# Smart Beamforming for Direct LEO Satellite Access of Future IoT
## 3 System Model
As the waveform design is beyond the scope of the paper, we will adhere to the simple case where \(N=M\), which embraces OFDM-like waveforms. To describe the RDMA scheme, we focus the attention on a given symbol. To ease the analytical tractability we use the following matrix notation \(\mathbf{x}_{k}^{m}=\left[x_{k}^{m}[0],\cdots,x_{k}^{m}[N-1]\right]\). The approach that is followed to create the signature is based on introducing redundant samples, which are a repetition of some specific samples. This operation can be done by applying a linear processing that results in a new sequence, i.e.,  
\[\bar{\mathbf{x}}_{k}^{m}=\mathbf{x}_{k}^{m}\mathbf{G}=\mathbf{x}_{k}^{m}\left[ \mathbf{I}_{N}\ \ \mathbf{C}_{p_{k}}\right]=\left[\mathbf{x}_{k}^{m}\ \ \mathbf{x}_{k}^{m}\mathbf{C}_{p_{k}}\right]=\left[\bar{ \mathbf{x}}_{k}^{m}[0],\cdots,\bar{\mathbf{x}}_{k}^{m}[N+L_{R}-1]\right] \tag{2}\]  
\[\mathbf{C}_{p_{k}}=\left[\mathbf{0}_{(p_{k}-1)L_{R}\times L_{R}}^{\intercal} \ \mathbf{I}_{L_{R}}\ \ \mathbf{0}_{N-p_{k}L_{R}\times L_{R}}^{\intercal}\right]^{\intercal}, \tag{3}\]  
for \(p_{k}\in\{1,\cdots,P\}\). Essentially, at the end of the sequence we append a subset of samples of length \(L_{R}\). Hence, the original signal is preserved. Now, the duration of the sequences is increased, resulting in:  
\[\bar{x}_{k}[n]=\sum_{m=0}^{N_{\text{S}}-1}\bar{x}_{k}^{m}[n-(N+L_{R})m]. \tag{4}\]  
The average energy per sample conforms to:  
\[P_{T}=\frac{1}{N+L_{R}}\mathbb{E}\Big{\{}||\bar{\mathbf{x}}_{k}^{m}||^{2} \Big{\}}. \tag{5}\]  
Notice that by repeating different parts of the sequence we can create different patterns. This feature can be exploited by the receiver to distinguish users, as long as different repetition patterns are used. This is equivalent to selecting different \(\mathbf{C}_{p_{k}}\) matrices, as \(\{\mathbf{C}_{1},\cdots,\mathbf{C}_{P}\}\) form an orthogonal basis. It is important to remark that the length of the sequence \(N\) and the length of the repeated part \(L_{R}\) will determine the number of different repetition schemes. The idea is illustrated in Figure 2.  
Before providing the details of the detection scheme, we formulate the system model. In LEO satellite communication systems at Ka-band, the terminal must establish directional transmission links to compensate for the high path loss experienced at the higher frequencies and to sustain an acceptable link quality. If the terminal is equipped with a global navigation satellite system (GNSS) receiver and is provisioned with the satellite ephemeris, then it can geometrically select the closest satellite and steer a beam towards it. At the other end of the link, the satellite receives multiple signals that come from different areas. In the uplink transmission, the purpose of the satellite is to harness on-board processing to discriminate users by designing directional beamformers. To this end, the satellite is equipped with a planar array of \(N_{\text{R}}\) elements, which are identical. The separation between adjacent antennas is denoted \(d\). For isotropic radiating elements and \(d=\lambda/2\), the 3-dB beamwidth of a planar array is \(\theta_{B}=\frac{100}{\sqrt{N_{\text{R}}}}\)[31]. Stacking column-wise the output of each antenna, the received signal becomes,  
\[\mathbf{y}[n]=\sum_{k=1}^{N_{\text{U}}}\mathbf{h}_{k}e^{j2\pi e_{k}n}\tilde{x} _{k}[n-\tau_{k}]+\mathbf{w}[n], \tag{6}\]  
where \(\mathbf{h}_{k}\in\mathbb{C}^{N_{\text{R}}\times 1}\), \(\tau_{k}\in\mathbb{N}\), and \(\epsilon_{k}\in\mathbb{R}\) correspond to the channel vector, the delay, and the carrier frequency offset, respectively, associated to the \(k\)-th user. Notice that multiple signals are received on shared time and frequency resources. The carrier frequency offset encompasses oscillator uncertainties and the uncompensated Doppler shift that results from the orbital motion, which can be characterized with the analytical expressions provided in [32]. In (6), we have assumed that the frequency misalignment produces a constant rotation during the packet transmission.