## 3 Transmitter and Receiver Structure
### Receiver Design
\begin{table}
\begin{tabular}{l l} \hline \hline
1: & **Input:**\(\mathbf{x}^{0},\mathbf{r}^{0},\mathbf{z}^{0},\mathbf{u}^{0},\mathbf{v}^{0}, \mathbf{w}^{0},\mathbf{\bar{H}}_{\mathcal{B}}^{u},\mathbf{y}_{g}^{u},\rho_{x}, \rho_{r},\rho_{z},Q\) \\
2: & \(f_{best}=\infty\) \\
3: & \(\mathbf{\Phi}\leftarrow\left(\left(\widetilde{\mathbf{H}}_{\mathcal{B}}^{\sim u }\right)\prescript{H\sim u}{}{}{\mathbf{H}_{\mathcal{B}}^{\sim u}}+(\rho_{x}+ \rho_{r}+\rho_{z})\mathbf{I}_{N_{f}N_{s}}\right)^{-1}\) \\
4: & **for**\(t=0,1,\ldots\)_Q-1_ \\
5: & \(\mathbf{s}^{(t+1)}\leftarrow\mathbf{\Phi}\left(\left(\widetilde{\mathbf{H}}_{ \mathcal{B}}^{u}\right)^{H}\mathbf{y}_{g}^{u}+\rho_{x}\left(\mathbf{x}^{(t)}- \mathbf{u}^{(t)}\right)+\rho_{r}\left(\mathbf{r}^{(t)}-\mathbf{v}^{(t)} \right)\right.+\rho_{z}\left(\mathbf{z}^{(t)}-\mathbf{w}^{(t)}\right)\) \\
6: & \(\left(\mathbf{x}^{i}\right)^{(t+1)}\leftarrow\prod_{\mathbb{S}_{0}}\left( \left(\mathbf{s}^{(t)}\right)^{(t+1)}+\left(\mathbf{u}^{i}\right)^{(t+1)}\right)\) \\
7: & \(\mathbf{r}^{(t+1)}\leftarrow\prod_{\mathcal{I}}\left(\mathbf{s}^{(t+1)}+ \mathbf{v}^{(t)}\right)\) \\
8: & \(\mathbf{z}^{(t+1)}\leftarrow\prod_{\mathcal{A}_{0}\prescript{N_{g}}}\left( \mathbf{s}^{(t+1)}+\mathbf{w}^{(t)}\right)\) \\
9: & \(I\leftarrow\text{supp}\left(\mathbf{x}^{(t+1)}\right)\cap\text{supp}\left( \mathbf{r}^{(t+1)}\right)\) \\
10: & **If**\(t=\)_Q-1_ **then** \\
11: & \(\widetilde{\mathbf{s}}_{g}^{u}[I]\gets 0\) \\  & \(\widetilde{\mathbf{s}}_{g}^{u}[I]\leftarrow\prod_{\mathcal{A}_{0}\prescript{N_ {g}N_{u}}}\left(\left(\left(\widetilde{\mathbf{H}}_{g}^{u}[,I]\right) \prescript{H\sim u}{}{}{\mathbf{H}_{\mathcal{B}}^{\sim u}};I\right)+2\sigma^{2} \mathbf{I}_{N_{q}N_{s}}\right)^{-1}\times\left(\widetilde{\mathbf{H}}_{g}^{u} [,I]\right)^{H}\mathbf{y}_{g}^{u}\) \\
12: & **else** \\
13: & \(\widetilde{\mathbf{s}}_{g}^{u}[I]\gets 0,\widetilde{\mathbf{s}}_{g}^{u}[I] \leftarrow\mathbf{z}^{(t+1)}[I]\) \\
14: & **end if** \\
15: & **If**\(f\left(\widetilde{\mathbf{s}}_{g}^{u}\right)<f_{best}\) **then** \\
16: & \(\widetilde{\mathbf{s}}_{g}^{u}[\overline{I}]\leftarrow 0\), \(\widetilde{\mathbf{s}}_{g}^{u}[I]\leftarrow\widetilde{\mathbf{s}}_{g}^{u}[I]\) \\
17: & \(f_{best}=f\left(\widetilde{\mathbf{s}}_{g}^{u}\right)\) \\
18: & **end if** \\
19: & \(\mathbf{u}^{(t+1)}\leftarrow\mathbf{u}^{(t)}+\mathbf{s}^{(t+1)}-\mathbf{x}^{(t+1)}\) \\
20: & \(\mathbf{v}^{(t+1)}\leftarrow\mathbf{v}^{(t)}+\mathbf{s}^{(t+1)}-\mathbf{r}^{(t+1)}\) \\
21: & \(\mathbf{w}^{(t+1)}\leftarrow\mathbf{w}^{(t)}+\mathbf{s}^{(t+1)}-\mathbf{z}^{(t+1)}\) \\
22: & **end for** \\
23: & **Output:**\(\widetilde{\mathbf{s}}_{g}^{u}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithm based in ADMM used for GSFIM detection [25]. used for 16 QAM (11.9375 bpcu). As expected, higher values of \(M\) require higher values of \(E_{s}/N_{0}\) to reach the reference BLER = \(10^{-1}\). A higher number of antennas per user increases the bpcu and decreases the \(E_{s}/N_{0}\) required to reach the reference BLER. Figure 4: Comparison of BER performances of PT-GSFIM, GSM and conventional BD MU-MIMO, with \(N_{u}\) = 8 and SE = 12 bpcu. Figure 5: BLER vs. (\(E_{s}/N_{0}\)) dB curves for PT-GSFIM in an MU scenario with \(N_{rx}\) = 5, 64 QAM (11.75 bpcu and 12.5 bpcu), 16 QAM (11.9375 bpcu) and 4 QAM (5.75 and 6.5 bpcu).