Location-Based Relay Selection in Full-Duplex Random Relay Networks

Abstract

Full-duplex relay (FDR) has attracted considerable interest in enhancing the performance of relay networks by utilizing resources more efficiently. In this paper, we propose a framework for full-duplex random relay networks (FDRRNs), where relay nodes equipped with full-duplex (FD) capability are randomly distributed within a finite two-dimensional region. We first derive the outage probability of an FDRRN and then identify the potential relay location that minimizes the outage probability. Furthermore, we introduce a low-complexity relay selection algorithm that selects the relay node nearest to the potential relay location. Finally, simulation results show that the proposed relay selection algorithm achieves performance comparable to that of the max-min relay selection algorithm.

1. Introduction

Relay networks are considered an efficient method for expanding coverage and ensuring reliable communication by utilizing multiple relay nodes for data transmission, thereby achieving spatial diversity [1,2,3,4]. Specifically, relay nodes can provide reliable channel links for users who suffer from severe channel impairment or fading due to unexpected interference or obstacles. In addition, by amplifying or regenerating weak signals, relay nodes enable these signals to reach remote users, which extends network coverage.

In particular, relay networks have been widely adopted in interference-limited environments, such as cognitive radio networks [5,6,7,8] and non-orthogonal multiple-access systems [9,10,11], to mitigate the effects of interference. Additionally, recent research on 6G systems has explored cooperative techniques to enhance coverage for non-terrestrial networks, including satellite communications [12], unmanned aerial vehicle networks [13,14,15], and device-to-device relaying [16].

However, these studies on relay systems focus on half-duplex (HD) mode, where each relay hop uses different temporal and spectral resources due to its simplicity. In other words, half-duplex relay (HDR) systems gain benefits at the expense of inefficient resource utilization.

To address the limitations of HDR, full-duplex relay (FDR) has attracted considerable attention across various fields [17,18]. In FDR, a relay node can transmit and receive signals simultaneously using the same frequency band, ideally doubling the spectral efficiency compared to HDR. One of the main hurdles in implementing FDR is managing strong self-interference (SI), which is induced at a node by its own transmitted signal while it receives another signal simultaneously. With the help of recent advances in self-interference cancellation (SIC), however, it has been demonstrated that SI on the same frequency band can be reduced to the noise level, thereby facilitating full-duplex (FD) transmission at the node [19,20,21,22,23,24].

Motivated by the progress of SIC techniques, extensive research on FDR has been conducted [25,26,27,28,29]. Specifically, the transceiver design for multiple-input multiple-output (MIMO) systems in FDR with amplify-and-forward (AF) relay mode was examined in [25]. In [26], the power control algorithm for FDR under various channel environments was studied. Additionally, a comparison between HDR and FDR was conducted in terms of latency in [27] and energy efficiency in [28]. In [29], a buffer-aware rate allocation and antenna selection algorithm was investigated to enhance the scheduling performance of FDR. However, the aforementioned works only consider channel variation while assuming that the locations and number of relay nodes are fixed. In practical networks, on the other hand, the locations and number of available relay nodes can vary and are difficult to determine accurately due to the mobility of nodes.

To capture the random nature of real networks, the performance of relay networks has been analyzed using stochastic geometry, where the relay nodes are randomly distributed within a finite two-dimensional plane [30,31,32,33]. In [30,31], the outage probabilities of the AF and decode-and-forward (DF) relay transmission protocols were explored when relay nodes were randomly distributed. Behnad et al. [32] analyzed the statistical performance of the distances among relay nodes under the generalized Gamma fading model. In addition, relay selection algorithms considering fairness and algorithms for cognitive relay networks were investigated in [33].

However, these studies were conducted under the assumption of HD mode in relay transmission, and no works considered FD mode in randomly distributed relay networks. In full-duplex random relay networks (FDRRNs), SI is a dominant factor influencing performance. Therefore, along with network interference, SI should be addressed in performance analysis and relay selection algorithms.

