OPTIMUM BEAMFORMING USING TRANSMIT ANTENNA ARRAYS Eugene Vsotsky Upamanyu Madhow Abstract - Transmt beamformng s a powerful means of ncreasng capacty n systems n whch the transmtter s eupped wth an antenna array, especally n systems n whch receve beamformng s not avalable, as s typcal n the base-to-moble \downlnk" n a cellular communcaton system. In ths paper, the problem of transmt beamformng s formulated as one of mnmzng the power radated by the base staton, subject to satsfyng ualty of servce reurements at the mobles. For a sngle cell system, a global mnmum s shown to exst, and an teratve algorthm that converges to t s provded. The soluton extends to accomodate receve beamformng at the mobles. It also extends to provde a mechansm for space-tme transmt lterng, whch explots the derences between the temporal as well as the spatal channels between the base staton and derent mobles. Fnally, a system wth multple source-destnaton pars (e.g., n an ad hoc network) s consdered. An teratve algorthm that outputs a convergent seuence of feasble ponts wth monontoncally decreasng costs s provded for ths case. 1 Introducton A fundamental lmtaton on the capacty of cellular and personal communcaton systems s the mutual nterference among smultaneous users. Transmt and receve beamformng usng antenna arrays can potentally crcumvent ths bottleneck by provdng solaton among users at derent locatons. In typcal cellular systems, t s reasonable to assume that the base staton s eupped wth an antenna array. However, the moble unts typcally have only one antenna element. In such a settng, transmt beamformng from the base staton to the mobles provdes a powerful method for ncreasng downlnk (base-to-moble) capacty. An mplct assumpton s that the base staton has feedback regardng the channel seen by each moble, whch n turn mples that these channels vary slowly. Thus, transmt beamformng s most eectve n applcatons nvolvng slowly movng moble unts. Ths paper provdes a method for computng the optmum weghts for transmt beamformng for the downlnk n a sngle cell. The soluton s then extended to several other stuatons: (a) optmum transmt and receve beamformng n a sngle cell; (b) opt- The authors are wth the Coordnated Scence Laboratory, Unversty of Illnos, 1308 W. Man, Urbana, IL 61801, USA. Ths work was supported by Motorola under the Unversty Partnershps n Research Programs. mum space-tme transmt lterng n a sngle cell, (c) transmt beamformng for multple source-destnaton pars (for whch the algorthm outputs a seuence of feasble ponts wth monotoncally decreasng costs). The basc problem posed s as follows: choose the beamformng weghts so as to mnmze the net power transmtted by the base staton, subject to each moble attanng a desred sgnal-to-nterference rato (SIR). The dea s to provde desred levels of solatons among mobles n a cell, whle mnmzng the nterference generated for other cells. Ths optmzaton problem was rst consdered, and solved n part, n [3]. The partal soluton n [3] conssts of an teratve algorthm that, assumng that there s a feasble soluton, converges to a feasble (but not necessarly optmal) pont. That s, t nds a set of beamformng weghts that guarantee that the SIR constrant at each moble s satsed. In ths paper, we provde a complete soluton to the problem addressed n [3]. In partcular, t s shown that, for a transformed verson of the orgnal optmzaton problem, the algorthm n [3] converges to the globally optmum beamformng weghts. Dependng on the system parameters, the net transmtted power reured wth the optmum weghts can be substantally smaller than that reured wth the feasble soluton obtaned n [3]. System Model Consder a sngle cell. The base staton has an M- element antenna array. Each moble has a sngle antenna element. Let h, 1, denote the M 1 channel from the base staton to the moble, where s the number of mobles. The base staton performs transmt beamformng and communcates smultaneously wth all mobles. The nstantaneous transmtted sgnal can then be expressed as x = =1 b w (1) where w s an M-dmensonal beamformng vector for the -th user and b s the transmtted data symbol for the -th user. At the -th moble, the nstantaneous receved sgnal can then be expressed as an nner product between the transmtted sgnal and the -th channel plus nose: r = j=1 b j w H j h + n 1 () 1
