Distributed relay assignment and power allocation for full-duplex two-way relaying in wireless sensor networks

Abstract

In this article, the joint relay assignment and power allocation for full-duplex two-way relaying in wireless sensor networks is considered. We investigated the weighted sum rate maximization problem under relay assignment constraints and total power constraints on each cooperative link, which turns out to be a mixed integer nonlinear programming problem. To tackle the challenging problem, we propose a distributed relay assignment and power allocation scheme by separating the original problem into two subproblems: relay assignment and power allocation. The relay assignment subproblem is solved by a stable matching approach based on Gale–Shapley algorithm, while the power allocation of each cooperative link is solved by sequential convex approximation method. Simulation results indicate the advantages of our proposed scheme.

Keywords

Full-duplex two-way relay power allocation relay assignment wireless sensor network matching

Introduction

Wireless sensor networks (WSNs) are employed over a wide range of applications, such as smart grid, intelligent agriculture, and health care.^1–3 The computation and energy limits of sensors bring some challenges for improving the spectrum efficiency in WSN. Relay is a widely adopted technology in Long-Term Evolution (LTE) which could significantly improve spectrum efficiency. Various types of relay have been investigated, including one-way relay and two-way relay. Two-way relay, which is based on physical layer network coding, enables two nodes exchanging signals via two time slots.⁴

However, full-duplex technology is another promising method in the next generation wireless system (5G) to enrich spectrum efficiency.^5–7 By alleviating self-interference efficiently, a full-duplex node can receive and transmit signals at the same time over the same frequency band. Various kinds of interference cancellation techniques include antenna separation, analog and digital cancellation, spatial domain cancellation, and time-domain cancelation.⁶

Combining full-duplex technology and two-way relaying, full-duplex two-way relaying is a kind of relaying which can accomplish information exchange between two nodes within only one time slots, while one-way half-duplex relaying, two-way half relaying, one-way full-duplex relaying requires four time slots, two time slots, and two time slots, respectively.⁷

Efficient wireless resource allocation can provide significant benefit for various kinds of relay system, including power allocation^8–9 and spectrum allocation.^10–11 Optimal power allocation (OPA) is investigated for full-duplex one-way amplify-and-forward (AF) relay⁸ and full-duplex two-way AF relay⁹, respectively, consisting of single user pair and single relay node. Resource allocation in full-duplex relaying system is investigated in Rodriguez et al.¹⁰ to optimize network level energy-efficiency, considering residual self-interference. Joint subcarriers and power allocation under individual power constraints is investigated in Yang et al.¹¹ to maximize throughput of full-duplex two-way relay networks.

In a relay network with multiple relays, efficient relay selection can effectively enhance the system performance.^12–14 In Liu et al.,¹² a relay selection for full-duplex relaying was proposed under Nakagami-m fading channels, considering direct link between source and destination. And, the outage performance was analyzed. In Li et al.,¹³ relay selection method in full-duplex two-way relay system based on best-worst-channel was proposed, and the close-form expressions of bit error ration, outage probability, and ergodic capacity were analyzed under the proposed relay selection method.

For a multi-user relay network with multiple relays, efficient relay assignment among users should be implemented to avoid different users selecting same relay, meanwhile to improve the system performance. In Cui et al.,¹⁵ OPA and relay assignment for multi-user full-duplex one-way relay network with multiple relays were discussed to maximize the weighted sum rate.

Based on the analysis above, although the full-duplex relay system has been investigated by many researchers, the distributed resource allocation of full-duplex two-way relaying in WSNs has rarely been addressed. Moreover, most of the methods presented above assume that global channel state information (CSI) can be derived and a centralized algorithm can be implemented in one central controller. In this article, we consider a distributed relay assignment and power allocation (DRAPA) in a WSN with multiple full-duplex two-way relays to maximize the weighted sum rate of the WSN under total power and relay assignment constraints. The main contributions of this article are summarized as follows:

We focus on relay assignment and power allocation for full-duplex two-way relaying in WSN, aiming to maximize weighted sum rate under total power constraint, considering the impact of the residual self-interference (RSI).

We propose a DRAPA, by solving relay assignment and power allocation separately. The relay assignment is transferred into a bipartite matching problem and solved by a distributed algorithm based on Gale–Shapley algorithm, which is easily carried out in a distributed manner. The power allocation is solved by sequential convex approximation method.

The rest of this article is organized as follows. Section “System model and problem formulation” describes the system model and problem formulation. In section “Solution of optimization problem,” the scheme of power allocation and relay assignment is described in detail. The performance analysis of the proposed scheme is described in section “Performance analysis.” Simulation results are shown in section “Simulation results.” Finally, we conclude this article in section “Conclusion.”

