Round Robin, Centralized and Distributed Relay Selection for Free Space Optical Communications

Abstract

In this paper, we derive the Symbol Error Probability (SEP) of Round Robin (RRS), Centralized (CRS) and Distributed Relay Selection (DRS) techniques for Free Space Optical communications. RRS consists to randomly select a node as relay with equal probability. It does not consider the SNR value for relay selection. In CRS, the SNRs of different relays are sent to a Central Node (CN) for detecting the relay with highest SNR. The CN selects the best relay with largest SNR. We propose a new DRS where each relay is allowed to transmit only when its SNR is higher than a given threshold T. The threshold T is optimized with the Gradient algorithm to guarantee the lowest SEP. The suggested DRS with optimal threshold allows close performance to CRS and better performance than RRS. DRS is less complex than CRS since no signalization is needed.

Appendices

Appendix A : SEP of a Direct Link

For M-PAM modulation, the SEP of the direct link for a given SNR $\gamma$ is written as

$$\begin{aligned} P_{e,MPAM}(\gamma )= & {} 0.5berfc\left( a\sqrt{\gamma /2 }\right) , \end{aligned}$$

(21)

$$\begin{aligned} b= & {} \frac{2(M-1)}{M}, \end{aligned}$$

(22)

$$\begin{aligned} a= & {} \sqrt{\frac{3\log _{2}(M)}{(M-1)(2M-1)}}. \end{aligned}$$

(23)

The average SEP of the S-$D_{i}$ link is written as

$$\begin{aligned} Pe_{direct}(D_{i})=0.5b\int _{0}^{+\infty }erfc\left( a\sqrt{\gamma /2 }\right) p_{\Gamma _{SD_{i}}}(\gamma )d\gamma . \end{aligned}$$

(24)

We have [32]

$$\begin{aligned} 0.5erfc(x/sqrt(2))=\frac{1}{\pi }\int _{0}^{\frac{\pi }{2}}e^{-\frac{x^{2}}{2\sin ^{2}(\theta )}}d\theta . \end{aligned}$$

(25)

We deduce

$$\begin{aligned} Pe_{direct}(D_{i})= & {} \frac{b}{\pi }\int _{0}^{+\infty }\int _{0}^{\frac{\pi }{2}}e^{-\frac{a^{2}\gamma }{2\sin ^{2}(\theta )}}d\theta p_{\Gamma _{SD_{i}}}(\gamma )d\gamma \nonumber \\= & {} \frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}\left[ \int _{0}^{+\infty }e^{- \frac{a^{2}\gamma }{2\sin ^{2}(\theta )}}p_{\Gamma _{SD_{i}}}(\gamma )d\gamma \right] d\theta . \end{aligned}$$

(26)

Let $M_{\Gamma _{SD_{i}}}(s)$ be the Moment Generating Function (MGF) of SNR $\Gamma _{SD_{i}}$

$$\begin{aligned} M_{\Gamma _{SD_{i}}}(s)=E(e^{-s\Gamma _{SD_{i}}})=\int _{0}^{+\infty }e^{-sx}p_{\Gamma _{SD_{i}}}(\gamma )dx. \end{aligned}$$

(27)

We deduce from (26) and (27) that

$$\begin{aligned} Pe_{direct}(D_{i})=\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{i}}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) d\theta . \end{aligned}$$

(28)

Using the Laplace Transform (LT) table, the MGF of SNR $\Gamma _{SD_{i}}$ is written as

$$\begin{aligned} M_{\Gamma _{SD_{i}}}(s)=\frac{1}{\left( 1+s\frac{\overline{\Gamma } _{SD_{i}}}{k}\right) ^{k}}. \end{aligned}$$

(29)

$\overline{\Gamma }_{SD_{i}}$ is the average SNR between S and $D_i$.

Using (28) and (29), we can write

$$\begin{aligned} Pes_{direct}(D_{i})=\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}\frac{1}{\left( 1+\frac{ a^{2}\overline{\Gamma }_{SD_{i}}}{k2\sin ^{2}(\theta )}\right) ^{k}} d\theta . \end{aligned}$$

(30)

This integral can be solved in closed form [32]

$$\begin{aligned} Pe_{direct}(D_{i})=\frac{b}{2}\frac{\sqrt{\frac{a^{2}\overline{ \Gamma }_{SD_{i}}}{2k}}}{\left( 1+\frac{a^{2}\overline{\Gamma } _{SD_{i}}}{2k}\right) ^{k+1/2}}\frac{\Gamma (k+1/2)}{\Gamma (k+1)}\text { } _{2}F_{1}\left( 1,k+1/2,k+1,\frac{k}{k+a^{2}\frac{\overline{\Gamma } _{SD_{i}}}{2}}\right) \end{aligned}$$

(31)

where $_2F_1(.,.,.,.)$ is the hypergeometric function.

