Scaling Up MIMO. with very large arrays] Thomas L. Marzetta, Ove Edfors, and Fredrik Tufvesson ]

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1 Scalng Up MIMO dynamc graphcs [ Opportuntes and challenges wth very large arrays] [ Fredrk Rusek, Danel Persson, Buon ong Lau, Erk G Larsson, Thomas L Marzetta, Ove Edfors, and Fredrk Tufvesson ] Multple-nput multple-output (MIMO) technology s maturng and s beng ncorporated nto emergng wreless broadband standards lke long-term evoluton (LTE) [] For example, the LTE standard allows for up to eght antenna ports at the base staton Bascally, the more antennas the Dgtal Object Identfer 009/MSP Date of publcaton: 5 December 0 transmtter/recever s equpped wth, and the more degrees of freedom that the propagaton channel can provde, the better the performance n terms of data rate or lnk relablty More precsely, on a quasstatc channel where a codeword spans across only one tme and frequency coherence nterval, the relablty of a pont-to-pont MIMO lnk scales accordng to - Prob(lnk outage) ` SNR nn t r where nt and nr are the numbers of transmt and receve antennas, respectvely, and sgnal-to-nose rato s denoted by SNR On a channel that vares rapdly as a IEEE SIGNAL PROCESSING MAGAZINE [40] january /3/$300 03IEEE

2 functon of tme and frequency, and where crcumstances permt codng across many channel coherence ntervals, the achevable rate scales as mn( nt, nr ) log( + SNR) The gans n multuser systems are even more mpressve, because such systems offer the possblty to transmt smultaneously to several users and the flexblty to select what users to schedule for recepton at any gven pont n tme [] The prce to pay for MIMO s ncreased complexty of the hardware [number of rado frequency (RF) chans] and the complexty and energy consumpton of the sgnal processng at both ends For pont-to-pont lnks, complexty at the recever s usually a greater concern than complexty at the transmtter For example, the complexty of optmal sgnal detecton alone grows exponentally wth nt [3], [4] In multuser systems, complexty at the transmtter s also a concern snce advanced codng schemes must often be used to transmt nformaton smultaneously to more than one user whle mantanng a controlled level of nteruser nterference Of course, another cost of MIMO s that of the physcal space needed to accommodate the antennas, ncludng rents of real estate Wth very large MIMO, we thnk of systems that use antenna arrays wth an order of magntude more elements than n systems beng bult today, say 00 antennas or more Very large MIMO entals an unprecedented number of antennas smultaneously servng a much smaller number of termnals The dsparty n number emerges as a desrable operatng condton and a practcal one as well The number of termnals that can be smultaneously served s lmted, not by the number of antennas, but rather by our nablty to acqure channel-state nformaton for an unlmted number of termnals Larger numbers of termnals can always be accommodated by combnng very large MIMO technology wth conventonal tme- and frequency-dvson multplexng va orthogonal frequency-dvson multplexng (OFDM) Very large MIMO arrays s a new research feld both n communcaton theory, propagaton, and electroncs and represents a paradgm shft n the way of thnkng both wth regards to theory, systems, and mplementaton The ultmate vson of very large MIMO systems s that the antenna array would consst of small actve antenna unts, plugged nto an (optcal) feldbus We foresee that n very large MIMO systems, each antenna unt uses extremely low power, n the order of mllwatts At the very mnmum, of course, we want to keep total transmtted power constant as we ncrease nt, e, the power per antenna should be? / nt But n addton we should also be able to back off on the total transmtted power For example, f our antenna array were servng a sngle termnal, then t can be shown that the total power can be made nversely proportonal to nt, n whch case the power requred per antenna would be? / n t Of course, several complcatons wll undoubtedly prevent us from Very large MIMO arrays s a new research feld n communcaton theory, propagaton, and electroncs and represents a paradgm shft n the way of thnkng wth regards to theory, systems, and mplementaton fully realzng such optmstc power savngs n practce: the need for multuser multplexng gans, errors n channel state nformaton (CSI), and nterference Even so, the prospect of savng an order of magntude n transmt power s mportant because one can acheve better system performance under the same regulatory power constrants Also, t s mportant because the energy consumpton of cellular base statons s a growng concern As a bonus, several expensve and bulky tems, such as large coaxal cables, can be elmnated altogether (The coaxal cables used for tower-mounted base statons today are up to 4 cm n dameter!) Moreover, very-large MIMO desgns can be made extremely robust n that the falure of one or a few of the antenna unts would not apprecably affect the system Malfunctonng ndvdual antennas may be hotswapped The contrast to classcal array desgns, whch use few antennas fed from a hghpower amplfer, s sgnfcant So far, the large-number-of-antennas regme, when nt and nr grow wthout bound, has mostly been of pure academc nterest, n that some asymptotc capacty scalng laws are known for deal stuatons More recently, however, ths vew s changng, and a number of practcally mportant system aspects n the large- ( nt, nr ) regme have been dscovered For example, [5] showed that asymptotcally as nt " 3 and under realstc assumptons on the propagaton channel wth a bandwdth of 0 Mz, a tme-dvson multplexng cellular system may accommodate more than 40 sngleantenna users that are offered a net average throughput of 7 Mb/s both n the reverse (uplnk) and the forward (downlnk) lnks, and a throughput of 36 Mb/s wth 95% probablty! These rates are achevable wthout cooperaton among the base statons and by relatvely rudmentary technques for CSI acquston based on uplnk plot measurements Several thngs happen when MIMO arrays are made large Frst, the asymptotcs of random matrx theory kck n Ths has several consequences Thngs that were random before, now start to look determnstc For example, the dstrbuton of the sngular values of the channel matrx approaches a determnstc functon [6] Another fact s that very tall or very wde matrces tend to be very well condtoned Also, when dmensons are large, some matrx operatons such as nversons can be done fast, by usng seres expanson technques (see the sdebar) In the lmt of an nfnte number of antennas at the base staton, but wth a sngle antenna per user, then lnear processng n the form of maxmumrato combnng for the uplnk (e, matched flterng wth the channel vector, say h) and maxmum-rato transmsson (beam- formng wth h h ) on the downlnk s optmal Ths resultng processng s remnscent of tme reversal (TR), a technque used for focusng electromagnetc or acoustc waves [7], [8] The second effect of scalng up the dmensons s that thermal nose can be averaged out so that the system s predomnantly lmted by nterference from other transmtters Ths s ntutvely IEEE SIGNAL PROCESSING MAGAZINE [4] january 03

