## 3 System Model
### Identifiability Conditions  
The identifiability for the constructed Tucker-2 model is an assignable problem for recovering the parameters to be estimated. In this paper, it is directly linked to the estimation of the signal matrix \(\mathbf{S}\) and channel matrix \(\mathbf{H}\) from the received signal tensor \(\underline{\mathbf{Y}}\). Conditions of parameter identifiability is given in the following theorem. **Theorem 1**. : _(Sufficient Conditions): Assuming that \(\mathbf{H}\) has independent and identically distributed (i.i.d.) entries, and \(\mathbf{S}\) has a full column rank. \(P_{1}\) denotes the number of nonzero elements in \(\mathbf{Q}\). Then sufficient conditions for identifiability of signal matrix \(\mathbf{S}\) and channel matrix \(\mathbf{H}\) are:_  
\[P_{1}\geqslant R\quad\text{and}\quad\min\left(M_{D},\ R\right)\geqslant M_{S}. \tag{20}\]  
Proof of Theorem 1.: From Equation (12) and Equation (13), necessary and sufficient conditions for identifiability of \(\mathbf{S}\) and \(\mathbf{H}\) requires that \(\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1}\) and \(\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2}\) have a full column rank, i.e. \[Rank\left(\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1}\right)=R \tag{21}\]  
\[Rank\left(\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2}\right)= M_{S}. \tag{22}\]  
Under the assumptions in Theorem 1 that \(\mathbf{H}\) has i.i.d. entries, \(M_{D}\geqslant M_{S}\) can ensure \(\mathbf{H}\) has a full column rank. Since \(\mathbf{I}_{P}\) and \(\mathbf{H}\) have a full column-rank, then \(\mathbf{I}_{P}\otimes\mathbf{H}\) has a full column rank, i.e., \(Rank\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)=PM_{S}\). Therefore, Equation (21) is satisfied if \(\mathbf{F}_{1}\) has a full column rank. We rewrite \(\mathbf{F}_{1}\) from Equation (15) as:  
\[\mathbf{F}_{1}=diag\left(vec\left(\mathbf{Q}^{T}\right)\right)\mathbf{B}_{1} \tag{23}\]  
where \(\mathbf{B}_{1}=\left[\mathbf{B}_{..1}^{T},\ldots,\mathbf{B}_{..P}^{T}\right]^ {T}\). Due to \(\mathbf{B}_{1}\) being a specially constructed matrix with different generators, any two rows (or two columns) of \(\mathbf{B}_{1}\) are linearly independent. If the number of non-zero rows of \(\mathbf{F}_{1}\) is greater than or equal to the number of columns of \(\mathbf{F}_{1}\) (i.e. \(P_{1}\geqslant R\)), then \(\mathbf{F}_{1}\) is full column rank. Thus, \(M_{D}\geqslant M_{S}\) and \(P_{1}\geqslant R\) can ensure that condition (21) is satisfied. Since \(\mathbf{S}\) has a full column rank, we can deduce that \(Rank\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)=PR\). Thus, condition (22) is satisfied if \(\mathbf{F}_{2}\) has a full column rank. We rewrite \(\mathbf{F}_{2}\) from Equation (16) as:  
\[\mathbf{F}_{2}=\mathbf{B}_{2}^{T}\mathbf{Q}_{P} \tag{24}\]  
where \(\mathbf{B}_{2}=\mathrm{blk}\left[\mathbf{B}_{..1},\ldots,\mathbf{B}_{..P}\right]\) and \(\mathbf{Q}_{P}=\left[D_{1}\left(\mathbf{Q}\right),\ldots,D_{P}\left(\mathbf{Q }\right)\right]^{T}\). Recall that \(\mathbf{Q}\) does not have an all-zero column, which means that \(\mathbf{Q}_{P}\) has a full column rank. We have that \(\mathbf{F}_{2}\) is full column rank if \(\mathbf{B}_{2}\) is full row rank. Since \(\mathbf{B}_{2}\) has the block diagonal structure and \(\mathbf{B}_{..p}\) has different generators, \(R\geqslant M_{S}\) ensures that \(\mathbf{B}_{2}\) has full row rank. Therefore, \(R\geqslant M_{S}\) can ensure that condition (22) is satisfied. This ends the proof. **Remark 1**. : _The conditions in Theorem 1 is sufficient but not necessary for parameter identifiability. Sufficient condition (21) and condition (22) also concern the ALS algorithm. In fact, identifiability of signal and channel parameters is possible in our simulation results when \(M_{S}>R\). Necessary conditions for parameter identifiability is based on the dimensions of \(\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1}\) and \(\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2}\).