A Feedback Reduction Technique for MIMO Broadcast Channels

Transcription

1 A Feedback Reducton Technque for MIMO Broadcast Channels Nhar Jndal Department of Electrcal and Computer Engneerng Unversty of Mnnesota Mnneapols, MN 55455, USA Emal: Abstract A multple antenna broadcast channel wth perfect channel state nformaton at the recevers s consdered. If each recever quantzes ts channel knowledge to a fnte number of bts whch are fed back to the transmtter, the large capacty benefts of the downlnk channel can be realzed. owever, the requred number of feedback bts per moble must be scaled wth both the number of transmt antennas and the system SNR, and thus can be qute large n even moderately szed systems. It s shown that a small number of antennas can be used at each recever to mprove the qualty of the channel estmate provded to the transmtter. As a result, the requred feedback rate per moble can be sgnfcantly decreased. I. INTRODUCTION In multple antenna broadcast (downlnk channels, capacty can be tremendously ncreased by addng antennas at only the access pont (transmtter []. owever, the transmtter must have accurate channel state nformaton (CSI n order to realze these multplexng gans. In frequency-dvson duplexed systems, tranng can be used to obtan channel knowledge at each of the moble devces (recevers, but obtanng CSI at the access pont generally requres feedback from each moble. 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. The feedback bts are determned by quantzng the channel vector to one of 2 B quantzaton vectors. A downlnk channel wth such a feedback mechansm was analyzed n [2][][4]. Whle only a few feedback bts suffce to obtan near-perfect CSIT performance n pont-to-pont MISO (multple-nput, sngle-output channels [5][6], consderably more feedback s requred n downlnk channels. In fact, the feedback load per moble 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 [2][4], t s shown that the followng scalng of feedback bts per moble B M db ( suffces to mantan a maxmum gap of db between perfect CSIT and lmted feedback performance when zero-forcng beamformng s used. Ths feedback load, however, can be prohbtvely large for even reasonable sze systems. In a 0 antenna system operatng at 0 db, for example, ths equates to 0 feedback bts per moble. In ths paper, we propose a method that sgnfcantly reduces the requred feedback load by utlzng a small number of receve antennas (denoted by N at each moble. The multple receve antennas are not used to ncrease the number of data streams receved at each moble, as they are n pont-topont MIMO systems, but nstead are used to mprove the qualty of the channel estmate provded to the transmtter. Each moble lnearly combnes the receved sgnals on ts N antennas to produce a scalar output, thereby creatng an ectve sngle antenna channel at each moble. Transmsson s then performed as n a multple transmt antenna, sngle receve antenna downlnk channel. owever, the cocents of the lnear combner at each moble are not arbtrary, but nstead are chosen to produce the ectve sngle antenna channel that can be quantzed wth mnmal error, thereby decreasng the quantzaton error for each moble. Increasng the number of receve antennas N ncreases the space of possble ectve channels, and thus leads to reduced quantzaton error. In a 0 antenna system operatng at 0 db, for example, ths method reduces the feedback from 0 bts per moble n the N scenaro to 25 bts and 2 bts, for N 2 and N, respectvely. 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. SYSTEM MODEL We consder a K recever multple antenna broadcast channel n whch the transmtter (access pont has M antennas, and each of the recevers has N antennas. The receved sgnal at the -th antenna s descrbed as: y h x + n,,...,nk (2 where h,h 2,...,h KN are the channel vectors (wth h C M descrbng the KN receve antennas, the vector x C M s the transmtted sgnal, and n,...,n NK are ndependent complex Gaussan nose terms wth unt varance. Note that recever has access to sgnals y,..., y N, recever 2 has access to y N+,...,y 2N, and the -th recever has access to y ( N+,..., y N. There s a transmt power constrant of,.e. we must satsfy E[ x 2 ]. We use to