3 clear for the uplnk, snce coherent averagng offered by a receve antenna array elmnates quanttes that are uncorrelated between the antenna elements, that s, thermal nose n partcular Ths effect s less obvous on the downlnk, however Under certan crcumstances, the performance of a very large array becomes lmted by nterference arsng from reuse of plots n neghborng cells In addton, choosng plots n a smart way does not substantally help as long as the coherence tme of the channel s fnte In a tme-dvson duplex (TDD) settng, ths effect was quantfed n [5], under the assumpton that the channel s recprocal and that the base statons estmate the downlnk channels by usng uplnk receved plots Fnally, when the aperture of the array grows, the resoluton of the array ncreases Ths means that one can resolve ndvdual scatterng centers wth unprecedented precson Interestngly, as we wll see later on, the communcaton performance of the array n the large-number-of-antennas regme depends less on the actual statstcs of the propagaton channel but only on the aggregated propertes of the propagaton such as asymptotc orthogonalty between channel vectors assocated wth dstnct termnals Of course, the number of antennas n a practcal system cannot be arbtrarly large owng to physcal constrants Eventually, when lettng nr or nt tend to nfnty, our mathematcal models for the physcal realty wll break down For example, the aggregated receved power would at some pont exceed the transmtted power, whch makes no physcal sense But long before the mathematcal models for the physcs break down, there wll be substantal engneerng dffcultes So, how large s nfnty n ths artcle? The answer depends on the precse crcumstances of course, but n general, the asymptotc results of random matrx theory are accurate even for relatvely small dmensons (even ten or so) In general, we thnk of systems wth at least 00 antennas at the base staton, but probably fewer than,000 Taken together, the arguments presented motvate entrely new theoretcal research on sgnal processng and codng and network desgn for very large MIMO systems Ths artcle wll survey some of these challenges In partcular, we wll dscuss ultmate nformaton-theoretc performance lmts, some practcal algorthms, nfluence of channel propertes on the system, and practcal constrants on the antenna arrangements Informaton Theory for Very Large MIMO Arrays Shannon s nformaton theory provdes, under very precsely specfed condtons, bounds on attanable performance of communcatons systems Accordng to the nosy-channel codng theorem, for any communcaton lnk there s a capacty or achevable rate, such that for any transmsson rate less than the capacty, there exsts a codng scheme that makes the errorrate arbtrarly small The classcal pont-to-pont MIMO lnk begns our dscusson, and t serves to hghlght the lmtatons of systems n whch the workng antennas are compactly clustered at both ends of the lnk Ths leads naturally nto the topc of multuser MIMO (MU-MIMO) whch s where we envson very large MIMO wll show ts greatest utlty The Shannon theory smplfes greatly for large numbers of antennas and t suggests capactyapproachng strateges Pont-to-pont MIMO Channel model A pont-to-pont MIMO lnk conssts of a transmtter havng an array of nt antennas, a recever havng an array of nr antennas, wth both arrays connected by a channel such that every receve antenna s subject to the combned acton of all transmt antennas The smplest narrowband memoryless channel has the followng mathematcal descrpton; for each use of the channel we have x = t Gs + w, () where s s the nt-component vector of transmtted sgnals, x s the nr-component vector of receved sgnals, G s the nr # nt propagaton matrx of complex-valued channel coeffcents, and w s the nr-component vector of recever nose The scalar t s a measure of the SNR of the lnk: t s proportonal to the transmtted power dvded by the nose-varance, and t also absorbs varous normalzng constants In what follows, we assume a normalzaton such that the expected total transmt power s unty, E" s, =, () where the components of the addtve nose vector are ndependent and dentcally dstrbuted (d) zero-mean and unt- varance crculary-symmetrc complex-gaussan random varables ( CN ( 0, )) ence f there were only one antenna at each end of the lnk, then wthn () the quanttes s, G, x and w would be scalars, and the SNR would be equal to t G In the case of a wde-band, frequency-dependent ( delayspread ) channel, the channel s descrbed by a matrx-valued mpulse response or by the equvalent matrx-valued frequency response One may conceptually decompose the channel nto parallel ndependent narrow-band channels, each of whch s descrbed n the manner of () Indeed, OFDM rgorously performs ths decomposton Achevable rate Wth d complex-gaussan nputs, the (nstantaneous) mutual nformaton between the nput and the output of the pont-topont MIMO channel (), under the assumpton that the recever has perfect knowledge of the channel matrx, G, measured n bts-per-symbol (or equvalently bts-per- channel-use) s t C = I(; x s) = log detcinr + GG m, (3) nt IEEE SIGNAL PROCESSING MAGAZINE [4] january 03

