Optimal routing and one hop routing for D2D communications in the presence of mutual interference

Abstract

The concept of device to device (D2D) communications allows to increase the capacity of cellular systems. In such systems, a D2D pair directly communicate without passing through the base station and use the same frequency band. Therefore, this communication generates interference to the base station. In some contexts, the D2D pair are not close to each other and relaying should be performed. In this paper, we investigate the performance of different routing protocols for D2D communications. The network is composed of D2D pair, multiple relays organized in clusters, a cellular user communicating with a base station. In such network, there is interference from the relays to the base station and interference from the cellular user to the relays. We call it mutual interference. We investigate optimal and suboptimal routing for D2D communications in the presence of interference. Optimal routing consists to select the path among available ones with the largest end-to-end signal to interference plus noise ratio. A sub-optimal one-hop routing is also proposed.

Appendix A

In the presence of interference from U cellular users, the SINR between two nodes X and Y is written as

$$\begin{aligned} \gamma _{X,Y}=\frac{E_{X}|h_{X,Y}|^{2}}{\sum _{i=1}^UE_{ci}|h_{ciY}|^{2}+N_{0}} \end{aligned}$$

(18)

where $E_{ci}$ is the transmitted energy per symbol of cellular user i, $h_{cY}$ is the channel coefficient between cellular user transmitter and node Y. $I=\sum _{i=1}^UE_{ci}|h_{ciY}|^{2}$ represents interference at node Y from U cellular user.

The CDF of the SINR is written as

$$\begin{aligned} P(\gamma _{X,Y}<\gamma )=P(E_{X}|h_{X,Y}|^{2}<\gamma (I+N_{0})) \end{aligned}$$

(19)

Since Rayleigh fading channels are assumed, $Z=E_{X}|h_{X,Y}|^{2}$ is exponentially distributed with mean $\sigma _{X,Y}^{2}=E_{X}E(|h_{X,Y}|^{2})$. We denote by $\sigma _{ci,Y}^{2}=E_{ci}E(|h_{ciY}|^{2})$ where E(.) is the expectation.

Therefore, the CDF can be written as

$$\begin{aligned} F_{\gamma _{X,Y}}(\gamma )= & {} P(Z<\gamma (N_{0}+I)) \nonumber \\= & {} \int _{N_{0}}^{+\infty }F_{Z}(\gamma u)f_{I}(u-N_{0})du \end{aligned}$$

(20)

where $F_{Z}(u)=P(Z<u)$ is the CDF of Z and $f_{I}(u)$ is the PDF of I.

I is the sum of exponential random variables. The PDF of I can be obtained using the Moment Generating Function (MGF) of sum of independent random variable that is the product of individual MGF. Using the results of [44], we have

$$\begin{aligned} f_{I}(u)=\sum _{i=1}^U \frac{res_i }{\sigma _{ci,Y}^{2}}e^{-\frac{u}{\sigma _{ci,Y}^{2}}} \end{aligned}$$

(21)

where $res_i$ are the residues given by

$$\begin{aligned} res_i=\prod _{k\ne i, k=1}^U \frac{\sigma _{ci,Y}^{2}}{\sigma _{ci,Y}^{2}-\sigma _{ck,Y}^{2}} \end{aligned}$$

(22)

We deduce

$$\begin{aligned} F_{\gamma _{X,Y}}(\gamma )= & {} \int _{N_{0}}^{+\infty }\left[ 1-e^{-\frac{ \gamma u}{\sigma _{X,Y}^{2}}}\right] \sum _{i=1}^U res_i e^{-\frac{(u-N_{0})}{\sigma _{ci,Y}^{2}}}\frac{1}{\sigma _{ci,Y}^{2}}du \nonumber \\= & {} 1-\sum _{i=1}^U res_i\frac{\sigma _{X,Y}^{2}}{\sigma _{X,Y}^{2}+\gamma \sigma _{ci,Y}^{2}} e^{-\frac{N_{0}\gamma }{\sigma _{X,Y}^{2}}} \end{aligned}$$

(23)

By a simple derivative, we obtain the PDF of SINR $\Gamma _{X,Y}$

$$\begin{aligned} f_{\gamma _{X,Y}}(\gamma )= & {} \sum _{i=1}^U res_i \left[ \frac{N_{0}}{\sigma _{X,Y}^{2}+\gamma \sigma _{ci,Y}^{2}}e^{-\frac{N_{0}\gamma }{\sigma _{X,Y}^{2}}} \right. \nonumber \\&\quad \left. +\frac{\sigma _{ci,Y}^{2}\sigma _{X,Y}^{2}}{\left( \sigma _{X,Y}^{2}+\gamma \sigma _{ci,Y}^{2}\right) ^{2}}e^{-\frac{N_{0}\gamma }{\sigma _{X,Y}^{2}}}\right] \end{aligned}$$

(24)