On secrecy performance of mixed α − η − μ and Málaga RF-FSO variable gain relaying channel

Abstract

With the completion of the standardization of fifth-generation (5G) networks, the researchers have begun visioning sixth-generation (6G) networks that are predicted to be human-centric. Hence, similar to 5G networks, besides high data rates, providing secrecy and privacy will be the center of attention by the wireless research community. To support the visions beyond 5G and 6G, in this paper we propose a secure radio frequency (RF)-free space optical (FSO) mixed framework under the attempt of wiretapping by an eavesdropper at the RF hop. We assume the RF links undergo $\alpha -\eta -\mu$ fading whereas the FSO link exhibits a unified Málaga turbulence model with pointing error. The secrecy performance is evaluated by deducing expressions for three secrecy metrics i.e. average secrecy capacity, secrecy outage probability, and probability of non-zero secrecy capacity in terms of univariate and bivariate Meijer’s G and Fox’s H functions. We further capitalize on these expressions to demonstrate the impacts of fading, atmospheric turbulence, and pointing errors and show a comparison between two detection techniques (i.e. heterodyne detection (HD) and intensity modulation with direct detection (IM/DD)) that clearly reveals better secrecy can be achieved with HD technique relative to the IM/DD method. The inclusion of generalized fading models at the RF and FSO hops offers the unification of several classical scenarios as special cases thereby exhibiting a more generic nature relative to the existing literature. Finally, all the analytical results are corroborated via Monte-Carlo simulations.

Appendices

Appendix

A Proof of ASC

For mathematical simplification, we have considered $\tilde{\alpha _r}=\tilde{\alpha _v}={\tilde{\alpha }}$. The final expressions of ${\mathcal {S}}_1$, ${\mathcal {S}}_2$, ${\mathcal {S}}_3$, and ${\mathcal {S}}_4$ are given below.

1.1 Calculation of ${\mathcal {S}}_1$

Using the identities (Prudnikov et al. 1988, Eqs. (8.4.3.1), (8.4.2.5)), (15a) can be given as

$$\begin{aligned} {\mathcal {S}}_1= \int _0^\infty \gamma ^{{\tilde{\alpha }} t_1} G_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} 0 \\ 0 \\ \end{array} \right] G_{0,1}^{1,0}\left[ u_{2}\gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ 0 \\ \end{array} \right] d\gamma . \end{aligned}$$

(23)

Now, applying (Prudnikov et al. 1988, Eqs. (2.24.1.1)) in (23) and performing the integration, we obtain

$$\begin{aligned} {\mathcal {S}}_1=(2 \pi )^{1-{\tilde{\alpha }}}G_{{\tilde{\alpha }},1+{\tilde{\alpha }}}^{1+{\tilde{\alpha }},{\tilde{\alpha }}}\left[ u_{2} \biggl \vert \begin{array}{c} \Delta ({\tilde{\alpha }}, -{\tilde{\alpha }}t_1) \\ 0,\Delta ({\tilde{\alpha }}, -{\tilde{\alpha }}t_1) \\ \end{array} \right] , \end{aligned}$$

(24)

where $\Delta (a,b)=\frac{b}{a}, \frac{b+1}{a},\ldots , \frac{b+a-1}{a}$.

1.2 Calculation of ${\mathcal {S}}_2$

We derive ${\mathcal {S}}_2$ by following the similar process as ${\mathcal {S}}_1$ and ${\mathcal {S}}_2$ can be expressed as

$$\begin{aligned} {\mathcal {S}}_2&= \int _0^\infty \gamma ^{{\tilde{\alpha }} {\mathcal {T}}} G_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} 0 \\ 0 \\ \end{array} \right] G_{0,1}^{1,0}\left[ \Im \gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ 0 \\ \end{array} \right] d\gamma \nonumber \\&=(2 \pi )^{1-{\tilde{\alpha }}}G_{{\tilde{\alpha }},1+{\tilde{\alpha }}}^{1+{\tilde{\alpha }},{\tilde{\alpha }}}\left[ \Im \biggl \vert \begin{array}{c} \Delta ({\tilde{\alpha }}, -{\tilde{\alpha }}{\mathcal {T}}) \\ 0,\Delta ({\tilde{\alpha }}, -{\tilde{\alpha }}{\mathcal {T}}) \\ \end{array} \right] , \end{aligned}$$

(25)

where ${\mathcal {T}}=t_1+t_2$ and $\Im =u_{2}+q_2$.

