Antenna Combnng for the IO Downlnk Channel arxv:0704.308v [cs.it] 0 Apr 2007 Nhar Jndal Department of Electrcal and Computer Engneerng Unversty of nnesota nneapols, N 55455, USA Emal: nhar@umn.edu Abstract A multple antenna downlnk channel where lmted channel feedback s avalable to the transmtter s consdered. If each recever has only a sngle antenna, the transmt antenna array can be used to transmt separate data streams to multple recevers only f the transmtter has very accurate channel knowledge,.e., f there s hgh-rate channel feedback from each recever. In ths work t s shown that channel feedback requrements can be sgnfcantly reduced f each recever has a small number of antennas and approprately combnes ts antenna outputs. A novel combnng method that mnmzes channel quantzaton error at each recever, and thereby mnmzes mult-user nterference, s proposed and analyzed. Ths technque s shown to outperform tradtonal technques such as maxmum-rato combnng because mnmzaton of nterference power s more crtcal than maxmzaton of sgnal power n the multple antenna downlnk. Analyss s provded to quantfy the feedback savngs, and the technque s seen to work well wth user selecton algorthms and s also robust to recever estmaton error. I. INTRODUCTION ult-user IO technques such as zero-forcng beamformng allow for smultaneous transmsson of multple data streams,.e., spatal multplexng, n the multple transmt antenna downlnk channel even when each recever (moble) has only a sngle antenna []. owever, the transmtter must have accurate channel state nformaton (CSI) n order to utlze such technques. In the practcally motvated fnte rate feedback model, each moble feeds back a fnte number of bts regardng ts channel nstantaton at the begnnng of each block or frame. When each moble has a sngle receve antenna, the feedback bts are determned by quantzng the channel vector to one of 2 B quantzaton vectors and feedng back the correspondng ndex. Whle a relatvely small number of feedback bts suffce to obtan near-perfect
CSIT performance n a pont-to-pont ISO (multple-nput, sngle-output) channel [2][3], consderably more feedback s requred n a multple transmt antenna, sngle receve antenna (per moble) downlnk channel. In fact, the per moble feedback must be scaled wth the number of transmt antennas as well as the system SNR n downlnk channels n order to acheve rates close to those achevable wth perfect CSIT. In [4], t s shown that the followng scalng of feedback bts B = P db, () 3 where P represents the SNR and the number of transmt antennas, suffces to mantan a maxmum gap of 3 db (equvalent to bps/z per moble) between perfect CSIT and lmted feedback performance n an d Raylegh block-fadng channel when the transmtter uses zero-forcng beamformng, a low-complexty but near-capacty technque for the IO downlnk. Ths feedback load can be qute large for even reasonable sze systems: for example, 0 feedback bts per moble are requred n a 4 transmt antenna system operatng at 0 db. In such a system the transmtter emts multple beams and uses ts channel knowledge to select beamformng vectors (.e., beam drectons) such that nulls are created at certan users: the beam ntended for the frst moble s chosen to create a null (or a near null) at all other mobles, and so forth. Inaccurate channel knowledge leads to naccurate nullng whch translates drectly nto mult-user nterference and reduced throughput. As a result, channel quantzaton error s the crtcal factor n the IO downlnk. In ths paper, we propose a novel receve antenna combnng technque, dubbed quantzaton-based combnng (QBC), that can be used by multple-antenna moble devces to reduce quantzaton error and thereby reduce the requred channel feedback load. Each moble lnearly combnes the receved sgnals on ts N antennas to produce a sngle output, thereby creatng an effectve sngle receve antenna channel at each moble. The resultng vector channel s quantzed and fed back, and transmsson s then performed as n a multple transmt, sngle receve antenna downlnk channel. When QBC s used, the combner weghts are chosen to produce the effectve sngle antenna channel that can be quantzed wth mnmal error. Tradtonal combnng technques such as the maxmum-rato based technque descrbed n [5] for pont-to-pont IO channels wth lmted channel feedback maxmze receved sgnal power but generally do not mnmze channel quantzaton error. Snce channel quantzaton error s so crtcal n the IO downlnk channel, quantzaton-based combnng leads to better performance by mnmzng quantzaton error (.e., nterference power) possbly at the expense of channel (.e., sgnal) power. Our analyss shows that ths combnng technque reduces the feedback requred from the expresson
2 n () to approxmately B N P db, (2) 3 where N s the number of antennas at each moble. Although feedback must stll be ncreased wth SNR, the rate of ths ncrease s decreased by approprately performng antenna combnng. For example, employng QBC at each moble reduces the feedback n a 4 transmt antenna system at 0 db from 0 bts to 7 bts f each moble has two antennas (N = 2). The remander of ths paper s organzed as follows: In Secton II we ntroduce the system model and some prelmnares. In Secton III we descrbe a smple antenna selecton method that leads drectly nto Secton IV where the much more powerful quantzaton-based combnng technque s descrbed n detal. In Secton V we analyze the throughput and feedback requrements of QBC. In Secton VI we compare QBC to other combnng technques (e.g., RC) and also consder the effect of user selecton and recever estmaton error, and fnally we conclude n Secton VII. Notaton: We use lower-case boldface to denote vectors, upper-case boldface for matrces, and the symbol ( ) for the conjugate transpose. The norm of vector x s denoted x. II. SYSTE ODEL AND PRELIINARIES We consder a K moble (recever) downlnk channel n whch the transmtter (access pont) has antennas, and each of the mobles has N antennas. The receved sgnal at the -th antenna s gven by: y = h x + n, =,...,NK (3) where h,h 2,...,h KN are the channel vectors (wth h C ) descrbng the KN receve antennas, x C s the transmtted vector, and n,...,n NK are ndependent complex Gaussan nose terms wth unt varance. Note that moble has access to sgnals y,...,y N, moble 2 has access to y N+,...,y 2N, and the -th moble has access to y ( )N+,...,y N. There s a transmt power constrant of P,.e. the nput must satsfy E[ x 2 ] P. We use to denote the concatenaton of the -th moble s channels,.e. = [h ( )N+ h N ]. We consder a block fadng channel wth ndependent Raylegh fadng from block to block and therefore the entres of each channel vector are d complex Gaussan wth unt varance. Each of the mobles s assumed to have perfect and nstantaneous knowledge of ts own channel, although we analyze the effect of relaxng ths assumpton n Secton VI-D. In ths work we study only the ergodc capacty, or the long-term average throughput. For smplcty of exposton we assume that the number of mobles
3 s equal to the number of transmt antennas,.e., K =. The results can easly be extended to the case where K <, and the proposed technque can be combned wth user selecton when K >, as dscussed n Secton VI-C. Furthermore, we assume that the number of moble antennas s less than the number of transmt antennas (N < ) because mult-user IO loses much of ts advantage over pont-to-pont technques when N. A. Fnte Rate Feedback odel In the fnte rate feedback model, each moble quantzes ts channel to B bts and feeds back the bts perfectly and nstantaneously to the transmtter at the begnnng of each block [5][6]. Vector quantzaton s performed usng a codebook C of 2 B -dmensonal unt norm vectors C {w,...,w 2 B}, and each moble quantzes ts channel to the quantzaton vector that forms the mnmum angle to t [5] [6]: and feeds the quantzaton ndex back to the transmtter. ĥ = arg mn sn 2 ( (h,w)), (4) w=w,...,w 2 B For analytcal tractablty, we study systems usng random vector quantzaton (RVQ) n whch each of the 2 B quantzaton vectors s ndependently chosen from the sotropc dstrbuton on the - dmensonal unt sphere and where each moble uses an ndependently generated codebook [2]. We analyze performance averaged over random codebooks; smlar to Shannon s random codng argument, there always exsts at least one quantzaton codebook that performs as well as the ensemble average. B. Zero-Forcng Beamformng After recevng the quantzaton ndces from each of the mobles, the AP can use zero-forcng beamformng (ZFBF) to transmt data to the users. Let us agan consder the N = scenaro, where the channels are the vectors h,...,h. When ZFBF s used, the transmtted sgnal s defned as x = = x v, where each x s a scalar (chosen complex Gaussan wth power P/) ntended for the -th moble, and v C s the beamformng vector for the -th moble. Snce the transmtter does not have perfect CSI, ZFBF must be performed based on the quantzatons nstead of the actual channels. The beamformng vectors v,...,v are chosen as the normalzed rows of the matrx [ĥ ĥ],.e., they satsfy v = for all and ĥ v j = 0 for all j. If all mult-user nterference s treated as addtonal nose, the resultng SINR at the -th recever s gven by: SINR = P h v 2 + j P h v j 2. (5)
4 The coeffcent that determnes the amount of nterference receved at moble from the beam ntended for moble j, h v j 2, s easly seen to be an ncreasng functon of moble s quantzaton error. C. IO Downlnk wth Sngle Antenna obles In [4] the IO downlnk channel wth sngle antenna mobles (N = ) s analyzed assumng that ZFBF s performed on the bass of (RVQ-based) fnte rate feedback. The basc result of [4] s that: [ R FB (P) R CSIT (P) log 2 ( ( )]) + P E sn 2 (ĥ,h ) (6) where R FB (P) and R CSIT (P) are the ergodc per-user throughput wth feedback and wth perfect CSIT, [ ( )] respectvely, and the quantty E sn 2 (ĥ,h ) s the expected quantzaton error. The expected quantzaton error can be accurately upper bounded by 2 B and therefore the throughput loss due to ( ) lmted feedback s upper bounded by log 2 + P 2 B, whch s an ncreasng functon of the SNR P. If the number of feedback bts (per moble) s scaled wth P accordng to: B = ( )log 2 P P db, 3 then the dfference between R FB (P) and R CSIT (P) s upper bounded by bps/z at all SNR s, or equvalently the power gap s at most 3 db. As the remander of the paper shows, quantzaton-based combnng sgnfcantly reduces the quantzaton error (more precsely, t ncreases the exponental rate at whch quantzaton error goes to zero as B s ncreased) and therefore decreases the rate at whch B must be ncreased as a functon of SNR. III. ANTENNA SELECTION FOR REDUCED QUANTIZATION ERROR In ths secton we descrbe a smple antenna selecton method that reduces channel quantzaton error. Descrpton of ths technque s prmarly ncluded for expostory reasons, because the smple concept of antenna selecton naturally extends to the more complex (and powerful) QBC technque. In pont-topont IO, the receve antenna wth the largest channel gan s selected, whle n the IO downlnk the antenna wth the smallest quantzaton error s selected. The vector channel correspondng to each receve antenna s separately quantzed, and the antenna wth mnmum error s selected. To be more explct, consder the operaton performed at the frst moble, whch has channel matrx = [h h N ] and a sngle quantzaton codebook consstng of 2 B quantzaton vectors w,...,w 2 B. The moble frst ndvdually quantzes the N vector channels h,...,h N that correspond to ts N receve antennas: ĝ l = arg mn sn 2 ( (h l,w)) l =,...,N, (7) w=w,...,w 2 B
