1. Introduction

Relay communication in wireless networks offers many benefits, including reliability enhancement and coverage extension. To meet data rate and bandwidth requirements, dual-hop mixed radio frequency (RF)/free space optical (FSO) relaying systems were proposed [1,2]. This kind of systems achieve inherent complementary benefits of both RF and FSO systems. Therefore, the study of dual-hop mixed RF/FSO relaying systems has been the focus of many works [1–4].

In dual-hop mixed RF/FSO relaying systems, a single relay can assist the communication between a source and a destination [1–14]. In the FSO hop, atmospheric turbulence and pointing errors are two major factors which need to be considered. To characterize the atmospheric turbulence, the range of applications and mathematical tractability of the models are often considered. In this end, gamma-gamma (GG) distribution is widely used for moderate-to-strong turbulence conditions [15], whereas $\mathcal {M}$ model is applicable for all irradiance conditions in homogeneous and isotropic turbulence [16]. Recently, the Fisher-Snedecor $\mathcal {F}$ distribution was introduced due to its mathematical tractability [17]. On the other hand, to characterize the pointing errors, the Beckmann distribution approximated by the modified Rayleigh distribution is widely used in the literature due to generalizability and mathematical tractability of the model [18,19]. In addition, two main detection modes, i.e., heterodyne detection (HD) and direct detection (DD), are often studied in FSO systems.

In the RF hop, popular fading models include the Rayleigh, Rician and Nakagami-m distributions. The $\kappa$-$\mu$ shadowed distribution is more general and can accurately describe the signal propagation for a line-of-sight (LOS) propagation scenario in the presence of shadowing [20]. In addition, due to aggressive frequency reuse, the system performance will be adversely affected by co-channel interferences (CCIs) [21–25]. These works studied the effects of CCIs on the performance of dual-hop relay networks, but they assumed that the network under consideration consists of a single relay.

On the other hand, multiple relays can improve the system performance through relay selection techniques. In multi-relay systems, the performance depends on the relaying protocol and the relay selection. There are two main protocols, i.e., decoded forwarding (DF) and amplify-and-forward (AF). In addition, the AF relaying is categorized as either fixed gain (FG) or variable gain (VG) relaying. As for relay selection, three schemes are studied, namely, opportunistic relay selection (ORS) [26], best relay selection (BRS), and partial relay selection (PRS) [27–30]. In ORS and BRS, the relay is selected according to the instantaneous global (two hop) channel state information (CSI), while in PRS, the relay is activated based on the only local (single-hop) CSI information [31]. In order to highlight the research gap between this work and existing literatures, some typical works of dual-hop mixed RF/FSO systems with multiple relays or CCIs have been summarized in Table 1. Due to brevity and space limitations, the summary of dual-hop mixed RF/FSO systems with single relay and without CCIs (see Table 1 in [9,14] for more details) will not be displayed in this work.

Table 1. Summary of dual-hop mixed RF/FSO systems with multiple relays or CCIs

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While previous works have investigated the impact of relaying protocols and relay selection schemes on the performance of multi-relay systems by using different channel models, the limitation of them is that the models they adopt lack generality and flexibility. Specifically, for PRS-aided RF and interference links, they simply assume Rayleigh and Nakagami-m distributions. However, these models don’t accurately describe the fading statistics of RF and interference links in practical applications such as device-to-device communications [36], 5G communications [37] and satellite relaying systems [38]. On the other hand, to characterize atmospheric turbulence along with pointing errors for different detection modes, numerous models were proposed and applied separately to different scenarios [15–19]. When considering different application scenarios, the model needs to be changed accordingly, increasing the complexity of the design. Therefore, in order to carry out a unified performance analysis, the PRS-aided RF link, the interference link, and the FSO link should adopt the generalized channel models that not only account for different application scenarios but also allow a more accurate description of the channel characteristics. However, the mathematical complexity of the generalized channel model leads to the difficulty in obtaining unified closed-form expressions for the end-to-end signal-to-noise ratio (SNR) or the end-to-end signal-to-interference plus noise ratio (SINR) statistics. To date, such a unified model in closed-form has not been reported.

