MIMO Systems With Quantized Covariance Feedback

A unified approach is developed for the study of multi-input, multi-output (MIMO) systems under fast fading with quantized covariance feedback. In such systems, the receiver computes, using perfect channel state information (CSI), the covariance matrices to be adopted by the transmitter corresponding to the current channel realization, and feeds back a quantized version of this information to the transmitter using finite, say Nf, bits per channel realization. Our analysis is based on a geometric framework we developed in a companion paper for manifolds of positive semi-definite (covariance) matrices with various trace and rank constraints. That analysis applies to MIMO systems in a unified manner to optimal as well as various suboptimal precoding methods for obtaining input covariances. These include, for example, the capacity achieving spatio-temporal water-filling strategy, the incorporation of just spatial water-filling and reduced-rank beamforming with power allocation in both cases, as well as MIMO systems with antenna selection. For a given system strategy, including the one that achieves capacity, the gap between the communication rates in the perfect and quantized covariance cases is shown to be O(2^-Nf/N), where N is the dimension of the particular Pn manifold used for quantization. This dimension depends on the precoding strategy adopted and strongly determines how quickly the communication rate with quantized covariance feedback approaches that with perfect CSI at the transmitter (CSIT). Our results show how the choice of covariance quantization manifold can be tailored to the available feedback rate in MIMO system design.