verse operation to the one at the transmitter, is performed at the receiver to esti- mate the transmit data symbols. Transformation of data symbols from the delay-Doppler to time-frequency domain at the transmitter and the corresponding inverse transformation at the receiver, clearly leads to full diversity gain across both time and frequency. The whole OTFS modulation and demodulation process based on OFDM for time-frequency transmission is shown in Figure 6.1. In the above setup, the equivalent channel between the transmitted and the received data symbols can be modeled by a two-dimensional time-invariant impulse response in the delay-Doppler domain. Soft detectors/equalizers, like some extensions to those presented in [5], may thus be used for near-optimal recovery of the transmitted information. In order to simplify such detectors, [4] has proposed that proper windows should be applied to the time-frequency signals at both the transmitter and receiver sides to sparsify the OTFS chan- nel. However, very little has been said about how such windows could be selected. The goal of this chapter is to introduce the main principles behind OTFS and provide a discrete time formulation that not only brings deep insights into OTFS but also leads to a simplified modem structure for such systems. Therefore, Section 6.2 explains the basics of OTFS where a generic multicarrier waveform is deployed for transmission of OTFS time-frequency samples. To simplify the derivations, OFDM is considered for time-frequency transmission of OTFS signal. Consequently, Section 6.3 covers SFFTinv Window Window LTVChannel Demod Figure 6.1 OTFS transmitter and receiver structure. 6.2 OTFS Principles OFDM-based OTFS systems. In Section 6.4, mathematical formulation of Section 6.3 is further expanded to provide a concise representation of the effect of each signal processing unit on the input-output relationship of OFDM-based OTFS systems in delay-Doppler domain. This leads to an interesting finding that leads to a substantial amount of saving i