Inspired by the aforementioned work, by concentrating on the generalized channel models, which capture a wide range of propagation conditions, this paper derives unified models to evaluate the performances of interference-limited and interference-free dual-hop mixed RF/FSO AF relaying systems with PRS, assuming that the PRS-aided RF link and the interference link undergo $\kappa$-$\mu$ shadowed fading while the FSO link unifies GG, $\mathcal {M}$, and $\mathcal {F}$ distributions for atmospheric turbulence along with pointing errors for different detection modes. These new unified models can account for different application scenarios compared to previous works. The major contributions of this work are listed as follows: considering interference-limited and interference-free scenarios, we derive new unified closed-form expressions for the cumulative distribution function (CDF). Using the derived CDF, we analyze the performance of VG AF relaying systems with PRS by developing exact closed-form expressions of the average bit-error-rate (BER) and ergodic capacity (EC). In addition, an asymptotic BER analysis is obtained and the diversity order is also provided.

2. System and channel models

2.1 Mixed RF/FSO system model

We consider a dual-hop mixed RF/FSO AF relaying system with a source $S$, a destination $D$ and ${{N}_{r}}$ relays. $S$ and $D$ have no direct link. The $i^{\rm th}$ relay node between $S$ and $D$ is denoted by ${{R}_{i}}$ for $i=1,\ldots, {{N}_{r}}$. The links $S-{{R}_{i}}$ and ${{R}_{i}}-D$ are RF and FSO links, respectively. Moreover, the selected relay suffers from $N$ independent and identically distributed (i.i.d) interfering signals. We further assume that a perfect CSI can be obtained for each relay. According to the PRS protocol, S selects the $k^{\rm th}$ worst relay ${{R}_{k}}$ based on the collected RF link CSI. After that, $S$ communicates with $D$ via ${{R}_{k}}$. The received signal at ${{R}_{k}}$ can be expressed as

To compensate for the fading of the $S-{{R}_{k}}$ link, the VG AF relaying is considered with the scaling gain at ${{R}_{k}}$ given by

The resulting end-to-end SINR can be expressed as [29]

2.2 RF channel model

The RF link experiences the $\kappa$-$\mu$ shadowed fading, the CDF of ${{\gamma }_{1\left ( \ell \right )}}$ can be reformulated as follows [39,40]

$\bullet$ If $\mu$ >$m$

$\bullet$ If $\mu$ $\le$ $m$

Based on partial relay selection protocol, the CDF of ${{\gamma }_{1\left ( k \right )}}$ can be written as [41]

However, Eq. (7) might be intractable which causes mathematical complexity in deriving unified expressions for the considered system. Therefore, we reformulate ${{F}_{{{\gamma }_{1\left ( \ell \right )}}}}\left ( \gamma \right )$ in Eq. (7) as Eq. (4) and Eq. (5), respectively, for subsequent derivations. In what follows, inserting Eq. (4) and Eq. (5) into Eq. (7) and using Eq. (1.111) in [42] and Eq. (26.4.9) in [43], we derive ${{F}_{{{\gamma }_{1\left ( k \right )}}}}\left ( \gamma \right )$ for $\mu$>$m$ and $\mu$ $\le$ $m$ as follows

$\bullet$ If $\mu$ >$m$

Proof : The detailed derivation of ${{F}_{{{\gamma }_{1\left ( k \right )}}}}\left ( \gamma \right )$ can be found in section 1 in Supplement 1.

Moreover, the relay receives i.i.d interfering signals from $N$ interfering terminals. Assuming that the instantaneous INR follows $\kappa$-$\mu$ shadowed distribution, i.e., ${{\gamma }_{I,i}}\sim \left ( {{\bar {\gamma }}_{I}},{{\kappa }_{I}},{{\mu }_{I}},{{m}_{I}} \right )$, the total instantaneous INR, ${{\gamma }_{I}}=\sum\limits _{i=1}^{N}{\frac {{{\left | {{h}_{i}} \right |}^{2}}{{P}_{i}}}{{{N}_{0}}}}$, follows another $\kappa$-$\mu$ shadowed distribution with scaled parameters, i.e., ${{\gamma }_{I}}\sim \left ( N{{{\bar {\gamma }}}_{I}},{{\kappa }_{I}},N{{\mu }_{I}},N{{m}_{I}} \right )$, with the PDF given by [39]

2.3 FSO channel model

When considering the joint effects of atmospheric turbulence and pointing errors, a unified PDF expression under both HD and DD technique cases can be expressed as

Table 2. Unified parameters for the PDF of ${{f}_{{{\gamma }_{2\left ( k \right )}}}}\left ( \gamma \right )$

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Proof : The detailed derivation of ${{f}_{{{\gamma }_{2\left ( k \right )}}}}\left ( \gamma \right )$ can be found in section 2 in Supplement 1.

