Performance of Cell-Free Massive MIMO Systems with MMSE and LSFD Receivers

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1 Performance of Cell-Free assve IO Systems wth SE and LSFD Recevers Elna Nayeb Unversty of Calforna San Dego, CA Alexe Ashhmn Bell Laboratores urray Hll, NJ Thomas L. arzetta Bell Laboratores urray Hll, NJ Bhasar D. Rao Unversty of Calforna San Dego, CA arxv: v [cs.it] 8 Feb 207 Abstract Cell-Free assve IO comprses a large number of dstrbuted sngle-antenna access ponts APs servng a much smaller number of users. There s no parttonng nto cells and each user s served by all APs. In ths paper, the upln performance of cell-free systems wth mnmum mean squared error SE and large scale fadng decodng LSFD recevers s nvestgated. The man dea of LSFD recever s to maxmze achevable throughput usng only coeffcents between APs and users. Capacty lower bounds for SE and LSFD recevers are derved. An asymptotc approxmaton for sgnal-to-nterference-plus-nose rato SINR of SE recever s derved as a functon of coeffcents only. The obtaned approxmaton s accurate even for a small number of antennas. SE and LSFD recevers demonstrate fve-fold and two-fold gans respectvely over matched flter F recever n terms of 5%-outage rate. I. INTRODUCTION In recent years assve IO mio has attracted consderable attenton as a canddate for the ffth generaton physcal layer technology [], [2]. Cell-free mio s a partcular deployment of mio systems wth a networ of randomly-located large number of sngle-antenna APs, where the geographcal area s not parttoned nto cells and each user s served smultaneously by all of the APs [3], [4]. Some of the advantages and lmtatons of the networs wth dstrbuted APs can be found n [3] [7]. In partcular n [3], [4] the performance of downln transmsson and upln transmsson wth F recever n cell-free mio systems have been studed. In ths paper we frst consder upln SE recever. We further propose a suboptmal SE recever called partal SE and demonstrate that t has vrtually optmal performance. In [7], the authors study upln performance of cellular mio systems wth dstrbuted antenna clusters n each cell. The authors consder SE and F recevers wth coordnaton among dstrbuted antenna clusters n each cell. In contrast we assume all dstrbuted APs coordnate wth each other to form the postcodng vectors and detect the sgnals transmtted by users. In [8] random matrx theory results are used to study performance of cellular mio systems. otvated by [8] we appled random matrx theory for dervng a tght approxmaton of the partal SE n cell-free systems as a functon of coeffcents wth cooperaton among dstrbuted APs. Snce partal The wor of E. Nayeb and B. Rao were supported by the Natonal Scence Foundaton under Grant CCF SE has almost optmal performance, our approxmaton s also very accurate for the optmal SE recever. Numercal results ndcate that the obtaned approxmaton s accurate even for a small number of APs and users. In [9] and [0], LSFD also nown as plot contamnaton postcodng was proposed for nterference reducton n cellular mio systems. In LSFD base statons cooperate, but only usng coeffcents. In ths wor we propose generalzaton of the LSFD recever for cell-free mio systems and derve the SINR expresson for t as a functon of coeffcents. We further derve an expresson for SINR of cell-free systems wth F recever n the regme when the number of users s constant and the number of APs grows wthout a lmt. Our result shows that n ths regme the system performance s lmted by the coherent nterference resultng from two or more users sharng the same plot sequence. In numercal experments we evaluate the system performance under ndependent and correlated shadow fadng models. Results show that SE and LSFD recevers provde sgnfcant gan over F recever. SE recever outperforms LSFD recever whle the latter has smaller complexty. The paper s organzed as follows. In secton II the system model and channel estmaton are descrbed. In