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1 Postpnt Ths s the accepted veson of a pape publshed n IEEE Tansactons on Vehcula Technology. Ths pape has been pee-evewed but does not nclude the fnal publshe poof-coectons o jounal pagnaton. Ctaton fo the ognal publshed pape (veson of ecod): Bandt, R., Bengtsson, M. () Dstbuted CSI Acquston and Coodnated Pecodng fo TDD Multcell MIMO Systems. IEEE Tansactons on Vehcula Technology Access to the publshed veson may eque subscpton. N.B. When ctng ths wo, cte the ognal publshed pape. Pemanent ln to ths veson:

2 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL Dstbuted CSI Acquston and Coodnated Pecodng fo TDD Multcell MIMO Systems Rasmus Bandt, Student Membe, IEEE, and Mats Bengtsson, Seno Membe, IEEE Abstact Seveal dstbuted coodnated pecodng methods exst n the downln multcell MIMO lteatue, many of whch assume pefect nowledge of eceved sgnal covaance and local effectve channels. In ths wo, we let the noton of channel state nfomaton (CSI) encompass ths nowledge of covaances and effectve channels. We analyze what local CSI s equed n the WMMSE algothm fo dstbuted coodnated pecodng, and study how ths equed CSI can be obtaned n a dstbuted fashon. Based on plot-asssted channel estmaton, we popose thee CSI acquston methods wth dffeent tadeoffs between feedbac and sgnalng, bachaul use, and computatonal complexty. One of the poposed methods s fully dstbuted, meanng that t only depends on ove-the-a sgnalng but eques no bachaul, and esults n a fully dstbuted jont system when coupled wth the WMMSE algothm. Naïvely applyng the WMMSE algothm togethe wth the fully dstbuted CSI acquston esults n catastophc pefomance howeve, and theefoe we popose a obustfed WMMSE algothm based on the well nown dagonal loadng famewo. By enfocng popetes of the WMMSE solutons wth pefect CSI onto the poblem wth mpefect CSI, the esultng dagonally loaded spatal fltes ae shown to pefom sgnfcantly bette than the naïve fltes. The poposed obust and dstbuted system s evaluated usng numecal smulatons, and shown to pefom well compaed wth benchmas. Unde centalzed CSI acquston, the poposed algothm pefoms on pa wth othe exstng centalzed obust WMMSE algothms. When evaluated n a lage scale fadng envonment, the pefomance of the poposed system s pomsng. I. INTRODUCTION MULTIPLE-ANTENNA coodnated pecodng s a pomsng technque fo mpovng spectal effcency n multcell multple-nput multple-output (MIMO) netwos, by sevng seveal spatally sepaated uses smultaneously n the same tme/fequency esouce bloc [], []. The cascade of physcal channels and pecodes ae the effectve channels expeenced by the eceves. By sutably selectng the pecodes, the downln weghted sum ate of the netwo can be maxmzed. The equements fo pactcal mplementaton of coodnated pecodng nclude channel estmaton [3] [], obustness aganst channel estmaton eos [6] [], and suffcently low complexty; pefeably acheved usng dstbuted methods [] [4]. In the multcell MIMO lteatue, thee ae seveal examples of dstbuted coodnated pecodng methods; see e.g. [] and Copyght (c) IEEE. Pesonal use of ths mateal s pemtted. Howeve, pemsson to use ths mateal fo any othe puposes must be obtaned fom the IEEE by sendng a equest to pubs-pemssons@eee.og. The authos ae wth the Depatment of Sgnal Pocessng, School of Electcal Engneeng, KTH Royal Insttute of Technology, SE- 44 Stocholm, Sweden. E-mals: ab4@th.se, mats.bengtsson@ee.th.se. efeences theen. These methods typcally eque nfomaton about the eceved sgnal covaance and local effectve channels at the nvolved nodes, and t s often assumed that ths nfomaton s pefectly nown. In ths wo, we denote the nfomaton about the eceved sgnal covaance and effectve channels as channel state nfomaton (CSI). We tae a systems pespectve and popose methods fo estmatng and acqung the necessay CSI at the nvolved nodes n a dstbuted fashon. The esouce allocaton s based on the WMMSE algothm [] fo dstbuted weghted sum ate optmzaton, because of ts low pe-teaton complexty and tactable fom. Due to poo pefomance when naïvely applyng the WMMSE algothm, we also popose some obustfyng pocedues, leadng to a obust and fully dstbuted jont coodnated pecodng and CSI acquston system. As the fst step n the system desgn, we succnctly descbe what nfomaton, n tems of weghts and CSI, that s needed fo the nodes of the netwo to pefom the pat n the WMMSE algothm. Thee s a multtude of concevable methods to obtan the necessay nfomaton at the nodes, e.g. usng vaous combnatons of channel estmaton, feedbac, sgnalng, bachaul, etc. We popose thee methods fo acqung the necessay CSI. Based on channel estmaton though plot tansmssons, feedbac, sgnalng, and bachaul use, the poposed CSI acquston methods coespond to dffeent tadeoffs between these technques. In patcula, one of the poposed CSI acquston methods s fully dstbuted, n the sense that the nodes of the netwo solely coopeate by means of ove-the-a sgnalng, thus equng no bachaul. A ey component of the poposed CSI acquston methods s the estmaton of the effectve channels. It s based on synchonous plot tansmsson n the downln, enablng the ecevng use equpments (UEs) to estmate both desed and ntefeng effectve channels [4]. Assumng tme-dvson duplex (TDD) opeaton and pefectly calbated tansceves [], [6], smla channel estmaton can be pefomed n the upln at the ecevng base statons (BSs). Ths s contay to fequency-dvson duplex opeaton, whee the BSs obtan the equed nfomaton by feedbac and bachaul sgnalng. When combnng the fully dstbuted CSI acquston wth the WMMSE algothm, the jont system s fully dstbuted. Naïvely applyng the ognal WMMSE algothm togethe wth the poposed fully dstbuted CSI acquston method leads to catastophc pefomance howeve. Ths s because the ognal algothm was not developed to be obust aganst The algothm taes ths name snce t s a Weghted Mnmzaton of the Mean Squaed Eo (WMMSE).

3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL mpefect CSI. We theefoe popose a obustfed WMMSE algothm whch etans the dstbutedness of the ognal algothm, contay to state of the at [7] []. We fomulate a wost-case WMMSE poblem, and solve an uppe bounded veson of the poblem. The esultng pecodes ae dagonally loaded, a technque whch s well nown fo ts obustfyng effect on beamfomes [7] [3]. The optmal amount of dagonal loadng s detemned by the wost-case channel estmaton eos, whose statstcs ae unfotunately unavalable n the poposed CSI acquston setup. Instead, we popose a pactcal method fo mplctly selectng the amount of dagonal loadng fo the pecodes. At the UEs, we show an nheent popety of the (spatal) eceve fltes and mean squaed eo (MSE) weghts obtaned fom the WMMSE algothm wth pefect CSI. When ths popety s enfoced onto the fltes wth mpefect CSI, the esultng eceve fltes ae also dagonally loaded. The obust MSE weghts have smalle egenvalues than the non-obust MSE weghts. Ths can be ntepeted as the eceves equestng lowe data ates when thee ae lage dscepances n the estmated CSI. A. Related Wo In [4], a ecpocal channel was exploted to dectly estmate the fltes maxmzng the sgnal-to-ntefeence-andnose ato, equng no othe sgnalng. Smla wo was pefomed n [4], whee non-lnea fltes also wee studed. ocusng on the ecpocty, and usng the eceve fltes as tansmt fltes n the upln, [] pefomed extensve smulatons fo a beam selecton appoach. Ou poposed effectve channel estmaton s smla to the busy bust methodology. o the sngle-steam multple-nput sngle-output (MISO) ntefeence channel, an analytcal method fo fndng the atemaxmzng zeo-focng beamfomes was deved n [3]. Salently, the method does not eque coss-ln CSI, and was consequently shown to be hghly obust aganst CSI mpefectons. In [4], decentalzed algothms based on WMMSE deas wee poposed, achevng faste convegence than the ognal WMMSE algothm n [], n addton to sgnalng stateges fo obtanng the necessay CSI. TDD ecpocty was assumed, and the UEs used combnatons of nte-cell and nta-cell effectve channel plot tansmssons. Contay to ou wo, pefect channel estmaton was assumed, and the decentalzed algothms stll eque some BS bachaul. Weghted sum ate maxmzaton by means of weghted MMSE mnmzaton was ognally poposed fo multuse MIMO systems n [6], whee the MSE weghts wee used to equate the Kaush-Kuhn-Tuce (KKT) condtons of the weghted MMSE poblem to the KKT condtons of the weghted sum ate poblem. Ths same method was dectly appled to multcell MIMO systems n [7], [7], but t was not untl [] that a goous connecton to the multcell weghted sum ate poblem was pesented. An eale wo s [6], whee the weghted MMSE optmzaton poblem was solved usng the same technque, but wthout explctly povdng the goous connecton to the weghted sum ate poblem. In [6], a obust WMMSE algothm was also suggested fo the case of nom bounded channel uncetanty asng fom lmted quantzed feedbac. Othe obustfed vesons of the WMMSE algothm, whee the contbuton of the downln channel estmaton eos n the nvolved covaance matces was aveaged out, wee poposed n [7], [8]. The same appoach was taen n [9], whee t was mentoned that ths coesponds to optmzng a lowe bound on the acheved pefomance, and n [] whee the lowe bound was explctly deved. The fltes wee n effect obustfed by dagonal loadng, whee the dagonal loadng factos wee detemned by the downln channel estmaton pefomance. The wo n [7] [] was manly focused on poposng obust WMMSE methods and thus the actual CSI acquston was not conclusvely studed, contay to ths pape. The majo assumpton n the system model of [7] [] s that downln channel estmaton s pefomed at the UEs, and that the downln channel estmates ae fed bac to the BSs. In ths wo we ae nteested n TDD channel estmaton and although the algothms n [7] [] could be appled n such a settng, dong so leads to some dosyncases that wll be detaled n Sec. IV-E. Due to the system model n [7] [], the nodes of the netwo eque feedbac of all fltes n all teatons of the algothm, leadng to a lage amount of feedbac whch would typcally be mplemented usng a centalzed CSI acquston nfastuctue. In ths pape, contay to [7] [], we ncopoate a detaled analyss of the CSI acquston component of the system, leadng up to a obust and dstbuted coodnated pecodng system. B. Contbutons The majo contbutons of ths wo ae as follows. We succnctly descbe the equed nfomaton fo the netwo nodes to pefom one WMMSE teaton. We popose thee CSI acquston methods whch povde the necessay nfomaton. The methods have vayng levels of dstbutedness and sgnalng needs. One of the poposed methods s fully dstbuted, meanng that t can be mplemented entely by ove-the-a sgnalng. o eslent pefomance aganst channel estmaton eos, we popose a obustfed, but stll dstbuted, WMMSE algothm to be appled togethe wth the poposed CSI acquston schemes. The obustness s due to dagonal loadng, and the level of dagonal loadng fo the pecodes s detemned mplctly by a pactcal pocedue. We dentfy and exploe new nheent popetes of the WMMSE algothm. When the popetes ae explctly enfoced onto solutons wth mpefect CSI, the esultng eceve fltes ae dagonally loaded. Pefomance s evaluated numecally, and t s shown that the poposed fully dstbuted system pefoms excellently compaed wth the naïve WMMSE algothm wth fully dstbuted CSI acquston. Wth centalzed CSI acquston, the poposed obust WMMSE algothm pefoms on pa wth exstng obust WMMSE algothms, whch howeve eque centalzed CSI acquston.

