signals in Gaussian noise as described in [] (see also references therein). The alication of this general estimatordetector idea to the frequenc non-s

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1 Mahesh K. Varanasi CISS 1997, Johns Hokins Universit, Baltimore, MD, March 19{1 A Sstematic Aroach to Noncoherent Detection for DPSK Modulation in Single-User Correlated Diversit Raleigh Fading Channels ith Alications to Post-Combining Decorrelative Multiuser Detection Abstract A correlated diversit Raleigh fading channel is dened as one that admits inter-diversit branch correlation in both the multilicative fading as ell as the additive noise rocesses and models a variet of diversit communication sstems oerating over Raleigh fading channels. In this aer e are interested in alications here receiver simlicit is aramount. The modulation method of choice for such alications is dierential hase shift keing so that noncoherent detection is ossible at the receiver ithout having to imlement comlex channel estimation algorithms. Previous ork in noncoherent detection for DPSK modulation is conned to ad-hoc strategies of the equal-gain combining te. We introduce a sstematic aroach for dealing ith noncoherent detection for DPSK modulation in correlated diversit Raleigh fading channels. For slol fading channels here the fading arameters in the diversit branches can be regarded as being essentiall identical over to successive smbol intervals, the generalized likelihood ratio test () is shon to ield an excellent solution. It has the advantage of not needing the knoledge of the statistics of the fading rocesses. A minimum error robabilit () detector is also derived that is alicable to slo as ell as fast fading channels. This detector requires a limited knoledge of the statistics of the fading rocesses. Exact bit error rates are obtained for the and the detectors. A comarative analsis ith a recentl roosed ad-hoc decorrelating equal gain combiner shos signicant erformance advantages for the and detectors. Alications to ost-combining decorrelativemultiuser detection are also considered. I. Introduction In this aer e are concerned ith the correlated diversit Raleigh fading (CDRF) channel. We dene the CDRF channel as one that admits inter-diversit branch correlation in both the multilicative fading as ell as the additive noise rocesses. The CDRF channel models a variet of diversit communication sstems including multiath diversit, time or frequenc diversit, receiver antenna diversit etc. Inter-diversit branch correlation in the fading rocesses ma result in the case of receiver antenna diversit The author is ith the Deartment of Electrical and Comuter Engineering at the Universit of Colorado in Boulder, CO 89. This ork as suorted in art b NSF grant NCR The results of this aer ere rst mentioned in a tutorial talk on Noncoherent Multiuser Detection b the author in the IEEE Communication Theor Worksho in Dustin, Florida, Aril, sstems for instance, hen sace limitations (such asona hand-held telehone) dictate that the antennas be saced closer to each other than hat is required for indeendent fading. In time or frequenc diversit sstems, it ma arise due to stringent dela or bandidth constraints, resectivel. The inter-diversit branch correlation in the additive noise ma result in the case of amultiath diversit sstem because of bandidth constraints. When the signalling aveform has a bandidth that is sucientl large comared to the coherence bandidth of the channel to result in multile resolvable aths at the receiver, but hoever, because the aveform ma still be constrained in bandidth and/or due to the distortion introduced b the channel, the various time translates of the aveform that arrive at the receiver ma not be mutuall orthogonal. In general, it can be said that the correlated diversit channel results hen sstems emlo diversit ith disciline. No resource (sace, time, bandidth) can be used as if it ere freel available. In ver slol (static) fading channels here the coherence time is on the order of a hundred times the smbol duration, the fading coecients remain constant over suf- cientl man smbols to allo for their accurate estimation at the receiver, thereb enabling coherent modulation and detection methods. For such slol fading, erfectl estimable channels, and under the assumtions that