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1 Chalmers Publication Library Performance analysis of large scale MU-MIMO with optimal linear receivers This document has been downloaded from Chalmers Publication Library CPL. It is the author s version of a wor that was accepted for publication in: 0 Swedish Communication Technologies Worshop, Swe-CTW 0, Lund, 4-6 October 0 Citation for the published paper: go, H. ; Matthaiou, M. ; Larsson, E. 0 "Performance analysis of large scale MU- MIMO with optimal linear receivers". 0 Swedish Communication Technologies Worshop, Swe-CTW 0, Lund, 4-6 October 0 pp Downloaded from: otice: Changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this wor, please refer to the published source. Please note that access to the published version might require a subscription. Chalmers Publication Library CPL offers the possibility of retrieving research publications produced at Chalmers University of Technology. It covers all types of publications: articles, dissertations, licentiate theses, masters theses, conference papers, reports etc. Since 006 it is the official tool for Chalmers official publication statistics. To ensure that Chalmers research results are disseminated as widely as possible, an Open Access Policy has been adopted. The CPL service is administrated and maintained by Chalmers Library. article starts on next page

2 Performance Analysis of Large Scale MU-MIMO with Optimal Linear Receivers Hien Quoc go, Michail Matthaiou, and Eri G. Larsson Department of Electrical Engineering ISY, Linöping University, Linöping, Sweden, s:{nqhien, Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden, Abstract We consider the uplin of multicell multiuser MIMO MU-MIMO systems with very large antenna arrays at the base station BS. We assume that the BS estimates the channel through uplin training, and then uses this channel estimate to detect the signals transmitted from a multiplicity of autonomous users in its cell. By taing the correlation between the channel estimate and the interference from other cells into account, we propose an optimal linear receiver OLR which maximizes the received signal-to-interference-plus-noise SIR. Analytical approximations of the exact and lower bound on the achievable rate are then derived. The bound is very tight, especially at large number of BS antennas. We show that at low SIR, maximalratio combing MRC receiver performs as well as OLR, however at high SIR, OLR outperforms MRC. Compared with the typical minimum mean-square error receiver, our proposed OLR improves systematically the system performance, especially when the interference is large. I. ITRODUCTIO MU-MIMO with very large antenna arrays at the BS represents an emerging hot area since it i offers more degrees of freedom; and ii offers increased power efficiency ] 3]. For such very large systems, the BS can serve a multiplicity of autonomous users in the same time-frequency resource, with each user getting high throughput. Since the number of BS antennas is large, the signal processing at the BS should be simple, e.g., using linear precoders for the downlin and linear detectors for the uplin. Among typical linear detectors at the BS, MRC is advantageous since it can be implemented in a distributed manner. From the system performance point of view, in 4], we showed that MRC performs as well as ZF and MMSE in the regime where each user has a throughput of an order of about bit per channel use, whilst MMSE is the optimal linear detector for single-cell systems. However, for multicell MU-MIMO systems, where the channel is estimated at the BS by using uplin pilots, it is unnown whether the typical MMSE receiver is optimal. In this paper, we shed some light onto this very interesting problem. Intuitively, the channel estimate is contaminated by the pilots transmitted from other cells. As a consequence, it correlates with the interference from other cells. Typical MMSE receivers do not tae this correlation into account and hence, MMSE receiver is not optimal for these systems. In this paper, we propose a novel OLR which taes the correlation between the channel estimate and the intercell interference into account. The paper maes the following specific contributions: We propose an OLR which exploits the correlation between the channel estimates and the interference from other cells, due to the pilot