Enhanced Max-Min SINR for Uplink Cell-Free Massive MIMO Systems

Transcription

1 Enhaned Max-Min SINR for Uplin Cell-Free Massive MIMO Systems Manijeh Bashar, Kanapathippillai Cumanan, Alister G. Burr,Mérouane Debbah, and Hien Quo Ngo Department of Eletroni Engineering, University of Yor, UK Large Networs and Systems Group, CentraleSupéle, Université Paris-Salay, Frane Mathematial and Algorithmi Sienes Lab, Huawei Tehnologies Co., Ltd., Frane Shool of Eletronis, Eletrial Engineering and Computer Siene, Queen s University Belfast, UK. mb1465, anapathippillai.umanan, alister.burr}@yor.a.u m.debbah@entralesupele.fr, merouane.debbah@huawei.om, hien.ngo@qub.a.u Abstrat In this paper, we onsider the max-min signal-tointerferene plus noise ratio SINR) problem for the uplin transmission of a ell-free Massive multiple-input multipleoutput MIMO) system. Assuming that the entral proessing unit CPU) and the users exploit only the nowledge of the hannel statistis, we first derive a losed-form expression for uplin rate. In partiular, we enhane or maximize) user fairness by solving the max-min optimization problem for user rate, by power alloation and hoie of reeiver oeffiients, where the minimum uplin rate of the users is maximized with available transmit power at the partiular user. Based on the derived losed-form expression for the uplin rate, we formulate the original user max-min problem to design the optimal reeiver oeffiients and user power alloations. However, this maxmin SINR problem is not jointly onvex in terms of design variables and therefore we deompose this original problem into two sub-problems, namely, reeiver oeffiient design and user power alloation. By iteratively solving these sub-problems, we develop an iterative algorithm to obtain the optimal reeiver oeffiient and user power alloations. In partiular, the reeiver oeffiients design for a fixed user power alloation is formulated as generalized eigenvalue problem whereas a geometri programming GP) approah is utilized to solve the power alloation problem for a given set of reeiver oeffiients. The performane of the proposed max-min SINR algorithm is evaluated in terms of the ahieved user uplin rate. Numerial results onfirm a three-fold inrease in system rate over existing shemes in the literature. Keywords: Cell-free Massive MIMO, onvex optimization, max-min SINR problem, geometri programming, generalized eigenvalue problem. I. INTRODUCTION Massive multiple-input multiple-output MIMO) is one of the most promising tehniques for 5th Generation 5G) networs due to its potential for signifiant rate enhanement and spetral as well as energy effiieny [1], []. This Massive MIMO tehnique an be implemented through three alternative ways [3]: i) single ell Massive MIMO, in whih a large number of users are simultaneously served by a single base station BS) with a muh larger number of antennas; ii) multi ell Massive MIMO, where the simulation area is divided into several ells with one BS in eah ell where the BSs jointly establish ommuniations with distributed users in the ells; and iii) ell-free Massive MIMO: there is no ell onept and the simulation area is not divided into ells. In ell-free Massive MIMO, randomly distributed aess points APs) jointly serve distributed users [3]. In this paper, we propose a max-min signal-to-interferene plus noise ratio SINR) approah for an uplin ell-free Massive MIMO system. In [4], the authors investigate the problem of max min SINR in a single-ell Massive MIMO system. A similar max-min SINR problem refereed to SINR balaning has been onsidered for ognitive radio networ in [5] [7]. In [8], the same max-min SINR problem is onsidered through appropriate user power alloation where a bisetion searh method is utilized to determine the optimal solution. However, a novel approah to signifiantly improve all users performane is proposed in this paper by designing optimal reeiver oeffiients and user power alloation. By employing maximal ratio ombining MRC) at the reeiver, we first derive a losed-form expression for the average uplin rate of the users. Based on these user power alloations and reeiver oeffiients, we formulate the orresponding maxmin SINR problem, whih is not jointly onvex in terms of the design parameters. In order to realize a solution for this non-onvex problem, we deompose the original problem into two sub-problems: reeiver oeffiient design and user power alloation. An iterative algorithm is proposed that suessively solves these two sub-problems while one of the design variables i.e., user power alloation or reeiver oeffiients) is fixed. The reeiver oeffiient design is formulated into a generalized eigenvalue problem [9] whereas a geometri programming GP) approah [10] is exploited to solve the user power alloation problem. The performane of the proposed sheme in terms of the user rate is signifiantly higher than that of the sheme proposed in [8]. The ontributions and the results of our wor are summarized as follows: 1) For the onsidered ell-free Massive MIMO system, we derive the average user rate in the uplin. ) Based on the derived user rate, we propose a novel max-min SINR approah to signifiantly improve the SINR performane in terms of the ahieved user rate. The original max-min problem formulation is not onvex and therefore we deompose the original problem into two sub-problems and propose an iterative algorithm to yield the optimal solution.

