Skip to main content
SpringerLink
Log in

Closed-form Analysis of RZF in Multicell Massive MIMO Over Correlated Rician Channel

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

Correlation exists in all practical channels, and each channel’s correlation level varies. Determining correlation matrices while estimating the channel and precoding is vital in providing spatial directivity to the signals. This paper considers the performance analysis of the downlink multi-cell multiuser Massive Multiple Input Multiple Output (mMIMO) system with user-specific correlation matrices for the Rician Fading channel. Using the Random Matrix Theory (RMT) results, the closed-form for the asymptotic analysis of ergodic downlink rate per user terminal (UT) is derived for the spatially correlated system with Regularized Zero-Forcing (RZF) scheme. A realistic multipath environment consisting of a dominant line-of-sight (LOS) path along the non-deterministic non-line-of-sight (NLOS) component with imperfect channel information and pilot contamination is considered for deriving the expression. The system is evaluated for the carrier frequency of \(30 GHz\) in the mmWave range with different parameters using Monte Carlo simulations to validate the derived expression. The numerical results show that the derived expression for ergodic user rate provides a tight approximation for the large-dimensional system and accurate results for small values. As for the Rician Factor, \(\kappa =0.5\), transmit power of \(10 dB\) and training power of \(6 dB\), the average user rate obtained for \(M=100\) and \(K=50\) is \(1.1233\) bits/sec/Hz/User/cell for simulated value and \(1.1243\) bits/sec/Hz/User/cell for theoretical value. The proposed correlated multicellular mMIMO system is shifted from the mmWave range to \(sub-6 GHz\) to compare it with the existing system in (Sanguinetti et al. in IEEE Transactions on Communications 67:1939–1955, 2018). The derived results show the improvement in system throughput for the proposed system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data Availability

The data generated and analysed during the current study of the given wireless communication multi-cell model is generated using the software MATLAB R2021a. It is not publicly available, as when the data is generated each time based on the program code (designed for the multi-cell mMIMO system) using the MATLAB software based on the system model parameters mentioned above, the data generated is based on the random results of the channel coefficients and user locations from the BSs in each cell. So, to check the system behaviour, multiple numbers of channel realizations and multiple system model setups are averaged to get the results mentioned in the manuscript. These results are describe using the figures and their respective values are mentioned in the table given in the manuscript.

References

  1. Paulraj, A. J., & Kailath, T. (1994). U.S. Patent No. 5,345,599. Washington, DC: U.S. Patent and Trademark Office. https://patents.google.com/patent/US5345599A/en

  2. Jensen, M. A., & Wallace, J. W. (2004). A review of antennas and propagation for MIMO wireless communications. IEEE Transactions on Antennas and propagation, 52(11), 2810–2824. https://doi.org/10.1109/TAP.2004.835272

    Article  Google Scholar 

  3. Marzetta, T. L. (2016). Fundamentals of massive MIMO. Cambridge University Press. https://doi.org/10.1017/CBO9781316799895

    Article  Google Scholar 

  4. Goldsmith, A., Jafar, S. A., Jindal, N., & Vishwanath, S. (2003). Capacity limits of MIMO channels. IEEE Journal on selected areas in Communications, 21(5), 684–702. https://doi.org/10.1109/JSAC.2003.810294

    Article  MATH  Google Scholar 

  5. Spencer, Q. H., Peel, C. B., Swindlehurst, A. L., & Haardt, M. (2004). An introduction to the multi-user MIMO downlink. IEEE communications Magazine, 42(10), 60–67. https://doi.org/10.1109/MCOM.2004.1341262

    Article  Google Scholar 

  6. Gesbert, D., Kountouris, M., Heath, R. W., Chae, C. B., & Salzer, T. (2007). Shifting the MIMO paradigm. IEEE signal processing magazine, 24(5), 36–46. https://doi.org/10.1109/MSP.2007.904815