5 and then selects the antenna wth the mnmum quantzaton error: j = arg mn l=,...,n sn2 ( (h l,ĝ l )), (8) and feeds back the quantzaton ndex correspondng to ĝ j. The moble uses only antenna j for recepton, and thus the system s effectvely transformed nto a IO downlnk wth sngle receve antennas. Due to the ndependence of the channel and quantzaton vectors, choosng the best of N channel quantzatons s statstcally equvalent to quantzng a sngle vector channel usng a codebook of sze N 2 B. Therefore, antenna selecton effectvely ncreases the quantzaton codebook sze from 2 B to N 2 B, and a system usng antenna selecton wth N antennas per moble and a codebook of sze 2 B (.e., B feedback bts per moble) acheves the same throughput as a sngle receve antenna downlnk channel wth B+log 2 N feedback bts per moble. Thus, antenna selecton reduces the requred feedback by log 2 N bts per moble as compared to a sngle receve antenna system. IV. QUANTIZATION-BASED COBINING In ths secton we descrbe the quantzaton-based combnng (QBC) technque that reduces channel quantzaton error by approprately combnng receve antenna outputs. We consder a lnear combner at each moble, whch effectvely converts each multple antenna moble nto a sngle antenna recever. The combner structure for a 3 user channel wth 3 transmt antennas ( = 3) and 2 antennas per moble (N = 2) s shown n Fg.. Each moble lnearly combnes ts N outputs, usng approprately chosen combner weghts, to produce a scalar output (denoted by y eff ). The effectve channel descrbng the channel from the transmt antenna array to the effectve output of the -th moble (y eff ) s smply a lnear combnaton of the N vectors descrbng the N receve antennas. After choosng combnng weghts the moble quantzes the effectve channel vector and feeds back the approprate quantzaton ndex. Only the effectve channel output s used to receve data, and thus each moble effectvely has only one antenna. The key to the technque s to choose combner weghts that produce an effectve channel that can be quantzed very accurately; such a choce must be made on the bass of both the channel vectors and the quantzaton codebook. Ths s qute dfferent from maxmum rato combnng, where the combner weghts and quantzaton vector are chosen such that receved sgnal power s maxmzed but quantzaton error s generally not mnmzed. Note that the antenna selecton technque descrbed n the prevous secton corresponds to choosng the effectve channel from the set (h,...,h N ), whle QBC allows the effectve channel to be an arbtrary lnear combnaton of these N vectors.
6 γ, eff y γ,2 2 γ 2, eff y 2 γ 2,2 3 γ 3, eff y 3 γ 3,2 Fg.. Effectve Channel for = K = 3, N = 2 System A. General Descrpton Let us consder the effectve receved sgnal at the frst moble for some choce of combner weghts, whch we denote as γ = (γ,,...,γ,n ). In order to mantan a nose varance of one, the combner weghts are constraned to have unt norm: γ =. The (scalar) combner output, denoted y eff, s: N N y eff = γ,l y l = γ,l (h l x + n l ) l= = l= ( N l= γ,l h l ) = (h eff ) x + n, x + N γ,l n k where n = N l= γ,l n k s unt varance complex Gaussan nose because γ =. The effectve channel vector h eff s smply a lnear combnaton of the vectors h,...,h N : N h eff = γ,l h l = γ. Snce γ can be any unt norm vector, h eff by h,...,h N,.e., n span( ). l= l= can be n any drecton n the N-dmensonal subspace spanned Because quantzaton error s so crtcal to performance, the objectve s to choose combner weghts that yeld an effectve channel that can be quantzed wth mnmal error, wthout any regard to the effectve channel norm. The quantzaton error correspondng to effectve channel h eff s mn sn 2 ( (h eff,w ) ). =,...,2 B Therefore, the optmal choce of the effectve channel s the soluton to: mn h eff mn sn 2 ( (h eff,w ) ), (9) =,...,2 B By well known propertes of d Raylegh fadng, the matrx s full rank wth probablty one [7].
7 where h eff s allowed to be n any drecton n span( ). Once the optmal effectve channel s determned, the combner weghts γ can be determned through a smple pseudo-nverse operaton. Snce the expresson for the optmum effectve channel gven n (9) conssts of two mnmzatons, wthout loss of optmalty the order of the mnmzaton can be swtched to gve: mn =,...,2 B mn h eff sn 2 ( (h eff,w ) ), (0) For each quantzaton vector w, the nner mnmzaton fnds the effectve channel vector n span( ) that forms the mnmum angle wth w : mn h eff sn 2 ( (h eff,w ) ). () By basc geometrc prncples, the mnmzng h eff s smply the projecton of w on span( ). The soluton to () s therefore the sne squared of the angle between w and ts projecton on span( ), whch s referred to as the angle between w and the subspace. As a result, the best quantzaton vector,.e., the soluton of (0), s the vector that forms the smallest angle between tself and span( ). The optmal effectve channel s the (scaled) projecton of ths partcular quantzaton vector onto span( ). In order to perform quantzaton, the angle between each quantzaton vector and span( ) must be computed. If q,...,q N form an orthonormal bass for span( ) and Q [q q N ], then sn 2 ( (w, span( ))) = Q w 2. Therefore, moble s quantzed channel, denoted ĥ, s: ĥ = arg mn w=w,...,w 2 B (w, span( )) = arg max Q w=w w 2. (2),...,w 2 B Once the quantzaton vector has been selected, t only remans to choose the combner weghts. The projecton of ĥ on span( ), whch s equal to Q Q ĥ, s scaled by ts norm to produce the unt norm vector s proj. The drecton specfed by s proj has the mnmum quantzaton error amongst all drectons n span( ), and therefore the effectve channel should be chosen n ths drecton Frst we fnd the vector u C N such that u = s proj, and then scale to get γ. Snce s proj s n span( ), u s unquely determned by the pseudo-nverse of : u = ( ) s proj, (3) and the combner weght vector γ s the normalzed verson of u : γ = u u. The quantzaton procedure s llustrated for a N = 2 channel n Fg. 2. In the fgure the span of the two channel vectors s shown along wth the chosen quantzaton vector h, ts projecton on the channel subspace, and the correspondng effectve channel.
