Combned Temporal and Spatal Flter Structures for CDMA Systems Ayln Yener WINLAB, Rutgers Unversty yener@wnlab.rutgers.edu Roy D. Yates WINLAB, Rutgers Unversty ryates@wnlab.rutgers.edu Sennur Ulukus AT&T Labs Research ulukus@research.att.com Abstract CDMA systems are nterference lmted and therefore effcent nterference management s necessary to enhance the capacty of a CDMA system. In ths paper, we consder combnng two effectve recever based nterference management strateges: multuser detecton (temporal flterng) and recever beamformng (spatal flterng). We formulate and examne the performance of several lnear flter structures whch are all based on mnmum mean squared error (MMSE) crtera, but dffer n how the MMSE problems are defned n the temporal and spatal domans,.e., jontly or n cascade. It s shown that whle the jont optmum MMSE flter acheves the maxmum sgnal-to-nterference rato (SIR) among all possble lnear flters, the constraned optmum MMSE flter whch results n a sngle temporal and sngle spatal flter, outperforms all combned sngleuser/multple-user approaches and cascaded optmzaton approaches ether unformly or asymptotcally. The constraned optmum MMSE flter s near-far resstant n all but very hghly loaded systems and enjoys low complexty. Introducton The demand for hgh capacty flexble wreless servces s ever-growng. CDMA shows promse n meetng ths demand and consequently W-CDMA [] has been proposed as a standard for the thrd-generaton (G) wreless systems. It s well known that CDMA systems are nterference lmted and suffer from near-far effect. The challenge to enhance the capacty of a CDMA system therefore les n nterference management. In ths work, we concentrate on the two commonly used nterference management methods: multuser detecton and recever beamformng. Both methods am at suppressng or cancellng the nterference usng recever sgnal processng; multuser detecton explots the temporal structure whereas beamformng explots the spatal structure of the nterference for nterference management [, ]. Multuser detecton [] performs temporal flterng of the receved sgnal to effectvely suppress the multple access nterference. The optmum multuser detector has been shown to be exponentally complex n the number of users, and a number of low complexty suboptmum recevers have been proposed followng ths development [, ]. Increasng the capacty of CDMA systems by employng antenna arrays at the base staton has been proposed n [], where the outputs of the multple antenna array elements are combned to make bt decsons for the user. In [], matched flter recevers are assumed n the temporal doman for each user and the array observatons are combned va a flter that s matched to the array response of the user,.e., sngle user processng s employed n both domans. Another method of capacty enhancement whch utlzes the spatal dversty s space-tme processng for CDMA whch tradtonally refers to recever beamformng (space processng) and multpath combnng (tme processng) []. The receved sgnals from dfferent paths and antennas are combned to better decode the desred user s bts. However, the nherent structure of the multple access nterference s not exploted,.e., no multuser detecton s employed [,, ]. A recent paper [] addresses the dervaton of the suffcent statstcs and the optmum and some lnear suboptmum multuser detectors when an antenna array s present at the recever for a multpath channel. In ths work, we wll nvestgate the possble recever flter structures when both multuser detecton and beamformng are employed to further ncrease the uplnk capacty of a CDMA system. Lnear processng s assumed n both the temporal and the spatal domans and the temporal-spatal flters are denoted by two-dmensonal matrx flters. Wthn ths framework, there are several possble flter structures. One can derve the jontly optmal temporal and spatal flter that mnmzes the mean squared error (MSE) between the nformaton bt and the flter output of a desred user. Snce ths jont MMSE flter may have hgh computatonal complexty, less complex flters whch nevertheless provde effcent nterference suppresson are of nterest. To serve ths purpose, recently, constraned optmum flters are proposed by forcng the jont recever flter matrx to be of rank [, ]. One can also construct cascaded flters where MSE optmzaton s carred over n temporal and spatal
