Energy Efficiency Optimization for Massive MIMO Backhaul Networks with Imperfect CSI and Full Duplex Small Cells

Abstract

In this paper, a massive MIMO enabled backhaul model for two-tier heterogeneous networks with in-band full-duplex transmission and imperfect CSI is provided. Because of a massive number of antennas at the macro base station and densification of small cells, circuit power consumption increases in the system. It is requisite to study energy efficiency of such backhaul networks. This paper aims at maximizing EE for optimal user association, spectrum allocation, and power allocation while accounting for QoS and backhaul constraints. A joint optimization problem is formulated as a non-convex mixed integer non-linear problem which provides an un-affordable computational complexity. To crack the problem efficiently, it is progressively divided into sub-problems until each one is convex and is solved separately to obtain the optimal solution. Furthermore, complexity of the proposed algorithm is evaluated and Jain’s fairness index is measured. Simulation results of the proposed distributive algorithm are shown and compared with that of the existing algorithm (Max SINR) to manifest the strength of the proposed algorithm. The impact of self-interference cancellation factor and channel estimation error on EE is clearly reflected by the numerical results. In addition, it is observed that using the proposed algorithm results in better EE than increasing the number of antennas.

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Notes

  1. 1.

    Similar to [24, 31], we are optimizing SBS transmission power only and therefore, the transmission power of MBS is kept constant during the complete optimization process. It will be considered in our future work.

  2. 2.

    Here, operator \([\cdot ]^{+}\) is defined as \(\left[ t\right] ^{+}=max\left( 0,t\right) .\)

References

  1. 1.

    Gupta, A., Jha, R.K. (2015). A survey of 5g network: Architecture and emerging technologies. IEEE Access, 3: 1206–1232. ISSN 2169-3536. doi:10.1109/ACCESS.2015.2461602.

  2. 2.

    Larsson, E.G., Edfors, O., Tufvesson, F., Marzetta, T.L. (2014). Massive mimo for next generation wireless systems. IEEE Communications Magazine. 52(2):186–195. ISSN 0163-6804. doi:10.1109/MCOM.2014.6736761.

  3. 3.

    Parikh, Jolly, & Basu, Anuradha. (2020). Technologies assisting the paradigm shift from 4g to 5g. Wireless Personal Communications. https://doi.org/10.1007/s11277-020-07053-3.

    Article  Google Scholar 

  4. 4.

    Wang, C., Haider, F., Gao, X., You, X., Yang, Y., Yuan, D., Aggoune, H.M., Haas, H., Fletcher, S., Hepsaydir, E. (2014) Cellular architecture and key technologies for 5g wireless communication networks. IEEE Communications Magazine. 52 (2): 122–130. ISSN 0163-6804. doi:10.1109/MCOM.2014.6736752.

  5. 5.

    Liu, G., Yu, F.R., Ji, H., Leung, V.C.M., Li, X. (2015a). In-band full-duplex relaying for 5g cellular networks with wireless virtualization. IEEE Network, 29 (6): 54–61 ISSN 0890-8044. doi:10.1109/MNET.2015.7340425.

  6. 6.

    Shweta Rajoria, Aditya Trivedi, and W. Wilfred Godfrey. A comprehensive survey: Small cell meets massive mimo. Physical Communication, 26: 40–49, February 2018. URL https://doi.org/10.1016/j.phycom.2017.11.004.

  7. 7.

    Kamel, M., Hamouda, W., Youssef, A. (2016) Ultra-dense networks: A survey. IEEE Communications Surveys Tutorials, 18 (4): 2522–2545, Fourthquarter. ISSN 1553-877X. doi:10.1109/COMST.2016.2571730.

  8. 8.

    Fratu, Octavian, Vulpe, Alexandru, Craciunescu, Razvan, & Halunga, Simona. (2014). Small cells in cellular networks: Challenges of future hetnets. Wireless Personal Communications, 78, 1613–1625. https://doi.org/10.1007/s11277-014-1906-9.

    Article  Google Scholar 

  9. 9.

    Jaber, M., Imran, M.A., Tafazolli, R., Tukmanov, A. (2016). 5g backhaul challenges and emerging research directions: A survey. IEEE Access, 4: 1743–1766. ISSN 2169-3536. doi:10.1109/ACCESS.2016.2556011.

  10. 10.

    Hou, X., Wang, X., Jiang, H., Kayama, H. (2016) Investigation of massive mimo in dense small cell deployment for 5g. In 2016 IEEE 84th Vehicular Technology Conference (VTC-Fall), pages 1–6. doi:10.1109/VTCFall.2016.7881053.