8 Fg. 2. Quantzaton procedure for a two antenna moble B. Algorthm Summary We now summarze the quantzaton-based combnng procedure performed at the -th moble: ) Fnd an orthonormal bass, denoted q,...,q N, for span( ) and defne Q [q q N ]. 2) Fnd the quantzaton vector closest to the channel subspace: ĥ = arg max Q w=w w 2. (4),...,w 2 B 3) Determne the drecton of the effectve channel by projectng ĥ onto span( ). = Q Q Q Q ĥ. (5) 4) Compute the combner weght vector γ : ( ) γ = s proj ( ) s proj. (6) s proj Each moble performs these steps, feeds back the ndex of ts quantzed channel ĥ, and then lnearly combnes ts N receved sgnals usng vector γ to produce ts effectve channel output y eff wth h eff ĥ = (h eff ) x+n = γ. Note that the transmtter need not be aware of the number of receve antennas or of the detals of ths procedure because the downlnk channel appears to be a sngle receve antenna channel from the transmtter s perspectve; ths clearly eases the mplementaton burden of QBC. V. TROUGPUT ANALYSIS Quantzaton-based combnng converts the multple transmt, multple receve antenna downlnk channel nto a multple transmt, sngle receve antenna downlnk channel wth channel vectors h eff,...,heff and channel quantzatons ĥ ĥ. After recevng the quantzaton ndces from each of the mobles, the transmtter performs zero-forcng beamformng (as descrbed n Secton II-B) based on the channel quantzatons. The resultng SINR at the -th moble s gven by: SINR = P (heff ) v 2 + j P (heff ) v j 2. (7)
9 We are nterested n the long-term average throughput, and thus the quantty E[log 2 ( + SINR )]. A. Prelmnary Calculatons We frst determne the dstrbuton of the quantzaton error and the effectve channel vectors wth respect to both the random channels and random quantzaton codebooks. Lemma : The quantzaton error sn 2 ( (ĥ,h eff )), s the mnmum of 2 B ndependent beta ( N,N) random varables. Proof: If the columns of N matrx Q form an orthonormal bass for span( ), then cos 2 ( (w j, span( )) = Q w j 2 for any quantzaton vector. Snce the bass vectors and quantzaton vectors are sotropcally chosen and are ndependent, ths quantty s the squared norm of the projecton of a random unt norm vector n C onto a random N-dmensonal subspace, whch s descrbed by the beta dstrbuton wth parameters N and N [8]. By the propertes of the beta dstrbuton, sn 2 ( (w j, span( )) = cos 2 ( (w j, span( )) s beta ( N,N). Fnally, the ndependence of the quantzaton and channel vectors mples ndependence of the 2 B random varables. h Lemma 2: The normalzed effectve channels eff h eff,..., h eff h eff are d sotropc vectors n C. h Proof: From the earler descrpton of QBC, note that eff, whch s the projecton of the best h eff = sproj quantzaton vector onto span( ). Snce each quantzaton vector s chosen sotropcally, ts projecton s sotropcally dstrbuted wthn the subspace. Furthermore, the best quantzaton vector s chosen based solely on the angle between the quantzaton vector and ts projecton. Thus s proj n span( ). Snce ths subspace s also sotropcally dstrbuted, the vector s proj s sotropcally dstrbuted s sotropcally dstrbuted n C. Fnally, the ndependence of the quantzaton and channel vectors from moble to moble mples ndependence of the effectve channel drectons. Lemma 3: The quantty h eff 2 s χ 2 2( N+). Proof: Usng the notaton from Secton IV-A, the norm of the effectve channel s gven by: h eff 2 = γ 2 u = u 2 = u 2 u 2 = sproj 2 u 2 = u 2, (8) where we have used the defntons h eff = γ and γ = u u, and the fact that u satsfes u = s proj. Therefore, n order to characterze the norm of the effectve channel t s suffcent to characterze u 2. The N-dmensonal vector u s the set of coeffcents that allows s proj, the normalzed projecton of the chosen quantzaton vector, to be expressed as a lnear combnaton of the columns of (.e., the channel vectors). Because s proj s sotropcally dstrbuted n span( ) (Lemma 2), f we change coordnates to any (N-dmensonal) bass for span( ) we can assume wthout loss of generalty that the projecton of
0 the quantzaton vector s [ 0 0] T. Therefore, the dstrbuton of u 2 s the same as the dstrbuton of. Snce the N N matrx [( ] s Wshart dstrbuted wth degrees of freedom, ths ), quantty s well-known to be χ 2 2( N+) ; see [9] for a proof. Accordng to Lemma 3, the norm of the effectve channel has the same dstrbuton as that of a ( N + )-dmensonal random vector nstead of a -dmensonal vector. An arbtrary lnear combnaton (wth unt norm) of the N channel vectors would result n another d complex Gaussan -dmensonal vector, whose squared norm s χ 2 2, but the weghts defnng the effectve channel are not arbtrary due to the nverse operaton. owever, ths reducton n channel norm only slghtly reduces performance and s compensated by the effect of reduced quantzaton error. B. Sum Rate Performance Relatve to Perfect CSIT In order to study the beneft of QBC, we compare the acheved sum rate, denoted R QBC (P), to the sum rate acheved usng zero-forcng beamformng on the bass of perfect CSIT n an transmt antenna, sngle receve antenna downlnk channel, denoted R CSIT (P). We use the sngle receve antenna downlnk wth perfect CSIT as the benchmark nstead of the N receve antenna perfect CSIT downlnk channel because the proposed method effectvely utlzes only a sngle receve antenna per moble. In a sngle receve antenna IO downlnk wth perfect CSIT, the zero-forcng beamformng vectors (denoted v ZF, ) can be chosen perfectly orthogonal to all other channels. Thus, the SNR of each user s as gven n (5) wth zero nterference terms n the denomnator. The resultng average rate s gven by: [ ( R CSIT (P) = E log 2 + P )] h v ZF, 2. The sum rate acheved wth QBC s gven by: R QBC (P) = E,W [ log 2 ( + )] P (heff ) v 2 + j P. (heff ) v j 2 Followng the procedure n [4], the rate gap R(P) s defned as the dfference between the per-user throughput acheved wth perfect CSIT and wth feedback-based QBC: R(P) R CSIT (P) R QBC (P). (9) Smlar to Theorem of [4], we can upper bound ths throughput loss: Theorem : The per-user throughput loss s upper bounded by: ( ) ( ) ) N+ R(P) log l 2 e + log 2 (+P E[sn 2 ( (ĥ,h eff ))] l= N+ Proof: See Appendx.
