# Multiple-Mode Orthogonal Time Frequency Space with Index Modulation
## 2 System Model
### The Receiver  
As shown in Figure 4, the received signal \(\mathbf{r}\in\mathbb{C}^{(M+N_{CP})N\times 1}\) over the time-varying channel is first rearranged as a matrix \(\mathbf{R}\in\mathbb{C}^{(M+N_{CP})\times N}\). Then, we removes the CP by multiplying \(\mathbf{R}\) with a CP removal matrix \(\mathbf{R}_{CP}\in\mathbb{C}^{M\times(M+N_{CP})}\) to obtain the MM-OTFS-IM symbols without CPs. Next, by applying the M-point FFT on each MM-OTFS-IM symbol, we obtain the received 2D block \(\mathbf{Y}^{FT}\) in the TF domain as  
\[\mathbf{Y}^{FT}=\mathbf{F}_{M}\mathbf{R}_{CP}\mathbf{R} \tag{6}\]  
Then, the SFFT is applied for \(\mathbf{Y}^{FT}\) to obtain the 2D data block \(\mathbf{Y}^{DD}\) in the DD domain as  
\[\mathbf{Y}^{DD}=\mathbf{F}_{M}^{H}\mathbf{Y}^{FT}\mathbf{F}_{N} \tag{7}\]  
In addition, the input-output relationship of the MM-OTFS-IM system can be written as  
\[\mathbf{y}^{DD}=\mathbf{H}\mathbf{x}^{DD}+\mathbf{z} \tag{8}\]  
where \(\mathbf{y}^{DD}=vec\{\mathbf{Y}^{DD}\}\in\mathbb{C}^{MN\times 1}\), \(\mathbf{x}^{DD}=vec\{\mathbf{X}^{DD}\}\in\mathbb{C}^{MN\times 1}\), \(\mathbf{H}\in\mathbb{C}^{MN\times MN}\) is the channel matrix in the DD domain, and \(\mathbf{z}\in\mathbb{C}^{MN\times 1}\) is the additive white Gaussian noise (AWGN). To recover the sending information from \(\mathbf{y}^{DD}\), the receiver first determines the constellation mode adopted by each grid in the subblock. To recover the index bits and the modulation bits according to the combined information of the constellation modes, a signal detection algorithm based on a distance matrix is proposed. Firstly, the minimum mean square error (MMSE) method is required to equalize the received signal before using the detection algorithm. \[\mathbf{\hat{x}}^{DD}=\mathbf{H}^{H}(\mathbf{H}^{H}\mathbf{H}+\frac{\mathbf{I }_{MN}}{\rho_{sur}})^{-1}\mathbf{y}^{DD} \tag{9}\]  
where \(\mathbf{\hat{x}}^{DD}\) is the MMSE equalized signal in the DD domain, which is the estimate of \(\mathbf{x}^{DD}\), \(\rho_{sur}\) is the average signal to noise ratio (SNR) in the DD domain, and \((\cdot)^{-1}\) denotes the inverse of a matrix. Then, we can obtain the estimate of \(\mathbf{\hat{x}}^{DD}\) as \(\mathbf{\hat{x}}^{DD}=invec\{\mathbf{\hat{x}}^{DD}\}\in\mathbf{x}^{DD}\). Figure 4: Block diagram of the MM-OTFS-IM receiver. \(\mathbb{C}^{M\times N}\), where the operation \(invec(\cdot)\) is the antioperation of \(vec(\cdot)\). Next, \(\hat{\mathbf{X}}^{DD}\) is divided into \(g\) subblocks as \(\hat{\mathbf{X}}^{DD}_{\beta}\), which are expressed as  
\[\hat{\mathbf{X}}^{DD}_{\beta}= \begin{bmatrix}\hat{x}^{\beta}_{0,0}&\cdots&\hat{x }^{\beta}_{0,n-1}\\ \vdots&\ddots&\vdots\\ \hat{x}^{\beta}_{m-1,0}&\cdots&\hat{x}^{\beta}_{m-1,n-1}\end{bmatrix}  \tag{10}\]  
The proposed signal detection algorithm is depicted in Algorithm 1, in which the input \(\hat{\mathbf{X}}^{DD}_{\beta}\) is first converted into a one-dimensional vector \(\mathbf{X}_{\beta}\in\mathbb{C}^{mn\times 1}\). Then, the modulation information matrix \(\mathbf{B}\) and distance matrix \(\mathbf{D}\) are recovered by a double-layer nested loop. The specific operation is as follows: we use constellation mode \(S_{j}\) to demodulate symbol \(X_{\beta,i}\) to obtain \(B_{i,j}\), which is the element at the position of the \(j\)-th row in the \(i\)-th column of matrix \(\mathbf{B}\). Then, \(S_{j}\) is used to demodulate \(B_{i,j}\). To calculate the distance between the modulated symbol and symbol \(X_{\beta,i}\), it obtains the element \(D_{i,j}\), which is the element at the position of the \(j\)-th row in the \(i\)-th column of matrix \(\mathbf{D}\). At the end of the cycle, the \(\mathbf{Hungarian}\{\cdot\}\) algorithm is used to process the distance matrix \(\mathbf{D}\) and obtain the index information \(\hat{\mathbf{S}}_{w^{\beta}}\) of the subblocks. When only one element can appear in each row and column of a matrix, the \(\mathbf{Hungarian}\{\cdot\}\) algorithm can find the position information of each element when the sum of elements is minimum.