# Mobile Fog Computing-Assisted Resource Allocation for Two-Hop SWIPT OFDM Networks

Authors: Xiaofei Di; Yu Zhang; Tong Liu; Shaoli Kang; Yue Zhao
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

The mobile fog computing-assisted resource allocation (RA) is studied for simultaneous wireless information and power transfer (SWIPT) two-hop orthogonal frequency division multiplexing (OFDM) networks, where a decode-and-forward (DF) relay first harvests energy from signals emitted by a source and then helps the source to forward information to its destination by using the harvested energy. Power splitting (PS) strategy is adopted at the relay and a different PS (DPS) receiver architecture is proposed, where the PS factors of all subcarriers are different. A RA problem is formulated to maximize the system’s achievable rate by jointly optimizing subcarrier pairing, power allocation, and PS factors. Since the RA problem is a nonconvex problem and is difficult to solve, an efficient RA algorithm is designed. As the wireless channels are fast time-varying, the computation is performed in mobile fog node close to end nodes, instead of remote clouds. Results demonstrate that the achievable rate is significantly increased by using the proposed RA algorithm. It is also found that the computation complexity of RA algorithm of DPS receiver architecture is much lower than the existing identical PS (IPS) receiver architecture, and thus the proposed DPS architecture is more suitable for computation-constrained fog system.


