Coherent opportunistic beamforming in multiuser wireless systems

Transcription

1 herent pprtunistic beamfrming in multiuser wireless systems Jae-Yun Yng-wan Lee Schl f Electrical Engineering IN, Seul Natinal University wana P. O. Bx 34, Seul 5-600, rea Abstract - In this paper, we prpse a beamfrming scheme that explits the advantages f pprtunistic beamfrming cherent beamfrming in multi-user envirnment. It is analytically shwn that the prpsed scheme achieves multiuser diversity gain beamfrming gain simultaneusly, prviding much better perfrmance ver the cnventinal nes. The perfrmance f the prpsed scheme is analyzed using an upper bund methd. Althugh the prpsed scheme invlves an additinal feedbac delay, the analytical result implies that the use f the prpsed scheme is quite effective unless the user mbility is t high. Finally, the analytic results are verified by cmputer simulatin. I. INTRODUTION In recent years, the capacity f wireless systems has been increased significantly with the develpment f tw ey technlgies; the use f multiple antennas nwn as multiinput multi-utput (IO) [,3] pprtunistic scheduling [4,5]. The use f multiple antennas at the transmitter /r the receiver enables t increase the data rate r lwer the transmissin errr rate fr a given signal-tnise pwer rati (SNR). On the ther h, in a multi-user envirnment, scheduler that cnsiders channel cnditins f users can remarably imprve the system efficiency by achieving s-called multi-user diversity (UD) gain. With channel nwledge at the transmitter, ne can achieve beamfrming gain (i.e., SNR spatial diversity gain) by using cherent beamfrming (als called maximum rati transmissin r transmit R) [6,7,3]. In frequency divisin duplex (FDD) systems, the channel nwledge cannt be explited at the transmitter withut feedbac signaling thrugh the uplin channel. Since the amunt f feedbac signaling increases in linear prprtin t the number f users, the cherent beamfrming is hardly applicable in multi-user envirnment. Althugh the use f quantizatin methds [8,9] can be applied t reduce the feedbac signaling burden, it may suffer frm the quantizatin nise as the number f users increases. Opprtunistic beamfrming is a multi-antenna technique that can increase the UD gain with the use f rm beamfrming [0]. Since it requires nly partial channel infrmatin (i.e., SNR) frm the users, it remarably reduces the feedbac signaling burden. rever, it can prvide This wr was in part supprted by the inistry f Infrmatin & mmunicatins, rea, under the Infrmatin Technlgy Research enter (ITR) Supprt Prgram. cherent beamfrming cnfiguratin with prbability f ne as the number f users increases t infinity [0]. wever, it may cnsiderably suffer frm perfrmance degradatin unless the number f users is sufficiently large, because it explits nly the UD gain. This prblem can be alleviated by emplying a receiver antenna selectin technique []. wever, the use f multiple receiver antennas may limit the flexibility f the receiver structure. In this paper, we cnsider a new beamfrming technique that explits bth the UD gain beamfrming gain simultaneusly in multi-user envirnment which can be achieved by cmbining the pprtunistic beamfrming cherent beamfrming. The prpsed scheme requires nly slightly larger feedbac infrmatin than pprtunistic beamfrming while prviding much better perfrmance ver the cherent beamfrming pprtunistic beamfrming. Althugh the prpsed scheme requires additinal feedbac delay which may degrade the perfrmance in a mbile envirnment, analytical result indicates that the use f the prpsed scheme is still desirable fr large cases f envirnments. This paper is rganized as fllws. Sectin II describes the system mdel in cnsideratin. Fllwing brief discussin f the cnventinal beamfrming techniques in Sectin III, the prpsed scheme is described its perfrmance is analyzed in Sectin IV. The perfrmance is verified by cmputer simulatin in Sectin V. Finally, Sectin VI cncludes this paper. II. SYSTE ODEL nsider a dwnlin envirnment with users, where the BS has transmitter antennas each user has a single receiver antenna. We assume the signal transmissin ver a wireless channel whse gains are described by independent zer mean cmplex Gaussian rm variables that all the users have the same average SNR γ experience independent fading. We als assume that the channel infrmatin is available at the user terminal with the use f cmmn pilt signal, but it is nt available at the BS withut feedbac signaling thrugh an uplin channel. When the signal is transmitted with beamfrming weight w, the received signal f user can be represented as y = hw s+ n () /06/$0.00 (c) 006 IEEE This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.