ISI is negligible, and that the L time translates of the received signalling aveform are orthogonal, and the fading coecients in the various aths fade indeendentl, the maximal ratio combiner or the RAKE receiver is otimal in that it minimizes error robabilit [7] [8]. The extension of the maximal ratio combiner to the correlated diversit channel is simle enough and quite ell understood (cf. [5] [8]). The coherent RAKE receiver is therefore stock in the trade of diversit communications engineering. When one relaxes the assumtion of the availabilit of erfect knoledge of the fading coecients, one of several aroaches can be adoted deending on given comlexit and erformance secications. The dnamic evolution of the fading channel states can be modeled b assuming the channel state vector to be a Gauss-Markov rocess. In the th detection of a smbolinthek time interval, the conditional mean estimate based on ast observations of the matched lter oututs can be obtained b using a decision-directed Kalman lter hich delivers not onl the channel state estimate but the associated error covariance matrix that contains information regarding the condence level that one can attribute to the estimates (cf. [1]). The detector then uses the channel state estimate as if it ere the correct value but it also otimall incororates the error covariance information th in making the decision on the k smbol. This aroach has its origins at least as far back as 1969 in Kailath's generalized likelihood formula for the detection of rather general random

2 signals in Gaussian noise as described in [] (see also references therein). The alication of this general estimatordetector idea to the frequenc non-selective Raleigh fading channel can be found in [1]. The frequenc non-selective channel ith indeendent diversit branches is considered in [4]. The estimator-detector aroach as introduced for multiuser channels for the rst time in [1] in the context of the frequenc non-selective Raleigh fading channel. We are concerned in this ork mainl ith alications here exlicit estimation algorithms are too exensive to imlement. Dierentiall hase shift keed modulation and noncoherent detection is a simle and attractive alternative in such alications. The rst non-trivial roblem on noncoherent multiuser detection of dierentiall encoded signals after the multiuser Gaussian channel ork in [1] arises for fading channels ith diversit as encountered in [14] and [] for frequenc-selective Raleigh fading snchronous multiuser channels. After ost-combining decorrelation, onl the oututs of the decorrelator of the user of interest are retained so that the multiuser channel is reduced to an eective single-user channel (ith a corresonding SNR enalt for decorrelating other users), but the authors of those aers hoever roose the ad-hoc equal gain combining rule to the eective single-user channel. In eect, fading and additive noise correlations are ignored, and the common-sense solution of the indeendent and identical diversit channel from [7] is used for the equivalent single-user correlated diversit channel. When the correlation is light as it is hen sread factors are chosen to be much larger than the number of users as in [14], this ad-hoc strateg orks quite ell. Hoever, bandidth ecient signalling is bound to tell a different stor. There are to reasons. Firstl, there is noise enhancement incurred due to the ost-combining oeration as described in [11]. Secondl, and this is the issue that e ill be concerned ith in this aer, there is an additional SNR enalt to be incurred for using the ad-hoc equal gain combiner. To address the latter roblem, it is therefore suf- cient to conne attention to the single-user correlated diversit channel. We advocate a sstematic aroach tothe noncoherent detection roblem here. Our aroach is based on ell knon ideas in detection theor such as the generalized likelihood ratio test and minimum robabilit of error detection (cf. [6]). The novelt is in the roblem formulations themselves. Where automobile velocities are sucientl lo and/or the data rates are sucentl high, it is realistic to assume the constanc of the fading channel arameters over to successive time intervals. We ill call this the slol fading channel. The general case of fast fading channel is also considered here it is not true that the fading