2 where eah element of n p,m, n p,m CN0, 1), denotes the noise at the mth antenna, p p represents the normalized signalto-noise ratio SNR) of eah pilot sequene whih we define in Setion V), and m is given by Figure 1. The uplin of a ell-free Massive MIMO system with K users and M APs. The dashed lines denote the uplin hannels and the solid lines present the bahaul lins from the APs to the entral proessing unit CPU). 3) The user power alloation and the reeiver oeffiient design sub-problems are solved through the GP approah and the generalized eigenvalue problem, respetively. 4) Finally, we present omplexity analysis of the proposed sheme. 5) Numerial results are provided to validate the superiority of the proposed algorithm in omparison with the sheme proposed in [8]. Outline: The reminder of the paper is organized as follows. Setion II desribes the system model. Setion III investigates the orresponding performane analysis. The proposed maxmin SINR approah is presented in Setion IV. Numerial results and omplexity analysis are provided in Setion V and Setion VI, respetively. Finally, Setion VII onludes the paper. II. SYSTEM MODEL We onsider uplin transmission in a ell-free Massive MIMO system with M randomly distributed single-antenna APs and K randomly distributed single-antenna users in the area, as shown in Fig. 1. The hannel oeffiient between the th user and the mth AP, g m, is modeled as [8] g m = β m h m, 1) where β m denotes the large-sale fading and h m CN0, 1) represents small-sale fading between the th user and the mth AP. A. Uplin Channel Estimation In order to estimate the hannel oeffiients in the uplin, the APs employ an minimum mean square error MMSE) estimator. All pilot sequenes used in the hannel estimation phase are olleted in a matrix Φ C τ K, where τ is the length of the pilot sequene for eah user and the th olumn, φ, represents the pilot sequene used for the th user. After performing a de-spreading operation, the MMSE estimate of the hannel oeffiient between the th user and the mth AP is given by [8] ĝ m = m τp p g m + K τp p g m φ H φ +φ H n p,m,) τpp β m m = K τp p =1 β m φ H φ +1. 3) The estimated hannels in ) are used by the APs to design the reeiver oeffiients and determine power alloations at users to maintain user fairness. In this paper, we investigate the ases of both random pilot assignment and orthogonal pilots in ell-free Massive MIMO. Here the term orthogonal pilots refers to the ase where unique orthogonal pilots are assigned to all users, while in random pilot assignment eah user is randomly assigned a pilot sequene from a set of orthogonal sequenes of length τ < K), following the approah of [8], [11]. B. Uplin Data Transmission In this subsetion, we onsider the uplin data transmission, where all users send their signals to the APs. The transmitted signal from the th user is represented by x = q s, 4) where s E s } = 1) and q denote respetively the transmitted symbol and the transmit power at the th user. The reeived signal at the mth AP from all users is given by y m = K ρ g m q s + n m, 5) =1 where n m CN0, 1) is the noise at the mth AP. In addition, MRC is employed at the APs. More preisely, the reeived signal at the mth AP, y m, is first multiplied with ĝm. The resulting ĝm y m is then forwarded to the CPU for signal detetion. In order to improve ahievable rate, the forwarded signal is further multiplied by a reeiver filter oeffiient at the CPU. The aggregated reeived signal at the CPU an be written as r = u m ĝmy m 6) = ρ K =1 u m ĝmg m q s + u m ĝmn m. By olleting all the oeffiients u m, m orresponding to the th user, we define u =[u 1,u,,u M ] T and without loss of generality, it is assumed that u =1. The optimal solution of u, q, for the onsidered max-min SINR approah is investigated in Setion IV. III. PERFORMANCE ANALYSIS In this setion, we derive the average user rate for the onsidered system model in the previous setion by following a similar approah to that in [8]. Note that the main differene between the proposed approah and the sheme in [8] is the new set of reeiver oeffiients whih are introdued at

