Full-Duplex DF Relaying with Parallel Hybrid FSO/RF Transmissions

, The ever-increasing demands for higher data rates in telecommunications networks, along with the signiﬁcant beneﬁts offered by wireless communications, have pushed the technological developments to new frontiers. Free-space optical (FSO) communications are among the technologies that can address these emerging demands, offering large and unregulated bandwidth. In addition, recent advances in radio frequency (RF) systems have enabled the development of in-band full-duplex (FD) radios with the employment of advanced self-interference cancellation techniques. Therefore, in this paper we study a robust FD relaying system comprised of parallel hybrid FSO/RF communication links, where the coordination of transmissions between the FSO/RF subsystems is carried out by the hard-switching protocol. The operation of the RF links is impaired by Nakagami- m fading, the residual self-interference due to the FD relaying operation, and the in-phase and quadrature-phase imbalance effect due to imperfect RF front-ends. As far as the FSO links are concerned, we consider the inﬂuence of the joint effects of atmospheric turbulence, beam wander and pointing errors. We ﬁrst derive analytical closed-form expressions for the outage probability of both subsystems as well as for the overall FD relaying hybrid system. Then, asymptotic expressions are extracted for the outage performance in the high signal-to-noise ratio regime. The achievable outage diversity orders of both FSO and RF subsystems are determined, which provide signiﬁcant insight into the design of such hybrid systems in a wide variety of operating conditions. Finally, the numerical results are presented using the derived outcomes, and signiﬁcant conclusions are drawn about FD relaying hybrid systems.


