Antenna Selection for Space-Time Communication with Covariance Feedback

Transcription

1 Antenna Selecton for Space-Tme Communcaton wth Covarance Feedback G.Barrac and U. Madhow School of Electrcal and Computer Engneerng Unversty of Calforna, Santa Barbara Santa Barbara, CA Emal: Abstract We consder space-tme communcaton for a cellular downlnk n whch the base staton (BS) transmtter has multple antennas, whle the moble recever has one or two. Only a subset of the avalable antennas at the BS are used for transmsson, thus reducng the number of RF chans and the complety of the baseband sgnal processng. It s assumed that the BS does not know the nstantaneous downlnk channel realzaton, but has estmates of the covarance of the spacetme channel: covarance nformaton can be easly obtaned n wdeband systems by averagng uplnk channel measurements over frequency. Based on optmzaton of a lower bound for capacty, we are able to provde rules of thumb for antenna selecton as a functon of the physcal channel characterstcs and the number of receve antennas. Ths procedure s shown to be optmal or near-optmal (relatve to ehaustve computaton of the best antenna subset) at moderate SNR. I. INTRODUCTION Space-tme communcaton, based on the use of multple antennas at the transmtter and/or recever has receved much attenton snce the large capacty gans promsed by the semnal work of Foschn 1 and Telatar 2. Whle most recent research on such Multple Input Multple Output (MIMO) systems assumes no channel feedback at the transmtter, t has been shown n recent work that even mperfect or ncomplete channel feedback results n sgnfcant gans 3, 4, 5. The startng pont of ths paper s our pror work 5, whch shows that, for Orthogonal Frequency Dvson Multpleed (OFDM) systems, the covarance of the space-tme channel can be obtaned mplctly by averagng uplnk channel measurements. We consder an outdoor wreless downlnk n whch the base staton (BS) has multple antennas and the moble has one or two antennas. Snce outdoor channels tend to have small angular spreads, knowledge of the channel covarance leads to large performance gans at reduced transcever complety 3, 4, 5. In ths paper, our focus s on complety reducton for MIMO systems wth covarance feedback. It s often prohbtvely epensve to nstall the large number of RF chans necessary to run a multple nput multple output (MIMO) system wth many antennas. A commonly proposed compromse s to have fewer RF chans than antennas, and to connect them to the best subset of avalable antennas. Thus, not only are MIMO capacty gans stll reaped, but also the cost s lower than the cost of a system equpped wth a full set of RF chans. Our specfc goal here s to choose a subset of antennas at the BS transmtter so as to mamze the ergodc capacty of an OFDM downlnk. Our choce of an nformaton-theoretc performance crteron s motvated by rapd advances n turbo-lke codes and teratve decodng, whch brng nformaton-theoretc lmts wthn reach. Due to the nvarance of the channel covarance across frequency 5, mamzng capacty for an OFDM system s equvalent to mamzng capacty for a sngle subcarrer. Thus, the problem of nterest reduces to choosng the subset of antennas at the transmtter that mamzes the ergodc capacty of a sngle narrowband channel, gven covarance feedback. Contrbutons: The purpose of antenna subset selecton s to create a channel egenvalue dstrbuton, such that, when the transmtter optmzes ts power allocaton across egenmodes, ergodc capacty s mamzed. We provde a lower bound on capacty whch can easly be mamzed to yeld a pseudooptmal egenvalue dstrbuton for the channel. Knowledge of ths dstrbuton n turn gudes our antenna subset selecton; the subset s chosen whose egenvalues most closely match the desred dstrbuton. Compared to brute force optmzaton, our approach provdes optmal or near-optmal performance at moderate SNR, whle gvng nsght nto