System model and problem formulation

System model

We consider a WSN, including M pairs of sensors, each of which wishes to change data with its partner, and N full-duplex two-way relays, as shown in Figure 1. We denote the sets of sensor nodes and relay nodes as S and R, respectively, S = {(A₁, B₁), (A₂, B₂), …, (A_M, B_M)}, R = {R₁, R₂, …, R_N}. We assume that there is no direct link between sensors in each pair and the data exchange has to be accomplished with the help of a relay assigned. In order to ensure that each sensor pair can be assigned at least one relay, we assume $M \leq N$ . All nodes operate in full-duplex (FD) mode and can simultaneously send and receive signals with the same frequency.

Figure 1.

Wireless sensor network with multiple full-duplex two-way relays.

The channel coefficient between A_m and R_n is represented by $h_{m, n}$ and that between B_m and R_n is represented by $g_{m, n}$ . All channels are assumed to be orthogonal to avoid co-channel interference. In addition, $h_{m, n}^{LI, A}$ , $h_{m, n}^{LI, B}$ , and $h_{m, n}^{LI, R}$ denote the channel gains of the self-interference channels at A_m, B_m, and R_n.

Assume sensors A_m and B_m exchange signals with cooperation of full-duplex two-way relay R_n. Denote the signal that transmitted by sensor A_m and B_m to R_n at time slot i as $x_{m, n}^{A} [i]$ and $x_{m, n}^{B} [i]$ , respectively. Let $P_{m, n}^{A}$ and $P_{m, n}^{B}$ represent the transmit power from A_m and B_m to R_n, respectively, and $P_{m, n}^{R}$ represents the transmit power from R_n to the mth sensor pair. Therefore, relay R_n receives a superposition signal, including desired signal from A_m, B_m, and self-interference. The signal received by R_n can be derived as

\begin{matrix} y_{m, n}^{R} [i] & = \sqrt{P_{m, n}^{A}} h_{m, n} x_{m, n}^{A} [i] \\ + \sqrt{P_{m, n}^{B}} g_{m, n} x_{m, n}^{B} [i] + v_{m, n}^{R} [i] + w_{m, n}^{R} [i] \end{matrix}

(1)

where $E [| x_{m, n}^{k} [i] |^{2}] = 1$ for k = A or B, $v_{m, n}^{R} [i]$ denotes the self-interference at R_n after imperfect interference cancellation, $v_{m, n}^{R} [i] = h_{m, n}^{LI, R} t_{m, n}^{R} [i]$ , $w_{m, n}^{R} [i]$ denotes the additive white Gaussian noise (AWGN) at the relay R_n, $w_{m, n}^{R} ~ CN (0, σ_{n, R}^{2})$ . $v_{m, n}^{R} [i]$ is modeled as AWGN with zero-mean and variance of $σ_{e, R}^{2}$ .¹³

Meanwhile, relay R_n broadcasts another signal $t_{m, n}^{R} [i]$ to sensors simultaneously, which is expressed as

t_{m, n}^{R} [i] = \sqrt{P_{m, n}^{R}} β_{m, n} y_{m, n}^{R} [i - 1]

(2)

where $β_{m, n}$ is the amplification factor of R_n, and $y_{m, n}^{R} [i - 1]$ denotes the received signal at time slot i − 1 of relay R_n. The amplification factor can be written as

β_{m, n} = \frac{1}{\sqrt{| h_{m, n} |^{2} P_{m, n}^{A} + | g_{m, n} |^{2} P_{m, n}^{B} + | h_{m, n}^{LI, R} |^{2} P_{m, n}^{R} + σ^{2}}}

(3)

Then, the signal received by node A_m and B_m from relay R_n at the ith time slot can be, respectively, expressed as

\begin{matrix} y_{m, n}^{A} [i] = h_{m, n} \sqrt{P_{m, n}^{R}} β_{m, n} \\ (\sqrt{P_{m, n}^{B}} g_{m, n} x_{m, n}^{B} [i - 1] + h_{m, n}^{LI, R} t_{m, n}^{R} [i - 1] + w_{m, n}^{R} [i - 1]) \\ + h_{m, n}^{LI, A} \sqrt{P_{m, n}^{A}} x_{m, n}^{A} [i] + w_{m, n}^{A} [i] \end{matrix}

(4)

and

\begin{matrix} y_{m, n}^{B} [i] = g_{m, n} \sqrt{P_{m, n}^{R}} β_{m, n} \\ (\sqrt{P_{m, n}^{A}} g_{m, n} x_{m, n}^{A} [i - 1] + h_{m, n}^{LI, R} t_{m, n}^{R} [i - 1] + w_{m, n}^{R} [i - 1]) \\ + h_{m, n}^{LI, B} \sqrt{P_{m, n}^{B}} x_{m, n}^{B} [i] + w_{m, n}^{B} [i] \end{matrix}

