Resource Allocation for Relay-Based OFDMA Power Line Communication System

Abstract

In this paper, we investigate the resource allocation for multi-user orthogonal frequency division multiple access (OFDMA) power line communication (PLC) systems, which satisfy both the total average transmit power constrained by PLC networks and the required interference limitations. First, we propose an optimal resource allocation scheme, with respect to joint consideration of the subcarrier and power constraint to maximize the throughput of the PLC system. Then, we divide the intractable problem into two individual subproblems to decrease the computational burden, which can avoid proceeding the mixed integer nonlinear programming (MINLP) problem. The proposed resource allocation scheme can ensure the optimality and efficiency of the system with rigorous proofs. The simulation results show that our proposed resource allocation scheme for the PLC system outperforms other algorithms, and is efficient with acceptable complexity.

1. Introduction

The orthogonal frequency division multiplexing (OFDM) combined with dynamic resource allocation technology can dynamically allocate the subcarriers to every user and allocate the transmit power on each subcarrier according to the channel state. Applying OFDM into the power line communication (PLC) system can effectively eliminate the multi-path, attenuation, time-varying and frequency selectivity of the power line channel, therefore, effectively improving the resource utilization, quality of service and data transmission rate [1,2,3,4,5]. Motivated by the above technique, OFDM-based PLC systems have become increasingly attractive.

Due to the long transmission distance of the multi-conductor cable, it is necessary to adopt a relay for signal amplification and forwarding in the PLC system. The diagram of the relay-aided PLC system is shown in Figure 1, we can see that the base station (BS) of the communication network is connected to the Internet through the access point, and the BS is connected to the power grid by the coupling circuit on the low voltage side of the power transformer. The signal of the BS is transmitted to each PLC gateway through the multi-conductor cable, and the gateway is connected to the user terminal through the power line. Therefore, the network is composed of outdoor broadband access and indoor broadband interconnection. The resource allocation in the relay-aided OFDM PLC system is a joint optimization problem, including the power control, subcarrier assignment, relay selection, etc., therefore, when referring to the relay-aided OFDM PLC system, it is usually considered challenging due to the complexity [6]. The essence of the problem is how to optimally allocate subcarriers to users and allocate power to subcarriers [7,8,9,10,11]. Recently, a variety of multi-user dynamic resource allocation algorithms have been proposed with different optimization objectives or constraints for different communication systems. Ref. [12] proposed a cooperative wireless power communication network protocol, in which the time allocation of hybrid access point downlink transmission and user uplink transmission were jointly optimized for two different aspects. Ref. [13] proposed a multiuser OFDM system named simultaneous wireless information and power transfer, which can transfer information and power separately on different subcarriers. The optimal design for resource allocation is investigated to maximize the sum rate with a minimum transferred power constraint. In [14], for OFDM downlink distributed multiple input single output (MISO) systems in frequency selective fading channels, two power allocation strategies are proposed to solve the power constraint of a single antenna unit. In [15], the energy-saving user scheduling and power control in downlink multicell multiuser orthogonal frequency division multiple access (OFDMA) networks is investigated only with the channel distribution information, which can solve the resource allocation problem subject to the minimum throughput constraint of each user, even if the channel gain is the same for all the subcarriers. In [16], a stochastic optimization programming in dynamic wireless self-backhaul small cell networks is formulated to investigate the spectral efficiency maximization by both considering the resource reuse and allocation, additionally, the random and finite traffic loads are also considered to keep the system stable. The resource allocation problem in two-tier heterogenous networks with massive MIMO is investigated in [17], which used cellular frequency bands and millimeter wave frequency bands for wireless backhaul links. A block diagonalization based precoding scheme is proposed to eliminate the multi-user and inter-tier interference and a power allocation optimization problem is formulated to maximize the downlink sum rate of the system subject to per-small cells clusters power and users’ quality of service requirements.