where n s an addtve whte Gaussan nose of varance, whch models other-cell nterference and background nose present n the system. The term b w H h represents the desred sgnal at the -th moble, whle P j6= b jwj H h represents the multple access nterference. Assumng that the data symbols b are uncorrelated, the SIR at the -th moble,, can be expressed as follows = jw H h j P j6= jwh j h j + 1 (3) 3 Optmum Transmt Beamformng The objectve s to mnmze the total transmtted power by the base staton, subject to ualty of servce reurements at the mobles, expressed as a mnmum SIR constrant at each moble,.e., SIR. Snce the total transmtted P power at the base staton can be expressed as jjw =1 jj, ths yelds the followng constraned optmzaton problem subject to 0 @ X j6= mn w 1:::w =1 1 jjw jj (4) jwj H h j + A? jw H h j 0 8 (5) the constrants have been rewrtten n terms of uadratc functons of the weght vectors. The followng mportant observatons can now be made. 1. Exstence of Global Mnmum: The cost functon n (4) s contnuous and coercve (.e., ts value converges to plus nnty along any seuence of ts arguments such that the norm of the arguments also converges to plus nnty; see [1], page 540, for a formal denton). Furthermore, the constrant set s closed. Thus, f the constrant set s nonempty (.e., a feasble soluton exsts), then, by the Weerstrass' Theorem (page 540 of [1]), a global mnmum exsts.. Actve Constrants: At the global mnmum the constrants (5) are actve (.e., they are satsed wth eualty). Ths can be shown by contradcton: f there s a global mnmum wth constrant n (5) nactve, then jjw jj can be reduced wthout volatng the constrants. Thus, we can obtan a feasble pont wth the cost functon smaller than the global mnmum, whch s a contradcton. 3. Normalzed Problem: The optmzaton problem depends only on the pars of parameters (h ; ), 1. Furthermore, an euvalent optmzaton problem s obtaned, f we rewrte (4) n terms of the normalzed parameter pars ( ~ h ; 1), 1, where ~ h = h =. We wll refer to ths euvalent optmzaton problem as the normalzed downlnk optmzaton problem. The Lagrangan for the normalzed downlnk optmzaton problem s gven by 0 1 X jjw jj + @ jwj H h ~ j + 1A? jw H h ~ j =1 =1 j6= (6) where, 1, are the correspondng Lagrange multplers, whch can be shown to be nonnegatve (p. 83 of [1]). Derentatng wth respect to the w, we obtan the followng euatons as the necessary condtons for optmalty. w? ~h ~h H w + X j6= j j ~h j ~h H j w = 0 8 (7) In addton, as noted earler, the optmal soluton must satsfy the feasblty constrants n (5) wth eualty, so that 0 1 X @ jwj H h ~ j + 1A = jw H ~h j 8 (8) j6= The key step n the development s to show that the optmum beamformng weghts for the normalzed downlnk problem are the same, up to scalng, as those correspondng to optmum jont power control and receve beamformng for a vrtual normalzed uplnk problem. The latter can be solved by an teratve algorthm n [], and ths algorthm can be moded to solve for the weghts and scalng factors for the downlnk problem of nterest. The precedng approach was already consdered n [3], where t was shown that the use of a vrtual uplnk as above yelds a feasble soluton to the downlnk problem, and provded an teratve algorthm for computng such a soluton. The somewhat surprsng result obtaned n ths paper s that, when the algorthm n [3] s appled to the normalzed downlnk problem, t actually yelds the optmum soluton. Dependng on the system parameters, ths may be much better than the feasble soluton obtaned n [3]. 3.1 Vrtual Uplnk Problem For the uplnk problem, the mobles seek to mnmze ther transmtted powers to attan SIR constrants at the base staton, takng nto account the fact that the base staton s eupped wth a receve antenna array that can suppress nterference, whle mantanng unty gan n the desred drecton. Thus, for 1, moble must determne ts transmtted power P, and the base staton must determne the beamformng vector ^w used to receve the sgnal from moble. The problem can therefore be formulated as follows mn ^w 1::: ^w ;P 1:::P =1 P (9)