The frst term n the expresson s the throughput loss due to the reduced norm of the effectve channel, whle the second (more sgnfcant) term, whch s an ncreasng functon of P, s due to quantzaton error. In order to quantfy ths rate gap, t clearly s necessary to bound the expected quantzaton error. By Lemma, the quantzaton error s the mnmum of 2 B d beta( N,N) random varables. Furthermore, a general result on ordered statstcs appled to the beta dstrbuton gves [8, Chapter 4.I.B]: E[sn 2 ( (ĥ,h eff ))] F ( 2 B ) where F X (x) s the CDF of a beta ( N,N) random varable, and F ( ) s ts nverse. The CDF s gven by: F X (x) = N =0 ( ) x N+ ( x) N + N X ( ) x N, N where the approxmaton s the result of keepng only the lowest order x term and droppng ( x) terms; ths s vald for small values of x. Usng ths we get the followng approxmaton: ( ) E[sn 2 ( (ĥ,h eff ))] 2 B N N. (20) N The accuracy of ths approxmaton s later verfed by our numercal results. Pluggng ths approxmaton nto the upper bound n Theorem we get: ( ) R(P) log l 2 e + log 2 (+P l= N+ ( N+ ) 2 B N ( ) ) N N If the number of feedback bts B s fxed, the resdual quantzaton error causes the system to become nterference lmted as the SNR s taken to nfnty (see [4, Theorem 2] for a formal proof when N = ). Thus, B must be approprately scaled wth the SNR n order to avod ths lmtaton. Indeed, f B s scaled such that the quantzaton error decreases as P, the rate gap n (2) can be kept constant. Such a scalng of feedback bts leads to a bounded throughput loss relatve to perfect CSIT, and also ensures that the full multplexng gan () s acheved. In order to determne ths scalng, we smply set the approxmaton of R(P) n (2) equal to a rate constant log 2 b and solve for B as a functon of P. Thus, a per-moble rate loss of at most log 2 b s mantaned f B s scaled accordng to: ( B N ( N)log 2 P ( N)log 2 c ( N)log 2 N+ N ( P db ( N)log 3 2 c ( N)log 2 N+ ) ( log 2 N ) log 2 ( N ), (2) ), (22) where c = b e (P ) l= N+ l. Note that a per user rate gap of log 2 b = bps/z s equvalent to a 3 db gap n the sum rate curves.
2 30 Capacty (bps/z) 25 20 5 0 5 Perfect CSIT Zero Forcng ~ 3 db Lmted FB (N=,2,3) 0 5 0 5 20 25 SNR (db) Fg. 3. Sum rate of = K = 6 downlnk channel The sum rate of a 6 transmt antenna downlnk channel ( = 6) s plotted n Fg. 3. The perfect CSIT zero-forcng curve s plotted along wth the rates acheved usng fnte rate feedback wth the feedback load scaled wth SNR accordng to (22) for N =,2 and 3; the actual feedback loads are plotted n Fgure 4 for the dfferent values of N. For N = 2 and N = 3 QBC s performed and the fact that the throughput curves are approxmately 3 db away from the perfect CSIT curve verfy the accuracy of the approxmatons used to derve the feedback scalng expresson n (22). In ths system, the feedback savngs at 20 db s 7 and 2 bts, respectvely, for 2 and 3 receve antennas. All numercal results n the paper are generated usng the method descrbed n Appendx II. VI. DISCUSSION AND EXTENSIONS In ths secton we hghlght the advantages of quantzaton-based combnng relatve to other combnng technques, evaluate the performance of QBC n conjuncton wth user selecton, and analyze the effect of nosy recever channel estmates. A. Comparson wth Sngle Receve Antenna Systems The frst pont of reference s the sngle receve antenna (N = ) IO downlnk. As dscussed n Secton II-C, scalng feedback accordng to B = 3 P db mantans a 3 db gap from perfect CSIT throughput. If compared wth the feedback scalng for QBC, descrbed n (22), notce that n both cases feedback must be ncreased lnearly wth SNR, but that the slope of ths ncrease s 3 when mobles have only a sngle antenna compared to a slope of N 3 for antenna combnng. If we compute the
3 35 Requred Feedback (bts) 30 25 20 5 0 5 N= N=2 N=3 0 0 5 0 5 20 SNR (db) Fg. 4. Feedback Requrements for = K = 6 downlnk channel dfference between the N = feedback load and the QBC feedback load, we can quantfy how much less feedback s requred to acheve the same throughput (3 db away from a sngle receve antenna downlnk wth perfect CSIT) f QBC s used wth N antennas/moble: QBC (N) = B B N N ( ) P db + log 3 2 (N )log N 2 e. The feedback reducton s an ncreasng functon of the SNR, and can be qute sgnfcant for even moderate SNR values and system szes; Fgure 4 llustrates these savngs when = 6. Perhaps a more useful method to understand ths feedback reducton s to consder the number of channel symbols devoted to feedback. Consder the feedback model ntroduced n [0] where each moble transmts ts feedback bts over an orthogonal AWGN channel wth the same SNR as the downlnk channel (P ); ths s reasonable when the average SNR s due prmarly to path loss. The capacty of ths feedback channel s log 2 ( + P) bts/symbol, and transmttng B = 3 P db ( )log 2 P bts requres approxmately channel symbols per moble. On the other hand, QBC requres ( N)log 2 P + O() bts and thus only N channel symbols are requred. As a result, quantzaton-based combnng effectvely reduces the number of channel symbols that must be reserved for feedback by N ; ths savngs s on a per-moble bass and thus can be substantal for moderate and large systems. B. Alternate Combnng Technques In addton to antenna selecton and QBC, another combnng method s to select the quantzaton vector that maxmzes receved power, as proposed n [5] for pont-to-pont IO channels. Ths method