The goal of this paper is to investigate the performance of full-duplex random relay networks (FDRRNs). We first derive the outage probability of a full-duplex random relay network (FDRRN) considering SI. We then identify the potential relay location with respect to SI and network interference. Lastly, we propose a location-based relay selection algorithm for full-duplex random relay networks (FDRRN)s with reduced complexity and present simulation results to validate the effectiveness of the proposed algorithm.

The remainder of this paper is organized as follows. In Section 2, the system model is described. The outage probability of an FDRRN is derived in Section 3. In Section 4, we propose a location-based relay selection algorithm that reduces complexity. Simulation results are presented in Section 5. Conclusions are given in Section 6.

2. System Model

We consider a spectrum-sharing relay network in which all nodes utilize the same spectrum for communication. This network is illustrated in Figure 1. Also, the notations used in this paper are summarized in

Table A1. Notations used throughout the paper.

Notation	Definition
$P_t$	Tx. power of each node
$H_{\mathbf{x}\mathbf{y}}$	Fading gain of the link between nodes x and y
$D_{\mathbf{x}\mathbf{y}}$	Distance between nodes x and y
$D_i$	Distance of the ith relay hop
$I_{SI}$	Residual SI
${\gamma }_i$	SIR of the ith relay hop
$\alpha$	Path-loss exponent
$\tau$	Target SIR of the FDRRN
${\mathbf{\Pi }}_i$	PPP for interfering node of the ith relay hop
${\mathbf{\Pi }}_{\mathrm{R}}$	PPP for candidate relay nodes
${\lambda }_i$	Spatial density of interfering node of the ith relay hop
${\lambda }_{\mathrm{R}}$	Spatial density of candidate relay nodes
$p_{\mathrm{s},i}$	The probability of successful transmission in the ith relay hop
$p_{\mathrm{o}}$	The outage probability of the FDRRN
$\mathbf{x}$	Location of transmitter node
$\mathbf{y}$	Location of receiver node
${\mathbf{x}}_{\mathrm{r}}$	Location of candidate relay node
${\mathbf{x}}_{\mathrm{r}}^{\ast }$	Potential relay location
${\mathbf{x}}_{\mathrm{o}}^{\ast }$	Optimal relay location

.

We assume that the transmitter node sends signals to the receiver node with equal transmission power. In addition, we consider path loss and Rayleigh fading for the propagation channel model.

Let us assume that the transmitter nodes are distributed according to a Poisson point process (PPP) with a spatial density of ${\lambda }_{\mathrm{tx}}$ over a finite two-dimensional plane [34]. Each transmitter node establishes a communication link with its corresponding receiver node. It is also assumed that some direct links between the transmitter node and receiver node experience outages due to unexpected channel impairments. To ensure a reliable communication link, those transmitter nodes exploit a relay node selected from a set of candidate relay nodes, which are distributed according to a PPP with a spatial density of ${\lambda }_{\mathrm{R}}$.

We assume that the relay nodes adopt the DF protocol to convey the data from the transmitter node to the receiver node. In addition, each relay node is equipped with SIC capability, which facilitates FD transmission. Hence, relay nodes can simultaneously transmit the signal received from the transmitter node to the receiver node while receiving the signal from the transmitter node by performing SIC.

During FD transmission, the relay node suffers from residual SI due to imperfect SIC. Regarding the residual SI, we consider a constant value in this paper for simplicity of analysis [35]. However, note that the analysis of its impact can be extended by averaging over the distribution of the residual SI.

Transmitting nodes, including transmitter nodes and relay nodes in FDRRNs, generate interference to the receiving nodes at each relay hop. Let us assume that the transmitting nodes of the first and second relay hops follow a PPP ${\mathbf{\Pi }}_1$ with a spatial density of ${\lambda }_1$, and a PPP ${\mathbf{\Pi }}_2$ with spatial densities of ${\lambda }_1$ and ${\lambda }_2$. Additionally, we assume that the interference is independent of different times and locations [36].

