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

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 optimumRA tomaximize 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.


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][5][6][7][8][9][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][12][13][14][15][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][18][19][20][21][22][23][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][26][27][28][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][31][32][33][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-pairbased 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 Section 3. Numerical experiments are shown in Section 4. Finally, Section 5 concludes the paper.

General Introduction of the Network and Protocol.
We consider an OFDMA downlink network as shown in Figure 1.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  Hz and with  subcarriers, which means that each OFDM symbol has a duration of / seconds.
For illustration purpose, the channel coefficient and noise-power normalized channel gain at any subcarrier  ( ∈ {1, . . ., }) from BS to RS, from RS to any user ,   and from BS to any user  are defined in Table 1, where  2 is assumed as the power of additive white Gaussian noise at each subcarrier.The 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 Section 2.2).The protocol is illustrated by Figure 2. In the following subsections, the transmission procedures for the direct and relay mode are explained in detail.

Transmission for a Subcarrier Pair in Relay
Mode.Suppose that a subcarrier  in slot-1 is paired with a subcarrier  in slot-2 to operate in relay mode, and this relay-link is assigned to user .Denote this subcarrier pair as (, , ), which is shown in Figure 3.Over this link, the transmission procedure is carried out as follows.
In slot-1, the BS broadcasts a symbol  to both the relay and the user with power  1 .The received signal at the relay is expressed as and that at the user is expressed as At the end of slot-1, the relay decodes   and recovers .To enable the relay to successfully decode the message bits, the maximum transmission rate is no greater than log 2 (1+ 1    ) bits/symbol.
In slot-2, the relay simply emits  at subcarrier  to the user with power  2 .The received signal at the user is expressed as At the end of slot-2, the source combines  ,1 and  ,2 with maximum-ratio combining (MRC) to maximize the received SNR.The final signal used for decoding at the user can be expressed as It can be shown that when  = √ 1 ℎ   * and  = √ 2 ℎ   * , the MRC is achieved [35].The maximum received SNR is 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 in the unit of bits/symbol.
Suppose that   is the sum power of  1 and  2 ; the optimums  1 and  2 for maximizing the rate are the optimum solution for max Using the same method as in [36], it can easily be shown that the optimums  1 and  2 are where Δ , =    −    .The maximum rate associated with the above solution is equal to

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 subcarrier  in either slot-1 or slot-2 is assigned to user  and operates in the direct mode, and BS uses power  for this subcarrier.Therefore, the average transmission rate over this direct link can be evaluated as

Energy-Efficient RA Algorithm Design
Before data transmission, we assume that the BS controller knows all CSI, that is, {   ,    ,    | ∀, ∀}.Based on the a priori knowledge, the BS runs an algorithm to find the optimum RA strategy to maximize the network EE.
(iv) Power allocation: how to allocate BS's and RS's transmission power for each subcarrier To design the RA algorithm, we proceed as follows.First, a mathematical description of the RA strategy and network EE is proposed in Section 3.1 Then, a Dinkelbach-methodbased 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.

Description of the RA Strategy and Network EE.
We first define the following variables to describe the RA strategy: is paired with subcarrier  in slot-2 when   = 1.
(iv)  1  ≥ 0 and  2  ≥ 0: respectively indicating the BS's transmission power for subcarrier  in slot-1 and  in slot-2 for the direct mode.
Let us define a RA strategy as S = {I, P}, where I collects all indicator variables and P collects all power variables.A feasible S must satisfy 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 given S, the network EE is formulated as where (S) represents the sum rate for the network: and (S) is the sum power consumption for the network: with  being the loss factor of the power amplifiers (PA) used by the BS and RS and  cir representing the total power consumption by the BS's and RS's circuit devices.12) , ( 13) , ( 14)

Dinkelbach-Method
It can be seen that solving problem (P1) is highly challenging due to the following reasons: (i) (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 by S ⋆ ) and the maximum EE (denoted by  ⋆ ), we make use of the Dinkelbach method.To be more specific, we define a parameter  and a function () as the optimum objective value for the following problem (P2): 12) , ( 13) , ( 14) , ( 18) , (19) whose optimum solution is S().
(1) compute   ,   Motivated by the above principle, the RA algorithm to solve (P1) for the EE maximized RA strategy is summarized in Algorithm 1.The iterative update of  has a superlinear convergence rate [37].We will elaborate on the design of Algorithm 2 to solve (P2) in Section 3.3.
Wireless Communications and Mobile Computing (ii) Otherwise,   (0) < 0 and  ⋆  > 0 must hold.Note that   () is increasing of .Once  > 0 satisfying   () = 0 is found,  and S  () satisfy the above two conditions and hence can be taken as  ⋆  and S(), respectively.We will find  > 0 satisfying   () = 0 with the bisection method.
To complete this subsection, the above procedures to solve (P2) are summarized in Algorithm 2 as follows.Algorithm 2 has a polynomial complexity with respect to .We will elaborate on the design of Algorithm 3 to solve (P3) in Section 3.4.