I. INTRODUCTION
W IRELESS communication systems have been at the forefront of many research efforts for over a century. As such, radio frequency (RF) communications have experienced tremendous development over the years. Nowadays, sophisticated RF systems have been developed aiming at high throughput performances with the highest possible spectral efficiency in accordance with the RF spectrum shortage and spectrum fees applied for its licensed frequency bands. Full-Duplex (FD) RF systems constitute a great example of such developments [2]. In-band FD systems can transmit and receive at the same frequency band concurrently, offering an exceptional way of utilizing efficiently the time and frequency resources. However, FD operation can be hindered by unavoidable self-interference (SI) due to signal leakage from the transmit to the receive port. Advanced analog and digital SI cancellation techniques have been developed [3], which are capable of reducing very efficiently the impact of the SI. However, residual SI (RSI) signal power can still hamper the performance of such systems. A very promising extension of FD operation is its incorporation into relaying architectures. Major performance benefits are offered by the utilization of FD relaying systems. Conventional RF relay architectures operate This work was co-funded by the European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation, under the projects EXCELLENCE/0918/0377 (PRIME) and INFRAS-TRUCTURES/1216/0017 (IRIDA). It was also supported by the European Cooperation in Science and Technology under COST Action NEWFOCUS (CA19111).
in half-duplex (HD) mode, i.e. transmit and receive either at different spectral bands or at different time slots (orthogonal channels) [4]. On the contrary, FD relaying provides very high spectral efficiency, where the relay node transmits and receives simultaneously at the same frequency band [4], [5], [6].
In addition, modern RF wireless transceivers incorporate up/down quadrature converters in order to accommodate the utilization of high spectral-efficient modulation schemes such as quadrature amplitude modulation (QAM) [7]. However, imperfections at the RF front-ends are inevitable, thus leading to performance limitations. Amplitude and phase mismatches between the in-phase (I) and quadrature-phase (Q) paths can arise due to imperfections in the local oscillator (LO) signal or any other stage involved in these paths, e.g. a low-pass filter [7], [8]. As a result, an imbalanced down-converted baseband signal can be corrupted by its image, thus leading to interference and signal-to-noise ratio (SNR) deterioration, an effect known as IQ imbalance (IQI) [9], [10].
On the other hand, the RF spectrum scarcity along with significant interference issues, due to the nature of the RF broadcast especially in the unlicensed bands, pose significant impediments for the implementation of reliable and efficient point-to-point RF links. Moreover, RF links experience the detrimental effects of multipath fading [11], [12]. As a viable alternative, free-space optical (FSO) communication systems operate in the unlicensed band of infrared frequencies. They offer larger and unregulated bandwidth, with FD capability, since FSO systems usually comprise one transmit and receive terminal [13]. Major challenges for wide terrestrial applications of FSO systems are their susceptibility to atmospheric effects, where turbulence-induced scintillation, fog attenuation and beam alignment disruptions degrade their performances [14], [15], [16]. As a consequence, FSO systems can reach dis-tances up to some kilometres, by considering the imposed eyesafety constraints on the transmitted optical power. In order to overcome these issues, multi-hop relay architectures, with decode-and-forward (DF) or amplify-and-forward protocols, are studied for their distance coverage extension [17], [18].
Hybrid parallel FSO/RF systems are deemed as a viable and very promising solution in establishing high throughput and very reliable links [19]. The transmission characteristics of RF and FSO systems are complementary to each other [20]. Fog can break down the FSO beam transmission, while RF waves remain unaffected [21]. On the contrary, high rainfall hinders high-frequency RF transmission, while having lesser effect on FSO communication [22], [23]. Thus, the combination of these two technologies can create favourable conditions for setting up high-capacity and high-availability wireless connections [24], [25]. Moreover, parallel hybrid FSO/RF systems with the employment of relay nodes, can offer extended coverage to longer distances [26]. Therefore, the implementation of a hybrid system with an FD RF relaying subsystem and a parallel dual-hop FSO subsystem offers highly efficient spectral use, extremely high availability and an increased wireless range.
Network topologies with parallel hybrid FSO/RF links and intermediate relay nodes have been extensively investigated in the literature so far. In [19], a source-destination hybrid FSO/millimeter wave RF link is discussed, wherein the hardswitching scheme is employed, by using single and dual FSO threshold implementations in Log-normal and Nakagami-m fading channels. In [25], the authors study a hybrid FSO/RF system with hard-switching operation and evaluate its performance in terms of the outage probability and bit error rate under the influence of Gamma-Gamma (GG) turbulence, pointing errors and Rician fading. In [26], the authors investigate optimal relay selection policies for the parallel hybrid RF/FSO relay channel with buffer-aided and non-buffer-aided relays. In [22], a switching-based cooperative DF relaying network with hybrid FSO/RF links and maximal ratio combining at the destination is investigated. Also, a DF relaying hybrid FSO/RF millimeter waves with selection combining is presented in [27], considering M(alaga) turbulence with pointing errors and Weibull fading. In [24], the authors study a backhaul network with relay nodes connected with parallel hybrid FSO/RF links dealing with the problem of minimizing its cost while satisfying data rate and reliability constraints. In [28], they examine a wireless mesh network with parallel hybrid FSO/RF links between the relay nodes, with transmission power and optical beam-width adaptations in order to meet specified quality of service requirements such as throughput and end-toend delay constraints. However, all the aforementioned works either do not discuss about the relaying operation or take into account HD relaying operation for the RF links. Motivated by the foregoing facts, in this paper for the first time, we examine the combination of an FD RF relaying subsystem with an FSO subsystem in a parallel hybrid relay topology.
More specifically, we investigate an FD DF relaying network, which consists of a source S, an FD DF relay R node and a destination D. The connection among the nodes is established with parallel hybrid FSO/RF links. For the routing of the information between the S − R and R − D links and the proper selection of the suitable subsystem for data transmission among the nodes, we consider the lowcomplexity hard-switching scheme [19], [21]. To the best of the authors knowledge, an FD DF relaying system with parallel hybrid FSO/RF links and hard-switching operation under the influence of Nakagami-m fading, RSI, IQI, atmospheric turbulence and pointing errors has not been investigated in the literature so far. Hence, we summarize the main contributions of this paper as follows.
• A detailed analysis of the system model is presented for an FD relaying setup with DF protocol consisted of parallel hybrid FSO/RF links. The considered dual-hop FD DF hybrid system operates under the hard-switching scheme. We take into account the Nakagami-m fading, the RSI and IQI for the FD RF relaying subsystem, while for the FSO subsystem we consider the joint effects of GG turbulence, beam wander and pointing errors. • We develop a novel channel model for the FSO links, where GG turbulence, beam wander and pointing errors are treated as three independent random variables. Analytical, closed-form expressions for the outage probability of the FSO and RF subsystems, as well as for the overall FD relaying hybrid system, are developed. • Asymptotic expressions for the outage probability in the high signal-to-noise ratio (SNR) regime are derived for both subsystems. The outage diversity order of both FSO and RF subsystems is determined. In this way, we gain valuable insight into the design of such systems and we highlight the significance of the treatment of the FSO channel model as a product of three independent random variables. • A variety of numerical results for the outage probability of the overall hybrid system is presented, taking into account the various effects that influence the FSO and RF subsystems. For the FD RF relaying subsystem, the effects of the RSI, the IQI and the generalized Nakagamim fading are taken into account. Concerning the FSO subsystem, we investigate its performance under a variety of turbulence conditions with pointing errors, where the beam wander manifests itself at certain conditions. It is shown that proper beam parameter selection is essential under certain turbulence conditions. Eventually, all the derived results and the whole system model analysis, are verified by Monte Carlo simulations.
The rest of this paper is organized as follows. Section II describes the system model for the RF subsystem as well as the FSO subsystem. In Section III, we derive closed-form expressions for the outage probability of the RF and FSO subsystems and for the whole hybrid dual-hop system. In addition, asymptotic, tractable expressions in the high SNR regime are provided for both subsystems. Numerical and simulation results are presented in Section IV and the paper concludes with Section V. Notation: P{X} and E{X} represent the probability and the expectation of X, respectively; (·) * is the complex conjugate; Γ(·), γ (·, ·) , and Γ (·, ·) denote the complete, lower incom-plete, and upper incomplete gamma function, respectively, K v (·) is the modified Bessel function of the second kind and order v, G m,n p,q (z | a p ; b q ) denotes the Meijer's G-function, and Φ (·, ·; z), q F p (a q ; b p ; z) represent the confluent and the generalized hypergeometric functions, respectively [29].