2 denote the concatenaton of the -th recever s channels,.e. [h ( N+ h N ]. For smplcty of exposton we assume that the number of mobles s equal to the number of transmt antennas,.e., K M. The results can easly be extended to the case where K < M, and the proposed technque can be combned wth user selecton when K > M. Furthermore, the number of receve antennas s assumed to be no larger than the number of transmt antennas,.e., N M. We consder a block fadng channel, wth ndependent Raylegh fadng from block to block. Each of the recevers s assumed to have perfect and nstantaneous knowledge of ts own channel. Notce t s not requred for mobles to know the channel of other mobles. In ths work we study only the ergodc capacty, or the average rates acheved over an nfnte number of blocks (or channel realzatons. A. Fnte Rate Feedback Model ere we brefly descrbe the feedback model for a sngle receve antenna (N. At the begnnng of each block, each recever quantzes ts channel (wth h assumed to be known perfectly at the -th recever to B bts and feeds back the bts perfectly and nstantaneously to the access pont. Vector quantzaton s performed usng a codebook C that conssts of 2 B M-dmensonal unt norm vectors C {w,...,w 2 B}, where B s the number of feedback bts. Each recever quantzes ts channel vector to the beamformng vector that forms the mnmum angle to t, or equvalently that maxmzes the nner product [7] [8]. Thus, user quantzes ts channel to ĥ, chosen accordng to: ĥ arg max h ww,...,w 2 B w ( arg mn sn 2 ( (h,w. (4 and feeds the quantzaton ndex back to the transmtter. It s mportant to notce that only the drecton of the channel vector s quantzed, and no magntude nformaton s conveyed to the transmtter. The quantzaton error can be thought of as ether the angle between the channel and ts quantzaton (h,ĥ or the quantty sn 2 ( (h,ĥ. In ths work we use random vector quantzaton (RVQ, n whch each of the 2 B quantzaton vectors s ndependently chosen from the sotropc dstrbuton on the M-dmensonal unt sphere [5]. To smplfy analyss, each recever s assumed to use a dfferent and ndependently generated codebook. We analyze performance averaged over all such choces of random codebooks. Random codebooks are used because the optmal vector quantzer for ths problem s not known n general and known bounds are rather loose, whereas RVQ s amenable to analyss and also provdes performance that s measurably close to the optmal [5]. B. Zero-Forcng Beamformng After recevng the quantzaton ndces from each of the mobles, the A can use zero-forcng beamformng (ZFBF to transmt data to the M users. Let us agan consder the N scenaro, where the channels are the vectors h,...,h M. Snce the transmtter does not have perfect CSI, ZFBF must be performed based on the quantzatons nstead of the actual channels. When ZFBF s used, the transmtted sgnal s defned as x M x v, where each x s a scalar (chosen complex Gaussan wth power /M ntended for the -th recever, and v C M s the beamformng vector for the -th recever. The beamformng vectors v,...,v M are chosen as the normalzed rows of the matrx [ĥ ĥm],.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 M h v 2 + j M h v j 2. (5 Note that the nterference terms n the denomnator are strctly postve because h ĥ,.e., due to the quantzaton error. III. EFFECTIVE CANNEL QUANTIZATION In ths secton we descrbe the proposed method to reduce the quantzaton error n the transmtter s estmate of the moble channels. We begn by frst descrbng a smple, antennaselecton method for reducng feedback, whch motvates the better performng ectve channel method. A smple method to utlze N receve antennas s to separately quantze each of the N channel vectors and then feed back the ndex of only the best of the N quantzatons. If, for example, antenna had the mnmum quantzaton error, the moble would only send the quantzaton ndex descrbng antenna and would only utlze the frst antenna when recevng. It s straghtforward to show that choosng the best of the N channel quantzatons, each from a codebook of sze 2 B, s equvalent to quantzng a sngle channel usng a codebook of sze N 2 B. Thus, f B feedback bts are sent by each moble, a system wth N antennas per moble wll perform