a multi - rank beamforming scheme in which the downlink channel is estimated and an optimal precoding matrix to be used by the mrbf transmitter is determined accordingly . the optimal precoding matrix is selected from a codebook of matrices having a recursive structure which allows for efficient computation of the optimal precoding matrix and corresponding signal to interference and noise ratio . the codebook also enjoys a small storage footprint . due to the computational efficiency and modest memory requirements , the optimal precoding determination can be made at user equipment and communicated to a transmitting base station over a limited uplink channel for implementation over the downlink channel .