Skip to main content
Log in

Uplink Sum-Rate Analysis of Massive MIMO System with Pilot Contamination and CSI Delay

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

We consider multi-cell multi-user massive MIMO system under correlated Rayleigh fading channels. Taking pilot contamination and CSI delay into consideration, we derive the equivalent channel model with MMSE channel estimation and one-tap prediction. Employing this equivalent channel model, the lower bound of the uplink sum-rate is derived, and its asymptotical performance is studied when the base station antenna number goes without bound. We find that if we schedule the \(k\)-th user of all cells who have the same prediction coefficient, the uplink sum-rate is the same as the one with no CSI delay when the number of BS antennas goes without bound at a much greater rate than the number of users. Simulation results show that the asymptotic approximation has good performance for large \(M\), and suggest that large antenna array can compensate for the decay due to CSI delay. Simulation results also verify our guess that CSI delay does not necessarilly decrease the uplink sum-rate due to the impact of pilot contamination.

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

Access this article

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

Similar content being viewed by others

References

  1. 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 

  2. 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 Signal Processing Magazine, 30(1), 40–60.

    Article  Google Scholar 

  3. Ngo, H. Q., & Larsson, E. G. (2012). Very large MIMO, massive MIMO, large-scale antenna systems. http://www.commsys.isy.liu.se/egl/vlm/vlm.html.

  4. Marzetta, T. L. (2006). How much training is required for multiuser MIMO? In Proceedings of IEEE asilomar conference on signals, systems and computers (ACSSC06), Pacific Grove, CA, USA (pp. 359–363).

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

    Article  Google Scholar 

  6. 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 

  7. Ngo, H. Q., Larsson, E. G., & Marzetta, T. L. (2011). Analysis of the pilot contamination effect in very large multicell multiuser MIMO systems for physical channel models. In Proceedings of IEEE international conference on acoustics, speech and signal processing (ICASSP11), Prague, Czech Republic (pp. 3464–3467).

  8. Hoydis, J., Brinkz, 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.

    Article  Google Scholar 

  9. Wang, D., Ji, C., Gao, X. Q., Sun, S., & You, X. (2013). Uplink sum-rate analysis of multi-cell multi-user massive MIMO system. In Proceedings of IEEE international conference on communications (ICC13), Budapest, Hungary (pp. 3997–4001).

  10. Zhou, B., Jiang, L., Zhao, S., & He, C. (2011). BER analysis of TDD downlink multiuser MIMO systems with imperfect channel state information. EURASIP Journal on Advances in Signal Processing, 2011, 104.

    Article  Google Scholar 

  11. Zhou, S., & Giannakis, G. B. (2004). How accurate channel prediction needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO channels? IEEE Transactions on Wireless Communications, 3(4), 1285–1294.

    Article  Google Scholar 

  12. Kay, S. (1993). Fundamental of statistical signal processing: Estimation theory. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  13. Hassibi, B., & Hochwald, B. M. (2003). How much training is needed in multiple-antenna wireless links? IEEE Transactions on Information Theory, 49(4), 951–963.

    Article  MATH  Google Scholar 

  14. Loyka, S. L. (2011). Channel capacity of MIMO architecture using the exponential correlation matrix. IEEE Communications Lettets, 5(9), 369–371.

    Article  Google Scholar 

  15. Couillet, R., & Debbah, M. (2011). Random matrix methods for wireless communications (1st ed.). New York, NY: Cambridge University Press.

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the Natural Science Foundation of China (NSFC) under Grant 61221002 and 61371113, the National Basic Research Program of China (973 Program 2013CB336600), National Key Special Program No. 2012ZX03 003005-003, 2013ZX03003003-005, the NSFC under Grants 61271205, and State Key Laboratory of Wireless Mobile Communications (CATT).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Juan Cao.

Appendix: Proof of Theorem 3

Appendix: Proof of Theorem 3

Proof

Using the matrix identity

$$\begin{aligned} \det \left( {{\varvec{I}} + {\varvec{AB}}} \right) = \det \left( {{\varvec{I}} + {\varvec{BA}}} \right) \end{aligned}$$

(31) can be rewritten as

$$\begin{aligned} {{\hat{C}}_{{\mathrm{LB}}}^\tau }&= {\log _2}\det \left( {{\varepsilon ^{ - \tau }}\left( {\sum \limits _{i = 1}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {{\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) }^{ - 1}}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}} + {{\varvec{I}}_K}} \right) \nonumber \\&\quad - {\log _2}\det \left( {{\varepsilon ^{ - \tau }}\left( {\sum \limits _{i \ne l}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {{\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) }^{ - 1}}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}} + {{\varvec{I}}_K}} \right) \end{aligned}$$
(34)