However, the PLC networks have not been designed for communication purposes; the existing resource allocation strategies for wireless networks are not suitable [18,19,20,21,22,23]. The PLC systems usually use the phase line to neutral line to transmit informations. The imbalance of the two wire to ground results in common mode current which greatly enhances the electromagnetic radiation. Meanwhile, the impedance mismatch between the power line and the load causes the signal reflection and form standing wave, which also generates strong electromagnetic radiation. Therefore, the power line is not a good medium for communication. The power line channel also has defects, such as frequency selective multipath fading with time-varying characteristics and color background noise, etc., which are the main difference between PLC systems and wireless communication systems. In order to deal with the poor channel conditions, the PLC system must apply an efficient resource allocation technique. In [24], a sub-carrier and power allocation algorithm is proposed for power spectrum limitation of indoor power line communication OFDM systems and the performance of the algorithm under various variable channel conditions is simulated. For the Homeplug AV standard in power line communication, two bit allocation algorithms with the largest system capacity under the condition of guaranteed bit error rate (BER) and target average BER are proposed [25,26]. In [5], the power allocation problem of a hybrid system composed of a cascaded power-line communications (PLC)/visible light communications (VLC) link in parallel to an radio frequency (RF) wireless link is investigated. This formulated and analyzed the problem of minimizing the total transmission power consumption under a defined quality of service constraint. [27] proposed a dynamic load-based PLC system model for energy efficiency maximization. By optimizing the paremeters including the load impedance, transmission power and subchannel allocation, they investigated power allocation of the multi-receiver PLC OFDM downlink network with non-white Gaussian noise channel. The simulation results showed that compared with the baseline scheme, the proposed system is more energy-efficient, and can greatly improve the energy efficiency by optimizing the impedance and adopting the subchannel allocation strategy.

The resource allocation problem in a relay-aided OFDMA PLC system usually involves a mixed integer nonlinear programming (MINLP) problem [28,29], which leads to an algorithmic complexity of an order of magnitude. The subcarriers and power are jointly optimized, which makes it difficult to obtain an optimal resource allocation strategy. Based on the single-user system model, this paper discusses the optimal resource allocation scheme in multi-user environment, and proves that the system can maximize the system throughput by using the optimal resource allocation strategy proposed in the paper. The specific contents of this paper are as follows: Firstly, for the case of power constraints of the transmission source or relay, the problem description of maximizing the throughput of the system under the constraints of average power is given; secondly, for the case of multi-user OFDMA, The method of separating subcarrier and power allocation is used to give the optimal resource allocation strategy under the average power constraint and the interference power constraint.

The remainder of this paper is organized as follows. Section 2 presents a mathematical description of the relay-based multi-user OFDMA downlink system model. Section 3 discusses the optimal subcarrier allocation and power control schemes in a PLC environment; Section 4 gives the numerical simulations and analysis; Conclusions are given in Section 5.

Notations:$\mathbb{E}\left\{·\right\}$ denotes the expectation w.r.t. all fading channels. $\mathcal{CN}\left(0,{\sigma }^2\right)$ denotes the complex Gaussian distribution with the mean being 0 and the variance being ${\sigma }^2$. ${\left[·\right]}^{\mathrm{T}}$ denotes the matrix transpose. ${\left[x\right]}^{+}$ denotes $max\{x,0\}$. $diag\{z_1,z_2,\cdots ,z_M\}$ denotes a diagonal matrix with the diagonal entries from $z_1,z_2,\cdots ,z_M$. $|·|$ denotes the absolute value of a real number or a modulus of a complex number. ${▽}^{2}f(·)$ denotes the Hessian matrix of $f(·)$.

2. System Model and Problem Formulation

2.1. System Model

Consider a downlink multi-user OFDMA PLC system with one BS, one non-regenerative relay and K users. Since we mainly focus on the resourse allocation strategy for K users under one relay, this paper adopt a two-hop protocol for simplify. Assume that there are N subcarriers in each OFDM symbol and one subcarrier can only be used by one user. In the first hop, the source transmits OFDM symbols $s_{n,k}$ with transmit power $P_{s_{n,k}}$ through the backward channel $h_1$ to the relay, where $n\in \{1,\cdots ,N\}$ is the subcarrier index and $k\in \{1,\cdots ,K\}$ is the user index. Then, the received signal of relay can be expressed as

$$y_{n,k}=h_1\sqrt{P_{s_{n,k}}}{s}_{n,k}+n^r,$$

(1)

where $n^r$ denotes the noise at the relay node. In the PLC system, the multi-source impulsive noise is considered as the main factor. According to the model of Middleton’s Class A [21,30], we can formulate the total color noise as the sum of Gaussian noise and the impulsive noise, and the probability density function of the color noise that with amplitude c can be given by

$$p(c)=\sum _{c=0}^{\infty }\frac{e^{-A}{A}^c}{c!}\frac{1}{\sqrt{2\pi }{\sigma }_c}exp(-\frac{z^2}{2{\sigma }_c^2}),$$

