Limited feedback for multi-carrier beamforming: A rate-distortion approach

The achievable rate of a wideband multi-input single-output channel with multi-carrier transmission is studied with limited feedback of channel state information (CSI). The set of sub-channel vectors are assumed to be jointly quantized and relayed back to the transmitter. Given a fixed feedback rate, the performance of an optimal joint quantization scheme can be characterized by the rate-distortion bound. The distortion metric is the average loss in capacity (forward rate) relative to the capacity with perfect channel state information at the transmitter and receiver. The corresponding rate distortion function gives the forward capacity as a function of feedback rate, and is determined explicitly by casting the minimization of mutual information in the rate-distortion problem as an optimal control problem. Numerical results show that when the feedback rate is relatively small, the rate-distortion bound significantly outperforms separate quantization of the state information of each sub-channel.