II. SYSTEM MODEL A. Topology
In this section, we elaborate on the system model for the individual FSO and RF subsystems, combined in a dual-hop FD hybrid set-up, as depicted in Figure 1. The system consists of a source S, an FD relay R node, employing the DF protocol, and a destination D. The source S transmits the information symbols either by using the FSO/RF subsystem according to the hard-switching scheme, with priority given to the larger bandwidth FSO link [19]. The hard-switching scheme is based upon the selection combining criterion [30], i.e. the activation of a subsystem is carried out with the selection of the highest SNR among them. Some notable algorithms for the implementation of the hard-switching protocol exist, which are the power hysteresis, the time hysteresis, the filtering method or a combination between them [21]. Besides the simplicity of the implementation of the hard-switching scheme, it requires one of the subsystems to be active at a time, compared to other combining approaches such as maximum ratio combining (MRC) or equal gain combining (EGC) [30]. In addition, an MRC or EGC would require both subsystems to be active and operating with the same data rate. Therefore, even if the FSO or RF subsystem is favourable for operation, it must be adapted to the performance capabilities of the other subsystem. Thus, an MRC or EGC can increase the reliability of the hybrid system, by concurrently exploiting two diversity branches, but diminishes the data throughput performance. Hence, we select the hard-switching scheme.
Thus, at a specific time slot, the subsystem of the S −R link with the highest SNR is selected to transmit the information towards R, while the other remains idle. At R, the received signal is decoded and forwarded towards D. In case where the S −R RF link is chosen for transmission, the detection process at R is influenced by the RSI due to the FD relaying operation and the IQI-impaired transmitted and received signal. On the contrary, if the S−R FSO link is active, the transmitted optical signal is corrupted by the atmospheric and misalignment effects, but is not affected by any type of interference. Note that the optical and the RF links do not interfere with each other, since they operate in completely different spectral bands. Also, for the RF subsystem, a direct line-of-sight S − D link does not exist or any scattered signal is strongly attenuated and communication between them is established only with the aid of the FD relaying scheme [5], [26]. Consequently, the retrieved data at R is transmitted to D through the chosen FSO/RF link based again on the hard-switching operation. In case the R − D RF link is activated for transmission, the RF transmitter at R and the destination RF receiver are impaired by the IQI effect. On the other hand, if the FSO R − D link is selected for transmission, the optical signal experiences again the degradation atmospheric effects and reaches D, where is detected and decoded. In the following subsections, the mathematical representation models for both RF and FSO subsystems are presented.

B. RF Subsystem
For the RF links of the FD relaying RF subsystem, we employ the standard additive white Gaussian noise (AWGN) model. In addition, we take into account the effect of IQI at the transmitters (TXs) and receivers (RXs) of the RF nodes. The IQ mismatches constitute an inevitable effect in any RF quadrature up/down-converter, which arises primarily due to imperfections in the LO signal and secondly from any other transceiver stage. So, when we assume IQ mismatches, the IQI baseband signal is formulated as with x X (t) being the desired signal and (·) * denotes the image signal. The coefficients K respectively, with the superscripts u/d signifying the up/downconversion processes and the subscript X ∈ (S, R, D). The parameters denote the amplitude and phase mismatches of the LO signals at the transceivers of each node [9], [10]. For instance, the mathematical representation of the IQI LO signal, used for downconversion, is described as where ω LO = 2πf RF with f RF being the RF carrier frequency. Ideally, only the first term would have been present in the LO signal. The existence of the second term can lead to interference from the image signal after the downconversion process, which in turn can deteriorate the signal-tointerference-plus-noise ratio (SINR) of the desired signal [8], [9]. None of the existing image-reject receiver architectures can deal completely with the IQ mismatches arising in the quadrature converters [8]. The severity of IQ mismatches is quantified by the image rejection ratio (IRR), defined as , whose values, for typical frontend integrated circuits, ranges from 20 to 40 dB [8], [9], [10].
The source S, provided that the RF link is active, transmits an IQI-impaired symbol x u S,IQI (t) with an average power E{|x S (t)| 2 } = P t . At the relay node R, considering frontend imperfections with IQI, the received signal is [31] where x u R,IQI (t) is the IQI-impaired transmitted signal from R again with an average transmitted power E{|x R (t)| 2 } = P t . The channel coefficient h SR is equal to h SR = ĥ SRhSR , where the individual terms are described as [26], [11], [12]  with k ∈ (SR, RD), where c = 3 × 10 8 m/s is the speed of light, G T X , G RX are the transceiver antenna gains, L k is the link distance. It is worthy to note that, in this work, we consider frequencies below 10 GHz, for which the rainfall does not have a significant impact 1 . Moreover, we assume the Nakagami-m model which has been proved a very accurate model for multipath propagation, encompassing as special cases the one-sided Gaussian (m = 1/2), Rayleigh (m = 1) and Rice distributions [11]. The channel coefficient h RR and the term n R (t) correspond to the RSI and the AWGN at R, respectively, which are circularly symmetric complex Gaussian random variables with zero mean and variance σ 2 RR and σ 2 n , respectively, i.e. h RR ∼ CN (0, σ 2 RR ) and n R (t) ∼ CN (0, σ 2 n ). For simplification reasons, we introduce the notations The received signal at D, considering IQI at the relay TX and the destination RX, is with h RD = ĥ RDhRD described similarly as in (4) and n D (t) ∼ CN (0, σ 2 n ). Next, taking into account the aforementioned analysis, we proceed with the evaluation of the SINRs for the S − R and R − D RF links, i.e. γ RF,SR and γ RF,RD , respectively. More precisely, for the S − R RF link, considering the RSI and IQI impairments, γ RF,SR is formulated as where S 1 , I 1 and I 2 are equal to , and For scenarios with a perfect matching between the IQ branches, the IQI effect becomes negligible, i.e. K u/d 1,X = 1 and K u/d 2,X = 0. As a result, the terms I 1 , I 2 become I 1 = |h RR | 2 , I 2 = 0 and hence, (6) simplifies to With regards to the R − D RF link, the transmitted IQIimpaired symbol from R reaches D, having propagated through the Nakagami-m distributed fading channel. The demodulation stage at the RX of D is also influenced by the IQI effect. Hence, for the case of TX/RX IQI-impaired nodes, the SINR of the R − D RF link is [10, Eq. (33)] where ,R ) * , and γ id,RD = P t |h RD | 2 /σ 2 n denoting the ideal SNR in the absence of IQI. When the IQI is negligible, we obtain ϑ 11 = 1, ϑ 12 = 0, ϑ 21 = 0, and ϑ 22 = 0, respectively, and as a result, (8) simplifies to γ RF,RD = γ id,RD .

