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.
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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).
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Appendix: Proof of Theorem 3
Appendix: Proof of Theorem 3
Proof
Using the matrix identity
(31) can be rewritten as
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
And because \({\log _2}\det \left( \cdot \right) \) is continuous function, we have
Similarly, as \(M \rightarrow \infty \), for the second term of the RHS of (34), we have
Defining
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).
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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
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DOI: https://doi.org/10.1007/s11277-014-1754-7