## 4 Semi-Blind Receiver
Under this model, the channel matrix \(\mathbf{H}^{(m)}\) is expressed as:

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]1where \(\alpha_{l}^{(m)}\) denotes the complex gain of \(l\)-th path, \(\theta_{l}^{(m)}\), and \(\phi_{l}^{(m)}\) are \(l\)-th's azimuth angles of arrival and departure (AoAs/AoDs) of the mobile station and base station, respectively. \(\Lambda_{MS}\left(\theta_{l}^{(m)}\right)\) and \(\Lambda_{BS}\left(\phi_{l}^{(m)}\right)\) are receive and transmit antenna array at a specific AoA and AoD, respectively. Finally, \(\mathbf{a}_{BS}\left(\phi_{l}^{(m)}\right)\) and \(\mathbf{a}_{MS}\left(\theta_{l}^{(m)}\right)\) are the steering vectors at the base station and mobile station, respectively. If uniform linear arrays are considered, the steering vectors \(\mathbf{a}_{BS}\left(\phi_{l}^{(m)}\right)\) and \(\mathbf{a}_{MS}\left(\theta_{l}^{(m)}\right)\) are respectively given by:

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]2 \[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]3

where \(\lambda\) denotes the signal wavelength, and \(d\) is the distance between two neighboring antenna elements. Similar to the analysis of Section 3.1, the received signal tensor \(\underline{\mathbf{Y}}^{(m)}\) of noiseless signal also satisfies the Tucker-2 model, and the proposed algorithm in Section 4 remains suitable for joint symbol and channel estimation at each mobile station. However, two points are important to note here. First, identifiability conditions of Theorem 1 are no longer applicable because of the low-rank property of \(\mathbf{H}^{(m)}\). However, we can deduce new identifiability conditions based on Equations (21) and (22), i.e., necessary and sufficient conditions for identifiability of \(\mathbf{S}\) and \(\mathbf{H}^{(m)}\) require that \(\left(\mathbf{I}_{P}\otimes\mathbf{H}^{(m)}\right)\mathbf{F}_{1}\) and \(\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2}\) have full column rank. For convenience of analysis, we assume that the antenna-to-slot allocation matrix is all-ones matrix. Then, we have the following theorem. **Theorem 2**. : _Assuming that the path gains of the low-rank channel \(\mathbf{H}^{(m)}\) are Rayleigh distributed, and \(N\) and \(R\) are large enough. Then sufficient conditions for identifiability of \(\mathbf{H}^{(m)}\) and \(\mathbf{S}\) are:_

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]4

Proof of Theorem 2.: The channel model \(\mathbf{H}^{(m)}\) is expressed as Equation (48). The rank of \(\mathbf{H}^{(m)}\) is \(L_{m}\), and the path gains of the \(\mathbf{H}^{(m)}\) are Rayleigh distributed. \(\mathbf{F}_{1}\) is a full rank matrix, which contains different generators. Consequently, \(\min\left(PM_{D},\ PL_{m},\ PM_{S}\right)\geqslant R\) ensures that \(\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1}\) have full column rank. Since \(\mathbf{H}^{(m)}\) is a low-rank, i.e., \(L_{m}<\min\left(M_{S},\ M_{D}\right)\), \(P\geqslant\frac{R}{L_{m}}\) can ensure \(\left(\mathbf{I}_{P}\otimes\mathbf{H}\right)\mathbf{F}_{1}\) has the full column rank. Since \(N\) and \(R\) are large enough, and \(\mathbf{S}\) has the random nature, the rank of \(\mathbf{S}\) is equal to \(N\) or \(R\). Moreover, \(\mathbf{F}_{2}\) is also a full rank matrix because of its special structure. We deduce that \(\left(\mathbf{I}_{P}\otimes\mathbf{S}\right)\mathbf{F}_{2}\) is full column rank if \(\min\left(PN,\ PR\right)\geqslant M_{S}\), i.e., \(P\geqslant\max\left(\frac{M_{S}}{N},\ \frac{M_{S}}{R}\right)\). Therefore, condition (51) can ensure identifiability of \(\mathbf{H}^{(m)}\) and \(\mathbf{S}\). This ends the proof of Theorem 2.