how the best subset of antennas s related to the channel characterstcs (.e., the number of multpath clusters, n the Saleh-Valenzuela framework 6) and the number of receve antennas. For eample, suppose the channel has a sngle multpath cluster (ths s common n many outdoor settngs). When there s a sngle receve antenna the chosen antenna subset should have adjacent elements. However, when there are two receve antennas, the antenna subset should be selected as two sets of adjacent elements, where the subsets are spaced as far apart as possble. Relaton to Prevous Work: The problem of selectng antennas n MIMO systems has receved consderable attenton n the lterature recently. Most of the work dealng wth transmt antenna selecton assumes that the transmtter has access to full channel state nformaton, and chooses the desred antennas accordngly 7, 8. In recent ndependent work on antenna selecton wth covarance feedback 9, the focus s on optmzng average probablty of error. We focus Globecom /04/$ IEEE

2 nstead on optmzaton of ergodc capacty as a more approprate measure for systems employng powerful error-correctng codes. Our work also dffers from 9 n that our rules of thumb for antenna subset selecton take eplct account of physcal channel parameters such as the number of clusters and the angular spread of each cluster. II. SYSTEM MODEL The BS s assumed to be far away from the moble, and at hgh enough alttude that there s lttle to no local scatterng around t. Thus, both uplnk and downlnk sgnals for a gven moble are restrcted to a farly narrow spatal cone, from the vewpont of the BS antenna array. In contrast, the moble s assumed to be n a rch scatterng envronment so that each of ts antennas sees uncorrelated channel responses. As n the classc Saleh-Valenzuela model 6, the channel response s decomposed nto clusters. Epermental measurements of outdoor channels ndcate that the number of clusters s small, usually one or two, and that the power delay profle (PDP) and power angle profle (PAP) for each cluster as seen by the BS can be modeled as eponental and Laplacan, respectvely. We assume the BS antenna array s a lnear array wth equdstant spacng of one half the carrer wavelength between antennas. Now, consder a narrowband channel wthn an OFDM system wth N T transmt antennas, and N R receve antennas. Assumng neglgble ICI, we can wrte ŝ = Hs + n (1) where s s the transmtted sgnal vector of length N T, ŝ s the receved data vector, H s the N R N T channel frequency response, and n s addtve whte Gaussan nose wth varance. 2 It follows from our earler work 10 that the rows of the channel matr, denoted H(l, :), are well-modeled as ndependent dentcally dstrbuted zero-mean proper comple Gaussan random vectors. In other words, the channel responses from each moble antenna to the BS antenna array are ndependent and dentcally dstrbuted: where H(l, :) CN(0, C) l (2) C = Ea(Ω)a(Ω) H (3) and a(ω) s the BS array response correspondng to angle of arrval/departure Ω. For a lnear array (our runnng eample here), we have a(ω) = a 1...a l..a NT T a l (Ω) = e j(l 1)2π d λ sn(ω) (4) where d s the antenna array spacng, and λ the carrer wavelength. The epectaton n (3) s taken over Ω, wth Ω dstrbuted accordng to the PAP. Lettng P be the power constrant, and s CN(0, Q), the ergodc capacty can be wrtten as C = ma E Q:trace(Q)=P logdet(i NR + 1 HQH H 2 (5) If the base staton knows the channel covarance matr C (and no other channel nformaton), the optmal Q 4, 3 s gven by Q 0 = UP 0 U H (6) where U s the untary egenvector matr of C: C = UΛU H (7) and P 0 s a dagonal matr of powers whose sum s equal to the mamum total power P. The optmal transmt strategy s thus to send along the egenmodes of the channel covarance matr C. We can then wrte C = ma logdet(i NR + 1 HUPU H H H = ma σ 2 n logdet(i NR ZΛPZ H where Z s a N R N T matr whose entres are..d. comple