  11. 11.

    S. Akbar, Y. Deng, A. Nallanathan, Elkashlan, M., Karagiannidis, G.K. (2017). Massive multiuser mimo in heterogeneous cellular networks with full duplex small cells. IEEE Transactions on Communications, 65 (11): 4704–4719. ISSN 0090-6778. doi:10.1109/TCOMM.2017.2728536.

  12. 12.

    Bjornson, Luca Sanguinetti Emil., & Kountouris, Marios. (2016). Deploying dense networks for maximal energy efficiency: Small cells meet massive mimo. IEEE, 34(4), 832–847.

    Google Scholar 

  13. 13.

    Bjonson, E., Kountouris, M., & Debbah, M. (2013). Massive mimo and small cells: Improving energy efficiency by optimal soft-cell coordination. ICT, 2013, 1–5. https://doi.org/10.1109/ICTEL.2013.6632074.

    Article  Google Scholar 

  14. 14.

    Wang, Xiaofei, Vasilakos, Athanasios V., Chen, Min, Liu, Yunhao, & Kwon, Ted Taekyoung. (2012). A survey of green mobile networks: Opportunities and challenges. Mobile Networks and Applications, 17(1), 4–20. https://doi.org/10.1007/s11036-011-0316-4.

    Article  Google Scholar 

  15. 15.

    Cai, S., Che, Y., Duan, L., Wang, J., Zhou, S., & Zhang, R. (2016). Green 5g heterogeneous networks through dynamic small-cell operation. IEEE Journal on Selected Areas in Communications, 34(5), 1103–1115. https://doi.org/10.1109/JSAC.2016.2520217.

    Article  Google Scholar 

  16. 16.

    Zhang, H., Huang, S., Jiang, C., Long, K., Leung, V. C. M., & Poor, H. V. (2017). Energy efficient user association and power allocation in millimeter-wave-based ultra dense networks with energy harvesting base stations. IEEE Journal on Selected Areas in Communications, 35(9), 1936–1947. https://doi.org/10.1109/JSAC.2017.2720898.

    Article  Google Scholar 

  17. 17.

    Adeogun, Ramoni O. (2018). A novel game theoretic method for efficient downlink resource allocation in dual band 5g heterogeneous network. Wireless Personal Communications, 101, 119–141. https://doi.org/10.1007/s11277-018-5679-4.

    Article  Google Scholar 

  18. 18.

    Van Chien, T., Nguyen Canh, T., Björnson, E., & Larsson, E. G. (2020). Power control in cellular massive mimo with varying user activity: A deep learning solution. IEEE Transactions on Wireless Communications, 19(9), 5732–5748. https://doi.org/10.1109/TWC.2020.2996368.

    Article  Google Scholar 

  19. 19.

    He, A., Wang, L., Elkashlan, M., Chen, Y., & Wong, K. (2015). Spectrum and energy efficiency in massive mimo enabled hetnets: A stochastic geometry approach. IEEE Communications Letters, 19(12), 2294–2297. https://doi.org/10.1109/LCOMM.2015.2493060.

    Article  Google Scholar 

  20. 20.

    Liu, D., Wang, L., Chen, Y., Zhang, T., Chai, K. K., & Elkashlan, M. (2015b). Distributed energy efficient fair user association in massive mimo enabled hetnets. IEEE Communications Letters, 19(10), 1770–1773. https://doi.org/10.1109/LCOMM.2015.2454504.

    Article  Google Scholar 

  21. 21.

    Zhou, Tian Qing, Jiang, Nan, Qin, Dong, & Li, Chunguo. (2017). Qos-aware balanced and unbalanced associations in massive mimo enabled heterogeneous cellular networks. Springer, 97, 5345–5366. https://doi.org/10.1007/s11277-017-4782-2.

    Article  Google Scholar 

  22. 22.

    Van Chien, T., Bjornson, E., & Larsson, E. G. (2016). Joint power allocation and user association optimization for massive mimo systems. IEEE Transactions on Wireless Communications, 15(9), 6384–6399. https://doi.org/10.1109/TWC.2016.2583436.

    Article  Google Scholar 

  23. 23.

    Youyun, Xu., Xia, Xiaochen, Ma, Wenfeng, Zhang, Dongmei, Kui, Xu., & Wang, Yurong. (2015). Full-duplex massive mimo relaying: An energy efficiency perspective. Wireless Personal Communications, 84, 1933–1961. https://doi.org/10.1007/s11277-015-2547-3.

