Two-way mixed VLC/RF system in the presence of randomly positioned relay

Abstract

This paper presented an adaptive two-way relay (TWR) based mixed visible light communication (VLC)/radio frequency (RF) network to improve the accessibility to the mobile user. The VLC link is constrained by a geometrical channel model based on intensity modulation (IM)/direct detection (DD), while the RF link is modelled by a nakagami-$\textit{m}$ fading distribution. It is considered that the VLC-link in the first-hop is decoded and forwarded (DF) by the relay (R) to the RF signal in the second-hop. Thereafter, selection combining (SC) and switch-and-examine combining (SEC) are implemented at R. The novel closed form expressions of outage probability (OP) and bit error rate (BER) for both SC and SEC schemes are derived and the performance results of both the schemes are compared. The SC is found to be $\sim 3\,\text {dB}$ more power efficient than the SEC at a target BER of $10^{-3}$. Moreover, the performance of this DF based TWR scheme is compared with other transmission schemes including amplify and forward TWR (AF-TWR), one-way relaying (OWR), uplink, downlink, direct VLC, and RF transmissions. The considered TWR scheme increases the spectral efficiency significantly as compared to OWR while maintaining the same outage performance. Furthermore, the performance of the system is determined based on different system parameters such as Nakagami-$\textit{m}$ fading parameter, path loss exponent, half illumination angle, switching threshold, VLC field of view (FOV) angle, distance between LED source and photo-detector (PD), and number of LED luminaries. In addition, all the theoretical results are obtained asymptotically at high signal-to-noise ratios (SNRs) in simple elementary functions, and that are validated by the simulation results.

Appendices

Appendix A

The Taylor series expansion of any differentiable function $X\left( t\right)$ at a value A can be written as

$$\begin{aligned} X\left( t\right) =\sum _{n=0}^{\infty }\frac{X^{\left( n\right) }\left( A\right) }{n!}\left( t-A\right) ^n \end{aligned}$$

(A.1)

Therefore, the CDF of the VLC link at $t = \gamma _{min}$ can be expressed in Taylor series format as

$$\begin{aligned} \begin{aligned} F_{\gamma _{RS_{i}}}\left( \gamma _{th}\right)&= 0+ \left( \frac{\varpi _{1}}{m_{rs}+3}\right) \left( \frac{\gamma _{th}-\gamma _{min}}{K_{mn}}\right) \frac{1}{{\bar{\gamma }}}-\left( \frac{\varpi _{1}}{m_{rs}+3}\right) \left( 1+\frac{1}{m_{rs}+3}\right) \\&\quad \times \left( \frac{\gamma _{th}-\gamma _{min}}{K_{mn}}\right) ^2\left( \frac{1}{{\bar{\gamma }}}\right) ^2+,..., \end{aligned} \end{aligned}$$

(A.2)

Similar procedure can be applied for $F_{\gamma _{S_{i}R}}\left( \gamma _{th}\right)$.

Furthermore, the CDF of the RF links at $t = 0$ can be expressed in Taylor series format as

$$\begin{aligned} \begin{aligned} F_{\gamma _{cd}}\left( \gamma _{th}\right)&= 0+,...,+\frac{1}{m_{cd}!}\left( \frac{m_{cd}\gamma _{th}}{{\bar{\gamma }}}\right) ^{m_{cd}}+,..., \end{aligned} \end{aligned}$$

(A.3)

Thereafter, by substituting the first dominant term of each Taylor series expansion in (20), the asymptotic expression of OP can be obtained as shown in (21).

Appendix B

By substituting (18) into (29) and using expansion of incomplete gamma function (Gradshteyn et al. 2007, eq. (8.352.4)) as

$\frac{\Gamma \left( m,m\gamma /{\bar{\gamma }}\right) }{\Gamma \left( m\right) } = \exp \left( -\frac{m\gamma }{{\bar{\gamma }}}\right) \sum _{k=0}^{m-1}\frac{\left( m\gamma /{\bar{\gamma }}\right) ^{k}}{k!}$, (29) can be rewritten as

$$\begin{aligned} \begin{aligned} {{\bar{P}}}_{b}&=\frac{q^{p}}{2\Gamma \left( p\right) }\int _{0}^{\infty }exp\left( -q\gamma \right) \gamma ^{p-1}\left. \Bigg [1- \exp \left( -\frac{m_{rd}\gamma }{{\bar{\gamma }}_{rd}}\right) \sum _{k=0}^{m_{rd}-1}\frac{1}{k!}\right. \left( \frac{m_{rd}\gamma }{{\bar{\gamma }}_{rd}}\right) ^{k} \\&\quad \times \exp \left( -\frac{m_{dr}\gamma }{{\bar{\gamma }}_{dr}}\right) \sum _{l=0}^{m_{dr}-1}\frac{1}{l!}\left( \frac{m_{dr}\gamma }{{\bar{\gamma }}_{dr}}\right) ^{l}\left. \Bigg \{\left( 1-\varpi _{1}\right) \right. \left. +\bigg (\varpi _{rs} \gamma ^{-\frac{1}{m_{rs}+3}} \bigg ) \right. - \left( 1-\varpi _{1}\right) \\&\quad \times \left. \Bigg ( \sum _{r=0}^{N}{N \atopwithdelims ()r}\left( -1\right) ^{r} \right. \left. \times \varpi _{1}^{N-r}\left( \varpi _{sr}\gamma ^{-\frac{1}{m_{sr}+3}}\right) ^{r}\Bigg )\right. - \varpi _{rs} \left. \times \Bigg (\sum _{r=0}^{N}{N \atopwithdelims ()r} \left( -1\right) ^{r} \right. \\&\quad \times \left. \varpi _{sr}^{r}\varpi _{1}^{N-r}\gamma ^{-\left( \frac{r}{m_{sr}+3}+\frac{1}{m_{rs}+3}\right) }\Bigg )\Bigg \} \Bigg ] d\gamma \right. \end{aligned} \end{aligned}$$

(B.1)

Following that, the final average BER is calculated in (30) using the fact that $F_{S_{i}R}\left( \gamma \right)$ and $F_{RS_{i}}\left( \gamma \right)$ are valid for $\gamma \in \left\{ \gamma _{min},\gamma _{max}\right\}$ and Gradshteyn et al. (2007, eq. (8.350.2)).