# Multiobjective Optimization Designs in Downlink NOMA Transmission Systems

Authors: Zhe Li
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

To further balance the high-capacity transmission requirement and the scarce spectrum resources status quo, we investigate the joint optimization of energy efficiency (EE) and spectral efficiency (SE) in a downlink multiuser NOMA system. While ensuring the user quality of service (QoS), a power allocation algorithm based on two-layer optimization is proposed by transforming the original multiobjective optimization problem into a univariate problem through the linear weighted sum method. Based on the optimal user power allocation coefficient solution obtained from the inner layer, the outer layer applies the bisection principle to search for the optimal system transmit power according to the uniqueness of the maximum value of pseudo-concave functions. Simulation results show that the proposed algorithm always converges within fewer iterations and can achieve the flexible tradeoff between EE and SE. This is of great significance for improving the energy utilization of actual scenarios with time-varying communication demands.


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## Body

## 1. Introduction

The application and popularization of 5G not only accelerates the development of the Internet of Things (IoT) but also promotes the emergence of many new communication services, which brings more strict performance requirements, such as better spectrum efficiency, denser network device access, and ultra-low transmission delay [1–3]. Taking into account the nearly exhausted frequency band resources, relevant scholars and research institutions begin to seek new access technologies to deal with the above problem [4]. Therefore, how to efficiently utilize the limited energy to meet the ever-increasing communication demand becomes a major topic in the current mobile communication field.Coincidentally, as a prominent wireless radio technique, nonorthogonal multiple access (NOMA) can transmit multiple users’ information simultaneously in the same frequency by introducing the power domain dimension, which generates excellent spectral efficiency (SE) [5, 6]. Especially, the key technology determining the performance of NOMA is power multiplexing [7]. Therefore, in the background of the high energy consumption and great demand for data rates in wireless communication system, it desperately needs more efficient power allocation schemes that consider altogether energy efficiency (EE), SE, and quality of service (QoS) to maintain the superiority of NOMA.Generally, the power allocation in NOMA is mainly discussed from two aspects, i.e., EE and SE. From EE vantage point, a fast convergence algorithm based on the Dinkelbach-Like theory was presented in [8], which contributes to obtain the optimal EE for a multiuser NOMA system. In [9], Z. Ma et al. investigated the problem about jointing user and subcarrier allocation while considering user fairness in a downlink multicarrier (MC) NOMA system. In [6], K. Cumanan et al. studied how the beamforming technology in MISO NOMA systems can better improve the EE. In addition, extensive researches have also been done from the SE aspect of NOMA. For instance, an effective resource schedule strategy is proposed to enhance the SE for a MC-NOMA system in [10]. In [11], a method for jointing optimal beamforming and power allocation all introduced to improve the throughput of the NOMA networks in multicell. In [12], a power allocation and spectrum sharing technology for the two-slot and secondary NOMA relay network is given to maximize the overall achievable rate of cellular users.However, the above-related works all consider the single-objective optimization (SOO) with EE or SE. In fact, the system metric often needs to be flexibly adjusted according to actual communication scenes. In some scenes with high real-time demands, the devices often need to quickly respond to the emergency requirements, such as medical care or alert information, which is SE critical [13]. But in other times, the devices can be served just to maintain its basic operation, and which requires optimal EE at this moment. Thus, it is significant to study how improve the energy utilization by weighing EE and SE in the real-time transmission scenarios. In [14], Y. Hei et al. conducted the parameter analysis and simulation verification for the EE-SE tradeoff in DCO-OFDM communication system. In [15], O. Aydin et al. studied the tradeoff about EE-SE in 5G multioperator heterogeneous networks. Additionally, the closed expressions for the relationship between EE and SE in CRN network were derived in [16], which also claimed that EE-SE are not contradictory in CRN networks and can achieve the optimal tradeoff. In NOMA system, the tradeoff about EE and SE for a single-carrier and multiuser network is investigated in [17], but it still adopted the constraint method, which makes EE as the main optimization objective, and SE as a constraint, in which the acquisition of EE’s value is more experience-based and less flexible. In [18], the EE-SE tradeoff in a NOMA system with multiuser pairs was studied, and it assumed that there was no interference between user pairs, which indicates that the power allocation is only based on every two users. An energy-efficient resource allocation technique for a hybrid time division multiple access—NOMA system—was proposed in [19], the available time for transmission is divided into several subtime slots, and a subtime slot is allocated to serve a group of users (i.e., cluster). A network network nonorthogonal multiple access (N-NOMA) technique for the uplink coordinated multipoint transmission (CoMP) was applied in [20] to improve the system throughput, and the numerical results are presented to show the accuracy of the analytical results and also demonstrate the superior performance of the proposed N-NOMA scheme.Based on the above researches, for improving the energy utilization, we consider both EE and SE as primary optimization goals simultaneously in a downlink multiuser NOMA system. By applying the linear weighted sum function method [21], the original multiobjective optimization (MOO) problem is firstly transformed into a SOO problem through a positive weighting factor. While satisfying users’ QoSs, we finally propose a dual-layer power allocation algorithm based on bisection searching. The optimal closed-form expressions for power allocation coefficient of all connected users and the optimal system transmit power can be, respectively, obtained in inner and outer layer optimization. Simulation results demonstrate the validity of the proposed algorithm.