1.3 Calculation of ${\mathcal {S}}_3$

Utilizing the identities (Prudnikov et al. 1988, Eqs. (8.4.3.1), (8.4.2.5)) in (15c), ${\mathcal {S}}_3$ can be expressed as

$$\begin{aligned} {\mathcal {S}}_3&= \int _0^\infty \gamma ^{{\tilde{\alpha }} t_1} G_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} 0 \\ 0 \\ \end{array} \right] G_{0,1}^{1,0}\left[ u_{2}\gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ 0 \\ \end{array} \right] G_{r+1,3r+1}^{3r,1}\left[ \frac{F \gamma }{u _r} \biggl \vert \begin{array}{c} 1,l_1 \\ l_2,0 \\ \end{array} \right] d\gamma . \end{aligned}$$

(26)

For deriving the closed form expression of ${\mathcal {S}}_3$, we must integrate (26) within the limit 0 to $\infty$ that is mathematically intractable. So, for obtaining ${\mathcal {S}}_3$ in closed-form, Meijer’s G function terms are converted into Fox’s H functions utilizing (Prudnikov et al. 1988, Eqs. (8.3.2.21)) as

$$\begin{aligned} {\mathcal {S}}_3&=\int _0^\infty \gamma ^{{\tilde{\alpha }} t_1} H_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} {[}0,1{]}\\ {[}0,1{]} \\ \end{array} \right] H_{0,1}^{1,0}\left[ u_{2}\gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ {[}0,1{]} \\ \end{array} \right] \nonumber \\&\quad \quad \times H_{r+1,3r+1}^{3r,1}\left[ \frac{F \gamma }{u _r} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \right] d\gamma . \end{aligned}$$

(27)

Again, for mathematical simplification, we assume $x=\gamma ^{{\tilde{\alpha }}}$. So, (27) can be expressed as

$$\begin{aligned} {\mathcal {S}}_3&=\frac{1}{{\tilde{\alpha }}}\int _0^\infty x^{ {\mathcal {M}}_1-1} H_{0,1}^{1,0}\left[ u_{2}x \biggl \vert \begin{array}{c} . \\ {[}0,1{]} \\ \end{array} \right] H_{1,1}^{1,1}\left[ x^{\frac{1}{{\tilde{\alpha }}}} \biggl \vert \begin{array}{c} {[}0,1{]}\\ {[}0,1{]} \\ \end{array} \right] \nonumber \\&\quad \quad \times H_{r+1,3r+1}^{3r,1}\left[ \frac{F x^{\frac{1}{{\tilde{\alpha }}}}}{u _r} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \right] dx. \end{aligned}$$

(28)

Now utilizing (Mittal and Gupta 1972, Eq. (2.3)), (Lei et al. 2017, Eq. (3)), the final expression of ${\mathcal {S}}_3$ is derived as

$$\begin{aligned}&{\mathcal {S}}_3=\frac{1}{{\tilde{\alpha }} {u_{2}}^{{\mathcal {M}}_1}}\nonumber \\&\quad \quad \quad \times H_{1,0;1,1;r+1,3r+1}^{1,0;1,1;3r,1}\left[ \begin{array}{c} {[}1-{\mathcal {M}}_1; \frac{1}{{\tilde{\alpha }}} , \frac{1}{{\tilde{\alpha }}}{]} \\ - \\ \end{array} \biggl \vert \begin{array}{c} {[}0,1{]} \\ {[}0,1{]} \\ \end{array} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \biggl \vert \frac{1}{{u_2}^{\frac{1}{{\tilde{\alpha }}}}}, \frac{F}{u_r {u_2}^{\frac{1}{{\tilde{\alpha }}}}} \right] , \end{aligned}$$

(29)

where ${\mathcal {M}}_1=\frac{1}{{\tilde{\alpha }}}+t_1$, $H_{m1,n1:m2,n2:m3,q3}^{p1,q1:p2,q2:p3,q3}[.]$ is the extended generalized bivariate Fox’s H function.