secton III, we nvestgate SE, partal SE, and LSFD recevers n upln transmsson. Fnally, numercal results are presented n secton IV. Throughout the paper dag a n denotes dagonal matrx wth a,, a n on ts dagonal. If S = {α,, α n } N n and S 2 = {σ,, σ m } N m, where α and σ s are n the ncreasng order, then [v ] S denotes the column vector [v α,, v αn ] T ; and [[v ]] S, S 2 denotes the n m matrx [ vα σ,, v α σm v αnσ,, v αnσm ] T. II. SYSTE ODEL AND CHANNEL ESTIATION We consder a geographcal area wth randomly dstrbuted sngle-antenna APs and K sngle antenna users, assumng that K. All APs are connected to a networ controller NC va an unspecfed bachaul networ. All APs and users are perfectly synchronzed n tme. The channel coeffcent between AP m and user s gven by g m = β m h m,

2 where β m s the coeffcent whch accounts for path loss and shadow fadng and h m CN 0, s the small scale fadng coeffcent. The coeffcents change slowly over tme and assumed to be nown at the NC. The small scale fadng coeffcents are..d. random varables that stay constant over a channel coherence nterval. We assume tme-dvson duplex TDD protocol,.e., all users synchronously send randomly assgned orthonormal plot sequences ψ,, ψ τ C τ, where ψ H ψ = δ to allow APs to estmate channel coeffcents, whch they further send to the NC. We consder short coherence nterval due to hgh user moblty and therefore τ s small and K > τ. Hence each plot s reused by several users, whch results n the plot contamnaton, [9], [0]. In [4], a greedy plot assgnment scheme n cell-free systems has been ntroduced whch s shown to mprove performance of cell-free system compared wth the random plot assgnment scheme. However, for smplcty we consder the random plot assgnment n the cell-free systems. All users are parttoned nto τ sets S,, S τ n a way that users n S use plot ψ. Let b be the ndex of the plot sequence transmtted by th th user. The receved sgnal n the frst step of the TDD protocol at the mth AP s y m = τ K g m ψ b + v m, = where s the upln transmt power of each user and v m C τ CN 0, s addtve Gaussan nose. AP m computes the SE estmate of g m as τβm ĝ m = + τ ψ H b S b β y m. m It can be verfed that ĝ m and the channel estmaton error g m = g m ĝ m are uncorrelated Gaussan random varables wth dstrbutons ĝ m CN 0, α m, g m CN 0, β m α m, where α m = τβ 2 m +τ S b β m. Note that ĝ m = βm β m ĝ m for every, S b. Therefore, t s enough for AP m to choose one user u S and send only the channel estmates ĝ mu, =,, τ to the NC. Let η denote the power coeffcent used by the th user to transmt upln data. For notaton convenence we defne A dag α m m, B dag β m m, K C B A, D η C + I. = III. UPLINK DATA TRANSISSION At the second step of the TDD protocol, users send data symbols and the mth AP receves y m = K η g m s + v m, = where v m CN 0, σz 2 s addtve nose and s s the data sgnal transmtted by the th user. The NC uses estmates ĝ m to form postcodng vectors v and obtans estmates of data sgnals ŝ = v H [y,, y ] T, =,, K. Usng the worst-case uncorrelated addtve nose, the upln achevable rate of the th user s R = E log 2 + SINR, wth SINR v = v H η v HĝĝH v, K η ĝ ĝ H + D v where ĝ = [ĝ,, ĝ ] T. Note that achevable SINR of the th user n s obtaned by tang nto account the channel estmaton error and plot contamnaton effect. A. SE Recever Frst, we consder SE recever, whch maxmzes SINR of each user. The SE vector of the th user s gven by v SE = η K η ĝ ĝ H + D ĝ. 2 = Note that the SE vector n 2 contans channel estmates of all users n the networ. Thus, t s optmal n the sense that t maxmzes SINR of each user. Whereas n cellular systems, the SE vector at cell l only contans channel vectors of cell l and the second-order statstcs of the channel coeffcents between base staton at cell l and all users n the networ [7], [8]. Achevable SINR of the th user wth SE recever s gven by SINR SE = SINR v SE gˆ H K = η ĝ ĝ H + D g ˆ = η gˆ H. 3 K = η ĝ ĝ H + D g ˆ The onte Carlo smulaton