## Yao Zhang , Haotong Cao , Meng Zhou and Longxiang Yang## |

Parameter | Value |
---|---|

[TeX:] $$N_{0} (noise power)$$ | [TeX:] $$290 \times \kappa \times B \times N F$$ |

[TeX:] $$\kappa, B, N F$$ | Boltzmann constant, 20 MHz, 9 dB |

[TeX:] $$\tau, \rho_{\mathrm{p}}, \rho_{\mathrm{u}}$$ | K / 2, 100, 100 mW |

[TeX:] $$\sigma_{\mathrm{sh}}$$ | 8 dB |

ated by using the Monte-Carlo simulation technique by averaging [TeX:] $$10^{3}$$ independent channel realizations. As we can see, the relative performance gap between (14) and (16) is marginal in all considered cases compared to the achievable sum rate, which indicates that our closed-form result in (16) is a good predictor to approximate the ergodic rate (14). In addition, we find the sum rate is an increasing function of M. This is because the larger M promises more antenna array gains, thereby bringing more degree-of-freedoms to resist the fading and interference.

Fig. 2 depicts the cumulative distribution functions (CDFs) of the achievable uplink per-user rate for different N under the EPC scheme, with M = 60 and K = 10: As expected, the ZF receiver outperforms the CB receiver except for the low-SINR regime. This is because in low SINR regime, although the ZF

receiver cancels the inter-user interference, it inevitably amplifies the thermal noise. It is also interesting to notice that, when N increases from 2 to 6, the merits of the ZF receiver in the low-SINR regime also appears. This finding shows that using additional antennas at the APs can compensate for the performance loss due to the additive noise. When implementing the ZF receiver, more users can enjoy the high-SINR service. For instance, when N = 6, more than 52% of users have reached a rate of 4 bits/s/Hz for the ZF receiver. But for the CB receiver, this ratio is only 10%. These above insights demonstrate the superiority of the ZF receiver.

In Fig. 3, the CDFs of the achievable uplink per-user rate for the ZF and CB receivers against different channel estimation error under the EPC scheme are plotted, with M = 60; K = 10 and N = 6. Here we represent different channel estimations by changing the value of [TeX:] $$\tau$$. Since the CPU randomly allocates pilots to the user, when [TeX:] $$\tau<K,$$ the smaller the [TeX:] $$\tau$$ is, the more likely the same pilot is to be used by more users, resulting in more serious pilot contamination. Note when [TeX:] $$\tau=K,$$ there is no pilot contamination. It is clear to see that the performance of the ZF receiver is dominated by the accuracy of the channel estimation. When we narrow the value of [TeX:] $$\tau$$ for the ZF receiver, the corresponding performance will drop dramatically. For instance, when [TeX:] $$\tau$$ is reduced from 10 to 8, the 5%-outage and median rates of the ZF curve have decreased 208.4% and 62.1%, respectively. This insightful observation reveals that the ZF receiver is sensitive to the channel detection errors. In addition, compared with the ZF receiver, the CB receiver is more robust on the channel estimation errors. Nevertheless, the CB receiver suffers from severe inter-user interference and hence, results in unsatisfactory rate performance.

Fig. 4 explores the total power consumptions of the ZF and CB receivers against the number of APs for different N under the EPC scheme, with K = 10. Here we adopt the generic power model given in [12, Section III]. As we can see, the total power cost of the ZF receiver is always higher than the CB receiver. While as the number of APs increases, the relative power gap between these two receivers increases. This is due to the fact that the ZF receiver requires extra power to suppress

interference from other users. In fact, since the merits of the ZF receiver gradually appear with the increasing of the number of APs, it is advisable to use more power to promise a huge performance gain.