3. Outage Probability of an FDRRN

In this section, the outage probability of an FDRRN is formulated. We begin by defining the instantaneous signal-to-interference ratio (SIR) for the ith relay hop, where the transmitter and receiver nodes are denoted as $\mathbf{x}$ and $\mathbf{y}$, respectively, as follows:

$$\begin{array}{c}\hfill {\gamma }_i={\displaystyle \frac{P_t{\mathsf{H}}_{\mathbf{x}\mathbf{y}}{\mathsf{D}}_{\mathbf{x}\mathbf{y}}^{-\alpha }}{I_{SI}+{\displaystyle \sum _{\mathsf{Z}\in {\mathbf{\Pi }}_i/\left\{\mathbf{X},\mathbf{Y}\right\}}}{P}_t{\mathsf{H}}_{\mathsf{Z}\mathbf{Y}}{\mathsf{D}}_{\mathsf{Z}\mathbf{Y}}^{-\alpha }}},\forall i\in \{1,2\},\end{array}$$

(1)

where $\alpha$ is the path-loss exponent, $P_t$ is the transmission power of the transmit node, and $I_{SI}$ means the residual SI after performing imperfect SIC. In addition, ${\mathsf{H}}_{\mathbf{x}\mathbf{y}}$ and ${\mathsf{D}}_{\mathbf{x}\mathbf{y}}$ denote the Rayleigh fading channel gain and the distance between the transmitter node $\mathbf{x}$ and receiver node $\mathbf{y}$, respectively.

Based on Equation (1), the success probability of a typical link for the ith relay hop can be given by [37]

$$\begin{array}{c}\hfill p_{\mathrm{s},i}\left(\tau \right)=exp\left\{-{\displaystyle \frac{I_{SI}{D}_i^{\alpha }\tau }{P_t}}\right\}exp\left\{-{\upsilon }_{\alpha }{D}_i^2{\tau }^{2/\alpha }{\lambda }_i\right\},\end{array}$$

(2)

where $\tau$ is the SIR threshold, $D_i$ is the distance of the ith relay hop, and ${\upsilon }_{\alpha }\stackrel{\Delta }{=}(2\pi /\alpha )\Gamma (2/\alpha )\Gamma (1-2/\alpha )$ with the gamma function $\Gamma \left(x\right)={\int }_0^{\infty }{y}^{x-1}{e}^{-y}dy$.

Proof. 

The successful transmission of the SIR in (1) for a Rayleigh fading channel can be expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathbb{P}({\gamma }_i\ge \tau )& =\mathbb{P}\left\{H_{\mathbf{x}\mathbf{y}}>{\displaystyle \frac{D_{XY}^{\alpha }\tau }{P_t}}\left(I_{SI}+I_i\right)\right\}\hfill \\ & ={\mathbb{E}}_{I_i}\left\{exp\left\{{\displaystyle \frac{D_{XY}^{\alpha }\tau }{P_t}}\left(I_{SI}+I_i\right)\right\}\right\},\hfill \end{array}\end{array}$$

(3)

where $I_i={\displaystyle \sum _{\mathsf{Z}\in {\mathbf{\Pi }}_i/\left\{\mathbf{X},\mathbf{Y}\right\}}}{P}_t{\mathsf{H}}_{\mathsf{Z}\mathbf{Y}}{\mathsf{D}}_{\mathsf{Z}\mathbf{Y}}^{-\alpha }$, the sum of the interference power from the transmitter nodes for the ith relay hop. Hence, by applying the Laplace transform for a general Poisson shot noise process with independent, identically distributed marks [34], we can obtain (2). □

Note that we assume that only the relay node has FD capability, so the residual SI at the receiver node of the second hop is zero. Consequently, the SIR and the probability of successful transmission for the second hop in (1) and (2) can be obtained with $I_{SI}=0$.

