(.)H and (.)* indicate transpose, conjugate trans- pose, and conjugate operations, respectively. f(n],g[n] > denotes the inner product of functions f[n] and g[n]. Finally, O, X, , and (())N represent elementwise multiplication, Kronecker product, 2D circular convolution, and modulo-N operations, respectively. 6.2 OTFS Principles OTFS modulation has two stages. First, it maps a set of data symbols with quadrature ampli- tude modulation (QAM) from delay-Doppler domain to time-frequency domain through a set of two-dimensional basis functions. The first stage also includes a time-frequency win- dowing operation. This can be implemented through an SFFT-¹ operation together with windowing procedure to be applied at the SFFT-¹ output. In the second stage, the resulting time-frequency samples are fed into a modulator to form the time-domain transmit signal using a set of orthogonal time-frequency basis functions. After the signal is passed through 6 Orthogonal Time-Frequency Space Modulation: Principles and Implementation the channel, the received signal needs to be mapped back to the delay-Doppler domain before detection. To this end, OTFS demodulator performs the reverse operations to those at the modulator. This procedure is shown in Figure 6.1. In the following, we cast the above explanation into mathematical formulation, paving the way for our further derivations in this chapter. We consider an OTFS system transmitting a block of M x N delay-Doppler QAM data symbols Xk,1, where k=0,...,M - 1 and 11=0,...,N - 1. The data symbols are from a zero mean independent and identically distributed (i.i.d) random process with the variance of unity. In OTFS modulation, the data symbols, Xk.1, are first modulated by a set of 2D basis functions, bk.1[m,n in the time-frequency domain as M-1 N-1 (6.1) where the time-frequency samples Ym,n for m = 0,...,M - 1,n = 0,...,N - 1 appear at the output of the SFFT-¹ block. Then, the time-frequency samples Ym,n are windowed through a 2D transmit window WTx[m,n] and are fed into the time-fre