Using Eq. (2.54) in [45], the CDF of ${{\gamma }_{2\left ( k \right )}}$ is derived by

3. Statistical characteristics

3.1 Cumulative distribution function

Since the SINR statistic in Eq. (3) is mathematically intractable, we use an upper bound as ${{\gamma }_{V}} \cong \min \left ( {{\gamma }_{1\left ( k \right )}}^{\rm eff},{{\gamma }_{2\left ( k \right )}} \right )$ as in [21,29]. Based on this upper bound, the CDF of ${{\gamma }_{V}}$ for the VG relaying system with PRS can be expressed as

$\bullet$ If $\mu$ >$m$

Proof : The detailed derivation of ${{F}_{{{\gamma }_{1\left ( k \right )}}^{\rm eff}}}\left ( \gamma \right )$ can be found in section 3 in Supplement 1.

Similiarly, when CCIs are neglected, the end-to-end SNR is reduced to ${{\gamma }_{V}} =\frac {{{\gamma }_{1\left ( k \right )}}{{\gamma }_{2\left ( k \right )}}}{{{\gamma }_{1\left ( k \right )}}+{{\gamma }_{2\left ( k \right )}}+1}$. To facilitate the mathematical treatment, we also use the upper bound as ${{\gamma }_{V}}\cong \min \left ( {{\gamma }_{1\left ( k \right )}},{{\gamma }_{2\left ( k \right )}} \right )$. This leads to the CDF of ${{\gamma }_{V}}$ as

In order to show effectiveness of the proposed system model, we make comparisons between our derived expressions (${{F}_{{{\gamma }_{1\left ( k \right )}}}}\left ( \gamma \right )$ and ${{F}_{{{\gamma }_{1\left ( k \right )}}^{\rm eff}}}\left ( \gamma \right )$) and that used in the existing literatures. It should be pointed out that ${{F}_{{{\gamma }_{1\left ( k \right )}}}}\left ( \gamma \right )$ in Eq. (23) and ${{F}_{{{\gamma }_{1\left ( k \right )}}^{\rm eff}}}\left ( \gamma \right )$ in Eq. (17) can be regarded as generalized distributions because they can describe different fading models by setting different parameter combinations. This provides the basis for further analysis of the performance of systems with PRS and BRS in the presence and absence of CCIs. As examples, in Fig. 1, ${{F}_{{{\gamma }_{1\left ( k \right )}}}}\left ( \gamma \right )$ and ${{F}_{{{\gamma }_{1\left ( k \right )}}^{\rm eff}}}\left ( \gamma \right )$ are plotted for different parameter combinations in comparison with existing models. From Fig. 1, we observed that the previously reported results in [29,31,47,48] are included in our results when PRS-aided RF link and interference link are modelled as Nakagami-m distributions (Note that K users are equivalent to ${{N}_{r}}$ relays in [47]). In addition, our models can also be used to analyze the system performance, when the RF and interference links are modelled as Rayleigh, Rician, $\kappa$-$\mu$, or Rician shadowed distributions.

 

Fig. 1. CDF of $\left ( a \right )$ ${{\gamma }_{1\left ( k \right )}}$ and $\left ( b \right )$ ${{\gamma }_{1\left ( k \right )}}^{\rm eff}$ for different parameter combinations in comparison with existing models.

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4. End-to-end performance metrics

In this section, we derive two important performance metrics: the average BER and the ergodic capacity for VG relaying systems with PRS in the absence and presence of CCIs.

