# Energy Efficiency Maximized Resource Allocation for Opportunistic Relay-Aided OFDMA Downlink with Subcarrier Pairing

Authors: Tao Wang; Chao Ma; Yanzan Sun; Shunqing Zhang; Yating Wu
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

This paper studies the energy efficiency (EE) maximization for an orthogonal frequency division multiple access (OFDMA) downlink network aided by a relay station (RS) with subcarrier pairing. A highly flexible transmission protocol is considered, where each transmission is executed in two time slots. Every subcarrier in each slot can either be used in direct mode or be paired with a subcarrier in another slot to operate in relay mode. The resource allocation (RA) in such a network is highly complicated, because it has to determine the operation mode of subcarriers, the assignment of subcarriers to users, and the power allocation of the base station and RS. We first propose a mathematical description of the RA strategy. Then, a RA algorithm is derived to find the globally optimum RA to maximize the EE. Finally, we present extensive numerical results to show the impact of minimum required rate of the network, the user number, and the relay position on the maximum EE of the network.


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## Body

## 1. Introduction

With rapid growth of multimedia services, requirements for high-speed wireless communications are growing fast. To meet these requirements, telecom operators arrange a large number of base stations, which lead to a high amount of energy consumption [1]. To address this issue, many scholars have proposed energy-saving methods to minimize total energy consumption for a variety of wireless communication systems, such as Device-to-Device (D2D) communications, wireless sensor networks (WSNs), and cellular networks [2, 3]. Recently, energy efficiency- (EE-) based optimization design, which aims to maximize the EE defined as the number of transmitted bits per Joule total energy consumption, has attracted much interest from academia and industry [4–10].Orthogonal frequency division multiple access (OFDMA) has been widely recognized as one of the dominant wireless technologies for high-data-rate wireless multimedia services. One of the main reasons behind this fact is that performance of OFDMA systems can be significantly improved by proper resource allocation (RA) when transmitter channel state information (CSI) is available [11–16]. Lately, relay-aided cooperation schemes have been widely used in combination with OFDMA networks to improve spectral efficiency. Under the constraint of guaranteeing users’ communication rate, some works designed RA algorithms to minimize the total transmission power of networks [17–24]. In [17], to minimize the maximum value between transmission power of the BS and transmission power of all the RSs, Muller et al. designed a RA algorithm for decode-and-forward (DF) relay-aided OFDMA networks. In [21], knowing the relay selection, Huang et al. proposed an optimization algorithm of subcarriers and power allocation to minimize the total power of BS and all the RSs. Chen et al. designed a strategy of user assignment for subcarriers, RS’s choice, and modulation scheme to minimize the total transmission power of networks for amplify-and-forward (AF) relay-aided downlink OFDMA networks [23]. The above works ignore the influence of circuit power of the BS and RSs, so these algorithms cannot ensure high EE of networks.It is interesting to further study how to improve EE of relay-aided OFDMA networks. When users lie outside the BS’s radio coverage, EE maximized RA problems for OFDMA networks using subcarrier-pair-based DF protocols have been addressed in [25–29]. In these works, every subcarrier in the first time slot is paired with a subcarrier in the second time slot for the relay-aided transmission. In most cases, BS can also transmit messages to users directly; designing EE maximized RA algorithm for flexible transmission protocols is more meaningful. In [30–34], the authors adopted more flexible transmission protocols and proposed EE maximized RA algorithms for downlink OFDMA networks when the total transmission power is constrained to be smaller than a prescribed value. In this case, when the network reaches the maximum EE, the total communication rate might be too small to meet the needs of users.In this paper, we focus on the optimum energy-efficient RA for downlink OFDMA with a RS using subcarrier-pair-based DF relaying, when the sum rate is constrained above a prescribed value. An opportunistic relay-aided transmission protocol is considered. User message bits are transmitted during two consecutive equal-duration time slots. In the first slot, the BS broadcasts OFDM symbols to RSs and users. In the second slot, subcarriers in direct mode can transmit to users directly; other subcarriers can be paired with subcarriers in first slot to transmit messages with the help of the RS. To be more specific, our contributions are summarized as follows:(i)
An EE maximized RA problem is formulated, and a polynomial-complexity algorithm is designed to find the optimum RA to maximize the EE based on the Dinkelbach method as well as the dual method to solve a subproblem.(ii)
Extensive numerical results are shown to exhibit the impact of system parameters (including minimum communication rate required by the network, RS deployment, and user number) on the network EE.The rest of this paper is organized as follows. In the next section, the transmission protocol of the network is described. After that, the EE maximized RA algorithm is developed in Section3. Numerical experiments are shown in Section 4. Finally, Section 5 concludes the paper.Notations. Cx=log2(1+x).


## 2. Network Model and Transmission Protocol

### 2.1. General Introduction of the Network and Protocol

We consider an OFDMA downlink network as shown in Figure1. The network under consideration consists of a BS, a RS, and multiple users. Both the BS and the RS adopt OFDMA scheme using the same frequency band of bandwidth B Hz and with K subcarriers, which means that each OFDM symbol has a duration of K/B seconds.Figure 1
The relay-aided OFDMA downlink network under consideration.For illustration purpose, the channel coefficient and noise-power normalized channel gain at any subcarrierkk∈1,…,K from BS to RS, from RS to any user u, and from BS to any user u are defined in Table 1, where σ2 is assumed as the power of additive white Gaussian noise at each subcarrier.Table 1
Channel coefficient and gain for subcarrierk∈1,…,K.
Channel for subcarrierk Coefficient Gain BS to RS h s r k G s r k = h s r k 2 σ 2 BS to useru h s u k G s u k = h s u k 2 σ 2 RS to useru h r u k G r u k = h r u k 2 σ 2The transmission protocol under consideration is carried out as follows. Each transmission needs two time slots denoted by slot-1 and slot-2, respectively. Each subcarrier in every slot can either operate in direct mode (it is used by the BS for transmission to a user directly) or be paired with a subcarrier in another slot to operate in relay mode (the relay helps the transmission as explained in Section2.2). The protocol is illustrated by Figure 2. In the following subsections, the transmission procedures for the direct and relay mode are explained in detail.Figure 2
The transmission procedure over the subcarrier pair(k,l,u).


