Energy Efficiency Maximization in the Uplink Delta-OMA Networks
Ramin Hashemi, Hamzeh Beyranvand, Mohammad Robat Mili, Ata Khalili, Hina Tabassum, Derrick Wing Kwan Ng
11 Energy Efficiency Maximization in the UplinkDelta-OMA Networks
Ramin Hashemi,
Student Member, IEEE,
Hamzeh Beyranvand
Member, IEEE , Mohammad Robat Mili
Member, IEEE , and Hina Tabassum,
Senior Member, IEEE . Abstract —Delta-orthogonal multiple access (D-OMA) has beenrecently investigated as a potential technique to enhance thespectral efficiency in 6G networks. D-OMA enables partialoverlapping of the adjacent sub-channels that are assigned todifferent clusters of users served by non-orthogonal multipleaccess (NOMA), at the expense of additional interference. Inthis paper, we analyze the performance of D-OMA in theuplink and develop a multi-objective optimization frameworkto maximize the uplink energy efficiency in a multi-cell networkenabled by D-OMA. Specifically, we optimize the subchanneland transmit power allocations of the users as well as theoverlapping percentage of the spectrum between the adjacentsub-channels. The formulated problem is a mixed binary non-linear programming problem; therefore, we first transformthe problem into a single-objective problem using Tchebyshevmethod. Then, we apply the monotonic optimization (MO) toexplore the hidden monotonicity of the objective function andconstraints, and reformulate the problem into a standard MOin canonical form. The re-formulated problem is then solvedby applying the outer polyblock approximation method. Ournumerical results show that D-OMA outperforms the conven-tional non-orthogonal multiple access (NOMA) and orthogonalfrequency division multiple access (OFDMA) when the adjacentsub-channel overlap and scheduling is optimized jointly.
Index Terms —Delta-OMA, multi-objective optimization, powercontrol, resource allocation.
I. I
NTRODUCTION
Delta orthogonal multiple access (D-OMA) has been re-cently considered as a potential variant of non-orthogonalmultiple access (NOMA) to enable massive multiple accessand enhanced spectral efficiency in 6G networks [1]–[3]. D-OMA exploits partial overlapping of adjacent sub-channelsthat are assigned to different clusters of users served byNOMA and thereby enhance spectral efficiency. That is,NOMA is a special case of D-OMA when there is no over-lapping of adjacent sub-channels. Clearly, the performanceof D-OMA critically depends on the number of users in aNOMA cluster, the fraction of overlapping spectrum, andsub-channel scheduling. It is noteworthy that while partialoverlapping of adjacent subchannels may enhance spectralefficiency, it can yield additional interference that can result
R. Hashemi is with the Centre for Wireless Communications (CWC),University of Oulu, 90014 Oulu, Finland. e-mail: (ramin.hashemi@oulu.fi).H. Beyranvand is with the Department of Electrical Engineering, AmirkabirUniversity of Technology, 424 Hafez Avenue, Tehran 15914, Iran, e-mail:([email protected]). M R. Mili is with the Department of Telecommu-nications and information processing, Ghent University, Belgium, e-mail:([email protected]). H. Tabassum is with the Lassonde Schoolof Engineering at York University, Canada (e-mail:[email protected]). in significant performance loss. Therefore, it is thus crucial tooptimize the scheduling, NOMA cluster size, and the fractionof overlapping spectrum efficiently.To date, many research works have considered optimizingthe performance of stand-alone NOMA or hybrid NOMA-OMA networks [4]–[6], and partial NOMA (P-NOMA) [7],[8]. In hybrid NOMA-OMA, the transmit power, time, andsub-channel resources to the users are determined optimallyto optimally exploit both NOMA and OMA. Compared to thetraditional NOMA or OMA, hybrid NOMA has a variety ofbenefits, including higher spectral efficiency than OMA, lesscomplexity