## 3 Spectrum-Saving Transmission Scheme  
It can be noticed from Section 2.2 that the HS and UTs take spectral resource separately. In this case, more UTs are not allowed to access the network when the spectrum is entirely occupied. Realizing that the receiving capability of the HS is much stronger than that of UTs, we propose a spectrum-saving transmission scheme, which is shown in Figure 3.  
In Figure 3, we can observe that UTs are allocated inside the frequency band occupied by the HS. That is, the remaining spectral resource can be saved. In this paper, terminals equipped at users are identical, and transmit the same type of service with different information. That is, all UTs' signals can be assumed to have the same power level. Under the proposed mechanism, the signal received by the satellite becomes  
\[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]  
Then, the satellite transmits the co-spectrum signal to the HS and UTs, whose received signals can be expressed as  
Figure 3: Spectrum allocation in the proposed scheme. \[y_{0} = \mathbf{h}_{0}^{\mathrm{H}}\mathbf{w}_{0,5}x+z\] \[= \mathbf{h}_{0}^{\mathrm{H}}\mathbf{w}_{0,5}(\mathbf{v}_{0,S}^{ \mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U,S}^{\mathrm{H}}( \sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}))+z,\] \[y_{1} = \mathbf{v}_{1}^{\mathrm{H}}(\mathbf{H}_{1}\mathbf{w}_{U,5}x+ \mathbf{z})\] \[= \mathbf{v}_{1}^{\mathrm{H}}(\mathbf{H}_{1}\mathbf{w}_{U,5}( \mathbf{v}_{0,5}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U,S} ^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}))+ \mathbf{z})\] \[\ldots,\] \[y_{M} = \mathbf{v}_{M}^{\mathrm{H}}(\mathbf{H}_{M}\mathbf{w}_{U,S}x+ \mathbf{z}) \tag{15}\] \[= \mathbf{v}_{M}^{\mathrm{H}}(\mathbf{H}_{M}\mathbf{w}_{U,S}( \mathbf{v}_{0,5}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U,S} ^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}))+ \mathbf{z}).\]  
To decode the UTs' signals from the co-spectrum signal \(x\), the receiver at the HS should be redesigned, which is shown in \[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]0  
In the first step, the HS demodulates \(s_{0}\) directly, where the signal-to-interference-noise ratio (SINR) can be derived as  
\[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]1 \[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]2  
After canceling \(s_{0}\), the \(M\) signals would be filtered in parallel at each UT's frequency band. Then, the \(M\) signals can be derived where the SNRs can be expressed as  
\[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]3 \[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]4 \[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]5  
Since \(s_{0}\) is stronger than \(s_{1},\ldots,s_{M}\), the receivers at UTs would demodulate the broadcast signal with SINRs given by  
\[x=\mathbf{v}_{0,S}^{\mathrm{H}}(\mathbf{g}_{0}s_{0}+\mathbf{n})+\mathbf{v}_{U, S}^{\mathrm{H}}(\sum_{i=1}^{M}\mathbf{G}_{i}\mathbf{w}_{i}s_{i}+\mathbf{n}) \tag{14}\]6 Structure of the receiver at the HS.