= diag(diag(Y)) and another matrix as = The M x M matrix Ye with all its diagonal elements being zero, and its non-diagonal elements, copied from the same positions of the matrix Y, produces the ICI. By substituting Y with + , (2.14) can be rearranged as (2.15) desired signal forward ISI backward ISI noise (6) Equalization and detection. The signal expressed in (2.15) can be equalized using the classic equalization methods with a trade-off between complexity and performance. The simplest method is one-tap equalization using diagonal matrix obtained as ak = which can be conducted independently on each subcarrier. However, one-tap equalization can only achieve sub-optimal performance due to the ignored intrinsic interference. More advanced equalization methods will be discussed in Section 2.3.4. 2.3 In-Band Interference Analysis and Discussion 2.3.1 Channel Diagonalization and In-Band Interference-Free Systems In interference-free systems, the channel can be fully diagonalized SO that an elegant one-tap equalization can be utilized to achieve optimal performance. In the sequel, we will derive the condition to achieve interference-free f-OFDM systems. The in-band interference signal to the k-th f-OFDM symbol can be decomposed into three components based on (2.15) as (2.16) forward ISI backward ISI The matrix Y pre produces the filtering/channel f-ISI. It is a strictly upper triangular matrix, of which only the top Np+Nch rows have nonzero elements. The proof can be found in Appendix 2.B. when Rcr is applied to remove the nonzero rows of Opre the length of CP can be chosen as (2.17) to force a zero-valued Y pre such that zero f-ISI is ensured. 2.3 In-Band Interference Analysis and Discussion The matrix generates the filtering b-ISI. It is a strict lower triangular matrix of which only the bottom N rows have nonzero elements. The proof is omitted to conserve space since it can be easily done in a similar manner to Opre Similarly, the length of CS can be chosen as Ncs Ng (2.18) to secure a zero b-ISI. When (2.17) and (