### 2.2. Transmission for a Subcarrier Pair in Relay Mode

Suppose that a subcarrierk in slot-1 is paired with a subcarrier l in slot-2 to operate in relay mode, and this relay-link is assigned to user u. Denote this subcarrier pair as (k,l,u), which is shown in Figure 3. Over this link, the transmission procedure is carried out as follows.Figure 3
The transmission procedure over the subcarrier pair(k,l,u).In slot-1, the BS broadcasts a symbolx to both the relay and the user with power P1. The received signal at the relay is expressed as(1)yr=P1hsrkx+nr,and that at the user is expressed as(2)yu,1=P1hsukx+nu,1.At the end of slot-1, the relay decodesyr and recovers x. To enable the relay to successfully decode the message bits, the maximum transmission rate is no greater than log2(1+P1Gsrk) bits/symbol.In slot-2, the relay simply emitsx at subcarrier l to the user with power P2. The received signal at the user is expressed as(3)yu,2=P2hrulx+nu,2.At the end of slot-2, the source combinesyu,1 and yu,2 with maximum-ratio combining (MRC) to maximize the received SNR. The final signal used for decoding at the user can be expressed as(4)z=αyu,1+βyu,2.It can be shown that whenα=P1hsuk∗ and β=P2hrul∗, the MRC is achieved [35]. The maximum received SNR is(5)SNRk,l,u=GsukP1+GrulP2.As a result, the maximum transmission rate over this relay link should be the minimum between the source-relay rate and the source-relay-user rate. It can be evaluated as(6)Cklu=minCSNRklu,CGsrkP1,in the unit of bits/symbol.Suppose thatP′ is the sum power of P1 and P2; the optimums P1 and P2 for maximizing the rate are the optimum solution for(7)maxP1,P2minP1Gsrk,P1Gsuk+P2Grul,s.t.P1+P2=P′,P1≥0,P2≥0.Using the same method as in [36], it can easily be shown that the optimums P1 and P2 are(8)P1=GrulΔu,k+GrulP′ifminGsrk,Grul>Gsuk,P′ifminGsrk,Grul≤Gsuk,P2=Gsrk-GsukΔu,k+GrulP′ifminGsrk,Grul>Gsuk,0ifminGsrk,Grul≤Gsuk,where Δu,k=Gsrk-Gsuk.The maximum rate associated with the above solution is equal to(9)
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### 2.3. Transmission for a Subcarrier in Direct Mode

Every subcarrier in either slot-1 or slot-2 can be assigned to operate in direct mode. In this mode, a direct link from the BS to a certain user is formed at this subcarrier. Suppose that a subcarrierk in either slot-1 or slot-2 is assigned to user u and operates in the direct mode, and BS uses power P for this subcarrier. Therefore, the average transmission rate over this direct link can be evaluated as(10)RP=B2KCGsukPbits/second.


## 2.1. General Introduction of the Network and Protocol

We consider an OFDMA downlink network as shown in Figure1. The network under consideration consists of a BS, a RS, and multiple users. Both the BS and the RS adopt OFDMA scheme using the same frequency band of bandwidth B Hz and with K subcarriers, which means that each OFDM symbol has a duration of K/B seconds.Figure 1
The relay-aided OFDMA downlink network under consideration.For illustration purpose, the channel coefficient and noise-power normalized channel gain at any subcarrierkk∈1,…,K from BS to RS, from RS to any user u, and from BS to any user u are defined in Table 1, where σ2 is assumed as the power of additive white Gaussian noise at each subcarrier.Table 1
Channel coefficient and gain for subcarrierk∈1,…,K.
Channel for subcarrierk Coefficient Gain BS to RS h s r k G s r k = h s r k 2 σ 2 BS to useru h s u k G s u k = h s u k 2 σ 2 RS to useru h r u k G r u k = h r u k 2 σ 2The transmission protocol under consideration is carried out as follows. Each transmission needs two time slots denoted by slot-1 and slot-2, respectively. Each subcarrier in every slot can either operate in direct mode (it is used by the BS for transmission to a user directly) or be paired with a subcarrier in another slot to operate in relay mode (the relay helps the transmission as explained in Section2.2). The protocol is illustrated by Figure 2. In the following subsections, the transmission procedures for the direct and relay mode are explained in detail.Figure 2
The transmission procedure over the subcarrier pair(k,l,u).


