Performance evaluation of IRS-assisted one-bit control-based mixed FSO-RF communication system

Abstract

In this paper, we investigate the performance of the novel one-bit control intelligent reflecting surface (IRS)-assisted mixed free-space optical (FSO)-radio frequency (RF) communication system, where an IRS is utilized over the RF hop to empower the end-to-end system performance. The FSO link is assumed to be affected by path loss, nonzero boresight pointing error, and atmospheric turbulence, which is modeled by generalized Malaga ($\mathcal {M}$)-distribution, whereas the multipath fading in RF link is modeled by Nakagami-m distribution. In particular, unified closed-form expressions for the outage probability (OP), bit-error-rate (BER), and ergodic capacity (EC) are derived for optical heterodyne detection (OHD) and intensity modulation with direct detection (IMDD) techniques. We also derive the achievable diversity order of the considered IRS-assisted system by obtaining the asymptotic OP and asymptotic BER expressions. In addition, we derive the asymptotic EC of the considered system. The numerical results show that the proposed IRS-assisted system significantly outperforms the conventional mixed FSO-RF system without IRS. Moreover, the impact of the number of reflecting elements, practical reflection amplitude, and controlling mechanism at the IRS is studied on the system performances.

Appendices

Appendix

Appendix A: Proof of Lemma 1

Considering the reflection amplitudes of each reflecting elements to be identical, i.e., $|\varpi _i|=|\varpi |$, $\forall i$, we can write (17) as $\gamma _{D}(\textbf{f}_\ell )\!=\rho _R\left| W_\ell \right| ^2$, where

$$\begin{aligned}&W_{\ell }=\frac{1}{\sqrt{D}}\underbrace{\sum \limits _{i=(\ell -1)D+1}^{\ell D} \!H_ie^{j\psi _i}}_{D\,\text {Terms}}\nonumber \\&=\frac{1}{\sqrt{D}}\underbrace{\sum \limits _{j=0}^{ D-1}\! H_{(\ell -1)D+1+j}e^{j\psi _{(\ell -1)D+1+j}}}_{D\,\text {Terms}}, \end{aligned}$$

(47)

where $j=i-(1-\ell )D-1$. Further, considering phase error as Generalized uniform distributed, obtaining the PDF of $W_\ell$ becomes difficult. Therefore, we obtain a lower bound of the performance of an IRS-assisted by considering phase error to be distributed as $U\sim (-\pi , \pi )$. Under, this assumption the PDF of $W_\ell$ can be obtained from [31] with some RV transformation as

$$\begin{aligned} f_{|W_{\ell }|^2}(v)\!=\!\sum _{r_0=0}^{m_g-1}\cdots \sum _{r_{D-1}=0}^{m_g-1}\!\prod \limits _{j=0}^{ D-1}\!\mathbb {A}_j \frac{2\Xi ^{\frac{(\nu +1)}{2}}}{(\nu \!-\!1)!}v^{\frac{\nu -1}{2}} \;K_{\nu -1}\!\left( 2\sqrt{\Xi v}\right) , \end{aligned}$$

(48)

where $\nu =D(m_g+ m_h-1)-\sum _{j=0}^{ D-1}r_j$, $\mathbb {A}_j=\frac{(m_h)_{m_g-\!1-r_j}(1\!-\!m_h)_{r_j}}{(m_g\!-\!1\!-\!r_j)!r_j!}$, and $\Xi =\frac{m_gm_hD}{\Omega _g\Omega _h}$. Moreover, the CDF of $|W_{\ell }|^2$ can be derived by $\int _{0}^{v}f_{|W_{\ell }|^2}(z)\,\mathrm{{d}}z$ along with the use of [25, Eq. 6.561.8] as

$$\begin{aligned} \mathbb {F}_{|W_{\ell }|^2}(v)\!=\!\sum _{r_0=0}^{m_g-1}\cdots \sum _{r_{D-1}=0}^{m_g-1}\prod \limits _{j=0}^{ D-1} \!\mathbb {A}_j\!\left[ \!1\!- \!\frac{2\Xi ^{\frac{\nu }{2}}}{(\nu \!-\!1)!}v^{\frac{\nu }{2}} \!\;K_{\nu }\!\left( 2\sqrt{\Xi v}\right) \right] , \end{aligned}$$

(49)

From the definition of $W_{\ell }$ in (47) and the assumption of $g_i$ and $h_i$ following Nakagami-m distribution with parameters $\left( m_g,\Omega _g/m_g\right)$ and $\left( m_h,\Omega _h/m_h\right)$, respectively, it is straightforward to write that $\mathbb {E}\left[ |W_{\ell }|^2\right] =\Omega _g \Omega _h$. Now we can write the CDF of $\gamma _{D}(\mathbf {f_\ell })$ given in (16) as

$$\begin{aligned} \mathbb {F}_{\gamma _{D}(\textbf{f}_{\ell })}(\gamma )\!=\!\Pr \!\left\{ \rho _R|W_{\ell }|^2\le \gamma \right\} \!=\mathbb {F}_{|W_{\ell }|^2}\left( \frac{\gamma }{\rho _R}\right) . \end{aligned}$$

(50)

Using (49) in (50), we get (1).

Appendix B: Proof of Lemma 4

From (39) $\mathcal {J}_1$ can be written as

$$\begin{aligned} \mathcal {J}_1=\mathbb {E}\left[ \ln {\left( 1+\Lambda \gamma ^{\text {FSO}}_q\right) }\right] =\mathbb {E}\left[ \ln {\left( 1+\Lambda \bar{\gamma }^{\text {FSO}}_qI^q\right) }\right] , \end{aligned}$$

(51)

where $\mathbb {E}[\cdot ]$ is the mean operator. Since $\ln {(1+x)}\approx \ln {(x)}$ when $x\rightarrow \infty$, the asymptotic expression of $\mathcal {J}_1$ in (51) considering high SNR conditions, i.e., $\bar{\gamma }_q^{\text {FSO}} \rightarrow \infty$ can be accurately lower-bounded as follows

$$\begin{aligned} \lim _{\bar{\gamma }_q^{\text {FSO}}\rightarrow \infty }\mathcal {J}_1 = \tilde{\mathcal {J}_1}=\ln {\left( \bar{\gamma }^{\text {FSO}}_q\right) }+\frac{1}{\Lambda }\int _{0}^{\infty }\frac{\mathbb {F}'_{I^q}(i)}{i}\mathrm{{d}}i, \end{aligned}$$

(52)

where $\mathbb {F}'_{I^q}(i)$ is the CCDF of $I_q$ which can be obtained using (41) by linear transformation as ${\mathbb {F}}'_{I^q}(i)= \mathbb {F}'_{\gamma ^{\text {FSO}}_q}\left( \bar{\gamma }^{\text {FSO}}_qi\right)$. Further, utilizing [27, (07.34.21.0086.01)] in (52), we get (45). Further, $\tilde{\mathcal {J}}_2=\lim _{\bar{\gamma }\rightarrow \infty }\mathcal {J}_2$, can be obtained from (43) by utilizing high SNR asymptotic expansion of Bessel function expression and then using [27, (07.34.21.0086.01)], we get (46).