4 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS 3 C. Notaton The opeatons ( ), ( ) H, ( ) T ae complex conjugate, Hemtan tanspose, and egula tanspose, espectvely. The opeatos T ( ),, ae the matx tace, Eucldean nom and obenus matx nom, espectvely. We denote the patal odeng of postve (negatve) semdefnte matces as ( ). The mth lagest egenvalue (sngula value) of Q s denoted m (Q) (s m (Q)). The zeo-mean and covaance Q complex symmetc Gaussan dstbuton s CN(, Q), and E ( ) denotes expectaton. Estmated quanttes ae denoted wth a hat ba and upln quanttes wth an aow a. The Konece delta s,j. II. DISTRIBUTED WEIGHTED SUM RATE OPTIMIZATION OR THE MULTICELL MIMO DOWNLINK Ou system model s a multcell system wth K t BSs, each sevng K c UEs, fo a total of K K t K c UEs. We ndex the BSs as {,...,K t }. The th seved UE of BS s ndexed by the pa of ndces (, ). o compactness, we wll often wte ths pa of ndces as. The system s opeatng usng coodnated pecodng,.e. each UE s only seved data fom one BS and the sgnals fom the othe BSs consttute ntecell ntefeence. When K c, nta-cell ntefeence s also obseved. The BSs ae equpped wth M t antennas each, the UEs have M antennas and ae seved N d data steams each 3. Communcaton taes place both n the downln and n the upln. We focus on optmzng pefomance n the downln, snce that typcally expeences heave taffc loads than the upln. The pesented method could equally well be appled n the upln howeve. In the downln, the multuse nteacton s descbed by the ntefeng boadcast channel. Denote a ealzaton of the flat-fadng MIMO channel between BS j and UE as H j and let each use s data sgnal x CN (, I Nd ) be lnealy pecoded by V C Mt N d. The eceved sgnal at UE s then y H V x + X H jv jl x jl + z, () (j,l)6(,) whee the last tem s a whte Gaussan nose tem z CN, I M. The sgnals {x } and {z } ae..d. ove uses. Gven these assumptons, the eceved ntefeence plus nose covaance matx fo UE s +n P (j,l)6(,) H jv jl Vj H l H H j + I. Assumng that the decodes n the UE temnals teat ntefeence as addtve nose, the achevable downln data ate fo UE s R log det I + V H H H +n H V. () Note that () s non-convex n {V }, snce the pecodes +n appea nsde. Ths non-convex dependence on the pecodes descbes the couplng between uses, and wll be the ey challenge n the optmzaton to come. Snce the focus of ths pape s the dstbuted mplementaton of multcell pocessng, we do not nvestgate jont tansmsson, whee seveal BSs jontly seve the UEs wth data. Such jont tansmsson eques sgnfcant bachaul between BSs, and s not amenable to fully dstbuted mplementaton. 3 The system model can easly be extended to scenaos whee the BSs seve dffeent numbe of UEs each and scenaos whee the nodes have dffeent numbe of antennas. One man assumpton n ths wo s that thee s a pefectly ecpocal upln channel avalable. That s, the channel n the upln fom UE j l to BS s H jl H T j l. Let x CN, be the tansmtted sgnal fom UE n the upln. The upln s descbed by the ntefeng multple access channel, and the eceved sgnal fo BS s then XK c XK t XK c y H T x + H T j l x j l + z, (3) j6 l whee z CN, t I Mt. o convenence, we wo wth the complex conjugate veson of the eceved sgnal. That s, the model we wll use fo the upln s: y y K c X XK t XK c H H x + H H j l x jl + z. (4) j6 l Wth the upln model n (4), the channel estmaton n Sec. III can be taloed to the needs of the weghted sum ate optmzaton, whch we detal n the next secton. A. Weghted Sum Rate Optmzaton Snce the CSI acquston to be poposed s taloed fo the WMMSE algothm [], we now befly summaze the algothm, as well as ntoduce some necessay notaton. By assgnng the UEs data ate weghts [, ], 8, the weghted sum ate s fomulated as P (,) R. Ths fomulaton descbes the ultmate pefomance of the system, but s just one way of fomng a system-level utlty fom the use ates []. The data ate weghts can be selected coespondng to use poty, e.g. to acheve a popotonally fa soluton [8]. In the followng, we wll assume that the weghts ae selected at the BSs. Let P be the sum powe constant fo BS. Wth the pecodes {V } as optmzaton vaables, the weghted sum ate optmzaton poblem s: X maxmze R {V } subject to (,) XK c T V V H apple P,,...,K t. Due to the non-convexty of (), ths s a non-convex optmzaton poblem. At least when M, the poblem s also NP-had [9]. We can theefoe only easonably stve to fnd a locally optmal soluton. By ntoducng addtonal optmzaton vaables {W } (actng le MSE weghts), t was shown n [] that () has the same global solutons as the followng weghted MMSE optmzaton poblem: mnmze {A },{V } {W } subject to X (T (W E ) log det (W )) (,) XK c T V V H apple P,,...,K t. () (6)