## 2.2. Transmission for a Subcarrier Pair in Relay Mode

Suppose that a subcarrierk in slot-1 is paired with a subcarrier l in slot-2 to operate in relay mode, and this relay-link is assigned to user u. Denote this subcarrier pair as (k,l,u), which is shown in Figure 3. Over this link, the transmission procedure is carried out as follows.Figure 3
The transmission procedure over the subcarrier pair(k,l,u).In slot-1, the BS broadcasts a symbolx to both the relay and the user with power P1. The received signal at the relay is expressed as(1)yr=P1hsrkx+nr,and that at the user is expressed as(2)yu,1=P1hsukx+nu,1.At the end of slot-1, the relay decodesyr and recovers x. To enable the relay to successfully decode the message bits, the maximum transmission rate is no greater than log2(1+P1Gsrk) bits/symbol.In slot-2, the relay simply emitsx at subcarrier l to the user with power P2. The received signal at the user is expressed as(3)yu,2=P2hrulx+nu,2.At the end of slot-2, the source combinesyu,1 and yu,2 with maximum-ratio combining (MRC) to maximize the received SNR. The final signal used for decoding at the user can be expressed as(4)z=αyu,1+βyu,2.It can be shown that whenα=P1hsuk∗ and β=P2hrul∗, the MRC is achieved [35]. The maximum received SNR is(5)SNRk,l,u=GsukP1+GrulP2.As a result, the maximum transmission rate over this relay link should be the minimum between the source-relay rate and the source-relay-user rate. It can be evaluated as(6)Cklu=minCSNRklu,CGsrkP1,in the unit of bits/symbol.Suppose thatP′ is the sum power of P1 and P2; the optimums P1 and P2 for maximizing the rate are the optimum solution for(7)maxP1,P2minP1Gsrk,P1Gsuk+P2Grul,s.t.P1+P2=P′,P1≥0,P2≥0.Using the same method as in [36], it can easily be shown that the optimums P1 and P2 are(8)P1=GrulΔu,k+GrulP′ifminGsrk,Grul>Gsuk,P′ifminGsrk,Grul≤Gsuk,P2=Gsrk-GsukΔu,k+GrulP′ifminGsrk,Grul>Gsuk,0ifminGsrk,Grul≤Gsuk,where Δu,k=Gsrk-Gsuk.The maximum rate associated with the above solution is equal to(9)
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## 2.3. Transmission for a Subcarrier in Direct Mode

Every subcarrier in either slot-1 or slot-2 can be assigned to operate in direct mode. In this mode, a direct link from the BS to a certain user is formed at this subcarrier. Suppose that a subcarrierk in either slot-1 or slot-2 is assigned to user u and operates in the direct mode, and BS uses power P for this subcarrier. Therefore, the average transmission rate over this direct link can be evaluated as(10)RP=B2KCGsukPbits/second.


## 3. Energy-Efficient RA Algorithm Design

Before data transmission, we assume that the BS controller knows all CSI, that is,Gsrk,Gsuk,Gruk∣∀k,∀u. Based on the a priori knowledge, the BS runs an algorithm to find the optimum RA strategy to maximize the network EE.To be more specific, the RA algorithm for the transmission protocol needs to be optimized:(i)
Subcarrier operation mode: how to decide whether each subcarrier should operate in direct or relay mode(ii)
Subcarrier pairing for relay mode: how to pair subcarriers in relay mode(iii)
Subcarrier assignment to users: how to allocate subcarriers to users(iv)
Power allocation: how to allocate BS’s and RS’s transmission power for each subcarrierTo design the RA algorithm, we proceed as follows. First, a mathematical description of the RA strategy and network EE is proposed in Section3.1 Then, a Dinkelbach-method-based RA algorithm, namely, Algorithm 1, is designed in Section 3.2. Moreover, algorithms called by Algorithm 1 are designed in Sections 3.3 and 3.4.Algorithm 1:The algorithm to solve (P1).
( 1 )  evaluate Gklu, ∀k,l,u; ( 2 )  θmin=0; θmax is set by a large value; θ=θmax; δ=10-3; ( 3 )  while  Fθ≤δ  do ( 4 )  solve (P2) for  S(θ)   using Algorithm 2; ( 5 )  update θ by minθmax,maxRSθ/PSθ,θmin; ( 6 )  end while ( 7 )  output θ as η⋆ and S(θ) as the optimum RA strategy.Algorithm 2:The algorithm to solve (P2).
( 1 )  μ=0; ( 2 )  solve (P3) for  Sθ(0) using Algorithm 3; ( 3 )  if  Sθ(0) is feasible for (P2)   then ( 4 )  output  Sθ(0) as  S(θ) ( 5 )  else ( 6 )  μmin=0; μmax is set as a sufficiently large value; ( 7 )  while  μmax-μmin>δ  do ( 8 )  μ=μmax+μmin2; ( 9 )  solve (P3) for  Sθ(μ) using Algorithm 3; ( 10 )  if  γθ(μ)>0  then ( 11 )  μmax=μ; ( 12 )  else if  γθ(μ)<0  then ( 13 )  μmin=μ; ( 14 )  else if  γθ(μ)=0then ( 15 )  break; ( 16 )  end if ( 17 )  end while ( 18 )  end if ( 19 )  Output  Sθ(μ) as  S(θ).Algorithm 3:The algorithm to find Sθ(μ).
( 1 ) compute pklu,pklab1,pklab2. ∀k,l,a,b,u with (25); ( 2 ) compute Aklu,Bklab,∀k,l,u,a,b; and Ckl,∀k,l; ( 3 ) solve (P5) for its optimum solution tkl⋆∣∀k,l with the Hungarian algorithm; ( 4 ) construct the optimum I for (P4) by assigning for every combination of k and l, all entries in tklu,tklab∣∀k,l to zero, except for the one with the metric equal to Ckl to tkl⋆. ( 5 )  SI=I,PI is output as Sθ(μ).