(2)

where ${\sigma }_c^2$ denotes the total color noise power, and

$${\sigma }_c^2={\sigma }_g^2+{\sigma }_I^2\frac{\frac{c}{A}+\mathsf{\Gamma }}{1+\mathsf{\Gamma }},\mathsf{\Gamma }=\frac{{\sigma }_g^2}{{\sigma }_I^2},$$

(3)

where ${\sigma }_g^2$, ${\sigma }_I^2$ denotes the Gaussian and impulsive noise power, respectively, A denotes the value of the detected impulsive noise. In the second hop, the relay amplifies the received signal with amplified factor ${\alpha }_n$, and retransmits the amplified signal to the destination through the forward channel $h_{n,k}$, the forward signal is given by

$$g_{n,k}={\alpha }_n{h}_1\sqrt{P_{s_{n,k}}}{s}_{n,k}+{\alpha }_n{n}^r.$$

(4)

Let $P_{r_{n,k}}$ denotes the transmit power on the relay node, and $P_{r_{n,k}}=\mathbb{E}\left(|g_{n,k}{|}^2\right)$, the factor ${\alpha }_n$ can be expressed as

$$|{\alpha }_n{|}^2=\frac{P_{r_{n,k}}}{|h_1{|}^2{P}_{s_{n,k}}+{\sigma }^2}.$$

(5)

After two-hop transmission, we can obtain the k-th user’s received signal as

$$d_{n,k}=h_1{h}_{n,k}{\alpha }_n\sqrt{P_{s_{n,k}}}{s}_{n,k}+h_2{\alpha }_n{n}^r+n_{n,k}^d,$$

(6)

where $n_{n,k}^d$ denotes the user node’s noise, and $n_{n,k}^d\sim \mathcal{CN}\left(0,{\sigma }^2\right)$. Based on the above description, we can obtain the k-th user’s information rate on the n-th subcarrier

$$R_{n,k}={log}_2\left(1+\frac{P_{s_{n,k}}{P}_{r_{n,k}}}{\frac{P_{s_{n,k}}{\sigma }^2}{|h_{n,k}{|}^2}+\frac{P_{r_{n,k}}{\sigma }^2}{|h_1{|}^2}+\frac{{\sigma }^4}{|h_1{|}^2{|h_{n,k}|}^2}}\right).$$

(7)

Consider the background noise and radio interferences in the PLC channel, the subcarrier signal might give rise to interference with other subcarriers. We can model the interference as out-of-band emissions, the interference from the m-th subcarrier to the n-th subcarrier can be expressed as

$$I_{n,m}=P_{s_{n,k}}{\eta }_n,$$

(8)

where ${\eta }_n$ denotes the interference coefficient [31,32], which is given by

$${\eta }_n={\int }_{f_{m,n}-\frac{▵f}{2}}^{f_{m,n}+\frac{▵f}{2}}{T}_s{\left(\frac{sin(\pi f{T}_s)}{\pi f{T}_s}\right)}^{2}df,$$

(9)

where $f_{n,m}=\left|f_m-f_n\right|$ denotes the spectral distance between the center frequency of subcarrier m and subcarrier n. $T_s$ denotes the symbol interval of OFDM and $▵f$ denotes the channel bandwidth.

2.2. Problem Formulation

In this section, we derive the optimal resource allocation problem in the relay-based OFDM PLC system. The maximum information rate is regarded as the object function, and both the transmit power constraint and interference limitation between the subcarriers should be guaranteed. Then the problem can be formulated as

$$\begin{array}{c}{\displaystyle \underset{P_{s_{n,k}},P_{r_{n,k}}}{max}R=\sum _{k=1}^K\sum _{n=1}^N{R}_{n,k},}\hfill \\ {\displaystyle \mathrm{s}.\mathrm{t}.\sum _{k=1}^K\sum _{n\in {\Omega }_{T_k}}{P}_{s_{n,k}}\le P_S,}\hfill \\ {\displaystyle \sum _{k=1}^K\sum _{n,m\in {\Omega }_{T_k}}{\eta }_n{P}_{s_{n,k}}\le I_{tol},}\hfill \\ {\displaystyle P_{s_{n,k}}\ge 0,\forall k,n,}\hfill \\ {\Omega }_{T_1}\bigcup \cdots \bigcup {\Omega }_{T_K}\subseteq {\Omega }_T,\hfill \\ {\Omega }_{T_i}\bigcup {\Omega }_{T_j}=\varnothing ,\forall i\ne j,i,j\in \{1,\cdots ,K\},\hfill \end{array}$$