of R SE = log 2 + SINR SE requres long averagng over small scale fadng coeffcents h m. Hence t s desrable to have an approxmaton of R SE as a functon of large scale fadng coeffcents only. The correlaton between the channel estmates.e., ĝ m = βm β m ĝ m for, S b does not allow us to use random matrx theory tools [, Theorem,2], [8] to acheve ths goal. Below we propose a partal SE recever whose performance s very close to the performance of the SE recever and allows us to overcome ths problem. B. Partal SE Recever Let I = S b { u,, u τ }, where u S s the ndex of a user from S whose selecton rule s dscussed later. The partal SE vector for user s then defned by v PSE = η I η ĝ ĝ H + / I E η ĝ ĝ H +D ĝ = η I η ĝ ĝ H +Q ĝ, 4

3 where Q = / I η B + I η C +I. Note that I contans all users that cause coherence nterference to user and one user from each non-coherent nterference group S,. Note that n mio systems, the coherent nterference s the domnant mparment whch lmts the system performance when number of antennas ncrease wthout bound. Therefore, n the partal SE recever we nclude channel vectors of all users that use the same plot sequence as user. The users u,, u τ should be chosen such that vectors ĝ, I n 4 have the maor contrbuton n 2 and hence 4 becomes close to 2. Numercal results show that a random selecton of users u,, u τ from the correspondng sets S,, S τ leads to poor performance see Fgure. A method for smart choce of these users can be formulated as followng u = arg max S β T β, =,, τ, 5 where β = [β,, β ] T. In other words, we choose user u S that s n the close vcnty of the th user. The SINR PSE can be obtaned by substtutng v PSE n. In the followng theorem we apply random matrx theory to obtan an asymptotc approxmaton of R PSE = log 2 + SINR PSE when and K grow nfntely large whle the rato /K s fnte. Ths asymptotc result s used as an approxmaton for fnte values of and K smlar to [] and [8] n whch the approxmatons are derved for ISO broadcast channel and cellular systems respectvely. Theorem. Assume matrces A, C =,, K have unformly bounded spectral norms. For the partal SE recever defned n 4, when and K grow large such that 0 < lm nf K lm sup K <, we have a.s. R PSE log 2 + SINR PSE 0,,K where SINR PSE s defned n 6 and all parameters n SINR PSE are summarzed n Table I. Note that the approxmaton SINR PSE n 6 s a functon of coeffcents only, and though t has a long formulaton, t can be easly calculated numercally for large values of and K. C. Large Scale Fadng Decodng Next, we propose the LSFD recever for cell-free systems. The man dea of LSFD s that only coeffcents are transmtted to NC from APs. Snce these coeffcents are ndependent of frequency and change about 40 tmes slower than small scale fadng coeffcents, LSFD allows one to reduce the bachaul traffc, whch can be very desrable n real lfe systems. Generalzed matrx nverson lemma and [8, Theorem,2] are used to derve the asymptotc approxmaton. Due to lac of space, dervatons are spped. δ t δ T [J] l TABLE I: Parameter defntons n Theorem. η tr A η A + +δ t Q δ [ δ ] T H δ t δ T γ Γ ν H N H I \S b lm t δt, wth δ 0 = I \S b η A + +δ Q 2 tr η η l A T A l T + δ l 2,, l I \ S b I \S b η tr A I + = I J [ η tr A T HT T HT + T I \{n} I \S b η A η A δ +δ t +δ 2 T ] + Q lm t δ t, wth δ 0 = η A +δ + Q I \{} ] [ η tr A /2 [[ η η tr A /2 [ η 2 tr A /2 2 [[ η η tr I \S b A /2 T S b ]] A /2 T S b, S b ] A /2 T H S b ]] A /2 T H S b, S b A /2 λ tr A /2 A /2 θ H 2 tr A T H 2Re + γ T Γ N H Γ γ T γ T Γ γ ν H T Γ γ By usng matched flter, the mth AP computes s m = ĝm y m for one user S b, and sends them to the NC. The NC computes s m = βm β m s m, S b and estmates data symbol s by usng lnear combnaton of all receved sgnals as followng ŝ = m= = K v m s m. 7 The NC computes postcodng coeffcents v m and power coeffcents η as a functon of coeffcents only. Lemma. The estmate of data symbol ŝ n 7 can be smplfed as ŝ = v H s, 8 where v = [v,, v ] T and s = [ s,, s ] T. The proof of Lemma follows drectly from the fact that assgnment v m = 0, S b, n 7 does not result n any performance loss, and s m = βm β m s m, S m. Theorem 2. Achevable SINR of the th user wth LSFD

