# Smart Beamforming for Direct LEO Satellite Access of Future IoT
## 4 Resource Sharing Beamforming Access
Under the assumption that signals from different users are statistically independent and that the noise is uncorrelated with the data, we can formulate the mean square error as:  
\[\mathbb{E}\Big{\{}||\mathbf{b}_{i}^{\mathrm{H}}(\mathbf{Y}_{d}\mathbf{C}_{i} -\mathbf{Y}_{r})||^{2}\Big{\}}=\mathbf{b}_{i}^{\mathrm{H}}(\mathbf{R}_{i}+ \mathbf{R}_{r})\mathbf{b}_{i}-\mathbf{b}_{i}^{\mathrm{H}}\Big{(}\mathbf{R}_{ir }+\mathbf{R}_{ir}^{\mathrm{H}}\Big{)}\mathbf{b}_{i} \tag{17}\]  
\[\mathbf{R}_{i}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{i}||^{2}\Big{\}}\mathbf{h}_{k} \mathbf{h}_{k}^{\mathrm{H}} \tag{18}\]  
\[\mathbf{R}_{r}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{p_{k}}||^{2}\Big{\}}\mathbf{h}_{ k}\mathbf{h}_{k}^{\mathrm{H}} \tag{19}\]\[\mathbf{R}_{ir}=\sum_{k=1}^{N_{\mathrm{U}}}\mathbb{E}\Big{\{}\mathbf{x}_{k} \mathbf{C}_{i}\mathbf{C}_{p_{k}}^{\mathrm{H}}\mathbf{x}_{k}^{\mathrm{H}}\Big{\}} \mathbf{h}_{k}\mathbf{h}_{k}^{\mathrm{H}}. \tag{20}\]  
To get the matrices, we rely on the fact that the noise samples are independent and identically distributed with \(\mathbb{E}\big{\{}\mathbf{W}\mathbf{W}^{\mathrm{H}}\big{\}}=(N+L_{R})\mathbf{ I}_{N_{\mathrm{T}}}\). In alignment with (7), the noise samples have unit variance. The optimal beamformer can be computed by setting the partial derivatives of the Lagrangian to zero, thus obtaining:  
\[(\mathbf{R}_{i}+\mathbf{R}_{r})\mathbf{b}_{i}=(1+\lambda)\Big{(}\mathbf{R}_{ ir}+\mathbf{R}_{ir}^{\mathrm{H}}\Big{)}\mathbf{b}_{i}, \tag{21}\]  
where \(\lambda\) is the Lagrange multiplier. The solution consists in computing the eigenvalue decomposition of matrices \((\mathbf{R}_{i}+\mathbf{R}_{r})\in\mathbb{C}^{N_{\mathrm{R}}\times N_{ \mathrm{R}}}\) and \(\big{(}\mathbf{R}_{ir}+\mathbf{R}_{ir}^{\mathrm{H}}\big{)}\in\mathbb{C}^{N_{ \mathrm{R}}\times N_{\mathrm{R}}}\). It can be verified that the mean square error is minimum for minimum \(\lambda\). Therefore, the optimal \(\mathbf{b}_{i}\) is selected as the eigenvector associated to the eigenvalue that is closest to \(1\). Another important results is that the minimum variance (MV) beamformer, namely:  
\[\mathbf{b}_{i}=\frac{\mathbf{R}_{i}^{-1}\mathbf{h}_{d}}{\mathbf{h}_{d}^{ \mathrm{H}}\mathbf{R}_{i}^{-1}\mathbf{h}_{d}}, \tag{22}\]  
is solution of (21), as long as the rank of \(\mathbf{R}_{ir}\) is one. This is tantamount to saying that there is just one user transmitting with the pattern \(\mathbf{C}_{i}\). Without loss of generality, the array vector in the direction of the desired user is denoted \(\mathbf{h}_{d}\), where \(d\in\{1,\cdots,N_{\mathrm{U}}\}\). Accordingly, the signal-to-interference-plus-noise ratio (SINR) can be formulated as:  
\[\mathrm{SINR}_{i}=\frac{|\mathbf{b}_{i}^{\mathrm{H}}\mathbf{h}_{d}|^{2}}{ \sum_{k\
eq d}|\mathbf{b}_{i}^{\mathrm{H}}\mathbf{h}_{k}|^{2}+\frac{|| \mathbf{b}_{i}||^{2}}{P_{\mathrm{T}}}}. \tag{23}\]  
To create the advantageous situation where the rank of \(\mathbf{R}_{ir}\) is one, \(P\geq N_{\mathrm{U}}\) must be satisfied. Thus, for a given \(N_{\mathrm{U}}\), the performance is enhanced by increasing the number of repetition patterns \(P\). It becomes evident that if \(P>>N_{\mathrm{U}}\), it is highly unlikely that two users transmit with the same repetition structure. In exchange, the complexity is increased, as it scales with \(P\). Figure 4: Scheme of the beamformer design at base band. The procedure can be extended to the situation where multiple terminals pick the same repetition pattern. Let \(N_{U_{i}}\) denote the number of terminals that employ \(\mathbf{C}_{i}\) to obtain the redundant symbols. Assuming that the number of active terminals \(N_{U_{i}}\) is correctly estimated, which will be tackled later on, the satellite must perform a specific signal processing for each terminal.