This is a repository copy of Mixed Quality of Service in Cell-Free Massive MIMO.

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1 This is a repository copy of ixed Quality of Service in Cell-Free assive IO. White Rose Research Online URL for this paper: Version: Accepted Version Article: Bashar,., Cumanan, K. orcid.org/ , Burr, A. G. orcid.org/ et al. more authors 018 ixed Quality of Service in Cell-Free assive IO. IEEE. ISSN Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Taedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by ing eprints@whiterose.ac.u including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.u

2 1 ixed Quality of Service in Cell-Free assive IO anijeh Bashar, Student ember, IEEE, Kanapathippillai Cumanan, ember, IEEE, Alister G. Burr, ember, IEEE, Hien Quoc Ngo, ember, IEEE, and H. Vincent Poor, Fellow, IEEE Abstract Cell-free massive multiple-input multiple-output IO is a potential ey technology for fifth generation wireless communication networs. A mixed quality-of-service QoS problem is investigated in the uplin of a cell-free massive IO system where the minimum rate of non-real time users is maximized with per user power constraints whilst the rate of the real-time users RTUs meet their target rates. First an approximated uplin user rate is derived based on available channel statistics. Next, the original mixed QoS problem is formulated in terms of receiver filter coefficients and user power allocations which can iteratively be solved through two subproblems, namely, receiver filter coefficient design and power allocation, which are dealt with using a generalized eigenvalue problem and geometric programming, respectively. Numerical results show that with the proposed scheme, while the rates of RTUs meet the QoS constraints, the 90%-liely throughput improves significantly, compared to a simple benchmar scheme. Index terms: Cell-free massive IO, geometric programming, max-min SINR, QoS requirement. I. INTRODUCTION The forthcoming 5th Generation 5G wireless networs will need to provide greatly improved spectral efficiency along with a defined quality of service QoS for real-time users RTUs. A promising 5G technology is cell-free massive multiple-input multiple-output IO, in which a large number of access points APs are randomly distributed through a coverage area and serve a much smaller number of users, providing uniform user experience [1]. The distributed APs are connected to a central processing unit CPU via high capacity bachaul lins [1] [4]. The problem of cell-free massive IO with limited bachal lins has been considered in [5] and [6]. In [1], max-min fairness power control is exploited, while paper [] studies the total energy efficiency optimization for cell-free massive IO taing into account the effect of bachaul power consumption. Different with previous wor, in this paper, we investigate a mixed qualityof-service QoS problem in which a set of RTUs requires a predefined rate and a max-min signal-to-interference-plusnoise ratio SINR is maintained between the non-real time users NRTUs. The RTUs are defined as the users of real time services such as audio-video, video conferencing, webbased seminars, and video games, which result in the need for wireless communications with mixed QoS [7]. We show that the cell-free massive IO system has the capability of satisfying QoS requirements of the RTUs while it can guarantee. Bashar, K. Cumanan and A. G. Burr are with the Department of Electronic Engineering, University of Yor, UK. {mb1465, anapathippillai.cumanan, alister.burr}@yor.ac.u. H. Q. Ngo is with the School of Electronics, Electrical Engineering and Computer Science, Queen s University Belfast, UK. hien.ngo@qub.ac.u. H. Vincent Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA. poor@princeton.edu. The wor of K. Cumanan and A. G. Burr was supported by H00-SC ARISE-015 under Grant excellent service for the NTTUs. The specific contributions of the paper are as follows: 1. An approximated SINR is derived based on the channel statistics and exploiting maximal ratio combining RC at the APs. We formulate the corresponding mixed QoS problem with a fixed QoS requirement i.e., SINR for RTUs, which need to meet their target SINRs whereas the minimum SINRs of the remaining users should be maximized.. The mixed QoS problem is not jointly convex. We propose to deal with this non-convexity issue by decoupling the original problem into two sub-problems, namely, receiver filter coefficient design and power allocation. 3. It is shown that the receiver filter design problem can be solved through a generalized eigenvalue problem [8] whereas the user power allocation problem can be formulated using standard geometric programming GP [9]. An iterative algorithm is developed to solve the optimization problem. The convergence of the proposed scheme is explored numerically. II. SYSTE ODEL We consider uplin transmission in a cell-free massive IO system with randomly distributed APs and K randomly distributed single-antenna users in the area. oreover, we assume each AP has N antennas. The channel coefficient vector between the th user and the mth AP, g m C N 1, is defined as g m = β m h m, where β m denotes the large-scale fading and h m CN0,1 represents small-scale fading between the th user and the mth AP [1]. All pilot sequences used in the channel estimation phase are collected in a matrix Φ C τ K, where τ is the length of pilot sequence for each user and the th column, φ, and represents the pilot sequence used for the th user. The minimum mean square error SE estimate of the channel coefficient between the th user and the mth AP is given by [1] ĝ m =c m τp p g m + K τp p g m φ H φ +W p,m φ, 1 where each element of W p,m, w p,m CN0,1, denotes the noise sequence at the mth antenna, p p represents the normalized signal-to-noise ratio SNR of each pilot symbol, and c m is given by c m = τppβ m τp p K =1 β m φh φ +1. In this paper, we consider 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, where s E{ s } = 1 and q denote the transmitted symbol and the transmit power at the th user. The N 1 received signal at the mth AP from all users is given by y m = ρ K =1 g m q s + n m, where each element of n m C N 1, n n,m CN0,1, is the noise at the mth AP and ρ refers to the normalized SNR.