### 3.1. Description of the RA Strategy and Network EE

We first define the following variables to describe the RA strategy:(i)
tklu∈0,1,∀k,l,u: indicating subcarrier k in slot-1 is paired with subcarrier l in slot-2 when tklu=1.(ii)
pklu≥0,∀k,l,u: indicating the total transmission power (i.e., P1+P2 as mentioned in Section 2.2) for the subcarrier pair (k,l,u).(iii)
tklab∈0,1,∀k,l,a,b: indicating that subcarrier k in slot-1 and subcarrier l in slot-2 are, respectively, allocated to user a and user b when tklab=1.(iv)
pklab1≥0 and pklab2≥0: respectively indicating the BS’s transmission power for subcarrier k in slot-1 and l in slot-2 for the direct mode.Let us define a RA strategy asS=I,P, where I collects all indicator variables and P collects all power variables. A feasible S must satisfy(11)tklu∈0,1,tklab∈0,1,∀k,l,u,a,b,(12)∑l∑utklu+∑abtklab≤1,∀k,(13)∑k∑utklu+∑abtklab≤1,∀l,(14)pklu≥0,pklab1≥0,pklab2≥0,∀k,l,u,a,b,where constraints (12) and (13) guarantee that each subcarrier in either slot-1 or slot-2 must operate in a single mode and be assigned to a single user.For givenS, the network EE is formulated as(15)ηS=RSPSbits/Joule,where R(S) represents the sum rate for the network:(16)RS=B2K∑klutkluCpkluGklu+∑klabtklabCpklab1Gsuk+Cpklab2Gsul,and P(S) is the sum power consumption for the network:(17)PS=1+α2∑klutklupklu+∑klabtklabpklab1+pklab2+Pcir,with α being the loss factor of the power amplifiers (PA) used by the BS and RS and Pcir representing the total power consumption by the BS’s and RS’s circuit devices.


### 3.2. Dinkelbach-Method-Based RA Algorithm Design

We consider the EE maximization problem when the sum rate must be greater than a prescribed valueRreq. This problem can be expressed as (P1):(18)maxSηSs.t.11,12,13,14RS≥Rreq.It can be seen that solving problem (P1) is highly challenging due to the following reasons:(i)
η(S) has a fractional structure, which is highly nonlinear.(ii)
(P1) is a mixed-integer problem containing both binary and continuous variables.To solve (P1) for the optimum RA (denoted byS⋆) and the maximum EE (denoted by η⋆), we make use of the Dinkelbach method. To be more specific, we define a parameter θ and a function F(θ) as the optimum objective value for the following problem (P2):(19)Fθ=maxSRS-θPSs.t.11,12,13,14,18,whose optimum solution is S(θ).Based on Dinkelbach method,η⋆ must satisfy F(η⋆)=0, and S(η⋆) is the optimum solution for problem (P1). Moreover, η⋆ can be found by iterative procedures as elaborated in [6].Motivated by the above principle, the RA algorithm to solve (P1) for the EE maximized RA strategy is summarized in Algorithm1. The iterative update of θ has a superlinear convergence rate [37]. We will elaborate on the design of Algorithm 2 to solve (P2) in Section 3.3.


### 3.3. Design of Algorithm2 to Solve (P2)

Using the same arguments as those in [38], we can show that (P2)’s duality gap is zero; therefore the dual method can be used to solve (P2). To this end, define(i)
μ as Lagrange multiplier for the constraint (18);(ii)
Lagrangian function(20)Lθμ,S=RS-θPS+μRS-Rreq;(iii)
Lagrange relaxation problem for (P2) as (P3):(21)maxSLθμ,Ss.t.11,12,13,14For (P3), we define its optimum solution asSθ(μ) and its dual function as dθ(μ)=Lθ(μ,Sθ(μ)). It can be shown that the dual function is convex of μ≥0, and(22)γθμ=RSθμ-Rreqis a subgradient of dθ(μ) satisfying(23)∀μ′,dθμ′≥dθμ+μ′-μγθμ.The key for the dual method is to find the dual optimum:(24)μθ⋆=arg⁡minμ≥0⁡dθμ,and then Sθ(μθ⋆) is the optimum solution for (P2).According to the dual method,μθ⋆ is the μ satisfying the following two conditions: (1)  μγθ(μ)=0 (i.e., the complementary slackness condition) and (2)  γθ(μ)≥0 (i.e., Sθ(μ) is feasible for (P2)). To find μθ⋆, we use the following method:(i)
First, we computeSθ(0). If γθ(0)≥0, μ=0 and Sθ(0) satisfy the above conditions. Therefore, Sθ(0) is the optimum for (P2).(ii)
Otherwise,γθ(0)<0 and μθ⋆>0 must hold. Note that γθ(μ) is increasing of μ. Once μ>0 satisfying γθ(μ)=0 is found, μ and Sθ(μ) satisfy the above two conditions and hence can be taken as μθ⋆ and S(θ), respectively. We will find μ>0 satisfying γθ(μ)=0 with the bisection method.To complete this subsection, the above procedures to solve (P2) are summarized in Algorithm2 as follows. Algorithm 2 has a polynomial complexity with respect to K. We will elaborate on the design of Algorithm 3 to solve (P3) in Section 3.4.