(10)

where $P_s$ denotes the source node’s average transmit power, $I_{tol}$ denotes the average interference power, ${\Omega }_{T_k}$ represents the subcarrier set for the k-th user, ${\Omega }_T$ represents the set of all subcarriers. In (10), constraints 1 and 2 represent the transmission power constraint of the source of the PLC system and the interference power constraint between subcarriers, respectively, constraint 3 represents the non-negative property of the power value, constraints 4 and 5 indicate that the subcarrier sets are disjointed. It obvious that problem (10) belongs to mixed integer nonlinear programming (MINLP), which usually leads to a $\mathcal{NP}$-hard problem. Since the information rate $R_{n,k}$ can be explicitly expressed as in (7), a general mathematical form for simplicity can be expressed as

$$\begin{array}{c}{\displaystyle \underset{x_n,{\Omega }_{T_k}}{max}f\left(\mathbf{x}\right)=\sum _{k=1}^K\sum _{n=1}^N{log}_2\left(1+\frac{x_n}{b_n{x}_n+c_n}\right),}\hfill \\ {\displaystyle \mathrm{s}.\mathrm{t}.\sum _{k=1}^K\sum _{n\in {\Omega }_{T_k}}{x}_n\le P_S,}\hfill \\ {\displaystyle \sum _{k=1}^K\sum _{n,m\in {\Omega }_{T_k}}{\eta }_n{x}_n\le I_{tol},}\hfill \\ {\displaystyle x_n\ge 0,\forall k,n,}\hfill \\ {\Omega }_{T_1}\bigcup \cdots \bigcup {\Omega }_{T_K}\subseteq {\Omega }_T,\hfill \\ {\Omega }_{T_i}\bigcup {\Omega }_{T_j}=\varnothing ,\forall i\ne j,i,j\in \{1,\cdots ,K\},\hfill \end{array}$$

(11)

where $x_n$ denotes the transmit power on the n-th subcarrier, $b_n$ and $c_n$ are constants determined by (7), which can be given by

$$b_n=\frac{{\sigma }^2}{h_{n,k}^2{P}_{r_{n,k}}},c_n=\frac{{\sigma }^2}{h_1^2}\left(1+\frac{{\sigma }^2}{h_{n,k}^2{P}_{r_{n,k}}}\right).$$

(12)

3. The Optimal Resource Allocation Strategy for a PLC System

In this section, we will give the optimal resource allocation algorithm by considering both the transmit power and interference power constraints [33,34,35,36]. Since problem (11) is a joint optimization problem, we can divide the complex problem into two subproblems to obtain the global optimal solution, i.e., the subcarrier allocation strategy and power allocation.

3.1. The Subcarrier Allocation Strategy

Assume the resource allocation strategy is any given, the power on subcarrier n is $P_n^{\prime }$, and the information rate can be written as $R_n^{\prime }(n\in {\Omega }_T)$. Let the n-th subcarrier licensed to the k-th PLC users that with the best forward channel, that is

$$k_n^{*}=arg\underset{k}{max}\{|h_{n,1}^2{|}^2,\cdots ,|h_{n,K}^2{|}^2\},m\in {\Omega }_T,$$

(13)

then, the information rate can be given as ${\tilde{R}}_n$ with the power $P_n^{\prime }$, submit ${\tilde{R}}_n$ to (7), we can obtian

$${\tilde{R}}_n\ge R_n^{\prime },n\in {\Omega }_T,$$

(14)

accordingly, the sum rate of the PLC system can be given by

$${\tilde{R}}_{sum}=\sum _{n\in {\Omega }_T}{\tilde{R}}_n\ge \sum _{n\in {\Omega }_T}{R}_n^{\prime }.$$

(15)

It can be seen that based on the subcarrier allocation scheme shown in (13), the maximum sum information rate can be achieved with the optimal power control strategy. In addition, by dividing the joint optimal problem into two subproblems, the complexity of the joint optimization problem can be descended. We can only consider the power allocation problem to obtain the global optimal solution.