## 2. System Model

In this paper, a downlink NOMA transmission scenario with multiple users is considered in Figure1, wherein one centrally located base station (BS) serves M single-antenna users simultaneously. The channel coefficient from the BS to the mth user (1≤m≤M) is represented by hm=gmd−α/2, where gm denotes the Rayleigh fading, d is the distance between the mth user and the BS, and α is the path-loss exponent. Without loss of generality, we assume that the BS can obtain the instantaneous user channel state information (CSI), and the channel gains of users are sorted as: 0<h12≤h22⋯≤hM2.Figure 1
System model.According to the protocol of NOMA, the BS sends a superimposed signal composed of all users’ information to each user through the code superposition technology. At the receiving end, the successive interference cancelation (SIC) is adopted to eliminate the inter-user interference. That is, each user first decodes signal of the user whose channel gain is lower than that of its own, and subtracts it from its received signal. Afterwards, it decodes own signal by treating the signal of users with a higher channel gain than itself as interference. Specifically, themth user first decodes the ith (i=1,2,⋯,m−1) user’s information and subtracts it from the superimposed signal it receives, and then the information of the jth (j=m+1,m+2,⋯,M) user is regarded as interference.Consequently, the achievable data rate of themth user can be written as:
(1)Rm=log21+amPThm2PThm2∑i=m+1Mai+σ2,m=1,⋯,M,where PT represents the maximum available transmit power at the BS, and am is the ratio of the allocated power of the mth user to PT.Furthermore, the sum rate of the system is expressed as:(2)R=∑m=1MRm.Assuming that the system has the unit bandwidth, the SE and EE can be defined as follows [16]:
(3)ηSE=R,(4)ηEE=RP+Pc,where Pc is the circuit power consumption, and P is the actual consumed transmit power by BS.