5 4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL Downln Upln TABLE I SUMMARY O CSI QUANTITY SHORTHANDS H V H + +n +n P (j,l)6(,) H jv jl Vj H l H H j + I G H H U s+ P (j,l) HH j l U j l U H j l H H j l TABLE II QUANTITIES THAT MUST BE SIGNALED IN ORDER OR EACH NODE TO PERORM ONE ITERATION O THE WMMSE ALGORITHM Covaance matx Effectve channel(s) Weght(s) UE BS {G } Kc {W / } Kc Note that the use potes { } Kc ae selected, and thus fully nown, at the sevng BSs. The {A } ae lnea eceve fltes, and E E x A H y x A H H y (7) I A H H V V H H H A + A H A s the MSE matx fo UE. uthe, H V V H H H + +n s the eceved sgnal and ntefeence plus nose covaance matx fo UE. The optmzaton poblem n (6) s stll non-convex ove the jont set {A, W, V }, but the ey beneft of (6) ove () s that t s ndependently convex n the blocs of vaables {A }, {W }, and {V }, when the emanng blocs ae ept fxed. uthe, a statonay pont can be found though altenatng mnmzaton 4 [3, Ch..7] ove the blocs []. Thee s a one-to-one coespondence between the statonay ponts of () and the statonay ponts of (6) [], and snce (6) optmzes a locally tght lowe bound of (), altenatng mnmzaton of (6) wll also convege to a statonay pont of () [3]. ) WMMSE Algothm fo Dstbuted Weghted Sum Rate Optmzaton: st, by fxng {W, V } n (6), t can easly be shown that the poblem decouples ove the UEs. The soluton s the well nown MMSE eceve A H V, 8. (8) Next, by fxng {A, V }, the poblem agan decouples ove the UEs. UE should theefoe solve mn W T (W E ) log det (W ), and the solutons ae W E I V H H H H V, 8, (9) whee the last equalty comes fom pluggng n A fom (8). nally, t emans to solve (6) fo {V }, whle eepng the UE vaables {A, W } fxed. The poblem decouples ove the BSs, and t can be shown that the emanng poblem 4 Ths technque s also nown as bloc coodnate descent o bloc nonlnea Gauss-Sedel n the lteatue. Algothm WMMSE Algothm [] (Pefect CSI) : epeat At UEs: : W I H 3: A, U p A W / At BSs: 4: nd µ whch satsfes P K c T V V H apple P : B ( + µ I) G, V p B W / 6: untl convegence cteon met, o fxed numbe of tes. fo BS s a quadatcally constaned quadatc pogam wth optmzaton vaables {V } Kc. The soluton s [] V ( + µ I) H H A W, 8, () whee s+ P (j,l) j l H H j l A j l W jl A H j l H jl s a sgnal plus ntefeence covaance matx fo BS n the upln. If P K c T V V H apple P s satsfed fo µ, the sum powe constant fo BS s nactve and the poblem s solved. Othewse, µ > s found such that P K c T V V H P holds. Ths can be done effcently usng e.g. bsecton []. When the pecodes have been found, a new teaton s commenced by agan optmzng ove {A }. Wth each update of {A }, {W } o {V }, the objectve value n (6) cannot ncease. The teatons thus contnue untl convegence, o fo a fxed numbe of teatons. ) Requed Local Infomaton fo the WMMSE Iteatons: In ode to clafy what nfomaton the CSI acquston schemes should povde, we ntoduce some shothands fo the quanttes nvolved n the WMMSE algothm. o UE, we defne a weghted eceve flte as U p A W / and denote the effectve downln channel as H V. The eceve flte can then be wtten as A. Symmetcally, n the upln fo UE, the pecode s V p B W / and the effectve upln channel s G H H U. nally, the component pecode s B ( + µ I) G. We summaze the shothands n Table I, and the WMMSE algothm wtten usng these shothands n Algothm. The WMMSE algothm opeates n two phases: one n whch the UEs fom the eceve fltes and weghts, and one n whch the BSs fom the pecodes fo the seved UEs. The optmzaton steps at the UEs and BSs ae completely decoupled, and as summazed n Table II, the nodes only eque local CSI and local weghts. Hence, the WMMSE algothm s an example of dstbuted esouce allocaton. In Sec. III, we wll descbe how the nodes can explot the channel ecpocty to obtan local CSI n a dstbuted fashon. III. DISTRIBUTED CSI ACQUISITION Accodng to Table II, the UEs eque nowledge about the effectve channel fom the sevng BSs, as well as the sgnal and ntefeence plus nose covaance matx. The BSs need to now the effectve upln channels {G } Kc to the UEs they seve, the coespondng MSE weghts {W / } Kc, and the upln sgnal plus ntefeence covaance matx. Seveal methods fo obtanng the

6 ... UE... UEs estmate and BS BS UE UE KtKc... UE BS Kt BS Kt UE UE UE KtKc... UE... BS BS c BSs estmate {G }K and Upln tanng V P... Kc X... BSs tansmt... Downln tanng BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS UEs tansmt U P g.. CSI estmaton n one subfame (cf. g. ). In each subfame, the downln channels ae estmated usng plots fom the BSs. Late, the upln plots ae estmated usng plots fom the UEs. Addtonally, the UEs feed bac W to the sevng BS usng an out-of-band feedbac ln. Subfame n- Upln plots Upln data BSs Subfame n Downln plots Downln data Guad tme UEs Subfame n+ Upln plots Upln data Downln plots BSs Downln data UEs g.. Schematc dawng of subfames. equed CSI at the nodes can be magned, usng vaous combnatons of channel estmaton, feedbac, sgnalng and bachaul. In ths secton, we wll popose thee CSI acquston methods, wth dffeent tadeoffs between these aspects. The channel estmaton n the poposed methods explots the ecpocty of the netwo, and uses plot tansmssons n both upln and downln. As the effectve channels change between teatons n the WMMSE algothm, we popose to pefom a tanng phase between one teaton and the next. A schematc dawng of the subfame stuctue that we envson can be seen n g.. The subfame s splt between plot tansmsson and data tansmsson, n both the upln and downln. Data tansmsson thus taes place between the flte updates of the algothm. The ato between upln and downln data tansmsson lengths could be flexbly allocated [3]. Befoe the teatve algothm has conveged, the data ates that ae achevable n the downln data tansmsson phase may be low, but not neglgble, as shown by the numecal esults n Sec. V-A. An llustaton of the channel estmaton n one subfame s shown n g.. In bloc fadng channels, the coheence nteval should be suffcently long such that the teatve algothm can pefom enough teatons to each good pefomance. The deployment scenao wll detemne the coheence tme of the channel, and the detals of the fame stuctue wll detemne the numbe of subfames that can be tansmtted wthn one coheence nteval. As a bef example, unde a bloc fadng channel wth cae fequency fc GHz and UE speed v 3 m/h, the coheence tme can be modeled as Tc fc vc 9 ms [33]. o futue G systems, the TDD swtchng peodcty s planned to be ms o less [3], [34], leadng to at least We emnd the eade that ou noton of CSI encompasses nowledge of the effectve channels and the covaance matces; see Tables I and II. 9 upln-downln teatons n one coheence nteval when the UEs ae slowly movng. In contnuous fadng channels, the poposed algothm would possbly nstead be able to tac the channel vaatons, assumng that they ae slow enough. In the est of the pape, we mae the assumpton that the channel s changng slowly enough fo the teatve algothm to each adequate pefomance. We now detal the dffeent CSI acquston methods, whch all ely on plot-asssted channel estmaton. When a statstcal chaactezaton of the channel s avalable, the MMSE channel estmato [] s typcally used. Hee we estmate the effectve channels, whch ae updated n each WMMSE teaton based on the cuent channel condtons. Obtanng a statstcal chaactezaton of the effectve channel s thus complcated. In the estmaton, we theefoe egad the effectve channels as detemnstc but unnown. Unde ths pespectve fom classcal estmaton theoy, t s easy to fnd the mnmum vaance unbased (MVU) estmato. A. ully Dstbuted CSI Acquston st, we see to estmate the effectve downln channel H V usng synchonous plot tansmssons. In the downln tanng phase, the BSs tansmt othogonal plot sequences6 P CNd Np,d pe use, such that P PH jl Np,d INd,jl. In ode to fulfll the othogonalty equement, Np,d K Nd. The eceved sgnal Y CM Np,d at UE s then X Y H V P + H j Vjl Pjl + Z. () (j,l)6(,) 6 The famewo can be extended to allow fo non-othogonal plots, but then a plot allocaton scheme must be set up to mnmze the poblem of plot contamnaton [3]. uthemoe, the esouce allocaton step should tae the plot contamnaton nto account.

7 6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL Notce that the powe allocated to the plots s the same as the powe allocated to the data symbols n (). Ths wll enable dstbuted and unbased estmaton of. Ths type of plot tansmssons, ntended to estmate the effectve channels, ae called UE-specfc efeence sgnals n the LTE standad [36]. Assumng that UE nows ts desgnated plot P, ths s a detemnstc paamete estmaton poblem n Gaussan nose. The MVU estmato of the effectve channel C M N d s then []: b N p,d Y P H H V + N p,d Z P H. () The MVU estmato s an unbased, effcent and asymptotcally consstent (n N p,d ) estmato of. In addton to nowng, UE also needs nowledge of C M M. Ths can be acheved by applyng the sample covaance estmato: b N p,d Y Y H X H jv jl Vj H l H H j + Z Z H N p,d (j,l) + X H jv jl P jl Z H N + Z P H j l Vj H l H H j. p,d (j,l) (3) Snce the only stochastc component of Y s Z, the estmato n (3) s unbased. The upln estmaton s pefomed n a smla manne as the downln estmaton. Now the UEs each tansmt a sgnal X U P, whee P C N d N p,u ae othogonal plots, such that P P H j l N p,u I Nd,j l. As wll be shown by Poposton n Sec. IV-D, U A W / apple N d /. In ode to maxmze the upln estmaton SNR, the scalng facto s set as 7 p P /N d, whee P s the maxmum tansmt powe of the UEs. The UE quanttes P and ae assumed to be nown at the BSs, such that they have pefect a ponowledge of. o ths setup, assumng synchonzed plot tansmssons fom the UEs, the eceved sgnal Y C Mt Np,u at BS dung the upln tanng phase s Y XK c H H U P + XK t K c X H H j l U jl P jl + Z. j6 l (4) The MVU estmato of the upln effectve channel G C Mt N d s bg N p,u Y P H H H U + N p,u Z P H. () uthemoe, the sgnal and ntefeence plus scaled nose s++n covaance matx C Mt Mt s estmated usng the sample covaance: bs++n Y Y H. (6) N p,u 7 Note that the UE dependent facto n U should not be emoved by the scalng, snce then cannot be estmated n a fully dstbuted fashon. If <, the full tansmt powe of UE cannot be used. The WMMSE algothm howeve needs an estmate of s+ s++n, wthout the nose covaance component of. In Sec. IV-C, we esolve ths ssue by modfyng the WMMSE algothm. When fomng the pecode n (), the poduct p H H A W G W / s needed. Instead of ndependently estmatng ths quantty n a second upln estmaton phase, we let UE feed bac W to ts sevng BS. Togethe wth (), BS can then fom b G W / and use that n (). The pont of ths pocedue s to avod sgnal cancelaton [37], whee a small msmatch between the estmate of G W / and the estmate of can have a lage detmental mpact on pefomance. If G and ae estmated usng the same plot tansmssons, as n () and (6), the covaance matx can be decomposed as b s++n b+n + G b G b H. Because of ths stuctue, thee s no msmatch between G b W / and b, and sgnal cancelaton does not occu [37]. It can be shown that R log det (W ). eedbac of the egenvalues of W theefoe consttutes a ate equest fo each data steam of UE, descbng what ate that steam can handle unde the cuent netwo condtons. Ths nfomaton s aleady fed bac to the sevng BS n a pactcal system. Recall that s fxed and nown at BS, and does theefoe not need to be fed bac. Rema. The CSI acquston poposed n ths secton s fully dstbuted ove BSs and UEs, n the sense that only ove-the-a sgnalng s equed. UE feeds bac W to ts sevng BS, but the BSs do not need to shae any nfomaton ove a BS bachaul. B. CSI Acquston wth Global Shang of Indvdual Scalng Paametes As noted n the pevous secton, and poved n Sec. IV-D, U apple N d /. The scalng facto was set based on ths to maxmze the upln tansmt powe. Howeve, unless the nequalty s met wth equalty and, the tansmt powe constant of that patcula UE s not met. Coespondngly, the upln estmaton SNR suffes fo that UE. If the equement of fully dstbuted estmaton of the upln covaance matx s dopped, and by ntoducng ndvdual scalng factos fo the UEs, the maxmum upln tansmt powe can always be used. In ths secton, we eep the downln estmaton the same as n Sec. III-A, but modfy the upln estmaton to maxmze the tansmt powe used. The BSs wll then need access to a bachaul netwo, whee nfomaton about the ndvdual scalng paametes can be shaed. Lettng X p P U U P, the effectve upln tansmt powe s maxmzed fo UE. The eceved sgnal at BS s then Y p X P H H j U jl l U jl P jl + Z. (7) (j,l) We now assume that the ndvdual scalng factos U ae fed bac fom the UEs to the sevng BSs, and then globally