domans ndependently n tandem, n both spatal-temporal and temporal-spatal orders. Whle the cascaded spataltemporal flter s smlar to the dea of cascade optmumspace/optmum-tme combner proposed n [], the temporal combner n our case s a chp combner that explots the temporal structure of the nterference composed of the temporal sgnatures of the nterferers, as opposed to a multpath combner n [] whch s a sngle user temporal processor. We consder all above flter structures and then gve analytcal performance comparsons among them. Whle the jont doman MMSE flter s clearly the SIR maxmzng temporal and spatal processor over all matrx flters, an nterestng observaton s that the constraned optmum temporal spatal processor outperforms all combned sngle user/multple user approaches and the cascaded approaches ether unformly or asymptotcally. System Model We consder a sngle cell DS-CDMA system where each user s assgned a unque sgnature sequence. For clarty of exposton, we assume a synchronous system wth processng gan G. At the base staton, an antenna array of K elements s employed. The receved sgnal at the output of the antenna array at the base staton s: r(t) = pj b j s j (t)a j + n(t) () where p j,b j, s j (t) and a j are the receved power, the bt, the (temporal) sgnature and the array response vector (spatal sgnature) of user j, respectvely. Both the temporal and spatal sgnatures of the users have unt energy, and the temporal sgnatures are of the followng form s j (t) = G l= s (l) j ψ(t (l )T c) () where ψ(t) s the chp waveform, T c s the chp duraton, and s (l) j = ±/ G. Chp matched flterng and samplng the receved sgnal r(t) at the lth chp nterval, we obtan a K-dmensonal observaton vector r l r l = pj b j a j s (l) j + n l () whch represents the chp matched fltered samples at the lth chp nterval across the K antenna array elements. Over one bt perod, T b = GT c, we collect a set of GK-dmensonal vectors {r l, l G} whch we can arrange n a G K matrx R =[r, r,, r G ] : R = pj b j s j a j + N () These KG observaton samples can also be seen as a collecton of K G-dmensonal vectors,.e., R = [z, z,, z K ], where z k denotes the observaton vector consstng G chp matched fltered samples at the output of the kth antenna element and s expressed as z k = pj b j s j a (k) j + n k () In (), N s the matrx that represents the spatally and temporally whte nose,.e., E[N kl N mn] =σ δ km δ ln, where ( ) denotes the conjugate of a complex number. We label user as the desred user and the other users as nterferers. Flter Structures The detecton of the nformaton bt of the desred user s done by takng the sgn of the real part of the decson statstc whch s found by combnng the entres of the observaton matrx R by usng a matrx flter X. Thus, the decson statstc, y, s the output of a two dmensonal lnear flter X : y = G K [X ] jl R jl = tr(x H R) () l= where tr( ) and ( ) H are the trace and the hermtan transpose operatons, respectvely. In what follows, we nvestgate the possble flter structures. The flter structures n Sectons.,., and. use sngle user processng n at least one of the temporal and spatal domans and are well-known. Next we derve two temporal-spatal flters, the flter structures n Sectons. and., whch use cascaded MMSE optmzatons n spatal and temporal domans. The structure of these recevers s to combne ether the receved chp samples at the output of each array element n the MMSE sense followed by a spatal flter that combnes the resultng vector n the MMSE sense (Secton.); or to combne all array outputs for each chp sample n the MMSE sense followed by the temporal MMSE combner for the resultng vector (Secton.). Next, n Sectons. and., we gve the jont temporalspatal MMSE flter structures. The dfference between the two jont MMSE structures s the fact that whle the jont optmum temporal-spatal MMSE n Secton. s the best flter n terms of mnmzng the MSE (and maxmzng the SIR) over all possble matrx flters, the constraned optmum temporal-spatal flter n Secton. s the MMSE flter when the flter space s constraned to contan matrx flters of rank only. The physcal nterpretaton of ths mathematcal constrant on the matrx flter s that t results n a separable flter wth a sngle temporal and a sngle spatal