Until now, we have investigated the rate performance of the ZF and CB receivers without any power optimization. The ZF receiver benefits from the cancellation of the inter-user interference, promising huge performance enhancement in the high- SINR regime. Next, we first evaluate the effectiveness of the Algorithms 1 and 3 in maximizing the total rate for these two receivers as well as their convergence rates. Fig. 5 depicts numerically the total rate versus the number of iterations, with [TeX:] $$M=60, K=10, N=6 \text { and } \bar{R}_{k}=1$$ bits/s/Hz. It can be easily concluded that both Algorithms 1 and 3 only need a few iterations to converge. The total rate tends to a stable value after 4 iterations, which demonstrates that our proposed algorithms have a fast converge rate. We then compare our Algorithms 1 and 3 with the EPC scheme to further highlight the effectiveness of our algorithms. Fig. 6 shows the relative performance gaps between the EPC and Algorithms 1 and 3 against the number of APs, with [TeX:] $$K=10, N=6 \text { and } \bar{R}_{k}=1$$ bits/s/Hz. It can be observed that, compared with the EPC scheme, Algorithms 1 and 3 have the ability to significantly improve the total rate and the rate improvement of the ZF receiver is better than the CB receiver. In addition, it is also interesting to notice for all considered cases, the relative performance gap between these two receivers increases as the number of APs increases. This is because as the number of APs increases, the channel hardening becomes more pronounced, thereby reducing the channel estimation errors. Benefiting from smaller channel estimation errors, the total antenna array has a stronger ability to resist the interuser interference, resulting in significant performance gains in terms of the sum rate. The same insight can also be obtained by using more antennas at each AP, see Fig. 2.

To conclude this paper, we examine the effectiveness of Algorithms 2 and 4. Table 2 presents the achievable uplink per-user rate of the proposed Algorithms 2 and 4, with [TeX:] $$M=60, K=10, N=6 \text { and } \bar{R}_{k}=1$$ bits/s/Hz. Note different channel realizations in Table 2 correspond to different APs’

Table 2.

Achievable uplink per-user rate (bits/s/Hz) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Receiver | Channel | User 1 | User 2 | User 3 | User 4 | User 5 | User 6 | User 7 | User 8 | User 9 | User 10 |

ZF | 1 | 9.881 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

2 | 9.505 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

3 | 7.859 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

CB | 1 | 3.277 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

2 | 4.032 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

3 | 2.936 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

and users’ locations (which are randomly generated in this simulation), thus leading to different large-scale fading coefficients. As expected, in all considered cases, the rate of user 1 has maximized after using the proposed algorithms while the rate of the remaining users can satisfy their QoS constraints. In addition, Table 2 also implies that the ZF receiver is always superior to the CB receiver in the high-SINR regime, which coincides with the conclusion drawn from Figs. 2 and 3.

In this paper, we investigated the uplink rate performance of cell-free mMIMO system with the ZF and CB receivers. A novel tight approximate rate expression for the ZF receiver was derived. Compared with the CB receiver, the ZF receiver benefited from eliminating the inter-user interference, thereby improving the system performance. In addition, by leveraging on the derived rate expressions, two power control algorithms were proposed, subjecting to the user power constraints and the QoS constraints. We found the total rate maximization algorithms have a fast convergence rate. Moreover, the proposed sum rate maximization algorithms can significantly enhance the sum rate for both ZF and CB receivers compared to the EPC scheme. Furthermore, we also observed the one user’s rate maximization algorithms work well in many respects.

Firstly, by defining

we obtain

Since [TeX:] $$-\log _{2}(1+1 / x)$$ is a convex function with respect to x and by virtue of the Jensen’s Inequality, we have

From (4) and (5), it remarks that the channel estimations of two users using the same pilot sequence are parallel. In other words, if [TeX:] $$\phi_{k}=\phi_{j}, \forall k \neq j$$, it yields

Expression (50) indicates that [TeX:] $$\hat{\mathbf{G}}$$ is rank-deficient, therefore the property of Wishart matrix can not be utilized.