contamination effect. Then we compare the performance between MRC, ZF, MMSE, and OLR receivers. We derive new approximating expressions for the statistical distributions of the received SIR. We then deduce closed-form approximations for the exact uplin achievable ergodic rate as well as a tight lower bound on this rate. This analysis provides a benchmar for all conventional linear receivers. Finally, we consider the asymptotic performance of OLR when the BS uses a large antenna array. Similar to other linear receivers, for OLR, the transmit power of each user can be made inversely proportional to the square-root of the number of antennas with no performance degradation. II. MULTICELL MU-MIMO SYSTEM MODEL Consider the uplin transmission of a multicell MU-MIMO system with L cells. Each cell includes one BS equipped with antennas, and K single-antenna users. We assume that L BSs share the same frequency band. The received vector at the lth BS is given by y l L i G li x i + n l where G li C K is the channel matrix between the lth BS and K users in the ith cell; x i C K is transmitted vector of K users in the ith cell the average power transmitted by each user is ; and n l C is the AWG vector, distributed as n l C0,I. Here we assume equal power allocation for all users. ote that this assumption does not affect the obtained results, and can provide a lower bound on the performance of practical systems, where power control is being implemented. The channel matrix, G li, models independent fast fading, path-loss attenuation, and log-normal shadow fading. Its elements g lim are given by g lim h lim βli, m,,..., where h lim C0, is the fast Rayleigh fading coefficient from the th user in the ith cell to the mth antenna of the lth BS, while

3 A. Channel Estimation To detect the signals transmitted from K users in its cell, the BS has to have nowledge of channel state information CSI. This CSI can be estimated at the BS. In practice, a standard way of estimating the channels is using uplin pilots. During a part of the coherence interval of the channel, all users simultaneously transmit pilot sequences of length τ symbols τ is smaller than the coherence time of the channel. In each cell, each user is assigned an orthogonal pilot sequence. We assume that the same set of pilot sequences is used in all cells. This assumption comes from the fact that, in practical cellular networs, the channel coherence time typically is not long enough to allow for orthogonality between the pilots in different cells. The MMSE channel estimates for the -th column of the channel matrix G li is given by 5] ĝg li β li β lj + g τp lj + w l 3 p τpp j j where w l C0,I, and p p is the transmit power of each pilot symbol. From 3, we see that ĝg li β li β ll ĝg ll,or ĜG li ĜG ll D i 4 where D i diag { βli β ll, β li β ll,..., β lik β llk }, and ĜG li ĝg li... ĝg lik ]. We can now decompose g li as g li ĝg li + ε li where ε ll is the estimation error. We have βli ĝg li C 0, L j β lj +/τp p I M and β li ε li C 0,β li L j β I M. lj +/τp p Since we use MMSE channel estimation, the channel estimate ĝg li is independent of the estimation error ε li 6]. B. Linear Receivers We consider linear detection schemes at the BS. Let A l be an K linear detection matrix which depends on the channel estimates ĜG li, i,..., L. From 4, A l can be represented as a function of ĜG ll. The lth BS processes its received signal by multiplying it by A H l as follows r l A H l y l A H l G li x i + A H l n l. 5 i Let a l be the th column of A l. Then the th element of r l is r l a H lĝg ll x l + + a H l K j ĜG li x i + a H l i l a H lĝg llj x lj + a H le ll x l E li x i + a H ln l 6 i l where x l is the th element of x l and E li ε li... ε lik ]. Since ĜG li and E li are independent, the SIR of the uplin transmission from the th user in lth cell to its BS is given by SIR a H lĝg ll K j a H lĝg llj +pu a H l R L ε a l + i l ah lĝg li + a l 7 where { } R ε E E li E H li γ li I M 8 and where γ li i i K β li βli β lj +. 