C. FSO Subsystem
In case that the FSO link is active, S activates the optical terminal and sends the information signal towards R. Intensity modulation with direct detection (IM/DD) mode [32] is employed and the AWGN model is considered for all the sources of noise entailed in the FSO links, e.g. shot noise, thermal noise etc. The optical signal is emitted by the source S and propagates through the atmospheric channel, wherein experiences all the aggravating effects related to the atmospheric medium, such as atmospheric turbulence, beam wander and pointing errors [14], [16], [15]. After the detection of the optical signal at R, the signal is re-encoded and transmitted from R to D by using the FSO/RF link, as it has already been described. The S − R and R − D FSO links can be characterized as identical since no interference issues take place at R due to the narrow optical beams employed [26]. Thus, the received signal at the input of the relay R or the destination D photodetector (PD) is described as [33] where s(t) is the transmitted information symbol from each node with an average transmitted optical power E{|s(t)| 2 } = P 0 , I k , k ∈ (SR, RD) is the total real-valued instantaneous channel coefficient of each hop, and n opt corresponds to the AWGN of the optical link n opt ∼ N (0, σ 2 n,opt ). The total channel coefficient, I k , can be represented as a product of three independent random variables, i.e. I k = I t,k I b,k I p,k , where I t,k corresponds to the atmospheric turbulence effect, I b,k to the beam wander, and I p,k to the pointing errors and geometrical loss. The instantaneous electrical SNR at the PD receiver output of each individual k-th FSO link is [32] and the average electrical SNR is defined as [32] with ρ k being the PD responsivity in A/W . Due to the independence between the channel coefficients, we obtain It is well-known that the probability density function (PDF) of the GG distribution accurately models the irradiance fluctuations ranging from weak to strong turbulence conditions. Its PDF is given as [34], [35] The term I l,k represents the deterministic attenuation of the optical signal due to scattering and absorption and is calculated by the Beer-Lambert law [36]. The expected value of I t,k is equal to E{I t,k } = I l,k [34]. The parameters a k , b k of the GG distribution, for plane wave propagation, are given as [37] where σ 2 R,k is the Rytov variance, calculated as σ 2 R,k = 1.23C 2 n,k κ 7/6 L 11/6 k with C 2 n,k denoting the refractive index structure parameter, which usually takes values in the range 10 −17 − 10 −13 m −2/3 for weak up to strong turbulence conditions. The parameter d k is defined as d k = 0.5D R κL −1 k with D R denoting the diameter of each PD aperture of the dual-hop FSO subsystem, L k being the link distance of the S − R, R − D links and κ = 2π/λ being the optical wavenumber.
The PDF of the pointing errors effect, due to terminal movements, is given as [15] with the parameters ξ k , A 0,k being defined as ξ k = W z,eq /σ s,k , A 0,k = [erf(υ)] 2 , with σ s,k being the standard deviation of the beam's radial displacement, υ k = √ 2πD R /4W z,k and erf(·) is the error function. The equivalent beam radius is calculated as W z,eq,k = √ πerf(υ k )W 2 z,k /2υ k exp −υ 2 k and is linked with the beam radius W z,k on the RX plane at distance L k [15]. Its value is calculated as W z,k = W k 1 + 1.63σ [14]. The parameter W 0 corresponds to the beam radius at each TX of the FSO subsystem and F 0 is the phase front radius of curvature, which for the case of a collimated beam F 0 → ∞ [14]. The expected value of I p,k is Additional irradiance fluctuations can be inflicted by the beam wander-induced jitter due to large-scale atmospheric turbulence [38]. The authors in [39] propose a beta distribution in order to characterize the effect of beam motion due to atmospheric turbulence, which is given as where the parameter β w,k is evaluated as [40] β w,k = 1 + 1 + σ 2 and is linked with the scintillation index σ 2 [35]. The parameter σ 2 pe,k refers to the jitter-induced pointing error variance due to beam wander and for the case of a collimated beam is calculated as [38,Eq. (14)], with C r being a scaling constant selected equal to 1.5π for a horizontal propagation path [16]. The parameter r 0,k = 0.16C 2 n,k κ 2 L k −3/5 is the Fried parameter and the expected value of I b,k is

