Joint Effects of Co-Channel Interferences and Pointing Errors on Dual-Hop Mixed RF/FSO Fixed-Gain and Variable-Gain Relaying Systems

The performance of a dual-hop mixed radio frequency (RF)/free space optical (FSO) system with co-channel interferences (CCIs) is studied for both fixed-gain and variable-gain amplify-and-forward(AF) relaying scenarios. The RF link and interference link undergo the <inline-formula><tex-math notation="LaTeX">$\kappa$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> shadowed fading and the FSO link experiences the Fisher-Snedecor <inline-formula><tex-math notation="LaTeX">${\mathcal F}$</tex-math></inline-formula> distribution with pointing errors. For the considered system, we derive new closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and average bit error rate (BER) of the end-to-end signal-to-interference-plus-noise ratio (SINR). In addition, efficient numerical expressions for the ergodic capacity are also obtained. Moveover, the asymptotic expressions for the CDF and average BER are derived. Our results can be regarded as a generalization of the previous results. Finally, numerical and simulation results are presented to verify the correctness of the proposed expressions.

relays along intermediate paths. In addition, the congestion of the backhaul networks has raised significant concern in the deployment of mmWave cellular networks [2], [3]. Recently, due to its immunity to interference, license-free spectrum, and high data rate, free space optical (FSO) communication technology has been considered as a promising solution for lowcost backhaul construction [4], [5]. In this perspective, mixed RF/FSO relaying systems represent an effective scenario for wireless networks.
Moreover, in the implementation of commercial FSO systems, the intensity modulation and direct detection (IM/DD) with on/off keying (OOK) has been widely used due to its simplicity and low cost. However, one of the main flaws of OOK is the requirement for adaptive detection threshold to achieve its optimal error rate performance. In contrast, the subcarrier intensity modulation (SIM) avoids the need for the adaptive threshold, so it has broad application prospects [31], [32], [33], [34]. Recently, SIM is widely implemented into the relays to modulate the intensity of the optical carrier.  On the RF side, small-scale fading is mainly considered. The Rayleigh, Rician, and Nakagami-m distributions are used to describe multipath fading scenarios. In fact, in wireless networks, the RF signals are adversely affected by shadowing caused by the user's body [35]. Therefore, a composite fading model that includes multipath fading and shadowing scenarios simultaneously is necessary to be studied. The κ-μ [10], η-μ [10], α-η-κ-μ [36], [37], [38], and Rician shadowed distributions [39] are used to describe the above scenarios. Interestingly, the κ-μ shadowed fading model including the one-side Gaussian, Rayleigh, Nakagami-m, Rician, κ-μ, and Rician shadowed distributions, is a generalized model [40].
In practical scenarios, co-channel interferences (CCIs) due to frequency reuse could severely limit the reception quality, especially on the RF reception [20]. However, in most of the above literatures, the CCIs incorporated in the RF link have been ignored by assuming interference-free relays. Although a handful of models considered CCIs in dual-hop mixed RF/FSO systems, as summaried in Table I, their applications are limited because either they adopt FSO models which are mathematically complex and lead to the difficulty of deriving closed-form expressions, or the RF models may not accurately describe composite multipath/shadowed signal. In particular, when RF and FSO links adopt simple, accurate, and generalized channel models, such as Fisher-Snedecor F, approximate Beckmann, κ-μ shadowed distributions, the effects of CCIs on dual-hop RF/FSO fixed-gain and variable-gain relaying systems have not been studied. Therefore, the effect of CCIs should be taken into account on the design and implementation of such relaying systems.
Motivated by the aforementioned discussions, we investigate the effects of CCIs on dual-hop mixed RF/FSO amplifyand-forward (AF) relaying systems, where the RF link experiences κ-μ shadowed fading and the FSO link is subject to Fisher-Snedecor F fading. We also include the effects of pointing errors in our models with the formalism recently derived [19]. In addition, we study systems of both gain mechanisms in the relay, fixed-gain and variable-gain, and compare the performance by assuming the same average gain. This work provides generalized, closed-form formalisms for analyzing systems of this kind when both CCIs and pointing errors are present.
Our main contributions of this work are as follows: For both fixed-gain and variable-gain relaying scenarios, we derive new closed-form expressions for the cumulative distribution function (CDF) and probability density function (PDF) of the end-toend signal-to interference-plus-noise ratio (SINR). Based on the above expressions, we derive new closed-form expressions for the average BER and provide efficient numerical expressions for the ergodic capacity.
The rest of the paper is organized as follows: In Section II, system and channel models are introduced. In Section III, we derive the PDF and CDF of the end-to-end SINR for both fixedgain and variable-gain relaying scenarios. In Section IV, we investigate the average BER and the ergodic capacity. Section V presents and discusses some numerical and analytical results. Finally, Section VI presents concluding remarks.

