expand the nth column of the matrix X. This leads to j2nni (6.14) Substituting (6.12) into (6.14), we get j2nni N-1 N-1 (6.15) where the M x 1 vector v(n) contains the noise samples at the OTFS demodulator output, WRx.c = and (6.16) Using (6.15), defining an MN X MN block circulant matrix (6.17) the MN x MN block diagonal windowing matrices WRx,C = WRx,c IN, and WTx, WTx,c X IN, the data stream at the output of the OTFS receiver, dx for ,MN - 1, can be obtained in a compact form as (6.18) where =vcX,d = = = [do,...,AMN_1]T = vec{X}, and = V(N - 1)]. From (6.18), one may realize that if the submatrices Hn in HBC are circulant, i.e. the channel is time-invariant within each OFDM symbol interval, multiplication of HBC,w = WRx.CHBC WTx.c to d realizes the 2D circular convolution of the windowed delay-Doppler channel impulse response HDD,w with the data matrix X. This is due to the fact that all the matrices WRx.c, HBC, and WTx,c are block circulant with circulant sub- matrices and their multiplication result in a matrix preserving their circulant properties. 6 Orthogonal Time-Frequency Space Modulation: Principles and Implementation Therefore, X can be written as (6.19) where the columns of the M X N matrix HDD,winclude the first columns of the matrices Hn that are windowed by the window functions WTx,c[m] and WRx.clm in the delay domain. In particular, the elements kl of HDD,w for k=0,...,M - 1 and 0, N - 1 are equal to WRx.CHIW1x.clko- When the channel is time-invariant within each OFDM symbol inter- val, we can reorder the matrices WRx,c' HBC, and WTx,c in (6.18) as This implies that given the transmit window in the delay domain is channel independent, it can be combined with the receive filter to simplify the transmitter. Based on Eq. (6.18), one may realize that channel effect in the delay-Doppler domain can be equalized using linear equalizers such as zero forcing (ZF) or minimum mean square error (MMSE). Since the ZF equalizer may suffer from the noise enhancement problem, MMSE equalization seems to be a more 