## 2 System Model
### Spectrum Isolation Transmission Method  
To avoid incurring extra interferences, the signals of HS and UTs are designed as spectrum-isolated, which is shown in Figure 2. With a total bandwidth \(B\) Hz, the HS takes \(\tau B\) Hz, and the rest is for the UTs. Here, \(0<\tau<1\) is the bandwidth factor of the HS. Figure 2: Spectrum allocation in conventional schemes. Let \(\mathbf{w}_{0,S}\) (\(\mathbf{v}_{0,S}\)) and \(\mathbf{w}_{U,S}\) (\(\mathbf{v}_{U,S}\)) denote the transmit (receive) beamforming vectors of the feeder and service beams at the satellite, respectively. The transmit and receive beamforming vectors at the \(i\)-th UT are represented by \(\mathbf{w}_{i}\) and \(\mathbf{v}_{i}\), respectively. Since the HS and UTs are allocated separately inside the whole frequency band, broadcasting and information collection could be carried out noninterferingly. The signals arriving at the satellite are given by  
\[x_{0}=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n })=\mathbf{v}_{0,S}^{\mathrm{H}}\mathbf{a}(\phi_{0},\theta_{0})\sqrt{E_{r}}l(d) \sqrt{G_{t}}\mathbf{b}^{\mathrm{H}}(\omega_{U}^{{}^{\prime}},\upsilon_{M}^{{}^ {\prime}})\mathbf{w}_{M}s_{M}+\mathbf{v}_{U,S}^{\mathrm{H}}\mathbf{n},\] \[x_{1}=\mathbf{v}_{U,S}^{\mathrm{H}}(\mathbf{G}_{1}\mathbf{w}_{1 }s_{1}+\mathbf{n})=\mathbf{v}_{U,S}^{\mathrm{H}}\mathbf{a}(\phi_{1},\theta_{ 1})\sqrt{E_{r}}l(d)\sqrt{G_{t}}\mathbf{b}^{\mathrm{H}}(\omega_{1}^{{}^{\prime }},\upsilon_{1}^{{}^{\prime}})\mathbf{w}_{1}s_{1}+\mathbf{v}_{U,S}^{\mathrm{H }}\mathbf{n},\] \[\ldots,\] \[x_{M}=\mathbf{v}_{U,S}^{\mathrm{H}}(\mathbf{G}_{M}\mathbf{w}_{M }s_{M}+\mathbf{n})=\mathbf{v}_{U,S}^{\mathrm{H}}\mathbf{a}(\phi_{M},\theta_{M })\sqrt{E_{r}}l(d)\sqrt{G_{t}}\mathbf{b}^{\mathrm{H}}(\omega_{M}^{{}^{\prime }},\upsilon_{M}^{{}^{\prime}})\mathbf{w}_{M}s_{M}+\mathbf{v}_{U,S}^{\mathrm{H }}\mathbf{n}, \tag{7}\]  
where \(s_{0}\) and \(s_{i}\) represent the broadcasting and \(i\)-th UT signals, respectively. The background noise is denoted as \(\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I})\), whose power can be found as \(\sigma^{2}\). Next, the satellite pushes forward the UTs' signals to the HS and broadcasts the HS's signals to the UTs. In this case, the \(M\) signals arriving at the HS can be calculated as  
\[y_{1,0} =\mathbf{h}_{0}^{\mathrm{H}}\mathbf{w}_{0,S}(\mathbf{v}_{U,S}^{ \mathrm{H}}(\mathbf{G}_{1}\mathbf{w}_{1}s_{1}+\mathbf{n}))+z\] \[=\rho_{0}\sqrt{G_{r,0}}l(d)\sqrt{E_{t}}\mathbf{a}^{\mathrm{H}}( \phi_{0}^{{}^{\prime}},\theta_{0}^{{}^{\prime}})\mathbf{w}_{0,S}(\mathbf{v}_ {U,S}^{\mathrm{H}}\mathbf{a}(\phi_{1},\theta_{1})\sqrt{E_{r}}l(d)\sqrt{G_{t}} \mathbf{b}^{\mathrm{H}}(\omega_{1}^{{}^{\prime}},\upsilon_{1}^{{}^{\prime}}) \mathbf{w}_{1}s_{1}+\mathbf{v}_{U,S}^{\mathrm{H}}\mathbf{n})+z,\] \[\ldots,\] \[y_{M,0} =\mathbf{h}_{0}^{\mathrm{H}}\mathbf{w}_{0,S}(\mathbf{v}_{U,S}^{ \mathrm{H}}(\mathbf{G}_{M}\mathbf{w}_{M}s_{M}+\mathbf{n}))+z\] Figure 2:0  
where \(z\sim\mathcal{CN}(0,\sigma^{2})\) denotes the background noise at the HS. For the UTs, we have  
Figure 2:1 Figure 2:2 Figure 2:3 Figure 2:4 Figure 2:5  
where \(\mathbf{z}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I})\) denotes the background noise at UTs. Considering the computing capability on satellites, beam steering is applied throughout the paper, where \(||\mathbf{w}||^{2}=||\mathbf{v}||^{2}=1\)[26]. Then, SNRs at the HS can be computed as  
Figure 2:6 Figure 2:7 Figure 2:8  
where \(P_{1}=\ldots=P_{M}=P\) represents the power of the UTs' signals.