3 the CPU to improve the ahievable user rates. The benefits of the proposed approah in terms of ahieved user uplin rate is demonstrated through numerial simulation results in Setion V. In deriving the ahievable rates of eah user, it is assumed that the CPU exploits only the nowledge of hannel statistis between the users and APs in deteting data from the reeived signal in 7). Without loss of generality, the aggregated reeived signal in 7) an be written as = M } ρe u m ĝmg m q s 7) r }} DS + M M }) ρ u m ĝmg m q E u m ĝmg m q } } BU K M + ρ u m ĝmg m q s + u m ĝmn m, } } IUI } } TN where DS and BU denote the desired signal DS) and beamforming unertainty BU) for the th user, respetively, and IUI represents the inter-user-interferene IUI) aused by the th user. In addition, TN aounts for the total noise TN) following the MRC detetion. The average SINR of the reeived signal in 7) an be defined by onsidering the worst-ase of the unorrelated Gaussian noise as follows [8]: SINR UP DS = E BU }+ K E IUI }+E TN }. 8) Based on the SINR definition in 8), the ahievable uplin rate of the th user is defined in the following theorem. Theorem 1. By employing MRC detetion at APs, the ahievable uplin rate of the th user in the Cell-free Massive MIMO system with K randomly distributed single-antenna users and M single-antenna APs is given by 9) defined at the beginning of the next page). Note that in 9), we have Γ =[γ 1,γ,,γ M ] T, u =[u 1,u,,u M ] T, 10a) 10b) Δ =[ γ 1β 1, γ β,, γ Mβ M ] T, β 1 β β M 10) R = diag [γ 1,γ,,γ M ], 10d) D = diag [β 1 γ 1,β γ,,β M γ M ], 10e) Proof: Please refer to Appendix A. IV. PROPOSED MAX-MIN SINR SCHEME In this setion, we formulate the umax-min SINR problem in ell-free massive MIMO, where the minimum uplin user rate between users is maximized while satisfying the transmit s Algorithm 1 Proposed Algorithm to Solve P 1 1. Initialize q 0) =[q 0) 1,q0),,q0) K ], i =1. Repeat 3. i = i Set q i) = q i 1) and find the optimal reeiver oeffiients U i) =[u i) 1, ui),, ui) K ] through solving the generalized eigenvalue Problem P in 1) 5. Compute q i+1) through solving Problem P 4 in 14). 6. Go ba to Step 3 and repeat until required auray. power onstraint at eah user. This max-min rate problem an be formulated as follows: P 1 :max q,u min R 11) =1,,K s.t. u =1,, 0 q p ) max,, where p ) max is the maximum transmit power available at user. Problem P 1 is not jointly onvex in terms of u and power alloation q,. Therefore, this problem annot be diretly solved through existing onvex optimization software. To tale this non-onvexity issue, we divide the original Problem P 1 into two sub-problems: reeiver oeffiient design i.e. u ) and the power alloation problem. To obtain a solution for Problem P 1, these sub-problems are alternately solved as explained in the following subsetions. A. Reeiver Coeffiients Design In this subsetion, we solve the reeiver oeffiient design problem to maximize the uplin rate of eah user for a given set of transmit power alloation at all users. These oeffiients i.e., u, ) an be obtained by interdependently maximizing the uplin SINR of eah user. Hene, the optimal oeffiients for all users for a given set of transmit power alloation an be determined by solving the following optimization problem: P : max u u H K ) u H q Γ Γ H u, q φh φ Δ Δ H + K =1 q D + 1 R ρ )u s.t. u =1,. 1) Problem P is a generalized eigenvalue problem [9], where the optimal solutions an be obtained by determining the generalized eigenvalue of the matrix pair A = q Γ Γ H and B = K q φh φ Δ Δ H + K =1 q D + 1 ρ R orresponding to the maximum generalized eigenvalue. B. Power Alloation In this subsetion, we solve the power alloation problem for a set of fixed reeiver oeffiients. The power alloation problem an be formulated into the following max-min problem: P 3 :max min SINR 13) q =1,,K