Next, leveraging on [21, Eq. (62)], [TeX:] $$1 /\left\|\mathbf{a}_{k}\right\|^{2}$$ can be approximated by

where

Since the norm of circularly symmetric complex Gaussian variable obeys Gamma distortion, it yields

The above expression allows us to obtain an approximate distribution of [TeX:] $$1 /\left\|\mathbf{a}_{k}\right\|^{2},$$ as

By virtue of the property of the Gamma function, we obtain

Plugging (56) into (49), (16) is obtained.

Yao Zhang received the B.S. degree in College of Computer Science & Technology from Qingdao University, China, in 2016. He is currently pursuing his Ph.D. degree at the Department of Communication and Information Engineering in Nanjing University of Posts and Telecommunications. His research interests include massive MIMO, especially power optimization in cell-free massive MIMO.

Haotong Cao received B.S. degree in Communication Engineering from Nanjing University of Posts and Telecommunications (NJUPT) in 2015. He is currently pursuing his Ph.D. Degree in NJUPT, Nanjing, China. He was a Visiting Scholar of Loughborough University, U.K. in 2017. He has served as the TPC member of multiple IEEE conferences, such as IEEE INFOCOM, IEEE ICC, IEEE Globecom. He is also serving as the reviewer of multiple academic journals, such as IEEE/ACM Transactions on Networking, IEEE Transactions on Network and Service Management and (Elsevier) Computer Networks. He has published multiple IEEE Trans./Journal/Magazine papers since 2016. His research interests include wireless communication theory, resource allocation in wired and wireless networks. He has been awarded the 2018 Postgraduate National Scholarship of China. He has been awarded 2019 IEEE ICC SecSDN Workshop Best Paper Award.

Meng Zhou received the M.S. degree from Northwest Normal University, Lanzhou, China, in 2018. He is presently pursuing the Ph.D. degree with the College of Communication and Information Engineering from the Nanjing University of Posts and Telecommunications (NJUPT), Nanjing, China. His broad interests in wireless communications, with a special interest of cell-free massive MIMO, resource allocation, physical layer security, HetNets in B5G and 6G.

Longxiang Yang is currently with the College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications (NJUPT), Nanjing, China. He is a Full Professor and Doctoral Supervisor of NJUPT. He is also the VicePresident of College of Telecommunications and Information Engineering, NJUPT. He has fulfilled multiple National Natural Science Foundation projects of China. He has authored and co-authored over 200 technical papers published in various journals and conferences. His research interests include cooperative communication, network coding, wireless communication theory, B5G and 6G mobile communication systems, ubiquitous networks and Internet of things.