3 III. PERFORANCE ANALYSIS In this section, in deriving the achievable rate of each user, it is assumed that the CPU exploits only the nowledge of channel statistics between the users and APs in detecting data from the received signal in. The aggregated received signal at the CPU can be written as r = u m ĝ H m y m. By collecting all the coefficients u m, m, corresponding to the th user, we define u = [u 1,u,,u ] T. To detect s, with the RC processing, the aggregated received signal in can be { rewritten as r = } ρe u m ĝ H m g m q s 3 DS + { } ρ u m ĝ H m g m q E u m ĝ H m g m q BU K + ρ u m ĝ H m g m q s + u m ĝ H m n m, } {{ } IUI } {{ } TN where DS and BU denote the desired signal DS and beamforming uncertainty BU for the th user, respectively, and IUI represents the inter-user-interference IUI caused by the th user. In addition, TN accounts for the total noise TN following the RC detection. The corresponding SINR can be defined by considering the worst-case of the uncorrelated Gaussian noise as follows [1]: DS SINR = E{ BU }+ K E{ IUI }+E{ TN }. 4 Based on the SINR definition in 4, the achievable uplin rate of the th user is defined in the following theorem: Theorem 1. The achievable uplin rate of the th user in the cell-free massive IO system with K randomly distributed single-antenna users and APs is given by 5 defined at the top of the next page. Proof: Please refer to Appendix A. Note that in 5, u = [u 1,u,,u ] T, and the following equations hold: Γ = [γ 1,γ,,γ ] T, γ m = ] τ p p p β m c m, Υ = diag [β 1 γ 1,, β γ, Λ = [ γ 1β 1, γ β,, γ β ] T, and R = β 1 β β diag[γ 1,,γ ], and γ m = E{ ĝ m } = τp p β m c m. IV. PROPOSED IXED QOS SCHEE We formulate the mixed QoS problem, where the minimum uplin user rate among NRTUs is maximized while satisfying the transmit power constraint at each user and the RTUs SINR target constraints. We assume users 1,,,K 1 are RTUs. The mixed QoS problem is given by s P 1 : max q,u min R, 6a =K 1+1,,K 0 q p max,, SINR t, = 1,,K 1 6b 6c where p max is the maximum transmit power available at user, and SINR t denotes the target SINR for the th RTU. Problem P 1 is not jointly convex in terms of u and power allocation q,. Therefore, it cannot be directly solved through existing convex optimization software. To tacle this non-convexity issue, we decouple Problem P 1 into two subproblems: receiver coefficient design i.e. u and the power allocation problem, which are explained in the following subsections. 1 Receiver Filter Coefficient Design: In this subsection, the problem of designing the receiver coefficient is considered. These coefficients i.e., u, are obtained by interdependently maximizing the uplin SINR of each user. Hence, the optimal receiver filter coefficients can be obtained through solving the following optimization problem: P : max u N u H q Γ Γ H u. N K q φh φ Λ Λ H +N K =1 q Υ +N ρ R u u H Problem P is a generalized eigenvalue problem [8], where the optimal solutions can be obtained by determining the generalized eigenvector of the matrix pair A = N q Γ Γ H and B = N K q φh φ Λ Λ H +N K =1 q Υ + N ρ R corresponding to the maximum generalized eigenvalue. Power Allocation: Next, we solve the power allocation problem for a given set of fixed receiver filter coefficients, u,. The optimal transmit power can be determined by solving the following mixed QoS problem: P 3 : max q min =K 1+1,,K 0 q p SINRUP, max,, SINR t. = 1,,K 1 7 8a 8b 8c Note that the max-min rate problem and max-min SINR problem are equivalent. Without loss of generality, Problem P 3 can be rewritten by introducing a new slac variable as P 4 : max t, 9a t,q 0 q p max,, t, = K 1 +1,,K, SINR t, = 1,,K 1 Proposition 1. Problem P 4 is a standard GP. 9b 9c 9d Proof: The SINR constraint 9c is not a posynomial functions in its form, however it can be rewritten into the following posynomial function: u H K q φh φ Λ Λ H + K =1 q Υ + 1 R ρ u < 1 q Γ Γ H u t.10 u H