3.2. The Optimal Power Control Scheme

Based on the above description, the optimization problem (11) can be degraded as

$$\begin{array}{c}{\displaystyle \underset{x_n}{max}f\left(\mathbf{x}\right)=\sum _{n\in {\Omega }_T}{log}_2\left(1+\frac{x_n}{b_n{x}_n+c_n}\right),}\hfill \\ {\displaystyle \mathrm{s}.\mathrm{t}.\sum _{n\in {\Omega }_T}{x}_n\le P_S,}\hfill \\ {\displaystyle \sum _{n\in {\Omega }_T}{\eta }_n{x}_n\le I_{tol},x_n\ge 0,\forall n,}\hfill \end{array}$$

(16)

In order to obtain the optimal solution of (16), we investigate the property of the objective function. The second-order partial derivative of the objective function $f(x)$ with N independent variables can be given as

$$\frac{{\partial }^{2}f(\mathbf{x})}{\partial x_n^2}=-c_n\frac{2x_n{b}_n(b_n+1)+2b_n{c}_n+c_n}{{(b_n{x}_n+c_n)}^2{(b_n{x}_n+x_n+c_n)}^2}<0,\forall n\in {\Omega }_T,$$

and

$$\frac{{\partial }^{2}f(\mathbf{x})}{\partial x_n\partial x_m}=0,\forall n,m\in {\Omega }_T,\mathrm{and}n\ne m,$$

we can see that the Hessian matrix ${\nabla }^{2}f(\mathbf{x})=\mathrm{diag}\left\{\frac{{\partial }^2\mathrm{f}(\mathbf{x})}{\partial {\mathrm{x}}_1^2},\cdots ,\frac{{\partial }^2\mathrm{f}(\mathbf{x})}{\partial {\mathrm{x}}_{\mathrm{N}}^2}\right\}$ is a diagonal matrix, and the diagonal elements are all negative, therefore, the Hessian matrix of objective function is a negative matrix, the objective function of problem (16) is strictly convex on the space spanned by $\{x_1,\cdots ,x_N\}$ [37,38,39]. Here we analyze the situation that the power control and interference constraint take the equal sign simultaneously, then, the problem can be formulated as

$$\begin{array}{c}{\displaystyle \underset{x_n}{max}f\left(\mathbf{x}\right)=\sum _{n\in {\Omega }_T}{log}_2\left(1+\frac{x_n}{b_n{x}_n+c_n}\right),}\hfill \\ {\displaystyle \mathrm{s}.\mathrm{t}.\sum _{n\in {\Omega }_T}{x}_n=P_S,}\hfill \\ {\displaystyle \sum _{n\in {\Omega }_T}{\theta }_m{x}_n=I_{tol},x_n\ge 0,\forall n,}\hfill \end{array}$$

(17)

where the constraint function $h_1(x_m)={\sum }_{m\in {\Omega }_T}{x}_m-P_S$ and $h_2(x_m)={\sum }_{m\in {\Omega }_T}{\theta }_m{x}_m-I_{th}$ is affine function, respectively. Therefore, the optimization problem (17) is convex optimization problem, which can be solved by the Karush-Kuhn-Tucher (KKT) conditions. In the following, based on the subcarrier allocation strategy, we discuss the optimal power control scheme with both the power and interference constraints by using the Lagrangian multiplier method.

The Lagrangian function of (17) is given by

$$L=f(\mathbf{x})+\sum _{n\in {\Omega }_T}\lambda x_n-{\nu }_1\left(\sum _{n\in {\Omega }_T}{x}_n-P_s\right)-{\nu }_2\left(\sum _{n\in {\Omega }_T}{\eta }_n{x}_n-I_{tol}\right).$$

(18)

where $\lambda$ is a constant. Let $\frac{\partial L}{\partial x_n}=0$, the derivative of (18) with respect to $x_n$ is given by

$$G_n(x_n)+\lambda -{\nu }_1-{\eta }_{n,m}{\nu }_2=0,$$

(19)

where

$$G_n(x_n)=\frac{c_n}{(b_n{x}_n+c_n)(b_n{x}_n+x_n+c_n)}.$$

(20)

according to the KKT (i) condition [40,41,42], i.e., $\lambda \ge 0$, we can obtain

$${\nu }_1+{\eta }_n{\nu }_2\ge G_n(x_n),$$

(21)

with KKT condition (ii), i.e., $\lambda x_n=0$, substituting (19) into (21), we can obtain

$$\left[{\nu }_1+{\eta }_{n,m}{\nu }_2-G_n(x_n)\right]x_n=0.$$

(22)

