Differential Feedback of MIMO Channel Gram Matrices Based on Geodesic Curves

This paper proposes a differential quantization strategy to be used in the feedback link of a multiple-input multiple-output (MIMO) communication system. This algorithm is applied to the channel Gram matrix using geodesic curves and exploiting the intrinsic geometry of positive definite Hermitian matrices. It also exploits the temporal correlation of the channel, and follows on average the gradient of the cost function associated to the transmitter design criterion. A full description of the algorithm, including the computational cost and a numerical analysis of the effect of delays and errors in the feedback link is presented. Simulation results show that the proposed algorithm improves other techniques based on the direct quantization of the channel response matrix or the quantization of the subspace spanned by the strongest eigenmodes of the MIMO channel, i.e., Grassmannian based techniques. The main drawback of Grassmannian based algorithms is that the transmitter is constrained to apply a uniform power allocation among spatial transmission modes, which is not forced in the algorithm proposed in this paper.