combner. The dfference between the jont MMSE structures n Sectons. and. and the cascaded MMSE structures n Sectons. and. les n the fact that the cascaded structures use temporal and spatal flters that are optmzed ndependently n each doman whle the jont structures are found by optmzaton n both domans smultaneously... Sngle User Temporal-Spatal Detector Ths s a sngle user based approach for both the spatal []. The decson statstc n ths case s y = s Ra = tr(a s R) leadng to X MF-MF = s a ().. Temporal MMSE Flter Spatal MF Ths approach uses multuser processng n temporal doman [] and sngle user processng n spatal doman. The decson statstc n ths case s y = c Ra = tr(a c R) leadng to where X MMSE-MF = c a () c = ( p p j a H a j s j s j I) + σ s ().. Temporal MF Spatal MMSE Flter Ths approach uses sngle user processng n the temporal doman combned wth mult user processng n spatal doman []. The decson statstc n ths case s y = s Rw = tr(w s R) leadng to X MF-MMSE = s w () where w = ( p p j (s s j ) a j a H j + σ I) a ().. Cascaded Temporal-Spatal MMSE Flter Assume that at the output of each antenna array we are allowed to desgn a separate temporal flter,.e., a lnear chp combner. Recall that the output of the kth antenna element s z k. We desgn K temporal flters c k,k =,...,K, such that each of the K resultng statstcs ỹ k = c H k z k = pj b j c H k s ja (k) j + c H k n k () has mnmum mean squared dfference from the desred bt, b. The soluton can be found as []: c k = ( p a (k) a (k) j p j s j s j + σ I) s () Note that what makes c k, the temporal MMSE flter at the output of the kth antenna, dfferent from c l, the temporal MMSE flter at the output of the lth antenna, are the dfferent spatal gans users have for dfferent antennas. Defnng the modfed gan at the output of the kth antenna for user j as ã (k) j =(c H k s j)a (k) j, from () we have ỹ k = pj b j ã (k) j +ñ k, k =,..., K () We can then combne {ỹ k } n the MMSE sense, and smlar to (), express the second stage of the cascaded flter as w = ( N ) p p j ã j ã H j + Λ ã () where Λ = dag{σ (c H k c k)} s the covarance matrx of ñ. The fnal bt decson s done by takng the sgn of the real part of y = w H ỹ. Note that to construct the overall recever we need to nvert KG G matrces and one K K matrx. To see how the overall cascaded flter can be expressed as matrx flter X TS-CMMSE and, observe that ỹ k = c H k z k = c H k Re k = tr(e k c H k R) () y = K k= ( K ) wkỹ k = tr wke k c H k R k= () where e k s a K vector whch has n ts kth entry and zeros elsewhere. Comparng () wth (), we fnd that X TS-CMMSE = whch can be of rank up to K. K w k c k e k () k=.. Cascaded Spatal-Temporal MMSE Flter Alternatvely, one can thnk of frst combnng all antenna array elements n each chp nterval, followed by a temporal combner. We frst desgn G spatal flters w l,l=,...,g, to combne the entres of r l n the MMSE sense w l = ( p s (l) (s (l) j ) p j a j a H j I) + σ a ()
Recall from () that (s (l) j ) =/G, for all l. Thus, defnng ŵ = p ( N G p ja j a H j ) + σ I a () we arrve at w l = s (l) ŵ for l =,...,G. At the output of the lth combner, the resultng statstc can be expressed as ŷ l = w H l r l = where ˆn l = w H l n l. Defnng ŝ (l) j ŷ l = pj b j ŵ H a j s (l) s (l) j +ˆn l () = ŵ H a j s (l) s (l) j, we have pj b j ŝ (l) j +ˆn l, l =,...,G () It remans to fnd the MMSE combner for ŷ. We can express ths second stage of the cascaded flter as c = ( N p p j ŝ j ŝ H j + G σ (ŵ ŵ)i) H ŝ () Notce that, as n the case of Secton., the nose covarance matrx s gven by dag{σ (wl T w l)}. Snce wl H w l = (/G)(ŵ H ŵ) for all l, the nose covarance matrx reduces to (σ /G)(ŵ H ŵ)i. The fnal bt decson s done by takng the sgn of the real part of y = c H ŷ. Note that to construct the overall recever we need to nvert one K K matrx to calculate ŵ and one G G matrx to calculate c. To see how the overall cascaded flter can be expressed as a matrx flter X ST-CMMSE, observe that and y = ŷ l = e l Rw l = s(l) tr(ŵ e l R) () G l= ( c l