Under the DF relay protocol, an outage event happens when the SIR of either the first or second hop link fails to meet the required SIR threshold. Hence, when a typical transmitter node sends its data to the receiver node through a relay node located at ${\mathbf{x}}_{\mathrm{r}}$, the outage probability of a two-hop FDRRN can be expressed as follows:

$$\begin{array}{c}\hfill p_{\mathrm{o}}\left({\mathbf{x}}_{\mathrm{r}}\right)=1-p_{\mathrm{s},1}\left(\tau \right)p_{\mathrm{s},2}\left(\tau \right).\end{array}$$

(4)

By substituting (2) into (4), we can express the outage probability of an FDRRN as follows:

$$\begin{array}{c}\hfill p_{\mathrm{o}}\left({\mathbf{x}}_{\mathrm{r}}\right)=1-exp\left\{-{\displaystyle \frac{I_{SI}{D}_1^{\alpha }\tau }{P_t}}-{\upsilon }_{\alpha }{\tau }^{2/\alpha }\left(D_1^2{\lambda }_1+D_2^2{\lambda }_2\right)\right\}.\end{array}$$

(5)

From (5), we can guess that the outage probability of an FDRRN is influenced by both network interference and the distance of each relay hop, which is determined by the location of the selected relay node. Based on this observation, we propose a relay selection algorithm for FDRRNs in the next section.

4. Location-Based Relay Selection in FDRRNs

In this section, we investigate relay selection algorithms for FDRRNs using (5). To begin, we analyze the relationship between the relay location and the outage probability of an FDRRN.

4.1. Analysis of Relay Location in FDRRNs

From (5), when the locations of the transmitter and receiver nodes are ${\mathbf{x}}_{\mathrm{tx}}=(x_{tx},y_{tx})$ and ${\mathbf{x}}_{\mathrm{rx}}=(x_{rx},y_{rx})$, respectively, the optimal relay location ${\mathbf{x}}_{\mathrm{o}}^{\ast }$ that minimizes the outage probability of an FDRRN is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill (x_o^{\ast },y_o^{\ast })& =arg\underset{(x,y)}{min}{p}_{\mathrm{o}}\left({\mathbf{x}}_{\mathrm{r}}\right)\hfill \\ & =arg\underset{(x,y)}{min}g(x,y),\hfill \end{array}\end{array}$$

(6)

where ${\mathbf{x}}_{\mathrm{r}}=(x,y)$ is the location of the candidate relay node and $g(x,y)$ can be given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g(x,y)=& {\displaystyle \frac{I_{SI}{D}_1^{\alpha }\tau }{P_t}}+{\upsilon }_{\alpha }{\tau }^{2/\alpha }\left(D_1^2{\lambda }_1+D_2^2{\lambda }_2\right)\hfill \\ & =\mu {\left({(x-x_{tx})}^2+{(y-y_{tx})}^2\right)}^{{\displaystyle \frac{\alpha }{2}}}\hfill \\ & +{\rho }_{\alpha }{\lambda }_1\left({(x-x_{tx})}^2+{(y-y_{tx})}^2\right)\hfill \\ & +{\rho }_{\alpha }{\lambda }_2\left({(x-x_{rx})}^2+{(y-y_{rx})}^2\right),\hfill \end{array}\end{array}$$

(7)

where $\mu ={\displaystyle \frac{I_{SI}\tau }{P_t}}$ and ${\rho }_{\alpha }={\upsilon }_{\alpha }{\tau }^{2/\alpha }$.

The optimal relay location refers to a specific position where, if the transmitter node selects a relay node located there, it can achieve optimal performance in terms of outage probability. Unfortunately, however, the generalized optimal point of (7) in terms of $\alpha$ cannot be determined. Therefore, we propose a simplified algorithm to identify a potential relay location that sub-optimally reduces the outage probability of an FDRRN when $\alpha$ is fixed.

Lemma 1.

