In this article, an important and widely used MATRIX DECOMPOSITION is introduced. This DECOMPOSITION, named the MATRIX polar DECOMPOSITION, is actually a generalization of the polar representation of complex numbers to matrices. Several important applications of MATRIX polar DECOMPOSITION in various fields are considered. Also, some iterative methods for computing the MATRIX polar DECOMPOSITION and their order of convergence are mentioned.

MIMO systems, and particularly massive MIMO systems, achieve high spectral efficiency by using a large number of antennas. An important issue in these systems is beamforming. In fully digital baseband beamforming, an RF chain is required for each antenna, which leads to high cost and power consumption. In analog beamforming, only one RF chain is used and beamforming is performed by using phase shifters. This method does not provide optimal spectral efficiency and thus, analog-digital hybrid methods for beamforming are considered. In this paper, a hybrid beamforming method is proposed in which the required number of RF chains is much less than fully digital method. The precoder and receiver filter are designed by maximizing the spectral efficiency. To this end, the optimal beamforming MATRIX (which contains the right singular vectors of the channel MATRIX) is approximated by the product of two analog and digital beamforming matrices. This approximation is improved by an iterative method. The criterion for the proximity of the two matrices is considered to be the Frobenius norm of their differences. In the receiver, the design of the hybrid beamforming is performed in a similar way, using the mean squares error criterion. Also, to improve the method, a gradient-based algorithm is proposed to further reduce the error. The simulation results show the performance superiority of the proposed method over similar methods as well as its less complexity.

Please click on PDF to view the abstract.

Theobjective in this paper to study and present a new iterative method possessing high convergence order for calculating the polar decompostion of a MATRIX. To do this, it is shown that the new scheme is convergent and has high convergence. The analytical results are upheld via numerical simulations and comparisons.