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Performance analysis of ZF receivers with imperfect CSI for uplink massive MIMO systems

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Abstract

We consider the uplink of massive multiple-input multiple-output systems in a multicell environment. Since the base station (BS) estimates the channel state information (CSI) using the pilot signals transmitted from the users, each BS will have imperfect CSI in practice. Assuming zero-forcing method to eliminate the multi-user interference, we derive the exact analytical expressions for the probability density function of the signal-to-interference-plus-noise ratio, the corresponding achievable rate, the outage probability, and the symbol error rate (SER) when the BS has imperfect CSI. An upper bound of the SER is also derived for an arbitrary number of antennas at the BS. Moreover, we derive the upper bound of the achievable rate for the case where the number of antennas at the BS goes to infinity, and the analysis is verified by presenting numerical results.

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Notes

  1. Assume that the k-th columns \(\mathbf {h}_{i\ell ,k}\) and \(\mathbf {h}_{j\ell ,k}\) of \(\mathbf {H}_{i\ell }\) and \(\mathbf {H}_{j\ell k}\), respectively, are mutually independent \(M\times 1\) vectors, whose elements are i.i.d. random variables with zero mean and unit variance. From the law of large numbers, we have \(\frac{1}{M}\mathbf {h}_{i\ell ,k}^{\dagger }\mathbf {h}_{i\ell ,k}\mathop {\rightarrow }\limits ^{a.s}1\) and \(\frac{1}{M}\mathbf {h}_{i\ell ,k}^{\dagger }\mathbf {h}_{j\ell ,k}\mathop {\rightarrow }\limits ^{a.s}0\) as \(M\rightarrow \infty \), where \(\mathop {\rightarrow }\limits ^{a.s}\) denotes the almost sure convergence.

  2. With \(\mathbf {h}_{i\ell ,k}\) and \(\mathbf {h}_{j\ell ,k}\) defined in (29), from Lindeberg–Levy central limit theorem, we have \(\frac{1}{\sqrt{M}}\mathbf {h}_{i\ell ,k}^{\dagger }\mathbf {h}_{j\ell ,k}\mathop {\rightarrow }\limits ^{d.}\mathcal {CN}(0,1)\) as \(M\rightarrow \infty \), where \(\mathop {\rightarrow }\limits ^{d.}\) denotes convergence in distribution.

  3. Note that \(\alpha _{i\ell }\) and \(\hat{\mathbf {G}}_{\ell \ell }\) in (17) are estimated by \(\alpha _{i\ell }=\mathrm {tr}\bigl (\mathbb {E}[\varvec{\xi }_{i\ell }\varvec{\xi }_{i\ell }^{\dag }]\bigr )\) and (9), respectively.

References

  1. 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.

    Article  Google Scholar 

  2. Gesbert, D., Kountouris, M., Heath, R. W., Chae, C.-B., & Salzer, T. (2007). Shifting the MIMO paradigm. IEEE Communications Magazine, 24(5), 36–46.

    Google Scholar 

  3. Andrews, J. G., Choi, W., & Heath, R. W. (2007). Overcoming interference in spatial multiplexing MIMO cellular networks. IEEE Transactions on Wireless Communications, 14(6), 95–104.

    Article  Google Scholar 

  4. Ngo, H. Q., Larsson, E. G., & Marzetta, T. L. (2013). Energy and spectral efficiency of very large multiuser MIMO systems. IEEE Transactions on Communications, 61(4), 1436–1449.

    Article  Google Scholar 

  5. Hoydis, J., Brink, S. T., & 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.

    Article  Google Scholar 

  6. Yang, H., & Marzetta, T. L. (2013). Performance of conjugate and zero-forcing beamforming in large-scale antenna systems. IEEE Journal on Selected Areas in Communications, 31(2), 172–179.

    Article  Google Scholar 

  7. Hanif, M.-F., Tran, L.-N., Tölli, A., & Juntti, M. (2014). Computationally efficient robust beamforming for SINR balancing in multicell downlink with applications to large antenna array systems. IEEE Transactions on Communications, 62(6), 1908–1920.

    Article  Google Scholar 

  8. Marzetta, T. L. (2010). Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Transactions on Wireless Communications, 9(11), 3590–3600.

    Article  Google Scholar 

  9. Huh, H., Caire, G., Papadopoulos, H. C., & Ramprashad, S. A. (2012). Achieving massive MIMO spectral efficiency with a not-so-large number of antennas. IEEE Transactions on Wireless Communication, 11(9), 3226–3239.

    Article  Google Scholar 

  10. Ozgur, A., Leveque, O., & Tse, D. (2013). Spatial degrees of freedom of large distributed MIMO systems and wireless ad hoc networks. IEEE Journal on Selected Areas in Communications, 31(2), 202–2014.

    Article  Google Scholar 

  11. Pitarokoilis, A., Mohammed, S. K., & Larsson, E. G. (2012). On the optimality of single-carrier transmission in large-scale antenna systems. IEEE Communications Letters, 1(4), 276–279.