in terms of successive interference cancellation(SIC) than NOMA, and reduced interference than NOMA.In [4], a user grouping and power allocation strategy basedon sum-rate maximization was proposed for both the uplinkand downlink NOMA. In [5], an energy-efficient power controland resource block assignment framework is presented for theuplink of a hybrid NOMA network with the quality of service(QoS) constraints. In [6], the downlink energy efficiency ofthe network is maximized by optimizing a user clusteringand power control framework in a hybrid NOMA system. Onthe other hand, P-NOMA partially overlaps the signals of theusers, where the fraction of the overlap is adjustable [7], [8].In particular, the partial overlapping is achieved by allowingboth users to transmit over the entire time slot, but there is apartial overlap of the spectrum shared by the two users.Different from the aforementioned variants of NOMA, inD-OMA, the spectrum overlapping is considered among twoNOMA clusters operating on adjacent sub-channels within agiven cell and the interference is controlled by optimizingeither the fraction of overlapping percentage or reducing thecluster size. Very recently, the authors in [1] have shownpreliminary results on the significance of D-OMA in thedownlink, compared to NOMA [9].In this paper, we provide a comprehensive framework toanalyze the performance of D-OMA in the uplink. Specifi-cally, we develop a multi-objective optimization frameworkto maximize the uplink energy efficiency of a multi-accesspoint (AP) network enabled by D-OMA. We optimize thesubchannel and transmit power allocations of the users as wellas the overlapping percentage of the spectrum between theadjacent sub-channels. The formulated problem is a mixedbinary non-linear programming problem; therefore, we firsttransform the problem into a single-objective problem usingTchebyshev method. Then, we apply the monotonic optimiza-tion (MO) framework to explore the hidden monotonicity ofthe objective function and constraints in order to reformulate a r X i v : . [ c s . I T ] F e b UE 𝐿 𝑁,𝑘 … Frequency P o w e r UE 𝐿 UE 𝐿 𝐵𝛿 … 𝐵 = 𝑊 𝑁 𝑊 Hz … … Partial ICI 𝐵𝛿 𝐵𝛿 𝐵𝛿 𝐵𝛿 𝑁−1,𝑘𝑟 𝐵𝛿 𝑁,𝑘 𝑙 𝐵 = 𝑊𝑁
Fig. 1: D-OMA and partial ICI illustration in AP k .the problem into a standard MO in canonical form. The re-formulated problem is then solved by the outer polyblockapproximation method. The numerical results show that D-OMA method outperforms the conventional NOMA and OMAwhen the adjacent sub-channel overlap and scheduling isoptimized jointly. In addition, our numerical results depictthe effectiveness of D-OMA considering two cases, (i) whenthe overlapping percentages are optimized individually oneach sub-channel denoted as per-subchannel optimized delta(POD), and (ii) when the overlapping parameter is fixed forall subchannels denoted as non-POD (NPOD).II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. System Model
Consider the UL of a cellular network where the D-OMAmethod is used by K access points (APs) to serve the users.The principle methodology of D-OMA scheme is illustratedin Fig. 1 in which the overall system bandwidth ( W Hz) isdivided into N subbands, and each subband is allocated toa sub-set of users (i.e., users in a specific NOMA cluster).Assume the number of users in subband n at AP k as L n,k ,and U k = { , , ..., U k } denotes the set of users in the coveragearea of AP k . Thus, we have ∑ Nn = L n,k = U k . We indicate theset of subbands by N and the set of all APs by K .In D-OMA method, an addition interference signal namedas partial inter-cluster interference (ICI) from the adjacent sub-channels incur. The adjacent subbands are interleaved from theright- and left-side by amount of B × δ rn,k Hz and B × δ ln,k Hz,respectively, where ≤ δ rn,k ≤ and ≤ δ ln,k ≤ denotethe overlap percentage of subband n for