###### Abstract  
Future wireless networks will require advance physical-layer techniques to meet the requirements of Internet of Everything ( IoE) applications and massive communication systems. To this end, a massive MIMO (m-MIMO) system is to date considered one of the key technologies for future wireless networks. This is due to the capability of m-MIMO to bring a significant improvement in the spectral efficiency and energy efficiency. However, designing an efficient downlink (DL) training sequence for fast channel state information (CSI) estimation, i.e., with limited coherence time, in a frequency division duplex (FDD) m-MIMO system when users exhibit different correlation patterns, i.e., span distinct channel covariance matrices, is to date very challenging. Although advanced iterative algorithms have been developed to address this challenge, they exhibit slow convergence speed and thus deliver high latency and computational complexity. To overcome this challenge, we propose a computationally efficient conjugate gradient-descent (CGD) algorithm based on the Riemannian manifold in order to optimize the DL training sequence at base station (BS), while improving the convergence rate to provide a fast CSI estimation for an FDD m-MIMO system. To this end, the sum rate and the computational complexity performances of the proposed training solution are compared with the state-of-the-art iterative algorithms. The results show that the proposed training solution maximizes the achievable sum rate performance, while delivering a lower overall computational complexity owing to a faster convergence rate in comparison to the state-of-the-art iterative algorithms. massive MIMO systems; CSI estimation; limited coherence time; sum rate maximization; spatial correlation; training sequence design; FDD; TDD; optimization on manifolds; iterative algorithm; gradient algorithm +
Footnote †: journal: Network  
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networks in addition to the innovation of new physical-layer techniques for the future generation of cellular systems. Massive MIMO (m-MIMO) is introduced as a key physical layer technique for the next generation wireless networks that is required to support the huge increasing demand for data traffic [3, 4, 5], whilst improving the energy efficiency [6]. In particular, the m-MIMO system is considered a potential solution for fifth generation (5G) wireless communications and beyond [7]. However, from an information theoretic point of view, the performance of the m-MIMO system depends on the accuracy and availability of the channel state information (CSI) estimation. Obtaining the CSI at the BS depends on the duplexing operation mode, i.e., either frequency division duplex (FDD) or time division duplex (TDD) mode. Despite the promising results of TDD operation with m-MIMO when it comes to the CSI estimation, the vast majority of the currently deployed cellular networks operates in FDD mode. For example, over 85% of the current commercial long term evolution (LTE) wireless mobile networks operate in FDD mode [8]. Therefore, this paper focuses on FDD systems, as the CSI can only be obtained by a dedicated training sequence in the downlink (DL). In particular, to obtain the CSI in FDD systems, the BS needs to optimize the DL training sequence and transmit it to the users [9, 10]. However, the expansion of a number of antenna elements at the BS in the m-MIMO system makes the DL training sequence optimization for fast CSI estimation, i.e., with limited coherence time, in FDD systems very challenging. To overcome the CSI estimation challenge in FDD m-MIMO systems, several studies have investigated the optimization of DL training sequences by considering the scenario where the users exhibit common spatial correlation at the BS under different system model assumptions; namely, refs. [11, 12, 13, 14, 15, 16, 17, 18] exploit correlations in the spatial domain and correlations both in the time and spatial domains, respectively.