## 2 System Model
### The Multi-Domain Channel  
(1) Time domain  
It can be seen from the above analysis that the OTFS system is based on the OFDM system with the addition of a pre-module (ISFFT transform) and a post-module (SFFT transform). Similarly, to acquire the time-domain received signal expression, we can begin with the time domain channel. It is known that the received signal in the time domain is obtained by convolution of the transmitted signal and the channel, and the relationship is as follows:  
\[r(n)=\sum_{l=0}^{L-1}g(n,l)s(n-l)+z(n),\ n\in[0,MN-1] \tag{7}\]  
where \(r(n)\) is the time-domain receiving symbol, and \(L\) represents the maximum delay of the channel, which is less than the CP length \(N_{\text{CP}}\), that is \(L<N_{\text{CP}}\), \(g(n,l)\) is the impulse response of the \(n\)-th sampling point in the \(l\)-th tap. It is noted that in order to avoid interference between OTFS transmitted data blocks, we add cyclic prefix (CP) at the vector header when OTFS symbols are converted to serial vectors. Therefore, the matrix relation of the time domain receiving vector can be expressed as follows:  
\[\mathbf{r=Gs+z} \tag{8}\]  
where \(\mathbf{s}\in\mathbb{C}^{MN\times 1}\) and \(\mathbf{r}\in\mathbb{C}^{MN\times 1}\) represent the received signal and the transmitted signal in the time domain, respectively; \(\mathbf{z}\in\mathbb{C}^{MN\times 1}\) represents the additive complex Gaussian white noise vector representing the mean of \(\mathbf{0}\) and the covariance matrix of \(\mathbf{\sigma^{2}I}\), and \(\mathbf{G}\in\mathbb{C}^{MN\times MN}\) represents the cyclic matrix composed of CIR, which is composed as follows:  
\[\mathbf{G=} \begin{bmatrix}g_{0}^{0}&0&\cdots&0&g_{L-1}^{0}&g_{L-2}^{0}&\cdots &g_{1}^{0}\\ g_{1}^{1}&g_{0}^{1}&0&\ddots&0&g_{L-1}^{1}&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&g_{L-2}^{k-2}\\ g_{L-2}^{k-1}&\ddots&g_{1}^{k-1}&g_{0}^{k-1}&0&\ddots&0&g_{L-1}^{k-1}\\ g_{L-1}^{k}&\ddots&\ddots&g_{1}^{k}&g_{0}^{k}&0&\ddots&0\\ 0&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&g_{L-1}^{MN-1}&g_{L-2}^{MN-1}&\cdots&g_{1}^{MN-1}&g_{0}^{MN-1}\\ \end{bmatrix}  \tag{9}\]  
For ease of representation, the element of \(\mathbf{G}\) in Equation (9) is denoted as \(g_{l}^{n}\), where \(\mathbf{g}\in\mathbb{C}^{MN\times L}\) is the time domain CIR. (2) Time-delay domain  
Let \(\widetilde{\mathbf{x}}\) and \(\widetilde{\mathbf{y}}\) denote the transmitted and received vectors in the time-delay domain, respectively, and the relationships between them and the time domain vectors are respectively given as follows:  
\[\widetilde{\mathbf{x}}=\mathbf{P^{\prime}s} \tag{10}\]\[\widetilde{\mathbf{y}}=\mathbf{P}^{\prime}\mathbf{r} \tag{11}\]  
Note that \(\mathbf{P}\in\mathbb{C}^{MN\times MN}\) is the row-column interleaving permutation matrix [22] and is defined as follows:  
\[\mathbf{P}= \begin{bmatrix}\mathbf{E}_{1,1}&\mathbf{E}_{2,1}&\cdots&\mathbf{E}_ {M,1}\\ \mathbf{E}_{1,2}&\mathbf{E}_{2,2}&\cdots&\mathbf{E}_{M,2}\\ \vdots&\ddots&\ddots&\vdots\\ \mathbf{E}_{1,N}&\mathbf{E}_{2,N}&\cdots&\mathbf{E}_{M,N}\end{bmatrix}  \tag{12}\]  
where \(\mathbf{E}_{i,j}\in\mathbb{C}^{M\times N}\) denotes  
\[\mathbf{E}_{i,j}(i^{\prime},i^{\prime})=\left\{ \begin{array}{ll}1,\text{ if }i^{\prime}=i\text{ and }j^{\prime}=j\\ 0,\text{ otherwise}\end{array} \right. \tag{13}\]  
Further, by submitting Equations (8) and (10) into Equation (11), the input-output relation of the time-delay domain can be expressed as follows:  
\[ \begin{array}{ll}\widetilde{\mathbf{y}}&=\mathbf{P}^{\prime}\mathbf{r}\\ &=\mathbf{P}^{\prime}\mathbf{G}\mathbf{s}+\mathbf{z}^{\prime}\\ &=\mathbf{P}^{\prime}\mathbf{G}\mathbf{\widetilde{x}}+\mathbf{z}^{\prime}\\ &=\widetilde{\mathbf{H}}\mathbf{\widetilde{x}}+\mathbf{z}^{\prime}\end{array}  \tag{14}\]  
where \(\widetilde{\mathbf{H}}=\mathbf{P}^{\prime}\mathbf{G}\mathbf{P}\in\mathbb{C}^ {MN\times MN}\) is the time-delay domain channel matrix.