Asymptotic performance of MIMO wireless channels with limited feedback

We consider a single-user, point-to-point communication system with M transmit and N receive antennas with independent flat Rayleigh fading between antenna pairs. The mutual information of the multi-input/multi-output (MIMO) channel is maximized when the transmitted symbol vector is a Gaussian random vector with covariance matrix Q. The optimal Q depends on how much channel state information is available at the transmitter. Namely, in the absence of any channel state information, the optimal Q is full-rank and isotropic, whereas with perfect channel knowledge, the optimal Q has columns which are the eigen-vectors of the channel, and has rank at most min {M, N}. We assume that the receiver can feed back B bits to the transmitter (per codeword). The feedback bits are used to choose the columns of Q from a random set of i.i.d. vectors. We compute the mutual information as a function of both B and the rank of Q. Our results are asymptotic in the number of antennas, and show how much feedback is needed to achieve a rate, which is close to the capacity with perfect channel knowledge at the transmitter.