## 3. Problem Formation

According to previous studies on EE and SE for NOMA systems, SE always grows with the increasing of the transmit power of the BS, while EE first increases and then begins to decrease after achieving its maximum value. This means that EE and SE are contradictory when the BS power is large enough, and also indicates that there is a tradeoff between EE and SE.In this section, we investigate the joint optimization of EE and SE to achieve a flexible tradeoff between the two. The primordial MOO problem can be formulated as:(5)P1:maxPηEEP,ηSEP,where0≤P≤PT.Unlike SOO problem, MOO problem can have multiple optimal solutions. ForP1, we suppose P∗ is the transmit power that enables EE to achieve its maximum value. When P∗≥PT, P1 only has one solution, i.e., the global optimal solution PT. When P∗<PT, the objective can be expressed as:
(6a)P2:maxPηEEP,ηSEP,s.t.P∗≤P≤PT.In order to addressP2, we first normalize ηSE and ηEE to make them additive and comparable, and then, referring to the common conversion method for MOO problems, the weighted sum function is adopted. Afterwards, a new system metric is defined as:
(7)WλEE,λSE=ω⋅λEE+1−ωλSE,where λEE=ηEE/ηEEmax and λSE=ηSE/ηSEmax. ηEEmax and ηSEmax are, respectively, the maximum value that EE and SE can achieve within the range of a given BS power. It is evident in (7) that the flexible tradeoff between EE and SE can be realized by adjusting ω (ω=ω0≤ω≤1), which is a preference factor.Moreover, substituting (3) and (4) into (7), we get
(8)λEE−SE=ω1ηEEmaxP+Pc+1−ω1ηSEmaxR.As a result,P2 can be further described as:
(9a)P3:maxλEE−SE,(9b)s.t.Rm≥Rmmin,1≤m≤M,(9c)P≤PTandP=∑m=1MamPT,where constraint (9a) denotes the minimum target rate of the mth user, which is required by its QoS and denoted as Rmmin for 1≤m≤M. (9b) shows the constraint for maximum transmit power of the BS.To ensure problem (9) is solvable, the maximum BS transmit power must be large enough to support all users’ QoS mentioned in (9a), which further introduces the minimum transmit power constraint. We define Pmin as the total minimum transmit power that can just satisfy all users’ QoS requirements, and thus, problem (9) is feasible only when P≥Pmin. Therefore, it is important to first establish the feasible range of P, and the derivation of which is discussed as follows.Denotepm as the allocated power of the mth user, and the problem of calculating Pmin can be described as
(10a)Pmin≜minpk,1≤m≤M∑m=1Mpm,(10b)s.t.pm≥Am∑i=m+1Mpi+σ2hm2,1≤m≤M,where Am=2Rmmin−1. And (10b) originates from the minimum target rate constraints in (9b). The solution to problem (10) can be found by using the following theorem.Theorem 1.
Denotedpmminm=1M as the optimal solution to the objective (10), which is given as
(11)Pmmin=Am∑i=m+1MPimin+σ2hm2,1≤m≤M.Proof.
We prove the theorem according to the contradiction principle. Assumingpm∗m=1M is a set of allocated power of all users and also is the optimal solution to problem (10) with at least one constraint in (10b) inactive. More generally, we then set the kth constraint as inactive, i.e.,
(12)pk∗>Ak∑i=k+1KPimin+σ2hk2.We next createPm∗∗m=1M as a new set by defining pm∗∗=pm∗ for k≠m and setpk∗∗ to the right-hand side (RHS) of (12). By observing the structure of the constraints in (10b), for an arbitrary m, we can find that the RHS of (10b) is a monotonically nondecreasing function of pi for 1≤i≤M. In the result, the setting of pk∗∗, whose value is less than pk∗∗, ensures that all the constraints in (10b) hold for the newly created set Pm∗∗m=1M. But then, we can get ∑m=1Mpm∗∗<∑MMM with the definition of Pm∗∗m=1M. It obviously contradicts to the assumption that Pm∗∗m=1M is the optimal solution to problem (10). Therefore, we can conclude that the function (10a) is minimized when all the constraints in (10b) are active. The proof is complete.Afterwards, when all constraints in (10b) are active, the optimal solution to objective (10), i.e.,Pm∗∗m=1M, can be calculated sequentially in the order m=M,M−1,⋯,1, this is because thatPmmin can be calculated by pi+1mini=m+1M, and pMmin=AMσ2/hm2is determinable. Subsequently, we obtainPmin=∑m=1Kpmmin and the feasible region of the actual consumed power is Pmin≤P≤PT.


## 4. Proposed Optimal Solutions

It is apparent thatP3 is nonconvex, so it is difficult to find the global optimal solution in polynomial time. Therefore, we decompose it into a dual problem, by searching the optimal power allocation in inner layer and optimal available transmit power in outer layer, the final solution of P3 can be obtained.