4.1 Average bit error rate

The average BER for many binary modulation formats can be represented as [49]

Substituting Eq. (17) into Eq. (24), and using Eq. (8.4.3.1) in [50], Eq. (07.34.21.0012.01) in [51], and Eq. (2.3) in [52], we obtain ${{\bar {P}}_{e}}^{V}$ for the VG relaying system with PRS as follows

$\bullet$ If $\mu$ >$m$

$\bullet$ If $\mu$ $\le$ $m$

Similarly, when CCIs are ignored, substituting Eq. (23) into Eq. (24) and performing the integration with the aid of Eq. (3.381.4) in [42] and Eq. (07.34.21.0012.01) in [51] yields:

$\bullet$ If $\mu$ >$m$

4.2 Asymptotic analysis of the average BER

Using Eq. (1.8.4) in [44] and Eq. (3.381.4) in [42], ${{\bar {P}}_{e}}^{V}$ can be asymptotically expressed at high SNR as

In Eq. (32), ${{P}_{asyV,2}}$ can be expressed as the following 3 cases:

a) When the FSO link experiences GG fading, ${{P}_{asyV,2}}$ can be derived as

b) When the FSO link is modelled as the ${\cal M}$ distribution, ${{P}_{asyV,2}}$ can be expressed as

c) When the FSO link follows the ${\cal F}$ distribution, ${{P}_{asyV,2}}$ can be obtained as

Using a method similar to the Ref. [53], from Eqs. (32)–(37), we can obtain the diversity order as $\min \left ( \frac {{{\xi }_{\bmod }}^{2}}{r}, \frac {{{\alpha }_{G}}}{r},\frac {{{\beta }_{G}}}{r},k\mu \right )$, $\min \left ( \frac {{{\xi }_{\bmod }}^{2}}{r},\frac {{{\alpha }_{M}}}{r},\frac {v}{r},k\mu \right )$, and $\min \left ( \frac {{{\xi }_{\bmod }}^{2}}{r},\frac {{{a}_{F}}}{r},k\mu \right )$ for GG, ${\cal M}$ and ${\cal F}$ distributions, respectively. Moreover, under the assumption that the interference power is scaled with the transmitted power, as the SNR approaches infinity, the INR approaches infinity. At high SNRs, the existence of CCIs leads to the occurence of BER floors and capacity ceilings, which reflects a zero diversity order.

Proof : The detailed derivation of the asymptotic BER can be found in section 4 in Supplement 1.

4.3 Ergodic capacity

From [12], the ergodic capacity can be represented as

Similarly, when CCIs are absent, inserting Eq. (23) into Eq. (38) and employing Eq. (2.3) in [52], ${{\bar {C}}^{V}}$ can be derived as follows

$\bullet$ If $\mu$ >$m$

$\bullet$ If $\mu$ $\le$ $m$

In order to demonstrate the effectiveness of our derived performance metrics, we compare our work with some existing literatures. We find that our results are consistent with the previous works, providing a generic description of this kind system and are inclusive about the previous works as special cases as follows: a) when taking the $\mathcal {F}$ distribution and setting ${{N}_{r}}=1$ and $k=1$ in the presence of CCIs, the BER and EC we derived can be simplified as the corresponding BER and EC in [25]; b) when taking the $\mathcal {M}$ distribution and setting ${{N}_{r}}=1$, $k=1$, $\mu =1$, ${{\mu }_{I}}=1$, $\kappa \to 0$, ${{\kappa }_{I}}\to 0$, $m\to \infty$ and ${{m}_{I}}\to \infty$, the BER we derived can be simplified as the corresponding BER in [23]; c) when taking the GG distribution and setting ${{N}_{r}}=1$, $k=1$, $\mu = \underline {m}$, $\kappa \to 0$ and $m\to \infty$, the BER and EC we derived can be simplified as the corresponding BER and EC in [6]; d) when taking the $\mathcal {F}$ distribution and setting ${{N}_{r}}=1$ and $k=1$ in the absence of CCIs, the BER and EC we derived can be simplified as the upper bounds of the corresponding BER and EC in [14].