Since $x_n$ denotes the power allocated on the n-th subcarrier, which is non-negative real number. Let $z_n\triangleq \frac{1}{c_n}$, we can obtain

$$\left\{\begin{array}{cc}{x}_n>0,{\nu }_1+{\eta }_n{\nu }_2=G_n(x_n)<G_n(0)=z_n,\hfill & \forall n\in {\mathsf{\Phi }}_1,\hfill \\ x_n=0,{\nu }_1+{\eta }_n{\nu }_2\ge G_n(0)=z_n,\hfill & \forall n\in {\mathsf{\Phi }}_2.\hfill \end{array}\right.$$

(23)

where ${\mathsf{\Phi }}_1\bigcup {\mathsf{\Phi }}_2={\Omega }_T$, ${\mathsf{\Phi }}_1\bigcap {\mathsf{\Phi }}_2=\varnothing$. When $x_m>0$, substituting (23) into (19), we can obtain

$$x_n=\frac{c_n}{b_n(b_n+1)}{\left[\sqrt{1+\frac{b_n(b_n+1)}{c_n({\nu }_1+{\eta }_{n,m}{\nu }_2)}}-b_n-1\right]}^{+},$$

(24)

where ${\left[x\right]}^{+}=\max \{0,x_n\}$, parameters ${\nu }_1,{\nu }_2$ are determined by the two equality equations of power and interference constraints. (24) gives the expression of the optimal solution for (17) and in what follows, we will prove there exists a unique solution that satisfies (24).

Assume the subcarrier sets ${\mathcal{N}}_1,{\mathcal{N}}_2\subseteq {\Omega }_T$, ${\overline{\mathcal{N}}}_1,{\overline{\mathcal{N}}}_2$ is the complement set, respectively. Let ${\mathcal{N}}_0={\mathcal{N}}_1\bigcap {\mathcal{N}}_2,{\mathcal{M}}_0={\mathcal{N}}_1\bigcup {\mathcal{N}}_2,{\mathcal{C}}_1={\mathcal{N}}_1-{\mathcal{N}}_0,{\mathcal{C}}_2={\mathcal{N}}_2-{\mathcal{N}}_0$, and suppose the power allocation solution of (24) is exist in subcarrier sets ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ simultaneously, we have

$$\left\{\begin{array}{c}{x}_{n\in {\mathcal{N}}_1}({\nu }_1,{\nu }_2)>0,x_{n\in {\overline{\mathcal{N}}}_1}({\nu }_1,{\nu }_2)\le 0,\hfill \\ x_{n\in {\mathcal{N}}_2}({\nu }_1^{\prime },{\nu }_2^{\prime })>0,x_{n\in {\overline{\mathcal{N}}}_2}({\nu }_1^{\prime },{\nu }_2^{\prime })\le 0.\hfill \end{array}\right\}$$

(25)

according to the assumption, there are two situations, i.e., ${\nu }_1>{\nu }_1^{\prime },{\nu }_2<{\nu }_2^{\prime }$ and ${\nu }_1<{\nu }_1^{\prime },{\nu }_2>{\nu }_2^{\prime }$, without loss of generality, we only consider the situation ${\nu }_1>{\nu }_1^{\prime },{\nu }_2<{\nu }_2^{\prime }$.

Calculate $x_n$ by (24), if the value is negative, let $x_n=0$, that is

$$x_n({\nu }_1,{\nu }_2)=0,n\in {\overline{\mathcal{N}}}_1,\mathrm{and}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime })=0,n\in {\overline{\mathcal{N}}}_2.$$

(26)

and ${\mathcal{C}}_2\subseteq {\overline{\mathcal{N}}}_1,{\mathcal{C}}_1\subseteq {\overline{\mathcal{N}}}_2$, which leads to

$$x_n({\nu }_1,{\nu }_2)=0,n\in {\mathcal{C}}_2,\mathrm{and}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime })=0,n\in {\mathcal{C}}_1.$$

Then, from ${\sum }_{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)={\sum }_{n\in {\mathcal{N}}_2}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime })=P_S$, we can derive that

$$\begin{array}{c}\hfill P_S=\sum _{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)+\sum _{n\in {\mathcal{C}}_2}{x}_n({\nu }_1,{\nu }_2)=\sum _{n\in {\mathcal{N}}_2}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime })+\sum _{n\in {\mathcal{C}}_1}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime }).\end{array}$$

(27)

According to the definition of set ${\mathcal{M}}_0$, we can obtain

$$\begin{array}{c}\hfill P_S=\sum _{n\in {\mathcal{M}}_0}{x}_n({\nu }_1,{\nu }_2)+\sum _{n\in {\mathcal{M}}_0}{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime }).\end{array}$$