dentcal to a sngle receve antenna system wth B+log 2 N feedback bts per moble. Thus, utlzng N receve antennas n ths smple manner decreases the feedback load by log 2 N bts per moble. A more sgnfcant decrease n feedback load can be acheved by consderng all possble lnear combnatons of the N receved sgnals, nstead of lmtng the system to selecton of one of the N sgnals. Consder the ectve receved sgnal at the frst recever after lnearly combnng the N receved sgnals by complex weghts γ (γ,,..., γ,n satsfyng γ : y N γ,ky k N γ,k(h k x + n k ( N γ,k h k (h x + n, x + n where h N γ,kh k γ and n N γ,k n k s unt varance complex Gaussan nose because γ. Snce any set of weghts satsfyng the unt norm can be

3 γ, y 2 γ,2 γ 2, y 2 span( h γ 2,2 γ, y h 2 h h proj γ,2 Fg.. Effectve Channel for M K, N 2 System Fg. 2. Quantzaton procedure for a two antenna moble chosen, h can be n any drecton n the subspace spanned by h,...,h N. Thus, the quantzaton error s mnmzed by choosng h to be n the drecton that can be quantzed best, or equvalently the drecton whch s closest to one of the quantzaton vectors. The combner structure for a user channel wth M and N 2 s shown n Fg.. Let us now more formally descrbe the quantzaton process performed at the frst moble. As descrbed n Secton II-A, the quantzaton codebook conssts of 2 B sotropcally chosen unt norm vectors w,...,w 2 B. In the sngle receve antenna (N, h scenaro, quantzaton s performed by choosng the quantzaton vector that has the smallest angle between tself and the channel vector h. When N >, we compute the angle between each quantzaton vector and the subspace spanned by the N channel vectors, and pck the quantzaton vector that forms the smallest such angle. Alternatvely, each quantzaton vector s projected onto the span of the N channel vectors, and the angle between the quantzaton vector and ts projecton s computed. If q,...,q N forms an orthonormal bass for span(h,...,h N (easly computable usng Gram- Schmdt, then the quantzaton s performed accordng to: ĥ arg mn (w, span(h,...,h N (6 N arg max w q k 2. (7 Let us denote the normalzed projecton of ĥ onto span(h,...,h N by the vector s proj. Notce that the drecton specfed by s proj has the mnmum quantzaton error amongst all drectons n span(h,...,h N. Thus, the ectve channel should be chosen n ths drecton,.e., we wsh to choose a unt norm vector γ such that h N j γ,jh j γ s n the drecton of the projected quantzaton vector s proj. 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 can be found by the pseudo-nverse of : u ( s proj, (8 and the cocent vector γ s the normalzed verson of u: γ u u. It s easy to check that h / 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 projecton of the best quantzaton vector onto ths subspace along wth the subsequent angular error. We now summarze the procedure for computng the quantzaton vector and the weghtng vector of the -th moble: Compute the channel quantzaton: ĥ arg mn (w, span( N arg max w q k 2. (9 where q,...,q N s an orthonormal bass for the span of the columns of. 2 roject the quantzaton vector onto the span of the channel vectors: N s proj qk(ĥ q k N qk(ĥ q k. Compute the weghtng vector γ : ( γ s proj ( (0 s proj. Each moble performs these steps, feeds back the ndex of ts quantzed channel, and then lnearly combnes ts N receved sgnals usng weghtng vector γ to get y (h x + n wth h γ. The proposed method fnds the ectve channel wth the mnmum quantzaton error wthout any regard to the resultng channel magntude (.e., h. Ths s reasonable because quantzaton error s the domnatng factor n lmted feedback downlnk systems, as we later see n the sum rate analyss. owever, t may be useful to later study alternatves that balance mnmzaton of quantzaton error wth maxmzaton of channel magntude. IV. SUM RATE ANALYSIS The ectve channel quantzaton procedure converts the multple transmt, multple receve antenna downlnk channel nto a multple transmt, sngle receve antenna downlnk channel wth channel vectors h,...,h M and channel quantzatons ĥ ĥm. In fact, the transmtter need not even be aware of the number of receve antennas, snce the multple receve antennas are used only durng quantzaton. After recevng the quantzaton ndces from each of the mobles, the transmtter performs zero-forcng beamformng