    Article  Google Scholar 

  7. Marzetta, T. L. (2006, October). How much training is required for multiuser MIMO?. In 2006 fortieth asilomar conference on signals, systems and computers. IEEE. pp. 359–363. https://doi.org/10.1109/ACSSC.2006.354768

  8. Marzetta, T. L. (2010). Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE transactions on wireless communications, 9(11), 3590–3600. https://doi.org/10.1109/TWC.2010.092810.091092

    Article  Google Scholar 

  9. Hoydis, J., Ten Brink, S., & Debbah, M. (2011, September). Massive MIMO: How many antennas do we need?. In 2011 49th Annual Allerton conference on communication, control, and computing (Allerton). IEEE. pp. 545–550. https://doi.org/10.1109/Allerton.2011.6120214

  10. 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. https://doi.org/10.1109/MCOM.2014.6736761

    Article  Google Scholar 

  11. Björnson, E., Larsson, E. G., & Marzetta, T. L. (2016). Massive MIMO: Ten myths and one critical question. IEEE Communications Magazine, 54(2), 114–123. https://doi.org/10.1109/MCOM.2016.7402270

    Article  Google Scholar 

  12. Min, K., Kim, T., & Jung, M. (2023). Performance analysis of multiuser massive MIMO with multi-antenna users: Asymptotic data rate and its application. ICT Express. https://doi.org/10.1016/j.icte.2023.01.003

    Article  Google Scholar 

  13. Hachem, W., Khorunzhiy, O., Loubaton, P., Najim, J., & Pastur, L. (2008). A new approach for mutual information analysis of large dimensional multi-antenna channels. IEEE Transactions on Information Theory, 54(9), 3987–4004. https://doi.org/10.1109/TIT.2008.928229

    Article  MathSciNet  MATH  Google Scholar 

  14. Wagner, S., Couillet, R., Debbah, M., & Slock, D. T. (2012). Large system analysis of linear precoding in correlated MISO broadcast channels under limited feedback. IEEE transactions on information theory, 58(7), 4509–4537. https://doi.org/10.1109/TIT.2012.2191700

    Article  MathSciNet  MATH  Google Scholar 

  15. Hoydis, J., Ten Brink, S., & Debbah, M. (2013). Massive MIMO in the UL/DL of cellular networks: How many antennas do we need? IEEE Journal on selected Areas in Communications, 31(2), 160–171. https://doi.org/10.1109/JSAC.2013.130205

    Article  Google Scholar 

  16. Fatema, N., Hua, G., Xiang, Y., Peng, D., & Natgunanathan, I. (2017). Massive MIMO linear precoding: A survey. IEEE systems journal, 12(4), 3920–3931. https://doi.org/10.1109/JSYST.2017.2776401

    Article  Google Scholar 

  17. Israr, A., Rauf, Z., Muhammad, J., & Khan, F. (2017). Performance analysis of downlink linear precoding in massive MIMO systems under imperfect CSI. Wireless Personal Communications, 96(2), 2603–2619. https://doi.org/10.1007/s11277-017-4314-0

    Article  Google Scholar 

  18. Zhang, W., Pan, C., Du, B., Chen, M., Gong, X., & Dai, J. (2014). Downlink SINR study in multiuser large scale antenna systems. Wireless personal communications, 79(2), 1539–1556. https://doi.org/10.1007/s11277-014-1945-2

    Article  Google Scholar 

  19. Lo, T. K. Y. (1999). Maximum transmission ratio. IEEE Transactions on Communications. doi, 10(1109/26), 795811.

    Google Scholar 

  20. He, J. (2010). Precoding and equalization for MIMO broadcast channels with applications in spread spectrum systems (Doctoral dissertation, Northeastern University). https://doi.org/10.17760/d20002063

  21. Müller, A., Couillet, R., Björnson, E., Wagner, S., & Debbah, M. (2015). Interference-aware RZF precoding for multicell downlink systems. IEEE Transactions on Signal Processing, 63(15), 3959–3973. https://doi.org/10.1109/TSP.2015.2423262