4 R =log 1+ u H K ) u H q Γ Γ H u q φ H φ Δ Δ H + K =1 q D + 1 ) ρ R u. 9) Table I COMPUTATIONAL COMPLEXITY OF DIFFERENT PROBLEMS Problems Required arithmeti operations Problem P, given by 1) OKM 3 ) Problem P 4, given by 14) OK 7 ) The sheme in [8] log tmax t min ɛ ) OK 4 ) s.t. 0 q p ) max,, Without loss of generality, Problem P 3 an be rewritten by introduing a new sla variable as P 4 : max t,q t 14) s.t. 0 q p ) max,, SINR t,. Proposition 1: Problem P 4 an be formulated into a GP. Proof: Please refer to Appendix B. Therefore, this problem an be effiiently solved through existing onvex optimization software. Based on these two sub-problems, an iterative algorithm is developed by alternately solving eah sub-problem in eah iteration. The proposed algorithm is summarized in Algorithm 1. V. CONVERGENCE ANALYSIS In this setion, the onvergene analysis of the proposed Algorithm 1 is provided. Two sub-problems are alternately solved to determine the solution to Problem P 1. At eah iteration, one of the design parameters is determined by solving the orresponding sub-problem while other design variable is fixed. Note that eah sub-problem provides an optimal solution for the other given design variable. At the ith iteration, the reeiver filter oeffiients u i), are determined for a given power alloation q i) and similarly, the power alloation q i+1) is updated for a given set of reeiver filter oeffiients u i),. The optimal power alloation q i+1) obtained for a given u i) ahieves an uplin rate greater than or equal to that of the previous iteration. In addition, the power alloation q i) is also a feasible solution in determining q i+1) as the reeiver filter oeffiients u i+1), are determined for a given q i). This reveals that the ahieved uplin rate monotonially inreases with eah iteration, whih an be also observed from the simulation results presented in Fig. 4. As the ahievable uplin max-min rate is upper bounded by a ertain value for a given set of per-user power onstraints, the proposed algorithm onverges to a partiular solution. Fortunately, the proposed Algorithm 1 onverges to the optimal solution, as we will prove by establishing the uplin-downlin duality in the following setion. VI. COMPLEXITY ANALYSIS Here, we provide the omputational omplexity analysis for the proposed Algorithm 1, whih solves a generalized eigenvalue problem P and a GP onvex optimization problem) P 4 at eah iteration. For the reeiver filter oeffiient design in P, given by 1), an eigenvalue solver requires approximately OKM 3 ) flops [1], [13]. In addition, a standard GP in Problem P 4, defined in 14), an be solved with omplexity equivalent to OK 7 ) [14, Chapter 10]. However, for the sheme in [8], the iterative bisetion searh method in [8] solves a Seond order one programming SOCP) at eah iteration. The omplexity of SOCP is OK 4 ) in eah iteration [15], [16]. Note that the total number of iterations to solve a bisetion searh method in [8] is given by log tmax tmin ɛ ), where ɛ referes to a predetermined threshold [10]. The number of arithmeti operations required for Algorithm 1 are provided in Table I. VII. NUMERICAL RESULTS AND DISCUSSION In this setion, we provide numerial simulation results to validate the performane of the proposed max-min SINR approah with different parameters. A ell-free Massive MIMO system with M APs and K single antenna users is onsidered in a D D simulation area, where both APs and users are randomly distributed. In the following subsetions, we define the simulation parameters and then present the orresponding simulation results. A. Simulation Parameters To model the hannel oeffiients between users and APs, the oeffiient β m is given by β m = PL m.10 σ sh z m 10 where PL m is the path loss from the th user to the mth AP, and 10 σ sh z m 10 denotes the shadow fading with standard deviation σ sh, and z m N0, 1) [8]. The noise power is given by P n = BW B T 0 W, where BW = 0 MHz denotes the bandwidth, B = represents the Boltzmann onstant, and T 0 = 90 Kelvin) denotes the noise temperature. Moreover, W = 9dB, and denotes the noise figure [8]. It is assumed that that Pp and ρ denote the pilot sequene and the uplin data, respetively, where P p = P p P n and ρ = ρ P n. In simulations, we set Pp = 100mW and ρ = 100mW. Similar to [8], we suppose the simulation area is wrapped around at the edges whih an simulate an

