: (3.10) The received signal You can be written as (3.11) (3.12) 3 Windowed OFDM for Mixed-Numerology 5G and Beyond Systems where is the L transmit matrix and L(i) = is the total signal length. In addition, g(l),v, for 0 <u<Zi and is an L vector with its n-th element being The L-dimensional Toeplitz matrix of the channel h is constructed with [h(0), E as the first row and 01x(L()-1). as the first column. Since the last samples of the trans- mitted symbols are ignored, the equivalent channel matrix can be con- structed by taking the first rows of the aforementioned Toeplitz matrix. The demod- ulated signal can be written as (3.13) herei = an Z(i) demodulated signal and W(i) is the corresponding Gaussian noise vector. The receiver window NON E is defined by (3.14) where 0<k<Z¹ and m<Ns is the demodulation vector with its n-th element equal to . According to (3.9), (3.13) can be simplified (3.15) where Hall = and Hi = dimensional CFR matrix. The noise variance matrix is = and = 3.2.2 System Model for Mixed Numerologies When it comes to mixed numerologies transmission, the system bandwidth is divided into several subbands where each of them is assigned with a different set of numerologies. A commonly accepted relationship between the SCS and/or CP length for different numerolo- gies [21-23] can be defined as f1i/2 = = (3.16) where Af(i) is the SCS for the subband i, and K is an integer. Figure 3.3 depicts a dual numerologies system with a GB, FGB, between them. Without loss of generality, we assume that the two subbands have the same sampling rate. Let us consider that the W-OFDM system is in a generalized synchronized scenario [4], where an integral least common mul- tiplier (LCM) symbol consists of N°1 samples. In addition, the subframe structures of dif- ferent numerologies in the system are aligned; thus, we have (3.17) 3.2 W-OFDM System Model Subband 1 with numerology 1 Subband 2 with numerology 2 Z - 1 Af(1) Guard band FGB Af(2) Figure 3.3 An example of mixed numerologies transmission with two subbands. Using (