quency modulator to form the OTFS transmit signal as M-1 N-1 (6.2) where are the windowed time-frequency samples and gm.nli] are the time-frequency basis functions of the transmitter. The received signal samples after transmission over a linear time-varying (LTV) channel can be obtained as (6.3) where h[i, l] is the channel impulse response with length L at time instant i, and v[i] ~ CN(0, 22) is the circularly symmetric complex additive white Gaussian noise (AWGN). To obtain data symbols that are affected by the channel in the delay-Doppler domain, the received signal, r[i], needs to be first projected onto the receiver's basis functions, fm.n[i], corresponding to time-frequency points m, k. (6.4) For inter-symbol interference-free communication, transmit and receive basis functions are designed to satisfy the perfect reconstruction criterion, i.e. (6.5) It is worth noting that if the underlying time-frequency modulation scheme is orthogonal, e.g. OFDM, fm.n ==mm However, for a non-orthogonal waveform, such as generalized frequency division multiplexing (GFDM) [8], the transmitter and receiver basis functions may not be the same. Recalling Figure 6.1, at the second stage of the OTFS demodulation, we form an M x N matrix, Z = [z(0), , z(N - 1)] where = its elements using 6.3 OFDM-Based OTFS a receive window WRx[m,n], and then perform a SFFT operation. This process can be rep- resented as M-1 N-1 (6.6) where k = 0, M - 1 and l=0, N - 1. OFDM-Based OTFS In this section, we consider OFDM for transmission of the generated time-frequency signals after SFFT-1 and windowing operations. Therefore, the signal samples ym,n are fed into the OFDM transmitter to form the OTFS transmit signal as S = ACPF, (6.7) where = 1)] is an M x N matrix containing the time-frequency win- dowed samples ym,n WTx[m,nlym,n on its mnth elements. The nth column of Y, i.e. y(n) = includes the frequency domain samples to be transmitted within the nth OFDM symbol. FM is the normalized M-point discrete Fourier transform (DFT) matrix with the elemen