Impact of CCI on performance analysis of downlink satellite-terrestrial systems: outage probability and ergodic capacity perspective

The evolution of non-orthogonal multiple access (NOMA) has raised many opportunities for massive connectivity with less latency in signal transmissions at great distances. We aim to integrate NOMA with a satellite communications network to evaluate system performance under the impacts of imperfect channel state information and co-channel interference from nearby systems. In our considered system, two users perform downlink communications under power-domain NOMA. We analyzed the performance of this system with two modes of shadowing effect: heavy shadowing and average shadowing. The detailed performance was analyzed in terms of the outage probability and ergodic capacity of the system. We derive closed-form expressions and performed a numerical analysis. We discover that the performance of two destinations depends on the strength of the transmit power at the satellite. However, floor outage occurs because the system depends on other parameters, such as satellite link modes, noise levels, and the number of interference sources. To verify the authenticity of the derived closed-form expressions, we also perform Monte-Carlo simulations.

most satellite networks have adopted the orthogonal multiple access (OMA) technique for the transmission and reception of data [3].
The major disadvantage of OMA is that it cannot meet the growing requirements of communications networks. Under OMA, efficient spectrum use and limitations on the number of users have become major challenges which diminish system performance. In the present paper, we consider non-orthogonal multiple access (NOMA) to tackle the challenges raised by OMA. Of the two NOMA categories, we applied Power-Domain NOMA since much of the research has proved this system as having promising features. In NOMA, signals are transmitted superimposed in the same resource block by varying the power level of each user according to their channel gain. To identify the required user signal at the receiver, the system applies successive interference cancellation (SIC) and thereby extracts the required signal. As mentioned, NOMA uses the same resource block for multiple users and thus increases the efficiency of spectrum use at a reasonable level of implemented complexity [4,5]. The NOMA technique has achieved significant attention from researchers around the globe and is a promising technology with advantages which can be exploited in 5 G communications. Numerous studies have been performed to compare the performance of the NOMA system to OMA. The main finding is that NOMA is efficient [6,7]. With an increase in spectrum efficiency, the benefits of NOMA performance can be more prevalent if NOMA is integrated with other techniques.
Various studies have introduced the NOMA technique in satellite communications [8][9][10][11][12][13][14][15][16]. In [11], the authors studied integration of the NOMA technique with multibeam satellite networks, while in [12], the authors investigated integration of NOMA with cognitive satellite networks to increase ergodic performance of the system. The performance of NOMA-hybrid satellite relay networks (HSRN) was studied in [13,14]. NOMA integrated cognitive HSRN has been studied to analyze outage performance [15]. The performance of a similar system studied in [13] was investigated with the effect of hardware impairments [16]. The work in [17][18][19] considered NOMA-based satelliteterrestrial networks to increase the efficiency of the spectrum by beamforming. In [20], a cooperative NOMA-HSRN was considered in which the user with better channel gain acted as a relay to the remaining users in the cluster. In [21], the authors studied the effect of imperfect channel state information (CSI) and channel impairments (CI) in a NOMA-based terrestrial mobile communications network (TMCN) which functioned with multiple relays. In [22], the authors considered NOMA-based integrated terrestrial satellite networks (ISTN) to study the effect of relaying configurations such as Amplify and Forward (AF) and Decode and Forward (DF). The authors in [29] investigated a system similar to the study in [22] and explored the effect of CI under a DF relay configuration.
In the context of NOMA-HSRN, the effect of co-channel interference (CCI) in all system models has rarely been addressed. In practice, NOMA-HSRN might experience a rich CCI situation, which is an important consideration in the deployment of NOMA and HSRN in wireless communications. It can be demonstrated that the aggressive reuse of spectrum resources leads to degraded performance because of the effect of CCI. As a result, it is more than simply an important priority consideration, as the performance of NOMA-HSRN is guaranteed only if CCI is taken into account. To the best -In contrast to other studies, our study focus on a complex NOMA-based terrestrial satellite relay network with two users on the ground and a relay which encounters interference from nearby sources. We provide system performance metrics by considering a Shadowed-Rician channel between the satellite and relay, and a Nakagamim channel between the relay and destination. -The system encounters HS and AS effects as a result of the shadowing channel links.
We analyzed and compared the system's performance under these two shadowing modes. -We derived the closed-form expressions for outage probability and ergodic capacity of both users in a dedicated NOMA user group. A performance gap is expected for these users depending on their demands and the portion of power allocated to each user. -Finally, we performed a numerical analysis and Monte-Carlo simulations for the derived expressions to verify their authenticity and analyze the system's performance. We illustrate the system's performance in HS mode by varying the interference links and mean square error of the channels.
The paper is organized as follows. Section 2 explains the system model and types of signal received from the users. Section 3 provides a performance analysis and describes the mathematical expressions for outage probability, ergodic capacity and diversity order of the system. Section 4 provides an analysis and simulations of the expressions obtained in Sect. 3. Section 5 concludes the paper with the attained results.

