Energy- and spectral-efficiency of zero-forcing beamforming in massive MIMO systems with imperfect reciprocity calibration: bound and optimization

In a time-division duplex (TDD) system with massive multiple input multiple output (MIMO), channel reciprocity calibration (RC) is generally required in order to cope with the reciprocity mismatch between the uplink and downlink channel state information. Currently, evaluating the achievable spectral efficiency (SE) and energy efficiency (EE) of TDD massive MIMO systems with imperfect RC (IRC) mainly relies on exhausting Monte Carlo simulations and it is infeasible to precisely and concisely quantify the achievable SE and EE with IRC. In this study, a novel method is presented for tightly bounding the achievable SE of massive MIMO systems with zero-forcing beamforming under IRC. On the basis of the analytical results, we demonstrate key insights for practical system design with IRC in three aspects: the scaling rule for interference power, saturation region of the SE, and the bound on the SE loss. Finally, the trade-off between spectral and energy efficiencies in the presence of IRC is determined with algorithms developed to optimize SE (EE) under a constrained EE (SE) value. The loss of optimal total SE and EE due to IRC is also quantified, which shows that the loss of optimal EE is more sensitive to IRC in a typical range of transmit power values.