subject to P P j6= P jj ^w H ~ h j j + jj ^w jj and ^w H ~ h = 1 8 (10) where the channels ~ h, the unty nose varance, and the desred SIR values,, are all chosen to be dentcal to those for the \normalzed" downlnk problem. As n the downlnk problem, t s easy to see that the optmum soluton must satsfy the constrants (10) wth eualty. By consderng the rst order necessary condtons for optmalty, t s easy to show that the global mnmzer of (9) satses the followng euatons ^w? P ~h ~h H ^w + X j6= P j ~h j ~h H j ^w = 0 8 (11) ^w H ~ h = 1 8 (1) In [], an teratve algorthm s proposed and s shown to converge, startng from any ntal condtons, to the unue xed pont whch s the global mnmzer of (9). It s straghtforward to show that (11) - (1) specfy the unue xed pont of the algorthm, P and, hence, for P, 1, satsfyng (11), we have P =1 = P uplnk, where P uplnk s the global mnmum of (9). In [3], the algorthm of [] s extended to obtan a feasble soluton for the downlnk problem by an approprate scalng of the optmum vrtual uplnk beamformng vectors. 3. Optmzaton Algorthm The algorthm for optmum transmt beamformng can be stated as follows. 1. Compute the normalzed channels ~ h = h = 8.. Apply the algorthm n [3] to the \normalzed" problem as follows Step 1: For 1 ^w n = arg mn w j6= subject to w H ~h = 1. Step : For 1 P n+1 = j6= Step 3: For 1 ~P n+1 = P n j jw H ~ h j j + jjw jj (13) P n j j( ^w n ) H ~h j j + jj ^w n jj (14) j6= ~P n j j( ^w n j ) H ~h j + (15) Proposton 1: The precedng algorthm converges to the global mnmum of the downlnk problem f there s a feasble soluton. The proposton mples that the followng uanttes converge: P n! P 1, P ~ n! P ~ 1, ^w n! ^w 1 8. The optmum transmt beamformng vectors are gven by w = ~P 1 ^w 1, 1. 3.3 Proof of Optmalty We rst state and prove the followng key lemma before provng Proposton 1. Lemma: Any beamformng vectors satsfyng the euatons (7) - (8) acheve the global mnmum of the optmzaton problem n (4) - (5). Proof of the Lemma: Let ( ; w ), 1, be any beamformng vector and egenvalue pars satsfyng (7) - (8). Some smple manpulatons yeld the followng relatonshp between the cost and the correspondng Lagrange multplers =1 = =1 jjw jj (16) From (8), observe that w H h ~ > 0 8. Thus, a set of complex scalars f ; 1 g can be found such that, settng ^w = w gves ^w? ~h ~h H ^w + X j6= j j ~h j ~h H j ^w = 0 8 (17) ^w H ~ h = 1 8 (18) Identfyng P = and comparng (17) - (18) wth (11) - (1), t becomes clear that play the role of the uplnk powers P satsfyng the xed pont euaton for the vrtual uplnk problem. Ths gves P =1 P = P uplnk, whch, together wth (16), mples that jjw =1 jj = P uplnk. Ths enables us to conclude that, for any feasble pont for the downlnk problem satsfyng (7) - (8), the downlnk cost functon euals the mnmum P uplnk of the uplnk cost functon. Snce the global mnmzer of the downlnk problem necessarly satses (7) - (8), we conclude that 1. P uplnk = P downlnk, where P downlnk s the global mnmum of the transmt beamformng optmzaton problem.. Any feasble pont satsfyng (7) - (8) s a global mnmzer of the transmt beamformng optmzaton problem. Ths establshes the Lemma. Proof of Proposton 1: Let (P 1 ; ^w 1 ; P ~ 1 ), 1 denote the lmts of the teratve algorthm, whch, by the results n [3], s guaranteed to converge f there s a feasble soluton to the downlnk problem We wll establsh that ~P 1 ^w 1, 1, satses (7) - (8), 3