When $\alpha =4$, the potential relay location ${\mathbf{x}}_r^{\ast }=(x^{\ast },y^{\ast })$ in the proposed algorithm can be expressed as follows:

$$\begin{array}{c}\hfill {\mathbf{x}}_r^{\ast }=\left(\begin{array}{c}{x}_{tx}+{\displaystyle \frac{\kappa }{6\sqrt[3]{2}\mu }}-{\displaystyle \frac{2\rho ({\lambda }_1+{\lambda }_2)}{\sqrt[3]{4}\kappa }},\hfill \\ y_{tx}+{\displaystyle \frac{\kappa }{6\sqrt[3]{2}\mu }}-{\displaystyle \frac{2\rho ({\lambda }_1+{\lambda }_2)}{\sqrt[3]{4}\kappa }}\hfill \end{array}\right),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill \kappa =\sqrt[3]{\sqrt{{\zeta }_l^2+4{(\rho \mu {\lambda }_1+\rho \mu {\lambda }_2)}^3}+{\zeta }_l},\end{array}$$

(9)

$$\begin{array}{c}\hfill {\zeta }_l=108\rho {\lambda }_2{l}_{rx}-108\rho {\lambda }_2{l}_{tx},l\in \{x,y\}.\end{array}$$

(10)

Proof. 

To derive the potential relay location that provides a sub-optimal solution in terms of minimizing the outage probability, we calculate the coordinates $x^{\ast }$ and $y^{\ast }$ separately. The potential relay location can then be expressed as follows:

$$\begin{array}{c}\hfill (x_o^{\ast },y_o^{\ast })=\left(arg\underset{x}{min}g\left(x\right),arg\underset{y}{min}g\left(y\right)\right),\end{array}$$

(11)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & g\left(x\right)=\mu {(x-x_{tx})}^4+{\rho }_4{\lambda }_1{(x-x_{tx})}^2+{\rho }_4{\lambda }_2{(x-x_{rx})}^2,\hfill \\ & g\left(y\right)=\mu {(y-y_{tx})}^4+{\rho }_4{\lambda }_1{(y-y_{tx})}^2+{\rho }_4{\lambda }_2{(y-y_{rx})}^2,\hfill \end{array}\end{array}$$

(12)

respectively.

Since $g\left(x\right)$ and $g\left(y\right)$ are convex functions of x and y, respectively, the potential relay location can be obtained by solving ${\displaystyle \frac{\partial g\left(x\right)}{\partial x}}=0$ and ${\displaystyle \frac{\partial g\left(y\right)}{\partial y}}=0$, respectively. □

Note that for different values of $\alpha$, we can simply extend the expressions by modifying $g\left(x\right)$ and $g\left(y\right)$ accordingly.

Even though (8) provides the closed form of the potential relay location, it is difficult to gain insight into the relationship between the potential relay location and network parameters. In this regard, we further simplify (8) to gain further insight by assuming that $\alpha =2$.

Lemma 2.

For the case when $\alpha =2$, the potential relay location that minimizes the outage probability of an FDRRN is given by

$$\begin{array}{c}\hfill {\mathbf{x}}_r^{\ast }=\left({\displaystyle \frac{(\mu +{\rho }_2{\lambda }_1)x_{tx}+{\rho }_2{\lambda }_2{x}_{rx}}{\mu +{\rho }_2{\lambda }_1+{\rho }_2{\lambda }_2}},{\displaystyle \frac{(\mu +{\rho }_2{\lambda }_1)y_{tx}+{\rho }_2{\lambda }_2{y}_{rx}}{\mu +{\rho }_2{\lambda }_1+{\rho }_2{\lambda }_2}}\right).\end{array}$$

(13)

Proof. 

From (6), the objective function $g(x,y)$ for $\alpha =2$ can be expressed as

$$\begin{array}{c}\hfill g(x,y)=(\mu +{\rho }_2{\lambda }_1)\left({(x-x_{tx})}^2+{(y-y_{tx})}^2\right)\end{array}$$

(14)

$$\begin{array}{c}\hfill +{\rho }_2{\lambda }_2\left({(x-x_{rx})}^2+{(y-y_{rx})}^2\right).\end{array}$$

(15)

Since $g(x,y)$ is a convex quadratic function with respect to x and y, the potential relay location can be obtained as in (13) by solving ${\displaystyle \frac{\partial g\left(x,y\right)}{\partial x}}=0$ and ${\displaystyle \frac{\partial g\left(x,y\right)}{\partial y}}=0$, respectively. □

Remark 1.