System model
In this section, we assume a satellite (S), a relay (R) and two users D i , i ∈ (1, 2) as in Fig. 1. All nodes are equipped with a single antenna, and the relay operates with the DF protocol. The relay node is also affected by N co-channel interference sources {I n } N n=1 . The link from S to D i is not available because of heavy shadowing [30]. h R denotes the channel coefficient from S to R and follow a Shadowed-Rician channel, h i denotes the channel coefficients from R to D i and follow a Nakagami-m channel, h nR denotes the channel coefficient of the link between the n-th interference source and relay and follows independent and non-identically distributed (i.ni.d.) Nakagami-m random variable (RVs). Under these conditions, the CSI procedure exhibits error. The estimation channel is expressed as [31] (1) h j =ĥ j + e j , where j ∈ {R, 1, 2} , e j is the error term for CN (0, σ 2 e j ) [32]. The power-domain assisted NOMA signal from the source transmits user signals superimposed in the same resource block by varying the power coefficient of each user according to their channel state information (CSI). At the receiver's end, perfect successive interference cancellation (SIC) is assumed to extract the desired signal from the superimposed signal. Imperfect CSI should therefore be studied in practical scenario. Satellite-terrestrial networks needs relay to empower signals before forwarding them to mobile users. The main reason of design a relay is to deal with signal transmission at long distances. The satellite needs to allocate suitable power level to expected users. The first user D 1 is assumed to be located at far distance and such weak signal needs higher power allocation. Meantime, the near user D 2 just acquire lower level of transmit power. In the first phase, S transmits the signal where P S is the transmit power, A 1 and A 2 are power allocation such that A 1 + A 2 = 1 and A 1 > A 2 assumed under the NOMA scheme. Then, the signal received at R is given as where P Cn is the transmit power of the n-th interference source and n R is the additive white Gaussian noise (AWGN) at R for CN (0, N 0 ) . The signal to interference plus noise ratio (SINR) is then used to decode x 1 and given as where ρ S = P S N 0 is the transmit SNR, ρ Cn = P Cn N 0 and γ C = N n=1 ρ Cn h nR 2 . Then, the SINR decoded x 2 is given as In the second phase, relay R forwards the signals to the ground users. The signal received at D i is given as where P R is the transmit power at R and n D i AWGN for CN (0, N 0 ) . It is noted that the other main parameters are listed in Table 2. The SINR which decodes x 1 at D 1 is given as where ρ R = P R N 0 , the SINR which decodes signal x 1 at D 2 is given as [32] (2) Applying SIC, the SINR which decodes its own signal x 2 at D 2 is computed according to For performance analysis, these SINRs provide important information which allows us to compute probabilities.

Performance analysis
In this section, we analyze the two main system metrics with the assumed channel models below.

