 the frequency offset of the subband. Let us define g(1) for 0 < n < N i as the transmit window (as can be seen in Figure 3.2), and assume that the window has a unit response in the middle N 4 - Tx_roff samples and smooth roll-off portions at the first and last N(i) Tx_roff samples. Thus, the first and last N(i) Tx_roff samples of transmit symbol will be weighted by the roll-off portions of the window [20], which results in the smooth transitions between adjacent symbols, thus reducing the OoBE. Note that if we set Tx_roff = 0, the W-OFDM system degenerates to the normal OFDM system. Assume that the time invariant channel for subband i has length LCH and the CIR is h ² = h(i) 1)]T. The received signal can be written as (3.3) 3.2 W-OFDM System Model where () is the additive white Gaussian noise (AWGN). By defining the analysis func- tion as (3.4) the demodulated m-th symbol on subcarrier k can be written as (3.5) In (3.4), qua denotes the receiver window, which is similar to the trans- mitter one, but with a shorter window length (as can be seen in Figure 3.2). Additionally, to maintain the system orthogonality, the first LR Zero elements of q(n) are set to zero; thus ISI can be avoided if the following condition is met: (3.6) Assume that (3.6) is satisfied and the receiver window has vestigial symmetry, i.e. (n)+q,(n+N) = 1,0 N(1) 1, (3.7) we have = s(k (3.8) where s(.) denotes the Dirac delta function. By substituting (3.1) and (3.3) into (3.5), (3.5) can be simplified as (derivation is provided in Appendix A): (3.9) where = the channel frequency response (CFR) of the k-th subcarrier in subband i, and w(Superscript(1)(k) is the corresponding AWGN with the variance 2 (k). Moreover, let us assume that the channel is flat for each subcarrier, such that there is no ISI and inter-channel interference (ICI) in the underlying W-OFDM systems; we can then focus on the INI investigation. The above signal modeling in (3.1)-(3.9) can be reformulated in matrix form. We can express the information symbols as an Z(i) x1 vector