# Performance Evaluation of a Full-Duplex Relaying-Enabled Satellite Sensor Network
## 2 System Model
By combining (1) and (2), and considering the FD concept, the sum of the received signal and loop interference at the satellite relaying with respect to infinite time slots with an iterative manner can be obtained as (see system model in [17] and [18])  
\[\sqrt{P_{2}}t\left[i\right] =\beta\sum_{j=1}^{\infty}\left(h_{l}\beta\right)^{j-1}\] \[\times\left(\sqrt{P_{1}}h_{1}x\left[i-j\tau\right]+n_{1}\left[i-j \tau\right]\right). \tag{3}\]  
Thus, the output signal power is calculated from (3) as  
\[P_{2}\mathrm{E}\left\{\left|t\left[i\right]\right|^{2}\right\} =\beta^{2}\sum_{j=1}^{\infty}\left(\left|h_{l}\right|^{2}\beta^{2 }\right)^{j-1}\left(P_{1}\left|h_{1}\right|^{2}+\sigma_{1}^{2}\right)\] \[=\beta^{2}\frac{P_{1}\left|h_{1}\right|^{2}+\sigma_{1}^{2}}{1- \left|h_{1}\right|^{2}\beta^{2}}. \tag{4}\]  
By considering the normalized condition \(\mathrm{E}\left\{\left|t\left[i\right]\right|^{2}\right\}=1\) to guarantee the transmit power of satellite relaying within the prescribed bound (In particular, we consider the normalized conditions \(\mathrm{E}\left\{\left|x\left[i\right]\right|^{2}\right\}=1\) and \(\mathrm{E}\left\{\left|t\left[i\right]\right|^{2}\right\}=1\) over the DVB-S2 constellations [12]. ), the gain factor is obtained by satisfying (The scaling factor \(\beta\) is to ensure that the power transmitted from the satellite remains within the prescribed bounds [15].) \[\beta=\left(P_{2}\middle/\left(P_{1}\left|h_{1}\right|^{2}+P_{2}\left|h_{1} \right|^{2}+\sigma_{1}^{2}\right)\right)^{1/2}. \tag{5}\]  
Finally, the received signal at the ES-2 can be expressed as  
\[y\left[i\right]=\sqrt{P_{2}}h_{2}t\left[i\right]+n_{2}\left[i\right], \tag{6}\]  
where \(h_{2}\) denotes the channel coefficient between satellite and ES-2, and \(n_{2}\left[i\right]\) is the noise component obeying \(\mathrm{E}\left\{\left|n_{2}\left[i\right]\right|^{2}\right\}=\sigma_{2}^{2}\). From (6), the instantaneous received power can be derived as  
\[\mathrm{E}\left\{\left|y\left[i\right]\right|^{2}\right\}=\beta^{ 2}P_{1}\left|h_{1}\right|^{2}\left|h_{2}\right|^{2}\] \[\sqrt{P_{2}}t\left[i\right]=\beta r\left[i-\tau\right]. \tag{2}\]0  
To this end, by reformulating the signal, residual loop interference and noise power in (7), the received signal-to-interference and noise ratio (SINR) can be expressed as  
\[\sqrt{P_{2}}t\left[i\right]=\beta r\left[i-\tau\right]. \tag{2}\]1  
Although some slow fluctuation would happen during on-board processing, but most of the channel components would be still static [15; 18; 19], by substituting (5) into (8) and involving the common assumption of non-fading residual interference channel, (8) can be further simplified as  
\[\sqrt{P_{2}}t\left[i\right]=\beta r\left[i-\tau\right]. \tag{2}\]2where signal-to-noise (SNR) level of uplink and downlink are \(\gamma_{1}=P_{1}\left|h_{1}\right|^{2}\Big{/}\sigma_{1}^{2}\), \(\gamma_{2}=P_{2}\left|h_{2}\right|^{2}\Big{/}\sigma_{2}^{2}\), respectively. The residual interference level \(\mathcal{\bar{\gamma}}_{I}=\mathrm{E}\left[\gamma_{I}\right]\) follows the well-known principles of echo cancellation, which can be expected to come from the complementary analog and digital cancellation efforts. To determine the on-board magnitude of the achievable SI cancellation levels, it is considered that the SI cancellation circuit is capable of matching the delay perfectly [14]. Hence, the remaining error component is introduced by the non-ideal estimated weight of the coupled SI signal. Due to the fact that \(h_{I}\) determines the strength of the SI, it can be explained as the residual SI channel after certain SI cancellation measures are applied.