al is i.i.d, the power of (k, m) can be calculated as (3.26) Accordingly, we can calculate the power of I(1)(k,m) as (3.27) 3.3 Inter-numerology Interference Analysis Without loss of generality, we set AF(1) = 0 and assume that the channel delay spread is smaller than the symbol duration. By substituting (3.2) and (3.4) into (3.26), we can have the following expression (derivations can be found in Appendix B): (3.28) where )g(n-5) denotes the equivalent window response taking both numerology 1 transmitter window, i.e. g(n - 5), and numerology 2 receiver window, i.e. q(n - T), into account. Moreover, Ak ¹ = k - k(u) is the spectral distance between sub- carrier k and k(u), where k(u) = for 0 u<Z(2). represents the subcarrier indexes of numerology 2 when its subcarriers are observed with a granularity of Af(1). Fur- thermore, = is the CFR on the u-th subcarrier of numerol- ogy 2. From (3.28), we can find that the INI power is a function of the overlapping window, the spectral distance between the victim and interfering subcarrier, and the CFR of the interfering subband. In Figure 3.4, we depict the overlapping windows in the multi numerologies W-OFDM system, where N2 is set to be 2. By taking into account the transmitter window of the previ- ous LCM symbol for numerology 2, i.e. gu (2) [n - (n2m - 1)N(2) there are N2 + 1 transmitter windows that may overlap with the receiver window of numerology 1. It can be observed that when the two subbands have the same numerology, the overlapping window is equiv- alent to the numerology 1 receiver window, and can satisfy the condition in (3.8). In this case, there is no INI generated. In addition, as indicated by (3.28), the value of p(1) (k,m) not related to the symbol index m under generalized synchronization scenario; thus, (k) is used instead of p(1) (k, m) in the following discussions. As a special case of W-OFDM, the overlapping window of the CP-OFDM system is shown in Figure 3.5, where g(2) (n) and qq (n) are changed to rectangular windows. Tx window Tx window numerolog