4 3 R = log 1+ u H u H N q Γ Γ H u N K q φh φ Λ Λ H +N K =1 q Υ + N ρ R u. 5 Algorithm 1 Proposed algorithm to solve Problem P 1 1. Initialize q 0 = [q 0 1,q0,,q0 K ], i = 1. Repeat, i = i+1 3. Set q i = q i 1 and determine the optimal receiver coefficients U i = [u i 1, ui,, ui K ] through solving the generalized eigenvalue Problem P in 7 4. Compute q i+1 through solving Problem P 4 in 9 5. Go bac to Step and repeat until required accuracy Cumulatice Distribution =100, N= K=5, K 1 =5 =40, N=, K=, K 1 = =100, N= K=5, K 1 =5 =40, N= K=, K 1 = =1, N=80, K=, K 1 = in uplin rate bits/s/hz Figure 1. The cumulative distribution of the per-user uplin rate, for = 100,K = 5,K 1 = 5 and = 40,K =,K 1 = with D = 1 m, τ = 0, and SINR t = 1. The solid curves refer to the proposed Algorithm 1, while the dashed curves present the case u m = 1, m,, and solve Problem P 4. By applying a simple transformation, 10 can be rewritten in form of q 1 K a q + K =1 b q +c < 1 t, which shows that the left-hand side of 10 is a posynomial function. The same transformation holds for 9d. Therefore, Problem P 4 is a standard GP convex problem. Based on two sub-problems, an iterative algorithm is developed by solving both sub-problems at each iteration. The proposed algorithm is summarized in Algorithm 1. V. NUERICAL RESULTS AND DISCUSSION To model the channel coefficients between users and APs, the coefficient 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 [1]. The noise power is given by P n = BW B T 0 W, where BW = 0 Hz denotes the bandwidth, B = represents the in-user rate bits/s/hz. Channel Channel Channel 3 Channel Number of iterations Figure. The convergence of the proposed Algorithm 1 for = 40, K =, K 1 =, N =, D = 1 m, SINR t = 1, and τ = 0. Boltzmann constant, and T 0 = 90 Kelvin denotes the noise temperature. oreover, W = 9dB, and denotes the noise figure [1]. It is assumed that that P p and ρ denote the transmit powers of the pilot and data symbols, respectively, where P p = P p P n and ρ = ρ P n. In simulations, we set Pp = 00 mw and ρ = 00 mw. A cell-free massive IO system is considered with 15 APs = 15 and 6 users K = 6 who are randomly distributed over the coverage area of size 1 1 m. oreover, each AP is equipped with N = 3 antennas and we set the total number of RTUs to K 1 =, and random pilot sequences with length τ = 5 are considered. Table I presents the achievable SINRs of the users while the target SINR for both RTUs is fixed as.3. The power allocations for all users and the max-min SINR values are obtained using the proposed Algorithm 1. It can be seen from Table I that both RTUs achieve their target SINR, while the minimum SINR of the rest of the users is maximized through using Algorithm 1 If the problem is infeasible, we set SINR = 0,. Fig. 1 presents the cumulative distribution of the achievable uplin rates for the proposed Algorithm 1 the solid curves and a scheme in which the received signals are not weighted i.e. we set u m = 1, m, and solve Problem P 4, which are shown by the dashed curves. As seen in Fig. 1, the median of the cumulative distribution of the minimum uplin rate of the users is significantly increased compared to the scheme with u m = 1, m, and solving ProblemP 4. As seen in Fig. 1, the performance i.e. the 10% outage rate of the proposed scheme is almost twice that of the case with u m = 1 m,. Note Table I TARGET SINRS AND THE POWER CONSUPTION OF THE PROPOSED SCHEE, WITH = 15, N = 3, K = 6, K 1 =, τ = 5, AND D = 1 K. Achieved SINR Power Allocation q Channels RTU1 RTU NRTU1 NRTU NRTU3 NRTU4 RTU1 RTU NRTU1 NRTU NRTU3 NRTU4 Channel Channel Channel Channel