(5)

where $w_{m, n}^{A}$ and $w_{m, n}^{B}$ denotes the AWGN at A_m and B_m, respectively, $w_{m, n}^{k} ~ CN (0, σ_{n, k}^{2})$ , for k = A or B. We suppose that $h_{m, n}^{LI, A} \sqrt{P_{m, n}^{A}} x_{m, n}^{A} [i]$ denotes the RSI modeled by AWGN with zero-mean and variance of $σ_{e, A}^{2}$ at A_m, and $h_{m, n}^{LI, B} \sqrt{P_{m, n}^{B}} x_{m, n}^{B} [i]$ is the RSI received at B_m. The instantaneous received signal-to-interference-and-noise ratio (SINR) at A_m and B_m can be expressed as

γ_{m, n}^{A} = \frac{| h_{m, n} |^{2} | g_{m, n} |^{2} P_{m, n}^{R} P_{m, n}^{B} β_{m, n}^{2}}{| h_{m, n} |^{2} P_{m, n}^{R} β_{m, n}^{2} φ_{R} + φ_{A}}

(6)

and

γ_{m, n}^{B} = \frac{| h_{m, n} |^{2} | g_{m, n} |^{2} P_{m, n}^{R} P_{m, n}^{A} β_{m, n}^{2}}{| g_{m, n} |^{2} P_{m, n}^{R} β_{m, n}^{2} φ_{R} + φ_{B}}

(7)

where $φ_{A} = σ_{e, A}^{2} + σ_{n, A}^{2}$ , $φ_{B} = σ_{e, B}^{2} + σ_{n, B}^{2}$ , and $φ_{R} = σ_{e, R}^{2} + σ_{n, R}^{2}$ .

Thus, the achievable rate of the mth sensor pair with the nth relay can be derived as

C_{m, n} = \log_{2} (1 + γ_{m, n}^{A}) + \log_{2} (1 + γ_{m, n}^{B}), \forall m

(8)

Problem formulation

In this article, we aim to maximize the weighted sum rate in the WSN by relay assignment on each sensor pair and the transmit power optimization on each node. We define $s_{m, n} \in {0, 1}$ as relay assignment indicator, where $s_{m, n} = 1$ represents that the mth sensor pair is assigned nth relay node, while $s_{m, n} = 0$ otherwise. Considering the quality-of-service (QoS) requirement of each sensor pair, the weight parameter $ρ_{m}$ is assigned to the mth sensor pair indicating its priority. The weighted sum rate of the WSN can be written as

C = \sum_{m = 1}^{M} \sum_{n = 1}^{N} ρ_{m} s_{m, n} C_{m, n}

(9)

In the total power constraint scenario, each cooperative link has its power limit. Thus, the power constraint should be

\sum_{n = 1}^{N} s_{m, n} (P_{m, n}^{A} + P_{m, n}^{B} + P_{m, n}^{R}) \leq P_{m}^{t}, \forall m

(10)

where $P_{m}^{t}$ represents the power limit of the mth cooperative link.

Assume each sensor pair can only select one relay for data exchange and each relay can be assigned at most one sensor pair in order to avoid conflict of relay assignment. Therefore, the optimization problem of the power allocation and relay assignment to maximize the weighted sum rate can be formulated as

P 1 : \max_{P^{A}, P^{B}, P^{R}, S} \sum_{m = 1}^{M} \sum_{n = 1}^{N} ρ_{m} s_{m, n} C_{m, n}

(11a)

s . t . C 1 : \sum_{n = 1}^{N} s_{m, n} = 1, \forall m

(11b)

C 2 : \sum_{m = 1}^{M} s_{m, n} \leq 1, \forall n

(11c)

C 3 : s_{m, n} \in {0, 1}

(11d)

C 4 : \sum_{n = 1}^{N} s_{m, n} (P_{m, n}^{A} + P_{m, n}^{B} + P_{m, n}^{R}) \leq P_{m}^{t}, \forall m

(11e)

C 5 : P_{m, n}^{A} \geq 0, P_{m, n}^{B} \geq 0, P_{m, n}^{R} \geq 0, \forall m, n

(11f)

where $P^{A} = {P_{m, n}^{A}}$ , $P^{B} = {P_{m, n}^{B}}$ , $P^{R} = {P_{m, n}^{R}}$ , and $S = {s_{m, n}}$ . The constraints C1, C2, and C3 represent of relay assignment conditions, C4 denotes maximum power constraint at each cooperative link, and C5 denotes available power limit at each node.