coecients remain constant even over to successive smbols intervals. Suchchannels arise in mobile communication at the usual automobile seeds and data rates due to the eect of Doler sread caused b the relative movement beteen transmitter and receiver. II. Discrete-Time Model for the Correlated Diversit Raleigh Fading Channel The discrete-time CDRF model secies the L- dimensional vector of observations for the zeroth and the revious smbol duration as q( 1) = Rc( 1) d ( 1) + n( 1) ; (1) q() = Rc() d () + n() : () The smbols d( 1) and d() are the dierentiall encoded smbols ith each belonging to an MPSK constellation jl M1 th F = fex M gl= of the k user. The information smbol b() F is related to the dierentiall encoded smbols according to the equation d() = d( 1) 1 b(). The matrix R is a correlation matrix and is assumed to be ositive denite in general and has the interretation, in frequenc-selective channels for instance, as the correlation matrix of the L ossibl non-orthogonal time-translates of the signalling aveform b integer multiles of the inverse of the signal bandidth. The signalling aveform is normalized to have unit energ so that the diagonal elements of R are all equal to unit. The additive noise vectors n( 1) ; n() and the fading channel coecient vectors c( 1) ; c() are zero-mean, comlex, circularl smmetric Gaussian random vectors. Moreover, n( 1) and n() each have an identical correlation matrix NR but are statisticall indeendent of each other. The correlation matrix E c() i c ( j) is denoted as 6ij. Wide sense stationarit of the fading rocess allos us to rite 6ii = 6() and 61 = 6 1 = 6(1). Moreover, in the slol fading channel, c( 1) = c() = c so that 6ij = 6 for the four values of ( i; j). In order to ensure that is the average received energ er smbol, the correlation matrix () is normalized such that Tr( () ) = R III. The Generalized Likelihood Ratio Test () Detector The roblem is to nd a good decision rule for the data smbol b() based onl on the observations q( 1) and q(). The matrix R is assumed to be knon. Hoever, since e are interested in noncoherent detection, the fading coecient realizations c( 1) and c() are unknon. In order not to have to estimate even the statistics of the fading rocesses, the correlation matrices 6 ij for ( i; j ) f1; g are assumed unknon as ell. The average received signal energ and the noise oer sectral densit height N are assumed to be not knon either. A direct alication of the generalized likelihood ratio test () to the roblem as stated does not ield meaningful results. Even though c( 1) and c() are not equal in the fast fading channel, one hoes that at least for the not-so-fast fading channels the are not likel to dier from each other ver much. Let us then roceed, for the urose of deriving the decision rule, to make the assumtion that these to vectors are identical to each other (but still unknon), and then al the rule. The analsis of the resulting scheme for the fast fading channel must of course consider that the correlation matrices 6ij are not all equal. It can easil be shon that the logarithm of the likelihood function of the air q( 1) and q() conditioned on c( 1) = c() = c and d( 1) and d() is maximized over the unknon vector b the vector ^ given as c c 1 c^ = 1 R ( q( 1) d ( 1) + q() d ()) () The resulting generalized likelihood function can be simli- ed under each hothesis corresonding to the M values of the air ( d ( 1) ;d()) in F F so that the air ( d ^( 1) ;d ^()) belongs to 8 9 arg max Re d( 1) d () q ( 1) R 1q() (4) (( d 1) ;d()) F F

3 We are onl interested in the M hotheses corresonding to the values of b(). In turn, b() is equal to the roduct d ( 1) d(). Fortunatel, the generalized likelihood func- tions are equal for all the M airs ( d ( 1) ;d()) hose roducts are equal so that the rule in (5) reduces to qjb 8 9 ^() b = arg max Re b () ( 1) () (5) b F The combining rule suggested b the above detector is that the matched lter outut vectors for the to relevant time intervals be combined in the bilinear form given as q ( 1) R 1q(). There is another interretation in that the combining rule suggests that the oututs be transformed b a noise hitening transformation R so that 1 = 1 = ith the transformed vectors ( m)= R q( m), the combining rule is one of equal-gain combining based on these noise hitened vectors. The