveform of the desired use? is not necessary, but it facilitates the initialization of the algorithm. (9) * it can be implemented in an asynchronous channel, with the only where the expectanon is with respect to the vector of transmitted bits b requirement that the timing of the desired user b acquired. The e and the noise vector n wluch as we saw has zero mean and covanance longer the allowed impulse response. the befter the p d o n n a i x e mamx equal to u2R Without invoking the Gaussian nature of it IS pos- will be, with a judicious truncation achieving almost the same per- sible w show that the linear MMSE detector replaces the inverse formance as a doubly infinite filter response. crosscorrelauon matrix R-’by the mamx The gradient descent algorithm shown in (12) is the simplest adap- tation law that minimizes (IO). Oper more complex, but faster. algo- rithms can be used instead, based, for example. on recursive least- squares or in laltice suumres (e.g. [291). Thus the linear MMSE has the aforementioned features of the decorrelat- ing detector. except that it requires knowledge of the received ampli- tudes. If eithex the background noise level or the kth user received 45 In addiuon to the aforemenhoned earher reference 1281. the adap- s m philosophy has been adopted in the synchronous case by uve hnear MMSE detector was proposed by Madhow and Homg [30]. Abdulrahman and Falconer 1431 and in a multipath QPSK multiaccess RapaJiC and VuCeUC [311 and Mdler [321, [331 The implementauon of channel by Abdulrahman. Falconer and Sheikh 1441, [451 which uses a (12) IS carried out with fiNte-di~~~enSiO~~al whose dimensionality vectors fractionally-spaced DFE detector whose feedforward and feedback IS equal to (or twice) the number of chps per symbol Several methods coefficients are adapted to nunimize mean-square emor using truning have been proposed in order to lower compleuty in systems with large sequences. Another adaptive multiuser detector based on DFE is experi- processing guns, for example the cychcally shfted filter bank of 1301. mentally demonstrated by Stojanovic and Zvonar for a channel with the replacement of simple tap delays by first-order low-pass filters in severe multipath [461. Kohno. Imai. Haton and Pasupathy [471 consider [341, and the symmemc dimension reducuon scheme in [3S] Lee 1361 a CDMA channel with limited bandwidth for which they design an adap- observes that the RIS algonthm is iU-con&uoned in near-far envuon- uve MMSE detector that uses decision-feedback to remove intersymbol mens with h h SNR. and proposes a transformanon of the chp g interference. The lint stage in that dewtor (which uses knowledge of all matched filter outputs to overcome th~sproblem Sigluficantspeed-up is the signature waveforms) performs preliminary decision which are then reponed with both the gradient descent and RLS implemenmons of the used in the adaptive stage. Rapajic and Vucetic 1311, [481 find no MMSE cntenon improvement over the adaptive MMSE detector by incorporating the pos- sibility of decision feedback. Adaptive versions of the multistage detec- Joint adapuve muluuser detecuon aqd Unnng recovery is acheved tors of Varanasi and Aazhang have been proposed by Chen. Sivesh and with an RLS algonthm by Zvonar and Brady I371 [381 and with a Bar-Ness [49] (in t e case of conventional tentative decisions) and [SO]. h steepest descent algorithm by Snuth and Miller 1391 [SI] (in the case of a decorrelating first stage). In those detectors, the fist stage is nonadaptive and requires knowledge of all the signature An interesung alternative to the nummizauon of mean-square error waveforms. However, the interferencecanceller is adaptive and does not has been proposed by Mandayam and Aazhang [NI It uses a stwhasuc require knowledge of amphtudes. The adaptation is camed out by gra- gradient algonthm to muunuze probahdity of error whch (for a linear dient descent of tbe energy of the difference berween y and the output of detector) can be wntten as the sum of Q-funcuons The gradient of t h ~ s the linear adaptive c n e l r (or a different penalty function in 1521). and acle penalty funcuon admits (via the chun rule) a closed-form expression therefore. it does nci require traimng sequences. Another adaptive two- For low background noise. and assuming that at each step of the adapta- stage multiuser detector based on soli tentative decisions is proposed by non the detector can be guaranteed to have positive asymptotic efficiency Brady and Catipovic 1531, which uses knowledge of