Solution of optimization problem

Optimization problem P1 is a mixed integer nonlinear programming, which can be solved by exhaustive searching over all variables, but the calculation is too complicated.^16–20 To reduce the computational complexity, we decompose the optimization problem into two subproblems: relay assignment and power allocation problem. Then, we propose distributed solutions to these two subproblems separately. Such decomposition method is widely used in resource allocation researches.¹¹ After decomposition, the relay assignment subproblem is considered with the maximum weighted sum rate of every possible sensor-relay pair. In power allocation subproblem, the optimal power allocated on each node is derived with any sensor-relay pair and the maximum weighted sum rate of every sensor-relay pair can be obtained.

Relay assignment

We first consider the relay assignment problem with fixed transmit power on each node. Considering the relay assignment constraints in equation (11), the relay assignment problem can be formulated as a 0–1 integer programming problem as follows

P 2 : \max_{s_{m, n}} \sum_{m = 1}^{M} \sum_{n = 1}^{N} ρ_{m} s_{m, n} C_{m, n}

(12a)

s . t . \sum_{m = 1}^{M} s_{m, n} \leq 1, \forall n

(12b)

\sum_{n = 1}^{N} s_{m, n} = 1, \forall m

(12c)

s_{m, n} \in {0, 1}

(12d)

The constraints of problem P2 indicate a one-to-one matching between sensor pairs set and relay node set, which can be transformed into a maximum weighted bipartite matching problem in a weighted bipartite graph. The two parts of weighted bipartite graph refer to sensor pair set and relay set in WSN, and the weighted edges refer to the achievable rate of each link.

For relay assignment in WSN, we construct a weighted bipartite graph $G = (V, E)$ , as shown in Figure 2, where V is the set of sensor pairs and set of relays, and E is the set of edges that connect all possible pairs of vertices.

Figure 2.

Weighted bipartite graph of sensor pairs and relays.

Denote $w_{m, n}$ as the weight on the edge between the sensor pair vertex m and the relay vertex n. The value of $w_{m, n}$ is equivalent to the weighted achievable rate $c_{m, n}$ , $w_{m, n} = ρ_{m} C_{m, n}$ , when the sensor pair m matches the relay n. We define weight matrix as

W = (\begin{matrix} w_{1, 1} & w_{1, 2} & \dots & \dots & w_{1, n} \\ w_{2, 1} & w_{2, 2} & \dots & \dots & w_{2, n} \\ \dots & \dots & \dots & \dots & \dots \\ w_{m, 1} & w_{m, 1} & \dots & \dots & w_{m, n} \end{matrix})

(13)

In the weighted bipartite graph, our goal is to match all vertices in the sensor pairs set and those in the relays set to maximize the sum of the weights of the matching. Each vertex in the sensor pairs set S can only match one of the vertices in the relays set R, and each vertex in the relay set R matches no more than one vertex in a set of sensor pairs S. Optimal assignment solution can be derived by Kuhn–Munkres algorithm to maximize the overall sum rate, but Kuhn–Munkres algorithm requires global channel state information and need to be implemented in a centralized manner.²¹

In this section, we proposed a distributed relay assignment method based on one-to-one stable matching approach. A stable matching is defined as a matching in which there are no blocking pairs. A blocking pair denotes a partnership formed by a sensor pair and a relay that are not matched with each other but prefer each other to be its partner. In the assignment, each sensor pair and each relay maintain a preference list. A preference list is defined as a list of partners ranged in descending order.

Inspired by Gale–Shapley algorithm,^22,23 we propose a distributed relay assignment method as follows:

Each sensor pair establishes its preference list based on the edge weight between itself and each relay. Also, each relay establishes its preference list based on the edge weight between itself and each sensor pair.

Each unmatched sensor pair proposes to its most preferred relay in its preference list.

Relays which receive one or more proposals accept its most preferred sensor pair as its candidate and reject other sensor pair.

Any sensor pair that has been previously rejected removes the relay that has rejected it from its preference list and proposes to its most preferred relays that have not rejected it yet.

The matching process would end when every sensor pair has already been matched with a relay or has been rejected by all relays in its preference list. When the sensor pair m matches the relay n, $s_{m, n} = 1$ .