(28)

Define two subcarrier sets ${\mathcal{D}}_1,{\mathcal{D}}_2$, which satisfy

$$\left\{\begin{array}{c}{x}_n({\nu }_1,{\nu }_2)\ge x_n({\nu }_1^{\prime },{\nu }_2^{\prime }),n\in {\mathcal{D}}_1,\hfill \\ x_n({\nu }_1,{\nu }_2)<x_n({\nu }_1^{\prime },{\nu }_2^{\prime }),n\in {\mathcal{D}}_2.\hfill \end{array}\right\},$$

(29)

where ${\mathcal{D}}_1\bigcup {\mathcal{D}}_2={\mathcal{M}}_0$, combined with (24), we can derived that

$${\eta }_{n\in {\mathcal{D}}_1}>{\eta }_{n\in {\mathcal{D}}_2}.$$

(30)

Additionally, from (27) we can obtain

$$\sum _{n\in {\mathcal{D}}_1}\left(x_n({\nu }_1,{\nu }_2)-x_n({\nu }_1^{\prime },{\nu }_2^{\prime })\right)=\sum _{n\in {\mathcal{D}}_2}\left(x_n({\nu }_1^{\prime },{\nu }_2^{\prime })-x_n({\nu }_1,{\nu }_2)\right).$$

(31)

therefore, from (31) and (30) we can obtain the expression given by

$$\sum _{n\in {\mathcal{D}}_1}{\eta }_n\left(x_n({\nu }_1,{\nu }_2)-x_n({\nu }_1^{\prime },{\nu }_2^{\prime })\right)>\sum _{n\in {\mathcal{D}}_2}{\eta }_n\left(x_n({\nu }_1^{\prime },{\nu }_2^{\prime })-x_n({\nu }_1,{\nu }_2)\right),$$

(32)

similarly, when ${\nu }_1<{\nu }_1^{\prime },{\nu }_2>{\nu }_2^{\prime }$, we can obtain

$$\sum _{n\in {\mathcal{N}}_1}{\eta }_n{x}_n({\nu }_1,{\nu }_2)<\sum _{n\in {\mathcal{N}}_2}{\eta }_n{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime }).$$

(33)

(32) and (33) imply that the following expression holds

$$\sum _{n\in {\mathcal{N}}_1}{\eta }_n{x}_n({\nu }_1,{\nu }_2)\ne \sum _{n\in {\mathcal{N}}_2}{\eta }_n{x}_n({\nu }_1^{\prime },{\nu }_2^{\prime }).$$

(34)

It can be seen that the above conclusion contradicts the previously held assumption. Therefore, there exists a unique solution for the optimization problem (24). Based on the above description, we can obtain the optimal power allocation scheme, however, the expression of the value L, ${\nu }_1$ and ${\nu }_2$ are not available. The details about the calculation of the parameters are given in Algorithm 1.

Algorithm 1 The optimal power control algorithm of problem (17).

1: Initialization: Let ${\nu }_1\in [{\nu }_{1,min},{\nu }_{1,max}]$, ${\nu }_2\in [{\nu }_{2,min},{\nu }_{2,max}]$, ${\nu }_{1,max}=min{G_{(i)}(P_s)}_{i\in {\Omega }_n}$, ${\nu }_{2,max}=min{G_{(i)}(0)}_{i\in {\Omega }_n}$, ${\nu }_{1,min}=0,{\nu }_{2,min}=0$. 2: Let  ${\nu }_1=\frac{{\nu }_{1,min}+{\nu }_{1,max}}{2},{\nu }_2=\frac{{\nu }_{2,min}+{\nu }_{2,max}}{2}$, calculate $x_n({\nu }_1,{\nu }_2)$ by (24), and obtain the subcarrier set  ${\mathcal{N}}_1=\{n\in {\Omega }_T|x_n>0\}$. 3: while$\frac{\left|{\sum }_{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)-P_S\right|}{P_S}>ϵ$ or $\frac{\left|{\sum }_{n\in {\mathcal{N}}_1}{\theta }_n{x}_n({\nu }_1,{\nu }_2)-I_{tol}\right|}{I_{tol}}>ϵ$ do 4:  while $\left({\sum }_{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)-P_S\right)\left({\sum }_{n\in {\mathcal{N}}_1}{\theta }_n{x}_n({\nu }_1,{\nu }_2)-I_{tol}\right)>0$ do 5:   If ${\sum }_{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)-P_S>0$, and ${\sum }_{n\in {\mathcal{N}}_1}{\theta }_n{x}_n({\nu }_1,{\nu }_2)-I_{tol}>0$, ${\nu }_{1,min}={\nu }_1$, otherwise, ${\nu }_{1,max}={\nu }_1$. 6:   Let ${\nu }_1=\frac{{\nu }_{1,min}+{\nu }_{1,max}}{2}$, calculate $x_n$ by (24), and obtain the set ${\mathcal{N}}_1$. 7:  end while 8:  If ${\sum }_{n\in {\mathcal{N}}_1}{x}_n({\nu }_1,{\nu }_2)>P_S$ and ${\sum }_{n\in {\mathcal{N}}_1}{\theta }_n{x}_n({\nu }_1,{\nu }_2)<I_{tol}$, ${\nu }_{1,min}={\nu }_1$, ${\nu }_{2,max}={\nu }_2$, otherwise, ${\nu }_{1,max}={\nu }_1$,  ${\nu }_{2,min}={\nu }_2$. 9:  Let ${\nu }_1=\frac{{\nu }_{1,min}+{\nu }_{1,max}}{2},{\nu }_2=\frac{{\nu }_{2,min}+{\nu }_{2,max}}{2}$, calculate $x_n({\nu }_1,{\nu }_2)$ by (24), obtain the set ${\mathcal{N}}_1$. 10: end while 11: Output $x_n$.