2 where h is the channel gain vectr f user whse elements are zer mean cmplex Gaussian rm variables with unit variance, s is the transmit symbl with average pwer f P n is additive white Gaussian nise (AWGN). We assume that the Frbenius nrm f w is equal t ne t preserve the ttal transmissin pwer ( i.e., w = ). III. ONVENTIONAL BEAFORING TENIUES A. Opprtunistic beamfrming We briefly review the pprtunistic beamfrming [0] fr easy understing f the prpsed scheme. Fig. depicts the prcessing cncept f the pprtunistic beamfrming, where the beamfrming weight w is generated in a rm manner, while preserving the transmissin pwer, i.e., ( ) / i i [ ] T w = = w w wl w () where w i represents the rm weight described as an independent identically distributed (i.i.d.) zer mean cmplex Gaussian rm variable the superscript T dentes transpse. Each user estimates the SNR fr a given beam reprts it t the BS. Then, the BS selects a user based n the scheduling plicy. In an independent Rayleigh fading channel, the effective channel f user h % = hw (3) can be described by a zer mean cmplex Gaussian rm variable with unit variance. Nte that h % has the same distributin as that in a single-input single-utput (SISO) Rayleigh fading channel. This implies that the pprtunistic beamfrming des nt prvide any perfrmance gain ver the SISO cnfiguratin in Rayleigh fading channel [0]. Assuming that the scheduler in the BS chses a user in the best channel cnditin, the effective channel gain f the selected user can be described as Γ = max { % }. (4) t t + Fig.. Prcedure f the pprtunistic beamfrming h =,, L, Thus, the channel capacity f the pprtunistic beamfrming in Rayleigh fading channel can be represented as { lg ( )} = E + γ Γ. (5) Since we cannt represent (5) in a clsed frm, we cnsider the use f an upper bund using Jensen s inequality as = lg + γ E Γ. (6) ( { }) Since the maximum f i.i.d. expnential rm variables has a mean value equal t the sum f harmnic numbers [4], (6) can be rewritten as = lg γ ( / i + + ) (7) lg + γ ( lg( + ) + ς 0.5/( + ) ) where ς ( ) is the Euler s cnstant. Althugh the pprtunistic beamfrming has the same capacity with the SISO system in Rayleigh fading channel, with the use f different beam weight at each scheduling time, the user may experience different SNR. Thus the pprtunistic beamfrming can prvide a fast fading cnditin even in a slwly varying channel, maing it easy t satisfy the qualityf-service (S) requirement fr delay sensitive traffic. B. herent beamfrming The cherent beamfrming technique generates the beam weight s that it maximizes the received SNR. The ptimum weight w fr user can be determined by [6] w = h / h (8) where the superscript dentes cnjugate transpse. Since the effective channel f user is described by h % = hw = h, (9) the effective channel gain Γ = h % can be described by a hi-square rm variable with degrees f freedm in Rayleigh fading channel [5]. Thus, the channel capacity f the cherent beamfrming is bunded by = E{ lg ( + γ Γ )} (0) lg + γ ( + ( ) ) where (-) represents the SNR gain achieved by the cherent beamfrming. Frm (7) (0), it can be seen that by using the cnventinal beamfrming schemes, ne can achieves either the UD gain r beamfrming gain. In the fllwing, we will cnsider a new beamfrming scheme that explits bth f them simultaneusly. IV. PROPOSED BEAFORING SEE Fig. illustrates the prcedure f the prpsed beamfrming scheme. Each user estimates the SNR by using cmmn pilt signal assuming the use f a beamfrming weight ptimized t its channel cnditin then reprts it t the BS. The BS selects a user based n the reprted SNR. Then the selected user reprts its ptimum beam weight t This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.

