  Future multi-input multi-output (MIMO) wireless communications systems will
use beamforming as a first-step towards realizing the capacity requirements
necessitated by the exponential increase in data demands. The focus of this
work is on beam alignment for time-division duplexing (TDD) systems, for which
we propose a number of novel algorithms. These algorithms seek to obtain good
estimates of the optimal beamformer/combiner pair (which are the dominant
singular vectors of the channel matrix). They are motivated by the power
method, an iterative algorithm to determine eigenvalues and eigenvectors
through repeated matrix multiplication. In contrast to the basic power method
which considers only the most recent iteration and assumes noiseless links, the
proposed techniques consider information from all the previous iterations of
the algorithm and combine them in different ways. The first technique
(Sequential Least-Squares method) sequentially constructs a least-squares
estimate of the channel matrix, which is then used to calculate the
beamformer/combiner pair estimate. The second technique (Summed Power method)
aims to mitigate the effect of noise by using a linear combination of the
previously tried beams to calculate the next beam, providing improved
performance in the low-SNR regime (typical for mmWave systems) with minimal
complexity/feedback overhead. A third technique (Least-Squares Initialized
Summed Power method) combines the good performance of the first technique at
the high-SNR regime with the low-complexity advantage of the second technique
by priming the summed power method with initial estimates from the sequential
method.