III. OUTAGE PROBABILITY ANALYSIS
In this section, we investigate the outage probability of the individual RF and FSO subsystems and for the FD DF relaying hybrid FSO/RF system. Firstly, we derive the outage probability of the RF subsystem for the S −R and R−D links and then for the corresponding FSO links. Lastly, we combine the derived outcomes to obtain the outage probability of the FD relaying hybrid system with parallel FSO/RF links.

A. RF Subsystem Outage Probability
On condition that the RF link is active, an outage occurs for the S − R link when the instantaneous γ RF,SR falls below a predefined threshold γ th,RF = 2 R th,RF − 1 with R th being the threshold achievable rate in bits per channel use (BPCU).
Considering that γ RF,SR of (6) depends on the Nakagami-m fading and the RSI, we formulate the following theorem for the S − R outage probability estimation.
Theorem 1. The outage probability of the S − R RF link, taking into account that γ RF,SR is a function of the channel gains |h SR | 2 and |h RR | 2 , is given by where γ th,RF = 2 R th,RF − 1 and the parameters A 1 , B 1 and B 2 are shown in Table I.

Proof. See Appendix A.
The derived, exact expression of (17), for the S − R outage probability, is quite cumbersome and does not provide direct insight into the S − R RF subsystem behaviour. In order to get a more simplified expression, we derive an asymptotic approximation in the following corollary, when Pt σ 2 n → ∞. Corollary 1. When the ratio Pt σ 2 n → ∞, the outage probability for the S − R RF link is asymptotically approximated as Proof. See Appendix B.
From the tractable approximation of (18), we observe that the S − R outage performance converges to an outage floor, dependent on the Nakagami parameters (m SR , Ω SR ), the RSI, the IQ mismatches. Thus, the S − R outage diversity order is zero. Likewise, for the R − D RF link outage probability evaluation, the following theorem is formulated.
Theorem 2. The outage probability for the R − D RF link is Proof. See Appendix C.
In addition, for the sake of completeness, we provide a tractable high SNR regime-based asymptotic approximation for the outage probability of the R − D RF link as follows.
Corollary 2. When the ratio Pt σ 2 n → ∞, the outage probability for the R − D RF link is given by It is observed that the R − D outage diversity order of the RF link is m RD . However, the outage behaviour of the total FD DF relaying RF subsystem at the high Pt σ 2 n regime is determined by the minimum diversity order and thus we conclude to the following remark.
Remark 1. The outage diversity order of the RF FD DF relaying subsystem is determined by the minimum diversity order between the S − R and R − D RF links. Therefore, the outage diversity order of the RF subsystem is zero (error floor).

B. FSO Subsystem Outage Probability
As it is already stated in Section II-C, since no interference occurs at R or from any other ambient source, the FSO subsystem consists of identical S − R, R − D links. Thus, the outage probability of the individual FSO links can be evaluated by having the cumulative distribution function (CDF) F γ F SO,k (x) of the instantaneous γ F SO,k . Since the total channel coefficient I k is a product of three independent random variables, i.e. I k = I t,k I b,k I p,k , we conclude to the following theorem for the outage probability estimation of the individual S − R and R − D FSO links.
Theorem 3. The outage probability for each individual FSO link, incorporating the GG distribution, the pointing errors effect, and the beam wander-induced jitter, is where γ th,F SO = 2 R th,F SO − 1.
Proof. See Appendix D.
In addition to the closed-form expression of (21), we provide an asymptotic analysis, which is valid when γ F SO,k → ∞. In the following corollary, we define the outage diversity order of the considered FSO system model and valuable engineering insight into the FSO subsystem behaviour at high γ F SO,k is obtained.

Corollary 3. The outage diversity order of the FSO link impaired by GG atmospheric turbulence, beam wander, and pointing errors is evaluated as
Proof. See Appendix E.