detector can therefore also be called the noise-hitening equal-gain combiner. In the secial case of binar modulation, the above detector further reduces as 9 9 In this section, e assume that the fading correlation matrices 6ij are knon at the receiver and so are and N. Hoever, the fading coecient realizations c( 1) and c() are of course still unknon. Starting ith the model in () e consider the roblem of obtaining the minimum robabilit of error decision ( d ^( 1) ;d ^()) that minimizes the robabilit that that air is in error and form the estimate of the transmitted smbol as ^() b = d ^ ( 1) d ^(). Let us de- T T T ne the L-dimensional vector q = [ q ( 1) q ()] and T let d = [ d( 1) d()]. Since the M realizations of d are equirobable, the maximum a osteriori test for the M -ar hothesis testing roblem reduces to the maximum likelihood rule so that ith f( qd j ) denoting the conditional robabilit densit function (df) of the random vector q conditioned on,ehave () ^() b = sgn f Re q ( 1) R 1q () : (6) It is interesting that in the secial case of the the uncorrelated diversit fading channel here the correlation matrix is the identit matrix, the detector reduces to the equalgain combiner. Thus e nohave a rigorous justication for the equal-gain combiner of [7] hich as roosed there b an imlicit aeal to common-sense. The ad-hoc solution roosed for this roblem in [14] (see also []) is that of equal gain combining based on the decorrelated oututs R q( 1) and R 1q() so that e have the 1 decorrelating equal-gain combiner ^() b = sgn f Re q ( 1) R q () : (7) IV. The Minimum Probabilit of Error () Detector d d^ = arg max f ( qd j ) (8) df q R 1q Noting that the conditional df of q under each hothesis is zero-mean comlex-valued Gaussian df that is identical for the M values of the vector d for hich d ( 1) d() are equal (to sa b()) and has a correlation matrix given as K () = R611R + R () R61R b() N N b R6 R R6 R R ; (9) e have a reduced M-ar hothesis test. It can be shon that, b using a determinental identit for the determinant of to bto block artitioned matrix, the determinant of the correlation matrix in (9) is indeendent of b() as ell, so that the roblem reduces to minimizing over all hotheses, the quadratic form that aears as the exonent in the exression for the M Gaussian densit functions ^() b = arg max qjb (1) b F ^() b = arg max Re fb () ( 1) () g (11) b F 1 N 1 1 N 1 8 = R R E + R R (1) here the matrix E is dened as N 1 E = R 61 (1) An alternative derivation of the rule in (11) can also be obtained b ursuing a estimator-detector decomosition 4 idea of [1]. Let us dene c~() = c() d( 1) so that the second exression in () becomes q() = Rc~() b() + n(). Letting c^() denote the conditional mean estimate of c~(), one can rite the equation for q() as q() = Rc^() b() + R (~() c c^()) b() + n() (14) The conditional mean estimate or the minimum meansquared error (MMSE) estimate of c~() based on the observation q( 1) can be obtained using the orthogonalit rincile hich states that the error in the estimate c~() c^() must be orthogonal to the estimate c^(). The result is () () q K 1 () q Using the matrix inversion lemma, it can be shon that the above decision rule further reduces to here the otimal combiner matrix N 1 c^() = R R q( 1) (15) The next ste is to vie (14) as reresenting a model for coherent detection ith knon fading coecients that are equal to the MMSE estimate c^() but recognize that the additive noise is R (~() c c^()) b() + n() so that its correlation matrix is RER + NR here E is the MMSE error covariance matrix hich e dened ith the benet of hindsight in (1). The minimal sucient statistic for b() is therefore N 1 1 c^ () E + R q() (16) and the minimum error robabilit rule based on this scalar statistic can be shon to reduce to the rule given in (11). V. Error Probabilit Analsis In this section, e restrict attention to binar modulation and obtain error robabilit exressions for the decorrelating equal-gain combiner (), the detector, and the detector. Fortunatel, it is ossible to give a single formula for all of these detectors since the general form of the decision rule for these detectors is given as q 8 ^() b = sgn f Re( q ( 1) q () g (17) 8q is dened as

