ion of the matrices and 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. As such, a 3-bit codebook for two transmit-antennas and a 6-bit codebook for four transmit-antennas iiiThe Euclidean norm of square matrix A is defined as ||A||2 = The spectral norm of matrix A is the largest singular value of A or the square root of the largest eigenvalue of the positive-semi-definite matrix i.e., ||A/l2 (AHA) where AH denotes the conjugate transpose of A. [67,68] CHAPTER 10 The IEEE 802. 16m Physical Layer (Part II) have been considered for IEEE 802.16m. Additional gain can be obtained using a differential codebook. 6. Low memory requirement is another design consideration for the MIMO codebooks. The IEEE 802.16-2009 standard codebooks were optimized exclusively for uncorrelated scenarios and the performance loss in other scenarios (correlated antennas, dual-polarized antennas) is significant. In order to improve the performance of the closed-loop MIMO schemes in various deployment scenarios, DFT-based transform codebooks and associated precoding matrices were investigated that were optimized for cross-polarized channels. An alternative approach to codebook-based precoding is to use differential codebook precoding, in which the transmitter computes the precoding matrix by using a differential codebook precoding matrix specified by the receiver and the previous precoding matrix used in the transmitter, thus Wk = Cmopt Wk-1 where each differential codebook precoding matrix Cm, m = 1,2, N'codebook is an N X N matrix. The receiver selects the optimum differential precoding matrix which maximizes the instantaneous capacity as follows: Optimum - Codebook - Index : Mopt = (10-44) where {C', C'2, , C'Ncodebook } = Fk-1} and r is the signal-to-noise codebook codebook ratio. The initial precoding matrix could be set as the first M columns of the N X N identity matr