According to [15, Theorem 3.4, Theorem 3.7], we know \(\frac{1}{M}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}}{\xrightarrow {M \rightarrow \infty }}{{\varvec{I}}_K}\), so we obtain

$$\begin{aligned}&\frac{1}{M}{\varepsilon ^{ - \tau }}\left( {\sum \limits _{i = 1}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) ^{ - 1}}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}} \nonumber \\&- {\varepsilon ^{ - \tau }}\left( {\sum \limits _{i = 1}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) ^{ - 1}}{{\varvec{I}}_K} {\xrightarrow {M \rightarrow \infty }} 0 \end{aligned}$$
(35)

And because \({\log _2}\det \left( \cdot \right) \) is continuous function, we have

$$\begin{aligned}&{\log _2}\det \left( {{\varepsilon ^{ - \tau }}\left( {\sum \limits _{i = 1}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {{\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) }^{ - 1}}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}} + {{\varvec{I}}_K}}\right) \nonumber \\&\quad \quad \quad - \sum \limits _{k = 1}^K {{{\log }_2}\left( {M{\varepsilon ^{ - \tau }}\frac{{\sum \nolimits _{i = 1}^L {{{\left| {\lambda _{l,i,k}^\tau } \right| }^2}} }}{{\sum \nolimits _{i = 1}^L {{\lambda _{l,i,k}}} + {\gamma _{\mathrm{P}}}}} + 1} \right) }{\xrightarrow {M \rightarrow \infty }} 0 \end{aligned}$$
(36)

Similarly, as \(M \rightarrow \infty \), for the second term of the RHS of (34), we have

$$\begin{aligned}&{\log _2}\det \left( {{\varepsilon ^{ - \tau }}\left( {\sum \limits _{i \ne l}^L {{\varvec{\varLambda }}_{l,i}^\tau {{\left( {{\varvec{\varLambda }}_{l,i}^\tau } \right) }^{\mathrm{H}}}} } \right) {{\left( {\sum \limits _{i = 1}^L {{{\varvec{\varLambda }}_{l,i}}} + {\gamma _{\mathrm{P}}}{{\varvec{I}}_K}} \right) }^{ - 1}}{{\varvec{H}}^{\mathrm{H}}}{\varvec{H}} + {{\varvec{I}}_K}} \right) \nonumber \\&\quad \quad \quad - \sum \limits _{k = 1}^K {{{\log }_2}\left( {M{\varepsilon ^{ - \tau }}\frac{{\sum \nolimits _{i \ne l}^L {{{\left| {\lambda _{l,i,k}^\tau } \right| }^2}} }}{{\sum \nolimits _{i = 1}^L {{\lambda _{l,i,k}}} + {\gamma _{\mathrm{P}}}}} + 1} \right) }{\xrightarrow {M \rightarrow \infty }} 0 \end{aligned}$$
(37)

Defining

$$\begin{aligned} {{\hat{C}}_{{\mathrm{LB,inf}}}}^\tau&\triangleq \sum \limits _{k = 1}^K {{{\log }_2}\left( {M{\varepsilon ^{ \!\!-\!\! \tau }}\frac{{\sum \nolimits _{i = 1}^L {{{\left| {\lambda _{l,i,k}^\tau } \right| }^2}} }}{{\sum \nolimits _{i = 1}^L {{\lambda _{l,i,k}}} \!\!+\!\! {\gamma _{\mathrm{P}}}}} \!+\! 1} \right) }- \sum \limits _{k = 1}^K {{{\log }_2}\left( {M{\varepsilon ^{ - \tau }}\frac{{\sum \nolimits _{i \ne l}^L {{{\left| {\lambda _{l,i,k}^\tau } \right| }^2}} }}{{\sum \nolimits _{i = 1}^L {{\lambda _{l,i,k}}} \!+\! {\gamma _{\mathrm{P}}}}} \!+\! 1} \right) } \nonumber \\&=\sum \limits _{k = 1}^K {{{\log }_2}\left( {1 + \frac{{{{\left| {\lambda _{l,l,k}^\tau } \right| }^2}}}{{\sum \nolimits _{i \ne l}^L {{{\left| {\lambda _{l,i,k}^\tau } \right| }^2}} + \frac{{\left( {{\varepsilon ^\tau } + {\gamma _{{\mathrm{UL}}}}} \right) }}{M}\left( {\sum \nolimits _{i = 1}^L {{\lambda _{l,i,k}}} + {\gamma _{\mathrm{P}}}} \right) }}} \right) } \end{aligned}$$
(38)

Using (36) and (37), we see that \({\hat{C}_{{\mathrm{LB}}}^\tau } - {\hat{C}_{{\mathrm{LB,inf}}}}^\tau {\xrightarrow {M \rightarrow \infty }} 0\).

Substituting \(\lambda _{l,i,k}^\tau = {\lambda _{l,i,k}}\rho _{l,i,k}^\tau \) into (38), we can derive (32).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, J., Wang, D., Li, J. et al. Uplink Sum-Rate Analysis of Massive MIMO System with Pilot Contamination and CSI Delay. Wireless Pers Commun 78, 297–312 (2014). https://doi.org/10.1007/s11277-014-1754-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-014-1754-7

Keywords

Navigation