AP k , and B is thebandwidth of each subband i.e. B = WN (see Fig. 1). Note thatwhen δ rn,k = δ ln,k = the subbands are completely overlappedsuch that the amount of interference power is maximum. Onthe other hand, each subband’s effective bandwidth denoted byB n = B ( + δ rn,k + δ ln,k ) is expanded three times at the expenseof additional ICI. Subsequently, the optimal values of δ rn,k and δ ln,k for ∀ n ∈ N and ∀ k ∈ K and cluster size should bedetermined efficiently to enhance the network sum rate (SR).Note that δ rn,k = and δ ln,k = for ∀ n ∈ N and ∀ k ∈ K correspond to the conventional NOMA. Let p m,k denote the transmitted data power from the UE m to the AP k . To model the subband allocation, we define thefollowing binary variable ρ nm,k = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , if UE m in cell k is associated with subband n, . o.w.Therefore, the received signal at AP k in subband n will beas given in (1) on the next page where w nk ∼ N C ( , σ ) is theadditive complex Gaussian white noise in the case of equallydivided subbands, in which the noise power is increased due tosubband expansion to ( + δ ln,k + δ rn,k ) σ , s m,k represents thetransmitted signal from UE m to the AP k with E [∣ s m,k ∣ ] = , δ rn,k and δ ln,k are the introduced parameters to control thepartial ICI at the subband n at AP k . B. Network Rate and Energy Efficiency
It is important to note that there are four types of partialICI on a given subband n , as illustrated in Fig. 1. That is, thepartial ICI in the left side is due to the expansion of subband n − towards right hand side which is controlled by δ rn − as well as the expansion of subband n to the left hand sidewhere it is controlled by δ ln . The same reason applies for thetwo partial ICI signals on the right hand side of subband n .Note that, δ rN and δ l do not exist and therefore their valueis zero. g k,nm,k ′ indicates the channel power gain between UE m in cell k ′ and AP k at subband n which is denoted as g k,nm,k ′ = h k,nm,k ′ √ β k,nm,k ′ , where h k,nm,k ′ is the small-scale fading coefficientassumed to be Rayleigh distributed and β k,nm,k ′ represents thelarge-scale fading and path loss. The channel gains are beingsorted at APs, i.e., ∣ g k,n ,k ∣ ≥ ∣ g k,n ,k ∣ ≥ ... ≥ ∣ g k,nL n,k ,k ∣ , ∀ k ∈ K and ∀ n ∈ N , to perform SIC, in order to extract the desired signalof all UEs. Therefore, the achievable rate of UE m associatedwith AP k at subband n is given by R nm,k = B n log ( + p m,k ∣ g k,nm,k ∣ I m,n,k IntraICI + I m,n,k InterICI + I m,n,k PartialICI + ˜ σ n,k ) , (2)where B = WN , B n = B ( + δ rn,k + δ ln,k ) , ˜ σ n,k = σ ( + δ ln,k + δ rn,k ) , and I m,n,k IntraICI , I m,n,k
InterICI and I m,n,k
PartialICI are defined asI m,n,k
IntraICI = U k ∑ m ′ = m + ρ nm ′ ,k p m ′ ,k ∣ g k,nm ′ ,k ∣ , (3)I m,n,k InterICI = K ∑ k ′ = ,k ′ ≠ k U k ′ ∑ m ′ = ρ nm ′ ,k ′ p m ′ ,k ′ ∣ g k,nm ′ ,k ′ ∣ , (4)I m,n,k PartialICI = K ∑ k ′ = U k ′ ∑ m ′ = ((√ δ ln + ,k ′ + √ δ rn,k ′ ) ∣ g k,n + m ′ ,k ′ ∣ ρ n + m ′ ,k ′ + (√ δ ln,k ′ + √ δ rn − ,k ′ ) ∣ g k,n − m ′ ,k ′ ∣ ρ n − m ′ ,k ′ ) p m ′ ,k ′ . (5)where I m,n,k IntraICI and I m,n,k
InterICI are the power of the intra-celland inter-cell ICI terms in (1), respectively. Additionally,I m,n,k
PartialICI refer to the power of partial ICI components as afunction of δ rn,k and δ ln,k . For simplicity let us define I m,n,k Total = I m,n,k IntraICI + I m,n,k InterICI + I m,n,k PartialICI + ˜ σ n,k as the total interference plus y nk,m = ρ nm,k s m,k √ p m,k g k,nm,k ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Desired Signal for UE m + U k ∑ m ′ = ,m ′ ≠ m ρ nm ′ ,k s m ′ ,k √ p m ′ ,k g k,nm ′ ,k ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Intra-cell ICI + K ∑ k ′ = ,k ′ ≠ k U k ′ ∑ m ′ = ρ nm ′ ,k ′ s m ′ ,k ′ √ p m ′ ,k ′ g k,nm ′ ,k ′ ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Inter-cell