1.4 Calculation of ${\mathcal {S}}_4$

Similar to ${\mathcal {S}}_3$, ${\mathcal {S}}_4$ is obtained as

$$\begin{aligned} {\mathcal {S}}_4&= \int _0^\infty \gamma ^{{\tilde{\alpha }} {\mathcal {T}}} G_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} 0 \\ 0 \\ \end{array} \right] G_{0,1}^{1,0}\left[ \Im \gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ 0 \\ \end{array} \right] G_{r+1,3r+1}^{3r,1}\left[ \frac{F \gamma }{u _r} \biggl \vert \begin{array}{c} 1,l_1 \\ l_2,0 \\ \end{array} \right] d\gamma \nonumber \\&=\int _0^\infty \gamma ^{{\tilde{\alpha }}{\mathcal {T}}} H_{1,1}^{1,1}\left[ \gamma \biggl \vert \begin{array}{c} {[}0,1{]}\\ {[}0,1{]} \\ \end{array} \right] H_{0,1}^{1,0}\left[ \Im \gamma ^{{\tilde{\alpha }}} \biggl \vert \begin{array}{c} . \\ {[}0,1{]} \\ \end{array} \right] \nonumber \\&\quad \; \times H_{r+1,3r+1}^{3r,1}\left[ \frac{F \gamma }{u _r} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \right] d\gamma \nonumber \\&=\frac{1}{{\tilde{\alpha }}}\int _0^\infty x^{ {\mathcal {M}}_2-1} H_{0,1}^{1,0}\left[ \Im x \biggl \vert \begin{array}{c} . \\ {[}0,1{]} \\ \end{array} \right] H_{1,1}^{1,1}\left[ x^{\frac{1}{{\tilde{\alpha }}}} \biggl \vert \begin{array}{c} {[}0,1{]}\\ {[}0,1{]} \\ \end{array} \right] \nonumber \\&\quad \; \times H_{r+1,3r+1}^{3r,1}\left[ \frac{F x^{\frac{1}{{\tilde{\alpha }}}}}{u _r} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \right] dx \nonumber \\&=\frac{1}{{\tilde{\alpha }} {\Im }^{{\mathcal {M}}_2}} H_{1,0;1,1;r+1,3r+1}^{1,0;1,1;3r,1}\left[ \begin{array}{c} {[}1-{\mathcal {M}}_2; \frac{1}{{\tilde{\alpha }}} , \frac{1}{{\tilde{\alpha }}}{]} \\ - \\ \end{array} \biggl \vert \begin{array}{c} {[}0,1{]} \\ {[}0,1{]} \\ \end{array} \biggl \vert \begin{array}{c} {[}1,1{]},{[}l_1,1{]} \\ {[}l_2,1{]}, {[}0,1{]} \\ \end{array} \biggl \vert \biggl \vert \frac{1}{{\Im }^{\frac{1}{{\tilde{\alpha }}}}}, \frac{F}{u_r {\Im }^{\frac{1}{{\tilde{\alpha }}}}} \right] , \end{aligned}$$

(30)

where ${\mathcal {M}}_2=\frac{1}{{\tilde{\alpha }}}+{\mathcal {T}}$.

B Proof of SOP

Plugging (8) and (11) into (17) leads to

$$\begin{aligned}&P_{out}^L(R_s)=1-\sum _{N_1=0}^{\infty } \sum _{N_2=0}^{\infty }\sum _{t_1=0 }^{W_1-1}u_{4} q_2 \theta ^{\tilde{\alpha _r} t_1}\int _0^\infty e^{-(u_{2}\theta \gamma ^{\tilde{\alpha _r}}+q_2\gamma ^{\tilde{\alpha _v}})}\\&\quad \quad \quad \quad \quad \quad \; \times {\gamma }^{q_3+\tilde{\alpha _r}t_1} \biggl (1-\sigma \sum _{{\tilde{m}}_d=1}^{\beta _d } c_d G_{r+1,3r+1}^{3r,1}\left[ \frac{F }{u _r}(\theta \gamma ) \biggl \vert \begin{array}{c} 1,l_1 \\ l_2,0 \\ \end{array} \right] \biggl )d\gamma . \end{aligned}$$

Here, for mathematical tractability, we consider $\tilde{\alpha _r}=\tilde{\alpha _v}={\tilde{\alpha }}$.