Lemma 2 reveals the influence of $I_{SI}$ and ${\lambda }_i$ on the potential relay location ${\mathbf{x}}_r^{\ast }$. As ${\lambda }_1$ increases, ${\mathbf{x}}_r^{\ast }$ moves toward the transmitter node. Conversely, as ${\lambda }_2$ increases, ${\mathbf{x}}_r^{\ast }$ moves closer to the receiver node. This behavior can be attributed to the fact that the interference at the ith hop transmission increases as ${\lambda }_1$ or ${\lambda }_2$ increases. To compensate for the impact of interference, the potential relay node adjusts its position accordingly, moving closer to either the transmitter node or receiver node to maintain the desired transmission condition. Furthermore, it can be observed that as $I_{SI}$ increases, ${\mathbf{x}}_r^{\ast }$ shifts toward the transmitter node. This indicates that when the SIC capability at the relay node is inadequate, the relay node should be positioned closer to the transmitter node to ensure more reliable communication by minimizing the impact of the residual SI.

Note that in this paper, we consider the cases where $\alpha =2$ and $\alpha =4$, representing free space and relatively lossy environments, respectively. Other values of the path-loss exponent can also be extended using the simplified algorithm.

Corollary 1.

For all values of α where $\alpha >2$, the proposed algorithm can determine the potential relay location ${\mathbf{x}}_r^{\ast }=(x^{\ast },y^{\ast })$.

Proof. 

For an arbitrary $\alpha$, $g\left(x\right)$ and $g\left(y\right)$ are given, respectively, by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & g\left(x\right)=\mu {(x-x_{tx})}^{\alpha }+{\rho }_4{\lambda }_1{(x-x_{tx})}^2+{\rho }_{\alpha }{\lambda }_2{(x-x_{rx})}^2,\hfill \\ & g\left(y\right)=\mu {(y-y_{tx})}^{\alpha }+{\rho }_{\alpha }{\lambda }_1{(y-y_{tx})}^2+{\rho }_{\alpha }{\lambda }_2{(y-y_{rx})}^2.\hfill \end{array}\end{array}$$

(16)

We can then obtain the second derivative of $g\left(x\right)$ with respect to x as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{2}g\left(x\right)}{\partial x^2}}=\alpha \left(\alpha -1\right){\left(x-x_{tx}\right)}^{\alpha -2}+{\rho }_{\alpha }{\lambda }_1+{\rho }_{\alpha }{\lambda }_2,\hfill \\ & {\displaystyle \frac{{\partial }^{2}g\left(y\right)}{\partial y^2}}=\alpha \left(\alpha -1\right){\left(y-y_{tx}\right)}^{\alpha -2}+{\rho }_{\alpha }{\lambda }_1+{\rho }_{\alpha }{\lambda }_2.\hfill \end{array}\end{array}$$

(17)

Note that $\alpha >2$, ${\rho }_{\alpha }{\lambda }_1>0$, and ${\rho }_{\alpha }{\lambda }_2>0$. Furthermore, the relay node should be positioned between the transmitter and receiver nodes, i.e., $x_{tx}<x<x_{rx}$ and $y_{tx}<y<y_{rx}$. As a result, we can obtain ${\displaystyle \frac{{\partial }^{2}g\left(x\right)}{\partial x^2}}>0$ and ${\displaystyle \frac{{\partial }^{2}g\left(y\right)}{\partial y^2}}>0$. This indicates that $g\left(x\right)$ and $g\left(y\right)$ each have a global minimum point, representing the potential relay location determined by the proposed algorithm. Consequently, we can conclude that the potential relay location identified by the proposed algorithm exists for all values of $\alpha$ where $\alpha >2$. □

Corollary 1 demonstrates the applicability of the proposed algorithm in practical scenarios, where the path-loss exponent varies with the environment, by determining the potential relay location based on the case when $\alpha =4$.