    Article  Google Scholar 

  12. Jose, J., Ashikhmin, A., & Marzetta, T. L. (2011). Pilot contamination and precoding in multi-cell TDD systems. IEEE Transactions on Wireless Communications, 10(8), 2640–2651.

    Article  Google Scholar 

  13. Dai, H., Molisch, A. F., & Poor, H. V. (2004). Downlink capacity of interference-limited MIMO systems with joint detection. IEEE Transactions on Wireless Communication, 3(2), 798–801.

    Article  Google Scholar 

  14. Tran, L. N., Juntti, M., Bengtsson, M., & Ottersten, B. (2013). Weighted sum rate maximization for MIMO broadcast channels using dirty paper coding and zero-forcing methods. IEEE Transactions on Communications, 61(6), 2362–2373.

    Article  Google Scholar 

  15. Cramer, H. (1970). Random variables and probability distributions. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  16. Ngo, H. Q., Matthaiou, M., Duong, T. Q., & Larsson, E. G. (2013). Uplink performance analysis of multicell MU-SIMO systems with ZF receivers. IEEE Transactions on Vehicular Technology, 62(9), 4471–4483.

    Article  Google Scholar 

  17. Yin, H., Gesbert, D., Filippou, M., & Liu, Y. (2013). A coordinated approach to channel estimation in large-scale multiple-antenna systems. IEEE Journal on Selected Areas in Communications, 31(2), 264–273.

    Article  Google Scholar 

  18. Fehske, A., Fettweis, G., Malmodin, J., & Biczok, G. (2011). The global footprint of mobile communications: The ecological and economic perspective. IEEE Communications Magazine, 49(8), 55–62.

    Article  Google Scholar 

  19. Brevis, P. G., Gondzio, J., Fan, Y., Poor, H. V., Thompson, J., Krikidis, I., et al. (2011). Base station location optimization for minimal energy consumption in wireless networks. In Proceedings of the 2nd green wireless communications & networks workshop (GreeNet 2011) (pp. 15–18). Hungary: Budapest.

  20. Larsson, E. G., Tufvesson, F., Edfors, O., & Marzetta, T. L. (2014). Massive MIMO for next generation wireless systems. IEEE Communications Magazine, 52(2), 186–195.

    Article  Google Scholar 

  21. Rusek, F., Persson, D., Lau, B. K., Larsson, E. G., Marzetta, T. L., Edfors, O., et al. (2013). Scaling up MIMO: Opportunities and challenges with very large arrays. IEEE Communication Magazine, 30(1), 40–46.

    Google Scholar 

  22. Ngo, H. Q., Larsson, E. G., & Marzetta, T. L. (2013). The multicell multiuser MIMO uplink with very large antenna and a finite-dimensional channel. IEEE Transactions on Communications, 61(6), 2350–2361.

    Article  Google Scholar 

  23. Tse, D. N. C., & Viswanath, P. (2005). Fundamentals of wireless communications. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  24. Gopalakrishnan, B., & Jindal, N. (2011). An analysis of pilot contamination on multi-user MIMO cellular systems with many antennas. In Proceedings of the IEEE international workshop signal process. Advances in wireless communications (SPAWC 2011), San Francisco, CA USA, pp. 381–385.

  25. Kay, S. M. (1993). Fundamentals of statistical signal processing: Estimation theory. Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

  26. Tulino, A. M., & Verdu, S. (2004). Random matrix theory and wireless communications. Foundations and Trends in Communication and Information Theory, 1(1), 1–182.

    Article  Google Scholar 

  27. Gore, D. A., Heath, R. W., & Paulraj, A. J. (2002). Transmit selection in spatial multiplexing systems. IEEE Communications Letters, 6(11), 1491–493.

    Article  Google Scholar 

  28. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of integrals, series, and products (7th ed.). San Diego, CA: Academic Press.

    Google Scholar 

  29. Simon, M. K., & Alouini, M.-S. (2000). Digital communication over fading channels: A unified approach to performance analysis. New York: Wiley.

    Book  Google Scholar 

  30. Yates, R. D., & Goodman, D. J. (2005). Probability and stochastic processes (2nd ed.). New York: Wiley.

    Google Scholar 

  31. Chiani, M., Dardari, D., & Simon, M. K. (2003). New exponential bounds and approximations for the computation of error probability in fading channels. IEEE Transactions on Wireless Communications, 2(4), 840–845.

    Article  Google Scholar 

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Acknowledgments

This work was supported by Basic Research Laboratories (BRL) through NRF grant funded by the Ministry of Science, ICT and Future Planning (MSIP) (No. 2015056354).

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  1. School of Electronic Engineering, Soongsil University, Seoul, 06978, Korea

    Van-Dinh Nguyen & Oh-Soon Shin

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  1. Van-Dinh Nguyen
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Correspondence to Oh-Soon Shin.

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Nguyen, VD., Shin, OS. Performance analysis of ZF receivers with imperfect CSI for uplink massive MIMO systems. Telecommun Syst 65, 241–252 (2017). https://doi.org/10.1007/s11235-016-0225-8

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