### 4.1. Inner Layer: Optimal Power Allocation Scheme

For a givenP within the feasible range Pmin≤P≤PT, P3 can be regarded as an optimization problem that only includes the power allocation factor am. Additionally, for any given ω, the part in square brackets on the RHS of (8) is a constant. At this time, problem P3 is equivalent to maximize the overall sum rate R.Moreover, substituting (1) into (2), R can recast as:
(13)R=log2PTh12∑i=1Mai+σ2+∑m=1M−1log2PThm+12∑i=m+1Mai+σ2−log2PThm2∑i=m+1Mai+σ2−log2σ2.In order to simplify the notation, we then define:(14)Γm≜PThm2,1≤m≤M,(15)θ≜∑i=1Mai=PPT,(16)βm≜∑i=m+1Mai,1≤m≤M−1,(17)Fmβm≜log2Γm+1βm+σ2−log2Γmβm+σ2.Based on the above definitions,R can be reformulated as:
(18)R=log21+Γ1θ/σ2+∑m=1M−1Fmβm.Here, we further emphasize thatθ=θ0≤θ≤1 is the power consumption factor, which denotes the ratio of the actually utilized power P to the maximum available BS transmit powe rPT. Therefore, the power allocation of the inner layer is to allocate parameter am∗θm=1M by treating θ as a constant, and its solution is a function of θ. Moreover, it can be easily observed that the first term in (16) is a constant for any given θ, and then the optimization problem for maximizing R is equivalent to:
(19a)P4:maxam,1≤m≤M∑m=1M−1Fmβm,s.t.Rm≥Rmmin,∑m=1Mam=θ,where the optimal power allocation coefficient of the mth user, i.e.,am∗θ, can be obtained by [22]:
(20)am∗θ=Am2Rmminθ−∑i=1m−1ai∗θ+σ2Phm2,m≠Mθ−∑i=1m−1ai∗θ,m=M.


### 4.2. Outer Layer: Optimal Available Transmit Power

In inner layer problem, we have obtained the optimal closed-form solution for the power allocation among users, in whichθ is the only variable. Furthermore, in outer layer, P3 can be transformed into a univariate optimization problem about θ.Based on the optimal power allocation demonstrated in (20), (8) can be rewritten as:
(21)λEE−SE′θ=QθRθ,where
(22)Qθ=ωηEEmaxθPT+PC+1−ωηSEmax,(23)Rθ=log21+Γ1θ/σ2+∑m=1M−1Fβm∗θ,and βm∗θ=∑i=m+1Mai∗θ, P=θPT.The problem in outer layer can be finally formulated as:(24a)P5:maxθλEE−SE′θ,(24b)s.t.Pmin/PT≤θ≤1..In order to obtain the optimal solution ofP5, we first introduce the following theorem:Theorem 2.
λEE−SE′θ is a strict pseudo-concave function with respect to θ.Proof.
Combining (22) and (23), λEE−SE′θ can be re-expressed as:
(25)λEE−SE′θ=ωRθηEEmaxθPT+PC+1−ωηSEmaxRθ,where the convexity of Rθ can be firstly verified. It is clear that the first term of Rθ, i.e., log21+Γ1θ/σ2, is a logarithmic function with respect to θ, which indicates that it is strictly concave. Moreover, the convexity of the second term, Fmβm∗θ, can be derived recursively from the convexity of Fmβm. The first-order derivative of Fmβm aboutβm is given as:
(26)dFmβmdβm=Γm+1−Γmσ2ln2Γm+1βm+σ2Γmβm+σ2≥0,and the second-order derivative can be further derived as:
(27)d2Fmβmdβm2=−Γm+1−ΓmDσ2ln2Γm+1βm+σ2Γmβm+σ22≤0,where D=Γm+1βm+σ2Γm+Γmβm+σ2Γm+1.Obviously, the second-order derivative ofFmβm is nonpositive, which reveals that Fmβm is concave with respect to βm for 1<m<M−1. On this basis, we can further come to the conclusion that Fmβm∗θ is also concave with respect to θ. This is due to the fact that ai∗θi=1M and βm∗θm=1M−1 are both affine mappings according to their linear expressions, which preserves the convexity of Fmβm∗θabout θ. As yet, actually, we have proved that Rθ is a strictly concave function with respect to θ, this is because it is the summation about log2Γ1θ/σ2+1 and Fmβm∗θ, which are all strict concave, and the convexity of functions is preserved by addition operations.Moreover, for arbitraryω, ηEEmax, and ηSEmax, both are deterministic quantities. Additionally, θPT+PC is an affine function of θ. Therefore, since the numerator and denominator of the first term on the RHS of (25) are, respectively, strict concave and affine, it can be easily concluded that the first term of λEE−SE′θ is strict pseudo-concave about θ. Clearly, the second term on the RHS of (25) is also strict concave. Finally, based on the additivity of concave functions, we can declare that λEE−SE′θ is a strict pseudo-concave function about θ. The proof is completed.For any pseudo-concave function, there is a unique maximizer [20], which guarantees the existence and uniqueness of the global optimal solution of objective (24a), i.e., the root of the equation dλEE−SE′θ/dθ=0, and the expression for dλEE−SE′θ/dθ is shown as:
(28)dλEE−SE′θdθ=−PTωηEEmaxθPT+Pc2Rθ+dRθdθQθ,where
(29)dRθdθ=1ln2ΓMdaM∗θ/dθΓMaM∗θ+σ2,dam∗θdθ=Am2Rmmin1−∑i=1m−1dai∗θdθ,m≠M1−∑i=1M−1dai∗θdθ,m=M.At this time, the outer layer optimization problem is transformed into searching the maximum value of the pseudo-concave functionλEE−SE′θ, which can turn to the bisection algorithm for help.