5. Numerical results

In this section, we carry out numerical simulations to verify the accuracy of our proposed expressions. Without loss of generality, we assume ${{\bar {\gamma }}_{1}}={{\bar {\gamma }}_{2}}$. Unless otherwise stated, the simulation parameters are given in Table 3. It should be noted that the parameter values we adopted for PRS-aided RF and interference links include cases that cannot be described by models in previous works [29,31,47,48]. The FSO link can be modeled as either GG, $\mathcal {M}$, or Fisher-Snedecor $\mathcal {F}$ distribution for different parameter combinations. Here, we take the GG distribution as an example. Notice that when setting the parameters $g=0$ and $\Omega '=1$ in the $\mathcal {M}$ distribution, this $\mathcal {M}$ distribution reduces to the GG distribution. Due to brevity and space limitations, the analysis using the $\mathcal {M}$ distribution as an example will not be presented in this work. Moreover, we also take the $\mathcal {F}$ distribution as an example and add results and discussions (see Supplement 1 for more details). In the MC simulation, the number of random samples is set to $4\times {{10}^{6}}$. GG, ${\cal M}$ and ${\cal F}$ random samples can be generated using Eqs. (10–11) in [15], Eq. (18) in [16] and Eq. (2) in [17], respectively, while $\kappa -\mu$ shadowed random samples can be generated using Eq. (1) in [20]. In addition, the pointing error samples can be obtained based on Eq. (4) in [19] and Eq. (9) in [56].

Table 3. Simulation parameter settings

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Figure 2 shows the variations of the average BER in the presence and absence of CCIs for different detection modes. As observed, the derived expressions (Eqs. (25)–(31)) coincide with the MC simulations, verifying the accuracy of our derivation. It can be seen that a higher BER performance is obtained for the HD technology $\left ( r=1 \right )$ compared to the DD technology $\left ( r=2 \right )$. As expected, the existence of CCIs leads to degradation of BER performance. At high SNRs, for the same detection mode, the BER converges to the same values regardless of the value of CCIs. This result indicates that at a fixed average interference power, the impacts of CCIs are negligible at high SNR. Moreover, we observe that the asymptotic results given in Eq. (32) coincide with the exact results derived in Eqs. (25)–(31) at high SNR.

 

Fig. 2. The effect of CCIs on the average BER of DBPSK modulation for different detection modes with ${{N}_{r}}=5$ and $k=1$.

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Figure 3 shows the variations of the average BER in the presence and absence of CCIs for different turbulence conditions in Fig. 3(a) and pointing errors in Fig. 3(b) when the best relay ($k={{N}_{r}}$) is available. We observe that the interference-free system illustrates better BER performance compared to the interference-limited system. In addition, the BER performance will deteriorate as the turbulence strength increases. At high SNRs, for the same turbulence strength, the BER converges to the same values regardless of the value of CCIs. This result shows that the effects of CCIs can be ignored at high SNR when the average interference power is fixed. Moreover, we compare the BER performance between the case with less interference and the case without interference. It is observed that the BER curves with negligible CCIs (e.g., ${{\bar {\gamma }}_{I}}= -30 \ {\rm dB}$) coincide with the BER curves without CCIs, indicating that when the CCIs approach zero, interference-limited cases (Eqs. (25–26)) converge to the interference-free cases (Eqs. (27–31)), which further confirms the accuracy of our expressions. In Fig. 3(b), we observe that the system with CCIs (e.g., ${{\bar {\gamma }}_{I}}= 15 \ {\rm dB}$) yields worse BER performance compared to the system without CCIs. It is clear from the figure that as pointing errors decrease, the average BER drops. For the same pointing errors, as the average SNR increases, the BER difference between the interference-free system and interference-limited system gradually decreases; at high SNRs, the BER converges to the same values regardless of the value of CCIs. Again, the results show that the effects of CCIs are negligible at high SNR.

 

Fig. 3. The effect of CCIs on the average BER of BPSK modulation with $k={{N}_{r}}=5$ for (a) different turbulence conditions and (b) different pointing errors.

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Figure 4 plots the average BER with respect to the average SNR for different number of interferers $N$ under both the best relay ($k={{N}_{r}}$) and the worst relay ($k=1$) scenarios by considering fixed average interference power in Fig. 4(a) and the average interference power proportional to the transmitted power in Fig. 4(b). In Fig. 4(a), we observe that, for the same $N$, selecting the best relay results in the best system performance for a wide range of SNRs. In addition, increasing $N$ leads to the BER performance degradation for a wide range of SNRs. At high SNRs, we note that the BER converges to the same values and no BER floor appears. This result shows that the CCIs and rank of the selected relay have no impact on performance at high SNRs. This is because when the interference power is fixed, as the average SNR increases, the influence of the FSO link on the average BER dominates, while the influence of RF links on the average BER can be ignored. In addition, by computing the corresponding parameters, we find that the FSO link parameter determines the diversity order, which further confirms that the FSO link dominates at high SNR. In Fig. 4(b), it is observed that by increasing the average SNR, the BER performance improves; however, as the average SNR increases, the BER curve saturates, and BER floors take place, meaning that further increasing the average SNR will not improve the system BER performance. This is because that interference-limited RF link becomes dominant. Note that in this case, the BER does not converge to the same values.

