## 3 System Model
### Constructed Tucker-2 Model  
Let us define \(c_{m_{S},r,p}=q_{p,m_{S}}b_{m_{S},r,p}\), where \(c_{m_{S},r,p}\) is the typical element of the compound tensor \(\underline{\mathbf{C}}\in\mathbb{C}^{M_{S}\times R\times P}\). So Equation (7) can be written as:  
\[y_{m_{D},n,p}=\sum_{m_{S}=1}^{M_{S}}\sum_{r=1}^{R}h_{m_{D},m_{S}}c_{m_{S},r,p}s _{n,r}+v_{m_{D},n,p}. \tag{8}\]  
By comparing Equation (4) with Equation (8), the received signal tensor \(\underline{\mathbf{Y}}\in\mathbb{C}^{M_{D}\times N\times P}\) of noiseless signals satisfies a Tucker-2 model, with the following correspondences:  
\[\left(\underline{\mathbf{G}},\ \mathbf{A}^{(1)},\ \mathbf{A}^{(2)}\right) \Leftrightarrow\left(\underline{\mathbf{C}},\ \mathbf{H},\ \mathbf{S}\right) \tag{9}\]  
\[\left(I_{1},I_{2},R_{1},R_{2},I_{3}\right)\Leftrightarrow\left(M_{D},N,M_{S},R, P\right). \tag{10}\]  
Using the mode-\(n\) product representation, the model (8) can be written as:  
\[\underline{\mathbf{Y}}=\underline{\mathbf{C}}\times_{1}\mathbf{H}\times_{2} \mathbf{S}+\underline{\mathbf{V}} \tag{11}\]  
where \(\mathbf{S}\) and \(\mathbf{H}\) represent the two loading matrices, and \(\underline{\mathbf{C}}\) is the core tensor. Let us define \(\mathbf{Y}_{\cdot\cdot p}\in\mathbb{C}^{M_{D}\times N}\), \(\mathbf{B}_{\cdot\cdot p}\in\mathbb{C}^{M_{S}\times R}\), \(\mathbf{C}_{\cdot\cdot p}\in\mathbb{C}^{M_{S}\times R}\) and \(\mathbf{V}_{\cdot\cdot p}\in\mathbb{C}^{M_{D}\times N}\) as the \(p\)-th matrix slice of \(\underline{\mathbf{Y}}\in\mathbb{C}^{M_{D}\times N\times P}\), \(\underline{\mathbf{B}}\in\mathbb{C}^{M_{S}\times R\times P}\), \(\underline{\mathbf{C}}\in\mathbb{C}^{M_{S}\times R\times P}\), and \(\underline{\mathbf{V}}\in\mathbb{C}^{M_{D}\times N\times P}\), respectively. We have \(\mathbf{C}_{\cdot\cdot p}=D_{p}\left(\mathbf{Q}\right)\mathbf{B}_{\cdot\cdot p}\). By defining \(\mathbf{Y}_{1}=\left[\mathbf{Y}_{\cdot\cdot\cdot}^{T},\ldots,\mathbf{Y}_{ \cdot\cdot p}^{T}\right]^{T}\in\mathbb{C}^{PM_{D}\times N}\), \(\mathbf{Y}_{2}=\left[\mathbf{Y}_{\cdot\cdot\cdot},\ldots,\mathbf{Y}_{\cdot \cdot p}\right]^{T}\in\mathbb{C}^{PN\times M_{D}}\), \(\mathbf{Y}_{3}=\left[\mathit{vec}\left(\mathbf{Y}_{\cdot\cdot\cdot 1}\right),\ldots, \mathit{vec}\left(\mathbf{Y}_{\cdot\cdot p}\right)\right]\in\mathbb{C}^{M_{D} \times N\times P}\), and \(\mathbf{Y}_{4}=\left[\mathit{vec}\left(\mathbf{Y}_{\cdot\cdot\cdot 1}^{T}\right),\ldots, \mathit{vec}\left(\mathbf{Y}_{\cdot\cdot p}^{T}\right)\right]\in\mathbb{C}^{ NM_{D}\times P}\), we can obtain four compact forms of the Tucker-2 model (11):  
\[\mathbf{Y}_{1} =\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1} \mathbf{S}^{T}+\mathbf{V}_{1} \tag{12}\] \[\mathbf{Y}_{2} =\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2} \mathbf{H}^{T}+\mathbf{V}_{2}\] (13) \[\mathbf{Y}_{3} =\left(\mathbf{S}\otimes\mathbf{H}\right)\mathbf{F}_{3}+\mathbf{V}_ {3}\] (14) \[\mathbf{Y}_{4} =\left(\mathbf{H}\otimes\mathbf{S}\right)\mathbf{F}_{4}+\mathbf{V}_ {4} \tag{15}\]with,  
\[ \begin{split}\mathbf{F}_{1}&=\left[\mathbf{B}_{\cdot 1}^{T}D_{1} \left(\mathbf{Q}\right),\ldots,\mathbf{B}_{\cdot p}^{T}D_{P}\left(\mathbf{Q} \right)\right]^{T}\\ \mathbf{F}_{2}&=\left[D_{1}\left(\mathbf{Q}\right) \mathbf{B}_{\cdot 1},\ldots,D_{P}\left(\mathbf{Q}\right)\mathbf{B}_{\cdot p} \right]^{T}\\ \mathbf{F}_{3}&=\ \left[vec\left(D_{1}\left(\mathbf{Q} \right)\mathbf{B}_{\cdot 1}\right),\ldots,vec\left(D_{P}\left(\mathbf{Q} \right)\mathbf{B}_{\cdot p}\right)\right]\\ \mathbf{F}_{4}&=\ \left[vec\left(\mathbf{B}_{\cdot 1}^{T}D_{1} \left(\mathbf{Q}\right)\right),\ldots,vec\left(\mathbf{B}_{\cdot p}^{T}D_{P} \left(\mathbf{Q}\right)\right)\right]\end{split}  \tag{16}\]  
and,  
\[