Channel model
Following the results in [33], the probability density function (PDF) of |ĥ R | 2 is formulated by , m R is the fading severity parameter, 2b R and Ω R denote multipath components and the average power of light of  can be obtained as The PDF and CDF of |h i | 2 are then, respectively, given as [35] and where m i and Ω i are the fading severity parameter and the average power, respectively, and Γ (., .) is the upper incomplete gamma function [46]. Moreover, the PDF of γ C is calculated with corresponding severity parameters {m Cn } N n and average powers {Ω Cn } N n . Therefore, we can express the PDF of γ C as [36,37] and [24] where the parameters m I and Ω I are obtained from moment based estimators. For this, we define Θ = I n=1 h nR 2 , and without loss of generality, we assume no power control is used, i.e., P Cn = P C or ρ Cn = ρ C . Then, we have where . From this, the exact moments of Θ can be obtained as

Outage probability of D 1
An outage event of D 1 is given when R and D 1 cannot detect x 1 correctly. Then, the outage probability of D 1 is given as where γ i = 2 2R i − 1 , and R i is the target rate.

Proposition 1
Here, the closed-form of B 1 is given as Next, using (6), B 2 is rewritten as Based on the CDF of ĥ i in (13), B 2 can be expressed as Finally, substituting (17) and (19) into (16), P D 1 can be obtained by (16)

Outage probability of D 2
The outage events of D 2 occurs when R and D 2 cannot detect x 2 correctly. Therefore, the outage probability of D 2 is given as

Proposition 2
The closed-form outage probability of P D 2 is obtained as Proof See Appendix B.

Diversity order
To gain some insight, we derive under the asymptotic outage probability of D i under a high SNR (ρ = ρ S = ρ R → ∞) . The diversity order is defined as [38] where P ∞ D i is the asymptotic outage probability of D i .

Proposition 3
The asymptotic outage probability of D 1 is given as Similarly, the asymptotic of D 2 can be expressed by The results in (24) and (25) refer to limits of outage performance in the region of high SNR. It can be predicted that the outage performance of two ground users (21)  encounters the lower bound even though we improve other system parameters. As discussed, the diversity is then zero.

Ergodic capacity of D 1
The ergodic capacity of x i is expressed as [39] where Q 1 = min Γ R→x 1 , Γ D 1 →x 1 .

Proposition 4
The closed-form ergodic capacity of x 1 is given as (27), where .
Proof See Appendix D.

Ergodic capacity D 2
Similarly, the ergodic capacity of x 2 is written as

Proposition 5
The closed-form ergodic capacity of x 1 is given as (29), where

Consideration on case of multiple antenna relay
In this section, we consider how a multiple antennas relay makes influences to performance of two users D i . In particular, a DF relay can be equipped with K R received antennas and K T transmit antenna. To represent mathematical expressions from now on, .., h K T i ] T denotes the K T × 1 channel vector between R and D i . In this first phase, the signal received at R with help (1) is given by where n R denote the vector of zero mean AWGN with variance N 0 and w R = h R ĥ R F . Then, the SINR is then used to decode x 1 and given as . Then, the SINR decoded x 2 is given as In the second phase, the received at D i is expressed as . The SINR which decodes x 1 at D 1 is given as , the SINR which decodes signal x 1 at D 2 is given as Similarly, the SINR which decodes its own signal x 2 at D 2 is computed according to

Statistical characterization
In this section, we consider ĥ R and ĥ i have independent and identically distributed (i.i.d.) entries as [43]. In addition, the PDF of η R can be expressed by [44] With the help of [46, Eq. 3.351.2], we have the CDF of η R as In addition, the CDF of η I is given by [44]

Proposition 6
The outage probability of D 1 is expressed as Similarly, the outage probability of D 2 can be expressed by

Simulation results and discussion
In this section, we set ρ C = 1 dB, ρ = ρ S = ρ R and the main parameters given in Table 3. The Shadowed-Rician fading parameters for the satellite link are taken from [40] and shown in Table 4. Additionally, the interference channels parameters were set and calculated according to the respective analytical curves in [24] and are shown in Table 5. Moreover, we using MATLAB for Monte Carlo simulations. Figure 2 shows the outage performance versus the ρ (dB) for different shadowing satellite links. We can observe that the performance of the system under AS is superior to the system under HS. That means satellite channel conditions contribute significantly to system performance at the ground users. We can also observe the difference in performance of the NOMA and OMA systems. In the OMA system, the gap between the (41)