II. SYSTEM AND CHANNEL MODELS
We consider a dual-hop mixed RF/FSO AF relaying system with CCIs, as shown in Fig. 1. In the considered system, the source node S communicates with the destination node D via a relay node R, where N sources of RF CCIs exist. In addition, the AF relaying node can convert RF signals into the FSO signal using SIM.

A. RF Channel Model
The received RF signal at the relay node R is expressed as where x is the information signal with an average power E[|x| 2 ] = P s , with E[·] being the expectation operator; h sr is the κ-μ shadowed fading coefficient; x i is the interference signal of the i-th interferer with an average power E[|x i | 2 ] = P i ; h i is the fading coefficient between the i-th interferer and the relay; n sr is the white Gaussian noise with zero-mean and variance of N 0 . The instantaneous SNR in the RF link is γ 1 =γ 1 |h sr | 2 , wherē γ 1 = P s N 0 is the average SNR. The PDF of γ 1 is given by [40] where Γ(·) is the gamma function; 1 F 1 (·; ·; ·) is the confluent hypergeometric function ((9.210.1) in [47]). However, 1 F 1 (·; ·; ·) leads to a very complex mathematical form. To this end, in [48], when μ and m take integer values, the κ-μ shadowed distribution is represented by a finite sum of powers and exponentials, which makes it extremely convenient to calculate. In addition, when the field measurement values are fitted to this distribution, the restriction on integer values for μ and m has little practical impact. The PDF of γ 1 can be rewritten as ( (12) in [48]) Using (3), (8.4.3.1) in [49], and (2.54) in [50], the CDF of γ 1 can be rewritten as where H ·,· ·,· (·) is the Fox's H function [51]. On the other hand, the relay receives N independently identically distributed (i.i.d.) interfering signals. Assume instantaneous interference-to-noise ratio (INR) follows κ-μ shadowed distribution, i.e., γ I,i ∼ (γ I , κ I , μ I , m I ), the to- follows another κμ shadowed distribution with scaled parameters, i.e., γ I ∼ (Nγ I , κ I , Nμ I , Nm I ), with the PDF given by

B. FSO Channel Model
When pointing errors are considered, the FSO link can be modeled by the Fisher-Snedecor F distribution with the PDF and CDF of the instantaneous SNR (γ 2 ) given by [19] and where G ·,· ·,· (·) is the Meijer's G function [47]; a and b represent turbulence parameters; μ r is the average electrical SNR, where r = 1 and r = 2 represent heterodyne detection (HD) and direct detection (DD), respectively. In particular, where ϕ = ξ mod 2 /(ξ mod 2 + 1); ξ mod 2 is associated with the equivalent beam width w Leq , the standard deviations for the horizontal (σ x ) and the elevation (σ y ) axes, and the fixed boresight errors for the horizontal (μ x ) and the elevation (μ y ) axes. The value of ξ mod 2 increases with the decrease of pointing errors, and ξ mod 2 → ∞ when pointing errors are absent.

C. End-to-End Sinr
The end-to-end signal-to-interference plus noise ratio (SINR) at the destination node D can be expressed as [21], [43] where C = 1/(G 2 N 0 ). When both fixed-gain and variable-gain relaying systems are assumed to have the same average gain as in [52], the scaling gain at the relay node R can be for the variable-gain relaying scenario, respectively. The end-to-end SINR for variable-gain relaying can be further simplified as where γ 1 eff = γ 1 /(γ I + 1) .