4 (as descrbed n Secton II-B based on the channel quantzatons. The resultng SINR at the -th recever s gven by: SINR M (h v 2 + j M (h v j 2. ( We are nterested n the long-term average sum rate acheved n ths channel, and thus the expectaton of M log( + SINR. Snce the beamformng vectors are chosen accordng to the ZFBF crteron based on the quantzed channels, they satsfy v for all and ĥv j 0 for all j. Quantzaton error, however, leads to msmatch between the ectve channels and ther quantzatons, and thus strctly postve nterference terms (of the form (h v j 2 n the denomnator of the SINR expresson. A. relmnary Calculatons In order to analyze the expected rate of such a system, the dstrbuton (over the random channels and quantzaton codebooks of the quantzaton error between h and ĥ and of the ectve channel must be characterzed. Lemma : The quantzaton error sn 2 ( (ĥ,h, s the mnmum of 2 B ndependent beta (M N, N random varables. roof: If q,...,q N denote an orthonormal bass for span(, cos 2 ( (w j, span( N w j q k 2 for any quantzaton vector. Snce the bass vectors and quantzaton vectors are sotropcally chosen, ths quantty s the squared norm of the projecton of a random vector n C M onto a random N-dmensonal subspace, whch s descrbed by the beta dstrbuton wth parameters N and M N [9]. By the propertes of the beta dstrbuton, sn 2 ( (w j, span( cos 2 ( (w j, span( s beta (M N, N. Fnally, the ndependence of the quantzaton vectors and the channels mples ndependence of the 2 B random varables. The followng lemma and conjecture characterze the dstrbuton of the ectve channel vectors. Lemma 2: The normalzed ectve channels h h,..., hm h M are d sotropc vectors n CM. roof: From the earler descrpton of ectve channel h quantzaton, note that h sproj, whch s the projecton of the best 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 s sotropcally dstrbuted n span(. Snce ths subspace s also sotropcally dstrbuted, the vector s proj s sotropcally dstrbuted n C M. Independence holds due to the ndependence of the quantzaton vectors and channel realzatons. Conjecture : The squared norm of the ectve channel s ch-squared wth 2(M N + degrees of freedom. Whle ths conjecture can be proven for the case when N M usng the fact that the dagonal entres of ( are each nverted ch-square wth two degrees of freedom when s square [0, Theorem.2.2], ths proof does not h yet extend to the scenaro where < N < M. owever, numercal results very strongly ndcate that the conjecture s true for all values of N and M. The clam s trvally true when N because h h when mobles have a sngle antenna. Furthermore, t s known that /(v v s ch-square dstrbuted wth 2(M N + degrees of freedom for any unt norm v [0]. If N M, there s zero quantzaton error but the resultng ectve channels have only two degrees of freedom. Ths scenaro s not relevant, however, because hgher rates can be acheved by smply transmttng to a sngle user usng pontto-pont MIMO technques, snce such a system has the same number of spatal degrees of freedom as the downlnk channel. B. Sum Rate erformance Relatve to erfect CSIT In order to study the ect of fnte rate feedback, we compare the sum rate acheved usng fnte rate feedback and ectve channel quantzaton (for N, denoted R FB (, to the sum rate acheved wth perfect CSIT n an M transmt, sngle receve antenna downlnk channel, denoted R ZF (. 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 ectvely utlzes a sngle receve antenna per moble for recepton, and thus cannot outperform a sngle receve antenna downlnk channel wth perfect CSIT, even n the lmt of an nfnte number of feedback bts. Furthermore, ths analyss allows us to compare the requred feedback load wth N > and the proposed method to the requred feedback load for downlnk channels wth sngle receve antennas, studed n [2][4]. Let us frst analyze the rates acheved n a sngle receve antenna downlnk channel usng ZFBF under the assumpton of perfect CSIT. If the transmtter has perfect CSIT, the beamformng vectors (denoted v ZF, can be chosen perfectly orthogonal to all other channels, thereby elmnatng all multuser nterference. 