traimng sequences a (so that the adapuve Lw operates in the region where the cost funcuon and signature waveforms in order to adapt to the channel paramems and IS convex), ths detector should converge to the opumum hnear muluuser refine a coarse initial estimate of tinung and phase. detector obtained by Lupas and VerdO 121 whch makes better use of the amplitudes than the MMSE detector 6. Blind Multiuser Detection 5. Tentative-Decision Based Multiuser Detectors The requirement of training sequences in the multiuser detectors surveyed above is a cumbersome one in multiuser communications. One of the simplest ideas in multiuser detection is that of successive Since transmiaers start and finish theu transmissionsasynchronously. the cancellation: d e m the dam of the strongest user with a convenuonal "birth" (or "death") of an interferer requires the recomputation of the detector and then subtract the signal due to that user from the received waveform, ?he process can then be repeated with the resulung adapuve receiver coefficients. Ohen. decision-directed operation of the waveform which contains no trace of the signal due to the strongest user adaptive detector is not robust enough to take care of those sudden assuming no mor was made in its demodulation. This techque has the changes, and the desired user must be asked to interrupt its data disadvantage that it requires extremely accurate e t m t o of the siain transmission so that a training sequence is transmiaed. received amplitudes, and unless the users can be ordered so that the In this w o n we will review a recent adaptive muluuser detector received amplitudes sausfy due to Honig. Madhow and VerdO [S41 whch has the following features: A,DA?D. DA, * it achieves optimal near-far resistance * (approumate) knowledge of the signanue waveform of the desued i ~ performance is actually worse than that of the decorrelaung detector s whxh requires no knowledge of the received amplitudes A related user is requued. technique is the mulustage detection of Varanasi and Aazhang where the first stage consists of a bank of convennonal detectors [41 or a decorrela- * the timing of the desired user must be acquired. tor [SI. the second stage assumes that the previous decisions are correct and simply cancels the corresponding signals from the received the received amplitudes need not be known or estimated. waveform, thereby resulung in a clear single-user channel in the event that previous decisions are indeed correct The decorrelaung decision- feedback detector of Duel-Hallen 1411 (and its adapnve version in [211) - the signature waveforms of the interferers need not be known incorporates feanues common U) both successive cancellauon and mulus- tage detecuon with a decorrelatingfront-end Similarly. it IS possible to - the unnng of intexfering users need not be acquired assume that decisions made about earlier bits in an asynchronous system are correct and thenfore they can be cancelled. as in convenuonal Training sequences are not required for any user single-user decision-feedback equalimon (DFE) The applicauon of tb~s idea to muluuser detecuon goes back to 1421 46 Therefore. we wdl see that it is possible to anam the same near-far resistance as the optimum receiver, the same asymptotic efficiency as the MOE(x6 - E [(A, b , - <y , s , + x,>)'l + A .; (17) decorrelating detector, and the same bit error rare as the linear MMSE Therefore, the x , that minimizes (16) is such that s I + x i is the MMSE detector with no more than the knowledge assumed by the conventional linear detector of Section 4. If the minimum output energy dewtor is single-user detector. Although the approach of this detector is reminis- the MMSE detector, what is the point of this alternative derivation? The cent of that of anchored minimum energy blind equalization proposed in o adaptive minimization of mean square mr requires training sequences. I551,the solution of (541 does not have a counterpart in single-user com- whereas the minimization of output energy does not Therefore. the munication, in contrast to the above multiuser detectors. With few minimization of the convex cost limction (16) lends i W to blind adap changes. it is possible to generalize the design and analysis of the blind tation. The simple method of projected gradient descent is adopted in multiuser detector below to the asynchronous case. [54] to show the following blind adaptation d e , which is guaranteed to The blind multiuser detector of [541 adapts a linear transformation converge globally: - of the observations whose impulse response is c , (assuming that the desired user is k 1). and outputs the decision where Z and Z are the outputs of the conventional single-user matched , filter and of the proposed linear transformation: Any linear multiuser detector can be written in a canonical arNmgoriol decoinposifloir: where The generalization of (18) to the asynchronous case is straightfor- ward. In fact, in order to write the key equation (17) we did not invoke any suucmre of the multiaccess inwference. In the asynchronous case. The only c , that cannot be represented i n this form are those for whch we can work with signals (or finite dimensional vectors) t a span only ht a one bit. 01 in order to improve p a f m a n c e . we c n lengthen the dura- tion of the linear transformation on both sides of the timelimited signal 5 , . As usual, it should be possible to speed up convergence speed at the expense of computational complexity by adopting an RlS-based method. but the decisions are scale invariant, and if c I were orthogonal to sI,the The foregoing simple blind adaptive multiuser detector, which as bit error rate would be 0.5. Thus, the freedom we lose in the decompo- we have seen, has no more requirements than the conventional detector. sition (14). is (like in marriage) a freedom we do not need to have. and yet, converges always to an optimally near-far resistant solution, is Let us focus attention on adapting x I . while preserving orthogonal- ideally suited to cope with transients due to i n i t i a l i o n . powering ity to s i . The energy (or more precisely, the second moment) of the out- onloff of interferers, or sudden changes in received power. The slower put of the linear uansformation variations that occur due to offset drift, slow fading, etc. could be fol- lowed more closely (albeif less robustly). by an MMSE adaptive detec- tor operating in decisiondirected mode, in lieu of training sequences. In practice, there wdl always be some mismatch between the received signature waveform s 1 of the desired user and the assumed has rhree additive independent componenrs the first due to the desired (nominal) waveform f , . So the natural question to invesagate is how user, the second due to the muluaccess interference. and the third due to robust will the blind multiuser detector be to mismatch? The answer the background noise The first component is transparent to the choice depends on the background noise leva. If i, is different from sI as well of r , Thus, by varying x I can can only change the energy of the as from the other interfering s i g n m e waveforms. one can always second and third components Accordingly. a very simple and sensible choose x , orthogonal to f l , so that f l + x l will be orthogonal to the sig- strategy is to choose x , that minimizes the output energy nals of all users: si, ' ' ' s This may require an x i with huge norm, but . , if U --t 0. then this will indeed be the solution that minimizes output energy. This means that in high SNR channels. the foregoing detector is not robust at all against mismatch in the nominal signal. In particular, as We would expect that if the background noise is comparatively small, long as 3, is different from sI. the asymptotic multiuser efficiency is the argument x 1 that minimizes (16) is such that it (almost) eliminates equal to zero. An increase in bafkground noise will have a robustifying the contribution of the multiuser interferers to the output, in other words effect In that case a hired signal suffering a small mismatch will not si + x i would approach the decorrelating detector. For higher back- be cancelled because that would quire au xi with very large energy. ground noise, x , would try to anenuare the conuibution of the multiac- and thus a correspondingly large colmibution to the output e m g y due to cess interference, but without becoming tw large in norm. and thus con- the background noise. Fomnately, we can acheve the same robustify- tributing a large component due to the background noise. We need to ing effects of background noise even in high SNR sifllations by simply speculate no further about the nature of the minimum output energy putting a constraint on the maximum allowable energy of x I . referred to detector. because it is easy to check that the output energy in (16) is a as surplus energy in [%I. The modified blind algorithm with con- translated version of the mean-square error: strained surplus energy is 41 uve linear transformation. A so-called radial-basis-functionneural