III. STATISTICAL CHARACTERISTICS
In this section, the PDF and CDF of the end-to-end SINR are derived for both fixed-gain and variable-gain relaying scenarios.

A. Fixed-Gain Relaying 1) Cumulative Distribution Function:
a) Exact Analysis: The CDF of γ of the considered dualhop RF/FSO fixed-gain relaying system can be obtained as (11) shown at the bottom of the next page, where H .,.:.,.:.,.
Proof: See Appendix A. Note that when ξ mod 2 → ∞, (11) is reduced to the CDF expression without pointing errors. b) Asymptotic Analysis: For infiniteγ 1 and μ r , using be obtained as (12) shown at the bottom of this page, where When the interference power is scaled with the transmitted power, the INR tends to infinity as the SNR tends to infinity. Under this assumption, at high SNRs, the presence of CCIs at relay node R results in the occurence of BER floors and capacity ceilings, which reflects the fact that the diversity order is G d = 0.
2) Probability Density Function: By differentiating (11) with respect to γ, the PDF of γ can be expressed as (13) shown at the bottom of this page. Similarly, when ξ mod 2 → ∞, (13) is reduced to the PDF expression without pointing errors.
Proof : See Appendix B.
B. Variable-Gain Relaying 1) Cumulative Distribution Function: a) Exact Analysis: Since the SINR statistic in (10) is mathematically intractable, we approximate it as γ = min(γ 1 eff , γ 2 ) as in [21], [42], [53], which is also the SINR statistics for the DF relaying system. Therefore, the approximate performance curves of the variable-gain AF relaying system are equivalent to those of the DF relaying system. The CDF of γ of the considered dual-hop RF/FSO variable-gain relaying system can be expressed as where where U (a, b, z) is the confluent hypergeometric function of second kind ((9.211.4) in [47]).
Proof : See Appendix C. b) Asymptotic Analysis: Similiarly, for infiniteγ 1 and μ r , the asymptotic CDF can be obtained as (16) shown at the bottom of the next page. From the asymptotic result, we can see that the diversity order is G d = 0.
2) Probability Density Function: By differentiating (14) with respect to γ and using (07.34.20.0001.01) in [54], the PDF of γ can be written as where IV. END-TO-END PERFORMANCE METRICS In this section, the average BER and ergodic capacity are analyzed for both fixed-gain relaying and variable-gain scenarios.
A. Fixed-Gain Relaying a) Exact Analysis: The average BER for different binary modulations is written as [55] where the parameters p and q represent different modulation schemes, as shown in Table I of [9]. Inserting (11) into (19), P e can be derived as (20) shown at the bottom of this page. Similarly, when ξ mod 2 → ∞,P e without pointing errors can be obtained.
Proof : See Appendix D. b) Asymptotic Analysis: At high SNRs, inserting (12) into (19) and using (07.34.21.0012.01) in [54], the asymptotic BER is derived as (21) shown at the bottom of this page. where [17], the ergodic capacity can be computed asC where c = 1 for HD; and c = e/(2π) for DD. Unfortunately, by substituting (11) or (12) into (22), such an integral is extremely complicated. Nevertheless, by performing the change of variables γ = tan θ and using a N p -point Gauss Chebyshev Quadrature (GCQ) rule formula ((25.4.39) in [56]),C can be easily estimated as where the abscissas x n and the weights w n are expressed as (22) and (23) in [57], respectively. It is noted that (23) requires only few terms to get a sufficient approximation accuracy.
Notice that at high SNRs, the interference-limited RF link becomes dominant, soP e asymptotically approaches an identical fixed error floor value for both fixed-gain and variable-gain relaying scenarios. From the asymptotic expressions, the error floor value is approximately equal to first two terms of (25) (first line of (25)).
2) Ergodic Capacity: Utilizing the same method similar to the fixed-gain relaying scenario, the ergodic capacity can be obtained asC