4 SINR PSE = θ D + S b \{} η λ 2 η λ 2 + η θ A + / I I \S b η θa + η trat 2 6 recever s gven by η v H SINR v = µ µ H v S b \{} η v Hµ µ H v + v HΛv, K = η α m β m + α m m where Λ [ = dag µ = ] τβ m β m +τ S β m b m. and We can show that the optmal v LSFD whch maxmzes SINR of each user s gven by v LSFD = η µ µ H + Λ µ. S b \{} The assocated SINR of the th user s gven by SINR LSFD = η µ H η µ µ H + Λ S b \{} µ. To obtan power coeffcents one can apply max-mn power allocaton problem wth per user transmt power constrants as followng max mn R LSFD = log 2 + SINR LSFD, 9a η s.t. η, =,, K. 9b Lemma 2. The obectve functon mn R LSFD η n 9a s a quasconcave functon of η = [η,, η K ] T and constrants 9b are convex. Snce the power allocaton problem 9 s quasconcave, bsecton method [2, Chapter 4.2.5] can be used to solve t. We wrap up ths secton by provdng the SINR expresson for F recever when the number of APs grows wthout lmt. Theorem 3. Achevable SINR of the th user for F recever,.e., v F = [,, ]T, wth unlmted number of APs and K = constant and ndependent coeffcents s gven by SINR v F where c m = a.s. η E βm c m 2 η E βm c m 2, 0 S b \{} τβ m +τ S b β m locaton of APs ndex m. and the expected value s over Note that the denomnator n 0 corresponds to power of the plot contamnaton related nterference. Smlar to the cellular mio systems, SINR of the th user usng F recever s lmted by the effect of plot contamnaton. However, unln cellular systems, n whch SINR depends on the coeffcents, SINR of cell-free system s a constant quantty n the lmt of an nfnte number of APs. IV. NUERICAL RESULTS We consder a square dense urban area of 2 2 m 2 wth randomly located APs and K randomly located users. The area s wrapped around to avod boundary effects. For large scale fadng coeffcents we consder a three-slope path loss model [3] as follows c 0 d 0.0 m c 0.0m < d 0.05 m β m = d 2 m, c 2 z m d 3.5 d > 0.05 m m where d m s the dstance n lometers between user and the AP m, and z m s the log-normal shadow fadng,.e., 0 log 0 z m N 0, σshad 2 wth σ shad = 8 db. For d > 0.05 m we use COST-23 Hata propagaton model 0 log 0 c 2 = log 0 f log 0 h B +. log 0 f 0.7h R.56 log 0 f 0.8, where f = 900 Hz s the carrer frequency, h B = 5 m s the AP antenna hght, and h R =.65 m s the user antenna hght. Parameters c and c 2 n are chosen n the way that path loss remans contnuous at boundary ponts. To model the correlaton between coeffcents caused by closely located users and/or APs, we use the correlaton model from [4] wth δ = 0.5 and d decorr = 0. m. The nose varance s σv 2 = 290 κ B NF, where κ, B, and NF are Boltzmann constant, bandwdth 20 Hz and nose fgure 9 db respectvely. We assume users transmt wth equal power η =, =,, K and = 200 mw. In fgure, CDFs of R SE, R PSE wth heurstc approach gven n 5, R PSE wth random user selecton, ˆR PSE = log 2 + SINR PSE, and R LSFD wth ndependent coeffcents are presented. The CDF of peruser throughput acheved by F recever [4] s also ncluded n the fgure for comparson. The horzontal lne corresponds to 5%-outage rates whch represents the smallest rate among 95% of the best users. One can observe that the asymptotc approxmaton of SE recever s very tght. SE and LSFD recevers provde respectvely 5.-fold and 2.6-fold gan over the F recever n terms of 5%-outage rate. Performance of the LSFD recever les between the smple F recever and SE recevers. Compared to the SE recever, LSFD reduces the overall complexty of the system. Fgure 2 shows 5%-outage and mean values of R SE, R PSE, ˆRPSE versus number of APs under ndependent and