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## Body

## 1. Introduction

In the past decade, cloud computing has emerged as a new paradigm. It enables computing, storage, and network managements to centralize in the clouds, which are referred to as data centers, cellular core networks, and so on. With the clouds, vast resources can be provided to resource-constrained devices to satisfy their requirements of computing and storage. However, there is an inherent limitation for cloud computing [1], i.e., the long propagation distance from the end user to the remote cloud center, resulting in very long latency.Recently, a new trend has been happening; that is, the computing is pushed to the network edge devices due to their progressively enhanced computation capacity. This is called mobile fog computing (MFC) or mobile edge computing (MEC) [2, 3], where the network edge devices perform computing tasks instead of remote clouds. Thanks to closer distance to end users, the latency is less and thus real-time tasks can be achieved via MFC, which is an effective supplement to cloud computing. MFC is applicable to delay sensitive tasks while cloud computing to sophisticated but delay-insensitive data processing work.One potential application of MFC is in Internet of Things (IoT), such as wireless sensor networks (WSN). In WSN, sensor nodes are responsible for data gathering, and sink nodes for collecting and preprocessing data from surrounding sensor nodes and then delivering data to the remote clouds, which perform further complicated data processing and information mining. For WSN, resource allocation (RA) is a key approach to improve system performance. It is carried out according to different channel states and deemed to be a real-time task, as the wireless channels are fast time-varying channels and RA needs to be processed rapidly to adapt the dynamic channels. So MFC is a more appropriate option than cloud computing.In the field of WSN, cooperative relay communication is deemed as an important technique, as it can guarantee that the far sensor nodes can complete communication with each other via intermediate relaying sensor nodes [4, 5]. On the other hand, orthogonal frequency division multiplexing (OFDM) is employed in wireless communication networks [6]. The combination of relay and OFDM is able to significantly enhance the performance of the system [7, 8].MFC-assisted cooperative relay systems [9, 10] and OFDM systems [11, 12] have attracted much attention and been widely investigated. In [9], a fog-enabled cooperative communication network was considered, where multiple fog nodes were configured to support two-hop transmissions, and the optimal system performance was achieved by designing time reuse patterns. In [10], cooperative fog computing for the Internet of Vehicles (IoV) was studied, where the cooperation of fog nodes was explored to enhance the system performance. In [11], a MFC-assisted multiuser OFDM network was considered, and the total consumed energy of mobile users was minimized by jointly optimizing subcarrier and CPU time allocation. In [12], the joint subcarrier and power allocation problem in an MFC-based OFDM system was investigated to minimize the maximal delay of all devices.Meanwhile, WSN are usually energy-constrained networks, and connecting sensor nodes to power grid is impossible sometimes. Batteries can be deployed in sensor nodes, but the batteries capacity is limited and may be hard to be replaced frequently. Recently, wireless power transfer has attracted much attention, in which energy-constrained devices can harvest energy by using wireless signals emitted by system nodes with sufficient energy source. Noting that wireless signal can simultaneously carry and transfer information and energy, this is deemed as the simultaneous wireless information and power transfer (SWIPT) [13].SWIPT has been widely investigated [14–24]. In [14], it was assumed that information decoding (ID) and energy harvesting (EH) are simultaneously carried out by using the received identical signals. Nevertheless, this is deemed not to be realized, and therefore some practical SWIPT receivers were also presented, such as time