3 the BS. Finally, the BS transmits data f the selected user using the ptimum beam weight. Nte that the pprtunistic beamfrming transmits the data using a beam generated with a rm weight. Since the scheduler nly needs the beam weight f the selected user, the amunt f additinal feedbac signaling burden is marginal, except additinal delay. The ptimum beam weight w f user fr a given channel vectr h can be determined by w = h h. () Assuming the scheduler selects a user with the largest channel gain, the effective channel gain f the selected user can be represented by Γ = max { h }. () B O t t + Fig.. Prcedure f the prpsed beamfrming =,, L, ere, the subscript B-O dentes the prpsed beamfrming scheme with the use f ptimum beam weight. The crrespnding channel capacity O f the prpsed scheme can be described by = E lg + γ Γ. (3) { ( )} B O B O Nte that Γ B O Γ are the maximum f i.i.d. hi-square rm variables with degrees f freedm, respectively. This implies that the prpsed scheme prvides a capacity larger than that f the pprtunistic beamfrming. In a slw fading channel, hwever, a scheduler that selects a user in the best channel cnditin may nt be desirable fr traffics with strict delay requirement because it may select the same user fr a lng time. Therefre, we cnsider anther realizatin f the prpsed scheme where each user estimates SNR with the use f rmly generated beam weight instead f the ptimum beam weight, prviding same delay S as the pprtunistic beamfrming. With the use f rm beam weight w, the effective channel f user can be written as h % = hw. (4) The ISO channel vectr h can be decmpsed by rthnrmal bases { O, O, L, O } as h = α O + α O + L + α O. (5) t + Since in spatially uncrrelated Rayleigh fading channel, each element f h can be represented as an i.i.d. zer mean cmplex Gaussian rm variable with unit variance, { α, α, L, α} can als be mdeled as i.i.d. zer mean cmplex Gaussian rm variables with unit variance fr any rthnrmal bases. Assuming that the first rthnrmal basis is w (i.e., O = w ), the effective channel can be represented as h % = ( αw + αo + L + αo ) w (6) = α. Let be the index number f the selected user. Then the effective channel gain f the selected user can be represented as α { α } = max =,, L,. (7) Since α is the maximum f i.i.d. expnential rm variables, it has the same distributin as the effective gain in the pprtunistic beamfrming. wever, the prpsed scheme uses the ptimum weight w= h / h fr the data transmissin, prducing an effective channel represented as h% = hw (8) = Γ + α + L + α. Thus, the channel capacity f the prpsed scheme can be described by { lg ( )} (,, R = E + γ Γ + α + L + α. (9) ere, the subscript B-R is used t indicate the prpsed scheme with the use f rm beam weight in the SNR calculatin prcedure. It can be shwn that ( γ E{ α L α }) lg + Γ B R,, = lg + γ + + ( ) R. i = i (0) Nte that the prpsed scheme achieves bth the UD gain / i beamfrming gain ( ) simultaneusly. A. Perfrmance cmparisn with the cnventinal beamfrming schemes We cmpare the perfrmance when the prpsed scheme prvides the same delay S with the pprtunistic beamfrming. We evaluate the perfrmance in terms f upper bunds B R,. Let G G be the perfrmance gains defined by G lg + γ + + ( ) R i i = = = lg + γ + i () This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.

4 lg + γ + + ( ) R i i = G = =. () lg ( + γ ) Since G / γ 0 G / γ 0, the perfrmance imprvement f the prpsed scheme ver the cnventinal nes increases as the SNR decreases. Using the Lpital's therem, it can be seen that the maximum perfrmance gain is btained as γ 0, i.e., ( ) lim G = + (3) γ 0 + i / i = ( ) / i i = lim G = +. (4) γ 0 On the ther h, it can be shwn that n perfrmance gain is achieved as γ, i.e., lim G = lim G =. (5) γ ( ) ( ) γ Thus, the prpsed scheme has a marginal gain ver the cnventinal schemes in gd channel cnditin, but it is quite effective in pr channel cnditin. In fact, frm the infrmatin theretic pint f view, it can be shwn that the prpsed scheme with the ptimum beam weight in the user selectin prcedure achieves the sum capacity f dirty paper cding (DP) DP as the average SNR appraches t zer, i.e., lim( O ) = DP. (6) γ 0 This is due t that the sum capacity f the DP cnverges t that f beamfrming technique where the BS transmits a single data stream t a user in the best channel cnditin as the average SNR appraches t zer []. Since the DP