1.1 Calculation of ${\mathcal {H}}_{1}$

Using (Gradshteyn and Ryzhik 2014, Eq. (3.326.2)), ${\mathcal {H}}_{1}$ is expressed as

$$\begin{aligned} {\mathcal {H}}_{1}&=\int _0^{\infty }{\gamma }^{q_3+{\tilde{\alpha }}t_1}e^{-\kappa \gamma ^{{\tilde{\alpha }}}}d\gamma ={\frac{\Gamma ({\mathcal {Z}}_1)}{ {\tilde{\alpha }}{\kappa }^{{\mathcal {Z}}_1}}}, \end{aligned}$$

(31)

where ${\mathcal {Z}}_1=\frac{q_3+\tilde{\alpha _r}t_1+1}{{\tilde{\alpha }}}$ and $\kappa =u_{2}\theta +q_2$.

1.2 Calculation of ${\mathcal {H}}_{2}$

${\mathcal {H}}_{2}$ in (18) is given as

$$\begin{aligned} {\mathcal {H}}_{2}&=\int _0^{\infty } {\gamma }^{q_3+{\tilde{\alpha }}t_1} e^{-\kappa \gamma ^{{\tilde{\alpha }}}} G_{r+1,3r+1}^{3r,1}\left[ \frac{F }{u _r}(\theta \gamma ) \biggl \vert \begin{array}{c} 1,l_1 \\ l_2,0 \\ \end{array} \right] d\gamma . \end{aligned}$$

(32)

Letting $I=\gamma ^{{\tilde{\alpha }}}$ and utilizing (Prudnikov et al. 1988, Eqs. (8.4.3.1) and (2.24.1.1)), ${\mathcal {H}}_{2}$ can be expressed as

$$\begin{aligned}&{\mathcal {H}}_{2} ={\frac{1}{{\tilde{\alpha }}}} \int _0^{\infty } I^{{\mathcal {Z}}_1-1} e^{-\kappa I} G_{r+1,3r+1}^{3r,1}\left[ \frac{F\theta }{u _r}I^{\frac{1}{{\tilde{\alpha }}}}\biggl \vert \begin{array}{c} 1,l_1 \\ l_2,0 \\ \end{array} \right] dI \nonumber \\&\quad \;\; ={\frac{(2\pi )^{(1-{\tilde{\alpha }})r}}{{\kappa }^{{\mathcal {Z}}_1}{\tilde{\alpha }}^{1-{\mathcal {Z}}_2}} } G_{{\tilde{\alpha }}r+{\tilde{\alpha }}+1,3{\tilde{\alpha }}r+{\tilde{\alpha }}}^{3{\tilde{\alpha }}r,{\tilde{\alpha }}+1}\left[ \frac{(F\theta )^{{\tilde{\alpha }} } }{{u_r}^{{\tilde{\alpha }}} \kappa {{\tilde{\alpha }}}^{2{\tilde{\alpha }}r}}\biggl \vert \begin{array}{c} x_1, 1-{\mathcal {Z}}_1,x_2 \\ x_3,0 \\ \end{array} \right] , \end{aligned}$$

(33)

where ${\mathcal {Z}}_2=\Delta (r,\varepsilon ^2)+ \Delta (r, \alpha _d)+ \Delta (r, {\tilde{m}}_d)-\Delta (r,\varepsilon ^2+1)-r$, $x_1=\Delta ({\tilde{\alpha }},1)$, $x_2=\Delta ({\tilde{\alpha }},l_1)$, and $x_3=\Delta ({\tilde{\alpha }},l_2)$.

At higher SNR, asymptotic analysis can be expressed by inverting the Meijer’s G function in (34) via utilizing (Springer 1979, Eq. (6.2.2)). For simplification we assume ${\tilde{\alpha }}=1$ and using (Ansari et al. 2015, Eq. 41), the ${\mathcal {H}}_{2}$ term of the lower bound of the SOP (asymptotic) is expressed as

$$\begin{aligned} {\mathcal {H}}_{2}^{\infty }&= \frac{1}{{\kappa }^{{\mathcal {Z}}_1}} \sum _{k=1}^{3r} {\left( \frac{{u_r}\kappa }{F \theta } \right) }^{L_{1,k}-1} \frac{\Pi _{l=1,l\ne k}^{3r}\Gamma (L_{1,k}-V_{1,l}) \Gamma (1-L_{1,k}) \Gamma (1+{\mathcal {Z}}_1-L_{1,k})}{\Gamma (2-L_{1,k}) \Gamma _{l=3}^{r+2} (L_{1,k}-L_{2,l})}. \end{aligned}$$

(34)