Appendix B : SEP of AF Relayed Link

The SNR of AF relaying is equal to

$$\begin{aligned} \Gamma _{SD_{j}D_{i}}=\frac{\Gamma _{SD_{j}}\Gamma _{D_{j}D_{i}}}{\Gamma _{SD_{j}}+\Gamma _{D_{j}D_{i}}+1}, \end{aligned}$$

(32)

The SNR of AF relaying can be upper bounded and lower bounded by

$$\begin{aligned} \Gamma _{SD_{j}D_{i}}^{low}=\frac{1}{2}\min (\Gamma _{SD_{j}},\Gamma _{D_{j}D_{i}})<\Gamma _{SD_{j}D_{i}}<\Gamma _{SD_{j}D_{i}}^{up}=\min (\Gamma _{SD_{j}},\Gamma _{D_{j}D_{i}}). \end{aligned}$$

(33)

1.1 B.1 Lower Bound of SEP

The CDF of $\Gamma _{SD_{j}D_{i}}^{up}$ is expressed as

$$\begin{aligned} P_{\Gamma _{SD_{j}D_{i}}^{up}}(x)=P(\min (\Gamma _{SD_{j}},\Gamma _{D_{j}D_{i}})<x). \end{aligned}$$

(34)

Assuming that the SNR of first and second hops are independent, we can write

$$\begin{aligned} P_{\Gamma _{SD_{j}D_{i}}^{up}}(x)= & {} 1-P(\Gamma _{SD_{j}}>x)P(\Gamma _{D_{j}D_{i}}>x) \nonumber \\= & {} 1-G^{up}\left( k,\frac{kx}{\overline{\Gamma }_{SD_{j}}}\right) G^{up}\left( k,\frac{kx}{ \overline{\Gamma }_{D_{j}D_{i}}}\right) \end{aligned}$$

(35)

The PDF of SNR of relaying link is deduced by a simple derivative of the CDF

$$\begin{aligned} p_{\Gamma _{SD_{j}D_{i}}^{up}}(x)= & {} \frac{1}{\Gamma (k)}\left[ \left( \frac{ k}{\overline{\Gamma }_{SD_{j}}}\right) ^k x^{k-1}e^{-\frac{kx}{\overline{\Gamma }_{SD_{j}}}}G^{up}\left( k,\frac{kx}{\overline{\Gamma }_{D_{j}D_{i}}}\right) +\right. \nonumber \\&+\left. \left( \frac{k}{\overline{\Gamma }_{D_{j}D_{i}}}\right) ^k x^{k-1}e^{- \frac{kx}{\overline{\Gamma }_{D_{j}D_{i}}}}G^{up}\left( k,\frac{kx}{\overline{ \Gamma }_{SD_{j}}}\right) \right] \end{aligned}$$

(36)

A LT of the PDF of SNR provides the MGF [32]

$$\begin{aligned}&M_{\Gamma _{SD_{j}D_{i}}^{up}}(s)=\left( \frac{k}{\overline{\Gamma }_{SD_{j}} }\right) ^{k}\left( \frac{k}{\overline{\Gamma }_{D_{j}D_{i}}}\right) ^{k} \frac{\Gamma (2k-1)}{k[\Gamma (k)]^{2}}\frac{1}{\left( \frac{k}{\overline{\Gamma } _{SD_{j}}}+\frac{k}{\overline{\Gamma }_{D_{j}D_{i}}}+s\right) ^{2k}} \nonumber \\&\quad \left[ _{2}F_{1}\left( 1,2k,k+1,\frac{\frac{k}{\overline{\Gamma }_{SD_{j}}}+s}{ \frac{k}{\overline{\Gamma }_{SD_{j}}}+\frac{k}{\overline{\Gamma } _{D_{j}D_{i}}}+s}\right) +_{2}F_{1}\left( 1,2k,k+1,\frac{\frac{k}{\overline{\Gamma } _{D_{j}D_{i}}}+s}{\frac{k}{\overline{\Gamma }_{SD_{j}}}+\frac{k}{\overline{ \Gamma }_{D_{j}D_{i}}}+s}\right) \right] , \end{aligned}$$

(37)

where $_{2}F_{1}$ is the Gauss’ Hypergeometric function.