C. Hybrid Dual-Hop FSO/RF Outage Probability
The overall outage probability of the dual-hop hybrid system can be written as [41] P out,tot = P out,SR + (1 − P out,SR ) P out,RD , where P out,SR , denotes the outage probability of the hybrid FSO/RF S − R link and P out,RD denotes the corresponding outage probability of the R − D link. Based on the hardswitching operation, the outage probability of each individual hop, is [19], [30] P out,k = P out,RF,k P out,F SO,k Thus, based on (24) we can deduce that the outage probability of the FD DF relaying hybrid system composed of parallel hybrid FSO/RF links, under hard-switching operation, is P out,tot =P out,RF,SR P out,F SO,SR + (1 − P out,RF,SR P out,F SO,SR ) × P out,RF,RD P out,F SO,RD , where P out,RF,SR is replaced by the expressions either of Theorem 1 or Corollary 1 for the exact or asymptotic case, P out,RF,RD from the expressions of Theorem 2 or Corollary 2, while P out,F SO,SR and P out,F SO,RD are given by the Theorem 3 or (42) for the exact or the asymptotic case, respectively.

IV. NUMERICAL RESULTS
In this section, we present the numerical results based on the aforementioned analysis and derivations. For the illustrated results, we assume different conditions among the S − R and R−D links for both FSO and RF subsystems, characterized by the following relations for the corresponding fading conditions m RD = m SR +3, Ω RD = Ω SR +5 dB and C 2 n,RD = C 2 n,SR /3. In addition, all the presented results are accompanied by Monte Carlo simulations using 10 6 realizations, which corroborate the analysis conducted in the previous sections. Unless otherwise stated, the values used for the various parameters, of the FSO and RF subsystems, are contained in Table II. Figure 2 illustrates the outage probability results for the FD DF relaying hybrid system as a function of the mean value of the RSI effect λ RR . We take into account weak and strong IQI impact on the RF subsystem front-ends at all the three nodes (S, R, D). For the RF links, we assume a moderate multipath fading environment, while for the FSO links we assume two cases of operation with two values for γ F SO,k under strong turbulence conditions. From the derived plots, we deduce the performance improvement of the outage probability P out,tot of the hybrid system. Compared to the cases when the dual-hop system is exclusively composed of FSO or RF links, P out,tot reaches acceptable values below 10 −3 . Specifically, when γ F SO,k = 20 dB, P out,tot reaches values below 10 −3 for a range of values for λ RR between −26 to −22 or −24 dB, depending on the IQI level. Consequently, when γ F SO,k = 30 dB, a significant performance improvement is observed. Note that P out,tot lies on the order of 10 −7 for λ RR = −26 dB and reaches values up to 10 −4 when λ RR = −6 dB. Furthermore, we also observe the impact of the IQI on the outage performance. As it is shown, the influence of the IQI becomes evident when the RSI is kept at minimum levels.
Otherwise, for λ RR ≥ −12 dB, we cannot observe significant difference due to the IQI, and the performance is determined mostly by the RSI, the multipath fading, the atmospheric turbulence and γ F SO,k . This clearly indicates the severity of the RSI and the importance to keep it at the minimum possible levels. Eventually, we can infer that the FSO links can counterbalance any performance degradation inflicted by the FD relaying operation of the RF links in case that high γ F SO,k can be attained. Next, in Figure 3, we present the outage probability performance for the dual-hop FD hybrid system as a function of the ratio Pt σ 2 n , along with the asymptotic ones in the high Pt σ 2 n regime. The IQ mismatches and the RSI are taken into account, with weak and strong influence. As far as the wireless RF channels are concerned, two different conditions are employed, corresponding to strong and weak multipath = 40 dB, P out,tot is on the order of 9×10 −5 with strong IQ mismatches, whereas for weak IQI is at 5 × 10 −4 . In addition, we can clearly notice the impact of multipath fading on the outage performance. For the case where (m SR , Ω SR ) = (3, 7 dB) and λ RR = −26 dB, we observe that the outage probability increases to 2 × 10 −3 or 5 × 10 −3 for weak or strong IQ mismatches, respectively. On the other hand, we observe again the detrimental impact of the RSI on the FD relaying hybrid FSO/RF system. For λ RR = −16 dB, the outage probability is on the order of 10 −2 , irrespective of the influence of multipath fading, RSI and IQI. Interestingly, the results indicate that the performance aggravation due to IQI becomes more evident when weak multipath fading and weak RSI are considered. Moreover, the availability of the FD relaying hybrid system under strong RSI, can be increased with higher γ F SO,k values for the FSO subsystem i.e. γ F SO,k > 20 dB and better RF wireless propagation conditions. Regarding the asymptotic results, we notice the tightness that provide in the high Pt σ 2 n regime compared to the exact expressions, while corroborating the remark that the outage performance reaches an error floor.
In Figure 4, outage probability results are shown versus the γ F SO,k along with the asymptotic ones in the high γ F SO,k regime. For the FSO links, we consider two cases for the turbulence strength with moderate and very strong turbulence conditions. Moreover, we select two values for W 0 , corresponding to narrow and wide beamwidths. We can observe that as γ F SO,k tends to values higher than 30 dB, the outage probability of the dual-hop FD hybrid link is improved essentially with attained values at 10 −4 and smaller. In addition, we must point out the differences that arise due to the selection of different beam radius sizes W 0 at the transmitters of each node. Specifically, for the moderate turbulence case, at high γ F SO,k values, we can observe the γ F SO,k gain achieved with the employment of W 0 = 6 cm. On the order of 10 −9 for P out,tot , the γ F SO,k gain is roughly 8 dB compared to the case with W 0 = 3 cm. On the other hand, when very strong turbulence conditions are considered, the corresponding achievable γ F SO,k gain is not so high.