4 where I ( x; s) denotes the mutual nformaton operator, Inr denotes the nr# nr dentty matrx, and the superscrpt denotes the ermtan transpose [9] The actual capacty of the channel results f the nputs are optmzed accordng to the water-fllng prncple In the case that GG equals a scaled dentty matrx, C s n fact the capacty To approach the achevable rate C, the transmtter does not have to know the channel, however t must be nformed of the numercal value of the achevable rate Alternatvely, f the channel s governed by known statstcs, then the transmtter can set a rate that s consstent wth an acceptable outage probablty For the specal case of one antenna at each end of the lnk, the achevable rate (3) becomes that of the scalar addtve complex Gaussan nose channel, C = log ^ +t G h (4) The mplcatons of (3) are most easly seen by expressng the achevable rate n terms of the sngular values of the propagaton matrx, G = UD W o, (5) where U and W are untary matrces of dmenson nr# nr and nt# nt respectvely, and D o s a nr # nt dagonal matrx whose dagonal elements are the sngular values, { o, o, g o mn( nt, nr) } The achevable rate (3), expressed n terms of the sngular values, mn( n, n ) t r to C = / log e +, o, (6), = nt s equvalent to the combned achevable rate of parallel lnks for whch the,th lnk has an SNR of to, / nt Wth respect to the achevable rate, t s nterestng to consder the best and the worst possble dstrbuton of sngular values Subject to the constrant [obtaned drectly from (5)] that mn( n, n ) t r / o, = Tr^GG h, (7), = where Tr denotes trace, the worst case s when all but one of the sngular values are equal to zero, and the best case s when all of the mn( nt, nr ) sngular values are equal (ths s a smple consequence of the concavty of the logarthm) The two cases bound the achevable rate (6) as follows: t $ Tr^GG h log c + m # C # mn( nt, nr) nt t $ Tr^GG h # log e + o (8) ntmn( nt, nr) If we assume that a normalzaton has been performed such that the magntude of a propagaton coeffcent s typcally equal to one, then Tr^GG h nn t r, and the above bounds smplfy as follows: t max( nt, nr) log ^+ tnrh # C # mn ( nt, nr) $ log c+ m (9) nt The rank- (worst) case occurs ether for compact arrays under lne-of-sght (LOS) propagaton condtons such that the transmt array cannot resolve ndvdual elements of the receve array and vce versa, or under extreme keyhole propagaton condtons The equal sngular value (best) case s approached when the entres of the propagaton matrx are d random varables Under favorable propagaton condtons and a hgh SNR, the achevable rate s proportonal to the smaller of the number of transmt and receve antennas Lmtng cases Low SNRs can be experenced by termnals at the edge of a cell For low SNRs, only beamformng gans are mportant and the achevable rate (3) becomes t $ Tr^GG h Ct " 0 nt ln tnr (0) ln Ths expresson s ndependent of nt, and thus, even under the most favorable propagaton condtons the multplexng gans are lost, and from the perspectve of achevable rate, multple transmt antennas are of no value Next, let the number of transmt antennas grow large whle keepng the number of receve antennas constant We furthermore assume that the row-vectors of the propagaton matrx are asymptotcally orthogonal As a consequence [0] GG c m Inr, () nt nt& nr and the achevable rate (3) becomes Cnt& nr log det^inr+ t $ Inrh () = nr $ log( + t), whch matches the upper bound (9) Then, let the number of receve antennas grow large whle keepng the number of transmt antennas constant We also assume that the column-vectors of the propagaton matrx are asymptotcally orthogonal, so G G c m Int (3) nr nr& nt The dentty det( I+ AA ) = det( I+ A A), combned wth (3) and (3), yelds C t = log detc I + G Gm nt tnr nt $ log c + m, (4) nt nr& nt nt whch agan matches the upper bound (9) So an excess number of transmt or receve antennas, combned wth asymptotc orthogonalty of the propagaton vectors, consttutes a hghly desrable scenaro Extra receve antennas contnue to boost the effectve SNR, and could n theory compensate for a low SNR and restore multplexng gans that would otherwse be lost as IEEE SIGNAL PROCESSING MAGAZINE [43] january 03

5 n (0) Furthermore, orthogonalty of the propagaton vectors mples that d complex-gaussan nputs are optmal so that the achevable rates (3) and (4) are n fact the true channel capactes MU-MIMO The attractve multplexng gans promsed by pont-to-pont MIMO requre a favorable propagaton envronment and a good SNR Dsappontng performance can occur n LOS propagaton or when the termnal s at the edge of the cell Extra receve antennas can compensate for a low SNR, but for the forward lnk ths adds to the complcaton and expense of the termnal Very large MIMO can fully address the shortcomngs of pont-topont MIMO If we splt up the antenna array at one end of a pont-topont MIMO lnk nto autonomous antennas, we obtan the qualtatvely dfferent MU-MIMO Our context for dscussng ths s an array of M antennas, for example, a base staton, whch smultaneously serves autonomous termnals (Snce we want to study both forward- and reverse-lnk transmsson, we now abandon the notaton nt and nr ) In what follows, we assume that each termnal has only one antenna MU-MIMO dffers from pont-to-pont MIMO n two respects: frst, the termnals are typcally separated by many wavelengths, and second, the termnals cannot collaborate among themselves, ether to transmt or to receve data Propagaton We wll assume TDD operaton, so the reverse-lnk propagaton matrx s merely the transpose of the forward-lnk propagaton matrx Our emphass on TDD rather than FDD s drven by the need to acqure channel state-nformaton between extreme numbers of servce antennas and much smaller numbers of termnals The tme requred to transmt reverse-lnk plots s ndependent of the number of antennas, whle the tme requred to transmt forward-lnk plots s proportonal to the number of antennas The propagaton matrx n the reverse lnk, G, dmensoned M#, s the product of an M# matrx,, whch accounts for smallscale fadng (e, whch changes over ntervals of a wavelength or less), and a # dagonal matrx, D / b, whose dagonal elements consttute a # vector, b, of large-scale fadng coeffcents, G = D / b (5) The large-scale fadng accounts for path loss and shadow fadng Thus, the kth column-vector of descrbes the small-scale fadng between the kth termnal and the M antennas, whle the k th dagonal element of D / b s the large-scale fadng coeffcent By assumpton, the antenna array s suffcently compact that all of the propagaton paths for a partcular termnal are subject to the same large-scale fadng We normalze the large-scale fadng coeffcents such that the small-scale fadng coeffcents typcally have magntudes of one For MU-MIMO wth large arrays, the number of antennas greatly exceeds the number of termnals Under the most favorable propagaton condtons, the column-vectors of the propagaton matrx are asymptotcally orthogonal, G G c m M = D b c m M Db D / / b M& M& (6) Reverse lnk On the reverse lnk, for each channel use, the termnals collectvely transmt a # vector of quadrature ampltude modulaton (QAM) symbols, qr, and the antenna array receves an M # vector, xr, xr = trgqr+ w r, (7) where wr s the M # vector of recever nose whose components are ndependent and dstrbuted as CN (,) 0 The quantty t r s proportonal to the rato of power dvded by nose- varance Each termnal s constraned to have an expected