Power allocation

After channel assignment, we optimize the power allocation on each cooperative link in this section. Supposing relay R_n is assigned to sensor A_m and B_m, the OPA in cooperative link A_m-R_n-B_m under total power constraints is formulated as

\begin{matrix} P 3 : \max_{P_{m, n}^{A}, P_{m, n}^{B}, P_{m, n}^{R}} C_{m, n} \\ s . t . P_{m, n}^{A} + P_{m, n}^{B} + P_{m, n}^{R} \leq P_{m}^{t}, \forall m, n \\ P_{m, n}^{A} \geq 0, P_{m, n}^{B} \geq 0, P_{m, n}^{R} \geq 0, \forall m, n \end{matrix}

(14)

The problem P3 is a non-convex programming problem duo to the non-convexity of objective function. To transform P3 into a convex problem, we introduce Lemma 1.

Lemma 1

A very tight lower bound of $lo g_{2} (1 + z)$ where $z \geq 0$ is given by $a lo g_{2} z + b$ . The coefficients a and b are chosen as

a = \frac{z_{0}}{1 + z_{0}}

(15a)

b = \log_{2} (1 + z_{0}) - \frac{z_{0}}{1 + z_{0}} \log_{2} z_{0}

(15b)

where z₀ is a positive value. At $z = z_{0}$ , the lower bound $a lo g_{2} z + b$ is equal to $lo g_{2} (1 + z)$ . The proof of Lemma 1 can be found in Zuo et al.²⁴ According to Lemma 1, the lower bound of sum rate C_m,n can be derived

{\underline{C}}_{m, n} = a_{1} \log_{2} (γ_{m, n}^{A}) + b_{1} + a_{2} \log_{2} (γ_{m, n}^{B}) + b_{2}

(16)

where a₁, b₁, a₂, and b₂ are the corresponding lower-bound coefficients based on a initial SINR. To solve problem P3, we first consider the following optimization problem

\begin{matrix} P 4 : \max_{P_{m, n}^{A}, P_{m, n}^{B}, P_{m, n}^{R}} {\underline{C}}_{m, n} \\ s . t . P_{m, n}^{A} + P_{m, n}^{B} + P_{m, n}^{R} \leq P_{m}^{t}, \forall m, n \\ P_{m, n}^{A} \geq 0, P_{m, n}^{B} \geq 0, P_{m, n}^{R} \geq 0, \forall m, n \end{matrix}

(17)

In order to solve the optimization problem P4, we implement variable replacement as follows: ${\hat{P}}_{m, n}^{A} = \log (P_{m, n,}^{A})$ , ${\hat{P}}_{m, n}^{B} = \log (P_{m, n,}^{B})$ , and ${\hat{P}}_{m, n}^{R} = \log (P_{m, n,}^{R})$ . Then, the optimization problem (15) can be converted as follows

\begin{matrix} P 5 : \max_{{\hat{P}}_{m, n}^{A}, {\hat{P}}_{m, n}^{B}, {\hat{P}}_{m, n}^{R}} {\hat{C}}_{m, n} \\ s . t . e^{{\hat{P}}_{m, n}^{A}} + e^{{\hat{P}}_{m, n}^{B}} + e^{{\hat{P}}_{m, n}^{R}} \leq P_{m}^{t}, \forall m, n \end{matrix}

(18)

where ${\hat{C}}_{m, n} = {\underline{C}}_{m, n} |_{P_{m, n}^{A} = e^{{\hat{P}}_{m, n}^{A}}, P_{m, n}^{B} = e^{{\hat{P}}_{m, n}^{B}}, P_{m, n}^{R} = e^{{\hat{P}}_{m, n}^{R}}}$ .

Optimization problem P5 is an equivalent transformation of problem P4, thus the solution of P4 can be obtained by solving P5. Problem P5 is a convex problem due to concavity of objective function and convexity of constraint.^19–20 We can solve problem P5 by interior point method using convex solving tools such as CVX. Since the objective function of problem P4 is a lower bound of objective function of problem P3, we develop an iterative algorithm for optimization problem P2 based on sequential convex approximation (SCA) method. The algorithm is summarized in Algorithm 1.

Algorithm 1. Power allocation.
1 Initialization: l = 0, $P_{m, n}^{A, (0)}$ , $P_{m, n}^{B, (0)}$ and $P_{m, n}^{R, (0)}$ is a feasible solution to problem P3; 2 Calculate ${\hat{P}}_{m, n}^{A, (0)}$ , ${\hat{P}}_{m, n}^{B, (0)}$ and ${\hat{P}}_{m, n}^{R, (0)}$ ; 3 Repeat 4 $l = l + 1; 4$ 5 Calculate the corresponding coefficients $a_{1}^{(l - 1)}$ , $b_{1}^{(l - 1)}$ , $a_{1}^{(l - 1)}$ , $b_{1}^{(l - 1)}$ based on $P_{m, n}^{A, (l - 1)}$ , $P_{m, n}^{B, (l - 1)}$ and $P_{m, n}^{R, (l - 1)}$ 6 Solve convex optimization problem P4, and obtain the optimal solution ${\hat{P}}_{m, n}^{A, (l)}$ , ${\hat{P}}_{m, n}^{B, (l)}$ and ${\hat{P}}_{m, n}^{R, (l)}$ ; 7 Calculate solution $P_{m, n}^{A, (l)}$ , $P_{m, n}^{B, (l)}$ and $P_{m, n}^{R, (l)}$ to problem P3; 8 Until convergence or maximum number of iterations 9 Return $P_{m, n}^{A, (l)}$ , $P_{m, n}^{B, (l)}$ and $P_{m, n}^{R, (l)}$ ;

Algorithm 1. Power allocation.