switching (TS) and power splitting (PS) in [13]. In TS receivers, ID and EH are performed in two different phases, respectively. In PS receivers, the wireless signal is split into two streams: one stream enters into the energy receiver to harvest energy, and the other enters into the information receiver to obtain information. In [16], these SWIPT receiver architectures were applied to cooperative two-hop network, and the achievable rate performance was investigated. These architectures have also been widely studied in OFDM systems; see, e.g., [17–20]. But, the existing work mainly concerns point-to-point OFDM systems. For example, in [17, 19], the performances of throughput and weighted sum-rate were investigated for multiuser OFDM networks. In [20], max-min fair resource allocation was studied for multigroup multicast OFDM systems. In [18], a new SWIPT receiver architecture was proposed, where one part of the subcarriers was used for ID, and the other part was used for EH.Recently, some work discussed the SWIPT-enabled two-hop OFDM system. In [21, 22], the authors considered amplify-and-forward (AF) relaying protocol and the achievable information rates were maximized for two-hop MIMO-OFDM AF relay system. In [23, 24], the SWIPT-enabled two-hop OFDM decode-and-forward (DF) relay system was considered, but the subcarrier pairing over the two hops was not involved. In [15], a PS receiver architecture was considered and a RA algorithm was proposed to improve the achievable rate; however, the complexity of algorithm was so high that it was hard to be applied to computation-constrained fog system.This paper investigates the SWIPT for a MFC-assisted two-hop OFDM network, in which a source node transmits information to a destination with the help of a DF relay. The source is assumed with fixed energy source, while the relay is an energy-constrained node and thus has to obtain energy from wireless signal emitted by the source and further forwards the information of source to destination.The main contributions of this paper are given as follows.Firstly, to achieve the simultaneous information and energy transmission, we adopt a different PS ratio PS (DPS) architecture, where a frequency selective power splitter splits the signal on each subcarrier into two streams and thus all subcarriers are of the different PS ratios, which can adaptively change. Further, a particular energy cooperation strategy is considered; i.e., the energy harvested on some subcarrier of the first hop is only used to forward the information received on the corresponding subcarrier, in order to reduce the excessive computational complexity. Unlike the existing work [15], it is assumed that PS receiver splits all subcarriers into two streams with identical PS ratio, which is called identical PS ratio PS (IPS) receiver architecture in this paper.Secondly, in order to explore the system performance limit of the proposed DPS architecture, a RA optimization problem is formulated to maximize the achievable rate of the system by jointly optimizing the subcarrier pairing (SP), the PS ratios, and the PA at both source and the relay. As the problem is nonconvex and hard to solve, a low-complexity efficient RA algorithm is designed by decomposing it into three separate subproblems. The related computation is operated at the source node, which is generally a sink node of higher computation capacity in WSN.Thirdly, extensive simulation experiments are performed to discuss the system performance. The results demonstrate that although there are some performance loss of achievable rate of the proposed DPS architecture compared with the existing IPS architecture, the computation complexity of DPS architecture is much lower than IPS architecture. So DPS architecture may be a better option for computation-constrained fog system.This paper is organized as follows. In Section2, the network architecture and system model are presented, and then RA optimization problem is formulated. In Section 3, an efficient RA algorithm is designed. Simulation results are shown in Section 4 to discuss the performance of the DPS receiver architecture and RA algorithm. In Section 5, this paper is summarized.