4 roughly corresponds to maxmum rato combnng, and selects the quantzaton vector accordng to: ĥ = arg max w=w w 2. (23),...,w 2 B Ths technque s referred to as RC. If beamformng vector w s used by the transmtter, then receved sgnal power s maxmzed by usng the maxmum rato combnng weghts γ = weghts yeld the followng effectve channel: h eff = γ = w w. w w [5]. These When the number of feedback bts s not very small, wth hgh probablty the quantzaton vector that maxmzes w 2 s the vector that s closest to the egenvector correspondng to the maxmum egenvalue of. To see ths, consder the maxmzaton of w when w s constraned to have unt norm but need not be selected from a fnte codebook. Ths corresponds to the classcal defnton of the matrx norm, and the optmzng w s n the drecton of the maxmum sngular value of. When B s not too small, the quantzaton error s very small and as a result the soluton to (23) s extremely close to 2. As a result, selectng the quantzaton vector accordng to the crtera n (23) s essentally equvalent to drectly fndng the quantzaton vector that s closest to the drecton of the maxmum sngular value of. To drectly perform ths, the moble frst selects the combner weghts γ such that the effectve channel h eff = γ s n the drecton of the maxmum sngular value, whch corresponds to selectng γ equal to the egenvector correspondng to the maxmum egenvalue of the N N matrx, and then fnds the quantzaton vector closest to h eff. The effectve channel norm satsfes h eff 2 = 2, whch can be reasonably approxmated as a scaled verson of a χ 2 2N random varable []. Therefore the norm of the effectve channel s large, but notce that the quantzaton procedure reduces to standard vector quantzaton, for whch the error s roughly 2 B. In Fgure 5, numercally computed values of the quantzaton error (log 2 (E[sn 2 ( (h eff,ĥ))]) are shown for QBC, antenna selecton, RC (correspondng to equaton 23), and drect quantzaton of the maxmum egenvector, along wth approxmaton 2 B as well as the approxmaton from (20), for a = 4, N = 2 channel. Note that the error of QBC s very well approxmated by (20), and the exponental rate of decrease of the other technques are all well approxmated by 2 B. Each combnng technque transforms the multple receve antenna channel nto a sngle receve antenna downlnk channel wth a possbly modfed effectve channel norm and a possbly modfed quantzaton error. These technques are summarzed n Table I. The key pont s that only QBC changes the exponent of the quantzaton error, whch determnes the rate at whch feedback must be ncreased wth SNR. As a
5 0 Log (base 2) of Quantzaton Error 2 3 4 5 6 7 8 RC Selecton Approxmaton Approxmaton QBC 9 2 4 6 8 0 2 4 Feedback Bts (B) Fg. 5. Quantzaton Error for Dfferent Combnng Technques ( = 4, N = 2) Effectve Channel Norm Quantzaton Error Sngle RX Antenna (N = ) χ 2 2 2 B/( ) Antenna Selecton χ 2 2 2 (B+log 2 N)/( ) RC max egenvalue 2 B/( ) QBC χ 2 2( N+) 2 B/( N) TABLE I SUARY OF COBINING TECNIQUES result, QBC requres N feedback channel symbols, whle all of the other technques requre symbols. When comparng these technques note that the computatonal complexty of QBC and RC are essentally the same: QBC and RC requre computaton of Q w 2 and w 2, respectvely. In Fgure 6 fve dfferent throughput curves for a 4 transmt antenna, 2 receve antenna ( = 4, N = 2) system are shown: ZF on the bass of perfect CSIT (ZF-CSIT), QBC, antenna selecton, RC, and the throughput of a N = system (no combnng). In ths plot, B s scaled accordng to (22),.e., roughly as ( N)log 2 P ; n the feedback model descrbed earler, ths s equvalent to usng approxmately N symbols per moble for feedback. As predcted by (22), the QBC throughput curve s approxmately a 3 db shft of the perfect CSIT curve. The other three throughput curves (antenna selecton, RC, and N = ) all lag sgnfcantly behnd, partcularly at hgh SNR. Ths s because the ( N)log 2 P scalng of feedback s smply not suffcent to mantan good performance f these technques are used. To be
6 8 Throughput (bps/z) 6 4 2 0 8 6 4 RC ZF CSIT QBC Ant. Selecton 2 Sngle Antenna 0 0 5 0 5 20 SNR (db) Fg. 6. Dfferent Combnng Technques: = 4, N = 2, B scaled wth SNR 8 6 Throughput (bps/z) 4 2 0 8 6 4 2 RC QBC 0 0 5 0 5 20 SNR (db) ZF CSIT Sngle Antenna Ant. Selecton Fg. 7. Dfferent Combnng Technques: = 4, N = 2, B = 0 more precse, the quantzaton error goes to zero slower than P whch results n a reducton n the slope (.e., multplexng gan) of these curves. In Fgure 7 throughput curves for the same system are shown for B = 0 (fxed). At low SNR s, RC actually performs better than QBC because sgnal power s more mportant than quantzaton error (.e., nterference power). owever, as the SNR s ncreased QBC begns to outperform RC because of the decreased quantzaton error.