In conclusion, it is revealed that the beam parameters can play a significant role under certain turbulence conditions, where beam wander-induced jitter can influence the FSO link performance. As far as the asymptotic results are concerned, we observe the perfect matching that offer compared to the exact results, especially when γ F SO,k > 40 dB.
Capitalizing on the above-mentioned observations for the FSO links, in Table III, we present a variety of results for the outage diversity order under various turbulence conditions. In this table we include the various turbulence conditions characterized by the C 2 n,k parameter, the dominant terms from the asymptotic expression of (42) and the exact outage probability from (21). Next, the percentage deviation of the dominant term value from the exact outage probability is evaluated and the corresponding values of the outage diversity orders are presented for the various conditions. For these results, we consider R th,F SO = 5 BPCU, while the rest parameters are drawn from Table II, along with γ F SO,k = 60 dB. It is noticed from Corollary 3, that the outage diversity order, for the considered FSO links, interchanges with respect to the turbulence conditions. Specifically, under weak and moderate atmospheric turbulence, the O d is determined by the parameter of the beam wander-induced jitter β w,k . On the other hand, for strong and very strong turbulence conditions, the O d is specified by the large-scale turbulence parameter a k of the GG distribution. The derived results verify the theory, which dictates that under weak turbulence conditions the beam wander can be quite evident, whereas in strong turbulence conditions beam wander is less severe and the beam is divided into a multitude of smaller spots. Therefore, we can deduce that from weak to moderate turbulence conditions, the beam wander effect must be taken seriously into consideration, with proper beam parameter selection when high throughput systems are designed, i.e. in the high γ F SO,k regime. In strong turbulence conditions, refractive and diffractive phenomena contribute essentially to the turbulence-induced scintillations.  As a consequence, proper link design must be carried out, taking into account the link distance and the beam parameters in order to overcome the atmospherically-induced effects and achieve high performance FSO links. It is worth pointing out that the O d can be determined also by the ξ k parameter of pointing errors, particularly when sway of the FSO terminals becomes prominent, a case investigated in Figure 7. In Figure 5, the outage probability performance is depicted as a function of the Nakagami fading parameter m SR . For the illustrated results, we assume strong turbulence conditions for the FSO links, with two values for γ F SO,k . For the FD RF relaying subsystem we assume two cases for the RSI, corresponding to weak and strong influence of the effect. From the derived plots, we observe the dependence of the outage performance of the FD relaying hybrid system on the γ F SO,k . For the higher γ F SO,k value, we observe a performance improvement of three orders of magnitude for all the illustrated cases. Concerning the RSI impact, we see again its detrimental impact on the FD relaying hybrid system. Specifically, when γ F SO,k = 20 dB and λ RR = −16 dB, the outage performance of the hybrid system is on the order of 10 −2 , irrespective of the IQI level. Lastly, we can clearly notice the impact of the IQI effect on the hybrid system's outage performance, when the RSI is weak and at high values of the Nakagami parameter, i.e. when multipath fading vanishes. Figure 6 shows the outage probability versus C 2 n,SR . We consider two cases of operation for the FSO links, characterized by the γ F SO,k . For the FD RF relaying subsystem, we consider strong multipath fading and IQI. We observe that in the weak turbulence regime, the outage performance, for the case of γ F SO,k = 20 dB, lies in acceptable levels. When C 2 n,SR ≥ 6 × 10 −15 m −2/3 (i.e. from moderate to strong turbulence), the outage probability exceeds the limit of 10 −3 for both cases of strong and weak RSI. On the contrary, when γ F SO,k = 30 dB, the outage performance remains at acceptable levels from weak to very strong turbulence conditions even when λ RR = −10 dB.
Eventually, Figure 7 depicts the outage probability versus the σ s,k parameter of the pointing errors. For the FSO links, we consider fixed γ F SO,k = 30 dB and two cases for the beam radius and the turbulence strength as well. For the FD RF relaying subsystem we assume two cases for the RF propagation environment, with strong and weak multipath fading, along with IQI influence. As it is shown, the beam size plays a major role in the overall performance of the hybrid system. At a first glance, we observe an unexpected phenomenon, where for the case of moderate turbulence conditions and narrow beam size we obtain the worst case performance, compared to the corresponding ones with strong turbulence, especially when σ s,k > 2 cm. This is mainly attributed to optical turbulence; the beam spreading due to turbulence acts in a beneficial way, compensating for the beam displacements due to pointing errors. On the other hand, in the moderate regime, the beam spreading is smaller and thus the irradiance fluctuations become immense as the spatial jitter increases, having a significant effect on the outage performance of the hybrid system. Concerning the RF wireless environment, we can see the performance improvement between the two considered cases, under both weak and strong atmospheric turbulence conditions. In general, under moderate turbulence, P out,tot achieves values below the limit of 10 −3 across the range of σ s,k ≤ 3.5 cm, whereas for the strong turbulence case, a spatial jitter of σ s,k ≤ 8 cm is tolerable.