## 2. Network Architecture and System Model

The considered network architecture is shown as in Figure1, which is divided into three layers, i.e., data layer, fog layer, and cloud layer. Data layer comprises data nodes, which are responsible for gathering data from surrounding environment. Cloud layer contains vast resources to store and process the amount of data from data layer; meanwhile, it also sends control information to data nodes to instruct their operations. Fog layer is a bridge between data layer and cloud layer, which means that on one hand, it is responsible for collecting and preprocessing data from data layer and delivering data to cloud layer to further process it; on the other hand, it is responsible for forwarding control information from cloud layer to data layer. The information between cloud layer and fog layer is transmitted on wired channels while the information between fog layer and data layer is transmitted on wireless channels.Figure 1
Network architecture of MFC-assisted two-hop OFDM system.In this paper, we consider the transmission of control information from cloud layer to data layer. The cloud layer first sends control information to fog layer; fog layer stores the information and then forwards it to data layer. The reason of introducing the fog layer instead of directly using cloud computing is that the wireless channels are deemed to be fast time-varying channels and thus information transmission and RA task have to be performed rapidly to adapt the dynamic channels. It is worth noting that due to the enhanced computing capacity, the fog layer has the capability of performing RA algorithm according to the channel state information (CSI).To study the information transmission from fog layer to data layer, a MFC-assisted two-hop OFDM network is considered, which consists of one source (S) in fog layer, and one destination (D) and one relay (R) in data layer, as shown in Figure1. S desires to send information to D with the help of R. No direct link exists between S and D. S is of steady energy supply by connecting to power grid in fog layer and PS denotes its power. R is an energy-constrained node operating in half-duplex mode and deploying DF relaying protocol, so it has to obtain energy from the signals of S and then uses the harvested energy to help S to forward information to D. The PS receiver architecture is adopted at R so that it can split the received RF signals into two streams to perform EH and ID, respectively. For such a SWIPT-enabled communication network, each transmission is based on frame of lengthT, which is divided into two subphases of equal length.In the first subphase, S sends OFDM symbols to R. The received signal at R on subcarrieri can be expressed as(1)yr,i=Ps,ihixi+zr,i,∀i∈1,…,N,where xi and hi, respectively, represent transmitted symbol and channel coefficient on subcarrieri andN is the number of subcarriers. zr,i represents the additive white Gaussian noise (AWGN) from the antenna on subcarrier i at R, which is of zero mean and variance σr,att2. Ps,i represents the transmission power at S on subcarrier i and satisfies(2)∑i=1NPs,i≤PS,Ps,i≥0,∀i.In the second subphase, using the stream for ID and EH, R, respectively, decodes the received information and harvests the energy and then reencodes the received information and forwards the reencoded information to D. Subcarrier pairing is adopted, so the information of the first hop received on subcarrieri can be transmitted on the subcarrier j in the second hop. The signal on subcarrier j received at D can be expressed as(3)yd,j=Pr,jgjxi+zd,j,∀j∈1,…,N,where gj is channel coefficient on subcarrier j at D, and Pr,j is the power on subcarrier j at R. zd,j is AWGN from the antenna on subcarrier j at D, which is of zero mean and variance σd,att2.To realize SWIPT, a PS receiver architecture is proposed, where all subcarriers are of different PS (DPS) ratios and can adaptively adjust, which is called DPS architecture. To implement DPS architecture, an analog adaptive passive frequency selective power splitter is required [21, 24]. In this paper, we consider a particular energy cooperation strategy which makes the computational complexity of the RA algorithm of the DPS architecture significantly decrease and is very meaningful for some communication scenarios where the processing capacity of communication nodes is limited, such as MFC-based WSN.Let the PS ratiosθiI and θiE represent the fraction of the signal power used for ID and EH received on subcarrieri, respectively, which satisfy the constraints of(4)θiI+θiE=1,θiI≥0,θiE≥0,∀i∈1,…,NThus, the harvested energy on subcarrieri at R is given by(5)Ei=T2ηθiEhi2Ps,i,where η denotes the EH efficiency.We consider such anenergy cooperationstrategy adopted in [21, 24], in which the energy harvested on subcarrier i of the first hop is only used to forward the information received on subcarrier i and thus the available power Pr,j on subcarrier j at R can be inferred as(6)Pr,j=EiT/2=ηθiEhi2Ps,i.The achievable information rate between S and D for DF relay system on a subcarrier pair(i,j) can be expressed as [4](7)Ri,jDPS=12min⁡log2⁡1+θiIhi2Ps,iθiIσr,att2+σr,proc2,log2⁡1+gj2Pr,jσd2,where σd2=σd,att2+σd,proc2 represents the total noise power of D on each subcarrier, σr,proc2 and σd,proc2, respectively, represent the power of signal processing noise on any subcarrier of R and D. In (7), the first part log21+θiIhi2Ps,i/θiIσr,att2+σr,proc2 represents the mutual information from S to R on subcarrieri, and the second part log21+gj2Pr,j/σd2 represents the mutual information from R to D on subcarrier j. The coefficient 1/2 in (7) is because each frame is composed of two subphases of equal length.Substituting (6) into (7), then we can write (7) as(8)Ri,jDPS=12min⁡log21+θiIhi2Ps,iθiIσr,att2+σr,proc2,log21+ηθiEhi2gj2Ps,iσd2.Thus, the achievable rate of the system can be expressed as(9)RDPSP,S,θ=∑i=1N∑j=1Nsi,jRi,jDPS,where θ=θiI,θiE,∀i is PS policy and satisfies the constraint (4). P=Ps,i≥0,∣∀i,j is power allocation (PA) policy and satisfies (2). S=si,j∈0,1∣∀i,j is SP policy, which represents that if the first hop subcarrier i is matched with the second hop subcarrierj, si,j =1; else si,j=0. Further, one first hop (second hop) subcarrier can only match with one second hop (first hop) subcarrier. That is,(10)∑j=1Nsi,j≤1,∀i,∑i=1Nsi,j≤1,∀j.With the objective of maximizing the achievable information rate of the system, by jointly optimizing the SP, the PA, and the PS ratio, the optimization problem is formulated as (P1):(11)maxP,S,θRDPSP,S,θs.t.2,4,10.