power of one, E" qrk, =, k =, f, (8) We assume that the base staton knows the channel Remarkably, the total throughput (eg, the achevable sumrate) of reverse lnk MU-MIMO s no less than f the termnals could collaborate among themselves [], C sum_r = log det^i +trg Gh (9) If collaboraton were possble, t could defntely make channel codng and decodng easer, but t would not alter the ultmate sum-rate The sum-rate s not generally shared equally by the termnals; consder for example the case where the slow fadng coeffcent s near-zero for some termnal Under favorable propagaton condtons (6), f there s a large number of antennas compared wth termnals, then the asymptotc sum-rate s C log det^i + MtrDbh = / log ^ + Mtb r kh (0) sum_rm& k = Ths has a nce ntutve nterpretaton f we assume that the columns of the propagaton matrx are nearly orthogonal, e, G G M $ Db Under ths assumpton, the base staton could process ts receved sgnal by a matched-flter (MF), G x r = t r G Gqr + G w M r trdb qr+ G w r () Ths processng separates the sgnals transmtted by the dfferent termnals The decodng of the transmsson from the kth IEEE SIGNAL PROCESSING MAGAZINE [44] january 03

6 termnal requres only the kth component of (); ths has an SNR of Mtb r k, whch n turn yelds an ndvdual rate for that termnal, correspondng to the kth term n the sum-rate (0) Forward lnk For each use of the channel the base staton transmts an M # vector, sf, through ts M antennas, and the termnals collectvely receve a # vector, x f, T xf = tfg s f + w, f () where the superscrpt T denotes transpose, and wf s the # vector of recever nose whose components are ndependent and dstrbuted as CN (,) 0 The quantty t f s proportonal to the rato of power to nose-varance The total transmt power s ndependent of the number of antennas, E" sf, = (3) The known capacty result for ths channel, see, eg, [] and [], assumes that the termnals as well as the base staton know the channel Let D c be a dagonal matrx whose dagonal elements consttute a # vector c To obtan the sum-capacty requres performng a constraned optmzaton, Csum_f = maxlogdet^im +tfgdg c h, { ck} subject to / ck =, ck $ 0, 6 k (4) k = Under favorable propagaton condtons (6) and a large excess of antennas, the sum-capacty has a smple asymptotc form, C sum_fm& = maxlogdet^i +tfdc G GD { ck} maxlogdet^i + MtfDcDbh { ck} / / c = max / log^ + Mtc f kbkh, (5) { ck} k = where c s constraned as n (4) Ths result makes ntutve sense f the columns of the propagaton matrx are nearly orthogonal, whch occurs asymptotcally as the number of antennas grows Then the transmtter could use a smple MF lnear precoder, s * / / f = GD - D b p qf, (6) M where qf s the vector of QAM symbols ntended for the termnals such that E" qf k =,, and p s a vector of powers such that / p = k = k The substtuton of (6) nto () yelds the followng: / / xf t fm D D b p qf+ wf, (7) whch translates nto an achevable sum-rate of / log M p, k ^ + tf kb = kh dentcal to the sum-capacty (5) f we dentfy p = c h Antenna and propagaton aspects of Very Large MIMO The performance of all types of MIMO systems strongly depends on propertes of the antenna arrays and the propagaton envronment n whch the system s operatng The complexty of the propagaton envronment, n combnaton wth the capablty of the antenna arrays to explot ths complexty, lmts the achevable system performance When the number of antenna elements n the arrays ncreases, we meet both opportuntes and challenges The opportuntes nclude ncreased capabltes of explotng the propagaton channel, wth better spatal resoluton Wth well-separated deal antenna elements, n a suffcently complex propagaton envronment and wthout drectvty and mutual couplng, each addtonal antenna element n the array adds another degree of freedom that can be used by the system In realty, though, the antenna elements are never deal, they are not always well separated, and the propagaton envronment may not be complex enough to offer the large number of degrees of freedom that a large antenna array could explot In ths secton, we llustrate and dscuss some of these opportuntes and challenges, startng wth an example of how more antennas n an deal stuaton mproves our capablty to focus the feld strength to a specfc geographcal pont (a certan user) Ths s followed by an analyss of how realstc (nondeal) antenna arrays nfluence the system performance n an deal propagaton envronment Fnally, we use channel measurements to address propertes of a real case wth a 8-element base staton array servng sx sngle-antenna users Spatal focus wth more antennas Precodng of an antenna array s often sad to drect the sgnal from the antenna array toward one or more recevers In a pure LOS envronment, drectng means that the antenna array forms a beam toward the ntended recever wth an ncreased feld strength n a certan drecton from the transmttng array In propagaton envronments where non-los components domnate, the concept of drectng the antenna array toward a certan recever becomes more complcated In fact, the feld strength s not necessarly focused n the drecton of the ntended recever, but rather to a geographcal pont where the ncomng multpath components add up constructvely Dfferent technques for focusng transmtted energy to a specfc locaton have been addressed n several contexts In partcular, t has drawn attenton n the form of TR where the transmtted sgnal s a tmereversed replca of the channel mpulse response TR wth sngle as well as multple antennas has been demonstrated lately n, eg, [7] and [3] In the context of ths artcle, the most nterestng case s multple-nput sngle-output, and here we speak of TR beamformng (TRBF) Whle most communcatons applcatons of TRBF address a relatvely small number of antennas, the same basc technques have been studed for almost two decades n medcal extracorporeal lthotrpsy applcatons [8] wth a large number of antennas (transducers) To llustrate how large antenna arrays can focus the electromagnetc feld to a certan geographc pont, even n a IEEE SIGNAL PROCESSING MAGAZINE [45] january 03