System parameters Values
Monte Carlo simulation   two curves shows that with increased SNR, system performance increases similarly to the NOMA system. The authenticity of the derived expressions is also evident from the strict match of the Monte Carlo simulations with the analytical simulations. Figure 3 represents how multiple antennas at relay contribute to improve the system performance at ground users. Once we design the relay with K T = K R = 3 , the big gap outage behavior can be observed compared with the case of the relay with K T = K R = 2 . The reason is that higher diversity from multiple antennas design could be strengthen signal received at ground users and hence outage performance could be improved thoroughly.  Figure 4 indicates the impact of imperfect CSI on outage probability when we change value of ρ for case of HS. We can see how performance could be affected by such CSI error by varying σ 2 . An increase in the value of σ 2 shows a reduction in the performance of the user, and for the lowest value of σ 2 , both users demonstrate better performance. As the SNR increases, the performance of both users continues to increase, while in similar conditions in Fig. 5, we varied the number of interfering links for both users by keeping σ 2 constant. Considering impact of CCI concern, the simulation shows that with a greater number of interference links, the performance of both users decreases. However, in all the links, the curves for each user meet at a saturated point at high SNR. We conclude that in the high SNR region, interference links do not have a great effect on user performance. Figure 6 shows the simulation for outage performance versus ρ (dB) with the different satellite links as in Fig. 2. The ergodic capacity rates of the message at D 1 are almost similar in both HS and AS modes, but for messaging at D 2 , the gap between the curves of ergodic capacity in both modes is comparatively very high. With a simultaneous increase in the SNR, the gap increase is unlike D 1 . Figures 7 and 8 indicate the several curves of ergodic capacity versus ρ (dB) under HS. We can see the impact of CSI error levels of σ 2 in Fig. 7 while Fig. 8 confirms how the number of CCI sources leads to degradation of performance in term of ergodic capacity. In Fig. 7, as we increase the value of σ 2 , the gap between the curves increases simultaneously at high SNR values. Figure 8 shows the ergodic capacity versus ρ (dB) varying N with satellite link under HS. Although gaps between the curves are evident at medium SNR values, the curves for both the users meet at a single point at higher SNRs, suggesting that at higher SNRs, a greater number of interference links does not show much differential effect on ergodic capacity rates.

Conclusion
We described the use of the NOMA technique for communication between a satellite to a relay and the relay to users. We investigated performance of the system in terms of outage probability and ergodic capacity under the effect of CCI at the relay. The performance gap between two destinations can be explained by the differences in power level allocated to the destinations. We also derived the closed-form expressions for outage probability and ergodic capacity at both users. We observed that with an increase in the channel error, both the outage probability and ergodic capacity of the users were significantly affected. The effect of CCI on both outage probability and ergodic capacity is more prominent when the SNR falls in range of 20-30 dB, whereas a greater number of interference links shows little effect in the high SNR region. More specifically, if, for example, the number of interference sources is 5, the outage performance of the system experiences a decrease of approximately 40% at a signal-to-noise ratio (SNR) of 30 dB at the satellite. Outage probability and ergodic capacity became saturated at SNRs of 50 dB and 45 dB, respectively. We also simulated and compared system performance under AS and HS modes for both outage probability and ergodic capacity.

Appendix A
Using (3), B 1 is written as can therefore be expressed as Substituting (11) and (12)   This completes the proof.

Appendix B
Let us denote the first and second terms of (21) as C 1 and C 2 , respectively. Using (4), C 1 can then be written as where ψ 1 = γ 2 ρ S A 2 . As in Proposition 1, we obtain C 1 as Next, C 2 is calculated as . Using (47) and (48), the closed-form outage probability of D 2 is obtained as (22). This completes the proof.

Appendix C
In the high SNR region, the CDF of |ĥ R | 2 and |ĥ i | 2 are, respectively, given s and Next, the asymptotic outage probability of D 1 is calculated as Then, B ∞ 1 is expressed by Using (49) and (14), we can rewrite B ∞ 1 as B ∞ 1 is thus obtained by Then, the term B ∞ 2 can be calculated by Substituting (54) and (55) into (51), we obtain (24). This completes the proof.