### 4.3. Power Allocation Scheme of Joint Optimization the EE and SE

In this section, we proposed the bisection searching-based power allocation algorithm for the joint optimization of EE and SE, which is shown in Algorithm1, and the process of it can be described as follows, where t is used to record the number of iterations before the algorithm converges.Algorithm 1: Bisection searching-based power allocation algorithm.
1.Input:Pmin,PT,t,ε;2.Initialization:θU=1, θL=Pmin/PT, t=1, ε=10−3;3.GivenθU=1, the optimal power allocation coefficient4.can be obtained by using (20), and then calculate QU:5.QU=dλEE−SE′/dθθ=θU;6.ifQu≥07.θ∗=θU;8.else9. GivenθL=Pmin/PT, the optimal power allocation10. can be obtained by using (20), and then calculateQL:11.QL=dλEE−SE′/dθθ=θL;12.ifQL≤013.θ∗=θL;14.else15.θM=θL+θU/2,QM=dλEE−SE′/dθθ=θM;16.whileQM≥ε17.ifQL⋅QM>018.θL=θM,QL=QM;19.else20.θU=θM,QU=QM;21.end22.t=t+1;23.θM=θL+θU/2, QM=dλEE−SE′/dθθ=θM;24.end while25..θ∗=θM;26.end27.end28.Output:θ∗PT, t.At the beginning, the BS calculates the minimum transmit powerPmin and then determines whether it is in the feasible range Pmin/PT≤θ≤1. If the condition is satisfied, we can further figure out the root of λEE−SE′θ, and the value of dλEE−SE′θ/dθ with the range Pmin/PT,1 can be divided into the following three cases:Case 1: IfdλEE−SE′θ/dθθ=1≥0, which indicates that λEE−SE′ keeps increasing in feasible range, and the optimal power consumption factor is θ∗=1.Case 2: IfdλEE−SE′/dθθ=Pmin/PT≤0, which indicates that λEE−SE′ keeps decreasing in feasible range. At this time, the optimal power consumption factor is θ∗=Pmin/PT.Case 3: IfdλEE−SE′/dθθ=Pmin/PT>0 and dλEE−SE′/dθθ=1<0, which indicates that λEE−SE′ first increases and then decreases. Thus, the optimal power consumption factor is θ∗∈Pmin/PT,1 and can be found by using the bisection searching method.