7 M-Element m/ ULA,600 m In Fgure, we show the resultng normalzed feld strength n a small 0m# 0m envronment around the recever to whch we focus the transmtted sgnal (usng MF precodng), for ULAs wth d = m/ of sze M = 0 and M = 00 elements The normalzed feld strength shows how much weaker the feld strength s n a certan poston when the spatal sgnature to the center pont s used rather 800 m than the correct spatal sgnature for that pont ence, the normalzed feld strength s 0 db at the center of both fgures, and negatve at all other ponts Fgure llustrates two mportant propertes of the spatal MF precodng: ) that the feld strength can be focused to a pont rather than n a certan drecton and ) that more antennas mprove the ablty to focus energy to a certan pont, whch leads to less 400 Scatterers [Fg] Geometry of the smulated dense scatterng envronment, wth 400 unformly dstrbuted scatterers n an 800# 800 m area The transmt M-element ULA s placed at a dstance of,600 m from the edge of the scatterer area wth ts broadsde pontng toward the center Two sngle scatterng paths from the frst ULA element to an ntended recever n the center of the scatterer area are shown narrowband channel, we use the smple geometrcal channel model shown n Fgure The channel s composed of 400 unformly dstrbuted scatterers n a square of dmenson 800m# 800m, where m s the sgnal wavelength The scatterng ponts (#) shown n the fgure are the actual ones used n the example below The broadsde drecton of the M-element unform lnear array (ULA) wth adjacent element spacng of d = m/ s pontng toward the center of the scatterer area Each sngle-scatterng multpath component s subject to an nverse power-law attenuaton, proportonal to dstance squared (propagaton exponent ), and a random reflecton coeffcent wth d complex Gaussan dstrbuton (gvng a Raylegh dstrbuted ampltude and a unformly dstrbuted phase) Ths model creates a feld strength that vares rapdly over the geographcal area, typcal of small-scale fadng Wth a complex enough scatterng envronment and a suffcently large element spacng n the transmt array, the feld strength resultng from dfferent elements n the transmt array can be seen as ndependent nterference between spatally separated users Wth M = 0 antenna elements, the focusng of the feld strength s qute poor wth many peaks nsde the studed area Increasng M to 00 antenna elements, for the same propagaton envronment, consderably mproves the feld strength focusng and t s more than 5 db down n most of the studed area Whle the example above only llustrates spatal MF precodng n the narrowband case, the TRBF technques explot both the spatal and temporal domans to acheve an even stronger spatal focusng of the feld strength Wth enough antennas and favorable propagaton condtons, TRBF wll not only focus power and yeld a hgh spectral effcency through spatal multplexng to many termnals It wll also reduce, or n the deal case completely elmnate, ntersymbol nterference In other words, one could dspense wth OFDM and ts redundant cyclc prefx Each base staton antenna would ) merely convolve the data sequence ntended for the kth termnal wth the conjugated, tmereversed M = 0 ULA M = 00 ULA verson of hs estmate for the 0 channel mpulse response to the kth termnal, ) sum the convolutons, - and 3) feed that sum nto hs antenna Agan, under favorable propagaton condtons, and a large number of antennas, - [db] -3 ntersymbol nterference wll decrease sgnfcantly -4 Antenna aspects # -5 It s common wthn the sgnal processng, 0 m communcatons, and nformaton (a) (b) theory communtes to assume that the transmt and receve antennas are sotropc and unpolarzed electromag- [Fg] Normalzed feld strength n a 0 # 0 m area centered around the recever to whch the beamformng s done Parts (a) and (b) show the feld strength when an M = 0 and an M = 00 ULA are used together wth MF precodng to focus the sgnal to a recever n netc wave radators and sensors, the center of the area respectvely In realty, such sotropc 0 m 800 m IEEE SIGNAL PROCESSING MAGAZINE [46] january 03

8 unpolar antennas do not exst, accordng to fundamental laws of electromagnetcs Nonsotropc antenna patterns wll nfluence the MIMO performance by changng the spatal correlaton For example, drectve antennas pontng n dstnct drectons tend to experence a lower correlaton than nondrectve antennas, snce each of these drectve antennas see sgnals arrvng from a dstnct angular sector In the context of an array of antennas, t s also common n these communtes to assume that there s neglgble electromagnetc nteracton (or mutual couplng) among the antenna elements nether n the transmt nor n the receve mode Ths assumpton s only vald when the antennas are well separated from one another In the rest of ths secton, we consder very large MIMO arrays where the overall aperture of the array s constraned, for example, by the sze of the supportng structure or by aesthetc consderatons Increasng the number of antenna elements mples that the antenna separaton decreases Ths problem has been examned n recent papers, although the focus s often on spatal correlaton and the effect of couplng s often neglected, as n [4] [6] In [7], the effect of couplng on the capacty of fxed length ULAs s studed In general, t s found that mutual couplng has a substantal mpact on capacty as the number of antennas s ncreased for a fxed array aperture It s concevable that the capacty performance n [7] can be mproved by compensatng for the effect of mutual couplng Indeed, couplng compensaton s a topc of current nterest, much drven by the desre of mplementng MIMO arrays n a compact volume, such as moble termnals (see [8] and references theren) One nterestng result s that couplng among copolarzed antennas can be perfectly mtgated by the use of optmal multport mpedance matchng RF crcuts [9] Ths technque has been expermentally demonstrated only for up to four antennas, though n prncple t can be appled to very large MIMO arrays [0] Nevertheless, the effectve cancellaton of couplng also brngs about dmnshng bandwdth n one or more output ports as the antenna spacng decreases [] Ths can be understood ntutvely n that, n the lmt of small antenna spacng, the array effectvely reduces to only one antenna Thus, one can only expect the array to offer the same characterstcs as a sngle antenna Furthermore, mplementng practcal matchng crcuts wll ntroduce ohmc losses, whch reduces the gan that s achevable from couplng cancellaton [8] Another ssue to consder s that due to the constrant n array aperture, very large MIMO arrays are expected to be mplemented n a two-dmensonal (-D) or three-dmensonal (3-D) array v s v s v s3 t t t3 Z s structure, nstead of as a lnear array as n [7] A lnear array wth antenna elements of dentcal gan patterns (eg, sotropc elements) suffers from the problem of front-back ambguty, and s also unable to resolve sgnal paths n both azmuth and elevaton owever, one drawback of havng a dense array mplementaton n -D or 3-D s the ncrease of couplng effects due to the ncrease n the number of adjacent antennas For the square array (-D) case, there are up to four