Gaussan random varables wth zero mean and unt varance, Λ s the N T N T dagonal matr of channel egenvalues, as defned by (7), and P s a dagonal matr of powers whose sum s equal to P. To obtan (8), we have used the fact that the BS knows the channel covarance and the optmal Q s thus of the form of (6). The last equalty n (8) comes about by wrtng H as the product of a whte matr and a colored one 4. III. PROBLEM FORMULATION We wsh to pck K T out of the avalable N T antennas at the BS for transmsson. Let S be the set of all subsets of {1, 2,...N T } that have K T elements. Our goal s to fnd the set of antennas that mamzes the ergodc capacty. In other words, we want to fnd s 0, whch s defned as s 0 = arg ma C(s) (9) where C(s) s the capacty when usng the transmt antennas as specfed by the set s. Let us defne H s to be the N R K T matr whose columns are the columns of H specfed by the ndces from s, and Z to be a N R K T matr whose entres are..d. comple Gaussan random varables wth zero mean and unt varance. Also, let C s be the K T K T matr obtaned by elmnatng the rows and columns of C whose ndces are not n s, and let Λ s be the K T K T egenvalue matr of C s. We then have C(s) = ma logdet(i NR + 1 H s QH H s Q:trace(Q)=P E = ma σ 2 n logdet(i NR + 1 ZΛ 2 s P Z H (8) (10) Snce the optmal powers are determned by the channel egenvalues, choosng the best antenna set s amounts to choosng the covarance matr C s whose egenvalues Λ s gve the hghest capacty. If we knew the optmal egenvalues, Λ 0, for a system wth K T BS antennas, we could pck s so that Globecom /04/$ IEEE

3 Λ s most closely appromates Λ 0. In the general case, fndng Λ 0 s as dffcult as fndng the best s va brute search, so we fnd a lower bound to capacty that approaches capacty at hgh SNR, and can be easly mamzed to yeld pseudo-optmal egenvalues, denoted Λ 0. Smulatons show that even for md range SNR, (e. 10 db), choosng the antenna set whose channel egenvalues match Λ 0 gves the best performance. For N R =1, Λ0 has one dagonal entry equal to the number of transmttng antennas K T, and the rest zero. For N R =2, Λ 0 has two equal dagonal entres whch sum to K T and the rest zero. These results are formalzed n the net secton. IV. OPTIMAL EIGENVALUES In ths secton, we fnd a smple lower bound for capacty. At hgh SNR, both ths lower bound and the upper bound usng Jensen s nequalty are mamzed by the same egenvalue dstrbuton, Λ 0. The lower bound reles on Lemma 1, as gven below. Lemma 1: Let{ } be dentcally dstrbuted non-negatve random varables, and let {a } be postve constants. Then, ( N ( N E log 1 a E log a N,a 0 =1 =1 (11) Proof: We would lke to show that ( N ( N 0 E log a E log 1 a (12) =1 =1 or, equvalently ( N =1 0 E log a N 1 =1 a (13) Lettng α a, the RHS of (13) can be wrtten as a ( N E log α (14) =1 1 where α =1. Usng Jensen s nequalty, we have that ( N ) N ( ) log α α log (15) 1 1 and thus ( N N ( E log α α E log (16) 1 1 N = α Elog( ) log( 1 (17) =0 (18) whch proves (13). Snce ths nequalty holds regardless of N, and for all a 0, the Lemma s seen to be true. We now present a lower bound for capacty that holds at all SNR. Theorem 1: A lower bound for the capacty of a MIMO system wth N R receve antennas and K T N R transmt antennas, gven that the transmtter knows the channel covarance, s as follows: ( ) C(Λ) ma log α s + Elog(ν 2 (19) P:trace(P)=P where S s the set of all possble subsets of {1, 2,...K T } wth N R elements, ν s the magntude of the determnant of a N R N R matr whose entres are CN(0, 1), and α s = s λ p. The symbols λ and p denote the th channel egenvalue, and the power sent along th egenmode, respectvely. In other words, λ, and p are the th dagonal elements of the matrces Λ and P, respectvely. Proof: The proof reles on the Cauchy-Bnet 1 formula and Lemma 1, and has been omtted due to space constrants. Corollary 1a: Consder a system wth one