## 4.1. Inner Layer: Optimal Power Allocation Scheme

For a givenP within the feasible range Pmin≤P≤PT, P3 can be regarded as an optimization problem that only includes the power allocation factor am. Additionally, for any given ω, the part in square brackets on the RHS of (8) is a constant. At this time, problem P3 is equivalent to maximize the overall sum rate R.Moreover, substituting (1) into (2), R can recast as:
(13)R=log2PTh12∑i=1Mai+σ2+∑m=1M−1log2PThm+12∑i=m+1Mai+σ2−log2PThm2∑i=m+1Mai+σ2−log2σ2.In order to simplify the notation, we then define:(14)Γm≜PThm2,1≤m≤M,(15)θ≜∑i=1Mai=PPT,(16)βm≜∑i=m+1Mai,1≤m≤M−1,(17)Fmβm≜log2Γm+1βm+σ2−log2Γmβm+σ2.Based on the above definitions,R can be reformulated as:
(18)R=log21+Γ1θ/σ2+∑m=1M−1Fmβm.Here, we further emphasize thatθ=θ0≤θ≤1 is the power consumption factor, which denotes the ratio of the actually utilized power P to the maximum available BS transmit powe rPT. Therefore, the power allocation of the inner layer is to allocate parameter am∗θm=1M by treating θ as a constant, and its solution is a function of θ. Moreover, it can be easily observed that the first term in (16) is a constant for any given θ, and then the optimization problem for maximizing R is equivalent to:
(19a)P4:maxam,1≤m≤M∑m=1M−1Fmβm,s.t.Rm≥Rmmin,∑m=1Mam=θ,where the optimal power allocation coefficient of the mth user, i.e.,am∗θ, can be obtained by [22]:
(20)am∗θ=Am2Rmminθ−∑i=1m−1ai∗θ+σ2Phm2,m≠Mθ−∑i=1m−1ai∗θ,m=M.


## 4.2. Outer Layer: Optimal Available Transmit Power

In inner layer problem, we have obtained the optimal closed-form solution for the power allocation among users, in whichθ is the only variable. Furthermore, in outer layer, P3 can be transformed into a univariate optimization problem about θ.Based on the optimal power allocation demonstrated in (20), (8) can be rewritten as:
(21)λEE−SE′θ=QθRθ,where
(22)Qθ=ωηEEmaxθPT+PC+1−ωηSEmax,(23)Rθ=log21+Γ1θ/σ2+∑m=1M−1Fβm∗θ,and βm∗θ=∑i=m+1Mai∗θ, P=θPT.The problem in outer layer can be finally formulated as:(24a)P5:maxθλEE−SE′θ,(24b)s.t.Pmin/PT≤θ≤1..In order to obtain the optimal solution ofP5, we first introduce the following theorem:Theorem 2.
λEE−SE′θ is a strict pseudo-concave function with respect to θ.Proof.
Combining (22) and (23), λEE−SE′θ can be re-expressed as:
(25)λEE−SE′θ=ωRθηEEmaxθPT+PC+1−ωηSEmaxRθ,where the convexity of Rθ can be firstly verified. It is clear that the first term of Rθ, i.e., log21+Γ1θ/σ2, is a logarithmic function with respect to θ, which indicates that it is strictly concave. Moreover, the convexity of the second term, Fmβm∗θ, can be derived recursively from the convexity of Fmβm. The first-order derivative of Fmβm aboutβm is given as:
(26)dFmβmdβm=Γm+1−Γmσ2ln2Γm+1βm+σ2Γmβm+σ2≥0,and the second-order derivative can be further derived as:
(27)d2Fmβmdβm2=−Γm+1−ΓmDσ2ln2Γm+1βm+σ2Γmβm+σ22≤0,where D=Γm+1βm+σ2Γm+Γmβm+σ2Γm+1.Obviously, the second-order derivative ofFmβm is nonpositive, which reveals that Fmβm is concave with respect to βm for 1<m<M−1. On this basis, we can further come to the conclusion that Fmβm∗θ is also concave with respect to θ. This is due to the fact that ai∗θi=1M and βm∗θm=1M−1 are both affine mappings according to their linear expressions, which preserves the convexity of Fmβm∗θabout θ. As yet, actually, we have proved that Rθ is a strictly concave function with respect to θ, this is because it is the summation about log2Γ1θ/σ2+1 and Fmβm∗θ, which are all strict concave, and the convexity of functions is preserved by addition operations.Moreover, for arbitraryω, ηEEmax, and ηSEmax, both are deterministic quantities. Additionally, θPT+PC is an affine function of θ. Therefore, since the numerator and denominator of the first term on the RHS of (25) are, respectively, strict concave and affine, it can be easily concluded that the first term of λEE−SE′θ is strict pseudo-concave about θ. Clearly, the second term on the RHS of (25) is also strict concave. Finally, based on the additivity of concave functions, we can declare that λEE−SE′θ is a strict pseudo-concave function about θ. The proof is completed.For any pseudo-concave function, there is a unique maximizer [20], which guarantees the existence and uniqueness of the global optimal solution of objective (24a), i.e., the root of the equation dλEE−SE′θ/dθ=0, and the expression for dλEE−SE′θ/dθ is shown as:
(28)dλEE−SE′θdθ=−PTωηEEmaxθPT+Pc2Rθ+dRθdθQθ,where
(29)dRθdθ=1ln2ΓMdaM∗θ/dθΓMaM∗θ+σ2,dam∗θdθ=Am2Rmmin1−∑i=1m−1dai∗θdθ,m≠M1−∑i=1M−1dai∗θdθ,m=M.At this time, the outer layer optimization problem is transformed into searching the maximum value of the pseudo-concave functionλEE−SE′θ, which can turn to the bisection algorithm for help.