and hence, by the Lemma, s the global mnmzer of the transmt beamformng problem. The transmt beamformng set ~P 1 ^w 1, 1, satses (8) by the result n [3], where t s shown that ~P 1 ^w 1, 1, s a feasble pont of (4) satsfyng the constrants wth eualty. We now establsh that (7) s satsed. As shown n [], (P 1 ; ^w 1 ), 1, s the global mnmzer of (9) and (P 1 ; ^w 1 ), 1, satses (11). Agan comparng (7) and (11), we observe that, by makng the substtuton P 1 = =, (7) can be recovered from (11). Thus, we conclude that, wth P 1 = playng the role of the Lagrange multpler n (7), for an arbtrary power scalng A, w = A ^w 1, 1, satses (7). In partcular, ~P 1 ^w 1, 1, satses (7). 4 Jont Transmt and Receve Beamformng It s shown that the soluton for sngle-cell transmt beamformng gven n the precedng secton extends mmedately to a stuaton where the mobles may also be eupped wth antenna arrays. However, several local mnma may exst for beamformng performed by multple source-destnaton pars, as mght be encountered n ad hoc topologes. In ths case, convergence of the algorthm to the global mnmum cannot be guaranteed. 4.1 A Sngle Cell Assume that the base staton s eupped wth an M- element antenna array, whle each moble s eupped wth an N-element antenna array. Whle the proposed soluton apples eually well to the downlnk and to the uplnk communcaton, consder downlnk communcaton for concreteness. The channel from the base staton to the -th moble s descrbed by an N M matrx denoted as H. Assume that the array response at a receve antenna from each transmt element at the base staton s the same (except for complex scalng). Ths assumpton s true f the scatterng envronment surroundng the receve antenna s not excessvely rch. In ths case, the channel matrx H can be assumed to be rank one, and can be decomposed nto an outer product as follows: H = h ~h T 1 (19) where h s an N 1 vector denotng the array response at the - th moble to the base staton and ~ h s an M 1 vector denotng the array response at the base staton to the - th moble. Let fw ; 1 g denote the transmt beamformng weghts used at the base staton and fv ; 1 g denote the receve beamformng weghts used at the mobles. Upon performng receve beamformng, the -th moble observes the followng sgnal r = j=1 b j v H h ~h T w j + v H n 8 (0) where n s an N 1 vector contanng uncorrelated samples of addtve whte Gaussan nose of varance present at each antenna element. Assumng that the data symbols b are uncorrelated, the SIR at the -th moble,, can be expressed as follows = jv H h ~h T w P j j6= jvh h ~h T w j j + jjv 8 (1) jj The jont transmt and receve optmzaton problem s formulated as mnmzng the total transmtted power by the base staton subject to SIR constrants at each of the mobles. To avod nullng out the desred user, the receve beamformer s reured to have unty gan n the desred drecton,.e., v H h = 1 8. Snce the sgnal destned for any moble orgnates from a sngle source, the base staton, t s easy to see that the optmum receve beamformers are smply spatal matched lters, gven by v = h =jjh jj 8. Wth ths choce of receve beamformng vectors, the eectve channels from the base staton to the mobles and the nose varances at the mobles become respectvely v H h ~h = ~ h 8, jjv jj = =jj~ h jj 8. The optmal transmt beamformng vectors can now be obtaned by applyng (13) - (15) to the transmt beamformng wth parameters ( ~ h ; =jj ~ h jj ), 1. 