7 30 Throughput (bps/z) 25 20 5 0 5 ZF CSIT QBC RC Ant. Selecton 0 0 5 0 5 20 SNR (db) Fg. 8. Combnng and User Selecton: = 4, N = 2, K = 20, B scaled wth SNR C. User Selecton The throughput of zero-forcng based IO downlnk channels can be sgnfcantly ncreased by transmttng to an ntellgently selected subset of mobles [2]. In order to maxmze throughput, users wth nearly orthogonal channels and wth large channel magntudes are selected, and waterfllng can be performed across the channels of the selected users to further ncrease throughput. In [3] a lowcomplexty greedy algorthm that selects users and performs waterfllng s proposed. In Fgure 8 the throughput of a 20 moble system wth 4 transmt antennas ( = 4) and 2 antennas per moble (N = 2) s plotted for a system usng the user selecton algorthm from [3]. The number of feedback bts s scaled wth SNR accordng to (22) and n addton to the purely drectonal nformaton captured by the quantzaton vector, we assume that perfect, (unquantzed) knowledge of the channel norm h eff s provded to the transmtter. 2 Although the throughput analyss of Secton V does not rgorously apply here, we see that the same behavor s observed even when user selecton s performed: the QBC throughput curve s approxmately 3 db away from the perfect CSIT throughput curve. The dfferent combnng technques are approxmately equvalent at low SNR s because wth hgh probablty the user selecton algorthm selects only one user at these SNR level s, but as the SNR s ncreased the QBC begns to sgnfcantly outperform the other technques because of the reduced quantzaton error. 2 Performance can be further ncreased by also feedng back the quantzaton error, but ths does not provde a sgnfcant beneft for the number of users consdered here.
8 D. Effect of Recever Estmaton Error In ths secton we analyze antenna combnng performed on the bass of mperfect recever CSI and show that QBC performs well even under realstc models for recever estmaton error. Note that the analyss here follows that gven n [0], where the effect of mperfect CSIR s analyzed for IO downlnk channels wth sngle antenna mobles and quantzed channel feedback. We consder the scenaro where a shared plot sequence s used to tran the mobles. If β downlnk plots are used (β plots per transmt antenna), channel estmaton at the -th moble s performed on the bass of observaton G = βp + n. The SE estmate of s Ĝ = βp +βp G, and the true channel matrx can be wrtten as the sum of the SE estmate and ndependent estmaton error: = Ĝ + e, (24) where e s whte Gaussan nose, ndependent of the estmate Ĝ, wth per-component varance ( + βp). After computng the channel estmate Ĝ, the moble performs QBC on the bass of the estmate Ĝ to determne the combnng vector γ. As a result, the quantzaton vector ĥ very accurately quantzes the vector Ĝγ, whch s the moble s estmate of the effectve channel output, whle the actual effectve channel s gven by h eff = γ. For smplcty we assume that coherent communcaton s possble, and therefore the long-term average throughput s agan E[log 2 ( + SINR )] where the same expresson for SINR gven n (7) apples 3. As a result, recever estmaton error further degrades the CSI provded to the transmtter. The general throughput analyss n Secton V stll apples, and n partcular, the rate gap upper bound gven n Theorem stll holds f the expected quantzaton error takes nto account the effect of recever nose. Rather than dealng drectly wth the expected quantzaton error E[sn 2 ( (ĥ,h eff ))], t s easer to nstead analyze the nterference term E [ (h eff ) v j 2], whch descrbes how much nterference moble receves from the sgnal ntended for moble j. If we return to the proof of Theorem, we can see that the rate gap upper bound can also be stated n terms of these nterference terms: ( ) ( + log 2 R log 2 e l= N+ l + P E [ (h eff ) v j 2]). (25) Usng the representaton of the channel matrx gven n (24), we can wrte the nterference term as: (h eff ) v j = ( γ ) v j = (Ĝ γ ) vj + (e γ ) v j. 3 We have effectvely assumed that each moble can estmate the phase and SINR at the effectve channel output. In practce ths could be accomplshed va a second round of plots that are transmtted usng the chosen beamformng vectors and receved at the effectve channel outputs, as descrbed n [0].
9 The frst term n the sum s statstcally dentcal to the nterference term when there s perfect CSIR, whle the second term represents the addtonal nterference due to the recever estmaton error. Because the nose and the channel estmate are each zero-mean and are ndependent we have: E [ [ (h eff ) v j 2] ) 2] [ (e ] = E (Ĝ γ vj + E γ ) 2 v j The frst term comes from the perfect CSIR analyss and s equal to the product of and the expected quantzaton error wth perfect CSIR. Because γ and v j are each unt norm and e s ndependent of these two vectors, the quantty (e γ ) v j s (zero-mean) complex Gaussan wth varance ( + βp). Usng the approxmaton for quantzaton error from (20) and pluggng nto (25), we then get the followng approxmaton to the rate gap when there s recever estmaton error: ( ) ( ( ) N+ R(P) log 2 e + log l 2 +P 2 B N l= N+ ( ) N N + β where we have approxmated ( + βp) (βp). Comparng ths expresson to (2) we see that estmaton error leads only to the ntroducton of an addtonal β ) term. If feedback s scaled accordng to (22) the rate loss f log 2 (b + β ) rather than log 2 (b). Snce downlnk plots are shared, ncreasng β s generally not too expensve from a system resource pont of vew; plots can also be transmtted at an ncreased power level as s done n Wax [4]. In Fgure 9 the throughput of a 4 moble system wth = 4 and N = 2 s plotted for perfect CSIT/CSIR and for QBC performed on the bass of perfect CSIR (β = ) and mperfect CSIR for β = and β = 2. Estmaton error causes non-neglgble degradaton, but the loss decreases rather quckly wth β., VII. CONCLUSION The performance of mult-user IO technque such as zero-forcng beamformng crtcally depend on the accuracy of the channel state nformaton provded to the transmtter. In ths paper, we have shown that an nnovatve receve antenna combnng technque can be used to reduce channel quantzaton error n fnte-rate feedback IO downlnk channels, and thus sgnfcantly reduce channel feedback requrements. Unlke tradtonal maxmum-rato combnng technques that maxmze receved sgnal power, the proposed quantzaton-based combnng technque mnmzes quantzaton error, whch translates nto mnmzaton of mult-user nterference power. Because mult-user IO systems are so senstve to nterference power, ths technque can sgnfcantly outperform standard combnng, partcularly at moderate and hgh SNR s.