9 τp p j ote that the above SIR can be approached if the message is encoded over many realizations of all sources of randomness in the model noise and channel estimate error 4]. III. OPTIMAL LIEAR RECEIVERS In this section, we derive a novel OLR which maximizes the received SIR given by 7. We then pursue a statistical characterization of the corresponding optimal SIR. Substituting 8 into 7, and after some algebraic manipulations, we obtain a H SIR lĝg ll a H l Ξ 0 a l where Ξ K j ĝg llj ĝg H llj+ L i l ĜG li ĜG H li + Using Cauchy-Schwarz s inequality, we have a H lĝg ll SIR a H l Ξ a l a H l Ξ a l a H l Ξ/ Ξ / ĝg ll a H l Ξ a l a H l Ξ/ Ξ / ĝg ll L i γ li+ I M. Ξ / ĝg ll. The equality holds when a l cξ ĝg ll, for any c C, c 0. ote that the choice of c does not affect the system performance. Therefore the optimal linear detector at the lth BS is given by a l cξ ĝg ll. Remar : ote that in typical MMSE receivers, the intercell interference is treated as uncorrelated noise, such that a l ĜG ll ĜG H { } ll +E E ll E H ll + i l { } E G li G H li + I M ĝg ll. 3 By contrast, our OLR taes the correlation between the channel estimate and the intercell interference into account. 60

4 This improves the system performance, especially in strong intercell interference conditions. Remar : From and 3, we can see that when the intercell interference is small, the typical MMSE approaches the OLR. Furthermore, it is well nown that the performances of MRC and ZF approach the performance of MMSE at low and high SRs, respectively. Therefore, the performance analysis of our proposed OLR is generic, i.e., it can be used to analyze the performance of systems with linear receivers such as MRC, ZF, and MMSE. ote that, to the best of the authors nowledge, even with MMSE receivers, there is no analysis of the considered multicell MU-MIMO system for finite. From, with OLR, the SIR of the uplin transmission from the th user is given by SIR opt, Ξ / ĝg ll. 4 Using a similar methodology as in 7], we obtain SIR opt, ρ 5 where L L ρ ĝg H ll ĜG li ĜG H li + γ li + I ĝg p ll. u i i From 4, we have L ρ ĝg H ll ĜG ll DĜG H ll + γ li + I ĝg p ll 6 u i where D L i D i. Using the matrix inversion lemma, and the property AB B A, where A and B are invertible square matrices, we obtain L ρ ĜG ll DĜG H ll + γ li + I ĜG ll ĜG H ll i ĜG H ll ĜG ll L i γ + D li + D ] DĜG H ll ĜG ll D L i γ + D li +. 7 Substituting 7 into 5, we obtain SIR opt, D ] ] + DĜG H ll ĜG ll D L +D i γ li+ pu D ] X D ] 8 X + where X ]. 9 D /ĜGH ll ĜG ll D / L + I i γ li+ K pu The probability density function PDF of X can be approximated by a Gamma distribution as follows 8] p X x xα e x/θ Γα θ α 0 where Γ is the Gamma function 9, Eq ], while K ++K μ α K ++K κ θ K ++K κ K ++K μ L i γ li + L i β li L i β li + τp p and where μ and κ are determined by solving the following equations: μ K κ K + K i,i K i,i τ i K τi K K i,i + K μ + τ i K τi K L j β lji L j β lji+ + K μ + τ i μ K + + K μ + where τ i τpp L By using the j γ lj+. pu approximate distribution of X, we can obtain the approximate distribution of SIR opt, as follows. Proposition : The approximate cumulative distribution function CDF and PDF of SIR opt, are respectively given by 3 and 4 shown at the bottom of the page, where γ a, x is the incomplete gamma function 9, Eq ]. F SIRopt, z p SIRopt, z Γα γ α,, otherwise D] α zα Γα θ α 0, otherwise θ D] z D] z D] exp D] z] α + θ D] z D] z D] 3 4 6

5 Proof: From 8, we have D ] F SIRopt, z Pr X D ] z X + D Pr ] D ] z X z Pr X, otherwise F X D] z D] z D] D] z D] z D], otherwise where F X x is the CDF of X which is given by x y α e y/θ F X x 0 Γα θ α dy α Γα γ, xθ 5 6 where the last equation is obtained by using 9, Eq ]. Substituting 6 into 5, we arrive at the desired result 3. The PDF of SIR opt, in 4 follows immediately from differentiating 3 with respect to z using the identity dγa,x dx x a e x 9, Eq ]. The distribution of SIR opt, enables us to further evaluate the system performance. For example, the above CDF expression can be used to study the outage probability of the system, i.e., the probability that the instantaneous SIR, SIR opt,, falls below a given threshold γ th. Furthermore, we can see that, regardless of the transmit power and channel realizations, the SIR for optimal linear receivers is always upper bounded by D], where D] L i l β li/β ll which represents the effect of interference from other cells. This result implies that with OLRs, the pilot contamination effect persists. IV. PERFORMACE AALYSIS FOR OPTIMAL LIEAR RECEIVERS A. Achievable Rate Given 8, the uplin achievable ergodic rate from the th user in the lth cell to its BS is given by R opt, E {log + SIR opt, } 7 D E {log ] + X } D ] 8 X + { } E log D ] + D ]. 