Adaptive multiuser detection Spread Spectrum Techniques and by benbenzhou


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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
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.

                                                                                   the reciprocal of the Y drmem of the invuse of the ucWscorreMon
-    d is the background noise power s p a l density

      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
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
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

     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
denoted by Pt(o) is defined as
                                                                                    -    the timing of all uwrs mustk acquired
                                                                                    *    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
                                                                                         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
                                      J,O) J,O)                        (5)
                                                                                                              Ply   -   A b + R%

                                                                                   Thus, in the hypahetical absence of hdrground noise, tbc decorrclating
                                                                                   lna transformation r e w v a s tbc rranunitred bits without multiuser
     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 :

-       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
                                                                                    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
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:

                                                                                    *    the data stream (training sequence) of the desired usex must be
                    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.
                                                                                    *     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
    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

      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].
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
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
(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
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

      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
                                                                                   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

                                                                                          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
                                                                                   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

                                                                                    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
                                                                                         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
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
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