1 Initialization: l = 0,

P_{m, n}^{A, (0)}

P_{m, n}^{B, (0)}

and

P_{m, n}^{R, (0)}

is a feasible solution to problem P3;
2 Calculate

{\hat{P}}_{m, n}^{A, (0)}

{\hat{P}}_{m, n}^{B, (0)}

and

{\hat{P}}_{m, n}^{R, (0)}

;
3 Repeat
4

l = l + 1; 4

5 Calculate the corresponding coefficients

a_{1}^{(l - 1)}

b_{1}^{(l - 1)}

a_{1}^{(l - 1)}

b_{1}^{(l - 1)}

based on

P_{m, n}^{A, (l - 1)}

P_{m, n}^{B, (l - 1)}

and

P_{m, n}^{R, (l - 1)}

6 Solve convex optimization problem P4, and obtain the optimal solution

{\hat{P}}_{m, n}^{A, (l)}

{\hat{P}}_{m, n}^{B, (l)}

and

{\hat{P}}_{m, n}^{R, (l)}

;
7 Calculate solution

P_{m, n}^{A, (l)}

P_{m, n}^{B, (l)}

and

P_{m, n}^{R, (l)}

to problem P3;
8 Until convergence or maximum number of iterations
9 Return

P_{m, n}^{A, (l)}

P_{m, n}^{B, (l)}

and

P_{m, n}^{R, (l)}

;

DRAPA scheme

After proposing solution for relay assignment and power allocation separately, we propose a DRAPA scheme, in which relay assignment under fixed power allocation is solved based on Gale–Shapley algorithm, then power optimization for each cooperative link is conducted among matched sensor pairs and relay nodes.

The main step of DRAPA scheme is described as follows: first, the sensors and relay nodes are pre-allocated with equal power under total power constraints. Let $P_{m, n}^{A} = P^{t} / 3$ , $P_{m, n}^{B} = P^{t} / 3$ , and $P_{m, n}^{R} = P^{t} / 3$ substitute the power into equation (8) to calculate the achievable rate $C_{m, n}$ . Then, the relay assignment can be solved by Gale–Shapley-based relay assignment algorithm. Each sensor pair is assigned a relay for cooperative communication under equal power allocation and a stable matching between sensor pairs and relays is derived.

Then, the transmit power for sensors and relay nodes of M cooperative links will be optimized. During the power allocation, we get M subproblems and each subproblem can be formulated as P3 solved by Algorithm 1. The weighted sum rate of the system can be derived by substituting the OPA and relay assignment into equation (9). To make it more clear, details of DRAPA scheme are presented in Algorithm 2.

Algorithm 2. Distributed relay assignment and power allocation (DRAPA).
1 Initialize $P_{m, n}^{A} = P^{t} / 3$ , $P_{m, n}^{B} = P^{t} / 3$ , $P_{m, n}^{R} = P^{t} / 3$ ; 2 Calculate the achievable rate $C_{m, n}$ and W according to (8); 3 Obtain optimal assignment $S^{}$ with respect to W by Gale–Shapley algorithm; 4 for m = 1:M 5 for n = 1:N 6 if $s_{m, n} = 1$ 7 solving power allocation using Algorithm 1; 8 end if 9 end 10 end 11 Substitute $(P_{m, n}^{A }, P_{m, n}^{B }, P_{m, n}^{R })$ into (8) and get the maximum achievable rate $C_{m, n}^{*}$ ; 12 calculate the maximum weighted sum rate of the system by (9);

Algorithm 2. Distributed relay assignment and power allocation (DRAPA).

