Complex-Value Recurrent Neural Networks for Global Optimization of Beamforming in Multi-Symbol MIMO Communication Systems

Multiple antennas at transmitter and receiver can be used to improve communication efficiency by canceling channel noises using the correlated information among the signals transmitted from different antennas. In this paper, a novel approach is proposed for this problem for another interesting case 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. It can be considered as a generalization of the vector optimization problem on Grassmannian manifold to that one a complex Stiefel manifold, which has not been well considered yet. 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. It is proved that the proposed algorithm converges to the global optimum. This algorithm can also be implemented into recurrent neural network to facilitate real-time computation. Its parallel structure can be realized using analog circuits. Furthermore, a modified gradient flow defined on the non-compact Stiefel manifold is also developed, which is robust against any initial condition error. The corresponding recurrent neural network is also discussed. Simulation experiments are included to demonstrate the advantages of the proposed algorithms.