prvides the maximum system capacity, it can be said that the prpsed scheme is the ptimum strategy in lw SNR envirnment. B. Perfrmance with feedbac infrmatin delay In practice, accurate channel infrmatin may nt be available at the transmitter. As the user mbility increases, the channel measurement delay can mae serius mismatch between the measured channel the actual ne due t the channel variatin. Let h() t = [ h, () t h, () t L h, ()] t be the channel vectr f user at time t. As illustrated in Fig., assume that the users reprt their SNR t the BS at time t, the selected user reprts the ptimum beam weight t the BS at time t +, finally the desired data is received t the user at time t +. The crrelatin cefficient f the channel can be written as ρ = Eh { ( t) h ( t+ )} ρ = Eh { ( t) h ( t+ )} ρ = Eh { ( t+ ) h ( t+ )} i, i, i, i, i, i, (7) where {, L, }, i {, L, } () dentes cmplex cnjugate. The crrelatin cefficient is usually depends n the mbility f the user. Fr example, in rich scattering envirnment, crrelatin cefficient with the time difference f can be represented as [7] ρ J ( ) = 0 π f d (8) where J () 0 is the zerth rder Bessel functin f the first ind f d is the maximum Dppler frequency. It is pssible t cmpletely describe the jintly Gaussian rm variables in terms f the first secnd rder statistics [5]. Thus, h ( t + ) h ( t + ) can be expressed in terms f h ( t ) as ( t+ ) = ρ ( ) t + ρ h h z (9) ρ ρ ρ ρ ρ ρ h( t+ ) = ρh () t + z+ ρ z (30) ρ ρ where z, z are zer mean cmplex Gaussian rm vectrs whse elements have unit variance h () t, z z are independent f each ther. Fr a selected user, the effective channel gain with delay can be represented as ( t ) h + Γ B R (, ) = E h ( t+ ). (3) h ( t + ) where h ( t + ) can be represented in term f h ( t + ) as h( t+ ) = αh( t+ ) + β z. (3) ere z is zer mean cmplex Gaussian rm vectr with each element f unit variance is independent f h ( t + ), α β are the cefficients calculated frm the crrelatin between h ( t + ) h ( t + ), the variance f h ( t + ), respectively. After sme derivatin, it can be shwn that ρ ρ + ρ i α = (33) ρ + i β = ( α ρ ρ ) α + i. (34) Then, (3) can be represented in terms f α β as This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.

5 Nrmalized capacity gain G (simulatin) G (simulatin) Nrmalized capacity gain G (simulatin) G G (simulatin) G SNR (db) Number f users () Fig. 3. Perfrmance gain f the prpsed beamfrming scheme when =4 =8 Fig. 4. Perfrmance gain assciated with when =4 SNR=0dB ( t ) h + Γ B R (, ) = E ( αh ( t+ ) + β z'' ) h ( t + ) (35) ( t ) h + = α E{ h( t+ ) } + ( β ) E z''. h ( t + ) Frm (9) the independence f z '' h ( t + ), the mean f the effective channel gain can be btained by { } E Γ ( ( B R, ) = α ρ β ) + +. (36) i Finally, the upper bund (, ) B R with feedbac delay is given by R (, ) = lg( + γe{ ΓB R (, ) }) (37) = lg + γ α ρ + + ( β ). i Nte that the UD gain is reduced by a factr f α ρ due t the additinal feedbac delay while the beamfrming gain is reduced by a factr f α. Similarly, it can easily be shwn that the pprtunistic beamfrming cherent beamfrming with feedbac delay have an upper bund represented as ( ) = lg + γ + ρ i = i (38) ( ) = lg + γ ( + ρ ( ) ). (39) Assume that =, which may be a very practical cnditin. Then, as the number f users decreases r the number f antennas increases, α β can be apprximated t ρ, i.e., β α ρ. In this case, it can be easily be shwn that R (, ) ( ) iff ( ρ ) (40) i Nrmalized capacity gain Number f transmitter antennas () B R (, ) ( ) frany. (4) Nte that R ( ) ( ) fr all ρ prvided that / i. V. SIULATION RESULTS G (simulatin) G (simulatin) Fig. 5. Perfrmance gain assciated with when =8 SNR=0dB T verify the design analysis f the prpsed scheme, we evaluate the perfrmance by cmputer simulatin assuming that all users have mutually independent channel with the same average SNR. We assume a ISO cnfiguratin with a Rayleigh fading channel. Fig. 3 depicts the perfrmance f the prpsed scheme in terms f SNR when =4 =8. It can be seen that the prpsed scheme prvides significant perfrmance gain especially at lw SNR. It can als be seen that the analytic results agree well with the simulatin results. Fig. 4 Fig. 5 depict the perfrmance gain f the prpsed scheme in term f the number f users transmitter antennas at SNR=0dB, respectively. It can be This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.