V. NUMERICAL RESULTS
Using the analysis presented in Section IV, numerical results compared with Monte Carlo (MC) simulations are presented for both fixed-gain and variable-gain relaying scenarios. Without loss of generality, we setγ 1 =γ 2 . Unless otherwise stated, the following parameters are considered. For the RF channel, the fading parameters set as follows: κ = 4, μ = 5, m = 8, and N = 2. As described in [41], we also set the interference power to be proportional to the transmitted power (P s /P i = 15 dB). For interference parameters, we set μ I = 1, κ I → 0, and m I → ∞, in which the κ-μ shadowed distribution is reduced to the Rayleigh distribution. For the FSO channel, the turbulence parameters are listed as: moderate turbulence (a = In Fig. 2, we plot the average BER versus average SNR with fixed values of ξ mod = 5.03, κ I = 4, μ I = 5, N = 5 and different values of m I for both fixed-gain and variable-gain relaying scenarios. The analytical results based on (20) and (24) coincide with the MC simulations, indicating the accuracy of our derivation. At high SNRs and under both gain scenarios, it can be noticed that increasing m I improves the BER performance as the interference links become more static with higher values of m I . As expected, the BER performance deteriorates significantlȳ   (21) and (25) match well with the exact results at high SNRs. In Fig. 3, we plot the average BER versus average SNR with fixed values of ξ mod = 5.03, κ I = 0.1, m I = 8, N = 5 and different values of μ I for both fixed-gain and variable-gain relaying scenarios. it can be noticed that the impact of cluster numbers of the interference links on the BER performance depends on the SNR. For example, at large SNR values and under both gain scenarios, less severe fading interference links (i.e., large μ I ) may lead to a better BER performance, otherwise, more severe fading interference links (i.e., small μ I ) may result in a worse BER performance. In addition, at high SNRs for both gain scenarios, we observe that BER floors happen in the presence of CCIs and do not appear in the absence of CCIs. In the operating region of an average SNR between 20 and 30 dB, the slopes of the BER curves for fixed-gain relaying scenario are about 3.42 and 3.61 for μ I = 1 and μ I = 5, respectively; the slopes of the BER curves for variable-gain relaying scenario are about 3.36 and 3.63 for μ I = 1 and μ I = 5, respectively.
In Fig. 4, we plot the average BER versus average SNR for both fixed-gain and variable-gain relaying scenarios with CCIs (P s /P i = 10 dB) under moderate and strong turbulence conditions. It can be seen that increasing the intensity of turbulence results in poor system performance for both gain scenarios. For comparison, the BER performance of the considered system without CCIs is also provided. As expected, the BER performance deteriorates significantly in the presence CCIs compared to no CCIs case. Moreover, at high SNRs for both gain  scenarios, we observe that BER floors happen in the presence of CCIs and do not appear in the absence of CCIs.
In Fig. 5, we plot the average BER versus average SNR for both fixed-gain and variable-gain relaying scenarios with different values of m. We observe that BER performance deteriorates with decreasing value of m. Expectedly, pointing errors reduce the system BER performance for a wide range of SNRs. Moreover, at high SNRs for both gain scenarios, we observe that BER floors happen and the BER performance converges to the same values for the same values of m, meaning that increasing the average SNR does not improve the system performance. This is because that at high SNRs, the interference-limited RF link becomes dominant considering that the interference power is scaled with the transmitted power. In addition, the occurrence of BER floors for large values of m may require a higher average SNR than that for small values of m. In Fig. 6, we plot the average BER versus average SNR for both fixed-gain and variable-gain  relaying scenarios with different values of μ. We also draw some conclusions similar to Fig. 5.
In Fig. 7, we plot the average BER versus average SNR for both fixed-gain and variable-gain relaying scenarios with different values of N . As expected, increasing the number of interferers results in higher BER for both gain scenarios. From this figure, we can oberve that the pointing error degrades the system BER performance for a wide range of SNRs. In addition, at high SNRs, we also observe that BER floors happen for both gain scenarios, and higher average SNR may be required for the occurrence of BER floors for small values of N . This is mainly due to the effect of CCIs.
In Fig. 8, we plot the ergodic capacity versus average SNR for both fixed-gain and variable-gain relaying scenarios with CCIs (P s /P i = 10 dB) under moderate and strong turbulence conditions. It can be seen that increasing the intensity of turbulence results in poor system performance for both gain  scenarios. In addition, for the evaluation of the ergodic capacity in (23) and (26), N p = 60 was used for enough accuracy. From this figure, we can observe that the numerical results match well with the MC simulations. For comparison, we also provide the ergodic capacity performance of the considered system in the absence of CCIs. As expected, the ergodic capacity performance deteriorates significantly in the presence CCIs compared to no CCIs case. Moreover, at high SNRs for both gain scenarios, we observe that capacity ceilings happen in the presence of CCIs and do not appear in the absence of CCIs.
In Fig. 9, we plot the ergodic capacity versus average SNR for both fixed-gain and variable-gain relaying scenarios with different interference power. As expected, with an increase of interference power, the ergodic capacity decreases for both gain scenarios. It can be observed that the pointing error reduces the ergodic capacity for a wide range of SNRs. In addition, at high SNRs, capacity ceilings appear for both gain scenarios, and the occurrence of capacity ceilings for lower interference power may require a higher average SNR than that for larger interference power. Furthermore, as the average SNR increases, the ergodic capacity first increases, then stabilizes and converges to the same values for the same interference power, because the interference-limited RF link dominates the system performance at high SNRs.
In addition, from Figs. 2, 3, 4, 5, 6, 7, 8, and 9, we note that in the presence of CCIs, at high SNRs, BER floors and capacity ceilings happen, and the BER and capacity performance will converge to the same values for the same RF link parameters. This is because that at high SNRs, the interference-limited RF link becomes dominant considering that the interference power is set to be proportional to the transmitted power. This also shows that diversity order of zero is obtained. We also find the variable-gain relaying scenario performs better than the fixed-gain relaying scenario for a wide range of SNRs when the RF link and the FSO link have the same average SNR. This phenomenon can be explained as follows. The relay node R amplifies the RF signal received from the first-hop link in different ways. Specifically, for the variable-gain relaying scenario, the scaling gain is adaptively changed according to the instantaneous channel state information (CSI) of the first-hop link in order to maintain a constant output power; in contrast, for the fixed-gain relaying scenario, the scaling gain is only dependent on the statistical CSI of the first-hop link. Therefore, the variable-gain relaying system always outperforms the fixed-gain relaying system at the expense of the processing complexity at the relay node R.