## 4.3. Power Allocation Scheme of Joint Optimization the EE and SE

In this section, we proposed the bisection searching-based power allocation algorithm for the joint optimization of EE and SE, which is shown in Algorithm1, and the process of it can be described as follows, where t is used to record the number of iterations before the algorithm converges.Algorithm 1: Bisection searching-based power allocation algorithm.
1.Input:Pmin,PT,t,ε;2.Initialization:θU=1, θL=Pmin/PT, t=1, ε=10−3;3.GivenθU=1, the optimal power allocation coefficient4.can be obtained by using (20), and then calculate QU:5.QU=dλEE−SE′/dθθ=θU;6.ifQu≥07.θ∗=θU;8.else9. GivenθL=Pmin/PT, the optimal power allocation10. can be obtained by using (20), and then calculateQL:11.QL=dλEE−SE′/dθθ=θL;12.ifQL≤013.θ∗=θL;14.else15.θM=θL+θU/2,QM=dλEE−SE′/dθθ=θM;16.whileQM≥ε17.ifQL⋅QM>018.θL=θM,QL=QM;19.else20.θU=θM,QU=QM;21.end22.t=t+1;23.θM=θL+θU/2, QM=dλEE−SE′/dθθ=θM;24.end while25..θ∗=θM;26.end27.end28.Output:θ∗PT, t.At the beginning, the BS calculates the minimum transmit powerPmin and then determines whether it is in the feasible range Pmin/PT≤θ≤1. If the condition is satisfied, we can further figure out the root of λEE−SE′θ, and the value of dλEE−SE′θ/dθ with the range Pmin/PT,1 can be divided into the following three cases:Case 1: IfdλEE−SE′θ/dθθ=1≥0, which indicates that λEE−SE′ keeps increasing in feasible range, and the optimal power consumption factor is θ∗=1.Case 2: IfdλEE−SE′/dθθ=Pmin/PT≤0, which indicates that λEE−SE′ keeps decreasing in feasible range. At this time, the optimal power consumption factor is θ∗=Pmin/PT.Case 3: IfdλEE−SE′/dθθ=Pmin/PT>0 and dλEE−SE′/dθθ=1<0, which indicates that λEE−SE′ first increases and then decreases. Thus, the optimal power consumption factor is θ∗∈Pmin/PT,1 and can be found by using the bisection searching method.