1 Initialize

P_{m, n}^{A} = P^{t} / 3

P_{m, n}^{B} = P^{t} / 3

P_{m, n}^{R} = P^{t} / 3

;
2 Calculate the achievable rate

C_{m, n}

and W according to (8);
3 Obtain optimal assignment

S^{*}

with respect to W by Gale–Shapley algorithm;
4 for m = 1:M
5 for n = 1:N
6 if

s_{m, n} = 1

7 solving power allocation using Algorithm 1;
8 end if
9 end
10 end
11 Substitute

(P_{m, n}^{A *}, P_{m, n}^{B *}, P_{m, n}^{R *})

into (8) and get the maximum achievable rate

C_{m, n}^{*}

;
12 calculate the maximum weighted sum rate of the system by (9);

Performance analysis

Complexity analysis

The computational complexity of the proposed DRAPA scheme will be analyzed in this subsection. For the Gale–Shapley algorithm used to solve the relay assignment problem, the complexity of the algorithm is $O$ (N²) in the worst case, but in average case analysis, the algorithm will runs in $O$ (NlogN).²⁰ After relay assignment, the power allocation problem is converted into M subproblems. We assume that each subproblem (power allocation on each cooperative link) is minimized within I iterations and the optimal power solution $(P_{m, n}^{A *}, P_{m, n}^{B *}, P_{m, n}^{R *})$ can be derived using Algorithm 1. The computational complexity for power allocation is $O (IM)$ .

Distributed implementation

Before relay assignment and power allocation, the channel gains need to be acquired at each node first. Since the self-interference channel on each node relies on the self-interference technology the node employs, we assume that each node knows the channel gain of its self-interference channel. In order to acquire the channel gains between sensors and relays, all sensors (A_m and B_m, m = 1,…, M) send pilot signals sequentially. Specifically, the channel gain of its self-interference channel is included in the pilot signals. Relay R_n receives the pilot signal from sensors A_m and B_m, then $h_{m, n}$ and $g_{m, n}$ can be measured and $h_{m, n}^{LI, A}$ and $h_{m, n}^{L I, B}$ can be obtained. Based on the channel state information acquired, the achievable rate of link A_m-R_n-B_m (m = 1, …, M), C_m,n can be calculated at relay n (n = 1, …, N), and the preference list of R_n can be established. Then, C_m,n can be broadcasted by relay R_n and received by sensors A_m and B_m. Note that R_n may only receive pilot signals from part of sensors and obtained part of channel gains, since pilot signals from sensor far away may not be received successfully. In this case, the preference list of R_n only includes part of sensor pairs. This differs with the centralized scheme in which all the channel gains are required for channel assignment and power allocation. In a centralized scheme, each node (sensor or relay) measures its CSI by receiving pilot signals transmitted by other nodes, then sends the CSI to a centralized controller sequentially. Thus, during the CSI acquisition, the overhead of the proposed distributed scheme is nearly half of that of centralized scheme.

After all relays broadcast their achievable rates, the preference list of sensor pair (A_m, B_m) can be established on sensor A_m, or B_m. Similarly, the preference list of each sensor pair may only include part of relays. During the matching process, each sensor pair proposes to its most preferred relay according to its preference list, which is a kind of local information on each node. Similarly, the relay decides to accept or reject the proposal it received according to its preference list. From the analysis above, we can notice that the proposed matching scheme for relay assignment can be easily implemented in a distributed manner.

In power allocation step, since relay R_n can obtain the channel state information between R_n and any sensor (A_m and B_m, m = 1, …, M), the OPA $(P_{m, n}^{A} *, P_{m, n}^{B} *, P_{m, n}^{R} *)$ can be computed at relay R_n. Then, the optimal power $P_{m, n}^{A} *$ and $P_{m, n}^{B} *$ will be broadcasted by relay R_n to sensors A_m and B_m, respectively. Although additional overhead is required for information sharing, the overhead in our proposed power allocation algorithm is still much less than that in a centralized power allocation scheme. To summarize, the power allocation problem on each link is implemented on the relay node, and the solution is broadcasted to sensor pair.

The effect of weight

In the WSN, we assume that sensors are with different priorities and assign each sensor pair a weight according to its priority. The result of matching in relay assignment will be affected by the weight of each sensor pair. Sensor pair with larger weight will be more preferred by each relay and is more likely to match a better relay. For example, suppose the mth sensor pair is a sensor for real-time applications and the weight parameter $ρ_{m}$ is larger. Thus, the mth sensor pair will be given priority to be assigned more appropriate relay to enhance the achievable rate $C_{m, n}$ and then get more improvement of weighted sum rate. Also, it will not cause significant performance degradation for sensor pairs with lower weight, due to the sensor does not require a high bit rate or critical delay requirement. For power allocation, the power allocation on each cooperative link will not be affected by its weight.

