Double reconfigurable intelligent surface-assisted wireless communication system for energy efficiency improvement over weibull fading channels

Abstract

In this paper, we propose a double reconfigurable intelligent surface (RIS)-assisted wireless communication system to improve energy efficiency when a direct link from source to destination is obstructed. First, we model a single RIS-assisted wireless communication system by deploying RIS midway between the source and destination link over the Weibull fading channel. Next, we consider the double-RIS system, in which RIS-1 (R₁) and RIS-2 (R₂) are near the source and destination respectively, with overall reflecting elements in the system equal to the single-RIS system. For both systems, we derive the closed-form expressions for the bounds (lower and upper) of ergodic capacity and exact closed-form expressions for outage probability. The accuracy of the presented theoretical framework is validated through Monte-Carlo simulations. From the comparison, we found that the double-RIS system surpasses the S-RIS system in terms of ergodic capacity and outage probability. Finally, we provide the ergodic capacity and energy efficiency analysis of both systems as a function of spectral efficiency by varying the position of RISs and notice that the double-RIS system with R₁ and R₂ near to source and destination is more energy efficient than the S-RIS system.

Appendix A

As mentioned earlier ${\gamma }_{D,1}$ follows NCCS with following mean, variance and second order moment [24],

$${\mathbb{E}}\left[{\gamma }_{D,1}\right]={\sigma }_{{Z}_{1}}^{2}+{{\lambda }_{{Z}_{1}}}^{2},$$

(24)

$${\mathbb{V}}ar\left[{\gamma }_{D,1}\right]=2{\sigma }_{{Z}_{1}}^{4}+4{\sigma }_{{Z}_{1}}^{2}{{\lambda }_{{Z}_{1}}}^{2},$$

(25)

$${\mathbb{E}}\left[{{\gamma }_{D,1}}^{2} \right]={\mathbb{V}}ar\left[{\gamma }_{D,1}\right]+ {{\mathbb{E}}\left[{\gamma }_{D,1}\right]}^{2}=3{\sigma }_{{Z}_{1}}^{4}+6{\sigma }_{{Z}_{1}}^{2}{{\lambda }_{{Z}_{1}}}^{2}+{{\lambda }_{{Z}_{1}}}^{4},$$

(26)

where ${\lambda }_{{Z}_{1}}=\sqrt{{\left({N}_{S}{\Omega }^{\frac{2}{\beta }}{\left(\Gamma \left(1+1/\beta \right)\right)}^{2}\right)}^{2}}$ is non-centrality parameter, and ${\sigma }_{{Z}_{1}}^{2}={N}_{S}{\Omega }^{\frac{4}{\beta }}({(\Gamma \left(1+2/\beta \right))}^{2}-\Gamma {\left(\left(1+1/\beta \right)\right)}^{4})$. By substituting ${\lambda }_{{Z}_{1}}$, ${\sigma }_{{Z}_{1}}^{2}$ in Eq. (24)–(26) and after some algebraic calculations we will get mean, second order moment and variance of ${\gamma }_{D,1}$ as in Eq. 9(a)–(c).

Similarly, by substituting non-centrality parameter ${\lambda }_{{Z}_{2}}=\sqrt{{\left(2{N}_{D}{\Omega }^{\frac{2}{\beta }}{\left(\Gamma \left(1+1/\beta \right)\right)}^{2}\right)}^{2}}$ and ${\sigma }_{{Z}_{2}}^{2}=2{N}_{D}{\Omega }^{\frac{4}{\beta }}\left[{\left(\Gamma \left(1+2/\beta \right)\right)}^{2}-{\left(\Gamma \left(1+1/\beta \right)\right)}^{4}\right]$ in Eq. (27)–(29) and after doing algebraic calculations we will get the mean, second order moment and variance of ${\gamma }_{D,2}$ as in Eq. 12(a)–(c).

$${\mathbb{E}}\left[{\gamma }_{D,2}\right]={\sigma }_{{Z}_{2}}^{2}+{{\lambda }_{{Z}_{2}}}^{2},$$

(27)

$${\mathbb{V}}ar\left[{\gamma }_{D,2}\right]=2{\sigma }_{{Z}_{2}}^{4}+4{\sigma }_{{Z}_{2}}^{2}{{\lambda }_{{Z}_{2}}}^{2},$$

(28)

$${\mathbb{E}}\left[{{\gamma }_{D,2}}^{2} \right]={\mathbb{V}}ar\left[{\gamma }_{D,2}\right]+ {{\mathbb{E}}\left[{\gamma }_{D,2}\right]}^{2}=3{\sigma }_{{Z}_{2}}^{4}+6{\sigma }_{{Z}_{2}}^{2}{{\lambda }_{{Z}_{2}}}^{2}+{{\lambda }_{{Z}_{2}}}^{4}.$$

(29)