adjacent antennas (located at the same dstance) for each antenna element, and n 3-D there are up to sx A further problem that s specfc to 3-D arrays s that only the antennas located on the surface of the 3-D array contrbute to the nformaton capacty [], whch n effect restrcts the usefulness of dense 3-D array mplementatons Ths s a consequence of the ntegral representaton of Maxwell s equatons, by whch the electromagnetc feld nsde the volume of the 3-D array s fully descrbed by the feld on ts surface (assumng suffcently dense samplng), and therefore no addtonal nformaton can be extracted from elements nsde the 3-D array Moreover, n outdoor cellular envronments, sgnals tend to arrve wthn a narrow range of elevaton angles Therefore, t may not be feasble for the antenna system to take advantage of the resoluton n elevaton offered by dense -D or 3-D arrays to perform sgnalng n the vertcal dmenson The complete sngle-user MIMO (SU-MIMO) sgnal model wth antennas and matchng crcut n Fgure 3 (reproduced from [3] wth permsson) s used to demonstrate the performance degradaton resultng from correlaton and mutual couplng n very large arrays wth fxed apertures In the fgure, Zt and Zr are the mpedance matrces of the transmt and receve arrays, respectvely, t and r are the exctaton and receved currents (at the th port) of the transmt and receve systems, respectvely, and vs and vr (Zs and Zl) are the source and load voltages (mpedances), respectvely, and vt s the termnal v t v t v t3 Z t G mc Transmtter Channel Recever [Fg3] Dagram of a MIMO system wth antenna mpedance matrces and matchng networks at both lnk ends (freely reproduced from [3]) Z r v r v r v r3 Z l rl r r3 IEEE SIGNAL PROCESSING MAGAZINE [47] january 03

9 voltage across the th transmt antenna port Gmc s the overall channel of the system, ncludng the effects of antenna couplng and matchng crcuts Recall that the nstantaneous capacty (from ths pont on, we shall for smplcty refer to the log- det formula wth d complex-gaussan nputs as the capacty to avod the more clumsy notaton of achevable rate ) s gven by (3) and equals [3] C t = log detcin + Gt Gt m, (8) nt mc mc mc force method [4] for a ULA consstng of three parallel dpole antennas: Zr( 005m ) = 7 9+ j j j j j j43 and 7 9+ j44-5-j98 Zr( 05m ) = - 5- j j j7 7-5-j j j43, 7 9+ j44 > 40 + j7 7-5-j j44 > where G t / / = rr ( Z + Z ) - GR -, (9) mc l l r t s the overall MIMO channel based on the complete SU-MIMO sgnal model, G represents the propagaton channel as seen by the transmt and receve antennas, and Rl = Re" Zl, Rt = Re" Zt, Note that t Gmc s the normalzed verson of Gmc shown n Fgure 3, where the normalzaton s performed wth respect to the average channel gan of a sngle-nput sngle-output (SISO) system [3] The source mpedance matrx Zs does not appear n the expresson, snce t Gmc represents the transfer functon between the transmt and receve power waves, and Zs s mplct n t [3] To gve an ntutve feel for the effects of mutual couplng, we next provde two examples of the mpedance matrx Zr, one for small adjacent antenna spacng (005m) and one for moderate spacng (05m) For a gven antenna array, Zt = Zr by the prncple of recprocty The followng numercal values are obtaned from the nduced electromotve Capacty Per Antenna Element [Bts/Channel Use] In MU-MIMO systems, the termnals are autonomous so that we can assume that the transmt array s uncoupled and uncorrelated It can be observed that the severe mutual couplng n the case of d = 005m results n off-dagonal elements whose values are closer to the dagonal elements than n the case of d = 05m, where the dagonal elements are more domnant Despte ths, the mpact of couplng on capacty s not mmedately obvous, snce the mpedance matrx s embedded n (9), and s condtoned by the load matrx Zl Therefore, we next provde numercal smulatons to gve more nsght nto the mpact of mutual couplng on MIMO performance In MU-MIMO systems, the termnals are autonomous so that we can assume that the transmt array n the reverse lnk s uncoupled and uncorrelated We remnd the reader that n MU-MIMO systems, we replace nt and nr wth and M, respectvely If the ronecker model [5] s assumed for the propagaton channel, G can be expressed as G / = Wr GIIDWt /, where W t and W r are the transmt and receve correlaton matrces, respectvely, and GIID s a matrx wth d / Raylegh entres [3] In ths case, W t = I and Zt s dagonal For the partcular case of M =, Fgure 4 shows a plot of the uplnk ergodc capacty (or average rate) per user, Cmc /, versus the antenna separaton for ULAs wth a fxed aperture of 5m at the base staton (wth up to M = = 30 elements) The correlaton but no couplng case refers to the MIMO channel G / = Wr GIIDWt /, whereas the correlaton and couplng case refers to the effectve channel matrx t Gmc n (9) The envronment s assumed to be unform -D angular power spectrum (APS) and the SNR s t = 0 db The total power s fxed and equally dvded among all users One thousand ndependent realzatons of the channel are used to obtan the average capacty For comparson, the correspondng ergodc capacty per user s also calculated for users and an M -element receve unform Correlaton and Couplng (ULA) Correlaton But No Couplng (ULA) Correlaton and Couplng (USA) Correlaton But No Couplng (USA) IID Raylegh Adjacent Element Spacng [m] [Fg4] Impact of correlaton and couplng on capacty per antenna over dfferent adjacent antenna spacng for autonomous transmtters M = and the apertures of ULA and USA are 5m and 5m# 5m, respectvely square array (USA) wth M = and an aperture sze of 5m# 5m, for up to M = 900 elements Rather than IEEE SIGNAL PROCESSING MAGAZINE [48] january 03