20 30 25 Throughput (bps/z) 20 5 0 5 Perfect CSIT & CSIR Combnng & Perfect CSIR 0 0 5 0 5 20 25 30 SNR (db) RX Error: Beta = 2 RX Error: Beta = Fg. 9. Combnng wth Imperfect CSIR: = 4, N = 2, K = 4, B scaled wth SNR Antenna combnng s just one method by whch multple moble antennas can be used n the IO downlnk. It s also possble to transmt multple streams to each moble, or to use receve antennas for nterference cancellaton f the structure of the transmtted sgnal s known to the moble. It remans to be seen whch of these technques s most benefcal n practcal wreless systems when channel feedback resources and complexty requrements are carefully accounted for. APPENDIX I PROOF OF TEORE Pluggng the rate expressons nto the defnton of (P), we have (P) = a + b where [ ( a = E log 2 + P )] h v ZF, 2 E,W log 2 + b = E,W log 2 + j P (heff ) v j 2 P (heff ) v j 2 j= To upper bound a, we frst defne the normalzed vectors h = h / h and note that the norm and drectons of h and of h eff h eff = h eff / h eff, and are ndependent. Usng ths, we derve the followng
2 lower bound to the second term n a : [ ( P E,W log 2 + (heff ) v j 2 E,W log 2 + P )] (heff ) v 2 j= h eff [ ( = E,W log 2 + P heff 2 v 2 )] = E [ log 2 ( + P X β h 2 h vzf, 2 )] (26) where X β s β( N+,N ). Snce the beamformng vector v ZF, s chosen orthogonal to the ( ) other channel vectors {h j } j, each of whch s an d sotropc vector, t s sotropc and s ndependent of the channel drecton h. Due to Lemma 2 the same s also true of v and h eff, and therefore we can substtute h vzf, 2 for (h eff ) v 2. Fnally, note that the product X β h 2 s χ 2 2( N+) because h 2 s χ 2 2, and therefore X β h 2 and h eff 2 have the same dstrbuton. Usng (26) we get [ ( a E log 2 + P )] [ ( )] h 2 h vzf, 2 P E log 2 + X β h 2 h vzf, 2 [ ( + P = E log h 2 h )] vzf, 2 2 P + X β h 2 h vzf, 2 [ ( P E log h 2 h )] vzf, 2 2 P X β h 2 h vzf, 2 E [log 2 (X β )] ( ) = log 2 e, l l= N+ where the last lne follows from E [log 2 (X β )] = E[log 2 χ 2 2 ] E[log 2 χ 2 2( N+) ] and the expressons n [7]. Fnally, we upper bound b usng Jensen s nequalty: b log 2 + E P (heff ) v j 2 j ( = log 2 + ( ) P E [ (h eff ) 2] E [ (h eff ) v j 2]) log 2 ( + P ( = log 2 + P N + ( )( N + ) E [ (h eff ) v j 2]) E [ sn 2 ( ( h eff,h ))] ). where the fnal step uses Lemma 2 of [4] to get E [ (h eff ) v j 2] = E [ sn 2 ( ( h eff,h ))].
22 APPENDIX II GENERATION OF NUERICAL RESULTS Brute force smulaton of RVQ s extremely computatonally complex because t requres generatng a codebook of 2 B quantzaton vectors and then multplyng each of these vectors wth a projecton matrx. For B larger than 5 or 20, ths becomes nfeasble wth standard processors and software (e.g., ATLAB). owever, the statstcs of RVQ can be exploted to exactly smulate the quantzaton process: ) Draw a realzaton of the quantzaton error Z, whch has a known CDF (Lemma ) that can be numercally nverted. 2) Draw a realzaton of the correspondng quantzaton vector accordng to: ( ) ĥ = Z u + Zs where u s chosen sotropcally n span( ) and s s chosen sotropcally n the nullspace of span( ), wth u, s ndependent. These steps exactly emulate step of quantzaton-based combnng at each moble, after whch combner weghts are chosen (whch determne the effectve channels) and the zero-forcng procedure s performed. Note that the same procedure can be used to smulate antenna selecton, quantzaton of the maxmum egenvector, and no combnng (N = ). owever, t does not appear that a smlar technque can be used for RC, and as a result all RC results are generated usng brute force RVQ. REFERENCES [] G. Care and S. Shama, On the achevable throughput of a multantenna Gaussan broadcast channel, IEEE Trans. Inform. Theory, vol. 49, no. 7, pp. 69 706, July 2003. [2] W. Santpach and. ong, Asymptotc capacty of beamformng wth lmted feedback, n Proceedngs of Int. Symp. Inform. Theory, July 2004, p. 290. [3] D. Love, R. eath, W. Santpach, and. ong, What s the value of lmted feedback for IO channels? IEEE Communcatons agazne, vol. 42, no. 0, pp. 54 59, Oct. 2004. [4] N. Jndal, IO broadcast channels wth fnte rate feedback, IEEE Trans. on Inform. Theory, vol. 52, no., pp. 5045 5059, 2006. [5] D. Love, R. eath, and T. Strohmer, Grassmannan beamformng for multple-nput multple-output wreless systems, IEEE Trans. Inform. Theory, vol. 49, no. 0, pp. 2735 2747, Oct. 2003. [6] K. ukkavll, A. Sabharwal, E. Erkp, and B. Aazhang, On beamformng wth fnte rate feedback n multple-antenna systems, IEEE Trans. Inform. Theory, vol. 49, no. 0, pp. 2562 2579, Oct. 2003. [7] A. Tulno and S. Verdu, Random matrx theory and wreless communcatons, Foundatons and Trends n Communcatons and Informaton Theory, vol., no., 2004. [8] A. K. Gupta and S. Nadarajah, andbook of Beta Dstrbuton and Its Applcatons. CRC, 2004.
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