## 3. Resource Allocation Design

In this section, we first describe our proposed resource allocation (RA) algorithm for problem P1 and then we shall prove that it is able to achieve the global optimal solution of problem P1.


### 3.1. The Proposed Resource Allocation

Our proposed RA is described as Algorithm1, which is divided into three separate subproblems. In what follows of this subsection (Section 3.1), the detailed process of each step in Algorithm 1 is described, and its global optimality is proven in Section 3.2.Algorithm 1:Resource allocation algorithm.
(1) Find the optimal S∗ from (12). (2) Calculate the optimal θ∗ with the obtained S∗ from (18). (3) Calculate the optimal P∗ with the obtained S∗ and θ∗ from (22).(1  ) The Optimal SP S ∗ . The proposed SP scheme is only based on the channel power gains. Firstly, according to the channel power gains of the two hops hi2 and gj2, the first hop subcarriers and the second hop subcarriers are, respectively, sorted from highest to lowest. Next, thekth first hop subcarrier is matched with thekth second hop subcarrier, which is equivalent to the optimal S∗ satisfying that(12)si,j=1,ifNumberhi2=Numbergj2,0,otherwise,whereNumber(ui2) represents the serial number of ui2 inside all uk2 for k∈{1,2,…,N} with an degressive sorting sequence for u∈{h,g}, respectively. This scheme is called the channel gain- (CG-) sorted SP scheme. The optimality of this scheme is given as follows.Lemma 1.
The optimal SP scheme of problem P1 is the CG-sorted SP scheme.Proof.
To prove this lemma, two-subcarrier case is first considered and proved. Then it is further extended to general multisubcarrier case. See the Appendix for details.(2  ) The Optimal PS θ ∗ with the Obtained S ∗ .First, it is easily found that the problem can be decomposed into N subproblems due to the independence of each subcarrier pair. For any given subcarrier pair (i,j), the subproblem can be expressed as (P2):(13)maxθiI,θiERi,js.t.θiI+θiE=1,θiI≥0,θiE≥0.To simplify the expressions, letA=hi2,B=ηgj2/σd2 and denote Ps,i as Pi. Since the subcarrier pair is fixed, we further drop the indexesi,jin this subsection. Thus the achievable information rate in (8) on a fixed subcarrier pair can be expressed as(14)R=12min⁡log21+AθIPθIσr,att2+σr,proc2,log21+ABθEP.It is easy to find that in (14) the first term, i.e., log21+AθIP/θIσr,att2+σr,proc2, is a monotonically increasing function of θI, and the second term, i.e., log2(1+ABθEP), is a monotonically decreasing function of θI, so, to obtain the optimal solution, the two terms should be equal. Meanwhile, using θI+θE=1, the optimal PS factor θI can be calculated, according to(15)Bσr,att2θI2+1-Bσr,att2+Bσr,proc2θI-Bσr,proc2=0.This is a quadratic equation and its two roots are given as(16)θI=-1-Bσr,att2+Bσr,proc2±1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2,where only the one satisfying the constraints in problem P2 can be considered as the optimal solution. Since θI+θE=1,θI,θE≥0, we have that 0≤θI,θE≤1.It is easy to observe that the one of the two roots(17)θI=-1+Bσr,att21Bσr,proc2-1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2is always less than 0, so it is discarded. For the other one, we can prove that it satisfies the above constraint.Thus, the optimal PS factors can be given by(18)θI∗=-1+Bσr,att2-Bσr,proc2+1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2.(3  ) The Optimal PA P ∗ with the Obtained S ∗ and θ ∗ .We have obtained the optimal PS factors θI∗ and θE∗. As the optimal PS factors are related to the channel gain of the second hop from (18), θiI∗, θiE∗ are represented as θi,jI∗,θi,jE∗ for given subcarrier pair (i,j). Since the two terms are equal in (8) for optimal θi,jI∗,θi,jE∗, (8) can be transformed as(19)Ri,j=12log21+hi2θi,jI∗Piθi,jI∗σr,att2+σr,proc2.We denotehi2θi,jI∗/θi,jI∗σr,att2+σr,proc2 as γi,j, and then (19) is transformed into(20)Ri,j=12log21+γi,jPi.So for givenS∗ and θ∗, the PA problem can be formulated as (P3):(21)maxP∑i,j∈SPRi,js.t.2,where SP is the set of subcarrier pairs. From (18), it can be easily found that the optimal PS factors are not related to PA and thus γi,j is also not related to PA. So this problem is a classical water-filling PA problem, and we can obtain its optimal solution as(22)Pi∗=1νln⁡2-1γi,j+,∀i,where [x]+=max⁡0,x, and ν is Lagrangian multiplier and can be solved using ∑i=1N1/νln⁡2-1/γi,j+=PS.


### 3.2. Global Optimum of Our Proposed RA

In this subsection, we shall prove that although Algorithm1 is divided into three separate subproblems, it can still achieve the global optimal solution of problem P1, and the result is given by the following theorem.Theorem 2.
The RA in Algorithm1 achieves the global optimum of problem P1.Proof.
To prove that the RA policy in Algorithm1 can achieve the global optimal solution of problem P1, we only need to prove that each step of Algorithm 1 maintains the global optimum. From Lemma 1, we have known that the CG-sorted SP scheme in step 1 of Algorithm 1 gives the globally optimal SP policy. The scheme is only related to channel gains, which does not require the knowledge of the optimal PS and PA. According to the derivation process in step 2 of Algorithm 1, the obtained PS is optimal under the given optimal SP, and it does not require the knowledge of optimal PA. Then in step 3 of Algorithm 1, the obtained PA is optimal under the given optimal SP and optimal PS. Since each step maintains the global optimum, Theorem 2 is proved.