8 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS 7 TABLE III EEDBACK AND ESTIMATION NEEDED OR THE DIERENT CSI ACQUISITION METHODS. Method Estmated at UE BS feedbac to seved UE Estmated at BS UE feedbac to sevng BS Shaed nfomaton ove BS bachaul ully dstbuted (Sec. III-A), Globally shaed ndvdual scale factos (Sec. III-B), {G } W, {G jl } U, W { U jl } Globally shaed fltes (Sec. III-C) {H j}, {V jl } {H jl } U, W {U jl }, {W jl }, {V jl } The use potes only need to be fed bac f/when they ae changed. The estmated quanttes fo the methods n Sec. III-A and Sec. III-B depend on the tansmt and eceve fltes, and must theefoe be e-estmated n evey subfame. The estmated quanttes fo the method n Sec. III-C do not change wthn one coheence bloc, and can theefoe be mpoved upon n evey subfame. shaed ove a BS bachaul. BS can then estmate the effectve channels fom UE j l as bg jl U j l pp N p,u Y P H j l H H ju jl + U j l pp N p,u Z P H j l. (8) Snce the scaled plots effectvely all have the same weght, the sample covaance estmato of n (6) cannot be used. Instead, we ely on the based estmato bs++n P b (j,l) G b jl G H j l. The bas s detemned by the factos U / p P and the plots {P jl }. Snce U apple N d /, the scalng factos could be quantzed ove, p N d /, 8. Ths estmaton scheme s smla to one poposed n [4], whee a scaled veson of A was used as the upln pecode. The MSE weghts W can then be dectly estmated at the sevng BSs, and do not need to be fed bac. Howeve, n ode fo the BSs to estmate b s++n n that estmaton scheme, they must exchange the MSE weghts fo the coespondng UEs ove the bachaul. In essence, educed ove-the-a feedbac has been taded fo moe bachaul use. Rema. The CSI acquston poposed n ths secton s fully dstbuted ove the UEs, but not ove the BSs. Each BS needs nowledge of the ndvdual scalng factos fo all UEs, nfomaton whch s shaed ove a BS bachaul. C. CSI Acquston wth Global Shang of Pecodes, Receve ltes and MSE Weghts Lastly, we pesent an CSI acquston scheme whch eles even futhe on feedbac, sgnalng and bachaul. We pesent ths method snce the state-of-the-at obust WMMSE algothms n [7] [] eque ths type of CSI acquston. In ths scheme, only the undelyng channels ae estmated explotng the ecpocty, and the fltes and MSE weghts must be sgnaled between all nodes. In the downln tanng, P j C Mt N p,d ae othogonal plots sent fom BS j such that P P H j N p,d I Mt,j. The eceved sgnal at UE s then Y P K c M t H P + X (j,l)6(,) Pj K c M t H jp j + Z. (9) Ths type of plot tansmssons ae called cell-specfc efeence sgnals n the LTE standad [36]. o the case of Method Sec. III-A Sec. III-B TABLE IV TOTAL ESTIMATION COMPLEXITY, PER ITERATION AND UE. Appoxmate numbe of flops (M N d + M )N p,d + M tn d N p,u + M t Np,u/Kc (M N d + M )N p,d +(M tn d N p,u + M t N d + M t )Kt Sec. III-C M M tk t(n p,d + N p,u)+(m M tn d + M N d + M )K +(MMtN d + Mt N d + Mt )Kt + M Raylegh fadng, vec (H j) CN(, I), the MMSE estmato [] s p Pj / (K c M t ) bh j Y P H j. () P N j p,d K cm t + Assumng that the nose vaance s nown, and that all pecodes {V jl } have been fed bac to UE, t can fom b b H V, b X (j,l) bh jv jl V H j l b H H j + I. () Wth the subfame stuctue n g., consecutve tanng phases can be used to monotoncally mpove the channel estmates n one coheence bloc of the channel. Ths can be done usng teatve technques, see e.g. [38, Ch..6]. In the upln, assumng feedbac of eceve fltes and MSE weghts, whch ae shaed among all BSs, G b and b ae fomed n a smla fashon as n (). Rema 3. The CSI acquston poposed n ths secton s centalzed. It eques sgnfcant sgnalng of fltes among BSs and UEs n evey subfame. In tems of estmatng the undelyng channels {H j}, t s howeve dstbuted ove the BSs and UEs. D. eedbac Requements, Computatonal Complexty and Quantzed MSE Weght eedbac We compae the feedbac equements of the poposed estmaton schemes n Table III. The estmaton matx opeaton complextes [39] ae shown n Table IV. The M M t K t (N p,d + N p,u ) tem n the Sec. III-C estmaton method flop count domnates all othe tems when the numbe of plots s lage. Ths effect s llustated n g. 8 of Sec. V-A3, whee the complextes of the estmatos ae vsually compaed wth the coespondngly acheved sum ates.

9 8 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL In the CSI acquston poposed n Sec. III-A and Sec. III-B, feedbac of the MSE weghts to the sevng BS s equed. In ode to be pactcal, the MSE weghts should be quantzed and fed bac. Snce the MSE matx W s Hemtan, t can be quantzed by quantzng ts egenvalue decomposton. The egenvectos can be quantzed usng e.g. Gassmannan subspace pacng [4]. Recallng that s ( ) denotes the lagest sngula value, we have the followng helpful lemma fo the quantzaton of the egenvalues: Lemma. The egenvalues of the MSE weghts ae bounded as apple n (W ) apple + Ps (H ), 8,n. Poof: The poof s gven n Appendx A. The squae oots of the egenvalues of W can theefoe be quantzed by fndng 8 a sutable scala quantze ove the nteval n Lemma. Afte UE has fed bac the quantzed egenvectos and the quantzed squae oots of the egenvalues, the sevng BS can then fom the econstucted squae oot of the MSE weght W c /. P As mentoned n Sec. III-A, R log det (W ) n log ( n (W )) can be seen as the data ate (summed ove data steams) fo UE. Quantzng n (W ) theefoe coesponds to mang a set of dscete ates avalable to the UE, coespondng to e.g. a set of dffeent modulaton and codng schemes. IV. INHERENT AND ENORCED ROBUSTNESS O WMMSE SOLUTIONS In ths secton, we popose some modfcatons to the ognal WMMSE algothm that lead to an algothm whch s obustfed aganst CSI estmaton eos. A. Naïve WMMSE Algothm wth Estmated CSI It s staghtfowad to naïvely feed the WMMSE algothm the estmated CSI fom one of the pesented CSI acquston methods. An example of the esultng pefomance can be seen n g. 3. The smulaton settngs ae descbed n detal n Sec. V-A. It s clea that the naïve applcaton of the WMMSE algothm wos modeately well fo the centalzed CSI acquston schemes, but the pefomance fo the fully dstbuted CSI acquston scheme catastophcally deteoates at hgh SNR. Thus, some fom of obustfcaton aganst CSI estmaton eos s necessay. B. Geneal Wost-Case Robustness WMMSE Poblem One appoach to obustfyng the optmzaton poblem n (6) s to mnmze the objectve functon unde the wost-case eo condtons: mn. {A } {W } {V } subject to max {uncetanty} (,) X (T (W E ) log det (W )) XK c T V V H apple P,,...,K t. () 8 The detals of desgnng such a quantze s outsde the scope of ths pape. Aveage sum ate [bts/s/hz] Pefect CSI Sec. III C Sec. III B Sec. III A 3 Downln SNR [db] g. 3. Sum ate pefomance when naïvely applyng the WMMSE algothm togethe wth the CSI acquston schemes. The scenao was a K t 3, K c, M t 4, M ntefeng boadcast channel wth N d. The channels wee..d. Raylegh fadng, and the upln SNR was set as SNR u P / t db fo all lns. Note that SNR d P t/ affects both the powe constant n the WMMSE algothm, as well as the estmaton pefomance n the downln estmaton, snce the downln plots ae pecoded wth the same pecodes as used n the data tansmsson. The poposed CSI acquston methods povde estmates both of the downln effectve channels, as well as of the upln effectve channels. Due to the defnton of these effectve channels, the geneal uncetanty set n () cannot be explctly defned n tems of the upln and downln estmaton eos smultaneously. o example, one of the tems n the objectve functon of () s W A H H V W / G H V W A H, whch cannot be wtten n tems of G and smultaneously. In the fothcomng altenatng mnmzaton, we wll theefoe solve () wth the CSI uncetanty elatng to the patcula bloc of vaables fo whch () s solved fo. That s, fo the pecodes the CSI uncetanty at the BSs wll be consdeed, wheeas fo the eceve fltes and MSE weghts, the CSI uncetanty at the UEs wll be consdeed. We now detal the altenatng mnmzaton solutons fo the thee blocs of (). C. Pecode Robustness st we fx {A, W } and solve () wth espect to the pecodes {V }. The optmzaton poblem can then be ntepeted as a local optmzaton poblem at each BS, gven that the CSI uncetanty n () comes fom the upln channel estmaton phase. We let the estmaton eos fo BS be e bs+ and G e G G b,,...,k c. o the local uncetanty set, we assume that the eos ae nom bounded as e apple " (BS) and G e W / apple (BS),,...,K c. Note that (BS) depends on W, whch s fxed. The local wost-case optmzaton poblem fo BS s then: mnmze {V } subject to max e apple" (BS) eg W / p Re apple (BS) T XK c T V H b s+ + e V W / bg + e G H V XK c T V V H apple P,,...,K t. (3)