### 3.4. Design of Algorithm3 to Solve (P3)

We show how to findSθ(μ) as follows:(i)
First, the optimumP for (P3) with fixed I is found and denoted by PI.(ii)
Second, defineSI={I,PI}. Then we find the optimum I to maximize Lθ(μ,SI) subject to the constraints on I in (P3).(iii)
Finally,SI corresponding to this optimum I can be taken as Sθ(μ).As for the first step, the elements in the optimumPI can be computed according to KKT conditions as follows [39]:(25)pklu=Λθ,μ,Gklu,∀k,l,upklab1=Λθ,μ,Gsuk,∀k,l,a,bpklab2=Λθ,μ,Gsul,∀k,l,a,b,where(26)Λθ,μ,G=μ+1B·log2e1+αθ·K-1G+As for the second step, it can readily be shown that(27)Lθμ,SI=-μRreq+∑klutkluAklu+∑klabtklabBklab,where(28)Aklu=μ+1B2KCGkluΛθ,μ,Gklu-θ1+α2Λθ,μ,GkluBklab=μ+1B2KCGsukΛθ,μ,Gsuk-θ1+α2Λθ,μ,Gsuk+μ+1B2KCGsulΛθ,μ,Gsul-θ1+α2Λθ,μ,GsulFinally, we find the optimumI for maximizing Lθ(μ,SI). This problem is equivalent to solving (P4):(29)maxI,tkl∣∀k,l∑kl∑uabtkluAklu+tklabBklabs.t.tklu≥0,tklab≥0,∀k,l,u,a,b,(30)∑ltkl=1,∀k,(31)∑ktkl=1,∀l,(32)tkl=∑utklu+∑abtklab,∀k,lNote that(33)∑uabtkluAklu+tklabBklab≤tklCklholds, where Ckl=maxmaxuAklu,maxa,bBklab. Call Aklu the metric for tklu and Bklab the metric for tklab; the inequality is tightened when all entries of tklu,tklab∣∀u,a,b are assigned to zero, except that the one with the metric equal to Ckl is assigned to tkl.Therefore, after problem (P5)(34)maxtkl∣∀k,l∑kltklCkls.t.30,31,tkl∈0,1,∀k,lis solved for its optimum solution tkl⋆∣∀k,l, an optimum (32) can be constructed by assigning for every combination of k and l all entries in tklu,tklab∣∀k,l to zero, except for the one with the metric equal to Ckl to tkl⋆.Most interestingly, (P5) is a standard assignment problem; hence every entry intkl⋆∣∀k,l is either 0 or 1 and tkl⋆∣∀k,l can be found efficiently by the Hungarian algorithm [40]. After knowing Hungarian algorithm, the optimum I can be constructed according to the way mentioned earlier.Motivated by the above principle, the method to solve (P2) is summarized in Algorithm3 as follows. The complexity of computing Aklu and Bklab is O(K2(U+U2)). We use the Hungarian algorithm to solve (P5); the complexity is O(K3). As a result, the complexity of Algorithm 3 is O(K2(U+U2+K)).


## 3.1. Description of the RA Strategy and Network EE

We first define the following variables to describe the RA strategy:(i)
tklu∈0,1,∀k,l,u: indicating subcarrier k in slot-1 is paired with subcarrier l in slot-2 when tklu=1.(ii)
pklu≥0,∀k,l,u: indicating the total transmission power (i.e., P1+P2 as mentioned in Section 2.2) for the subcarrier pair (k,l,u).(iii)
tklab∈0,1,∀k,l,a,b: indicating that subcarrier k in slot-1 and subcarrier l in slot-2 are, respectively, allocated to user a and user b when tklab=1.(iv)
pklab1≥0 and pklab2≥0: respectively indicating the BS’s transmission power for subcarrier k in slot-1 and l in slot-2 for the direct mode.Let us define a RA strategy asS=I,P, where I collects all indicator variables and P collects all power variables. A feasible S must satisfy(11)tklu∈0,1,tklab∈0,1,∀k,l,u,a,b,(12)∑l∑utklu+∑abtklab≤1,∀k,(13)∑k∑utklu+∑abtklab≤1,∀l,(14)pklu≥0,pklab1≥0,pklab2≥0,∀k,l,u,a,b,where constraints (12) and (13) guarantee that each subcarrier in either slot-1 or slot-2 must operate in a single mode and be assigned to a single user.For givenS, the network EE is formulated as(15)ηS=RSPSbits/Joule,where R(S) represents the sum rate for the network:(16)RS=B2K∑klutkluCpkluGklu+∑klabtklabCpklab1Gsuk+Cpklab2Gsul,and P(S) is the sum power consumption for the network:(17)PS=1+α2∑klutklupklu+∑klabtklabpklab1+pklab2+Pcir,with α being the loss factor of the power amplifiers (PA) used by the BS and RS and Pcir representing the total power consumption by the BS’s and RS’s circuit devices.


## 3.2. Dinkelbach-Method-Based RA Algorithm Design

We consider the EE maximization problem when the sum rate must be greater than a prescribed valueRreq. This problem can be expressed as (P1):(18)maxSηSs.t.11,12,13,14RS≥Rreq.It can be seen that solving problem (P1) is highly challenging due to the following reasons:(i)
η(S) has a fractional structure, which is highly nonlinear.(ii)
(P1) is a mixed-integer problem containing both binary and continuous variables.To solve (P1) for the optimum RA (denoted byS⋆) and the maximum EE (denoted by η⋆), we make use of the Dinkelbach method. To be more specific, we define a parameter θ and a function F(θ) as the optimum objective value for the following problem (P2):(19)Fθ=maxSRS-θPSs.t.11,12,13,14,18,whose optimum solution is S(θ).Based on Dinkelbach method,η⋆ must satisfy F(η⋆)=0, and S(η⋆) is the optimum solution for problem (P1). Moreover, η⋆ can be found by iterative procedures as elaborated in [6].Motivated by the above principle, the RA algorithm to solve (P1) for the EE maximized RA strategy is summarized in Algorithm1. The iterative update of θ has a superlinear convergence rate [37]. We will elaborate on the design of Algorithm 2 to solve (P2) in Section 3.3.