ICI (1) + K ∑ k ′ = U k ′ ∑ m ′ = ((√ δ ln + ,k ′ + √ δ rn,k ′ ) g k,n + m ′ ,k ′ ρ n + m ′ ,k ′ + (√ δ ln,k ′ + √ δ rn − ,k ′ ) g k,n − m ′ ,k ′ ρ n − m ′ ,k ′ ) s m ′ ,k ′ √ p m ′ ,k ′ ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Partial ICI + √ + δ ln,k + δ rn,k w nk , noise power. The total SR of network in bps/Hz is given bySR = N ∑ n = K ∑ k = U k ∑ m = ρ nm,k R nm,k . It is inferred that by appropriatelychoosing δ rn,k and δ ln,k for ∀ n ∈ N , ∀ k ∈ K , the SR will beincreased as well. To the best of our knowledge, the joint opti-mization of SR and sum power (SP) in terms of finding optimalvalue of δ , p , and ρ have not been investigated before wherewe will discuss it in the next subsequent sections. Note that ρ , p and δ are the vector representation of the variables p k,m for ∀( m, k ) ∈ U k × K and ρ nm,k for ∀( m, k, n ) ∈ U k × K × N and δ rn,k and δ ln,k for ∀( n, k ) ∈ N \ N × K , respectively.Our aim is to optimize the total energy efficiency (EE)which is EE = SRSP + CP where CP = K ∑ k = U k ∑ m = p circuit m,k denotes thetotal circuit power consumption with p circuit m,k for UE m in cell k and SP = K ∑ k = U k ∑ m = p m,k is the total transmitted data power.It can be easily proved that a problem with the objective ofmaximizing EE (which is a ratio of total rate to the powerconsumption) is equivalent to a multi-objective optimizationwhich the objectives are maximizing total rate (the nominatorof the EE) and minimizing total power consumption (thedenominator of the EE) [10]. Therefore, invoking this property,we formulate a multi-objective optimization problem in thenext section. C. Multi-objective Problem Formulation
In this section, we formulate an optimization frameworkwhere the objective is to jointly maximize the SR and min-imize the total transmitted data power, i.e. SP. The problemformulation is described as follows P1 ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ max ρ , p , δ SR = N ∑ n = K ∑ k = U k ∑ m = ρ nm,k R nm,k min ρ , p , δ SP = K ∑ k = U k ∑ m = p m,k (6a) s.t. C1: N ∑ n ′ = ρ n ′ m,k R n ′ m,k ≥ ρ nm,k R QoS m , ∀( m, k, n ) ∈ U k × K × N , C2: ≤ δ ln,k ≤ , ∀ n ∈ N , n ≠ , ∀ k ∈ K , (6b)C3: ≤ δ rn,k ≤ , ∀ n ∈ N , n ≠ N, ∀ k ∈ K , (6c)C4: U k ∑ m = ρ nm,k ≤ L n,k , ∀ n ∈ N , ∀ k ∈ K , (6d)C5: N ∑ n = ρ nm,k = , ∀ m ∈ U k , ∀ k ∈ K , (6e) C6: p m,k ≤ P max m , ∀ m ∈ U k , ∀ k ∈ K , (6f)C7: ρ nm,k ∈ { , } , ∀ m ∈ U k , ∀ k ∈ K , ∀ n ∈ N . (6g)where C1 denotes the QoS constraint guaranteeing minimumrate of each user, C2 and C3 are the amount of allowed inter-cluster partial ICI overlapping percentage. The constraint C4shows that the total number of users in subband n at AP k is L n,k and C5 states that each UE must be allocated to only onesubband, C6 is the maximum transmission power constraintfor UE m . Problem P1 is a type of mixed integer nonlinearand non-convex optimization which is intractable to solve. Inorder to find Pareto-optimal solutions for P1 , we inspire theTchebyshev approach [11], [12] which is investigated in thenext section comprehensively.III. P ROBLEM T RANSFORMATION AND S OLUTION
A. Problem Transformation
Inspired from weighted max-min formulation for multi-objective optimizations, to convert P1 into a single objectiveoptimization problem we employ Tchebyshev method as itprovides complete Pareto-optimal solutions [12]. Henceforth,by applying this method, the multi-objective optimizationproblem P1 is be transformed as follows: P2 min ρ , p , δ ,λ λ (7) s.t. ˜ C1: ω ( U ∗ − U ( ρ , p , δ )) ≤ λ, ˜ C2: ω ( U ( ρ , p , δ ) − U ∗ ) ≤ λ, C1–C7 . where λ is an auxiliary parameter, ω i for i ∈ { , } are the non-negative weights generally set by a decision maker where ω + ω = and U ( ρ , p , δ ) = SR, U ( ρ , p , δ ) = SP. Furthermore, U ∗ i is the utopia point [12], for i ∈ { , } obtained by