10 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS 9 The soluton to the nne optmzaton poblem of (3) can be found by extendng the esults of [], [3] to the multuse matx case. By uppe boundng the optmal value of the nne optmzaton poblem usng the tangle nequalty 9 and the submultplcatvty of the obenus nom, the (pessmstc) obust optmal pecode fo UE s V ob p bs+ + " (BS) + (BS) V ob + µ! I! b G W/ (4) As befoe, µ s the Lagange multple fo the sum powe constant. Note that the obust pecode n (4) s dagonally loaded by a constant facto " (BS), a data dependent facto (BS) / V ob, and the Lagange multple µ. Dagonal loadng s well nown to obustfy beamfomes n vaous settngs, and a lage body of lteatue has studed ts obustfyng effects; see e.g. [7] [3]. In ode to constuct the obust pecode n (4), the paametes " (BS) and (BS) must be nown. o the fully dstbuted CSI estmaton n Sec. III-A, the effectve channel eo G e follows a zeo-mean Gaussan dstbuton wth nown covaance, and (BS) can thus be selected such that eg W / apple (BS) holds wth some pobablty. The statstcs of the covaance eo e howeve depend on the fltes {U }, whch ae unnown at BS. Snce the optmal amount of dagonal loadng s unnown, we theefoe popose to dsegad " (BS) and (BS), and let the facto µ handle all the dagonal loadng. To compensate fo the mssng " (BS) and (BS), we mplctly amplfy µ usng a scalng pocedue. ) Implctly Selectng the Dagonal Loadng Paamete: When applyng dagonal loadng fo obustness, a heustc often used n the lteatue [3] s to select a fxed loadng level aound db ove the nose level. Instead, we popose a data dependent method fo selectng the dagonal loadng paamete mplctly. We note that, gven estmates b s++n, G b and fed bac W, the pecodes n the WMMSE algothm ae fomed le V p b s++n. + µ I bg W /. () The fom of () and (4) ae smla, and t can theefoe be concluded that µ alone acts as the dagonal loadng fo the naïve WMMSE pecode. The facto µ theefoe obustfes the soluton, and the amount of dagonal loadng s detemned by b s++n, G b W / and P. In ode to atfcally amplfy the facto µ, we now ntoduce a scalng pocedue. We let apple apple be a scalng facto, and modfy the WMMSE algothm as follows: ) In the pecode optmzaton at BS (step 4 n Algothm ), let the sum powe constant be P. The esultng pecodes fom () ae denoted {V ( ) }, and wll have equal o hghe dagonal loadng level than the ognal pecode n (), snce µ s nonnceasng n the sum powe constant value. 9 Ths elaxes the poblem such that e s the wost fo each UE smultaneously. Ths s equvalent to eplacng the exstng covaance constant wth e apple " (BS),,...,K c, and changng the objectve accodngly. Aveage sum ate [bts/s/hz] 4 3 SNR u {,,...,, 3} db 3 Scaled Sum Powe Constant P t [db] g. 4. Sum ate pefomance when vayng, fo P t and and vayng SNRu. The sold maes epesent the pefomance fo mn( P/ t P t /, ). The scenao s the same as n g. 3. ) om scaled pecodes V opt, and use these fo downln plot and data tansmsson. Ths scalng ensues that the coect tansmt powe s used. p V ( ) 3) At the UEs, pefom the estmaton gven the pecodes {V }, gvng { b( ) } and { b ( ) }. 4) Scale the estmates as b b ( ), b p b( ), and use { b } and { b } to fom eceve fltes and MSE weghts. Ths scalng s necessay n ode fo the WMMSE algothm at the UEs to be awae of what the ognal pecodes V ( ) wee. The same scalng s used at all BSs, and theefoe the sgnalto-ntefeence atos of the coss-lns ae not affected. In g. 4, we plot the mpact of selectng dffeent. A smple selecton that appeas to wo well s mn P/ t P t/ ) Removng the Nose Component of b s++n : Compang () wth (), t can be noted that the covaance matx should be b s+, and not b s++n. The nose poton of b s++n t wll on aveage be I, but smply subtactng that mght mae the esultng matx ndefnte. Instead, we modfy µ to allow fo negatve values; ths s the same as seeng µ as the dffeence of a non-negatve Lagange multple wth an estmate of the nose powe. Specfcally, we allow µ mn t, M t b s++n value detemnng how close to sngula b s++n D. Receve lte and MSE Weght Robustness, whee s some constant + µ I can be. Wth smla notaton and assumptons as n (3), the local wost-case optmzaton poblem fo the eceve flte at UE s mn. {A } max e apple" (UE) e W / Re T apple (UE) T W I + A H b + e A W b + e H A, whose (pessmstc) soluton [3] s A ob b + " (UE) + (UE) A ob W /!. (6) I! b. (7)

11 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL nally, the coespondng (pessmstc) obust MSE weght s! W b H b + " (UE) + (UE) A ob W / I! b A (8) Agan, the optmal level of dagonal loadng s unnown. Ths s because " (UE) depends on the statstcs of the covaance eo e b, whch n tun depend on the unnown pecodes {V }. We theefoe popose to ndectly apply dagonal loadng at the UEs nstead, based on the followng obsevaton: Poposton. The eceve flte A and MSE weght W obtaned n the UE sde optmzaton of the WMMSE algothm wth pefect CSI satsfes A W / +n T apple N d /. If the effectve channel s fully contaned n an ntefeence-fee subspace of dmenson N s apple N d, then asymptotcally A W /! N s/ as the SNR gows lage. Poof: The poof s gven n Appendx B. The fst pat of ths poposton has an mpotant connecton to the upln tanng stage n the fully dstbuted estmaton scheme (Sec. III-A). Snce U p P /N d p A W / s actng as the upln tanng stage pecode, A W / detemnes the effectve UE tansmt powe, and hence the upln estmaton SNR. The second pat shows that A W / also ndcates whethe pefect ntefeence algnment s acheved fo UE. ) Enfocng Poposton onto WMMSE Solutons wth Impefect CSI: Poposton elates to pefect CSI, but the nequalty may not hold fo the naïve solutons n (8) and (9) wth mpefect CSI. In ode to obustfy the algothm, we theefoe explctly mpose the constant on the UE sde optmzaton poblem wth mpefect CSI. The poblems stll decouple ove uses, and the poblem each UE should solve s mnmze A,W T W I + A H b A Re W bh A subject to A W / apple N d /. log det (W ) Poposton. The soluton to (9) s A opt b + opt I b W opt I bh b + opt I b. (9) If A opt W opt / apple N d / holds fo opt, the constant s not actve and the soluton has the same fom as the ognal soluton n Sec. II-A. Othewse, opt can be found by bsecton ove, such that A opt W opt / N d /. Poof: The poof s gven n Appendx C. Inteestngly, explctly mposng Poposton as a constant n (9) coesponds to dagonal loadng of the eceve flte A opt, gvng t the same fom as A ob. Lewse, W opt has Aveage sum ate [bts/s/hz] Pefect CSI Robust UE and BS Robust BS Robust UE Nave WMMSE 3 Downln SNR [db] g.. Sum ate pefomance when selectvely applyng the obustfyng measues n Sec. IV, togethe wth the fully dstbuted CSI acquston n Sec. III-A. o compason puposes, the scenao s the same as n g. 3. the same fom as W ob. The mpotant dffeence s that A opt and W opt do not depend on unnown paametes, wheeas can be appled as ealzable poxes fo the unealzable A ob and W ob n the obust WMMSE algothm to be poposed. By nceasng, the equested ate log det W opt s deceased. A lage would occu when thee ae obvous dscepances n the estmated CSI, such that A W / apple N d / s fa fom beng fulflled wthout the dagonal loadng. We vsualze the obustfyng effects n g. fo the same smulaton settngs as n g. 3. The obustfyng measues ae effectve, and esult n up to a facto sum ate gan ove the naïve WMMSE algothm. A ob and W ob do. Thus, A opt E. Robustfed WMMSE Algothm and W opt We now combne the dagonal loadng obustfcatons n Sec. IV-C and Sec. IV-D (.e. V opt, A opt, and W opt ) to fom a RoBustfed WMMSE algothm (RB-WMMSE); see Algothm. Ths algothm can be combned wth any of the channel estmaton pocedues outlned n Sec. III, and the jont system s fully dstbuted f the CSI acquston s dstbuted. The exstng obust WMMSE algothms n [7] [] also gan the obustness fom dagonal loadng, obtaned by optmzng a lowe bound on pefomance. Although not beng dectly taloed fo TDD channel estmaton, these algothms can be appled togethe wth the centalzed CSI acquston method poposed n Sec. III-C. In dong so, an mplct assumpton on the channel estmaton eos n the upln and downln s made howeve. Snce these algothms only have a noton of downln channel estmaton eos, they ae unawae of the upln channel estmaton eos n the TDD channel estmaton. Thus, the mplct assumpton that the channel estmaton eos n the downln and upln ae dentcal s made. The pefomance of ths appoach s studed n Sec. V-A. The algothms n [7] [] need the statstcs of the CSI uncetanty, whch s vey complcated to deve fo the CSI acquston methods n Sec. III-A and Sec. III-B snce those methods estmate the effectve channels.

