ts [FMlpq for p,q = 0,...,M , - 1. ACP = is the CP addition matrix where the MCPXM matrix GCP includes the last MCP rows of the identity matrix I'M and MCP is the CP length. Finally, the (M + MCP) X N matrix S includes the OFDM time domain transmit signals on its columns. After parallel to serial conversion S, the OTFS transmit signal at the baseband can be formed as S = vec {S}. The received signal samples after transmission over an LTV channel with length L can be obtained as was shown in Eq. (6.3). Assuming MCP L, the received OFDM symbols are free of inter-symbol interference. Thus, the received OFDM symbol n at the output of the OFDM demodulator can be written as z(n) = v(n) = FH,FHy(n)+v(n) = Huy(n)+v(n), (6.8) where Hn = RCPHACP, RCP = [0MxMcp"I is the CP removal matrix, Hn = FMH,FH, and Hn is the LTV channel matrix realizing the convolution operation on the samples of the nth OFDM symbol. The vector v(n) contains the noise samples at the OFDM demodulator output and z(n) = As mentioned in Section 6.2, stacking blocks of N OFDM symbols at the output of the demodulator, we form an M x N matrix Z, window its elements, and perform a SFFT oper- ation to obtain the received data symbols at the delay-Doppler domain. This procedure is mathematically shown in Eq. (6.6). It is shown in [4] that the signal samples, XK,1, at the output of OTFS receiver can be obtained as 2D circular convolution of the QAM data sym- bols Xk.1 and the channel impulse response in the delay-Doppler domain. More details and derivations on the baseband channel response in the delay-Doppler domain will be pre- sented in Section 6.4. 6 Orthogonal Time-Frequency Space Modulation: Principles and Implementation 6.4 Channel Impact To study the channel impact, in this section, we expand further the formulation that was presented in Section 6.3. This leads to deeper insights into OTFS while paving the way for the derivation of a simplified modem structure in Section 6.5. As the first step, let us form an M X N matrix Y that includes the time-freque