## 3.3. Design of Algorithm2 to Solve (P2)

Using the same arguments as those in [38], we can show that (P2)’s duality gap is zero; therefore the dual method can be used to solve (P2). To this end, define(i)
μ as Lagrange multiplier for the constraint (18);(ii)
Lagrangian function(20)Lθμ,S=RS-θPS+μRS-Rreq;(iii)
Lagrange relaxation problem for (P2) as (P3):(21)maxSLθμ,Ss.t.11,12,13,14For (P3), we define its optimum solution asSθ(μ) and its dual function as dθ(μ)=Lθ(μ,Sθ(μ)). It can be shown that the dual function is convex of μ≥0, and(22)γθμ=RSθμ-Rreqis a subgradient of dθ(μ) satisfying(23)∀μ′,dθμ′≥dθμ+μ′-μγθμ.The key for the dual method is to find the dual optimum:(24)μθ⋆=arg⁡minμ≥0⁡dθμ,and then Sθ(μθ⋆) is the optimum solution for (P2).According to the dual method,μθ⋆ is the μ satisfying the following two conditions: (1)  μγθ(μ)=0 (i.e., the complementary slackness condition) and (2)  γθ(μ)≥0 (i.e., Sθ(μ) is feasible for (P2)). To find μθ⋆, we use the following method:(i)
First, we computeSθ(0). If γθ(0)≥0, μ=0 and Sθ(0) satisfy the above conditions. Therefore, Sθ(0) is the optimum for (P2).(ii)
Otherwise,γθ(0)<0 and μθ⋆>0 must hold. Note that γθ(μ) is increasing of μ. Once μ>0 satisfying γθ(μ)=0 is found, μ and Sθ(μ) satisfy the above two conditions and hence can be taken as μθ⋆ and S(θ), respectively. We will find μ>0 satisfying γθ(μ)=0 with the bisection method.To complete this subsection, the above procedures to solve (P2) are summarized in Algorithm2 as follows. Algorithm 2 has a polynomial complexity with respect to K. We will elaborate on the design of Algorithm 3 to solve (P3) in Section 3.4.


## 3.4. Design of Algorithm3 to Solve (P3)

We show how to findSθ(μ) as follows:(i)
First, the optimumP for (P3) with fixed I is found and denoted by PI.(ii)
Second, defineSI={I,PI}. Then we find the optimum I to maximize Lθ(μ,SI) subject to the constraints on I in (P3).(iii)
Finally,SI corresponding to this optimum I can be taken as Sθ(μ).As for the first step, the elements in the optimumPI can be computed according to KKT conditions as follows [39]:(25)pklu=Λθ,μ,Gklu,∀k,l,upklab1=Λθ,μ,Gsuk,∀k,l,a,bpklab2=Λθ,μ,Gsul,∀k,l,a,b,where(26)Λθ,μ,G=μ+1B·log2e1+αθ·K-1G+As for the second step, it can readily be shown that(27)Lθμ,SI=-μRreq+∑klutkluAklu+∑klabtklabBklab,where(28)Aklu=μ+1B2KCGkluΛθ,μ,Gklu-θ1+α2Λθ,μ,GkluBklab=μ+1B2KCGsukΛθ,μ,Gsuk-θ1+α2Λθ,μ,Gsuk+μ+1B2KCGsulΛθ,μ,Gsul-θ1+α2Λθ,μ,GsulFinally, we find the optimumI for maximizing Lθ(μ,SI). This problem is equivalent to solving (P4):(29)maxI,tkl∣∀k,l∑kl∑uabtkluAklu+tklabBklabs.t.tklu≥0,tklab≥0,∀k,l,u,a,b,(30)∑ltkl=1,∀k,(31)∑ktkl=1,∀l,(32)tkl=∑utklu+∑abtklab,∀k,lNote that(33)∑uabtkluAklu+tklabBklab≤tklCklholds, where Ckl=maxmaxuAklu,maxa,bBklab. Call Aklu the metric for tklu and Bklab the metric for tklab; the inequality is tightened when all entries of tklu,tklab∣∀u,a,b are assigned to zero, except that the one with the metric equal to Ckl is assigned to tkl.Therefore, after problem (P5)(34)maxtkl∣∀k,l∑kltklCkls.t.30,31,tkl∈0,1,∀k,lis solved for its optimum solution tkl⋆∣∀k,l, an optimum (32) can be constructed by assigning for every combination of k and l all entries in tklu,tklab∣∀k,l to zero, except for the one with the metric equal to Ckl to tkl⋆.Most interestingly, (P5) is a standard assignment problem; hence every entry intkl⋆∣∀k,l is either 0 or 1 and tkl⋆∣∀k,l can be found efficiently by the Hungarian algorithm [40]. After knowing Hungarian algorithm, the optimum I can be constructed according to the way mentioned earlier.Motivated by the above principle, the method to solve (P2) is summarized in Algorithm3 as follows. The complexity of computing Aklu and Bklab is O(K2(U+U2)). We use the Hungarian algorithm to solve (P5); the complexity is O(K3). As a result, the complexity of Algorithm 3 is O(K2(U+U2+K)).


## 4. Numerical Experiments and Discussions

We will first introduce system setup for numerical experiments, as well as two benchmark protocols for comparison purpose. Then, results and discussions are presented to show the impact of different parameters on the network EE.


