# Smart Beamforming for Direct LEO Satellite Access of Future IoT
## 4 Resource Sharing Beamforming Access
As Appendix A proves, for large scale antenna arrays, this boils down to selecting the \(N_{U_{i}}\) eigenvectors associated to the \(N_{U_{i}}\) eigenvalues that are closest to 1. Each of them tends to the MV beamformer (22) in the direction of the terminal that is targeted. The beauty of the proposed beamforming technique is that the terminals can be distinguished and separated, as long as they are not located in the same spot, without estimating the channel nor making a beam scan. In practical situations, \(\mathbf{R}_{i}\), \(\mathbf{R}_{r}\), and \(\mathbf{R}_{ir}\) must be estimated by their corresponding sample-covariance matrices. This entails a complexity increase with respect to the fixed grid beamforming network, which is a simpler solution. However, the numerical results presented in Section 5 show that the computational complexity translates into a significant SINR increase with respect to the fixed beamforming approach. Sticking to the system model formulated in (14), the estimated covariance matrices can be computed as:  
\[\mathbf{\hat{R}}_{i}=\frac{1}{N_{\mathrm{S}}}\sum_{m=0}^{N_{\mathrm{S}}-1} \mathbf{Y}_{m,1}\mathbf{C}_{i}\mathbf{C}_{i}^{\mathrm{H}}\mathbf{Y}_{m,1}^{ \mathrm{H}} \tag{24}\]  
\[\mathbf{\hat{R}}_{r}=\frac{1}{N_{\mathrm{S}}}\sum_{m=0}^{N_{\mathrm{S}}-1} \mathbf{Y}_{m,2}\mathbf{Y}_{m,2}^{\mathrm{H}} \tag{25}\]  
\[\mathbf{R}_{i}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{i}||^{2}\Big{\}}\mathbf{h}_{k} \mathbf{h}_{k}^{\mathrm{H}} \tag{18}\]0  
from the observations:  
\[\mathbf{R}_{i}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{i}||^{2}\Big{\}}\mathbf{h}_{k} \mathbf{h}_{k}^{\mathrm{H}} \tag{18}\]1  
\[\mathbf{R}_{i}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{i}||^{2}\Big{\}}\mathbf{h}_{k} \mathbf{h}_{k}^{\mathrm{H}} \tag{18}\]2  
To corroborate that the proposed beamforming technique allows to increase the number of terminals without collision, which is the key point of the paper, we have evaluated the beamformer response in different azimuth and elevation angles. We have defined the beampattern as \(G(\theta_{d},\phi_{d})=|\mathbf{b}_{i}^{\mathrm{H}}\mathbf{s}_{d}|^{2}\), for \(0\leq\theta_{d}\leq 90^{\circ}\) and \(0\leq\phi_{d}\leq 360^{\circ}\). In Figure 5 we have depicted the beampattern, for \(N_{\mathrm{U}}=10\), \(d=\lambda/2\), and \(N_{x}=N_{y}=32\). In particular, we have represented the MV beamformer with perfect CSIR and the proposed blind beamformer. For the sake of clarity, the highest value of the beampattern is normalized to 1. Regarding the format of the signal, the terminals employ the OFDM modulation. The frame consists of \(M=64\) subcarriers, which only 58 are active, and \(N_{\mathrm{S}}=400\) OFDM symbols. Information is conveyed in QPSK symbols. To create the repetition pattern, users can pick any of the \(P=4\) orthogonal signatures. Hence, the redundancy spans \(L_{R}=16\) samples. The rest of the system parameters are taken from \[\mathbf{R}_{i}=L_{R}\mathbf{I}_{N_{\mathrm{R}}}+\sum_{k=1}^{N_{\mathrm{U}}} \mathbb{E}\Big{\{}||\mathbf{x}_{k}\mathbf{C}_{i}||^{2}\Big{\}}\mathbf{h}_{k} \mathbf{h}_{k}^{\mathrm{H}} \tag{18}\]3 In the case represented in Figure 5, the user of interest is located at \((\theta,\phi)=(24.91^{\circ},7.89^{\circ})\) or, equivalently \(u=0.4172,v=0.0578\). It is worth underlining that 3 out of the 9 interfering users select the same repetition pattern. In Figure 5 we only plot the first eigenvector, which corresponds to the beamformer that points to the terminal of interest in this simulation. Graphically comparing both schemes, we can resolve that the proposed beamformer technique provides a reasonably good estimate of the MV beamformer.