agonal matrix generated by repeating A matrix N times. [a] denotes the largest integer smaller than or equal to a. g(t) denotes the conjugate of the function g(t). 1N is an Nx1 vector with all its elements being 1. A(i,j) refers to the i-th row and j-th column of the matrix A. The operator * denotes the linear convolution of two vectors, and x represents Kronecker product. W-OFDM System Model In a W-OFDM system, time-domain windowing operation is performed at both the transmitter and the receiver. Instead of the rectangular window used in the traditional OFDM system, W-OFDM implements windows with soft edges on both sides, which results in much sharper side-lope decay in its frequency domain. Figure 3.2 illustrates the time-domain window processing in a W-OFDM system. It can be seen that although the edges of the window further expand each symbol, the overhead of the signal is not increased because of the transmit roll-off portion (or edge transition region). Unless otherwise stated, the raised cosine (RC) window suggested by Bala et al. [18] will be considered for both transmitter and receiver in this section. 3 Windowed OFDM for Mixed-Numerology 5G and Beyond Systems Tx Roll-off Tx Roll-off portion N. (i) portion Tx roff Tx window Tx_roff Rx window _Zero N (i) Rx_roff NRx_roff Figure 3.2 Time-domain window processing in transmitter and receiver. 3.2.1 Single Numerology System Model Let us establish a single numerology W-OFDM system first. Assume that the user n is allo- cated with Z(i) continuous subcarriers in subband i. Then we can write the discrete-time baseband W-OFDM signal as (3.1) where xu.v denotes the v-th independent identically distributed (i.i.d) complex modulation W-OFDM symbol on the u-th subcarrier. In addition, it is assumed that xu.v has zero mean and unit power. Similarly, gu,(n) is defined as the synthesis function for the u-th subcarrier of the v-th symbol, and can be written as (3.2) where CP is the transmitted symbol duration by taking the CP length N(i) CP into account, and AF(i) denotes