VI. CONCLUSION
We have investigated the performance of dual-hop mixed RF/FSO relaying systems under both fixed-gain and variablegain relaying scenarios with CCIs at the relay node. The RF and FSO links are modeled by the κ-μ shadowed distribution and the Fisher-Snedecor F distribution with pointing errors, respectively. We derived new closed-form expressions and asymptotic expressions for the CDF and average BER of the considered system. We also presented efficient numerical expressions for the ergodic capacity by using GCQ rule formula. We have verified the analytical results through the MC simulations. These new results allow us to investigate the impacts of number of interferers, interference power, cluster numbers, and shadowing parameter, atmospheric turbulence, pointing errors, and different forward protocols on the considered system's performance. In addition, we found that BER floors and capacity ceilings appear for the interference-limited case and don't appear for the interference-free case at high SNRs. We also found that the variable-gain relaying scenario outperforms fixed-gain relaying scenario for a wide range of SNRs in our system. This work provides closed-form and generalized models for the design and analysis of dual-hop mixed RF/FSO relaying systems when both CCIs and pointing errors are present.

DISCLOSURES
The authors declare no conflicts of interest.

DATA AVAILABILITY
Data underlying the results presented in this paper were generated using the equations herein.
APPENDIX A DERIVATION OF (11) The CDF of γ can be expressed as Inserting (4), (5), and (7) into (A.1), we get (A.2) shown at the bottom of this page.