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: E [R ZF (] E [log ( + M h v ZF, 2 ]. Snce the beamformng vector v ZF, s chosen orthogonal to the (M other channel vectors {h j } j, each of whch s an d sotropc vector, the beamformng vector s also an sotropc vector, ndependent of the channel vector h. Because the ectve channel vectors are sotropcally dstrbuted (Lemma 2, the same s true of the beamformng vectors and the ectve channel vectors when the proposed method s used. If the number of feedback bts s fxed, the rates acheved wth fnte rate feedback are bounded even as the SNR s taken to nfnty. Thus, the number of feedback bts must be approprately scaled n order to avod ths lmtaton. Furthermore, t s useful to consder the scalng of bts requred to mantan a desred rate (or power gap between perfect CSIT and lmted feedback. Thus, we study the rate gap

5 at asymptotcally hgh SNR, denoted as R. Some smple algebra yelds the followng upper bound to R: 0 R lm E,W[R ZF ( R LF (] [ lm E log ( + M ] h v ZF 2 ( E,W [log + ] M (h v 2 + E,W log + M (h v j 2 j Capacty (bps/z erfect CSIT Zero Forcng ~ db Lmted FB (N,2, The dfference of the frst two terms s the rate loss due to the reduced ectve channel norm (Conjecture and can be computed n closed form usng the expectaton of the log of ch-square random varables, gvng a loss of a log 2 e M l+ l. The fnal term s the rate loss due to the quantzaton error and can be upper bounded usng Jensen s nequalty and some of the technques from [2][4] to gve: R a + log E,W M (h v j 2 [ a + log + + j ( M N + M ] E[sn 2 ( (ĥ,h ] We now utlze Lemma to estmate the quantzaton error. If we let X be a beta(m N, N random varable, the CCDF of X can be accurately approxmated for x as r(x x ( M N ( x (. Snce the quantzaton error s one mnus the maxmum of 2 B such random varables, we use extreme value theory and fnd x such that r(x x 2 B to get the followng approxmaton for the quantzaton error: E[sn 2 ( (ĥ,h Thus we have R log 2 e M ] 2 B l+ l + ( ( M N + log + M ( M 2 B ( M If we set ths quantty equal to a desred rate gap r > 0 and solve for the requred scalng of B as a functon of the SNR (n db we get: B M N db (M Nlog 2 c (2 ( ( M M (M Nlog 2 log M N + 2, where c 2 r e ( M l+ l. Note that a per user rate gap of r bps/z s equvalent to a db gap n the sum rate curves. If we compute the dfference between ths expresson and the feedback load requred when N (gven n ( and a db gap s desred (r, we can get the followng Fg.. SNR (db Sum rate of M K 6 downlnk channel approxmaton for the feedback reducton as a functon of the number of moble antennas N: FB (N N ( M db + log 2 (N log 2 e. For N 2, the feedback savngs s gven by: FB (2 db + log 2 (M log 2 e. The sum rate of a 6 transmt antenna downlnk channel s plotted n Fg.. The perfect CSIT zero-forcng curve s plotted along wth the rates acheved usng fnte rate feedback wth the feedback load scaled as specfed n (2 for N, 2 and. Notce that the rates acheved for dfferent numbers of transmt antennas are nearly ndstngushable, and all three curves are approxmately db shfts of the perfect CSIT curve. In ths system, the feedback savngs at 20 db s 7 and 2 bts, respectvely, for 2 and receve antennas. 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 , July 200. [2] N. Jndal, MIMO broadcast channels wth fnte rate feedback, n roceedngs of IEEE Globecom, []. Dng, D. Love, and M. Zoltowsk, Multple antenna broadcast channels wth partal and lmted feedback, 2005, submtted to IEEE Trans. Sg. roc. [4] N. Jndal, MIMO broadcast channels wth fnte rate feedback, March 2006, submtted to IEEE Trans. Inform. Theory. [5] W. Santpach and M. ong, Asymptotc capacty of beamformng wth lmted feedback, n roceedngs of Int. Symp. Inform. Theory, July 2004, p [6] D. Love, R. eath, W. Santpach, and M. ong, What s the value of lmted feedback for MIMO channels? IEEE Communcatons Magazne, vol. 42, no. 0, pp , Oct [7] 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 , Oct [8] K. Mukkavll, 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 , Oct [9] A. K. Gupta and S. Nadarajah, andbook of Beta Dstrbuton and Its Applcatons. CRC, [0] R. J. Murhead, Aspects of Multvarate Statstcal Theory. Wley, 982.