6 Spectral efficiency (bps/z) Spectral efficiency (bps/z) (simulatin) U (simulatin) U (simulatin) B-R U B-R f d (nrmalized delay) (a) When SNR=0dB (simulatin) U (simulatin) U (simulatin) B-R U B-R f d (nrmalized delay) (b) When SNR=0dB Fig. 6. Perfrmance with feedbac signaling delay when =4 =6 seen that the perfrmance gain f the prpsed scheme ver the cherent beamfrming increases as the number f users increases by expliting the UD gain. On the ther h, as the number f transmitter antennas increases, the prpsed scheme has increased perfrmance gain ver the pprtunistic beamfrming by expliting the beamfrming gain. Fig. 6 depicts the perfrmance assciated with feedbac signaling delay when =. It can be seen that the prpsed scheme can prvides perfrmance gain ver the cnventinal schemes unless the nrmalized delay is larger than 0.5 (this value is smewhat assciated with system parameters, e.g., ). Fr IS-856 system where is abut tw time slts (i.e., 3.33ms) [8], this nrmalized delay 0.5 crrespnds t a mbility f abut 5m/h at.9 Gz carrier frequency. This simulatin result indicates that the prpsed scheme is quite effective fr users in lw mbility (e.g., nmadic envirnments). VI. ONLUSIONS We have prpsed an imprved beamfrming scheme by expliting the advantages f pprtunistic beamfrming cherent beamfrming. By using a beam weight ptimized t the selected user, the prpsed scheme can achieve the beamfrming gain as well as the UD gain, withut nticeable increase f feedbac signaling burden. The perfrmance f the prpsed scheme is analytically evaluated using an upper bund methd verified by cmputer simulatin. The simulatin results shw that the prpsed scheme prvides the perfrmance gain ver the cnventinal beamfrming schemes, particularly significant in lw SNR envirnment. They als shw that the prpsed scheme is quite effective unless the user mbility is t high in spite f an additinal delay fr the prcessing. REFERENES [] G. J. Fschini. J. Gans, On limits f wireless cmmunicatins in a fading envirnment when using multiple antennas, Wireless Persnal mmun., vl. 6, n. 3, pp , June 998. [] S.. Alamuti, A simple transmit diversity technique fr wireless cmmunicatins, IEEE J. Select. Areas mmun., vl. 6, n. 8, pp , Oct [3] V. Tarh, N. Seshadri, A. R. alderban, Space-time cdes fr high data rate wireless cmmunicatin: Perfrmance criterin cde cnstructin, IEEE Trans. Infrm. Thery, vl. 44, pp , ar [4] Xin Liu, Edwin. P. hng, N. B. Shrff, A Framewr fr Opprtunistic Scheduling in Wireless Netwrs, mputer Netwrs, vl. 4, pp , ar. 003 [5] W. Rhee, W. Yu, J.. iffi, Utilizing multiuser diversity fr multiple antenna systems, in Prc. f Wireless mmun. Netwr. nf., vl., pp , Sept [6] T.. Y. L, aximum rati transmissin, IEEE Trans. mm., vl. 47, pp , Oct [7] X. Feng. Leung, A new ptimal transmit receive diversity scheme, in Prc. f IEEE PARI00, vl., pp , Aug. 00. [8].. uavilli, A. Sabharwal, E. Erip, B. Aazhang, On beamfrming with finite rate feedbac in multiple antenna systems. IEEE Trans. Infrm. Thery, vl. 49, pp , Oct [9] D. J. Lve, R.W. eath, Jr., T. Strhmer, Grassmannian beamfrming fr multiple-input multiple-utput wireless systems. IEEE Trans. Infrm. Thery, vl. 49, pp , Oct [0] P. Viswanath, D. N.. Tse R. Laria, Opprtunistic beamfrming using dumb antennas, IEEE Trans. Infrm. Thery, vl. 48, n. 6, pp , June 00. [] L. Zan S. A. Jafar, mbined pprtunistic beamfrming receive antenna selectin, in Prc. f Wireless mmun. Netwr. nf., vl., pp , ar [] N. Jindal, ulti-user cmmunicatin systems: capacity, duality, cperatin, Ph. D. dissertatin, Department f electrical engineering, Stanfrd university, July 004. [3] A. Paulraj, R. Nabar, D. Gre, Intrductin t Space-Time Wireless mmunicatins, ambridge University Press, 003. [4] G. L. Stuber, Principles f bile mmunicatin, luwer academic publishers, 996. [5] J. G. Prais, Digital mmunicatins, cgraw-ill, 4-th ed., 00. [6] E. Vistsy U. adhw, Optimum beamfrming using transmit antenna arrays, IEEE Prc. Vehicular Technlgy nference 999, vl., pp , ay 999. [7] W.. Jaes, icrwave bile mmunicatins, New Yr, NY, BEE Press, 993. [8] D. Tse P. Viswanath, Fundamentals f Wireless mmunicatin, ambridge University Press, 005. This full text paper was peer reviewed at the directin f IEEE mmunicatins Sciety subject matter experts fr publicatin in the IEEE I 006 prceedings.