ŷl = tr ŵ ( G l= ) c l s(l) e ) l R Then, comparng () wth (), we fnd that whch s of rank. () ( G X ST-CMMSE = c l s (l) e l )ŵ () l=.. Optmum Temporal Spatal MMSE Flter The optmum matrx flter n temporal and spatal domans whch mnmzes the MSE between y and b s X O-MMSE =argmn X E [ tr(x H R) b ] () The optmzaton problem () can be converted to an optmzaton problem wth vector varables for easer manpulaton []. The problem then becomes a straghtforward extenson of the standard MMSE problem. Its soluton s [,,,, ]: x = ( p p j q j q H j I) + σ q () where q j s the temporal-spatal sgnature of user j and s constructed by stackng columns of s j a j as a long vector of sze KG. The matrx flter X O-MMSE s constructed by takng every G elements of x and puttng as a column to X O-MMSE. The jont MMSE flter requres a possbly large matrx (KG KG) to be nverted whch can be computatonally costly, or the correspondng adaptve mplementaton may be slow. Ths s the reason why we consder the less complex jont MMSE flter n the next secton... Constraned Optmum Temporal Spatal MMSE Flter To reduce complexty of the temporal and spatal flterng wth lttle sacrfce n performance, [,] proposed fndng the optmum matrx flter n a constraned class of matrx flters. The proposed constraned class s rank matrx flters, or the separable temporal-spatal flters,.e., the flters that can be expressed as X = cw. We can fnd the jont optmal flter par n the MMSE sense for ths constraned class. The optmzaton problem n () becomes [ c, w] =argmn c,w E [ c H Rw b ] () The resultng [ c, w] par yelds the matrx flter X CO-MMSE = c w () Ths matrx flter s suboptmal for the optmzaton problem gven n () snce t s found n a constraned X space. The MSE functon n () can be expressed as MSE = p j c H s j w H a j + σ (c H c)(w H w) p R{(c H s )(w H a )} + () where R{ } denotes the real part of a complex number. The mnmzer of () does not have a closed form expresson []. Further, the MSE functon s not jontly convex n c and w, although t s convex n each varable (c,orw), when the other varable s fxed. Thus, standard teratve optmzaton algorthms cannot guarantee convergence to global mnma. However, an alternatng mnmzaton algorthm
[] s gven n [] that s observed to have good convergence propertes. We restate the algorthm here for convenence. Consder fxng the value of one of the flters; say w s fxed to w. It s then possble to fnd the flter, ĉ, that maxmally decreases the MSE functon n (). The soluton s analogous to the MMSE detector [], where user j s receved ampltude s modfed such that t s p j ( w H a j ). Denote ths flter as ĉ = MMSE( w): ĉ = ( p ( w H a ) p j w H a j s j s H j + σ w I) s () The same argument can be made for the case where c s fxed to c and the spatal flter s found to maxmally decrease the MSE, ŵ = MMSE( c): ŵ = ( p ( c H s ) p j c H s j a j a H j + σ c I) a () Now, consder the followng algorthm. Startng wth the flter par c(), w() and keepng w() fxed, one can fnd c() = MMSE(w()). Ths operaton decreases the MSE defned n (). Then keepng c() fxed, one can fnd w() = MMSE(c()). Ths operaton further decreases the MSE n (). Iteraton n +of ths two step teratve algorthm for user s gven below. c(n +) = MMSE(w(n)) () w(n +) = MMSE(c(n +)) () Note that the order n whch c and w are updated could be reversed. After each two-step teraton gven by () and (), the MSE n () monotoncally decreases. The algorthm s provably convergent and the convergence pont s expermentally observed to be the optmum par [ c, w] where the MSE s mnmzed and the SIR of the user s maxmzed []. Performance Comparson An mportant performance comparson crteron s the bt error rate (BER). Unfortunately, for general system parameters, t s dffcult to derve analytcal results for the BER renderng ths comparson ntractable. Commonly the BER s expressed as a functon of the SIR by applyng a Gaussan approxmaton to the total nterference. It was reported n [] that ths approxmaton s partcularly accurate when MMSE recevers are employed. For a general matrx flter X, the MSE and the SIR are related as (for detals, see the Appendx of []) =+SIR(X) () mn α MSE(αX) Thus, wth an approprate scalng, the MSE and the SIR produced by a flter