## 5. Simulation Results

The validity of Algorithm1 is verified using computer simulations. We assume that the distance d from all users to the BS is the same and d=80m. Besides, we also suppose that all users have the same QoS requirements, which means that Rmmin=Rmin=1bps/Hz for 1≤m≤M. Finally, the settings of other parameters are shown in Table 1. Note that if the maximum available transmit power of the BS cannot support all users’ minimum rate constraints, we default that the system does not work.Table 1
Simulation parameters.
ParametersCharacteristicNumber of usersM=2,3,4Noise powerσ2=‐70dBmPath-loss exponentα=3Maximum available transmit power of the BSPT=5~30dBmCircuit power consumptionPc=28.30dBmFigure2 reveals the EE versus SE with variable PT, and the variation range of PT is 5 ~ 30 dBm. As PT increases, it can be seen that EE first increases with SE, but when EE reaches its optimal value, EE begins to go down while we continue to increase the PT, and the system power utilization will decrease. This indicates that EE and SE are contradictory when the available BS power is large enough, and there is a tradeoff between the two.Figure 2
The EE versus SE with differentM and Pc..Figure3 shows the EE versus PT, in which M is set to 2, 3, 4. And “max-SE” means that when the system focuses on maximizing SE, it always applies the full power allocation strategy among users. However, from the EE’s point of view, it is obviously not optimal. Because when the power is larger than P∗, EE will increase firstly and then begin to decrease with the increasing of the power. “Opt-EE” means that in the proposed joint optimization algorithm, the system treats EE as an indicator when ω=1, and the system power can always be maintained at the optimal value that maximizes EE.Figure 3
The EE versusPT of the BS with different M.Figure4 depicts the relationship between the optimal transmit power θ∗PT and the weighting factor ω. The maximum available transmit power of the BS is set to 30 dBm, i.e., PT=30dBm, and M is set to 3. It can be seen that as ω increases, θ∗PT first remains unchanged at the PT, and then gradually approaches the P∗. This is because when ω is smaller, the system metric is equivalent to SE, and BS uses as much transmit power as possible to maximize the SE. With the increasing of ω, the system metric gradually transforms to be EE. Therefore, θ∗PT gradually approaches P∗ that maximizes EE.Figure 4
Optimal transmit powerθ∗PT versus ω.Figure5 reveals θ∗PT versus t with different M in the proposed joint optimization algorithm, where PT is set to 25 dBm. It can be firstly observed that the actual consumed power grows with the increasing of M. This is because the BS needs more power to provide guaranteeing for the increased users’ QoS, but actually, the computational complexity is not significantly enhanced. Because the proposed joint algorithm can always achieve convergence around 7 iterations. More importantly, we can find that the value of θ∗PT with ω=0.8 is obviously smaller than that ω=0.2. This is due to that the main system metric is the SE, and at this time, the BS always tends to use all power to maximize the system capacity. Oppositely, the main system metric is EE when ω=0.8, the system always tries to keep the EE at the optimal value, and the value is always less than the power value required to maximize the SE.Figure 5
Optimal transmit powerθ∗PT versus t with different M.Figure6 investigates the λEE and λSE, and the joint objective function WλEE,λSEvarying with ω. It can be seen that EE gradually increases with ω, while the SE gradually goes down. However, when the value of ω is close to zero, both EE and SE remain unchanged, which is in consistent with the trend of θ∗PT demonstrated in Figure 3. In particular, when ω=0, the maximum SE can be achieved, and the EE reaches the optimal value when ω=1.Figure 6
TheλEE, λSE, and WλEE,λSE versus ω.


## 6. Conclusion

In order to better maintain the system performance and reduce energy consumption, we study the joint optimization of EE and SE in a downlink NOMA system in this paper. We first define a new system performance measurement indicator and then transform the original MOO problem into a SOO problem. Under the condition that guarantees users’ QoSs, a two-layer bisection searching-based power allocation algorithm is presented finally. Simulation results verify that the proposed power allocation scheme can achieve the opportune tradeoff between EE and SE by flexibly adjusting the weighting factor in the given indicator function from zero to 1. The method is universal and suitable for improving the energy utilization of actual scenarios with time-varying communication demands.



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*Source: 7268083-2022-07-20.xml*