10 advocatng the practcalty of 900 users n a sngle cell, ths assumpton s only ntended to demonstrate the lmtaton of aperture-constraned very large MIMO arrays at the base staton to support parallel MU-MIMO channels As can be seen n Fgure 4, the capacty per user begns to fall when the element spacng s reduced to below 5m for the USAs, as opposed to below 05m for the ULAs, whch shows that for a gven antenna spacng, packng more elements n more than one dmenson results n sgnfcant degradaton n capacty performance Another dstncton between the ULAs and USAs s that couplng s n fact benefcal for the capacty performance of ULAs wth moderate antenna spacng (e, between 05m and 07m), whereas for USAs the capacty wth couplng s consstently lower than that wth only correlaton The observed phenomenon for ULAs s smlar to the behavor of two dpoles wth decreasng element spacng [8] There, couplng nduces a larger dfference between the antenna patterns (e, angle dversty) over ths range of antenna spacng, whch helps to reduce correlaton At even smaller antenna spacngs, the angle dversty dmnshes and correlaton ncreases Together wth loss of power due to couplng and mpedance msmatch, the ncreasng correlaton results n the capacty of the correlaton and couplng case fallng below that of the correlaton only case, wth the crossover occurrng at approxmately 05m On the other hand, each element n the USAs experences more severe couplng than that n the ULAs for the same adjacent antenna spacng, whch nherently lmts angle dversty Even though Fgure 4 demonstrates that both couplng and correlaton are detrmental to the capacty performance of very large MIMO arrays relatve to the d case, t does not provde any specfc nformaton on the behavor of t Gmc In partcular, t s mportant to examne the mpact of correlaton and couplng on the asymptotc orthogonalty assumpton made n (6) for a very large array wth a fxed aperture n an MU settng To ths end, we assume that the base staton serves = 5 sngle-antenna termnals The channel s normalzed so that each user termnal has a reference SNR t / = 0 db n the SISO case wth conjugatematched sngle antennas As before, the couplng and correlaton at the base staton s the result of mplementng the antenna elements as a square array of fxed dmensons 5m# 5m n a channel wth unform -D APS The number of elements n the receve USA M vares from 6 to 900, to support one dedcated channel per user The average condton number of Gt mcg t mc/ s gven n Fgure 5(a) for,000 channel realzatons Snce the Average Condton Number Average Rate Per User [Bts/Channel Use] propagaton channel s assumed to be d n (9) for smplcty, Db = I Ths mples that the condton number of Gt mcg t mc/ should deally approach one, whch s observed for the d Raylegh case By way of contrast, t can be seen that the channel s not asymptotcally orthogonal as assumed n (6) n the presence of couplng and correlaton The correspondng maxmum rate for the reverse lnk per user s gven n Fgure 5(b) It can be seen that f couplng s gnored, spatal correlaton yelds only a mnor penalty, relatve to the d case Ths s so because the transmt array of dmensons 5m# 5m s large enough to offer almost the same number of spatal degrees of freedom ( = 5) as n the d case, despte the channel not beng asymptotcally orthogonal On the other hand, for the realstc case wth couplng and correlaton, addng more receve elements nto the USA wll eventually result n a reducton of the achevable rate, despte havng a lower average condton number than n the correlaton but no couplng case Ths s attrbuted to the sgnfcant power loss through couplng and mpedance msmatch, whch s not modeled n the correlaton only case Real propagaton measured channels When t comes to propagaton aspects of MIMO as well as very large MIMO the correlaton propertes are of paramount nterest, snce those together wth the number of antennas at the termnals and base staton determnes the orthogonalty of the propagaton channel matrx and the possblty to separate dfferent users or data streams In conventonal MU-MIMO systems the rato of number of base staton antennas and antennas (a) Correlaton and Couplng Correlaton But No Couplng IID Raylegh M Correlaton and Mutual Couplng Correlaton But No Couplng IID Raylegh M (b) [Fg5] Impact of correlaton and couplng on (a) asymptotc orthogonalty of the channel matrx and (b) max sum-rate of the reverse lnk, for = 5 IEEE SIGNAL PROCESSING MAGAZINE [49] january 03

11 at the termnals s usually close to one, at least t rarely exceeds two In very large MU-MIMO systems, ths rato may very well exceed 00; f we also consder the number of expected smultaneous users,, the rato at least usually exceeds ten Ths s mportant because t means that we have the potental to acheve a very large spatal dversty gan It also means that the dstance between the null-spaces of the dfferent users s usually large, and, as mentoned before, that the sngular values of the tall propagaton matrx tend to have stable and large values Ths s also true n the case where we consder multple users where we can consder each user as a part of a larger dstrbuted, but uncoordnated, MIMO system In such a system each new user consumes a part of the avalable dversty Under certan reasonable assumptons and favorable propagaton condtons, t wll, however, stll be possble to create a full rank propagaton channel matrx (6) where all the egenvalues have large magntudes and show a stable behavor The queston s now what we mean by the statement that the propagaton condtons should be favorable? One thng s for sure as compared to a conventonal MIMO system, the requrements on the channel matrx to get good performance n very large MIMO are relaxed to a large extent due to the tall structure of the matrx It s well known n conventonal MIMO modelng that scatterers tend to appear n groups wth smlar delays, angle-ofarrvals, and angle-of-departures, and they form so-called clusters Usually the number of actve clusters and dstnct scatterers are reported to be lmted, see, eg, [6], also when the number of physcal objects s large The contrbutons from ndvdual multpath components belongng to the same cluster are often correlated whch reduces the number of effectve Prob(v # Abscssa) Meas 6 8 Meas 6 6 IID 6 8 IID Ordered Egenvalues of G G [db] [Fg6] CDFs of ordered egenvalues for a measured 6# 8 large array system, a measured 6# 6MIMO system and smulated d 6# 6and 6# 8 MIMO systems Note that for the smulated d cases, only the CDFs of the largest and smallest egenvalues are shown for clarty scatterers Smlarly t has been shown that a cluster seen by dfferent users, so called jont clusters, ntroduces correlaton between users also when they are wdely separated [7] It s stll an open queston whether the use of large arrays makes t possble to resolve clusters completely, but the large spatal resoluton wll make t possble to splt up clusters n many cases There are measurements showng that a cluster can be seen dfferently from dfferent parts of a large array [8], whch s benefcal snce the correlaton between ndvdual contrbutons from a cluster then s decreased To exemplfy the channel propertes n a real stuaton we consder a measured channel matrx where we have an ndoor 8-antenna base staton consstng of four stacked double polarzed 6-element crcular patch arrays, and sx sngleantenna users Three of the users are ndoors at varous postons n an adjacent room and three users are outdoors but close to the base staton The measurements were performed at 6 Gz wth a