Adaptive cooperation for free space optical communications

Abstract

In this paper, we propose a new Adaptive Cooperation (AC) protocol for Free Space Optical (FSO) communications. Two versions of the protocol are presented. The first protocol uses the Instantaneous Signal to Noise Ratio (ISNR) to activate the relay only when the corresponding instantaneous throughput is better than that of the direct link. The second protocol uses the Average SNR (ASNR) to activate the relay when its average throughput is larger than that of the direct link. The proposed AC protocol has been extended to include Adaptive Modulation and Coding (AMC). Our results are valid for FSO communications in the presence of Gamma Gamma atmospheric turbulence.

Appendices

Appendix A : PDF of SNR of AF relaying when atmospheric turbulence follows a Gamma Gamma Distribution

The SNR of AF relaying between the source S, relay R and destination D is upper bounded by

$$\begin{aligned} \Gamma _{SRD}<\Gamma _{SRD}^{up}=min(\Gamma _{SR},\Gamma _{RD}) \end{aligned}$$

(51)

If $\Gamma _{SR}$ and $\Gamma _{RD}$ are independent, the Cumulative Distribution Function (CDF) of $\Gamma _{SRD}^{up}$ is given by

$$\begin{aligned} F_{\Gamma _{SRD}^{up}}(x)= & {} 1-[1-F_{\Gamma _{SR}}(x)][1-F_{\Gamma _{RD}}(x)]\nonumber \\= & {} F_{\Gamma _{SR}}(x)+F_{\Gamma _{RD}}(x)-F_{\Gamma _{SR}}(x)F_{\Gamma _{RD}}(x)\nonumber \\ \end{aligned}$$

(52)

We deduce the PDF of $\Gamma _{SRD}^{up}$ :

$$\begin{aligned} f_{\Gamma _{SRD}^{up}}(x)= & {} f_{\Gamma _{SR}}(x)+f_{\Gamma _{RD}}(x)\nonumber \\&\quad -f_{\Gamma _{SR}}(x)F_{\Gamma _{RD}}(x)-F_{\Gamma _{SR}}(x)f_{\Gamma _{RD}}(x) \end{aligned}$$

(53)

$f_{\Gamma _{SR}}(x)$ and $f_{\Gamma _{RD}}(x)$ are the PDF of SNR of first and second hop expressed similarly to (25), $F_{\Gamma _{SR}}(x)$ is the CDF of SNR expressed as

$$\begin{aligned} F_{\Gamma _{SR}}(x)= & {} \frac{{\overline{\Gamma }}_{SR}^{\frac{\alpha +\beta }{2}}(\alpha \beta )^{\frac{\alpha +\beta }{2}}}{2^{\alpha +\beta -2}\Gamma (\alpha )\Gamma (\beta )}\nonumber \\&\quad D_{\alpha +\beta -1,\alpha -\beta }\left( 2\sqrt{\frac{\alpha \beta x}{{\overline{\Gamma }}_{SR}}}\right) \end{aligned}$$

(54)

where

$$\begin{aligned} D_{a,b}=\int _0^{+\infty }u^aK_b(u)du. \end{aligned}$$

(55)

$F_{\Gamma _{RD}}(x)$ is the CDF of $\Gamma _{RD}$ expressed similarly to (54) where we have to replace ${\overline{\Gamma }}_{SR}=E(\Gamma _{RD})$ by ${\overline{\Gamma }}_{RD}=E(\Gamma _{RD})$.

Appendix B

When there are K relays, we choose the relay node $R_{sel}$ that has the best SNR:

$$\begin{aligned} \Gamma _{SR_{sel}D}=\underset{1\le k\le K}{max}(\Gamma _{SR_kD}) \end{aligned}$$

(56)

Assuming that $\Gamma _{SR_{k}D}$ are independent, the CDF of $\Gamma _{SR_{sel}D}$ is given by

$$\begin{aligned} F_{\Gamma _{SR_{sel}D}}(x)=\prod _{k=1}^KF_{\Gamma _{SR_{k}D}}(x). \end{aligned}$$

(57)

We deduce the PDF of $\Gamma _{SR_{sel}D}$

$$\begin{aligned} f_{\Gamma _{SR_{sel}D}}(x)= & {} \sum _{n=1}^Kf_{\Gamma _{SR_{n}D}}(x)\nonumber \\&\quad \prod _{k=1, k\ne n}^KF_{\Gamma _{SR_{k}D}}(x). \end{aligned}$$

(58)