4. Multple Source-Destnaton Pars Assume that the system conssts of source-destnaton pars whch nterfere wth each other. Ths scenaro may apply ether to an ad hoc network, or to a multple cell system n a cellular network. Denote by H j the N M channel from the source of the j-th par to the destnaton of the -th par, where, as above, H j can be decomposed as an outer product of two column vectors. Unlke n the case of a sngle source (or base staton), the spatal sgnatures of nterference and of the desred transmsson do not le n the same complex subspace. Hence, the matched lterng soluton for the receve beamformng vectors s suboptmum. The problem has a large number of xed ponts, each correspondng to a derent way of suppressng nterference. Roughly speakng, each recever has the choce of nullng out one or more of the sgnals from nterferng transmtters, whle each transmtter has the choce of nullng out one or more of the recevers. The followng algorthm, whch terates between (locally) optmal transmt and receve beamformng, can be shown to converge to one such xed pont wth the convergent seuence havng monotoncally decreasng costs. Step 1: Perform receve beamformng for 1 v n+1 = arg mn v j6= jv H h j ~h T j(w n j ) j + jjv jj () 4
subject to v H h = 1. Step : Perform transmt beamformng for 1 fw n+1 ; 1 g = arg mn w 1:::w subject to =1 jjw jj (3) P j6= j(vn+1 ) H h j ~h T jw j j + jjvn+1 jj? j ~ h T w j 0 8 (4) Note that the precedng algorthm, f appled to the problem of jont transmt and receve beamformng n a sngle cell, converges to the global mnmum n one teraton. In general, t can be shown that the algorthm produces a convergent seuence of transmt beamformng weghts whch are feasble and have non-ncreasng costs. Convergence to the global mnmum s guaranteed f, and only f, there exsts a unue xed pont of the algorthm. However, multple xed ponts were observed even for the case of two transmt-receve pars n our experments. 4.3 Optmum Space-Tme Transmt Flterng For temporally dspersve channels, temporal lterng n addton to beamformng may be used to obtan nterference suppresson. For a sngle cell, the optmum transmt beamformng soluton provded n Secton 3 can be generalzed to allow transmt lterng n both space and tme. Due to space lmtatons, attenton s restrcted to a smple example to llustrate the deas nvolved. Consder a sngle base staton eupped wth an M-dmensonal antenna array, and two mobles, each eupped wth a sngle antenna element. Assume that the channel to each moble conssts of two multpath rays wth a delay spread of one tme unt. Each space-tme channel can be descrbed n the z-transform doman as follows H (z) = a 0 1 + a 1 1z?1, = 1;, where a j s the M-dmensonal spatal channel to the -th moble along the j-th multpath. The transmtter performs spacetme lterng of the outgong bt seuence. For smplcty of llustraton, restrctng the temporal span of the beamformng lters to two, the transmt beamformng lters can be expressed as follows: W (z) = v 0 +v1 z?1, = 1;, where v j s the j-th M-dmensonal temporal component of the -th lter. The z-transform of the receved vector at the -th moble s as follows R (z) = H (z)? B 1 (z)w H 1 (z) + B (z)w H (z) + N (z), where B (z) s the z-transform of -th user's bt stream. The followng expresson can now be obtaned for the receved sample at the rst moble at t = 1: r 1 [1] = [a 1 1; a 0 1] H? b 1 [0][v 1 1; 0] + b 1 [1][v 0 1; v 1 1]+ b 1 [][0; v 0 1] + b [0][v 1 ; 0] + b [1][v 0 ; v1 ] + b [][0; v 0 ] + n 1 [1] (5) where [a 1 1; a 0 1] denotes a column vector wth a 1 1 stacked on top of a 0 1. Ths expresson can be rewrtten as a correlaton between [v1; 0 v1] 1 and [v; 0 v], 1 whch play the role of eectve transmt beamformng vectors, and the acyclc shfts of [a 1 1; a 0 1], whch play the role of eectve channels. Assumng, wthout loss of generalty, that the sample r 1 [1] s used as a decson statstc for the bt b 1 [1], we obtan SIR expressons dentcal n form to (3). The optmzaton problem of mnmzng the sum of suared Eucledean norms of [v 0; v1 ], = 1;, subject to the SIR constrants can now be solved by applyng (13) - (15). 