4.2. Location-Based Relay Selection Algorithm for FDRRNs

In conventional relay systems, the max-min relay selection algorithm is regarded as an optimal solution from a capacity perspective in terms of the DF relay protocol [38]. In this scheme, the relay node that maximizes the minimum relay link between the first and second relay hops is selected. Following this approach, we can adopt a similar approach for FDRRNs as follows:

$$\begin{array}{c}\hfill {\mathbf{x}}_{\mathrm{s}}=arg\underset{{\mathbf{x}}_{\mathrm{r}}\in {\mathbf{\Pi }}_{\mathrm{R}}}{max}min\left\{{\gamma }_1,{\gamma }_2\right\}.\end{array}$$

(18)

In addition, the optimal relay selection scheme based on the outage probability can be obtained by selecting the relay node that minimizes the outage probability of an FDRRN as follows:

$$\begin{array}{c}\hfill {\mathbf{x}}_{\mathrm{s}}=arg\underset{{\mathbf{x}}_{\mathrm{r}}\in {\mathbf{\Pi }}_{\mathrm{R}}}{min}{p}_{\mathrm{o}}\left({\mathbf{x}}_{\mathrm{r}}\right).\end{array}$$

(19)

Note that while (18) and (19) can achieve good performance, they require a substantial amount of information, including instantaneous channel state information (CSI) and signal-to-noise ratio (SNR) details. Acquiring this information in a real environment is challenging and can lead to a significant signaling overhead, which ultimately degrades spectral efficiency. In particular, for FDR systems, obtaining the instantaneous status of SIC is even more challenging. Therefore, we propose a low-complexity algorithm that utilizes the potential relay location determined in (8). Based on Lemma 1, we propose a location-based relay selection algorithm for FDRRNs, which selects the relay node that is closest to the potential relay location from among the candidate relay nodes, as follows:

$$\begin{array}{c}\hfill {\mathbf{x}}_{\mathrm{s}}=arg\underset{{\mathbf{x}}_{\mathrm{r}}\in {\mathbf{\Pi }}_{\mathrm{R}}}{min}{D}_{{\mathbf{x}}_{\mathrm{r}}^{\ast }{\mathbf{x}}_{\mathrm{r}}}.\end{array}$$

(20)

It is important to note that the proposed algorithm requires only location information, which can be obtained from network-level parameters. As a result, it achieves the desired performance with reduced system overhead.

5. Simulation Results

This section evaluates the performance of an FDRRN based on the proposed relay selection algorithm under various conditions. In addition, we used MATLAB for all simulations in this section. Unless otherwise specified, the system parameters listed in

Table 1. Simulation parameters.

Parameter	Value	Parameter	Value
$\alpha$	4	$\tau$	3
${\lambda }_1$, ${\lambda }_2$	${10}^{-4}$	${\lambda }_{\mathrm{R}}$	${10}^{-3}$
$D_{\mathbf{x}\mathbf{y}}$	10 m	Cell radius	300 m
$I_{SI}$	−110 dB	$P_t$	1

were used for the simulation.

In the simulation, candidate relay nodes and interfering nodes were uniformly distributed according to their spatial densities within a cell with a radius of 300 m. In addition, a normalized transmission power of 1 was used for all transmitting nodes. For reliable FD transmission, a SIC level of approximately $-110$ dB is required, so we set $I_{SI}$ to $-110$ dB [22].

Figure 2 depicts the potential relay locations for different residual SI values. As discussed in Remark 1, the potential relay location shifts closer to the transmitter as the residual SI increases. This is due to the fact that as the residual SI increases, the performance of the first relay hop degrades. Consequently, to compensate for the degraded performance of the first hop, the potential relay location moves toward the transmitter. In other words, when the capability of SIC in the relay node is not reliable, it is desirable to select the relay node near the transmitter.