V. CONCLUSION
In this work, we investigate an FD DF relaying communication system with parallel hybrid FSO/RF links. The FD RF relaying subsystem is impaired by the RSI and the IQI effect at the front-ends of each node and the FSO subsystem operates under the joint influence of GG atmospheric turbulence, beam wander and pointing errors. Under these assumptions, the outage performance of the dual-hop FD DF hybrid FSO/RF system is analyzed. Novel, tractable closed-form expressions for the outage probability are derived along with their asymptotic approximations in the high SNR regime. The asymptotic approximation results unveiled the vital characteristics of operation of the hybrid communication system, and significant insight into the design of FD DF relaying systems with parallel hybrid FSO/RF links is provided. Specifically, we showed that for weak turbulence conditions, the beam wander affects the FSO link performance and proper selection of beam parameters can counterbalance its impact on the FD relaying hybrid system. On the contrary, we observe that in strong turbulence conditions the beam parameter selection does not play a significant role and the performance of the hybrid system is degraded essentially under such conditions. Moreover, for the FD RF relaying subsystem we demonstrate that its outage probability performance reaches an outage floor, which is strongly dependent on the RSI, the Nakagami parameters and the IQ mismatch. We conclude that major performance limitations are primarily caused by the atmospheric turbulence, Nakagami-m fading, and RSI. The impact of the IQI becomes evident in scenarios of weak RSI and finally, the pointing errors influence the performance when the spatial jitter increases substantially. APPENDIX A PROOF OF THEOREM 1 Firstly, we introduce some approximations for (6) of γ RF,SR . Knowing that the inequality |α| 2 + |β| 2 2R(αβ * ) holds, i.e. |α + β| 2 ≈ |α| 2 + |β| 2 , we obtain and where the parameters A 1 , B 1 and B 2 are shown in Table I. Consequently, we evaluate the conditional probability P γ RF,SR ≤ γ th |h SR | 2 , given that |h SR | 2 is known.
is obtained by replacing the γ RF,SR from (6), as depicted in (26). Similarly, due to the fact that h SR ∼ Nakagami (m SR , Ω SR ), |h SR | 2 is gamma distributed as Thus, the outage probability of the S − R RF link is calculated by using the integral formulae of (27), concluding to the corresponding closed-form expression.

APPENDIX B PROOF OF COROLLARY 1
For the asymptotic approximation of (17), when Pt and we obtain (18).
By replacing the argument of (31) into (30), we obtain the closed-form expression of P out,RF,RD as presented in (19).

APPENDIX D PROOF OF THEOREM 3
In [43], the Mellin Transform (MT) is utilized to study the distribution of products of independent random variables. So, for the case of the channel state of FSO links, which is a product of three independent random variables, I k = I t,k I b,k I p,k , we apply the following multiplicative property where M (f x (x) | s) = The MT for the distribution of the pointing errors, given in (14), is calculated as while from (15), the MT for the PDF of the beam wander effect, which is a beta distribution, is [43] M f I b,k (I b,k ) | s = β w,k Γ (s + β w,k − 1) Γ (s + β w,k ) .
By the use of the multiplicative property of (32), we obtain Taking the inverse MT of M (f I k (I k ) | s) as f I k (I k ) = 1 2πj c+j∞ c−j∞ M (f I k (I k ) | s) I −s k ds, we obtain the PDF of the total channel coefficient I k as where using [44,Eq. (6.422.19)], we derive Accordingly, by applying [44, Eq. (9.31.2)], we obtain f I k (I k ) = β w,k ξ 2 k a k b k A 0,k I l,k Γ (a k ) Γ (b k ) × G 4,0 2,4 a k b k I k A 0,k I l,k β w,k , ξ 2 The corresponding CDF is derived as [45,Eq. (26)] Combining (10) and (11), we can express the instantaneous γ F SO,k as γ F SO,k = γ F SO,k (E{I k }) 2 I 2 k . The CDF of γ F SO,k is derived following a simple random variable transformation, concluding to a k b k E{I k } A 0,k I l,k x γ F SO,k 1, β w,k + 1, ξ 2 k + 1 ξ 2 k , β w,k , a k , b k , 0 .
Thus, the outage probability of the FSO links can be calculated as presented in Theorem 3.

APPENDIX E PROOF OF COROLLARY 3
In order to derive an asymptotic approximation of (21), we utilize the expansion formula for the Meijer's Gfunction in terms of the generalized hypergeometric function q F p (a q ; b p ; z) as given in [44,Eq. (9.303)], where z = a k b k E{I k } A 0,k I l,k x γ F SO,k . On condition that γ F SO,k → ∞, i.e. z → 0, then q F p (a q ; b p ; z) → 1 and (21) is approximated as depicted in (42), where asymptotically the outage probability behaves as P out,F SO,k ≈ O c γ F SO,k −O d , with O c and O d denoting the coding gain and the outage diversity order, respectively [5]. Thus, we observe that the outage diversity order varies between four parameters which are a k 2 , b k 2 , β w,k 2 and ξ 2 k 2 , concluding to Corollary 3.