    Article  Google Scholar 

  24. 24.

    Zhang, H., Liu, H., Cheng, J., & Leung, V. C. M. (2018). Downlink energy efficiency of power allocation and wireless backhaul bandwidth allocation in heterogeneous small cell networks. IEEE Transactions on Communications, 66(4), 1705–1716. https://doi.org/10.1109/TCOMM.2017.2763623.

    Article  Google Scholar 

  25. 25.

    Tabassum, H., Sakr, A. H., & Hossain, E. (2016). Analysis of massive mimo-enabled downlink wireless backhauling for full-duplex small cells. IEEE Transactions on Communications, 64(6), 2354–2369. https://doi.org/10.1109/TCOMM.2016.2555908.

    Article  Google Scholar 

  26. 26.

    Wang, N., Hossain, E., & Bhargava, V. K. (2016). Joint downlink cell association and bandwidth allocation for wireless backhauling in two-tier hetnets with large-scale antenna arrays. IEEE Transactions on Wireless Communications, 15(5), 3251–3268. https://doi.org/10.1109/TWC.2016.2519401.

    Article  Google Scholar 

  27. 27.

    Liu, Y., Lu, L., Li, G. Y., Cui, Q., & Han, W. (2016). Joint user association and spectrum allocation for small cell networks with wireless backhauls. IEEE Wireless Communications Letters, 5(5), 496–499. https://doi.org/10.1109/LWC.2016.2593465.

    Article  Google Scholar 

  28. 28.

    Korpi, D., Riihonen, T., Sabharwal, A., & Valkama, M. (2018). Transmit power optimization and feasibility analysis of self-backhauling full-duplex radio access systems. IEEE Transactions on Wireless Communications, 17(6), 4219–4236. https://doi.org/10.1109/TWC.2018.2821682.

    Article  Google Scholar 

  29. 29.

    Hao, W., & Yang, S. (2018). Small cell cluster-based resource allocation for wireless backhaul in two-tier heterogeneous networks with massive mimo. IEEE Transactions on Vehicular Technology, 67(1), 509–523. https://doi.org/10.1109/TVT.2017.2739203.

    Article  Google Scholar 

  30. 30.

    Han, Q., Yang, B., Miao, G., Chen, C., Wang, X., & Guan, X. (2017). Backhaul-aware user association and resource allocation for energy-constrained hetnets. IEEE Transactions on Vehicular Technology, 66(1), 580–593. https://doi.org/10.1109/TVT.2016.2533559.

    Article  Google Scholar 

  31. 31.

    Lohani, S., Hossain, E., & Bhargava, V. K. (2016). On downlink resource allocation for swipt in small cells in a two-tier hetnet. IEEE Transactions on Wireless Communications, 15(11), 7709–7724. https://doi.org/10.1109/TWC.2016.2606394.

    Article  Google Scholar 

  32. 32.

    Uwaechia, Anthony Ngozichukwuka, Mahyuddin, Nor Muzlifah, Ain, Mohd Fadzil, Latiff, Nurul Muazzah Abdul., & Zabah, Nor Farahidah. (2019). Compressed channel estimation for massive mimo-ofdm systems over doubly selective channels. Physical Communication, https://doi.org/10.1016/j.phycom.2019.100771.

    Article  Google Scholar 

  33. 33.

    Wang, C., Au, E.K.S., Murch, R.D., Lau, V.K.N. (2016). Closed-form outage probability and ber of mimo zero-forcing receiver in the presence of imperfect csi. In 2006 IEEE 7th Workshop on Signal Processing Advances in Wireless Communications, pages 1–5. doi:10.1109/SPAWC.2006.346359.

  34. 34.

    Hao, Y., Ni, Q., Li, H., & Hou, S. (2017). On the energy and spectral efficiency tradeoff in massive mimo-enabled hetnets with capacity-constrained backhaul links. IEEE Transactions on Communications, 65(11), 4720–4733. https://doi.org/10.1109/TCOMM.2017.2730867.

    Article  Google Scholar 

  35. 35.

    Sanguinetti, L., Moustakas, A. L., & Debbah, M. (June 2015). Interference management in 5g reverse tdd hetnets with wireless backhaul: A large system analysis. IEEE Journal on Selected Areas in Communications, 33(6), 1187–1200. https://doi.org/10.1109/JSAC.2015.2416991.

    Article  Google Scholar 

  36. 36.

