Global Optimal Beamforming for Multi-Symbol MIMO Communications

Multiple antennas at transmitter and receiver can be used to improve communication efficiency. However, such improvement can only be achievable if the antennas are coordinated appropriately. There are a rich set of research works contributed for this purpose. However, each of them is only applicable to a special case that can be simplified, for example, into a convex optimization problem or a vector optimization problem on a Grassmannian manifold. In this paper, another interesting case is considered where multiple symbols are used to make the best use of the multiple antenna channel. Such an issue cannot be converted into a convex optimization problem. Instead, it can be considered as a generalization of the vector optimization problem on Grassmannian manifold to that on a complex Stiefel manifold. The proposed algorithm is based on the gradient search on a complex Stiefel manifold of a non-convex problem to maximize the system signal to noise ratio. With appropriately defined Riemannian metric on this manifold, a neat formula has been developed for the gradient function. This proposed algorithm is guaranteed to converge to the global optimum. In addition, it can also be implemented into recurrent neural network to facilitate real-time computation. Its parallel structure can be realized using analog circuits. Furthermore, it is proved that the proposed algorithm is robust against initial condition error. Theoretical analysis and simulation experiments are included to demonstrate the advantages of the proposed algorithm.