12 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS Algothm RB-WMMSE Algothm (Estmated CSI) : epeat At UEs: : Plot tansmsson fom BSs: estmate b ( ) usng one of the methods n Sec. III. 3: Rescale b b ( ), b p b( ) 4: nd to satsfy A W / and b ( ) apple N d / : W I bh b + I b 6: A b + I b, U p A W / At BSs: 7: Plot tansmsson fom UEs: estmate b s++n and G b usng one of the methods n Sec. III. 8: Obtan W / though feedbac. t 9: nd µ mn, M b s++n t to satsfy P K c T V ( ) V ( ),H : B ( ) b s++n apple P + µ I bg, V ( ) : Scale V p V ( ) : untl fxed numbe of teatons V. PERORMANCE EVALUATION p B ( ) W / Pefomance of the poposed system s evaluated by means of numecal smulatons. Two scenaos ae studed: ) A canoncal ntefeng boadcast channel, wthout lage scale fadng. Ths model s elevant n local envonments whee the nte-cell ntefeence powe levels ae on pa wth the desed powe levels. ) A lage scale 3-cell netwo, wth path loss, shadow fadng, and small scale fadng. Ths models a possble lage scale deployment scenao, whee only cell-edge uses ae sgnfcantly affected by nte-cell ntefeence. In both scenaos, we study a case wth K t 3BSs, each sevng K c uses wth N d data steams each. The numbe of antennas wee M t 4and M. The BSs tansmt powe was P P t fo all BSs, and the UEs tansmt powe was P fo all UEs. Unless othewse stated, the RB-WMMSE P/ BS powe scalng was set as mn t P t/,, based on the fndngs n g. 4. o numecal stablty, we let the constant such that M b s++n t + µ I, 8, n the RB-WMMSE algothm. The UE data ate weghts wee fo all UEs. Tuncated dscete oue tansfom (DT) matces of appopate dmensons wee used fo the plot matces P and P, as well as fo the ntal pecodes. As a baselne pefomance measue, we used sngle-use egenpecodng and watefllng wth channels estmated by the MMSE estmato n Sec. III-C. Wth the sngle-use pocessng, we show the pefomance unde tme-dvson multple access (TDMA), as well as unde nonothogonal concuent tansmssons fom all BSs smultaneously ( uncoodnated tansmsson ). In the spt of epoducblty, we povde the full Matlab smulaton pacage as open souce. It s avalable fo download at [4]. Aveage sum ate [bts/s/hz] 3 TDMA (MMSE estm.) WMMSE (pefect CSI) RB WMMSE (Sec. III A) RB WMMSE (Sec. III B) Nave WMMSE (Sec. III A) Uncood. tans. (MMSE estm.) 3 Subfame numbe g. 6. Convegence compason of the dffeent methods fo K t 3,K c,m t 4,M,N d, SNR d db and SNR u db. A. Canoncal ntefeng boadcast channel o the smulatons wth the canoncal channel the channel model was..d. Raylegh fadng on all antenna-pas n the system such that [H j] mn CN(, ). Ths models a settng whee each ntefeng ln on aveage s equally stong as the desed channel. We assume a suffcently long coheence nteval, such that the channels do not change between teatons. We let each BS seve K c UEs, a settng whch s feasble fo ntefeence algnment [4]. o faness when compang estmaton schemes, we let N p,d K t M t and N p,u K M. The esults wee aveaged ove ndependent Monte Calo ealzatons. ) Convegence: st, we nvestgate the aveage convegence behavou of the RB-WMMSE algothm wth SNR d P t / db and SNR u P / t db. The esults n g. 6, ndcate that the RB-WMMSE algothm needs on the ode of teatons to convege, whch s consstent wth the fndngs of []. We do howeve note that a sgnfcant facton of the fnal pefomance s acheved afte just aound to teatons. In the followng, we theefoe let the algothms teate fo teatons. ) Sum Rate vs. Sgnal-to-Nose Rato: Next, we study the sum ate when the downln and upln SNRs ae vaed. Recall that the downln SNR affects both the downln data tansmsson, as well as the downln estmaton pefomance (cf. Sec. III-A). The upln SNR only affects the upln estmaton pefomance. We compae wth MaxSINR [43], fo whch we actvely tun off two uses n ode not to oveload the algothm. The esults fo the fully dstbuted CSI acquston (Sec. III-A) ae shown n g. 7a. The RB-WMMSE algothm consstently pefoms bette than MaxSINR, and bette than TDMA fo suffcently hgh upln SNR. The esults fo the CSI acquston wth global shang of fltes and MSE weghts (Sec. III-C) ae shown n g. 7b. Hee we also compae wth the lowe bound optmzaton method of [7] [], whch eques ths fom of centalzed CSI estmaton. We elax the equement of downln and upln estmaton eos beng dentcal (cf. Sec. IV-E). In g. 7b, t can be seen that the RB-WMMSE algothm exhbts smla pefomance as the lowe bound optmzaton method of [7] []. The sum ates n g. 7b ae hghe than the coespondng sum ates n g. 7a. Ths s because the mpoved channel estmaton pefomance, due to the pefect feedbac of fltes, and that

13 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL 3 Aveage sum ate [bts/s/hz] 4 WMMSE (pefect CSI) RB WMMSE MaxSINR TDMA Uncood. tansmsson 3 WMMSE (pefect CSI) Lowe bound opt. WMMSE [7]-[] RB WMMSE ( ) MaxSINR TDMA Uncood. tansmsson 3 SNRu db SNRu db Aveage sum ate [bts/s/hz] 4 3 SNRu db SNRu db Downln SNR [db] 3 (a) ully dstbuted CSI acquston (Sec. III-A) Downln SNR [db] 3 (b) CSI acquston wth globally shaed fltes and MSE weghts (Sec. III-C) 9 8 Sec. III C Sec. III B Sec. III A 4 Sum ates 3 7 Estmaton flop counts 6 4 RB-WMMSE flop count Numbe of plots Np,d Np,u 9 g. 8. Compason of complexty and sum ate pefomance afte the th teaton fo Kt 3, Kc, Mt 4, M canoncal ntefeng boadcast channel wth Nd, SNRd db, and SNRu db. the estmates of the channels ae mpoved n evey teaton, as descbed n Sec. III-C. 3) Sum Rate and Complexty vs. lop Count: o the case wth SNRd db, and SNRu db, we vay the numbe of plots Np,d Np,u and study the esultng pefomance and complexty of the system. The esults can be seen n g. 8. The moe complex CSI acquston methods pefom slghtly bette n the sum ate sense. The centalzed CSI acquston fom Sec. III-C eques patculaly many flops, snce t estmates all ntefeng channels. 4) Quantzed MSE Weght eedbac: So fa, the feedbac of the MSE weghts was assumed to be pefect. We now study pefomance of the system, usng quantzed MSE weghts, whle vayng the numbe of feedbac bts used. o the case wth fxed upln SNRu 3 db, we vay the SNRd and the numbe of quantzaton bts. Each UE had an wthweghts unfomly quantzed on h ndvdual codeboo P s (H ), log + db. The pefomance s shown n g. 9. o hghe downln SNR, moe bts ae needed fo good pefomance. o hgh esoluton quantzaton, the pefomance s equal to that of pefect feedbac. Aveage sum ate [bts/s/hz] 3 Numbe of loflops Aveage sum ate [bts/s/hz] g. 7. Sum ate afte the th teaton fo Kt 3, Kc, Mt 4, M canoncal ntefeng boadcast channel wth Nd. 4 Pefect feedbac Quantzed feedbac 3 SNRd 3 db SNRd db SNRd db SNRd db Numbe of quantzaton bts 7 8 g. 9. Sum ate as a functon of quantzaton accuacy fo Kt 3, Kc, Mt 4, M canoncal ntefeng boadcast channel wth Nd and SNRu 3 db. B. Lage scale 3-cell netwo The esults pesented so fa descbe pefomance n a settng whee the desed sgnal and ntefeng sgnals had equal aveage powe levels. In ealstc deployments, e.g. macocell setups, lage scale fadng such as path loss and shadow fadng ae pesent howeve, leadng to a moe heteogeneous settng. In ode to nvestgate the pefomance fo such a settng, we study a scenao whee the BSs ae located at the vetces of an equlateal tangle, and the antenna boe sghts ae amed towads the cente of the tangle (see g. ). Ths scenao models thee ntefeng sectos n a lage hexagonal macocell deployment. In patcula, we assume a setup whee factonal fequency euse s combned wth coodnated pecodng, such that cell cente and cell edge uses ae seved on othogonal subbands [44]. Snce the cell cente uses typcally have vey hgh sgnal-to-ntefeence atos (SIRs), they can be seved well usng sngle cell technques. The cell edge uses expeence low SIRs howeve, and thus multcell coodnated pecodng s a futful tansmsson stategy fo these uses. Snce ou focus s on coodnated pecodng, ou smulatons only study the pefomance of the cell edge uses. The smulaton paametes (see Table V) can be descbed as a smplfed veson of the 3GPP Case [4], [46], whee the