can be related, and the flter that mnmzes the MSE also maxmzes the SIR. Note that the SIR, and therefore the BER when defned n terms of the SIR, are nsenstve to the scalng of the lnear recever flter. From the arguments above, t s clear that the optmum MMSE recever of Secton. outperforms all other recever structures mentoned n Sectons. through., as well as the constraned optmum MMSE recever n Secton., n terms of both the MSE and the SIR. The reason for ths s that the flter n Secton. s chosen to mnmze the MSE over all possble matrx flters. It only remans to compare the performance of the constraned optmum MMSE recever of Secton. wth the recever structures n Sectons. through.. Frst we observe from (), (), (), () that the flters X MF-MF, X MMSE-MF, X MF-MMSE, X ST-CMMSE are of rank. Gven that the constraned optmum MMSE, X CO-MMSE, mnmzes the MSE and therefore maxmzes the SIR among all possble rank matrx flters, we conclude that the constraned optmum MMSE recever flter outperforms all of these suboptmum recever flters. In fact, the teratve algorthm descrbed by ((),()) can be started at any of the temporal-spatal flter pars that defne X MF-MF, X MMSE-MF, X MF-MMSE or X ST-CMMSE. Snce each teraton ncreases the SIR and decreases the MSE monotoncally, wth each teraton, the performance of the resultng flter par s better than the prevous one and the convergence pont temporal-spatal flter par, X CO-MMSE, outperforms the startng pont flter par. The cascaded temporal-spatal MMSE flter, X TS-CMMSE, n Secton. can have rank up to K, just as the jont optmum MMSE flter of Secton. whch can have rank up to mn{k, G}. Thus, there could be cases under whch X TS-CMMSE performs better than the constraned optmum flter; see Secton. However, the fact that X TS-CMMSE has hgher rank than X CO-MMSE does not necessarly guarantee that t yelds a lower MSE or a hgher SIR than X CO-MMSE. Ths s also demonstrated n Secton. Thus, nether X CO-MMSE nor X TS-CMMSE performs unformly better than the other; dependng on the system parameters (N, G, K, spatal and temporal sgnatures of the users, etc.) one may outperform the other. It s possble to compare the two flters n the asymptotc regme when the background nose σ goes to zero, or equvalently the receved powers of the nterferng users go to nfnty. It s well-known that the MMSE recever reduces to a decorrelatng recever as the background nose power goes to zero or the receved powers of the nterferng users go to nfnty []. The decorrelatng recever [] s a
lnear multuser detector whch suppresses the multaccess nterference totally. Ths s done by projectng the desred user s sgnal onto the subspace that s orthogonal to the sgnal space spanned by the nterferng users. The decorrelaton operaton s ndependent of the receved powers of the users and only depends on ther sgnature sequences. The multaccess nterference s suppressed totally f the desred user s sgnature sequence s lnearly ndependent of the nterferng sgnatures. Nevertheless, the decorrelatng detector exsts even when the sgnature sequences of the users are not lnearly ndependent; n ths case one needs to use the Moore-Penrose generalzed nverse of the cross correlaton matrx as opposed to ts drect nverse []. The decorrelator s ndependent of the receved powers n ths case as well []. Recall now that n the calculaton of the cascaded temporal-spatal MMSE recever flter X TS-CMMSE, frst, K temporal MMSE recever flters are found and that these K MMSE recevers, c k s, are dfferent due to the fact that the receved powers of the users are dfferent at each antenna array element; see (). Ths s because the actual receved powers of the users are multpled wth the square magntudes of the antenna gans a (k) j at the outputs of dfferent antenna elements. Snce n the nterference lmted regme the MMSE recevers go to decorrelators and snce decorrelators are ndependent of the receved powers of the users, all K temporal recever flters become dentcal,.e., c k = c for all k. Note that ths s true even when the cross correlaton matrx s not nvertble and the Moore-Penrose generalzed nverse s used. When the temporal flters at the output of all antenna array elements are the same, the