5 4 that the authors in [1] considered a max-min SINR problem defining only power coefficients and without QoS constraints for RTUs. Hence, the dashed curves in Fig. 1 refer to the scheme in [1] along with QoS constraints. oreover, note that the case with = 1 and N = 80 refers to the singlecell massive IO system, in which all service antennas are collocated at the center of cell. As the figure demonstrates the performance of cell-free massive IO is significantly better than the conventional single-cell massive IO system. Fig. demonstrates numerically the convergence of the proposed Algorithm 1 with 0 APs = 0 and 0 users K = 0 and random pilot sequences with length τ = 15. At each iteration, one of the design parameters is determined by solving the corresponding sub-problem while other design variables are fixed. Assume that at theith iteration, the receiver filter coefficients u i, are determined for a fixed power allocation q i and the power allocation q i+1 is obtained for a given set of receiver filter coefficients u i,. The optimal power allocation q i+1 obtained for a given u i achieves an uplin rate greater than or equal to that of the previous iteration. As a result, the achievable uplin rate monotonically increases at each iteration, which can be also observed from the numerical results presented in Fig.. VI. CONCLUSIONS We have investigated the mixed QoS problem with QoS requirements for the RTUs in cell-free massive IO, and proposed a solution to maximize the minimum user rate while satisfying the SINR constraints of the RTUs. To realize the solution, the original mixed QoS problem has been divided into two sub-problems and they have been iteratively solved by formulating them into a generalized eigenvalue problem and GP. APPENDIX A: PROOF OF THEORE 1 The desired signal for the user is given by DS = { ρe u mĝ H m g } m q = N ρq u mγ m. Hence, DS = ρq N. u mγ m oreover, the term E{ BU } can be obtained as { E BU } = ρe{ u m ĝ H m g m q 11 { } ρe u m ĝ H m g m q =ρn q u mγ m β m, where the last equality comes from the analysis in [1, Appendix A]. The term E{ IUI } is obtained as E { IUI } = ρe u m ĝ H m g m q = ρq E c m u m g H m w m A +ρτp pe q K Hgm c m u m g mi φ H φ i. 1 i=1 B Since w m = φ H W p,m is independent from the term g m similar to [1, Appendix A], the term A in 1 immediately is given by A = Nq c m u m β m.the term B in 1 can be obtained as B = τp pq E c m u m g m φ H φ C K Hgm + τp pq E c m u m g mi φ H φ i. 13 i D The first term in 13 is given by C = Nτp p q φ H φ + N q φ H φ c mu mβ m u m γ m β m β m, 14 where the last equality is derived based on the fact γ m = τpp β m c m. The second term in 13 can be obtained as D = N τp p q u mc m β m β m Nq Nτp p q u mc mβ m u mc mβm φ H φ. 15 Finally we obtain E{ IUI } = Nρq u mβ m γ m +N ρq φ H φ β m u m γ m. 16 β m { The total noise for the user is given by E TN } = { E u mĝ H m n m } = N u m γ m. Finally, SINR of user is obtained by 5, which completes the proof. REFERENCES [1] H. Q. Ngo, A. Ashihmin, H. Yang, E. G. Larsson, and T. L. arzetta, Cell-free massive IO versus small cells, IEEE Trans. Wireless Commun., vol. 16, no. 3, pp , ar [] H. Q. Ngo, L. Tran, T. Q. Duong,. atthaiou, and E. G. Larsson, On the total energy efficiency of cell-free assive IO, IEEE Trans. Green Commun. and Net., vol., no. 1, ar [3]. Bashar, K. Cumanan, A. G. Burr,,. Debbah, and H. Q. Ngo, Enhanced max-min SINR for uplin cell-free massive IO systems, in Proc. IEEE ICC, ay 018, pp [4]. Bashar, K. Cumanan, A. G. Burr,. Debbah, and H. Q. Ngo, On the uplin max-min SINR of cell-free massive IO systems, IEEE Trans. Wireless Commun., submitted. [5]. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, and. Debbah, Cellfree massive IO with limited bachaul, in Proc. IEEE ICC, ay 018, pp [6] A. G. Burr,. Bashar, and D. aryopi, Cooperative access networs: Optimum fronthaul quantization in distributed massive IO and cloud RAN, in Proc. IEEE VTC, Jun. 018, pp [7] D. Feng, C. Jiang, G. Lim, L. J. Cimini, G. Feng, and G. Y. Li, A survey of energy-efficient wireless communications, IEEE Communications Surveys and Tutorials, vol. 15, no. 1, pp , 001. [8] G. Golub and C. V. Loan, atrix Computations, nd ed. Baltimore, D: The Johns Hopins Univ. Press, [9] S. P. Boyd, S. J. Kim, A. Hassibi, and L. Vandenbarghe, A tutorial on geometric programming, Optim. Eng., vol. 8, no. 1, pp , 007.