5 1.5 Figure. The umulative distribution of the per-user uplin rate, with orthogonal and random pilots for M =60, K =0and D =1m. Min-user uplin rate bits/s/hz) Channel 1 Channel Channel 3 Channel 4 Channel Number of iterations Figure 4. The onvergene of the proposed max-min SINR approah for M =60, K =0, τ =10, and D =1m. Cumulative distribution The sheme in [4] Proposed sheme Orthogonal pilots Random pilot assignment, τ= Min-user uplin rate bits/s/hz) Figure 3. The umulative distribution of the per-user uplin rate, with random pilots for M = 100, K =40, τ =0, and D =1m. area without boundaries. Hene, the square simulation area has eight neighbors. We evaluate the rate of the system over 300 random realizations of the loations of APs, users and shadowing. B. Simulation Results In this subsetion, we investigate the effet of the max-min SINR problem on the system performane. Fig. ompares the umulative distribution of the ahievable uplin rates for our proposed algorithm with the power alloation sheme in [8], for three ases of orthogonal pilots, random pilots with τ =10and τ =5for the length of pilot sequene. In Fig., M =60APs and K =0users are randomly distributed through the simulation area of size 1 1 m.as the figure shows, the performane of the proposed sheme is almost three times than that of the sheme in [8]. In Fig. 3, we ompare the performane of the proposed maxmin SINR approah with the sheme in [8] for the ase of M = 100 APs, K = 40 users and τ = 0 as the length of the pilot sequene. Fig. 3 shows the superiority of the proposed iterative algorithm over the power alloation sheme in [8]. Moreover, Fig. 3 demonstrates that the rate of the proposed max-min SINR approah is more onentrated around the median. Fig. 4 investigates the onvergene of the proposed max-min SINR algorithm for a set of different hannel realizations. The figure shows that the proposed algorithm onverges after a few iterations, while the minimum rate of the users inreases with the iteration number. VIII. CONCLUSIONS We have onsidered the max-min optimization problem in ell-free Massive MIMO systems, and propose an effiient solution that maximizes the smallest of the uplin rate of the users. We propose to divide the original max-min problem into two sub-problems whih an be iteratively solved by exploiting generalized eigenvalue problem and GP. The simulation results showased the effetiveness of the proposed sheme in terms of maximising the smallest of the uplin rate of the users ompared with existing shemes. APPENDIX A PROOF OF THEOREM 1 The desired signal for the user is given by M } DS =E u m ĝmg m q Hene, = M q u m γ m. 15) M ) DS = q u m γ m. 16)

6 Moreover, the term E BU } an be obtained as E BU } = ρe u m ĝ mg m q M } ρe u m ĝmg m q = ρ q u m E ĝmg m E ĝmg m } }) M = ρq u mγ m β m, 17) where the last equality omes from the analysis in [8, Appendix A], Appendix A] and using the following fat; γ m = E ĝ m } = τp p β m m. The term E IUI } is obtained as E IUI } = ρe u m ĝmg m q = ρe m u m g m q K ) τpp g mi φ H φ i +φ H n p,m i=1 = ρq E m u m g m ñ m 18) }} A + ρτp p E q K ) m u m g m g mi φ H φ i, i=1 }} B where the third equality in 18) is due to the fat that for two independent random variables X and Y and EX} =0, we have E X + Y } = E X } + E Y } [8, Appendix A]. Sine ñ m = φ H n p,m CN0, 1) is independent from the term g m similar to [8, Appendix A], the term A in 18) immediately is given by A = q M m u m β m.the term B in 18) an be obtained as B = τp p q E m u m g m φ H φ }} 19) C K + τp p q E m u m g m g mi φ H φ i. i } } D The first term in 19) is given by C = τp p q E m u m g m φ H φ =τp p q φ H φ E φ H φ M n m = τp p q φ H φ M mu mβ m + τp pq m n u m u n g m g n M mu mβm + q φ H φ M u m γ m β m β m ), 0) where the last equality is derived based on the fat γ m = τpp β m m. The seond term in 19) an be obtained as K D = τp p q E m u m g m g mi φ H φ i i M K = τp p q mu mβ m β mi φ H φ i. 1) i Hene, 18) an be written as and E IUI } = q + τp p q φ H φ + τp p q mu mβ m } } C 1 M mu mβm K mu mβ m β mi φ H φ i i + q φ H φ M C = τp p q φ H φ u m γ m β m β m M mu mβm ) ), 3) + τp p q mu mβ m β mi φ H φ i. 4) i } } C 3