recever flter X TS-CMMSE becomes a rank flter. The MSE acheved by X TS-CMMSE, MSE TS-CMMSE, s larger than that of the constraned optmum MMSE flter, MSE CO-MMSE, smply because X CO-MMSE s the flter that yelds the mnmum MSE among all rank matrx flters. Thus, lm σ MSETS-CMMSE lm σ MSECO-MMSE () Equvalently, usng (), the SIRs acheved by these two flters n ths nterference lmted regme s compared as lm σ SIRTS-CMMSE lm σ SIRCO-MMSE () Hence, the constraned optmum MMSE flter outperforms the cascaded temporal-spatal MMSE flter of Secton. asymptotcally. Results and Conclusons We consder a sngle cell CDMA system, the base staton of whch employs a lnear antenna array []. The temporal sgnatures and users postons whch n turn determne the spatal sgnatures are created randomly, and kept fxed for Resultng SIR of the desred user C Fgure. N =,K =,G=. the duraton of the experment. We plot the output sgnal-tonterference rato (SIR) for the desred user (n lnear scale) versus the receved sgnal-to-nose ratos (SNRs) of all nterferers (n db scale). The desred user s SNR s db. Consder frst a system wth processng gan G =, K =array elements and N =users. Fgure shows the output SIR of the desred user. As expected, wth only a sngle nterferer present, all flters perform well even under very severe near-far condtons where the nterferer s power s as much as db above the desred user s. The only excepton s the temporal-spatal matched flter whch s wellknown to be not near-far resstant. The more nterestng observaton about ths system s better observed n Fgure. Recall that we concluded n Secton that the constraned optmum MMSE flter does not necessarly outperform the cascaded temporal-spatal MMSE flter of Secton., t only s as good or better than the cascaded temporal-spatal MMSE flter asymptotcally. Indeed, n Fgure, we see that the cascaded temporal-spatal MMSE outperforms the constraned optmum MMSE flter. When the system becomes nterference lmted, both detectors have dentcal performance. The constraned optmum MMSE flter outperforms all other flters, except for the optmum MMSE flter whch s the SIR maxmzer among all matrx flters. For the rest of ths secton, the system consdered has K =antenna array elements and G =processng gan. We wll examne the performance of the flter structures for ths system under dfferent loadng condtons. We frst consder N = users. Fgure shows the output SIR of the desred user. The constraned optmum MMSE flter outperforms all flters, except for the optmum MMSE flter. Note that, for ths system, user s temporal sgnature sequences are lnearly ndependent and thus, when the system s nterference lmted, the flters that perform
Resultng SIR of the desred user....... C Resultng SIR of the desred user C.. Fgure. Fgure magnfed. Fgure. N =,K =,G=. temporal MMSE frst,.e., the temporal MMSE flter-spatal matched flter of Secton. and the cascaded temporalspatal MMSE flter of Secton. end up decorrelatng all nterferers n temporal doman. Specfcally, the temporal MMSE flter at the output of each antenna becomes a decorrelator, c k = c, for all k =,..., K, for the the cascaded temporal-spatal MMSE flter. In ths case, the output statstcs of the frst stage of the cascaded temporal-spatal MMSE flter are nterference free,.e., () can be expressed as ỹ = p b (c H s )a + ñ () where ñ k s the enhanced nose at the output of the kth antenna and the components of ñ are ndependent. Thus, the second stage spatal MMSE combner w n () becomes the spatal matched flter, a, whch explans why the temporal MMSE-spatal matched flter and the cascaded temporalspatal MMSE flter have dentcal performance asymptotcally. It s also notable that constraned optmum MMSE detector becomes a temporal-spatal decorrelator and chooses to suppress some of the nterferers n temporal doman and others n spatal doman such that t gets the best asymptotc SIR among such temporal-spatal decorrelators. Next we consder N =users. The output SIR of the desred user s plotted n Fgure. There are stll enough temporal dmensons for users to be decorrelated n the temporal doman,.e., N G, even f they can not all be suppressed n the spatal doman. As a result, all flters that employ MMSE combnng n the