9 /X + By using the approximate distribution of X given by 0, we derive a closed-form approximation for the uplin achievable ergodic rate as follows. Proposition : The uplin achievable ergodic rate of the th user in the lth cell for an OLR can be approximated as R opt, log e α Γα G,3 3, θ,,, 0 log e Γα G,3 3, θ θ D] α,, 30, 0 where G m,n p,q x α,...,αp β,...,β q denotes the Meijer s-g function 9, Eq. 9.30]. Proof: From 9, we have R opt, E{log X +} E { log D ] } X + log x + xα e x/θ Γα θ α dx Define 0 0 I a, α, θ log D ] x α e x/θ x + Γα θ α dx. 3 0 log ax +x α e θx dx. 3 To evaluate the integral I a, α, θ, we first use 0, Eq ] to express log ax + in terms of a Meijer s G-function as follows log ax + log eg,, ax,. 33, 0 Then, using 0, Eq..4.3.], we obtain I a, α, θ log eθ α G,3 a α,, 3, θ. 34, 0 Substituting 34 into 3, we arrive at the desired result 30. In addition to the result given by Proposition, we now propose an analytical lower bound on the achievable ergodic rate which is easier to evaluate. By the convexity of log /x and using Jensen s inequality, we have R opt, R opt,, where R opt, log D ] + D ] E { +X }. 35 Proposition 3: The lower bound on the uplin achievable ergodic rate of the th user in the lth cell for an OLR can be approximated as R opt, log D ] + D ] F 0 α, ; ; θ 36 where p F q is the generalized hypergeometric function 9, Eq. 9.4.], while α and θ are given by and, respectively. Proof: From 0, we have { } x α e x/θ E X + 0 x + Γα θ α dx e/θ θ α Γ α, θ 37 where the last equality is obtained by using 9, Eq ] and Γa, x x ta e t dt being the upper incomplete gamma function 9, Eq ]. Typically, the value of α is very large and the value of θ is very small. This causes a numerical instability in the evaluation of e/θ θ α Γ α, θ. 6

6 Spectral Efficiency bits/s/hz 8,0 4,0 0,0 6,0,0 8,0 4,0 β MRC ZF MMSE OLR 0, SR db Fig.. Spectral efficiency for MRC, ZF, typical MMSE, and optimal linear receivers 0and β. Spectral Efficiency bits/s/hz 6,0,0 8,0 4,0 β 5 MRC ZF MMSE OLR 0, SR db Fig.. Spectral efficiency for MRC, ZF, typical MMSE, and optimal linear receivers 0and β 5. For this reason, we transform this formula in terms of the generalized hypergeometric function as follows: e /θ θ α Γ α, θ F 0 α, ; ; θ 38 where 38 stems from, Eqs and ]. Substituting 38 into 37, we obtain the desired result in Proposition 3. B. Large Analysis Without significant loss of generality, we assume p p. With E u /, from 8, when grows large, we have SIR opt, a.s. D ] + D ] τe u L i β li + τe u τeuβ ll L i,i l β li + 39 where a.s. denotes almost sure convergence, and the limit is obtained by using the law of large numbers. We can see that with OLR and very large antenna arrays at the BS, we can reduce the transmit power proportionally to / while maintaining a desired quality-of-service. Furthermore, we can see that the above result coincides with the asymptotic results of MRC, ZF, and MMSE in 4]. This means that when the number of BS antennas is large, MRC, ZF, and MMSE perform almost as well as OLR. V. UMERICAL RESULTS Consider a cellular networ with L 7cells. Each cell serves 0 users K 0. The large-scale fading matrix D i,i,..., 7, is chosen as shown at the bottom of the page, where β is the intercell interference factor which represents the effect of the interference from other cells. Unless otherwise stated, we choose β. Furthermore, we choose T 96, τ 0, p p, and define SR. We consider the spectral efficiency sum-rate in bits/s/hz in cell which is To examine the performance in a practical scenario, the large-scale fading coefficients are chosen as follows: we consider hexagonal cells which have a radius of 000 meters. We assume that the users are located randomly and uniformly in the cell and the distance between the user and the base station is greater than 00 meters. The large-scale fading is modelled via a log-normal distribution with a standard deviation of 8 db and the path loss exponent is equal to 4. By simulation, one realization of large-scale fading is chosen. D 0 3 diag9.08, 0.399, , , 0.550, 3.8, 0.730,.49, , 87.99] D 0 3 β diag0.035, 0., 0.00, 0.506,.007, 0.386, 0.046, 0.009, , ] D β diag.699, 0.008, , 0.8, 0.008, , 0.039, 0.039, 0.000, ] D β diag0.0684, 0.030, 0.036, 0.069, 0.540, 0.977, , , ,.0689] D β diag.5533, 0.089, 0.930, 0.0, , 0.699, 0.039, 0.305, 0.87, 0.307] D β diag0.040, 0.345, 0.034,.3590, , 0.00, 0.056, 0.073, 0.079, 0.360] D β diag.5309, , 0.39, , 0.00, , 0.489, 0.797,.8758, ]. 63