7 For the last term of 4), we have K C 3 = τp p q mu mβ m β mi φ H φ i i M K = τp p q u m m β m m β mi φ H φ i = τp p q τp p q u m mβ m i=1 ) φ H φ u m m β m β m q u m mβ m u m mβ m φ H φ, 5) where in the last step, we used equation 3). As a result, C 1 + C = M τp p q u m mβ m β m. Then finally we have E IUI } M ) = ρq u mβ m γ m 6) + ρq φ H φ M u m γ m β m β m ). The total noise for the user is given by E TN } M =E u m ĝmn m = u mγ m,7) where the last equality is due to the fat that the terms ĝ m and n m are unorrelated. Finally, by substituting 16), 17), 7) and 7) into 8), SINR of th user is obtained by 9), whih ompletes the proof of Theorem 1. APPENDIX B PROOF OF PROPOSITION 1 The standard form of GP is defined as follows [10]: P 5 :min f 0 x) 8) s.t. f i x) 1, i =1,,m, g i x) =1, i =1,,p, where f 0 and f i are posynomial and g i are monomial funtions. Moreover, x = x 1,,x n } represent the optimization variables. The SINR onstraint in 14) is not a posynomial funtion in its form, however it an be rewritten into the following posynomial funtion: u H K ) q φh φ Δ Δ H + K =1 q D + 1 R p u ) < 1 u H q Γ Γ H u t,. 9) By applying a simple transformation, 9) is equivalent to the following inequality: K K a q + b q + < 1 t, 30) q 1 =1 where a = uh φh φ Δ Δ H )u u H Γ Γ H )u, b = uh D u u H Γ Γ H )u, u and finally = H R u pu H Γ Γ H )u. The transformation in 30) shows that the left-hand side of 9) is a polynomial funtion. Therefore, the power alloation Problem P 4 is a standard GP onvex problem), where the objetive funtion and onstraints are monomial and polynomials, whih ompletes the proof of Proposition 1. REFERENCES [1] A. Zappone, L. Sanguinetti, G. Bai, E. A. Jorswie, and M. Debbah, Energy-effiient power ontrol: a loo at 5G wireless tehnologies, IEEE Trans. Signal Proess., vol. 64, no. 7, pp , Apr [] E. Björnson, E. A. Jorswie, M. Debbah, and B. Ottersten, Multiobjetive signal proessing optimization: the way to balane onfliting metris in 5G systems, IEEE Signal Proess. Mag., vol. 31, no. 6, pp. 14 3, Ot [3] H. Yang and T. L. Marzetta, Capaity performane of multiell largesale antenna systems, in Pro. IEEE Allerton Conferene, Ot. 013, pp [4] H. Sifaou, A. Kammoun, L. Sanguinetti, M. Debbah, and M. Alouini, Max min SINR in large-sale single-ell MU-MIMO: asymptoti analysis and low-omplexity transeivers, IEEE Trans. Signal Proess., vol. 65, no. 7, pp , Apr [5] K. Cumanan, R. Krishna, L. Musavian, and S. Lambotharan, Joint beamforming and user maximization tehniques for tehniques for ognitive radio networs based on branh and bound method, IEEE Trans. Wireless Commun., vol. 9, no. 10, pp , Ot [6] K. Cumanan, L. Musavian, S. Lambotharan, and A. B. Gershman, SINR balaning tehnique for downlin beamforming in ognitive radio networs, IEEE Signal Proess. Lett., vol. 17, no., pp , Feb [7] K. Cumanan, Y. Rahulamathavan, S. Lambotharan, and Z. Ding, MMSE based beamforming tehniques for relay broadast hannel, IEEE Trans. Veh. Tehnol., vol. 6, no. 8, pp , Ot [8] H. Q. Ngo, A. Ashihmin, H. Yang, E. G. Larsson, and T. L. Marzetta, Cell-free Massive MIMO versus small ells, IEEE Trans. Wireless Commun., vol. 16, no. 3, pp , Mar [9] G. Golub and C. V. Loan, Matrix Computations, nd ed. Baltimore, MD: The Johns Hopins Univ. Press, [10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University Press, 004. [11] H. Ahmadi, A. Farhang, N. Marhetti, and A. MaKenzie, A game theoreti approah for pilot ontamination avoidane in Massive MIMO, IEEE Wireless Commun. Lett., vol. 5, no. 1, pp. 1 15, Feb [1] C. Chien, H. Su, and H. Li, Joint beamforming and power alloation for MIMO relay broadast hannel with individual SINR onstraints, IEEE Trans. Veh. Tehnol., vol. 63, no. 4, pp , 004. [13] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. Baltimore, MD, USA: Johns Hopins Univ. Press, [14] Y. Nesterov and A. Nemirovsy, Interior-point polynomial methods in onvex programming. Studies in Applied Mathematis, SIAM, Philadelphia, [15] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, Appliations of seond-order one programming, Linear Algebra Appliat, pp , Nov [16] J. F. Sturm, Using SeDuMi 1.0, a MATLAB toolbox for optimization over symmetri ones, Optim. Meth. Softw., vol. 11, p , Aug