### 4.1. System Setup and Benchmark Protocols

In numerical experiments, we consider the downlink OFDMA network with a RS exhibit in Figure4. U represents the number of users in service and they are randomly distributed in a circular region of radius R km. The RS is located between the BS and the user-region (UR) center, and the BS-RS distance is d km. The distance between the BS and the user-region center is D km. The bandwidth of the system is B Hz, and the OFDMA uses K subcarriers. The network parameters are listed in Table 2.Table 2
Network Parameters.
Meanings Parameters Values Bandwidth of the network B 20 MHz Subcarriers’ number K 32 Loss factor of the PA α 0.3 Circuit power of BS and RS P c i r 500 w Radius of UR R 0.2 km Distance between BS and the UR center D 1.5 kmFigure 4
The downlink OFDMA system considered in numerical experiments.The channels are independent of each other and are generated in the same way as in [7]. For every user u, the impulse response of the source-to-u channel is modeled as a tapped delay line with L=6 taps, which are independently generated from circularly symmetric complex Gaussian distributions with zero mean and variance equal to 1/Ldsu/dref-4, where dref=1km and dsu represents the source-to-u distance. The source-to-relay and relay-to-u channels are generated in the same way, with each tap having variance as 1/Ldsr/dref-4 and 1/Ldru/dref-4, respectively, where dru represents the relay-to-u distance. The CSI hsrk∣∀k,  hsuk∣∀k,u, and hruk∣∀k,u are computed by making K-point FFT over the impulse response of the associated channels.In order to illustrate the benefit of optimized subcarrier pairing and opportunistic relaying, we also consider two other benchmark protocols, namely, BP-1 and BP-2. BP-1 is similar to the considered protocol, except that subcarrierk in slot-1 can only be paired with subcarrier k in slot-2 in relay mode. BP-2 is a simplified version of the considered protocol, and the simplification lies in the fact that each subcarrier in every slot should be allocated to users in direct mode. The RA algorithms for both BP-1 and BP-2 can be derived in the same way as that for the considered protocol, and therefore the derivation is omitted for the sake of clarity.The complexity of the three algorithms is shown in Table3. It can be seen that the proposed algorithm has the highest complexity, while the BP-2 algorithm has the lowest complexity.Table 3
Complexity comparison.
Algorithm Complexity Proposed algorithm O K 2 U + U 2 + K BP-1 O K U + U 2 BP-2 O ( K U )


### 4.2. Impact ofRreq on the Optimum EE and Corresponding Sum Rate

To show the influence ofRreq on the EE, we choose U=10 and d=0.6km and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when Rreq increases from 0 to 40 Mbits/s. The results are shown in Figure 5.Figure 5
The average EE as the minimum rate changes.
(a) (b) (c)Compared with BP-1 and BP-2, we can see that the proposed protocol and algorithm always correspond to a higher average EE as shown in Figure5(a). Since BP-2 does not utilize opportunistic relaying as the proposed protocol and BP-1, it is reasonable that BP-2 corresponds to much lower average EE. The proposed protocol can achieve higher EE than BP-1, because a subcarrier in every slot can be paired with the other slot’s subcarrier freely.Figure5(a) also shows that the average EE of these methods decreases with the increase of Rreq. This is because the feasible set of the problem shrinks with the increase of Rreq. From Figures 5(a), 5(b), and 5(c), when Rreq<20 Mbits/s and the average EE reaches the optimum value, the average communication rate is larger than Rreq, the average total power remains stable, and the average EE of the network maintains high value. When Rreq≥20 Mbits/s and the average EE reaches the optimum value, the average communication rate is equal to Rreq, the average total power increases rapidly, and the average EE of the network decreases. The above phenomenon indicates that the restricted condition R(S)≥Rreq influences the choice of the optimum solution.


### 4.3. Impact of Relay Position on the Optimum EE and Corresponding Sum Rate

To show the impact of relay position on the EE, we chooseU=10 and Rreq=20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when d increases from 0.2 to 1.2km. The results are shown in Figure 6.Figure 6
The average EE as the relay position changes.It is shown that the proposed protocol leads to a higher average EE than the BP-1 and BP-2 for every relay position. Moreover, the average EE improves as the RS moves towards the middle region between the BS and the users. This can be interpreted as follows. In theory, the optimum EE enhances if∀k,l,u,Gklu is more likely to take a high value. Note that Gklu takes a high value only if both Gsrk and Grul are much higher than Gsuk. When RS lies in the middle between the BS and the users’ region, it is more likely to have Gsrk and Grul, both much greater than Gsuk, and thus Gklu is more likely to take a high value. Moreover, BP-2 is a direct transmission protocol; the RS does not help to transmit signals. It is reasonable that the average EE remains steady when the relay position changes.


### 4.4. Impact of User Number on the Optimum EE and Corresponding Sum Rate

To show the impact of user number on the EE, we choosed=0.6km and Rreq=20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when U increases from 5 to 30. The results are shown in Figure 7.Figure 7
The average EE as the user number changes.From Figure7, we see that the average EEs of the three methods increase with the increase of user number. This is because when the number of users in the network increases, the subcarrier assignment has more flexibility. The numbers of Aklu and Bklab increase with the increase of user number, which can improve the probability of Ckl taking a larger value. In this way, the average EE of the network will be improved.