Simulation results

In this section, the simulation results are provided to evaluate the DRAPA schemes proposed. During all simulations, the variances of all noise are assumed to be 1, $σ_{n, A}^{2} = σ_{n, B}^{2} = σ_{n, R}^{2} = 1$ , and the variances of all RSI are assumed to be 1, $σ_{e, A}^{2} = σ_{e, B}^{2} = σ_{e, R}^{2} = 1$ . We compare the performance of DRAPA with three benchmark algorithms: distributed relay assignment and equal power allocation (DRA-EPA), with random relay assignment optimal power allocation (RRA-OPA), and random relay assignment equal power allocation (RRA-EPA). In DRA-EPA, relays are assigned according to the relay assignment scheme proposed in section “Relay assignment” and power is equally allocated. In RRA-OPA, relays are assigned randomly and then power is allocated according Algorithm 1. In RRA-EPA, relays are assigned randomly and power is equally allocated on each node.

Figure 3 illustrates the weighted sum rate performance of different schemes versus total power $P^{t}$ with M = 5 and N = 10. From Figure 3, our proposed DRAPA scheme outperforms DRA-EPA, RRA-OPA, and RRA-EPA algorithms. The simulation results show that the weighted sum rates of all schemes are significantly increased with $P^{t}$ increasing, which is caused by higher multiplexing order in FD system.

Figure 3.

Weighted sum rate versus total transmit power.

Figure 4 shows the weighted sum rate performance of different schemes versus number of relays with M = 5, P^t = 5 dB. The number of relays changes from 5 to 30. From the simulation results, the weighted sum rate of DRAPA and DRA-EPA is gradually increased with the increase in the number of relays, while for RRA-OPA and RRA-EPA schemes, the weighted sum rate almost remains the same. This is due to that with the increase in the N, each sensor pair can match a better of relay for cooperative communication by relay assignment. However, relays are randomly matched to transmit signals in RRA-OPA and RRA-EPA, and the redundant relay cannot improve the weighted sum rate of the system. In addition, compared with DRAPA and DRA-EPA schemes, the performance improved approximation is 9.7% by OPA. The weighted sum rate achieved by the proposed DRAPA scheme is 1–3 times larger than RRA-OPA scheme by relay assignment.

Figure 4.

Weighted sum rate versus number of relays.

Figure 5 presents the weighted sum rate performance of different schemes versus number of sensor pairs with N = 30, P^t = 5 dB. The number of sensor pairs changes from 5 to 30. It is clear that the weighted sum rate of all schemes gradually increases with the increase in the number of sensor pairs. Specifically, with the increase in the number of sensor pairs, the performance of DRAPA and DRA-EPA schemes is growing faster than that of RRA-OPA and RRA-EPA schemes. Simulation results show that the DRAPA scheme always maintains the highest performance due to the optimization of relay assignment and power allocation. The weighted sum rate of the proposed DRAPA scheme is 1–2 times larger than that of RRA-OPA scheme by relay assignment, and the performance improvement of DRAPA scheme over DRA-EPA scheme is about 7.2% by power allocation.

Figure 5.

Weighted sum rate versus number of sensor pairs.

Figure 6 illustrates the effect of residual self-interference on the weighted sum rate under different schemes, in which the x-axis denotes the variance of RSI. The figure reveals that the weighted sum rates of all schemes are gradually decreased with the increase in the RSI variance. The performance of the DRAPA scheme is rapidly decreased when the variance of RSI is larger than 1 dB.

Figure 6.

Weight sum rate versus residual self-interference.

The sensor pairs’ achievable rate versus total power $P^{t}$ with different weights is shown in Figure 7. The x-axis denotes the total power, while y-axis denotes the first sensor pair’s achievable rate $C_{1, n}$ . $ρ_{1}$ represents the weight of the first sensor pair, while the weight of other sensor pairs is set to 1. It can be seen that $C_{1, n}$ increases with the increase in the $P^{t}$ . Note that when the power is fixed, the sensor pairs’ rate also increases slightly with the increase in the weight. Because the first sensor pair will match a relay with higher priority when $ρ_{1}$ is increased to 2 or 3. In addition, the performance of RRA-OPA remains the same with $ρ_{1} = 1$ and $ρ_{1} = 2$ , which implies that different weights will not affect the achievable rate of each sensor pair.

Figure 7.

Sensor pairs’ achievable rate with different weights.

Conclusion

In this article, the joint relay assignment and power allocation problem for full-duplex two-way relay in WSNs is investigated, aiming to maximize the weighted sum rate of the system with the total power constraint. The problem is formulated as a mixed integer nonlinear optimization problem. We propose a distribute solution DRAPA by optimizing transmit power for each node and assigning optimal relay to sensor pairs for cooperative communication separately. Simulation results show that DRAPA scheme can achieve great performance gains comparing with other algorithms.

Footnotes

Academic Editor: Shi-Jinn Horng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Fundamental Research Funds for the Central Universities (under grant no. 2014QNB46).

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