### 3.3. Complexity Analysis

The complexity of step 1 of Algorithm1 depends on the adopted sorting method, which is ONlog⁡N if the quick-sort method is applied. Moreover, the complexity of step 2 of Algorithm 1 is O(N), and the complexity of step 3 of Algorithm 1 is also ON (the water-filling over the sorted γi,j) [25]. Thus, the total computational complexity can be expressed by ONlog⁡N. For comparison, the computational complexity of RA algorithm for IPS architecture in [15] is ONlog⁡N+MNN+2q, where M is the number of loops in the algorithm [15], so it can be found that the computational complexity of proposed DPS architecture’s RA algorithm is on the order of MN+2q/log⁡N less than the IPS architecture. So the proposed DPS architecture may be more proper for MFC-assisted networks, where the devices are of lower computation capacity.


## 3.1. The Proposed Resource Allocation

Our proposed RA is described as Algorithm1, which is divided into three separate subproblems. In what follows of this subsection (Section 3.1), the detailed process of each step in Algorithm 1 is described, and its global optimality is proven in Section 3.2.Algorithm 1:Resource allocation algorithm.
(1) Find the optimal S∗ from (12). (2) Calculate the optimal θ∗ with the obtained S∗ from (18). (3) Calculate the optimal P∗ with the obtained S∗ and θ∗ from (22).(1  ) The Optimal SP S ∗ . The proposed SP scheme is only based on the channel power gains. Firstly, according to the channel power gains of the two hops hi2 and gj2, the first hop subcarriers and the second hop subcarriers are, respectively, sorted from highest to lowest. Next, thekth first hop subcarrier is matched with thekth second hop subcarrier, which is equivalent to the optimal S∗ satisfying that(12)si,j=1,ifNumberhi2=Numbergj2,0,otherwise,whereNumber(ui2) represents the serial number of ui2 inside all uk2 for k∈{1,2,…,N} with an degressive sorting sequence for u∈{h,g}, respectively. This scheme is called the channel gain- (CG-) sorted SP scheme. The optimality of this scheme is given as follows.Lemma 1.
The optimal SP scheme of problem P1 is the CG-sorted SP scheme.Proof.
To prove this lemma, two-subcarrier case is first considered and proved. Then it is further extended to general multisubcarrier case. See the Appendix for details.(2  ) The Optimal PS θ ∗ with the Obtained S ∗ .First, it is easily found that the problem can be decomposed into N subproblems due to the independence of each subcarrier pair. For any given subcarrier pair (i,j), the subproblem can be expressed as (P2):(13)maxθiI,θiERi,js.t.θiI+θiE=1,θiI≥0,θiE≥0.To simplify the expressions, letA=hi2,B=ηgj2/σd2 and denote Ps,i as Pi. Since the subcarrier pair is fixed, we further drop the indexesi,jin this subsection. Thus the achievable information rate in (8) on a fixed subcarrier pair can be expressed as(14)R=12min⁡log21+AθIPθIσr,att2+σr,proc2,log21+ABθEP.It is easy to find that in (14) the first term, i.e., log21+AθIP/θIσr,att2+σr,proc2, is a monotonically increasing function of θI, and the second term, i.e., log2(1+ABθEP), is a monotonically decreasing function of θI, so, to obtain the optimal solution, the two terms should be equal. Meanwhile, using θI+θE=1, the optimal PS factor θI can be calculated, according to(15)Bσr,att2θI2+1-Bσr,att2+Bσr,proc2θI-Bσr,proc2=0.This is a quadratic equation and its two roots are given as(16)θI=-1-Bσr,att2+Bσr,proc2±1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2,where only the one satisfying the constraints in problem P2 can be considered as the optimal solution. Since θI+θE=1,θI,θE≥0, we have that 0≤θI,θE≤1.It is easy to observe that the one of the two roots(17)θI=-1+Bσr,att21Bσr,proc2-1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2is always less than 0, so it is discarded. For the other one, we can prove that it satisfies the above constraint.Thus, the optimal PS factors can be given by(18)θI∗=-1+Bσr,att2-Bσr,proc2+1-Bσr,att2+Bσr,proc22+4B2σr,att2σr,proc22Bσr,att2.(3  ) The Optimal PA P ∗ with the Obtained S ∗ and θ ∗ .We have obtained the optimal PS factors θI∗ and θE∗. As the optimal PS factors are related to the channel gain of the second hop from (18), θiI∗, θiE∗ are represented as θi,jI∗,θi,jE∗ for given subcarrier pair (i,j). Since the two terms are equal in (8) for optimal θi,jI∗,θi,jE∗, (8) can be transformed as(19)Ri,j=12log21+hi2θi,jI∗Piθi,jI∗σr,att2+σr,proc2.We denotehi2θi,jI∗/θi,jI∗σr,att2+σr,proc2 as γi,j, and then (19) is transformed into(20)Ri,j=12log21+γi,jPi.So for givenS∗ and θ∗, the PA problem can be formulated as (P3):(21)maxP∑i,j∈SPRi,js.t.2,where SP is the set of subcarrier pairs. From (18), it can be easily found that the optimal PS factors are not related to PA and thus γi,j is also not related to PA. So this problem is a classical water-filling PA problem, and we can obtain its optimal solution as(22)Pi∗=1νln⁡2-1γi,j+,∀i,where [x]+=max⁡0,x, and ν is Lagrangian multiplier and can be solved using ∑i=1N1/νln⁡2-1/γi,j+=PS.