    Auer, G., Giannini, V., Desset, C., Godor, I., Skillermark, P., Olsson, M., et al. (2011). How much energy is needed to run a wireless network? IEEE Wireless Communications, 18(5), 40–49. https://doi.org/10.1109/MWC.2011.6056691.

    Article  Google Scholar 

  37. 37.

    E. Bjornson, L. Sanguinetti, J. Hoydis, and M. Debbah. (2014) Designing multi-user mimo for energy efficiency: When is massive mimo the answer? In 2014 IEEE Wireless Communications and Networking Conference (WCNC), https://doi.org/10.1109/WCNC.2014.6951974.

  38. 38.

    Garcia-Saavedra, A., Serrano, P., Banchs, A., & M. Hollick (2011) Energy-efficient fair channel access for ieee 802.11 wlans. In, IEEE international symposium on a world of wireless. Mobile and Multimedia Networks. https://doi.org/10.1109/WoWMoM.2011.5986436.

    Article  Google Scholar 

  39. 39.

    J. Nocedal and Numerical Optimization S. Wright. Numerical Optimization, volume 2nd ed. New York, NY, USA: Springer, 2006. New York, NY, USA: Springer, 2 edition, 2006.

  40. 40.

    A. Mutapcic S. Boyd (2007). Sub-gradient methods. in Notes for EE392o. Stanford, CA, USA: Stanford Univ. Press,

  41. 41.

    Aly EL Gamal V V Venugopal (2018). Interference Management in Wireless Networks: Fundamental Bounds and the Role of Cooperation. Cambridge University Press https://doi.org/10.1017/9781316691410.

  42. 42.

    Omidvar, N., Liu, A., Lau, V., Zhang, F., Tsang, D. H. K., & Pakravan, M. R. (2018). Optimal hierarchical radio resource management for hetnets with flexible backhaul. IEEE Transactions on Wireless Communications, 17(7), 4239–4255. https://doi.org/10.1109/TWC.2018.2809745.

    Article  Google Scholar 

  43. 43.

    D. Chiu R. Jain and W. Hawe. (1984) A quantitative measure of fairness and discrimination for resource allocation in shared systems. techreport, Digital Equipment Corporation

  44. 44.

    Irving Ron. Beyond the Quadratic Formula, volume 43. American Mathematical Soc., 2013.

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Appendices

Steps for Calculating \(\Delta p_{j}\)

Let \(f_{1}\) be the optimization function given in (27). Then, first and second order partial derivatives of \(f_{1}\) with respect to \(p_{j}\) are derived as below

$$\begin{aligned} \frac{{\partial f_{1}}}{{\partial {p_j}}} =&\sum \limits _{i = 1}^K {\sum \limits _{j = 1}^M {\frac{{{G_{i,j}}}}{{\ln 2\left( {I + \sigma _n^2} \right) .\left( {1 + {\Upsilon _{i,j}}} \right) .{{\log }_2}\left( {1 + {\Upsilon _{i,j}}} \right) }}} \, + \sum \limits _{j = 1}^M {\frac{{{U_j}}}{{{\varepsilon _j}{P_j}}}} } \\ \nonumber +&\sum \limits _{j = 1}^M {\frac{{{\lambda _2}{U_j}\left( {1 - \eta } \right) }}{{\left( {1 + {\Upsilon _j}} \right) }}\left( {\frac{{ - {\zeta _j}{p_m}{G_{j,0}}}}{{{{\left( {{\zeta _j}{p_j} + {I_s}} \right) }^2}}}} \right) } + \,\sum \limits _{i = 1}^K {\sum \limits _{j = 1}^M {\frac{{{\lambda _2}\left( {1 - \eta } \right) {G_{i,j}}}}{{\left( {I + \sigma _n^2} \right) .\left( {1 + {\Upsilon _{i,j}}} \right) }}} } \end{aligned}$$
(35)