5 CDF SE Partal SE Approxmaton l Partal SE Partal SE wth Random User Selecton LSFD atched Flterng 5% Outage Rate per user throughput bts per second per hertz Fg. : CDFs of the achevable per-user rates for LSFD and SE recevers wth = 000, K = 50, and τ = 0. bts per second per hertz SE Partal SE Approxmaton l Partal SE ean rate, ndependent ean rate, correlated 4 5% outage rate, ndependent 2 5% outage rate, correlated number of access ponts Fg. 2: 5%-Outage and mean rates versus for correlated and ndependent wth K = 6 and τ = 4. correlated shadow fadng. One can observe that n all consdered scenaros the partal SE s vrtually optmal and partal our approxmaton ˆR SE s very accurate. The shadow fadng correlaton sgnfcantly affects the system performance. The CDFs of per-user rates for dfferent number of APs and users are plotted n Fgure 3. The rato between APs and users s constant n all cases,.e., /K = 8 and K /τ = 4. We observe that the 5%-outage rate of SE and partal SE recevers ncrease as the networ sze ncreases. V. CONCLUSION In ths paper we studed the upln performance of cellfree systems wth SE and LSFD recevers. A suboptmal SE recever, whch s more tractable to study the asymptotc behavor of the cell-free systems, s ntroduced. Rates acheved by SE, partal SE, and asymptotc approxmaton are very close. The asymptotc approxmaton s very accurate even for small number of APs and users. LSFD CDF SE Partal SE Approxmaton l Partal SE = 28 K = 6 τ = 4 = 256 K = 32 τ = 8 = 52 K = 64 τ = per user throughput bts per second per hertz Fg. 3: CDFs of the achevable per-user rates for SE recevers wth dfferent number of APs and users. recever n cell-free systems s ntroduced. LSFD recever depends only on the coeffcents. SE and LSFD recevers demonstrate sgnfcant gan over F recever. There s a consderable gap between SE and LSFD recevers. REFERENCES [] T. L. arzetta, Noncooperatve cellular wreless wth unlmted numbers of base staton antennas, Wreless Communcatons, IEEE Transactons on, vol. 9, no., pp , 200. [2] F. Ruse, D. Persson, B. K. Lau, E. G. Larsson, T. L. arzetta, O. Edfors, and F. Tufvesson, Scalng up IO: Opportuntes and challenges wth very large arrays, IEEE Sgnal Processng agazne, vol. 30, no., pp , Jan [3] E. Nayeb, A. Ashhmn, T. arzetta, and H. Yang, Cell-free massve IO systems, n Aslomar Conference on Sgnals, Systems and Computers, 205. [4] H. Q. Ngo, A. Ashhmn, H. Yang, E. G. Larsson, and T. L. arzetta, Cell-free massve mmo: Unformly great servce for everyone, arxv preprnt arxv: , 205. [5] D. Gesbert, S. Hanly, H. Huang, S. S. Shtz, O. Smeone, and W. Yu, ult-cell IO cooperatve networs: A new loo at nterference, IEEE Journal on Selected Areas n Communcatons, vol. 28, no. 9, pp , Dec [6]. Sawahash, Y. Kshyama, A. ormoto, D. Nshawa, and. Tanno, Coordnated multpont transmsson/recepton technques for lte-advanced [coordnated and dstrbuted mmo], IEEE Wreless Communcatons, vol. 7, no. 3, pp , Jun [7] K. T. Truong and R. W. Heath, The vablty of dstrbuted antennas for massve IO systems, n 203 Aslomar Conference on Sgnals, Systems and Computers. IEEE, 203, pp [8] J. Hoyds, S. Ten Brn, and. Debbah, assve mmo n the ul/dl of cellular networs: How many antennas do we need? Selected Areas n Communcatons, IEEE Journal on, vol. 3, no. 2, pp. 60 7, 203. [9] A. Ashhmn and T. arzetta, Plot contamnaton precodng n multcell large scale antenna systems, n Informaton Theory Proceedngs ISIT, 202 IEEE Internatonal Symposum on. IEEE, 202, pp [0] A. Adhary, A. Ashhmn, and T. L. arzetta, Upln nterference reducton n large scale antenna systems, n Informaton Theory ISIT, 204 IEEE Internatonal Symposum on. IEEE, 204, pp [] S. Wagner, R. Coullet,. Debbah, and D. Sloc, Large system analyss of lnear precodng n correlated ISO broadcast channels under lmted feedbac, Informaton Theory, IEEE Transactons on, vol. 58, no. 7, pp , 202.

6 [2] S. Boyd and L. Vandenberghe, Convex optmzaton. Cambrdge unversty press, [3] A. Tang, J. Sun, and K. Gong, oble propagaton loss wth a low base staton antenna for nlos street mcrocells n urban area, n Vehcular Technology Conference, 200. VTC 200 Sprng. IEEE VTS 53rd, vol.. IEEE, 200, pp