## 4.1. System Setup and Benchmark Protocols

In numerical experiments, we consider the downlink OFDMA network with a RS exhibit in Figure4. U represents the number of users in service and they are randomly distributed in a circular region of radius R km. The RS is located between the BS and the user-region (UR) center, and the BS-RS distance is d km. The distance between the BS and the user-region center is D km. The bandwidth of the system is B Hz, and the OFDMA uses K subcarriers. The network parameters are listed in Table 2.Table 2
Network Parameters.
Meanings Parameters Values Bandwidth of the network B 20 MHz Subcarriers’ number K 32 Loss factor of the PA α 0.3 Circuit power of BS and RS P c i r 500 w Radius of UR R 0.2 km Distance between BS and the UR center D 1.5 kmFigure 4
The downlink OFDMA system considered in numerical experiments.The channels are independent of each other and are generated in the same way as in [7]. For every user u, the impulse response of the source-to-u channel is modeled as a tapped delay line with L=6 taps, which are independently generated from circularly symmetric complex Gaussian distributions with zero mean and variance equal to 1/Ldsu/dref-4, where dref=1km and dsu represents the source-to-u distance. The source-to-relay and relay-to-u channels are generated in the same way, with each tap having variance as 1/Ldsr/dref-4 and 1/Ldru/dref-4, respectively, where dru represents the relay-to-u distance. The CSI hsrk∣∀k,  hsuk∣∀k,u, and hruk∣∀k,u are computed by making K-point FFT over the impulse response of the associated channels.In order to illustrate the benefit of optimized subcarrier pairing and opportunistic relaying, we also consider two other benchmark protocols, namely, BP-1 and BP-2. BP-1 is similar to the considered protocol, except that subcarrierk in slot-1 can only be paired with subcarrier k in slot-2 in relay mode. BP-2 is a simplified version of the considered protocol, and the simplification lies in the fact that each subcarrier in every slot should be allocated to users in direct mode. The RA algorithms for both BP-1 and BP-2 can be derived in the same way as that for the considered protocol, and therefore the derivation is omitted for the sake of clarity.The complexity of the three algorithms is shown in Table3. It can be seen that the proposed algorithm has the highest complexity, while the BP-2 algorithm has the lowest complexity.Table 3
Complexity comparison.
Algorithm Complexity Proposed algorithm O K 2 U + U 2 + K BP-1 O K U + U 2 BP-2 O ( K U )


## 4.2. Impact ofRreq on the Optimum EE and Corresponding Sum Rate

To show the influence ofRreq on the EE, we choose U=10 and d=0.6km and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when Rreq increases from 0 to 40 Mbits/s. The results are shown in Figure 5.Figure 5
The average EE as the minimum rate changes.
(a) (b) (c)Compared with BP-1 and BP-2, we can see that the proposed protocol and algorithm always correspond to a higher average EE as shown in Figure5(a). Since BP-2 does not utilize opportunistic relaying as the proposed protocol and BP-1, it is reasonable that BP-2 corresponds to much lower average EE. The proposed protocol can achieve higher EE than BP-1, because a subcarrier in every slot can be paired with the other slot’s subcarrier freely.Figure5(a) also shows that the average EE of these methods decreases with the increase of Rreq. This is because the feasible set of the problem shrinks with the increase of Rreq. From Figures 5(a), 5(b), and 5(c), when Rreq<20 Mbits/s and the average EE reaches the optimum value, the average communication rate is larger than Rreq, the average total power remains stable, and the average EE of the network maintains high value. When Rreq≥20 Mbits/s and the average EE reaches the optimum value, the average communication rate is equal to Rreq, the average total power increases rapidly, and the average EE of the network decreases. The above phenomenon indicates that the restricted condition R(S)≥Rreq influences the choice of the optimum solution.


## 4.3. Impact of Relay Position on the Optimum EE and Corresponding Sum Rate

To show the impact of relay position on the EE, we chooseU=10 and Rreq=20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when d increases from 0.2 to 1.2km. The results are shown in Figure 6.Figure 6
The average EE as the relay position changes.It is shown that the proposed protocol leads to a higher average EE than the BP-1 and BP-2 for every relay position. Moreover, the average EE improves as the RS moves towards the middle region between the BS and the users. This can be interpreted as follows. In theory, the optimum EE enhances if∀k,l,u,Gklu is more likely to take a high value. Note that Gklu takes a high value only if both Gsrk and Grul are much higher than Gsuk. When RS lies in the middle between the BS and the users’ region, it is more likely to have Gsrk and Grul, both much greater than Gsuk, and thus Gklu is more likely to take a high value. Moreover, BP-2 is a direct transmission protocol; the RS does not help to transmit signals. It is reasonable that the average EE remains steady when the relay position changes.


## 4.4. Impact of User Number on the Optimum EE and Corresponding Sum Rate

To show the impact of user number on the EE, we choosed=0.6km and Rreq=20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when U increases from 5 to 30. The results are shown in Figure 7.Figure 7
The average EE as the user number changes.From Figure7, we see that the average EEs of the three methods increase with the increase of user number. This is because when the number of users in the network increases, the subcarrier assignment has more flexibility. The numbers of Aklu and Bklab increase with the increase of user number, which can improve the probability of Ckl taking a larger value. In this way, the average EE of the network will be improved.


## 5. Conclusions

We have addressed an EE maximized RA problem for cooperative OFDM transmission using the improved DF protocol with optimized subcarrier pairing when the network’s communication rate is larger than a required value. The subcarrier-pair-based opportunistic DF relay-aided protocol has two operation modes: direct mode and relay mode. This scheme improves the flexibility of the communication network. Subcarriers can choose the mode that can improve the network’s EE to send messages. Based on the above protocol, we formulate the optimization problem of maximizing the network’s EE.The problem is polynomial complexity, so we solve it with the following three steps. In the first step, we eliminate the fractional structure with the help of Dinkelbach method and transfer problem (P1) into problem (P2). In the second step, we get the Lagrangian function by using the dual method. In the third step, we use KKT conditions and Hungarian algorithm to solve the Lagrangian function. Then we can get the RA algorithm of maximizing the network’s EE. Numerical experiments show that the proposed RA algorithm can improve the EE of the downlink OFDMA networks. And the experiments also illustrated the impact of minimum required communication rate, relay position, and the user number. Theoretical analysis has been presented to interpret what is observed in numerical experiments.



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*Source: 9046847-2018-05-29.xml*