Scaling Laws of Optimal Training Lengths for TDD Massive MIMO Systems

In this study, the scaling laws of optimal training lengths were investigated for time-division duplexing massive multiple-input multiple-output systems. First, the generalized asymptotic achievable rate was derived by taking into account both uplink and downlink training. From the general form, the achievable rates of zero-forcing (ZF) and matched filter (MF) precoders are directly obtained by using the mean and variance of the effective channel. Then, we analyzed the rate gaps of ZF and MF with respect to uplink and downlink training, respectively. According to the analysis of rate gaps, it is shown that the achievable rates are more dominated by the uplink training than downlink training for both ZF and MF. The joint optimization problem for training lengths maximizing the spectral efficiency is formulated as a function of rate gaps. To study the scaling of optimal training lengths with the system parameters, we derived the optimal training lengths as a closed-form expression by using an approximation for the logarithm function. From the analysis, it is shown that the optimal training lengths decrease as the number of antennas increases, and increase as the coherence block length increases. In addition, the optimal training lengths increase as the transmit power increases for ZF, whereas it does not change for MF. Finally, we investigated the superior region for noncoherent detection (which means only statistical information is used for demodulation without downlink training) with the system parameters by comparing the spectral efficiency with and without downlink training.