14 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS 3 TABLE V SIMULATION PARAMETERS OR LARGE SCALE 3-CELL NETWORK (ADAPTED ROM 3GPP CASE [4],[46]) m UE UE UE BS UE UE 3 UE 3 m BS BS 3 m Cell edge g.. Cell layout fo lage scale 3-cell netwo, hee dsplayed wth K c UEs pe cell. small scale fadng s..d. Raylegh fadng, and whee we only study one subcae. The pupose of the smulaton study s to nvestgate the mpact on sum ate pefomance, when the numbe of cell edge uses pe cell K c s vaed. o each of the two smulatons to be descbed, we geneated ndependent use dops (ncludng shadow fadng), whee the UEs wee dopped unfomly at andom n the cell, but neve fathe than m fom the cell edge. o each use dop, ndependent small scale fadng ealzatons wee geneated. We used the fully dstbuted estmaton method n Sec. III-A fo channel estmaton n the RB-WMMSE algothm and MaxSINR, and we assumed pefect feedbac fo the MSE weghts. In ode to have the same estmaton pefomance egadless of K c, we fxed N p,d N p,u 3 3. We used the same baselne methods as descbed n Sec. V-A. o lage K c, all baselne methods would pefom pooly due to the hgh ntefeence levels expeenced at the cell edge. We theefoe coupled the baselne methods wth a use selecton pocedue that detemned whch UEs to seve. Befoe descbng the man esults of the smulaton study, we fst detal the use selecton pocedue pefomance. ) Sum Rate vs. Numbe of Uses Selected fo Tansmsson: In ode to study how many uses to select fo tansmsson fo the baselne methods, we pefomed smulatons whee K c uses wee dopped pe cell, and the numbe of uses selected fo tansmsson was vaed. The use selecton was based solely on the channel stength to the sevng BS. The esults ae plotted n g. a. It can be seen that pefomance fo MaxSINR s maxmzed when uses ae selected fo tansmsson. o TDMA and uncoodnated tansmsson, pefomance s maxmzed when only a sngle use s selected fo tansmsson n each cell. The pefomance of the RB-WMMSE algothm s the hghest when 3 uses ae selected fo tansmsson n each cell, but pefomance only Inte ste dstance Max. dstance, UE to cell edge Path loss Penetaton loss BS antenna gan [47] UE antenna gan Shadow fadng m m log (dstance [m]) db db mn db 3 Small scale fadng..d. CN(, ) Bandwdth? BS tansmt powe?, 3 db Lognomal wth std. dev. 8 db Hz P t 8. dbm UE eceve nose powe? 3. dbm (9 db N) UE tansmt powe? P 4.8 dbm BS eceve nose powe? t 7. dbm ( db N)? We only study one subcae n a MHz system wth 6 subcaes. The total tansmt powes ae thus Pt tot 46dBm and P tot 3dBm. slghtly dops as moe uses ae selected. Ths s because the RB-WMMSE algothm, just le the WMMSE algothm, s able to mplctly pefom use selecton n the teatons. The fact that the pefomance s almost constant when moe than 3 uses ae selected fo tansmsson n each cell suggests that the RB-WMMSE algothm s able to fnd a good local soluton to the weghted sum ate poblem. ) Sum Rate vs. Total Numbe of Uses pe Cell: We now study system pefomance as a functon of the total numbe of uses pe cell K c. Gven the esults n the pevous secton, we select at most uses fo tansmsson pe cell fo MaxSINR. Smlaly fo TDMA and uncoodnated tansmsson, dsegadng the obvous unfaness of such a stategy, we only select use pe cell. o the RB-WMMSE and WMMSE algothms, we do not explctly pefom any use selecton, but athe let the algothms pefom the own mplct use selecton n the teatons. The esults of the smulaton can be seen n g. b. o the baselnes wth use selecton, the mpoved pefomance wth K c s due to the nceased multuse dvesty. The RB-WMMSE algothm s also able to haness ths ncease n multuse dvesty. The lage gap between the pefect CSI case, and the RB-WMMSE algothm wth fully dstbuted CSI acquston (Sec. III-A) s due to the low upln SNR n ths scenao. The RB-WMMSE algothm pefoms slghtly wose than the MaxSINR wth use selecton, fo lage K c. It has howeve been vefed that ths gap can be closed by combnng the RB-WMMSE algothm togethe wth explct use selecton, sevng at most 3 uses pe cell (cf. g. a). Hee we howeve show the esults wthout explct use selecton, n ode to dsplay the self-elant pefomance of the algothm. The uncoodnated tansmsson stategy wos faly well n tems of sum ate, Along the cell edge, the aveage upln SNR anges between 4.9 db and.8 db. The coespondng aveage downln SNR anges between 7. db and 4. db. Note that the aveage SNRs ae detemned both by the dstance to the BS, as well as the angle to the antenna boe sght.

15 4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL Aveage sum ate [bts/s/hz] WMMSE (pefect CSI) RB WMMSE MaxSINR (pefect CSI) MaxSINR TDMA Uncood. tansmsson WMMSE (nave) Aveage sum ate [bts/s/hz] 4 3 WMMSE (pefect CSI) MaxSINR (pefect CSI) RB WMMSE Uncood. tansmsson MaxSINR TDMA WMMSE (nave) Numbe of uses selected fo tansmsson pe cell and subcae (a) Vayng the numbe of uses selected fo tansmsson, gven K c uses pe cell Total numbe of uses pe cell and subcae (b) Vayng the total numbe of uses pe cell K c. At most uses pe cell wee seved by MaxSINR. o TDMA and uncoodnated tansmsson, only use pe cell was seved. g.. Sum ate afte the th teaton fo the lage scale scenao wth fully dstbuted CSI acquston (Sec. III-A). K c but as noted eale, coesponds to a hghly unfa stuaton whee only one use s seved pe cell. VI. CONCLUSIONS Many dstbuted coodnated pecodng algothms have been poposed n the lteatue, but few of these wos study the ssue of obustness aganst mpefect CSI. The wos that do study ths mpotant ssue can howeve not be coupled wth dstbuted CSI acquston, thus leadng to centalzed CSI acquston equng lage amounts of BS bachaul usage. To ou nowledge, the pesent pape s the fst that poposes a obust, and yet stll fully dstbuted, coodnated pecodng algothm. In dong so, thee CSI acquston methods have been poposed, and the coespondng equements n tems of channel estmaton, feedbac, and sgnalng have been llumnated. The obustfcaton of the algothm comes fom usng nheent popetes of the WMMSE solutons and applyng a data dependent scalng pocedue, leadng to a ealzable algothm whch does not depend on unnown paametes of the CSI uncetanty. Compaed wth the centalzed state-ofthe-at methods, ou system pefoms smlaly, wth the majo dffeence that ou system can be mplemented n a fully dstbuted manne. When evaluated n a macocell setup, the poposed system also pefoms well. A. Poof of Lemma APPENDIX +n o UE t holds that I, wth equalty f the UE does not expeence any ntefeence. Thus, the MSE weght fo that UE satsfes W I + V H H H +n H V (3) I + V H H H H V. (3) Dc Now ntoduce the spectal nom D max c s (D). Then, fo all c such that c, we have that c H V H H H H V c applev c H H H V c s (H ) apple V c s (H ) c V s (H ) applev s (H ) apple P s (H ). (3) Thus, V H H H H V apple P s (H ), and the uppe bound then dectly follows fom (3). o the lowe bound, note that I I + V H H H +n H V W. B. Poof of Poposton Decompose H +n + and let C H +n and D H +n. We have that A and W I + C. Pluggng n, A W / T A W A H T A H A W T H (I + C ). Applyng the matx nveson lemma to, t can be shown that H (I + C ) D (I + C ) afte smplfcatons. Thus, A W / T (I + C ) D +n T I + H +n H +n. Applyng the matx nveson lemma bacwads, we then get A W / T +n. uthe, A W / (a) T (I + C ) D apple T C T H +n H +n h Let e +n / e H T e e H +n e T e e H e e H +n D