net- work is proposed in 1611 for singleuser equalizationand investigated in [621 in synchronous multiuser detection. The number of nodes is exponential with the number of users, and the decision stvisuc IS a where 0 < p < 1 Note that the convenuonal single-user receiver hnear combination of nonlinear transformauons of the observables. corresponds to a bhnd muluuser detector with zero s y l u s energy. while allowing unlimited surplus energy makes the detecto? non-robust agunst Miyajima Hasegawa and Haneishi [631 propose a Hopfleld neural desired signal mismatch in high SNR channels A good choice for the network for synchronous multiuser detection using the likelihwd func- surplus energy is the energy necessary to elrminate the interfenng sig- Uon as the energy function to be minimized. The weights of the net- nals. which turns out to be -I plus the reciprocal of the near-far resis- work are nonadaptive and equal to the crosscorrelations times the lance of a desired user with signal ji and interfering signals s2 sx In corresponding amplitudes. both of which are assumed known. When the general, it is necessary that the nominal signal 3, be closer to s l than to true minimum of the funcuon is found. the decisions are opt”. the space spanned by the interfenng signals Provided this is sausfied Although the network does not always converge to the global minimum, and the blind detector has reached a stage in its convergence where the this approach has shown promse in the soluuon of other NP-complete bit error rate is not too high, the assumed nonunal can be refined by combinatorial optimization problems. It is shown empirically in [63] correlating the received waveform with the decisions of the user of that the probability of convergence to spunous local mimma increases interest f with the number O users. the background noise level. or when the interfenng signals are weak. However. t e achieved bit error rate is h near-optimum. Finally we mention the application of Kohonen’s Self-Organinng Map to synchronous muluuser detection to be presented at this confer- ence [@]. This algorithm works with a matched filter bank front-end, The estimator in (22) converges. by the law of large numbers. to a and thus, it assumes knowledge of the signature waveforms; however it scaled version of the received signature waveform of the desued user sI. does not require the use of training sequences or knowledge of ampli- tudes in order to adapt the decision boundaries of the detector. There have been other efforts in blind mulnuser detecuon. O a and d Sat0 1561 consider a muludimensional generalization of the conventional single-user blind equalizauon methods that attempt to minimze a non- cqyex funcoon of the output The channel model can be specialized to synchronous CDMA, however since the qualim in [561 does not use References knowledge of any signature waveforms or data, bit error rate perfor- mance would be poor for weak users. Convergence IS (as in the single- 1. S. Verdu, “Minimum Probability of Error for Asynchronous Gaus- user case) nor guaranteed using this method. Soon and Tong [571 sian Multiple-Access Channels,” IEEE Trans. on Information develop a blind idenuficauon algorithm for a synchronous noiseless mul- Theory, vol. IT-32, pp. 85-96. Jan. 1986. uuser Channel. which requires introducing a different amount of correla- 2. R. Lupas and S. Verdu. “hnear Muluuser Detectors for Synchro- uon in the data modulated by each user. The method is based on the nous Code-Divsion Muluple-AccessChannels.” IEEE Trans. Infor- singular value decomposition of the estimated covariance of the vector of mation Theory, vol. IT-35, pp. 123-136, Jan. 1989. observables (obtiuned by fracuonal sampling). Paris[58] proposes a 3. R. Lupas and S. Verdu. “Near-Far Resistance of Multiuser Detec- bhnd seft-tumng maximum hkelihood sequence estimator which, in pnn- tors in Asynchronous Channels,” IEEE Trans. Comunicarions. ciple, could be used for optimum asynchronous muluuser detecuon vol. COM-38. M r 1990. a. without pnor knowledge of amplitudes and crosscorrelauons. 4. M. K. Varanasi and B. Aazhang, “Mulustage Detection in Asyn- chronous Code-Division Multiple-Access Communications,’’ IEEE Trans. Communicarions. vol. 38. pp. 509-Sl9, Apr. 1990. 5. M. K. Varanasi and B. Aazhang , “Near-Optimum Detection in 1. Neural Network Multiuser Detectors Synchronous Code-Division MulupleAccess Systems,” IEEE Trans Communications, pp. 725-736.. May 1991 6. A. 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