Finite-rate feedback of input covariance matrices in MIMO systems

We analyze feedback of the optimal input covariance matrix Q from a channel-aware multi-antenna receiver to a multi-antenna transmitter using N_f bits per block. The matrix Q is allowed to have real or complex entries and its rank is allowed to be either fixed or variable. By unravelling the geometry of the quantization spaces involved, we obtain the normalized volume of geodesic balls and use these to evaluate the distortion suffered in quantizing information using different code books. The difference in ergodic capacity between the finite and infinite rate feedback cases is bounded as O(2^-Nf/N) where N is the dimension of the quantization manifold. The results apply to both the MIMO link and the MIMO MAC channels, and do not depend on the specific distribution of the channel matrix.