## 3.2. Global Optimum of Our Proposed RA

In this subsection, we shall prove that although Algorithm1 is divided into three separate subproblems, it can still achieve the global optimal solution of problem P1, and the result is given by the following theorem.Theorem 2.
The RA in Algorithm1 achieves the global optimum of problem P1.Proof.
To prove that the RA policy in Algorithm1 can achieve the global optimal solution of problem P1, we only need to prove that each step of Algorithm 1 maintains the global optimum. From Lemma 1, we have known that the CG-sorted SP scheme in step 1 of Algorithm 1 gives the globally optimal SP policy. The scheme is only related to channel gains, which does not require the knowledge of the optimal PS and PA. According to the derivation process in step 2 of Algorithm 1, the obtained PS is optimal under the given optimal SP, and it does not require the knowledge of optimal PA. Then in step 3 of Algorithm 1, the obtained PA is optimal under the given optimal SP and optimal PS. Since each step maintains the global optimum, Theorem 2 is proved.


## 3.3. Complexity Analysis

The complexity of step 1 of Algorithm1 depends on the adopted sorting method, which is ONlog⁡N if the quick-sort method is applied. Moreover, the complexity of step 2 of Algorithm 1 is O(N), and the complexity of step 3 of Algorithm 1 is also ON (the water-filling over the sorted γi,j) [25]. Thus, the total computational complexity can be expressed by ONlog⁡N. For comparison, the computational complexity of RA algorithm for IPS architecture in [15] is ONlog⁡N+MNN+2q, where M is the number of loops in the algorithm [15], so it can be found that the computational complexity of proposed DPS architecture’s RA algorithm is on the order of MN+2q/log⁡N less than the IPS architecture. So the proposed DPS architecture may be more proper for MFC-assisted networks, where the devices are of lower computation capacity.


