ncy samples Ym,n at the output of the SFFT-¹ block as its elements. In this chapter, we consider the transmit and receive windows to be separable functions of time and frequency indices. This makes our derivations simple while leading to an in-depth understanding of the windowing effect on the channel, in either delay or Doppler dimension. = WTx,c[m] and WTx,r[n] perform windowing operation on the columns and rows of Y, respectively. It is worth noting that the time-frequency windowing operation is equivalent to filtering in the delay-Doppler domain. Thus, the transmit windowing operation is applied through the multiplication of two diagonal matrices to Y from left and right, respectively, i.e. WTx,c = g{[WTx,c[0], WTx,c[M - 1]]} and W,r = diag{[WTx.r[0], WTx,r[N 1]]}. This pro- cedure can be represented as Y = W1x,c YWTx,r = (6.9) From (6.9) the nth column of the matrix Y can be obtained as (6.10) where WTx,r(n) is the column of WTx,r The SFFT-¹ operation that was shown in (6.1) can be represented as Y = FMXFH, (6.11) where X is an M X N matrix containing the transmit QAM data symbols Xk.1 on its klth ele- ments. Substituting Eqs. (6.10) and (6.11) in (6.8), the received OFDM symbol n at the OFDM demodulator output can be expanded as z(n) = = v(n) (6.12) where WTx.c = FWTx.com and fn(n) is the nth column of the N-point DFT matrix. After packing the vectors z(n) for n=0, ...,N - 1 on the columns of the matrix = [z(0), z(N 1)] and performing windowing and SFFT operations based on Eq. (6.6), the received signal samples at the output of OTFS receiver can be packed in an M x N matrix X as (6.13) with the elements XK.1. Similar to the transmitter, we consider a separable win- dow at the receiver, i.e. WRx[m, n] = WRx.clm]WRx.rIn] where WRx.c = diag{[WRx, [0], ... ,WRx.cIM - 1]]} and WRx,r = diag{[WRx.r[0],..., WRx.r[N - 1]]} window the columns and rows of Z, respectively. To derive a clear-cut representation of the channel 6.4 Channel Impact effect on the transmit data symbols, Xk,1, at the output of the receiver, we 