ements of each constitu- ent matrix or vector from a small binary set, for example, a four alphabet ( +1 1, + j) set, which eliminates the need for matrix or vector multiplication. In addition, nested property of the codebooks can further reduce the complexity of CQI calculation when rank adaptation is performed. Base station may perform rank overriding which results in significant CQI mismatch, if the codebook structure cannot adapt to it. A nested property with respect to rank over- riding can be exploited to mitigate the mismatch effects. Power amplifier balance is taken into consideration when designing codebooks with constant modulus property, which may eliminate unnecessary increase in PAPR. 4. Good performance for a wide range of propagation scenarios, for example, uncorrelated, correlated, and dual-polarized channels, is expected from the codebook design algo- rithms. A DFT-based codebook is optimal for linear array with small antenna spacing since the vectors match with the structure of the transmit array response. In addition, with an optimal selection of the matrices and the entries of the codebook (rotated block diagonal structure), significant gains can be obtained in dual-polarized scenarios. 5. Low feedback and signaling overhead are desirable from operation and performance perspective. 6. Low memory requirement is another design consideration for the MIMO codebooks. Let us consider multi-user MIMO systems and briefly study how precoding is applied in those scenarios. In the downlink direction of a precoded MU-MIMO system (alternatively known as BC in the literature) with NTX transmit antennas at the base station and one receive antenna at the kth mobile station the input-output relationship can be written as = ,Nuser, where = is the NTX X 1 vector of weighted transmitted symbols Si, yk and Nk are the received signal and noise, respectively, hk is the kth NTX X 1 channel vector, where matrix H = (h1,h2, h Nuser The Euclidean norm of square matrix A is defined as The spectral norm of matrix A is the