7 defined as S τ T K R 40 where T is the coherence time interval in symbols and R is the achievable uplin ergodic rate of the th user in cell. Figures and show the spectral efficiency for OLR, typical MMSE, ZF, and MRC, with different intercell interference factors β and β 5, respectively. Here we choose 0. We can see that at low SR and hence, low spectral efficiency, MRC performs as well as OLR, and better than ZF. However, at high SR, ZF performs better than MRC. OLR always performs the best across the SR range. In particular, we can see that when the effect of intercell interference increases, the performance gap between OLR and typical MMSE increases, thereby revealing the optimality of the proposed receiver. This is due to the fact that the OLR taes the correlation between intercell interference and the channel estimate into account. To validate the tightness of our analytical approximation, we consider the approximately bounded and the exact spectralefficiencies of OLR as Fig. 3. The Analytical Approximation curves are obtained by using Proposition 3, and the Simulation curves are generated from the outputs of a Monte- Carlo simulation using 8. We can see that our bound is very tight, especially at large. At high SR, we cannot improve the system performance by using more power. The reason is that when we increase the transmit power then the interference from other cells also increases, and hence the system is interference-limited. To improve the system performance, instead of increasing the transmit power, we can deploy more antennas at the BS. We can see that when increases the spectral efficiency increases significantly. VI. COCLUSIO In this paper, we proposed a novel OLR for multicell MU- MIMO systems where the channel is estimated using uplin training. Our proposed OLR taes the correlation between the channel estimate and the interference into account. Therefore, the OLR is always better than MRC, ZF, or typical MMSE receivers, and its performance is very close to the latter. We derived the statistical distribution of the received SIR for this OLR. We then deduced an analytical approximation of the exact and lower bound on the achievable ergodic rate. The latter approximation is very tight, especially at large number of BS antennas. Furthermore, as for MRC, ZF, and MMSE, we showed that for OLR, large antenna arrays enable us to cut the transmit power of each user proportionally to /. Spectral Efficiency bits/s/hz K τ 0 T 96 Simulation Analytical Approximation SR db Fig. 3. Spectral-efficiency for the optimal linear receiver L 7, K τ 0, andt 96. REFERECES ] T. L. Marzetta, oncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol. 9, no., pp , ov. 00. ] F. Ruse, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Signal Process. Mag., to appear. Online]. Available: 3] J. Jose, A. Ashihmin, T. L. Marzetta, and S. Vishwanath, Pilot contamination and precoding in multi-cell TDD systems, IEEE Trans. Wireless Commun., vol. 0, no. 8, pp , Aug. 0. 4] H. Q. go, E. G. Larsson, and T. L. Marzetta, Energy and spectral efficiency of very large multiuser MIMO systems, IEEE Trans. Commun., submitted. Online]. Available: 5] J. Hoydis, S. ten Brin, and M. Debbah, Massive MIMO in the UL/DL of cellular networs: How many antennas do we need? IEEE J. Sel. Areas Commun., 0, accepted. 6] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Englewood Cliffs, J: Prentice Hall, ]. Kim, Y. Lee, and H. Par, Performance analysis of MIMO system with linear MMSE receivers, IEEE Trans. Wireless Commun., vol. 7, no., pp , ov ] L. Hong and A. G. Armada, Bit error rate performance of MIMO MMSE receivers in correlated Rayleigh flat-fading channels, IEEE Trans. Veh. Technol., vol. 60, no., pp , Jan. 0. 9] I. S. Gradshteyn and I. M. Ryzhi, Table of Integrals, Series, and Products, 7th ed. San Diego, CA: Academic, ] A. P. Prudniov, Y. A. Brychov, and O. I. Marichev, Integrals and Series. ew Yor: Gordon and Breach Science, 990, vol. 3. ] Wolfram, The Wolfram functions site. Online]. Available: ACKOWLEDGMET The wor of H. Q. go and E. G. Larsson was supported in part by the Swedish Research Council VR, the Swedish Foundation for Strategic Research SSF, and ELLIIT. The wor of M. Matthaiou was supported in part by the Swedish Governmental Agency for Innovation Systems VIOVA within the VI Excellence Center Chase. 64