receve antenna, N R = 1, and more than one transmt antenna, K T 1. The lower bound to capacty s mamzed when there s one channel egenmode that contans all the channel energy, and the BS sends all ts power along ths mode. More precsely, dag( Λ 0 )=K T, 0, 0, 0... (20) where dag( ) denotes the dagonal elements of ( ) and Λ 0 s the dagonal egenvalue matr that mamzes the lower bound n (19),.e.: ( Λ 0 = arg ma Λ:trace(Λ) K T ma P:trace(P) P ) α s (21) As before, S s the set of all possble subsets of {1, 2,...K T } wth N R elements, α s = s γ, and γ = λ p. Corollary 1b: Consder a system wth two receve antennas, N R =2, and more than two transmt antennas, K T 2. The lower bound to capacty s mamzed when there are two equ-powered channel egenmodes that contan all the channel energy. More precsely, dag( Λ KT 0 )= 2, K T, 0, 0, 0... (22) 2 We note that, for N R =1, t has been shown 11 that Λ 0,as defned n (20), s the true optmal egenvalue dstrbuton: t mamzes capacty, and not just a lower bound to capacty, at all SNR. V. ANTENNA SELECTION Once Λ 0 s known, the subset of antennas whose egenvalues most closely match Λ 0 can be chosen for transmsson. Ths s shown, va smulatons, to be an effectve polcy n the net secton. However, the above results can also be used to obtan general rules of thumb for the antenna selecton problem. For eample, f N R =2, and we wsh to chose four BS antennas for transmsson, n one cluster channels wth farly narrow PAPs, the antennas should be chosen as two sets 1 see Globecom /04/$ IEEE

4 of two such that the dstance between the sets s mamzed and the dstance wthn the sets s mnmzed. In other words, f there are eght equally spaced antennas labeled (n order) as 1,.., 8, then antennas 1, 2, 7 and 8 should be selected. The ntuton behnd ths statement, as well as a substantatng proposton, s gven below. When N R = 2, Theorem 2 says that, deally, we want the channel energy concentrated n two equal strength egenmodes. If K T = 4, and we could pck two sets of two antennas such that the correlaton wthn the set s hgh whle the correlaton between the sets s low, the channel covarance matr C s would be appromately C s 1 ρ 0 0 ρ ρ 0 0 ρ 1 (23) where ρ s large ( ρ close to 1). The egenvalues of (23) are {λ 1,λ 1,λ 2,λ 2 }, where λ 1 and λ 2 are the egenvalues of 1 ρ (24) ρ 1 If ρ s large, then λ 1 2 and λ 2 0. Thus, dag(λ s ) dag( Λ 0 ), whch s the desred stuaton. In general, the correlaton between adjacent antennas s hgh, whle the correlaton between antennas that are far apart s low. Hence, t s ntutvely appealng that pckng two sets of adjacent antennas, wth the sets spaced as far apart as possble, gves dag(λ s ) dag( Λ 0 ). Note that f λ 2 s suffcently small, then the optmal transmt strategy s to send all avalable power along the two domnant egenmodes correspondng to the egenvalue λ 1. For one cluster channels, the correlaton between antennas decreases monotoncally wth dstance. If the PAP s farly narrow, adjacent antennas (spaced a half wave length apart) are hghly correlated whle antennas spaced more than few wavelengths apart are nearly uncorrelated. Thus, for the set s = {1, 2,N T 1,N T }, Λ s Λ 0. (Of course, N T, the number of antennas at the BS, needs to be suffcently large so that the two sets of antennas are separated by several wavelengths.) In the followng proposton, we consder the smplfed stuaton where the correlaton matr C s s real. We show that for N R =2, K T =4, and N T 4, f the channel has a sngle cluster, and the BS s restrcted to transmttng along only two of the four channel egenmodes, then the optmal antenna selecton s s = {1, 2,N T 1,N T }. We assume hgh SNR. Also, we focus on transmt strateges nvolvng only two egenmodes for several reasons. Frstly, ths creates a 22 MIMO system, whch reduces both transmtter and recever complety. Secondly, n the deal case, where two of the egenvalues are