# Hybrid Precoder Using Stiefel Manifold Optimization for Mm-Wave Massive MIMO System
###### Abstract  
Due to the increasing demand for fast data rates and large spectra, millimeter-wave technology plays a vital role in the advancement of 5G communication. The idea behind Mm-Wave communications is to take advantage of the huge and unexploited bandwidth to cope with future multigigabit-per-second mobile data rates, imaging, and multimedia applications. In Mm-Wave systems, digital precoding provides optimal performance at the cost of complexity and power consumption. Therefore, hybrid precoding, i.e., analog-digital precoding, has received significant consideration as a favorable alternative to digital precoding. The conventional methods related to hybrid precoding suffer from low spectral efficiency and large processing time due to nested loops and the number of iterations. A manifold optimization-based algorithm using the gradient method is proposed to increase the spectral efficiency to be near optimal and to speed up the processing speed. A comparison of performances is shown using the simulation outcomes of the proposed work and those of the existing techniques. hybrid precoder; millimeter wave; MIMO; Stiefel manifold; manifold optimization +
Footnote †: journal: Applied  
architecture [12] in which the RF chain from the analog precoder is connected to a subgroup of antennas. The design of the Mm-Wave system is configured as the design of two sub-problems, i.e., precoding on the transmitter side and combining on the receiver side [11; 12; 13; 14]. The performance of the millimeter-wave system in terms of spectral efficiency can be maximized by solving a convex problem in which the Euclidean distance is to be minimized between the hybrid precoder (\(\mathbf{F}_{RF}*\mathbf{F}_{BB}\)) and the digital precoder (\(\mathbf{F}_{opt}\)). \(\mathbf{F}_{RF}\) represents the analog RF precoder matrix; \(\mathbf{F}_{BB}\) is the digital precoder matrix at the baseband; and \(\mathbf{F}_{opt}\) is the optimal precoder matrix obtained by applying singular value decomposing on the channel matrix. The analog RF precoder makes use of phase shifters that inflict supplementary element-wise unit modulus constraints. To solve this problem, an algorithm, known as the orthogonal matching pursuit (OMP) algorithm [11; 12], is employed for the spatial structure of Mm-Wave systems having a large array of antennas, thus transforming it into a sparse reconstruction problem. However, this solution deprecates the performance of Mm-Wave systems due to the limitation on space of analog precoding. Lastly, an algorithm based on the gradient approach was proposed to maximize mutual information at the cost of high complexity [15; 16]. Ref. [14] proposed a manifold optimization (MO) technique based on the alternating minimization (AltMin) algorithm for improving the spectral efficiency to be near optimal. This solution has a nested-loop structural design between \(\mathbf{F}_{BB}\) and \(\mathbf{F}_{RF}\) that reduces the speed of convergence and also increases complexity. Therefore, ref. [17] proposed PE-AltMin, an alternative low-complexity algorithm. However, this algorithm decreases the spectral efficiency by increasing the number of transmitting RF chains. Mm-Wave is a recent technology in the field of wireless communication. To prevent severe loss of propagation of the Mm-Wave channel, a hybrid precoder and combining transceivers are used for Mm-Wave massive MIMO systems [18; 19; 20; 21]. The RF signal in analog precoding is used for energy-harvesting purposes [22; 23; 24; 25; 26]. The concept of cognitive radio can be used for the detection of primary users to improve the energy efficiency [27]. Due to slow variation in the spatial-channel matrix (SCM), as compared with the instantaneous value of complete CSI, ref. [28] proposed a unique spatial-channel algorithm for sub-6 GHz using an SCM. The SCM is a hard problem in the massive MIMO framework due to the massive antenna at the base station.