4. Numerical Results

In this section, we show the numerical results of the proposed resource allocation strategy in the practical PLC environment. Assume the relay lies in the middle of the source and users, and we use the following parameters: The frequency band is $0-20$ MHz, the subcarrier band is $\mathsf{\Delta }f=62.5$ KHz. The total subcarrier number is $N=128$. Substituting the OFDM symbol interval $\mathsf{\Delta }T=\frac{1}{\mathsf{\Delta }f}$ into (9) to obtain the normalized interference power ${\eta }_n$ between channel n and channel m. With the proposed dynamic resource allocation algorithm, we can obtain the interference coefficient, interference power and optimal transmit power, as shown in Figure 2. It can be seen that the subcarrier $(1,3,6,7)$ has a large interference coefficient, and no power allocated on these subcarriers. Meanwhile, the subcarrier $(9,10,11,17)$ has small interference coefficient (${\eta }_{17}=0.011$) and more power is allocated for signal transmission, that is, as the interference increases, the power gradually decreases, and vice versa. In addition, the PLC channel status can also affect the power allocation result, we can see that the interference coefficient ${\theta }_{18}=0.023<{\theta }_{14}=0.036$. However, the optimal transmit power allocated on 14 is larger than 18. Similary, due to the PLC channel status, the subcarrier $(12,13,15)$ did not assign power to transmit signals. Based on the above discussion, we can conclude that the optimal power allocation is affected by both the interference coefficient and the channel status.

Figure 3 depicts the PLC system throughput with the growth of the interference threshold $I_{tol}$ under different transmit powers. It is obvious that when the interference threshold $I_{tol}$ takes a smaller value, the system throughput increases with the growth of $I_{tol}$ no matter the value of transmit power. With the further increase of $I_{tol}$, the system throughput is constrained by the transmit power and the larger power the higher throughput. In addition, the throughput under different transmit power $P_s$ no longer increases till the accordingly points of $I_{tol}$, since with the growth of $I_{tol}$, the interference constraint ineffective to the resource allocation.

In Figure 4, the system throughput is curved with the growth of interference threshold $I_{tol}$, and the transmit power is $P_s=10$ dBW. Different resource allocation strategies have been shown in Figure 4, including the subcarrier pre-assignment method of FDMA and the optimal subcarrier allocation scheme (Opt. subcarrier), and the power control policy (Opt.PA and Equal PA). It can be seen that the proposed scheme achieves the maximum system throughput comparing with all the other schemes.

5. Conclusions

This paper focuses on the relay-based multi-user PLC downlink system, which can satisfy the total power constraint and the interference limitations required by the users. An optimal resource allocation scheme is proposed to maximize the system throughput with respect to the joint subcarrier and power constraints. In order to avoid the comuputational burden caused by the MINLP, we divide the joint optimization problem into two individual subproblems with rigorous proofs. The numerical results show that the global optimal solution can still obtained and the simulation results confirm that the proposed resource allocation scheme outperforms other schemes that mentioned in this paper.