large, and the other two are close to zero, sendng along only two egenmodes s the optmal transmt strategy. Thrdly, for narrow power angle profles, most antenna sets s S acheve optmal results usng only two egenmodes. Proposton 1: Consder a system wth N R =2, where 4 out of N T BS antennas are to be selected for transmsson,.e. K T =4. Also, assume that the four antennas are pcked as two sets of two antennas, where the two sets have equal dstances between ther consttuent antennas. For a one cluster channel, the correlaton between antennas decreases monotoncally as the antennas are spaced further apart. Assumng that C s s a real matr, we can wrte 1 1 C s (25) 1 1 where and le n the nterval 0, 1 and depend on the set s chosen. Of course, 1 snce the correlaton must decrease wth antenna spacng, and 0 snce the elements n C s le n 0, 1. In the hgh SNR regme, f the BS only sends along the two most domnant egenmodes, t can be shown that capacty s mamzed f and are as small as possble. Ths corresponds to pckng two sets of adjacent antennas, where the sets are spaced as far apart as possble. VI. SIMULATION RESULTS In ths secton we consder a system where N T =8, and K T =4; we wsh to select the set of four antennas out of eght whch gves the hghest ergodc capacty. We smulate results for N R = 2, for both sngle and double cluster channels. The SNR s set at 10 db, and we assume an equally spaced antenna array at the BS wth a spacng of one half the carrer wavelength. Our smulatons show that the set of antennas whose egenvalues most closely match Λ 0 n the L 1 norm gves optmal or close to optmal results. Results for a one cluster channel wth N R =2are shown n Fgure 1. Plotted s the ergodc capacty vs the angular spread of the PAP for varous sets of four antennas. The PAP s Laplacan centered at 0 o, wth standard devaton σ d ; angular spread s defned as 2σ d. The upper sold lne plots the mamum achevable ergodc capacty vs. angular spread where the mamum s taken over all possble sets of four transmsson antennas, the dashed lne plots the capacty acheved when the frst four antennas are used, and the crcles plot the ergodc capacty when antennas 1, 2, 7 and 8 are used. The crcles also correspond to the s S, denoted s, whose egenvalues are closest to dag( Λ 0 ) n the L 1 norm. For convenence, let us denote s 1 as the set of the frst four antennas, and s 2 as the set {1, 2, 7, 8}. Clearly, at small angular spreads, there s a large dscrepancy between usng s 1 and s 2, ndcatng that proper antennas selecton can result n large gans. Also, t can be seen that set s 2 gves the best results for all angular spreads ecept for the last. Proposton 1 ndcates that for 1 cluster channels where N R =2, and the BS s restrcted to transmsson along two drectons, set s 2 should be used. The ntuton s that for narrow angular spreads, nothng s lost by restrctng the transmt strategy n ths way snce two of the channel egenvalues wll be small for Globecom /04/$ IEEE

5 all the antenna sets s S. Indeed, s 2 s the optmal antenna set when the PAP has a narrow spread. As the spread ncreases, the optmal transmt strategy for many s S s to use 3 or 4 egenmodes, and hence f such strateges are allowed, Proposton 1 can no longer be used to choose the optmal s. Note that even though Proposton 1 was derved for hgh SNR, the results stll apply at SNR= 10dB. The results of Corollary 1 are also relevant for ths SNR. Specfcally, s, marked by crcles, gves optmal, or close to optmal, results. Fgure 2 shows results when N R =2and there are two clusters. Agan, s gves optmal or near optmal results. Results for both s 1 and s 2 are plotted n dashed lnes, wth the former beng dfferentated wth + s. It can be seen that no set works unformly well for all scenaros. ergodc capacty ma guess mn 6.2 1,2,7,8 1,2,3, center of 2nd cluster *π/32 radans ma guess & 1,2,7,8 mn 1,2,3,4 Fg. 2. Results for