Figure 3 illustrates the outage probability of an FDRRN as a function of the spatial density of the interference when the relay node positioned at the potential relay location ${\mathbf{x}}_{\mathrm{r}}^{\ast }$ is selected. The numerical and simulation results are represented by blue circles and red lines, respectively. As shown in the figure, the numerical results show a similar trend to the simulation results, with only a negligible difference. Moreover, we can see that as the spatial density of the interfering node at each relay hop increases, the overall performance of the FDRRN decreases due to increased interference.

Figure 4 displays the outage probability of an FDRRN as a function of the density of the candidate relay nodes for different residual SI values. It can be observed that as the residual SI increases, the outage probability increases. In Figure 4, we can also see that as the density of the relay nodes increases, the outage probability decreases. This is attributed to the fact as the number of candidate relay nodes increases, the probability of a relay node being positioned closer to the potential relay location also increases. Consequently, the selected relay node is more likely to provide reliable communication.

Figure 5 shows the outage probability of an FDRRN for the proposed, max-min, and optimal relay selection algorithms at different densities of candidate relay nodes. We can observe the performance differences among the various relay selection algorithms. While the performance of the proposed algorithm is slightly degraded compared to the max-min and optimal algorithms due to the limited information available for relay node selection, it is worth noting that the proposed algorithm still achieves comparable performance with reduced information.

Figure 6 presents the outage probability of an FDRRN for different path-loss exponents. Note that the potential relay locations of the proposed algorithm, when $\alpha =3.5$ and $\alpha =4.5$, were calculated using MATLAB R2024a functions under the simulation setting. We can see that as the path-loss exponent decreases, the outage probability increases. This is due to the fact that as the path loss decreases, not only does the strength of the desired signal increase but so does the network interference, which significantly impacts the FDRRN’s performance.

We can also observe a similar performance trend between the proposed and optimal algorithms, with a consistent performance gap.

This is because the proposed algorithm can mitigate location-dependent impacts, such as path loss and interference, which are dominant factors in the FDRRN’s performance. As a result, it can achieve performance close to that of the optimal solution. Consequently, based on Corollary 1, it is feasible to determine the potential relay location for any general path-loss exponent.

Figure 7 illustrates the outage probability of an FDRRN with different SIR thresholds. The figure shows that the outage probability improves as the relay density increases due to the benefits of enhanced spatial diversity. Furthermore, the outage probability decreases with a lower SIR threshold, as this leads to more relaxed conditions for determining outage events. From these results, we can conclude that selecting an appropriate SIR threshold is essential, depending on the system requirements.

Figure 8 depicts the achievable spectral efficiency of an FDRRN for the various relay selection schemes. The achievable spectral efficiency is defined as the product of the successful transmission probability and the spectral efficiency of the DF relay protocol [39] as follows:

$$\begin{array}{c}\hfill \varsigma =p_{\mathrm{o}}\left({\mathbf{x}}_{\mathrm{s}}\right)\times min\left\{{log}_2\left(1+{\gamma }_1\right),{log}_2\left(1+{\gamma }_2\right)\right\}.\end{array}$$

(21)

We can see that the achievable spectral efficiency increases with higher relay density due to enhanced selection diversity. Moreover, similar to the outage probability, the achievable rate of the proposed algorithm shows slightly poorer performance compared to the optimal and max-min algorithms because the proposed algorithm utilizes less information in the relay selection process.

6. Conclusions

We evaluated the performance of an FDRRN in terms of outage probability. We first derived the outage probability for the FDRRN and then identified the potential relay location that minimizes the outage probability. Based on these insights, we proposed a location-based relay selection scheme designed to reduce the selection complexity of FDRRNs. By demonstrating that the proposed algorithm can achieve performance comparable to that of the optimal solution, our work highlights the importance of considering location-dependent parameters, such as interference and path loss, in the design of efficient relay selection algorithms. Furthermore, this paper paves the way for several future works, including the investigation of multiple relay selection and selection diversity order in FDRRNs.