16 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS (b) apple max T an( )N d apple N d +n +n M N d +n apple N d whee s a an-n d pojecton matx. The nequalty (a) s due to the tace beng an nceasng functon on the cone of postve defnte matces and the fact that D / (I + C ) D / D / C D /. The nequalty (b) holds snce e e H e e H s a an-n d pojecton matx. Now assume thee ae N s apple N d ntefeence-fee dmensons, and that the effectve channel s fully contaned n those. Let the egenvalues of be {apple s++n m } and the +n egenvalues of be {apple +n m }. Let the egenvalues be odeed such that apple s++n m apple +n m fo all m {,...,M N s }. Consequently, apple s++n m apple s m + and apple +n m fo all m {M N s +,...,M }. Hee, {apple s m} ae the eceved sgnal powes n the ntefeence-fee subspace. Then, +n XM T M X m N s apple +n m apple +n m {z } Intefeence dmensons XM mm N s+ apple s m m + apple +n m (apple s m + )! XM apple s++n m mm N apple s m + s+ {z } Intefeence-fee dmensons XM mm N s+ N s as the {apple s m} gow lage wth espect to. C. Poof of Poposton It can be shown that all feasble ponts ae egula, and thus any mnmze of ths non-convex poblem satsfes the KKT condtons [3, Ch. 3.3.]. The expessons ae obtaned by fomng the Lagangan L (A, W, ) T W I + A H b A Re W bh A log det (W ) + T A W A H N d, settng the complex patal devatves to zeo, and applyng the matx nveson lemma. It now emans to fnd the optmal opt. If the constant s satsfed fo opt, the poblem s solved and the fom s dentcal to the soluton n Sec. II-A. Othewse, let b L L H and b +n L +n +n L +n,h be egenvalue decompostons. Then as can be seen n (33), A W / s deceasng n, and the opt whch satsfes the nequalty constant wth equalty can be found by bsecton. A natual statng pont fo the lowe value n the bsecton s lowe. 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17 6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED OR PUBLICATION, APRIL T A W / T A W A H T b H b + I b I + b H b +n + I b L H b b H L ( + I) + +n + I hl H b b H L XM m XM m mm + ( + I) L H b bh L +n [ ] mm + hl H b b H L mm [ ] mm + + XM XM m p / L +n,h b b H L ( + I) L H b b H L +n +n + I [ ] mm + +n + pp / h L H b b H L +n mp +n + I / (33) [] S. Voobyov, A. Geshman, and Z.-Q. Luo, Robust adaptve beamfomng usng wost-case pefomance optmzaton: a soluton to the sgnal msmatch poblem, IEEE Tans. Sgnal Pocess., vol., no., pp , eb. 3. [] S. Shahbazpanah, A. Geshman, Z.-Q. Luo, and K. M. Wong, Robust adaptve beamfomng fo geneal-an sgnal models, IEEE Tans. Sgnal Pocess., vol., no. 9, pp. 7 69, Sep. 3. [3] K. Zaf, S. Shahbazpanah, A. Geshman, and Z.-Q. Luo, Robust blnd multuse detecton based on the wost-case pefomance optmzaton of the MMSE eceve, IEEE Tans. Sgnal Pocess., vol. 3, no., pp. 9 3, Jan.. [4] M. Xu, D. Guo, and M. L. Hong, Jont b-dectonal tanng of nonlnea pecodes and eceves n cellula netwos, axv: [cs.it], Aug. 4. [] B. Ghme, G. Aue, and H. Haas, Busy bust enabled coodnated multpont netwo wth decentalzed contol, IEEE Tans. Weless Commun., vol., no., pp ,. [6] S. Chstensen, R. Agawal, E. Cavalho, and J. Coff, Weghted sumate maxmzaton usng weghted MMSE fo MIMO-BC beamfomng desgn, IEEE Tans. Weless Commun., vol. 7, no., pp , Dec. 8. [7]. Nego, S. P. Shenoy, I. Ghau, and D. T. M. Sloc, Weghted sum ate maxmzaton n the MIMO ntefeence channel, Poc. IEEE Int. Symp. Pesonal, Indoo, Moble Rado Commun. (PIMRC ), pp ,. [8]. P. Kelly, A. K. Maulloo, and D. K. H. Tan, Rate contol fo communcaton netwos: Shadow pces, popotonal faness and stablty, J. Opeatonal Reseach Soc., vol. 49, no. 3, pp. 37, 998. [9] Y.-. Lu, Y.-H. Da, and Z.-Q. Luo, Coodnated beamfomng fo MISO ntefeence channel: Complexty analyss and effcent algothms, IEEE Tans. Sgnal Pocess., vol. 9, no. 3, pp. 4 7, Ma.. [3] D. Betseas, Nonlnea pogammng. Athena Scentfc, 6. [3] M. Razavyayn, M. Hong, and Z. Luo, A unfed convegence analyss of bloc successve mnmzaton methods fo nonsmooth optmzaton, SIAM J. Optmzaton, vol. 3, no., pp. 6 3, 3. [3] T. A. Levanen, J. Psanen, T. Kosela, J. Talvte, and M. Valama, Rado nteface evoluton towads G and enhanced local aea communcatons, IEEE Access, vol., pp. 9, 4. [33] N. Jndal and A. Lozano, A unfed teatment of optmum plot ovehead n multpath fadng channels, IEEE Tans. Commun., vol. 8, no., pp , Oct.. [34] E. Lähetangas, K. Pajuos, E. Tola, G. Beadnell, I. Hajula, and J. Vhälä, On the TDD subfame stuctue fo beyond 4G ado access netwo, n utue Netwo and Moble Summt, Jul. 3, pp.. [3] J. Jose, A. Ashhmn, T. L. Mazetta, and S. Vshwanath, Plot contamnaton and pecodng n mult-cell TDD systems, IEEE Tans. Weless Commun., vol., no. 8, pp. 64 6,. [36] E. Dahlman, S. Pavall, and J. Söld, 4G LTE/LTE-Advanced fo Moble Boadband. Academc Pess,. [37] H. Cox, Resolvng powe and senstvty to msmatch of optmum aay pocessos, J. Acoustcal Soc. Ameca, vol. 4, no. 3, pp , 973. [38] S. M. Kay, undamentals of Statstcal Sgnal Pocessng: Estmaton Theoy. Pentce Hall, 993. [39] L. N. Tefethen and D. Blau III, Numecal Lnea Algeba. SIAM, 997. [4] D. Love and R. Heath, Lmted feedbac untay pecodng fo spatal multplexng systems, IEEE Tans. Inf. Theoy, vol., no. 8, pp ,. [4] R. Bandt, Smulaton Matlab code. [Onlne]. Avalable: [4] T. Lu and C. Yang, On the feasblty of lnea ntefeence algnment fo MIMO ntefeence boadcast channels wth constant coeffcents, IEEE Tans. Sgnal Pocess., vol. 6, no. 9, pp. 78 9, 3. [43] K. Gomadam, V. R. Cadambe, and S. Jafa, A dstbuted numecal appoach to ntefeence algnment and applcatons to weless nteeence netwos, IEEE Tans. Inf. Theoy, vol. 7, no. 6, pp ,. [44] L.-C. Wang and C.-J. Yeh, 3-cell netwo MIMO achtectues wth sectozaton and factonal fequency euse, IEEE J. Sel. Aeas Commun., vol. 9, no. 6, pp. 8 99, Jun.. [4] 3GPP, TR.84, Physcal laye aspects fo evolved unvesal teestal ado access (Release 7), 3GPP, Tech. Rep., 6. [46], TR 36.84, uthe advancements fo E-UTRA physcal laye aspects (Release 9), 3GPP, Tech. Rep.,. [47], TR.996, Spatal channel model fo multple nput multple output (MIMO) smulatons (Release ), 3GPP, Tech. Rep.,. Rasmus Bandt (S ) was bon n Uppsala, Sweden, n 98. He eceved the Tech.Lc. degee n Electcal Engneeng fom KTH Royal Insttute of Technology, Stocholm, Sweden, n 4 and the M.Sc. degee n Engneeng Physcs fom Uppsala Unvesty, Uppsala, Sweden, n. As pat of hs M.Sc. pogam, he spent the academc yea 7/8 at Queen s Unvesty, Kngston, Canada and the summe of 9 as an IAESTE Inten at CMC Mcosystems, Kngston, Canada. He s cuently wong towads the Ph.D. degee n Electcal Engneeng at the KTH Royal Insttute of Technology. Hs eseach nteests nclude statstcal sgnal pocessng fo weless communcatons as well as esouce allocaton and ntefeence algnment fo ntefeence management. M. Bandt s a Publcty Cha of the 6th IEEE Intenatonal Woshop on Sgnal Pocessng Advances n Weless Communcatons (SPAWC ).

18 BRANDT AND BENGTSSON: DISTRIBUTED CSI ACQUISITION AND COORDINATED PRECODING OR TDD MULTICELL MIMO SYSTEMS 7 Mats Bengtsson (M SM 6) eceved the M.S. degee n compute scence fom Lnöpng Unvesty, Lnöpng, Sweden, n 99 and the Tech. Lc. and Ph.D. degees n electcal engneeng fom the KTH Royal Insttute of Technology, Stocholm, Sweden, n 997 and, espectvely. om 99 to 99, he was wth Ecsson Telecom AB Kalstad. He cuently holds a poston as Assocate Pofesso at the Sgnal Pocessng depatment, School of Electcal Engneeng, KTH. Hs eseach nteests nclude statstcal sgnal pocessng and ts applcatons to communcatons, mult-antenna pocessng, coopeatve communcaton, ado esouce management, and popagaton channel modellng. D. Bengtsson seved as Assocate Edto fo the IEEE Tansactons on Sgnal Pocessng 7-9 and was a membe of the IEEE SPCOM Techncal Commttee 7-.