4 here = R for the, R 1 for the detector, and 8 for the rule here 8 is dened as in (1). The decision statistic Re fq ( 1) q() g can also be ritten as the quadratic form q5qhere 5 is a L-dimensional matrix hich hen ritten as a block artition matrix of square L-dimensional matrix elements, the diagonal blocks 5ii = and the o-diagonal blocks are identical ith 5ij =. The robabilit of error can be exressed as the robabilit ofthe error event described b the condition q 5q < conditioned on the transmitted bit b() being equal to +1. Since q is a comlex zero mean Gaussian random variable ith a correlation matrix given as in (15), the decision variable is a Hermitian quadratic form in comlex Gaussian random variables for hich the characteristic function is given in Aendix B of [8] as q5q E ex( s ) = = X i 1 1 det I + skqjb() 5 1 Q L i =1 (1 + s : i) Consequentl, the robabilit oferror can be exressed as the sum of the residues of the artial fraction exansion of the above characteristic function corresonding to negative eigenvalues so that ith i denoting the residues, e have P = i : () i; < VI. Numerical Results In this section e resent numerical results. For the single-user channel examles, the matrix R is chosen to be jkj l a four-dimensional Toelitz matrix Rkl = : 8. For the multiuser channel, the KL dimensional correlation matrix R as dened in [11] is generated ith six Gold sequences of length 1 including the ISI-IUI mask described therein. There are four aths for each user so that e have a 4- dimensional R matrix. For the multiuser channel examle, e let R in the model of this aer be the inverse of the uer-left four-dimensional matrix of the inverse of R. Figure 1 demonstrates the error robabilities of the decorrelating equal-gain combiner, the detector and the detector for the slol fading channel as a function of the signal-to-noise ratio =N in db. The matrix 6 of fading correlations is chosen to reresent four indeendent diversit channels but ith unequal energies so that 6 = diagf: 56: : 6; : 1; : 6 g. Notice the substantial erformance gains achieved ith the and the detectors over the detector. Moreover, the dierence beteen the and the detectors do not aear to be signicant enough to arrant the use of the rule since the latter needs information about the fading channel statistics and the average signal to noise ratio. There as hoever a more noticeable ga beteen the erformances of those to detectors for correlated 6 channels. The coherent maximal ratio combiner error rate is also shon in Figure 1 to indicate the loss incurred in using noncoherent detection as oosed to the more energ ecient BPSK signalling method ith coherent detection. Of course, there is an extra cost of channel arameter estimation associated ith the (18) (19) This robabilit of error can be numericall comuted for an of the, the, and the decision detectors b substituting the aroriate value of. latter scheme. Morover, the eects of imerfect estimation are eectivel ignored. Figure shos the error robabilities of the ostcombining decorrelative multiuser detectors for the slol fading indeendent but unequal energ diversit channel (same distribution of energ as in gure 1) that are based on the, and combining rules. Notice again that the and the rules signicantl outerform the. The higher error rates at a given SNR in Figure relative to Figure 1 are attributable to the noise enhancement that results from the ost-combining decorrelation oeration. For more on ho to avoid this noise enhancement, the reader is referred to [11]. Figure deicts a comarative analsis of the and the detectors for identical indeendentl fast fading channels ith equal fade rates in the four diversit branches. The fade rate is characterized b the value of the correlation coecient here 6(1) = 6 ( ) = : 5 I. The further the value of is from unit, the faster the fade rate, hich in ractical terms can be thought of as higher relative velocit beteen transmitter and receiver in a mobile radio channel and/or loer data rates. The fade rate is chosen as a arameter and the error robabilit is deicted as a function of the received SNR. Notice that both detectors exhibit an error oor as SNR values increase because of the domination of bit errors due to channel estimation errors relative to additive noise. For loer SNRs, the still does better than the but for high SNRs the outerforms the ith its limiting error rate being much loer than that of the detector. This