bandwdth of 50 Mz In total, we consder an ensemble of 00 snapshots (taken from a contnuous movement of the user antenna along a 5 0 m lne) and 6 frequency ponts, gvng us n total 6,00 narrow-band realzatons It should be noted, though, that they are not fully ndependent due to the nonzero coherence bandwdth and coherence dstance The channels are normalzed to remove large-scale fadng and to mantan the small-scale fadng The mean power over all frequency ponts and base staton antenna elements s unty for all users In Fgure 6, we plot the cumulatve dstrbuton functons (CDFs) of the ordered egenvalues of G G (the left-most sold curve corresponds to the CDF of the smallest egenvalue, etc) for the 6 # 8 propagaton matrx ( Meas 6 # 8 ), together wth the correspondng CDFs for a 6# 6 measured conventonal MIMO ( Meas 6 # 6 ) system (where we have used a subset of sx adjacent copolarzed antennas on the base staton) As a reference, we also plot the dstrbuton of the largest and smallest egenvalues for a smulated 6 # 8 and 6# 6 conventonal MIMO system ( d 6 # 8 and d 6# 6 ) wth d complex Gaussan entres Note that, for clarty of the fgure, the egenvalues are not normalzed wth the number of antennas at the base staton and therefore there s an offset of 0 log0 ( M) Ths offset can be nterpreted as a beamformng gan In any case, the relatve spread of the egenvalues s of more nterest than ther absolute levels It can be clearly seen that the large array provdes egenvalues that all show a stable behavor (low varances) and have a relatvely low spread (small dstances between the CDF curves) The dfference between the smallest and largest egenvalue s only around 7 db, whch could be IEEE SIGNAL PROCESSING MAGAZINE [50] january 03

12 compared wth the conventonal 6# 6MIMO system where ths dfference s around 6 db Ths egenvalue spread corresponds to that of a 6# 4 conventonal MIMO system wth d complex Gaussan channel matrx entres eepng n mnd the crcular structure of the base staton antenna array and that half of the elements are cross polarzed, ths number of effectve channels s about what one could antcpate to get One mportant factor n realstc channels, especally for the uplnk, s that the receved power levels from dfferent users are not equal Power varatons wll ncrease both the egenvalue spread and the varance and wll result n a matrx that stll s approxmately orthogonal, but where the dagonal elements of G Ghave varyng mean levels, specfcally, the D b matrx n (6) Transcevers We next turn our attenton to the desgn of practcal transcevers A method to acqure CSI at the base staton begns the dscusson Then we dscuss precoders and detecton algorthms sutable for very large MIMO arrays Acqurng CSI at the base staton To do multuser precodng n the forward lnk and detecton n the reverse lnk, the base staton must acqure CSI Let us assume that the frequency response of the channel s constant over NCoh consecutve subcarrers Wth small antenna arrays, one possble system desgn s to let the base staton antennas transmt plot symbols to the recevng unts The recevng unts perform channel estmaton and feed back, partal or complete, CSI va dedcated feedback channels Such a strategy does not rely on channel recprocty (e, the forward channel should be the transpose of the reverse channel) owever, wth a lmted coherence tme, ths strategy s not vable for large arrays The number of tme slots devoted to plot symbols must be at least as large as the number of antenna elements at the base staton dvded by NCoh When M grows, the tme spent on transmttng plots may surpass the coherence tme of the channel Consequently, large antenna array technology must rely on channel recprocty Wth channel recprocty, the recevng unts [TABLE ] SNR and SINR expressons for a collecton of standard precodng technques SNR and SINR expressons as M, " 3, M / = a Precodng Technque Benchmark: Intererence-free system Perfect CSI tfa Zero Forcng tf( a - ) Matched Flter Vector Perturbaton Imperfect CSI ta f p tfa tf + tf + tar f - a c - m, a M 79 6 a N/A To do multuser precodng n the forward lnk and detecton n the reverse lnk, the base staton must acqure CSI p tf( a - ) ( - p ) tf + send plot symbols va TDD Snce the frequency response s assumed constant over NCoh subcarrers, NCoh termnals can transmt plot symbols smultaneously durng one OFDM symbol nterval In total, ths requres NCoh / tme slots (we remnd the reader that s the number of termnals served) The base staton n the kth cell constructs ts channel estmate Gt T kk, subsequently used for precodng n the forward lnk, based on the plot observatons The power of each plot symbol s denoted tp Precodng n the forward lnk: Collecton of results for sngle-cell systems User k receves the kth component of the composte vector T xf = G s f + w f The vector sf s a precoded verson of the data symbols qf Each component of sf has average power t f /M Further, we assume that the channel matrx G has d CN (,) 0 entres In what follows, we derve SNR/sgnal-to-nterference-plus-nose rato (SINR) expressons for a number of popular precodng technques n the large system lmt, e, wth M, " 3, but wth a fxed rato a = M / The obtaned expressons are tabulated n Table Let us frst dscuss the performance of an ntererence-free (IF) system that wll subsequently serve as a benchmark reference The best performance that can be magned wll result f all the channel energy to termnal k s delvered to termnal k wthout any nteruser nterference In that case, termnal k receves the sample xfk x / M fk = g, k qfk+ wfk, = M Snce `/ g k M ", M " 3,, =, j and E" qfkqf k, = tf /, the SNR per recevng unt for IF systems converges to ta f as M " 3 We now move on to practcal precodng methods The conceptually smplest approach s to nvert the channel by means of the pseudonverse Ths s referred to as zero-forcng (ZF) precodng [9] A varant of zero forcng s block dagonalzaton [30], whch s not covered n ths artcle Intutvely, when M grows, G tends to have nearly orthogonal columns as the termnals are not correlated due to ther physcal separaton Ths assures that the performance of ZF precodng wll be close to that of the IF system owever, a dsadvantage of ZF s that processng cannot be done dstrbutedly at each antenna separately Wth ZF precodng, all data must nstead be collected at a central node that handles the processng Formally, the ZF precoder sets s G q f = = G GG q c c T + ) T ) - ( ) f ( ) f, where the superscrpt + denotes the pseudonverse of a matrx, T + e, ( G ) = G ) T ( G G ) - ), and c normalzes the average power n IEEE SIGNAL PROCESSING MAGAZINE [5] january 03