5 Numercal Results In ths secton, we present numercal results obtaned for a system wth a ve-element lnear antenna array and sx moble users, dstrbuted at dstnct angles and dstances throughout the cell. In a cellular system, the nose varance at a gven moble s manly due to nterference from adjacent base statons (not explctly modeled), and can therefore depend on the locaton of the moble wthn ts cell. To llustrate such locaton eects, we employ the followng model. Let moble be at an angle and dstance d ;1 from the desred base staton and, dstance d ; from an nterferng base staton. Assume that the strength of the channel and nose varance at the mobles are determned by the path loss from the desred and nterferng base statons, respectvely. Assumng 1=d 4 law for dependence of path loss on dstance and =4 antenna spacng, we have: h (k) = p 1 5d exp(j (k? 1) sn ), ;1 = 1=d 4 ;, 1 k 5; 1 6, where h (k) s the k-th component of the -th channel vector and s the wavelength. To generate Fgure 1, t was assumed that d ;1 +d ; = 1 8 and the followng numercal values were used d 1;1 = 0:, d ;1 = 0:4, d 3;1 = 0:3, d 4;1 = 0:5, d 5;1 = 0:45, d 6;1 = 0:45. We now apply the algorthm n (13)-(15) to the transmt beamformng problem wth these parameters. db 0 0 40 60 80 100 Beamformer angular response theta_1 theta_ theta_3 theta_4 theta_5 theta_6 10 0 0 40 60 80 100 10 140 160 180 degrees Fgure 1: Plot of W 3 () vs. Fgure 1 dsplays, as a functon of, the magntude suared of the normalzed correlaton between the 3- rd beamformng vector and the lnear array response, 5
7 Comparson of the algorthms References Gan (db) 6 5 4 3 1 0 1 1 3 4 5 6 7 8 9 10 nose_{ 1}/nose_ Fgure : Plot of P feas =P mn vs?1 =. whch we term the beamformer angular response and denote by W () for the -th moble. Then W () = j( ^w1 )H h()j jj ^w 1 jj jjh()jj (6) where h()(k) = p 1 5 exp(j (k?1) sn ), 1 k 5. The angles of arrval for the sx users are shown usng dashed vertcal lnes. From Fgure 1, note that the beamformer places a lobe n the angular drecton correspondng to the desred moble and suppresses other users. Note that the lobe placed n the drecton of the desred moble s wde and s robust, to a certan degree, aganst channel estmaton errors. Fgure dsplays a comparson between the global mnmum of the transmt optmzaton problem, P mn, and a feasble pont, P feas, obtaned by applyng the orgnal teratve algorthm n [3] (wthout normalzaton). The parameters are slghtly derent from those n Fgure 1. In partcular, the nose varances are assumed to form a decreasng seuence, wth?1 xed at some number larger than one, for = ; :::; 6, and = 1 1=d 1;. All other parameters are unchanged. Fgure dsplays P feas =P mn as a functon of?1 =, 6. Note that, f?1= = 1, 6,.e. the nose varances are the same at all mobles, no normalzaton s necessary, and the performance of the two algorthms s dentcal. As the nose varances at the mobles become more dsparate (as would be typcal for mobles dstrbuted unformly wthn a cell), observe that P feas =P mn ncreases sgncantly, whch mples that sgncant savngs for total transmtted power are obtaned by the smple modcaton of the algorthm n [3] proposed here. Numercal results for the extensons n Sectons 4 and 5 wll be presented at the conference. [1] D.P. Bertsekas, Nonlnear Programmng. Belmont, Massachusetts: Athena Scentc, 1995. [] F. Rashd-Farrokh, L. Tassulas, and.j.r. Lu, \Jont Optmal Power Control and Beamformng n Wreless Networks Usng Antenna Arrays," IEEE Transactons on Communcatons, vol. 46, no. 10, pp. 1313-133, October 1998. [3] F. Rashd-Farrokh,.J.R. Lu, and L. Tassulas, \Transmt Beamformng for Cellular Communcaton Systems," Proc. 31st Conf. Info. Scences and Systems, Johns Hopkns, Baltmore, pp. 9-97, March 1997. [4] D. Gerlach, A. Paulraj, \Adaptve Transmttng Antenna Arrays wth Feedback," IEEE Sgnal Processng Letters, vol. 1, pp. 150-15, October 1994. [5] D. Gerlach, A. Paulraj, \Adaptve Transmttng Antenna Methods for Multpath Envronments," Proc. 1994 GlobeCom Conference, pp. 45-49. [6] J.-H. Chang, L. Tassulas, and F. Rashd-Farrokh, \Jont Transmtter and Recever Beamformng for Maxmum Capacty n Spatal Dvson Multaccess," Proc. 35-th Annual Allerton Conf. on Communcaton, Control and Computng, pp. 93-101, September 1997. 6