 

Fig. 4. BER of BPSK modulation versus the average SNR for different number of interferers with (a) ${{\bar {\gamma }}_{I}}=15\ {\rm dB}$ and (b) ${{{P}_{s}}}/{{{P}_{i}}}=10\ {\rm dB}$.

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Figure 5 illustrates the variations of the ergodic capacity in the presence and absence of CCIs when $k={{N}_{r}}$ is available, and when only $k=1$ is available. The analytical results based on (Eqs. (39–41)) match with the MC simulations, indicating the accuracy of our derivation. As observed, the system without CCIs shows better ergodic capacity performance than the system with CCIs. In addition, the best relay performs much better than the worst relay. Furthermore, compared to the system without CCIs, the difference in the ergodic capacity between the best relay and the worst relay increases when CCI is present. This result shows that implementing the PRS technique can combat the effects of CCIs.

 

Fig. 5. The effect of CCIs on the ergodic capacity for different rank of the selected relay with ${{N}_{r}}=5$ and ${{\bar {\gamma }}_{I}}=15\ {\rm dB}$.

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Figure 6 plots the ergodic capacity with respect to i) rank of the selected relay $k$ and ii) number of relays ${{N}_{r}}$ for fixed average interference power in Fig. 6(a) and the average interference power proportional to the transmitted power in Fig. 6(b). In Fig. 6(a), we observe that the increasing the interference power leads to decrease in the ergodic capacity. When $k$ is changed from 1 to 5 for ${{N}_{r}}=5$, $k=5$ outperforms $k=1$ in the system performance. Therefore, the rank of the selected relay has a significant impact on the system performance. When ${{N}_{r}}=k=1$, there will be no relay selection procedure. Compared to the system with single relay, when PRS with ${{N}_{r}}=k=5$ is implemented, the ergodic capacity increases. In fact, when increasing ${{N}_{r}}$, a stronger CSI is used for relay selection. Therefore, to serve the dense cells without throughput perturbation, the number of relays must be large enough to primarily handle possible power shortage/outage for further communications [29]. In Fig. 6(b), we observe that when $k$ is changed from 1 to 5 for ${{N}_{r}}=5$, $k=5$ has better capacity performance than $k=1$. Compared to the system with single relay for ${{N}_{r}}=k=1$, the ergodic capacity increases when implementing PRS with ${{N}_{r}}=k=5$. As the interference power increases, the ergodic capacity decreases. Furthermore, at high SNRs, capacity ceilings happen, meaning that increasing the average SNR further does not yield performance improvements. This phenomenon indicates that the diversity order is zero.

 

Fig. 6. Ergodic capacity versus the average SNR for different number of relays and rank of the selected relay with (a) ${{\bar {\gamma }}_{I}}=\left \{ 5,\ 15 \right \}\ {\rm dB}$ and (b) ${{{P}_{s}}}/{{{P}_{i}}}\;=\left \{ 5,\ 15 \right \}\ {\rm dB}$.

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6. Conclusion

We have investigated the performance of a dual-hop mixed RF/FSO VG AF relaying system with PRS in the absence and presence of co-channel interferences, where PRS-aided RF, interference, and FSO links are all characterized as generalized channel models. Specifically, the PRS-aided RF link and the CCIs at the relay node are all modeled by $\kappa$-$\mu$ shadowed distributions, whereas the FSO link unifies GG, $\mathcal {M}$, and Fisher-Snedecor $\mathcal {F}$ distributions for atmospheric turbulence along with pointing errors for different detection modes. For the considered system, we first derived unified closed-form expressions for the CDF with and without CCIs for different application scenarios. Based on the above expressions, both interference-free and interference-limited receptions at the relay node were studied by deriving new expressions for the average BER and ergodic capacity. Moreover, the asymptotic expressions of the average BER were also derived. We have verified the analytical results through the MC simulations. The results are consistent with previous findings, but due to adopting generalized channel models, our derived results are more general than previously reported results.