- 1 H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, T. L. Marzetta, "Cell-free massive mimo versus small cells,"
*IEEE Trans. Wireless Commun.*, vol. 16, no. 3, pp. 1834-1850, Mar, 2017.doi:[[[10.1109/TWC.2017.2655515]]] - 2 H. Yang, T. L. Marzetta, "Energy efficiency of massive mimo: Cellfree vs. cellular,"
*in Proc.IEEE VTC*, pp. 15-64, June, 2018.custom:[[[-]]] - 3 H. Q. Ngo, L. Tran, T. Q. Duong, M. Matthaiou, E. G. Larsson, "On the total energy efficiency of cell-free massive mimo,"
*IEEE Trans.Green Commun.Netw.*, vol. 2, no. 1, pp. 25-39, Mar, 2018.doi:[[[10.1109/TGCN.2017.2770215]]] - 4 E. Nayebi, A. Ashikhmin, T. L. Marzetta, H. Yang, B. D. Rao, "Precoding and power optimization in cell-free massive mimo systems,"
*IEEE Trans.WirelessCommun.*, vol. 16, no. 7, pp. 4445-4459, July, 2017.doi:[[[10.1109/TWC.2017.2698449]]] - 5 Y. Li, G. A. A. Baduge, "NOMA-aided cell-free massive MIMO systems,"
*IEEE Wireless Commun. Lett.*, vol. 7, no. 6, pp. 950-953, Dec, 2018.doi:[[[10.1109/LWC.2018.2841375]]] - 6 Q. Huang, A. Burr, "Compute-and-forward in cell-free massive MIMO: Great performance with low backhaul load,"
*in Proc. IEEE ICC Workshops*, pp. 601-606, May, 2017.doi:[[[10.1109/ICCW.2017.7962724]]] - 7 M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, M. Debbah, "Cellfree massive MIMO with limited backhaul,"
*in Proc. IEEE ICC*, pp. 1-7, May, 2018.custom:[[[-]]] - 8 P. Liu, K. Luo, D. Chen, T. Jiang, "Spectral efficiency analysis in cell-free massive MIMO systems with zero-forcing detector," May, 2018.custom:[[[https://arxiv.org/abs/1805.10621]]]
- 9 J. Zhang, Y. Wei, E. Björnson, Y. Han, X. Li, "Spectral and energy efficiency of cell-free massive MIMO systems with hardware impairments,"
*in Proc.IEEE WCSP*, pp. 1-6, Oct, 2017.doi:[[[10.1109/wcsp.2017.8171057]]] - 10 M. Bashar, K. Cumanan, A. G. Burr, M. Debbah, H. Q. Ngo, "Enhanced max-min SINR for uplink cell-free massive MIMO systems,"
*in Proc.IEEE ICC*, pp. 1-6, May, 2018.custom:[[[-]]] - 11 T. H. Nguyen, T. K. Nguyen, H. D. Han, V. D. Nguyen, "Optimal power control and load balancing for uplink cell-free multi-user massive MIMO,"
*IEEE Access*, vol. 6, pp. 14462-14473, Feb, 2018.doi:[[[10.1109/ACCESS.2018.2797874]]] - 12 Y. Zhang, H. Cao, M. Zhou, L. Yang, "Power optimization for energy efficiency in cell-free massive MIMO with ZF receiver,"
*in Proc. IEEE ICACT*, pp. 366-371, Feb, 2019.doi:[[[10.23919/ICACT.2019.8702035]]] - 13 T. L. Marzetta, E. G. Larsson, H. Yang, H. Q. Ngo,
*Fundamentals of MassiveMIMO*, Cambridge, U.K.: Cambridge Univ. Press, 2016.custom:[[[-]]] - 14 Z. Luo, S. Zhang, "Dynamic spectrum management: Complexity and duality,"
*IEEE J.Sel.TopicsSignalProcess*, vol. 2, no. 1, pp. 57-73, Feb, 2008.doi:[[[10.1109/JSTSP.2007.914876]]] - 15 A. Beck, A. Ben-Tal, L. Tetruashvili, "A sequential parametric convex approximation method with applications to nonconvex truss topology design problems,"
*J.Glob.Optim.*, vol. 47, no. 1, pp. 29-51, May, 2010.doi:[[[10.1007/s10898-009-9456-5]]] - 16 G. Scutari, F. Facchinei, L. Lampariello, "Parallel and distributed methods for constrained nonconvex optimization-Part I: Theory,"
*IEEE Trans.SignalProcess.*, vol. 65, no. 8, pp. 1929-1944, Apr, 2016.custom:[[[-]]] - 17 H. Tuy,
*Convex Analysis and Global Optimization*, MA, Boston: Kluwer, 1998.custom:[[[-]]] - 18 S. Boyd, L. Vandenberghe,
*Convex Optimization*, Cambridge, U.K.: Cambridge Univ. Press, 2004.custom:[[[-]]] - 19
*M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.0 beta, Sept. 2013. Available:*, http://cvxr.com/cvx - 20 M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, H. V. Poor, "Mixed quality of service in cell-free massive MIMO,"
*IEEE Commun. Lett.*, vol. 22, no. 7, pp. 1494-1497, July, 2018.doi:[[[10.1109/LCOMM.2018.2825428]]] - 21 J. Wang, L. Dai, "Asymptotic rate analysis of downlink multi-user systems with co-located and distributed antennas,"
*IEEE Trans. Wireless Commun.*, vol. 14, no. 6, pp. 3046-3058, June, 2015.doi:[[[10.1109/TWC.2015.2399921]]]