    Article  MathSciNet  MATH  Google Scholar 

  22. Rusek, F., Persson, D., Lau, B. K., Larsson, E. G., Marzetta, T. L., Edfors, O., & Tufvesson, F. (2012). Scaling up MIMO: Opportunities and challenges with very large arrays. IEEE signal processing magazine, 30(1), 40–60. https://doi.org/10.1109/MSP.2011.2178495

    Article  Google Scholar 

  23. Peel, C. B., Hochwald, B. M., & Swindlehurst, A. L. (2005). A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: Channel inversion and regularization. IEEE Transactions on Communications, 53(1), 195–202. https://doi.org/10.1109/TCOMM.2004.840638

    Article  Google Scholar 

  24. Mostafa, M., Newagy, F., & Hafez, I. (2021). Complex regularized zero forcing precoding for massive MIMO systems. Wireless Personal Communications, 120(1), 633–647. https://doi.org/10.1007/s11277-021-08482-4

    Article  Google Scholar 

  25. Dikmen, O., Pakyurek, M., & Kulac, S. (2023). A novel pilot power allocation method based on energy efficiency maximization in massive MIMO systems for future generation wireless technologies. International Journal of Communication Systems, 36(3), e5384. https://doi.org/10.1002/dac.5384

    Article  Google Scholar 

  26. Liu, P., Kong, D., Ding, J., Zhang, Y., Wang, K., & Choi, J. (2021). Channel estimation aware performance analysis for massive MIMO with Rician fading. IEEE Transactions on Communications, 69(7), 4373–4386. https://doi.org/10.1109/TCOMM.2021.3073111

    Article  Google Scholar 

  27. Kazemi, M., Göken, Ç., & Duman, T. M. (2023). Robust joint precoding/combining design for multiuser MIMO Systems with calibration errors. IEEE Transactions on Wireless Communications. https://doi.org/10.1109/TWC.2022.3232140

    Article  Google Scholar 

  28. Riadi, A., Boulouird, M., & Hassani, M. M. R. (2021). ZF and MMSE detectors performances of a massive MIMO system combined with OFDM and M-QAM modulation. Wireless Personal Communications, 116(4), 3261–3276. https://doi.org/10.1007/s11277-020-07848-4

    Article  Google Scholar 

  29. Yang, D., Xu, J., Xu, W., Huang, Y., & Lu, Z. (2023). Secure communication for spatially correlated ris-aided multiuser massive MIMO systems: Analysis and optimization. IEEE Communications Letters. https://doi.org/10.1109/LCOMM.2023.3236325

    Article  Google Scholar 

  30. Tse, D., & Viswanath, P. (2005). Fundamentals of wireless communication. Cambridge University Press. https://doi.org/10.1017/CBO9780511807213

    Article  MATH  Google Scholar 

  31. Zhang, Q., Jin, S., Wong, K. K., Zhu, H., & Matthaiou, M. (2014). Power scaling of uplink massive MIMO systems with arbitrary-rank channel means. IEEE Journal of Selected Topics in Signal Processing, 8(5), 966–981. https://doi.org/10.1109/JSTSP.2014.2324534

    Article  Google Scholar 

  32. Zhang, J., Dai, L., Zhang, X., Björnson, E., & Wang, Z. (2015). Achievable rate of Rician large-scale MIMO channels with transceiver hardware impairments. IEEE Transactions on Vehicular Technology, 65(10), 8800–8806. https://doi.org/10.1109/TVT.2015.2504428

    Article  Google Scholar 

  33. Sanguinetti, L., Kammoun, A., & Debbah, M. (2017, March). Asymptotic analysis of multicell massive MIMO over Rician fading channels. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE. (pp. 3539–3543). https://doi.org/10.1109/ICASSP.2017.7952815

  34. Sanguinetti, L., Kammoun, A., & Debbah, M. (2018). Theoretical performance limits of massive MIMO with uncorrelated Rician fading channels. IEEE Transactions on Communications, 67(3), 1939–1955. https://doi.org/10.1109/TCOMM.2018.2884003