shouldn't be surrising because the detector as derived under the assumtion of slo fading ( = 1). Even the ad-hoc is b chance more robust than the detector is to the model mismatch. Can the error oor of the be further loered? Ho does it comare ith the rule for fast fading? Figure 4 shos the error rate comarison beteen the detector and the for identical indeendentl fast fading channels ith equal fade rates and ith fade rate as arameter. Notice that the outerforms the for the loer SNR values as did the detector and as SNR values are increased, the and the detectors seem to be converging to the same error oor levels. It turns out this is attributable to the assumtions of identical indeendent and equal fade rates of the articular diversit channel under consideration. Figure 5 disels an hoe that the simle ma alas rival erformance for high SNRs. Figure 5 considers indeendent but not identicall fast fading channels (the energ role is the same as in Figure 1) and moreover, the fade rates associated ith various diversit branches are distinct. To dierent sets of fade rates are considered as shon in the gure. Notice that in the situation here the strongest (rst) channel fades the fastest (denoted b L in gure), the reaches an error oor that is much loer than that achieved b the. The case here the strongest branch fades the sloest results in ver similar error oors for the and the. Figure 6 deicts the limiting values (as SNR!1) of the bit error rates as a function of the fade rate (assumed equal for all diversit branches) for the, the, and the detectors in the case of heavil correlated diversit branches. The limit of the bit error rate of the has been found analticall, and those of the and the

5 have been comuted for sucientl high SNRs at hich there is negligible variation of error rate ith changes in SNR. The abscissa is 1 in db so that fade rates increase from left to right. There is a ver substantial ga beteen the and the or the in this case. This examle serves as an illustration of the fact that there is nothing inherent in DPSK modulation and its noncoherent detection that gives rise to high values of error oors as has been suggested in the literature ([7] suggests that DPSK is suitable for channels here the channel fading coecients remain constant over to successive smbol intervals but this comment ishoever not aroriate for the detector). VII. Conclusions A sstematic aroach as introduced to obtain noncoherent detectors of DPSK modulated signals transmitted over correlated slo or fast fading diversitchannels. For the slo fading channel, the generalized likelihood ratio test () detector is derived hereas the minimum error robabilit () detector is obtained for both slo and fast fading channels. Estimator-detector interretations ere given for both detectors. The error rate analsis as carried out for each detector under ver general conditions for binar modulation. Our numerical comutations reveal that the detector far out-erforms the reviousl roosed decorrelating equal-gain combiner for slol fading channels here the fading coecients over to successive intervals are assumed to be constant. The rule is of course the best but its gain over the rule is small enough for slol fading channels, articularl for indeendent diversit channels, that the extra knoledge of the fading statistics and average signal to noise ratio that are required to imlement the detector, ma not arrant the use of the detector over the detector. On the other hand, in fast fading channels, the outerforms the rule in high signal-to-noise ratio regions and this is attributable to the fact that the rule is derived under the mis-assumtion of a slol fading channel. Hoever, the rule can in turn signicantl outerform the rule articularl in cases here the fading rocesses in the various diversit branches are sucientl correlated and/or in the case of indeendent diversit branches hen the energ in the diversit branches are dierent and/or hen the fading rates associated ith the diversit branches are dierent from each other. In articular, the error rate oors for the rule in such situations can be signicantl loer than that for the and the detectors. High error oors for fast fading channels are not endemic to noncoherent detection ith DPSK modulation but a characteristic of an ad-hoc detector such as the or the detector