The SEP can be lower bounded by

$$\begin{aligned} Pe_{relayed}(D_{i})>\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{j}D_{i}}^{up}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) d\theta . \end{aligned}$$

(38)

1.2 B.2 Upper Bound of SEP

Let $U=\Gamma _{S,D_{j},D_i}^{low}$ and $V=\Gamma _{S,D_{j},D_i}^{up}$. We have U=V/2.

We deduce

$P_U(u)=P_V(2u)$,

$p_U(u)=2p_V(2u)$,

$M_U(s)=E(e^{-sU})=M_V(s/2)$,

The SEP can be upper bounded by

$$\begin{aligned} Pe_{relayed}(D_{i})<\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{j}D_{i}}^{low}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) d\theta =\frac{b}{ \pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{j}D_{i}}^{up}}\left( \frac{a^{2}}{ 4\sin ^{2}(\theta )}\right) d\theta . \end{aligned}$$

(39)

where $M_{\Gamma _{SD_{j}D_{i}}^{up}}(s)$ provided in (37).

Appendix C : SEP of Combined Direct and Relayed Links

The SNR at $D_{i}$ when the relayed and direct links are combined is written as

$$\begin{aligned} \Gamma _{coop}=\Gamma _{SD_{i}}+\Gamma _{SD_{j}D_{i}}. \end{aligned}$$

(40)

The PDF of $\Gamma _{coop}$ is equal to

$$\begin{aligned} p_{\Gamma _{coop}}(x)=(p_{\Gamma _{SD_{i}}}*p_{\Gamma _{SD_{j}D_{i}}})(x), \end{aligned}$$

(41)

Therefore, the MGF of $\Gamma _{coop}$ is written as

$$\begin{aligned} M_{\Gamma _{coop}}(s)=M_{\Gamma _{SD_{i}}}(s)M_{\Gamma _{SD_{j}D_{i}}}(s). \end{aligned}$$

(42)

Using lower and upper bounds of SNR (33), we can write

$$\begin{aligned} \Gamma _{coop}^{low}=\Gamma _{SD_{i}}+\Gamma _{SD_{j}D_{i}}^{low}<\Gamma _{coop}<\Gamma _{coop}^{up}=\Gamma _{SD_{i}}+\Gamma _{SD_{j}D_{i}}^{up} \end{aligned}$$

(43)

Therefore, the SEP is upper bounded by

$$\begin{aligned} Pe_{coop}(D_{i})<\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{i}}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) M_{\Gamma _{SD_{j}D_{i}}^{low}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) d\theta . \end{aligned}$$

(44)

The SEP is lower bounded by

$$\begin{aligned} Pe_{coop}(D_{i})>\frac{b}{\pi }\int _{0}^{\frac{\pi }{2}}M_{\Gamma _{SD_{i}}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) M_{\Gamma _{SD_{j}D_{i}}^{up}}\left( \frac{a^{2}}{2\sin ^{2}(\theta )}\right) d\theta . \end{aligned}$$

(45)

Appendix D

We have

$$\begin{aligned} \frac{\partial Pe(D_i)}{\partial T}= & {} -0.5p_{\Gamma _{SD_{j}}}(T)\int _{0}^{T}p_{\Gamma _{coop}}(x)berfc(a\sqrt{x/2 })dx\nonumber \\&+0.5[1-P_{\Gamma _{SD_{j}}}(T)]p_{\Gamma _{coop}}(T)berfc(a\sqrt{T/2 }) \nonumber \\&+0.5p_{\Gamma _{SD_{j}}}(T)\int _{0}^{T}p_{\Gamma _{SD_{i}}}(x)berfc(a\sqrt{x/2})dx \nonumber \\&+0.5P_{\Gamma _{SD_{j}}}(T)p_{\Gamma _{SD_{i}}}(T)berfc(a\sqrt{T/2})\nonumber \\&-0.5p_{\Gamma _{SD_{j}}}(T)\int _{T}^{+\infty }p_{\Gamma _{SD_i}}(x)berfc(a\sqrt{x/2 })dx \nonumber \\&-0.5[1-P_{\Gamma _{SD_{j}}}(T)]p_{\Gamma _{SD_{i}}}(T)berfc(a\sqrt{T/2}) \nonumber \\&+0.5p_{\Gamma _{SD_{j}}}(T)\int _{T}^{+\infty }p_{\Gamma _{max}}(x)berfc(a\sqrt{x/2})dx \nonumber \\&-0.5P_{\Gamma _{SD_{j}}}(T)p_{\Gamma _{max}}(T)berfc(a\sqrt{T/2}) \end{aligned}$$

(46)