This paper studies the performance of quantized massive multiple-input multiple-output (MIMO) systems with superimposed pilots (SP), using linear minimum mean-square-error (LMMSE) channel estimation and maximum ratio combining (MRC) detection. In contrast to previous works, arbitrary-bit analog-to-digital converters (ADCs) are considered. We derive an accurate approximation of the uplink achievable rate considering the removal of estimated pilots. Based on the analytical expression, the optimal pilot power factor that maximizes the achievable rate is deduced and an expression for energy efficiency (EE) is given. In addition, the achievable rate and the optimal power allocation policy under some asymptotic limits are analyzed. Analysis shows that the systems with higher-resolution ADCs or larger number of base station (BS) antennas need to allocate more power to pilots. In contrast, more power needs to be allocated to data when the channel is slowly varying. Numerical results show that in the low signal-to-noise ratio (SNR) region, for 1-bit quantizers, SP outperforms time-multiplexed pilots (TP) in most cases, while for systems with higher-resolution ADCs, the SP scheme is suitable for the scenarios with comparatively small number of BS antennas and relatively long channel coherence time.

In this letter, we elucidate the ergodic capacity of multiple-input multiple-output (MIMO) systems with M-ary phase-shift keying (MPSK) modulation and time-multiplexed pilots in frequency-flat Rayleigh fading environment. With linear minimum mean square error (LMMSE) channel estimation, the optimal pilots design is presented. For mathematical tractability, we derive an easy-computing closed-form lower bound of the channel capacity. Based on the lower bound, the optimal power allocation between the data and pilots is also presented in closed-form, and the optimal training length is investigated by numerical optimization. It is shown that the transmit scheme with equal training and data power and optimized training length provides suboptimal performance, and the transmit scheme with optimized training length and training power is optimal. With the latter scheme, in most situations, the optimal training length equals the number of the transmit antennas and the corresponding optimal power allocation can be easily computed with the proposed formula.