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We study the impact of phase noise on the downlink performance of a multi-user multiple-input multiple-output system, where the base station (BS) employs a large number of transmit antennas M. We consider a setup where the BS employs Mosc freerunning oscillators, and M/Mosc antennas are connected to each oscillator. For this configuration, we analyze the impact of phase noise on the performance of the zero-forcing (ZF), regularized ZF, and matched filter (MF) precoders when M and the number of users K are asymptotically large, while the ratio M/K = _ is fixed. Large-Scale Analysis of Linear Massive MIMO Precoders in the Presence of Phase Noise   568 AWGN, and interference [15]. Furthermore, we define the effective SINR, and discuss the achievable rates for the GO setup. A. Received Signal Model For the RZF precoder in (7), the received signal at the kth UE in (6) becomes In (8), we have introduced the following definitions: ˆH 0,−k = [ˆh0,1, . . . ,ˆh0,k−1,ˆh0,k+1, . . . ,ˆh0,K], P−k = diag {p1, . . . , pk−1, pk+1, . . . , pK}, and cτ,−k =[cτ,1, . . . , cτ,k−1, cτ,k+1, . . . , cτ,K]T. Furthermore, Isig C and IT int CM−1×1 denote the scaling factors associated with the desired symbol and the interfering symbols at the kth UE, respectively. The factor Isig is simplified in the following proposition. Proposition 1: Consider an RZF precoded downlink transmission from a BS having M antennas to K single-antenna UEs employing TDD in the presence of oscillator phase noise. Let α > 0, M/K = β, β ≥ 1, and q0 [0, 1]. Assume that 1 M ˆH H ˆH has uniformly bounded spectral norm for allM. Then, the desired signal factor Isig, for M,K → ∞, can be simplified to Proof: Please refer to Theorem 1 for a discussion on the proof, and to [15, Proposition 1] for a detailed proof. Remark 1: The terms t and ξ in (9) depend on α and β, and capture the channel hardening effect [1], [4] that results from the averaging of the random fading channels when RZF precoding is used, and M,K → ∞. The term TPN in (10) captures the effects of phase noise variations at the BS between the training and the data transmission phases, and is given by A. Sanyasi Rao and L.Ramesh 569  For the CO and the DO setups, the MISO system model in (6) and (8) becomes an equivalent SISO phase noise channel in (16) [15]. However, when 2 ≤ Mosc < ∞, (16) still corresponds to a MISO phase noise channel, since TPN in (14) depends on the random phase noise variations of the multiple oscillators at the BS. B. Effective SINR and Achievable Rates For the CO and the DO setups, we define the effective SINR based on the SISO phase noise channel in (16) as [15] Based on the effective SINR in (17), an upper bound for the achievable rate of the kth UE for the CO and the DO setups for a given block of symbols (i.e., given τ) is [15], [16] This upper bound, which corresponds to the AWGN channel capacity, is generally tight for lowto-medium SINR values. Another upper bound for the achievable rate for the CO and the DO setups, which is generally tight at high SINR, was derived by Lapidoth et al. [13], and is given by IV. SINR ANALYSIS In this section, we introduce Theorem 1, which provides the analytical form for the effective SINR achievable at a given UE for the RZF precoder. Theorem 1: Large-Scale Analysis of Linear Massive MIMO Precoders in the Presence of Phase Noise   570 Consider an RZF precoded downlink transmission from a BS having M antennas to K singleantenna UEs employing TDD in the presence of oscillator phase noise. Let α > 0, β ≥ 1, q0 [0, 1], and SINRk denote the effective SINR at the kth UE. Then, almost surely, and the effective SINR associated with the kth UE for the GO setup is given as V. SIMULATION RESULTS In this section, the analytical results for the RZF precoder presented in Section IV are verified by comparing them against the results obtained from MC simulations. Even though the analytical results are derived for M,K → ∞, in the sequel, we observe that these results concur with those from simulations for finite values of M and K. We simulate the system model specified in (6) using the RZF precoder, and numerically evaluate the effective SINR in (22). Then, the achievable rate in the downlink for a given UE is computed using (21) for all values of Mosc, unless otherwise stated. Recall that this evaluation only depends on the SINR achieved at the UE, and does not account for the reduced variance A. Sanyasi Rao and L.Ramesh 571  VI. CONCLUSIONS In this work, we derived the effective SINR of the RZF precoder for the GO setup in the presence of phase noise. We showed that the effect of phase noise on the SINR can be expressed as an effective reduction in the CSI quality available at the BS. Importantly, the SINR degrades as the number of oscillators, Mosc, increases. This is because as Mosc increases, the desired signal power decreases, and the interference power increases. Furthermore, we showed that the variance of the random phase variations caused by the BS oscillators reduces with increasing Mosc. We demonstrated that the SINR obtained is tight and agrees with that obtained from simulations with remarkable accuracy for interesting, and practical values of M and K. Finally, we observed that for the RZF precoder, the CO setup has a higher achievable rate than the DO setup when β is small, while the DO setup outperforms the CO setup when the SNR at the UE is low and β is large. REFERENCES [1] F. Rusek, D. Persson, K. L. Buon, 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., vol. 30, no. 1, pp. 40-60, Jan. 2013. [2] G. Caire and S. Shamai, “On the Achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1691-1706, Jul. 2003. [3] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528-541, Mar. 2006. [4] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large system analysis of linear precoding in correlated MISO broadcast channels under limited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509-4537, Jul. 2012. Large-Scale Analysis of Linear Massive MIMO Precoders in the Presence of Phase Noise   572 [5] A. M ̈uller, A. Kammoun, E. Bjornson, and M. Debbah, “Linear precoding based on truncated polynomial expansion – part I: large-scale single-cell systems,” IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 815-830, Oct. 2014. 2013. AUTHOR 1:Allanki Sanyasi Rao, received his Bachelors Degree in Electronics & Communication Engineering in 1997, Masters Degree in Digital Systems & Computer Electronics in 2012. He is having 18 years of teaching experience. Currently he is Associate Professor at Balaji Institute of Technology& Sciences, Warangal, India. His research interest includes Wireless Communications, Signal Processing and Network Security & Cryptography. AUTHOR 2:Lavudya Ramesh completed his b.tech in balaji institute of engineering and sciences(2014)and pursuing M.Tech in balaji institute of technology and sciences.

Large-scale analysis of linear massive mimo precoders in the presence of phase noise