## 4 Semi-Blind Receiver
Remove the scaling ambiguity: \(\mathbf{\hat{S}}^{(final)}=\frac{\mathbf{S}^{(\infty)}}{\delta_{1,1}



where \(\mathbf{g_{us}}\in\mathbb{C}^{NR\times 1}\) and \(\mathbf{g_{u_{H}}}\in\mathbb{C}^{M_{D}M_{S}\times 1}\) are respectively given by:

\[\mathbf{g_{us}}=\mathbf{J_{us}^{H}J_{us}\mathbf{u_{S}}-J_{us}^{H}\mathbf{y}} \tag{43}\]

\[\mathbf{g_{u_{H}}}=\mathbf{J_{u_{H}}^{H}J_{u_{H}}u_{H}-J_{u_{H}}^{H}\mathbf{y}}. \tag{44}\]

In Algorithm 1, the estimated matrix \(\mathbf{H}^{(\infty)}\) is projected onto a low rank estimated matrix \(\mathbf{H}_{new}^{(\infty)}\) by the SVP based algorithm when \(L<\min\left(M_{S},M_{D}\right)\). Here \(\mathbf{H}_{new}^{(\infty)}\) is calculated as \(\mathbf{H}_{new}^{(\infty)}=SVP\left(\mathbf{H}^{(\infty)}\right)=\sum\limits _{l=1}^{L}\beta_{l}\mathbf{U}_{l}^{(C)}\left(\mathbf{V}_{l}^{(C)}\right)^{H}\), where \(\beta_{l}\) denotes the \(l\)-th largest singular value of \(\mathbf{H}^{(\infty)}\), \(\mathbf{U}_{l}^{(C)}\) and \(\mathbf{V}_{l}^{(C)}\) are the corresponding left and right singular vectors. The overall complexity of the optimized LM algorithm mainly depends on the per-iteration complexity and the numbers of iterations. The per-iteration complexity of this algorithm can be estimated as \(\mathcal{O}\left(\left(NR+M_{D}M_{S}\right)^{3}\right)\). Since the antenna-to-slot allocation matrix and the coding tensor are fixed and known at the receiver, the convergence of the optimized LM algorithm is usually achieved in only a few iterations. The average number of iterations for the optimized LM algorithm will be further analyzed in Section 6. ## 5 Extension to Multi-User Massive Mimo Systems

In the following section, we show that the developed algorithm can be applied to multi-user massive MIMO systems with hybrid precoding architecture for joint symbol and channel estimation. We consider a fully-connected hybrid precoding architecture, which is the typical model of massive MIMO systems. The base station communicates with \(M\) users simultaneously, and each mobile station is equipped with \(M_{D}\) antennas. The base station is equipped with \(M_{S}\) antennas and \(M_{RF}\) independent radio frequency chains to transmit \(R\) streams for \(M_{D}\) receive antennas in each mobile station. In the considered downlink system, each symbol \(s_{n,r}\) is coded by a three-dimensional baseband code \(b_{m_{RF},r,p}^{(M)}\) followed by a radio frequency code \(e_{m_{S},m_{RF}}\) in the base station. At the \(m\)-th (\(m=1,\ldots,M\)) mobile station, the discrete-time baseband signal at the \(m_{D}\)-th receive antenna is written as:

\[y_{m_{D},n,p}^{(m)}=\sum\limits_{m_{S}=1}^{M_{S}}\sum\limits_{m_{RF}=1}^{M_{ RF}}\sum\limits_{r=1}^{R}h_{m_{D},m_{S}}^{(m)}q_{p,m_{S}}e_{m_{S},m_{RF}}b_{m_{ RF},r,p}s_{n,r}+v_{m_{D},n,p}^{(m)} \tag{45}\]

where \(e_{m_{S},m_{RF}}\) and \(h_{m_{D},m_{S}}^{(m)}\) are \((m_{S},m_{RF})\)-th and \((m_{D},m_{S})\)-th elements of the radio frequency precoder matrix \(\mathbf{E}\in\mathbb{C}^{M_{S}\times M_{RF}}\) and the massive MIMO channel matrix \(\mathbf{H}^{(m)}\in\mathbb{C}^{M_{D}\times M_{S}}\), respectively. \(y_{m_{D},n,p}^{(m)}\) is the typical element of the received signal tensor \(\underline{\mathbf{Y}}^{(m)}\in\mathbb{C}^{M_{D}\times N\times P}\). Then Equation (45) can be rewritten as:

\[y_{m_{D},n,p}^{(m)}=\sum\limits_{m_{S}=1}^{M_{S}}\sum\limits_{r=1}^{R}h_{m_{ D},m_{S}}^{(m)}c_{m_{S},r,p}s_{n,r}+v_{m_{D},n,p}^{(m)} \tag{46}\]

where,

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]0

Following [33; 34], we also adopt a geometric channel model with \(L_{m}\) scatterers between the base station and the \(m\)-th mobile station, \(L_{m}=1,\ldots,L_{M}\).