temporal doman have nonzero SIRs asymptotcally. However, the constraned optmum MMSE flter, by choosng the approprate users to suppress n the spatal or temporal domans, acheves hgher SIR over all flters except the optmum MMSE flter. The next system to be consdered has N =users and results are shown n Fgure. Snce the number of nterferers are larger than both the processng gan and the number Resultng SIR of the desred user C Fgure. N =,K =,G=. of array elements, all nterferers cannot be suppressed n a sngle doman, thus the combned sngle user/multple user flter structures,.e., X MMSE-MF and X MF-MMSE, are not nearfar resstant. For ths example, the cascaded structures,.e., X ST-CMMSE and X TS-CMMSE are not near-far resstant ether, snce n the nterference domnated regme, each stage tres to suppress all nterferers ndependently n cascade. When an nterference suppressor s desgned by consderng both domans jontly, as n the case of constraned optmum and optmum detectors, X CO-MMSE, and X O-MMSE, near-far resstance s acheved. The last example we consder s a very hghly loaded system wth N =users. The purpose of ths experment s to show the dfference between the optmum MMSE flter and the constraned optmum MMSE flter. We observe from
Resultng SIR of the desred user C Resultng SIR of the desred user C Fgure. N =,K =,G=. Fgure. N =,K =,G=. Fgure that although the constraned optmum MMSE flter, X CO-MMSE, results n acceptable SIR values n near-far stuatons, e.g. an output SIR of (db) when all nterferers powers are db hgher than the desred user, t s not near-far resstant. Ths s smply due to the fact that the constraned optmum MMSE flter can suppress up to G users n the temporal doman and K users n the spatal doman. Thus, for ths example, when N >, the constraned optmum MMSE flter s not able to suppress all the nterference and the output SIR t produces approaches when the nterferers powers approach nfnty. The optmum MMSE detector on the other hand can suppress up to KG users and for ths example s near-far resstant. References [] F. Adach, M. Sawahash, and H. Suda. Wdeband DS- CDMA for next-generaton moble communcaton systems. IEEE Comm. Mag., ():, September. [] X. Bernsten and A. M. Hamovch. Space-tme optmum combnng for CDMA communcatons. Wreless Personal Communcatons, ():,. [] J. A. Fessler and A. O. Hero. Space-alternatng generalzed expectaton-maxmzaton algorthm. IEEE Trans. Sgnal Proc., ():, October. [] R. Kohno, H. Ima, M. Hator, and S. Pasupathy. Combnaton of an adaptve array antenna and a canceller of nterference for drect-sequence spread-spectrum multple-access system. IEEE JSAC, ():, May. [] R. Lupas and S. Verdú. Lnear multuser detectors for synchronous code-dvson multple-access channels. IEEE Trans. Info. The., ():, January. [] U. Madhow and M. L. Hong. MMSE nterference suppresson for drect-sequence spread-spectrum CDMA. IEEE Trans. Comm., ():, December. [] R. A. Monzngo and T. W. Mller. Introducton to Adaptve Arrays. Wley,. [] A. F. Nagub, A. J. Paulraj, and T. Kalath. Capacty mprovement wth base-staton antenna arrays n cellular CDMA. IEEE Trans. Vehc. Tech., ():, August. [] A. J. Paulraj and C. B. Papadas. Space-tme processng for wreless communcatons. IEEE Sgnal Proc. Mag., ():, November. [] H. V. Poor and S. Verdú. Probablty of error n MMSE multuser detecton. IEEE Trans. Info. The., ():, May. [] V. G. Subramanan and U. Madhow. Blnd demodulaton of drect-sequence CDMA sgnals usng an antenna array. In CISS,. [] S. Verdú. Multuser Detecton. Cambrdge Unversty Press,. [] X. Wang and H. V. Poor. Space-tme multuser detecton n multpath CDMA channels. IEEE Trans. Sgnal Proc., ():, September. [] T. Wong, T. M. Lok, J. S. Lehnert, and M. D. Zoltowsk. A lnear recever for drect-sequence spread-spectrum multple-access systems wth antenna arrays and blnd adaptaton. IEEE Trans. Info. The., ():, March. [] A. Yener, R. D. Yates, and S. Ulukus. Interference management for CDMA systems through power control, multuser detecton, and beamformng. IEEE Trans. Comm.,. Submtted. http://www.wnlab.rutgers.edu/ yener. [] A. Yener, R. D. Yates, and S. Ulukus. Jont power control, multuser detecton and beamformng for CDMA systems. In IEEE VTC, May.