and

$$\begin{aligned} \frac{{{\partial ^2}{f_1}}}{{\partial p_{i,j}^2}} =&\sum \limits _{i = 1}^K {\sum \limits _{j = 1}^M {\frac{{ - G_{i,j}^2}}{{{{\left( {I + \sigma _n^2} \right) }^2}{{\left( {1 + {\Upsilon _{i,j}}} \right) }^2}{{\left( {\log \left( {1 + {\Upsilon _{i,j}}} \right) } \right) }^2}}}.\left( {1 + \log \left( {1 + {\Upsilon _{i,j}}} \right) } \right) } \,}\nonumber \\ -&\sum \limits _{j = 1}^M {\frac{{{\lambda _2}{U_j}\left( {1 - \eta } \right) }}{{{{\left( {1 + {\Upsilon _j}} \right) }^2}}}\times {{\left( {\frac{{{\zeta _j}{p_m}{G_{j,0}}}}{{{{\left( {{\zeta _j}{p_j} + {I_s}} \right) }^2}}}} \right) }^2} }\times \left( {1 - \frac{{2\left( {1 + {\Upsilon _j}} \right) \left( {{\zeta _j}{p_j} + {I_s}} \right) }}{{{p_m}{G_{j,0}}}}} \right) \nonumber \\ -&\sum \limits _{i = 1}^K {\sum \limits _{j = 1}^M {\frac{{{\lambda _2}\left( {1 - \eta } \right) G_{i,j}^2}}{{{{\left( {I + \sigma _n^2} \right) }^2}{{\left( {1 + {\Upsilon _{i,j}}} \right) }^2}}}} }- \sum \limits _{j = 1}^M {\frac{{{U_j}}}{{{{\left( {{\varepsilon _j}{P_j}} \right) }^2}}}} \end{aligned}$$
(36)

where \(I={{p_m}{G_{i,0}} + \sum \nolimits _{j' = 1/j}^M {{p_{j'}}{G_{i,j'}}} }\) and \(I_{s}=\sum \nolimits _{j' = 1/j}^M {{p_{j'}}{G_{i,j'}}} + {\sigma _n^2}\). Then, from the Newton’s method [39], \(\Delta {p_{j}}\) can be evaluated as follows

$$\begin{aligned} \Delta {p_{j}} = \left| {{{\partial f_{1}} \over {\partial {p_{j}}}}} \right| \big / \left| {{{{\partial ^2}f_{1}} \over {\partial p_{j}^2}}} \right| . \end{aligned}$$
(37)

Procedure for Calculating the Value of \(\eta\)

Let’s proceed with \(f_{2}\) provided in (31). \(f_{2}\) is differentiated with respect to \(\eta\) and is equated to zero. Then, the quadratic equation in \(\eta\) is attained as follows

$$\begin{aligned} A{\eta ^2} + \left( {\sum \limits _{i = 1}^K {{y_{i,0}}} + \sum \limits _{i = 1}^K {\sum \limits _{j = 1}^M {{y_{i,j}}} - A} } \right) \eta - \sum \limits _{i = 1}^K {{y_{i,0}}} = 0, \end{aligned}$$
(38)

where \(A = \sum \nolimits _{i = 1}^K {{\lambda _1}\left( {{y_{i,0}}{a_{i,0}}L - \sum \nolimits _{j = 1}^M {{y_{i,j}}{a_{i,j}}} } \right) }\).

Afterward, the roots of the aforementioned equation can be obtained by using the Quadratic Equation Method [44] as follows

$$\begin{aligned} \begin{array}{l} \eta = \\ {{ - \left( {\sum \nolimits _{i = 1}^K {{y_{i,0}}} + \sum \nolimits _{i = 1}^K {\sum \nolimits _{j = 1}^M {{y_{i,j}}} - A} } \right) \pm \sqrt{{{\left( {\sum \nolimits _{i = 1}^K {{y_{i,0}}} + \sum \nolimits _{i = 1}^K {\sum \nolimits _{j = 1}^M {{y_{i,j}}} - A} } \right) }^2} + 4 \times A \times \sum \nolimits _{i = 1}^K {{y_{i,0}}} } } \over {2 \times A}}. \end{array} \end{aligned}$$
(39)

If \(B={\left( {\sum \nolimits _{i = 1}^K {{y_{i,0}}} + \sum \nolimits _{i = 1}^K {\sum \nolimits _{j = 1}^M {{y_{i,j}}} - A} } \right) }\) then, (39) is redrafted as

$$\begin{aligned} \eta = {{ - B \pm \sqrt{{B^2} + 4 \times A \times \sum \nolimits _{i = 1}^K {{y_{i,0}}} } } \over {2 \times A}}. \end{aligned}$$
(40)

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Rajoria, S., Trivedi, A. & Godfrey, W.W. Energy Efficiency Optimization for Massive MIMO Backhaul Networks with Imperfect CSI and Full Duplex Small Cells. Wireless Pers Commun (2021). https://doi.org/10.1007/s11277-021-08231-7

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Keywords

  • Backhauling
  • Heterogeneous networks
  • Massive MIMO
  • Optimization
  • Small cell