    Article  Google Scholar 

  35. Kammoun, A., Sanguinetti, L., Debbah, M., & Alouini, M. S. (2019). Asymptotic analysis of RZF in large-scale MU-MIMO systems over Rician channels. IEEE Transactions on Information Theory, 65(11), 7268–7286. https://doi.org/10.1109/TIT.2019.2923401

    Article  MathSciNet  MATH  Google Scholar 

  36. Björnson, E., Hoydis, J., & Sanguinetti, L. (2017). Massive MIMO networks: Spectral, energy, and hardware efficiency. Foundations and Trends® in Signal Processing, 11(3–4), 154–655. https://doi.org/10.1561/2000000093

    Article  Google Scholar 

  37. Özdogan, Ö., Björnson, E., & Larsson, E. G. (2019). Massive MIMO with spatially correlated Rician fading channels. IEEE Transactions on Communications, 67(5), 3234–3250. https://doi.org/10.1109/TCOMM.2019.2893221

    Article  Google Scholar 

  38. Sanguinetti, L., Björnson, E., & Hoydis, J. (2019). Toward massive MIMO 2.0: Understanding spatial correlation, interference suppression, and pilot contamination. IEEE Transactions on Communications, 68(1), 232–257. https://doi.org/10.1109/TCOMM.2019.2945792

    Article  Google Scholar 

  39. Kumar, V., Amudala, D. N., & Budhiraja, R. (2023). Analysis and optimization of hardware impaired fd rician-faded mmimo systems. IEEE Transactions on Communications. https://doi.org/10.1109/TCOMM.2023.3252545

    Article  Google Scholar 

  40. Couillet, R., & Debbah, M. (2011). Random matrix methods for wireless communications. Cambridge University Press.

    Book  MATH  Google Scholar 

  41. Hachem, W., Kharouf, M., Najim, J., & Silverstein, J. W. (2012). A CLT for information-theoretic statistics of non-centered gram random matrices. Random Matrices: Theory and Applications, 1(02), 1150010. https://doi.org/10.1142/S2010326311500109

    Article  MathSciNet  MATH  Google Scholar 

  42. Hachem, W., Loubaton, P., Najim, J., & Vallet, P. (2013). On bilinear forms based on the resolvent of large random matrices. In Annales de l’IHP Probabilités et statistiques, 49(1), 36–63. https://doi.org/10.1214/11-AIHP450

    Article  MathSciNet  MATH  Google Scholar 

  43. Hoydis, J. (2012). Random matrix theory for advanced communication systems (Doctoral dissertation, Supélec). https://theses.eurasip.org/theses/653/random-matrix-theory-for-advanced-communication/

  44. 3GPP. (2018). Study on channel model for frequencies from 0.5 to 100 GHz. 3rd Generation Partnership Project (3GPP), Tech. Rep., 38. https://www.etsi.org/deliver/etsi_tr/138900_138999/138901/14.00.00_60/tr_138901v140000p.pdf

Download references

Funding

This work is supported by DST-Inspire, Ministry of Science and Technology, Government of India (DST/INSPIRE Fellowship/[IF190244]). Author Harleen Kaur is awarded by this INSPIRE Fellowship.

Author information

Authors and Affiliations

  1. Electronics & Communication Engineering Department, Thapar Institute of Engineering & Technology, Patiala, Punjab, 147004, India

    Harleen Kaur & Ankush Kansal

Authors
  1. Harleen Kaur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ankush Kansal
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed equally to this work and have read and approved the final manuscript.

Corresponding author

Correspondence to Harleen Kaur.

Ethics declarations

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaur, H., Kansal, A. Closed-form Analysis of RZF in Multicell Massive MIMO Over Correlated Rician Channel. Wireless Pers Commun 131, 1685–1719 (2023). https://doi.org/10.1007/s11277-023-10518-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-023-10518-w

Keywords

Search

Navigation