designed for the slol fading channel. The erformance of the detector derived here must be considered as the standard against hich the best achievable error rates ith coherent detection are comared. The question of comarison ith the loest achieveable error rates ith the resumabl more oer ecient BPSK modulation and coherent detection is concetuall straightforard. Coherent detection must be accomlished ith elaborate channel estimation algorithms (cf. [1] [1] [9]). Our develoment of the detector-estimator interretation of the detector in this aer allos us to ut the detector and the coherent estimator-detector receivers in a uni- ed frameork. The dierence is that the error covariance E matrix resulting from the MMSE estimation of the fading arameters based on the revious observation in the case of the noncoherent rule is relaced b the error covariance that results from an estimator such as a Kalman lter that uses all ast observations rather than one revious observation. Such estimators are decision-directed and suscetible to the hase reversal henomenon (cf. [1] hich suggests coherent detection ith dierentiall encoded data). Moreover, the more elaborate channel estimation algorithms associated ith coherent detection require far more than just the knoledge of 6() and 6(1). In contrast, there is no error roagation roblem ith the noncoherent detector and moreover, the simlicit resulting from not having to exlicitl use a channel estimation algorithm ill, in ractice, eigh heavil in favor of the noncoherent rule obtained in this aer. Acknoledgements The author ould like to thank Eric Fain for erforming the numerical comutations and for creating the lots of this aer. References [1] R. Haeb and H. Mer, \A Sstematic Aroach to Carrier Recover and Detection of Digitall Phase Modulated Signals on Fading Channels," IEEE Trans Commun., 7: , Jul [] H. C. Huang and S. C. Schartz, \A Comarative Analsis of Linear Multiuser Detectors for Fading Multiath Channels," Proc of 1994 IEEE Globecom, , December [] T. Kailath, \The General Likelihood-Ratio Formula for Random Signals in Gaussian Noise," IEEE Trans. on Inform. Theor, IT-15:, , Ma [4] P. Y. Kam, \Otimal Detection of Digital Data Over the Nonselective Raleigh Fading Channel ith Diversit Recetion," IEEE Trans Commun., 9: , Februar, [5] F. Ling, \Matched-Filter Bound for Time-Discrete Multiath Raleigh Fading Channels, IEEE Trans Commun., 4: // , FebMar/Ar [6] V. H. Poor, Introduction to Detection and Estimation, Sringer-Verlag, [7] J. G. Proakis, Digital Communications, Third Ed., Mc- Gra Hill, NY, [8] M. Schartz, W. R. Bennett and S. Stein, Communication Sstems and Techniques," An IEEE Press Classic Reissue, 1996, originall a McGra Hill Publication, [9] M. Stojanovic and Z. Zvonar, \Linear Multiuser Detection in Time-Varing Multiath Fading Channels," Proc CISS, Princeton, NJ, , March [1] M. K. Varanasi, \Noncoherent Detection in Asnchronous Multiuser Channels," IEEE Trans. on Inform. Theor, IT- 9:1, , Januar 199. [11] M. K. Varanasi, \Grou Detection for Snchronous CDMA Communication Over Frequenc-Selective Fading Channels," Proc 1st Annual Allerton Conf on Communication, Control, and Comuting, Set 9-Oct 1, 199, [1] S. Vasudevan and M. K. Varanasi, \Multiuser Detection over Snchronous CDMA Fading Channels ith Memor," Proc 6th CISS, Johns Hokins Universit, March 199. [1] H. Wu and A. Duel-Hallen, \Multiuser Detection ith Dierentiall Encoded Data for Mismatched Flat Raleigh Fading CDMA Channels," Proc CISS, Princeton, NJ,. -7, March [14] Z. Zvonar and D. Brad, \Coherent and Dierentiall Coherent Multiuser Detectors for Asnchronous CDMA Frequenc- Selective Channels," Proc Milcom., 199.

6 Fig. 1: comarison of the,, and detectors and the coherent MRC for the slol fading channel Fig 4: comarison of and detectors for indeendent identical diversit channels ith fade rate as a arameter MRC 1 - rho= Fig : comarison of the ost-combining decorrelative multiuser detectors based on the,, and combining rules Fig 5: comarison of and detectors for indeendent but unequal energ diversit channels for to distinct sets of unequal multiath fade rates L1={ } L={ } L L L L1 1-1 Fig : comarison of the and the detectors for the fast fading channel Fig. 6: The limit of as N for the,, and detectors as a function of 1 fade rate for highl correlated fading rho= ρ (db)