Design of Algorithm 3 to Solve (P3).
We show how to find S  () as follows: (i) First, the optimum P for (P3) with fixed I is found and denoted by P I .
(ii) Second, define S I = {I, P I }.Then we find the optimum I to maximize   (, S I ) subject to the constraints on I in (P3).
(iii) Finally, S I corresponding to this optimum I can be taken as S  ().
As for the first step, the elements in the optimum P I can be computed according to KKT conditions as follows [39]: where As for the second step, it can readily be shown that where is solved for its optimum solution { ⋆  | ∀, }, an optimum (32) can be constructed by assigning for every combination of  and  all entries in {  ,   | ∀, } to zero, except for the one with the metric equal to   to  ⋆  .Most interestingly, (P5) is a standard assignment problem; hence every entry in { ⋆  | ∀, } is either 0 or 1 and { ⋆  | ∀, } 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 Algorithm 3 as follows.The complexity of computing   and   is ( 2 ( +  2 )).We use the Hungarian algorithm to solve (P5); the complexity is ( 3 ).As a result, the complexity of Algorithm 3 is ( 2 ( +  2 + )).

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.

System Setup and Benchmark Protocols.
In numerical experiments, we consider the downlink OFDMA network with a RS exhibit in Figure 4.  represents the number of users in service and they are randomly distributed in a circular region of radius  km.The RS is located between the BS and the user-region (UR) center, and the BS-RS distance is  km.The distance between the BS and the user-region center is  km.The bandwidth of the system is  Hz, and the OFDMA uses  subcarriers.The network parameters are listed in Table 2.
The channels are independent of each other and are generated in the same way as in [7].For every user , the impulse response of the source-to- channel is modeled as a tapped delay line with  = 6 taps, which are independently generated from circularly symmetric complex Gaussian distributions with zero mean and variance equal to (1/)(  / ref ) −4 , where  ref = 1 km and   represents the source-to- distance.The source-to-relay and relay-tou channels are generated in the same way, with each tap having variance as (1/)(  / ref ) −4  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 subcarrier  in slot-1 can only be paired with subcarrier  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 Table 3.It can be seen that the proposed algorithm has the highest complexity, while the BP-2 algorithm has the lowest complexity.

Impact of 𝑅 req on the Optimum EE and Corresponding Sum
Rate.To show the influence of  req on the EE, we choose  = 10 and  = 0.6 km and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when  req increases from 0 to 40 Mbits/s.The results are shown in Figure 5.
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 Figure 5(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.
Figure 5(a) also shows that the average EE of these methods decreases with the increase of  req .This is because the feasible set of the problem shrinks with the increase of  req .From Figures 5(a), 5(b), and 5(c), when  req < 20 Mbits/s and the average EE reaches the optimum value, the average communication rate is larger than  req , the average total power remains stable, and the average EE of the network maintains high value.When  req ≥ 20Mbits/s and the average EE reaches the optimum value, the average communication rate is equal to  req , the average total power increases rapidly, and the average EE of the network decreases.The above phenomenon indicates that the restricted condition (S) ≥  req influences the choice of the optimum solution.

Impact of Relay Position on the Optimum EE and Corre-
sponding Sum Rate.To show the impact of relay position on the EE, we choose  = 10 and  req = 20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when  increases from 0.2 to 1.2 km.The results are shown in Figure 6.
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 ∀, , ,   is more likely to take a high value.Note that   takes a high value only if both    and    are much higher than    .When RS lies in the middle between the BS and the users' region, it is more likely to have    and    , both much greater than    , and thus   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.

Impact of User Number on the Optimum EE and Corre-
sponding Sum Rate.To show the impact of user number on the EE, we choose  = 0.6 km and  req = 20 Mbits/s and then evaluate the average optimum EE for every protocol over 1000 random channel realizations, when  increases from 5 to 30.The results are shown in Figure 7.
From Figure 7, 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   and   increase with the increase of user number, which can improve the probability of   taking a larger value.In this way, the average EE of the network will be improved.

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 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.

Algorithm 3 :
The algorithm to find S  ().

Figure 4 :
Figure 4: The downlink OFDMA system considered in numerical experiments.

Figure 5 :
Figure 5: The average EE as the minimum rate changes.

Figure 6 :
Figure 6: The average EE as the relay position changes.

Figure 7 :
Figure 7: The average EE as the user number changes.
,   ) , ∀, , ,   )Finally, we find the optimum I for maximizing   (, S I ).   +     ) +     ) ≤    (33)holds, where   = max{max    , max ,   }.Call   the metric for   and   the metric for   ; the inequality is tightened when all entries of {  ,   | ∀, , } are assigned to zero, except that the one with the metric equal to   is assigned to   .