## 4. Simulation Results

In this section, some simulation results are given to illustrate the performance of the presented DPS receiver architecture and RA algorithm. The noise powers are assumed follows:σr2att=σr2proc=-33 dBm and σd2=-30 dBm. The three network nodes (S, R, and D) are assumed to be placed on a straight line. The distance from S to D is reference distance and represented by d0, where d0 = 10 m. The location of R is expressed as dr/d0, where dr denotes the distance from S to R. hi and gj are, respectively, obtained from the distribution as(23)hi~CN0,1L1+drα,gj~CN0,1L1+d0-drα,where α is the path loss factor and set to be 3 and L is the number of taps and set to be 4.Firstly, we discuss the performance of our proposed Algorithm1. For comparisons, the three other methods are also simulated, i.e., (1) OPAwoSP method: Optimal PA without SP; (2) EPAwSP method: Equal PA with SP; (3) EPAwoSP method: Equal PA without SP. EH efficiencyη= 1. In Figure 2, we plot the achievable rates versus the total power PS. It is easily seen that our proposed Algorithm 1 is superior to the three other methods.Figure 2
Achievable information rate versusPS with N=64 and R located at the midpoint between S and D for DPS architecture.We also show the effect of EH efficiencyηon the achievable rate in Figure 3 and it can be found that whenη= 1, the achievable rate is maximum. In the following simulations, to discuss the SWIPT-enabled system’s performance limit, EH efficiency is always set to beη= 1.Figure 3
Achievable information rate versus EH efficiencyηwith N=64 and R located at the midpoint between S and D for DPS architecture.Secondly, to figure out the system performance of the DPS receiver architecture, we compare the achievable information rates of the proposed DPS architectures and IPS architecture in [15]. For IPS architecture, let ρ be the PS ratio used for ID, and the rest 1-ρ part is used for EH; ρ should satisfy(24)0≤ρ≤1.The energy obtained by R is ER=T/2η1-ρ∑i=1Nhi2Ps,i and the available power of R is PR=ER/T/2=η(1-ρ)∑i=1Nhi2Ps,i. So the available powerPr,j on subcarrier j at R satisfies (25)∑j=1NPr,j≤η1-ρ∑i=1Nhi2Ps,i.The achievable rate from S to D on each subcarrier pair (i,j) can be expressed as(26)Ri,jIPS=12min⁡log21+ρhi2Ps,iρσr,att2+σr,proc2,log21+gj2Pr,jσd2,and thus the achievable rate of the system is given by(27)RIPSP,S,ρ=∑i=1N∑j=1Nsi,jRi,jIPS.To maximize the achievable rate of the system, an optimization problem is formulated as(28)maxP,S,ρRIPSP,S,ρ=∑i=1N∑j=1Nsi,jRi,js.t.2,8,24,25.The solution of the problem is given in [15]. In addition, conventional non-SWIPT two-hop OFDM system is also compared in order to show the difference between SWIPT-enabled and non-SWIPT systems. For the non-SWIPT system, we use the optimal RA algorithm proposed in [8].In Figures4 and 5, the achievable rates of the DPS/IPS architectures and non-SWIPT system with respect to PS and dr/d0 are given, respectively. In these two figures, it can be found that the achievable rate of the non-SWIPT system is higher than SWIPT-enabled IPS/DPS architectures, and DPS architecture is worse than IPS architecture.Figure 4
Comparison of achievable information rates of DPS architecture, IPS architecture, and non-SWIPT system versusPS with N=4 and R located at the midpoint between S and D.Figure 5
Comparison of achievable information rates of DPS architecture, IPS architecture, and non-SWIPT system versus relay location withPS=30dBm and N=4.Moreover, from Figure5, one can find that, for the SWIPT-enabled system, when R is placed close to S or D, the system can obtain the better performance, and the proposed DPS architecture agrees with the existing IPS architecture. One can also find that, for conventional non-SWIPT system, the achievable information rate achieves maximum when R is placed at the midpoint on the line from S to D.Finally, we also compare the average running time of IPS and DPS architectures in Figure6. It shows that the running efficiency of the DPS architecture is far superior to the IPS architecture, which agrees with the analysis of computational complexity in Section 3.3, and thus, for computation-constrained MFC system, DPS architecture is a better option although there are some loss of the achievable rate compared with IPS architecture.Figure 6
Comparison of running time of IPS and DPS architectures versus the number of subcarriersN with R located at the midpoint from S to D and PS=10dBm.


## 5. Conclusion

This paper investigated SWIPT for MFC-assisted two-hop OFDM network and proposed DPS receiver architectures. To study the system achievable rate limit, an efficient RA algorithm was given. In simulations, it was found that the achievable rate of the DPS architecture is worse than the existing IPS architecture; however, the computation complexity of DPS architecture is much lower than IPS architecture. So DPS architecture may be a better option for computation-constrained MFC system.



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