r does not lead to channel shortening effect. 6.5 Simplified Modem Structure This degrades channel sparsity in the delay dimension and hence can be harmful. How- ever, Doppler spread can lead to a great amount of leakage all along the Doppler dimension. Therefore, windowing/filtering with a well-localized filter in the Doppler dimension is very effective in reducing the leakage and making the channel sparse. On the basis of the above discussion, utilization of a rectangular window along the fre- quency domain, while using a window with smooth corners along the consecutive OFDM symbols in time leads to a sparse channel response. Hence, the 2D delay-Doppler channel impulse response keeps its minimum delay spread, i.e. L, while becoming quite sparse in the Doppler dimension. Furthermore, from (6.15) and (6.16), one may realize that the trans- mit and receive windows applied to N consecutive OFDM time symbols can be combined and utilized only at the receiver side. Accordingly, the modem structure that was discussed in Section 6.3 is reduced to an OTFS modulator without windowing, i.e. Y = Y in (6.7), and an OTFS demodulator with windowing/filtering only along the Doppler dimension. As mentioned earlier, the SFFT-¹ operation in (6.11) can be represented as Y = FMXFH N (6.21) Inserting Y from (6.21) into (6.7), one may realize that the IDFT operation at OFDM modu- lator and the M-point DFT operation in SFFT-¹ block cancel out. Consequently, (6.7) boils down to S = AcpXFH. (6.22) Thus, OTFS modulator can be implemented through a similar structure to OFDM including M number of N-point IDFT operations and a CP addition block. The IDFT operations are performed on the rows of the data block X and the CP addition block appends a CP to the beginning of every M samples. This structure is shown in Figure 6.2 where dk is the high N-point N-point Addition N-point Figure 6.2 Simplified OFDM-based OTFS modulator structure. 6 Orthogonal Time-Frequency Space Modulation: Principles and Implementation rate QAM data stream before gettin