a two cluster channel when N R =2.The top lne s the best capacty possble wth antenna selecton, and the crcles represent the best guess for the optmal antenna set usng the dstance from dag( Λ 0 ) n the 1 norm as the decson metrc. The dotted lne s the lowest capacty possble when usng antenna selecton. ergodc capacty at moderate to hgh SNR depends strongly on the number of receve antennas, we note that at low SNR, prevous work 5, 12 mples that, regardless of the number of receve antennas, we should select the subset that concentrates as much of the channel energy as possble along a sngle egenmode angular spread (degrees) Fg. 1. Results for a one cluster channel when N R =2.The top lne s the best capacty possble wth antenna selecton, and the crcles represent the best guess for the optmal antenna set usng the dstance from dag( Λ 0 ) n the 1 norm as the decson metrc. The dotted lne s the lowest capacty possble when usng antenna selecton. VII. CONCLUSION The use of a larger number of antennas than RF chans provdes a sgnfcant degree of fleblty that can provde large performance gans, when the subset of antennas used for transmsson s carefully selected. Of course, such selecton requres covarance feedback about all the antennas nvolved, whch requres ether usng all of the antennas on the uplnk (from whch we antcpate obtanng the covarance feedback), or of devsng an approprate antenna swtchng strategy that enables estmaton of the full covarance matr. Note that the number of RF chans on the uplnk and downlnk may dffer, dependng on the relatve cost of low nose amplfers (for recepton) and power amplfers (for transmsson). We have shown that optmzng usng a lower bound on capacty yelds ecellent results at moderate SNR, and that consderable gans are obtaned from pckng the optmal subset of antennas, compared to an arbtrary choce. We epect the performance to be more senstve to SNR for a large number of receve antennas (N R > 2). Also, whle the optmal subset REFERENCES 1 G. Foschn, Layered space-tme archtecture for wreless communcaton n a fadng envronment when usng mult element antennas, Bell Labs techncal Journal, vol. 1, pp , March E. Telatar, Capacty of mult-antenna gaussan channels, European Transactons of Telecommuncatons, vol. 10, pp , December E. Vsotsky and U. Madhow, Space-tme transmt precodng wth mperfect channel feedback, IEEE Transactons on Informaton Theory, vol. 47, pp , September S. Jafar, S. Vshwanath, and A. Goldsmth, Channel capacty and beamformng for multple transmt and receve antennas wth covarance feedback, n IEEE Conference on Communcaton, G. Barrac and U. Madhow, Wdeband space-tme communcaton wth mplct channel feedback, n Seventh Internatonal Symposum on Sgnal Processng and ts Applcatons, A. Saleh and R. Valenzuela, A statstcal model for ndoor multpath propagaton, IEEE Journal on Selected Areas n Communcatons, vol. 5, pp , February A. Molsch, M. Wn, and J. Wnters, Reduced-complety transmt/receve dversty systems, IEEE Transactons on Sgnal Processng, vol. 51, pp , November M. Wn and J. Wnters, Vrtual branch analyss of symbol error probablty for hybrd selecton/mamal-rato combnng n raylegh fadng, IEEE Transactons on Communcatons, vol. 49, pp , November D. Gore and A. Paulraj, Statstcal MIMO antenna sub-set selecton wth space-tme codng, n ICC, vol. 1, pp , May G. Barrac and U. Madhow, Characterzng outage capacty for spacetme communcaton over wdeband wreless channels, n Aslomar Conference on Sgnals, Systems, and Computers, H. Boche and E. Jorsweck, Optmum power allocaton, and complete characterzaton of the mpact of correlaton on the capacty of mso systems wth dfferent cs at the transmtter, n IEEE Internatonal Symposum of Informaton Theory, S. Jafar and A. Goldsmth, On optmalty of beamformng for multple antenna systems wth mperfect feedback, n IEEE Internatonal Symposum of Informaton Theory, Globecom /04/$ IEEE