solvingthe single-objective problems. The problem P2 is still non-convex due to binary constraint C7 and non-convex constraintC1. To resolve it, first, we relax the binary constraint C7 then,add a regulation term; next, we combine p m,k and ρ nm,k andintroduce new constraint ˜ p nm,k ≤ ρ nm,k P max m to be replaced withC6, where it is equal to zero when ρ nm,k = . Therefore, theproblem formulation will be P3 max ρ , ˜ p , δ ,λ − λ + N ∑ n = K ∑ k = U k ∑ m = α ( ρ nm,k − ρ nm,k )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Regulation term (8) s.t. ˜ C1, ˜ C2, C1–C5 , ˜ C6: ˜ p nm,k ≤ ρ nm,k P max m , ∀ k ∈ K , ∀ m ∈ U k , ∀ n ∈ N , ˜ C7: ρ nm,k ∈ [ , ] , ∀ k ∈ K , ∀ m ∈ U k , ∀ n ∈ N . where a regulation term is added to the objective functionwith parameter α ≫ that forces the relaxed variables ρ nm,k to be approximately close to zero or one. In other words, theparameter α controls the importance of the regulation termpenalty in the objective function, however its value during thesimulations shall be chosen properly, since for high amount of α the λ will be ignored. Note that ˜ p is the vector representationof the new variables ˜ p nk,m for ∀( m, k, n ) ∈ U k × K × N . B. Solution Approach (Monotonic Optimization)
In this section, our aim is to convert the problem P3 intoa monotonic optimization framework [13]–[15]. By taking alook at the problem P3 , it is observed that the objective andthe constraints are not strictly increasing, however they can bewritten in terms of difference of increasing functions (DIF).Because the the objective and the constraints in P3 do notimplicitly indicate monotonicity, our aim is to reformulate P3 to explore some hidden monotonicity. To do so, we first definethe following functions q + ( ρ ) = N ∑ n = K ∑ k = U k ∑ m = α ( ρ nm,k ) , q − ( ρ , λ ) = N ∑ n = K ∑ k = U k ∑ m = αρ nm,k + λ,q + ( ˜ p , δ , λ ) = ω W N ∑ n = K ∑ k = U k ∑ m = B n log ( ˜ p nm,k ∣ g k,nm,k ∣ + I m,n,k Total ) + λ,q − ( ˜ p , δ ) = ω W N ∑ n = K ∑ k = U k ∑ m = B n log ( I m,n,k Total ) + ω U ∗ ,q + ( ˜ p , δ ) = min m,n,k ⎧⎪⎪⎨⎪⎪⎩ B n W log ( ˜ p nm,k ∣ g k,nm,k ∣ + I m,n,k Total )+ N ∑ n ′ = ,n ′ ≠ n K ∑ k ′ = ,k ′ ≠ k U k ′ ∑ m ′ = ,m ′ ≠ m ( B n ′ W log ( I m ′ ,n ′ ,k ′ Total ) + ρ n ′ m ′ ,k ′ R QoS m ′ )⎫⎪⎪⎬⎪⎪⎭ ,q − ( ˜ p , δ ) = N ∑ n ′ = K ∑ k ′ = U k ∑ m ′ = ( B n ′ W log ( I m ′ ,n ′ ,k ′ Total ) + ρ n ′ m ′ ,k ′ R QoS m ′ ) ,q + ( λ ) = λ + ω U ∗ , q − ( ˜ p ) = ω K ∑ k = U k ∑ m = ˜ p nm,k q + ( ˜ p , ρ ) = min k,m,n { ρ nm,k P max m + K ∑ k ′ = ,k ′ ≠ k U k ∑ m = ,m ′ ≠ n ˜ p nm,k } ,q − ( ˜ p ) = K ∑ k = U k ∑ m = ˜ p nm,k , where we observe that q ± ( . ) , q ± ( . ) , q ± ( . ) , q ± ( . ) , q ± ( . ) are in-creasing functions. Now, the constraints can be written in theform of difference of increasing functions. To do so, we define ˜ p max = P max m , ρ max = for ∀ k ∈ K , m ∈ U k and ∀ n ∈ N and δ max = , ∀ k ∈ K , n ∈ N ; moreover we set λ max = Λ . Now theproblem P4 is an optimization problem where DIFs are usedin the objective as well as constraints. Here, we convert theproblem into a monotonic optimization framework. In orderto proceed, we define the auxiliary variables t, w, l, u, v then,the problem P4 will be transformed into P5 max ρ , ˜ p , δ ,λ,t,w,l,u,v q + ( ρ ) + t TABLE I: Simulation parameters.
Parameter Value Parameter Value K = . +
38 log ( d ) d ≥ [m] N L R QoS m , ∀ m U k σ − dBm/Hz Shadowing variance 4 dB P max
200 mW Receiver noise figure (NF) 3 dB s.t.
N1: ≤ t + q − ( ρ , λ ) ≤ q − ( ρ max , Λ ) , N2: ≤ t ≤ q − ( ρ max , Λ ) − q − ( , ) , ˜ C1: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Co-N1: q + ( ˜ p , δ , λ ) + w ≥ q − ( ˜ p max , δ max ) , N3: ≤ w ≤ q − ( ˜ p max , δ max ) − q − ( , ) , N4: ≤ w + q − ( ˜ p , δ ) ≤ q − ( ˜ p max , δ max ) , C1: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Co-N2: q + ( ˜ p , δ ) + l ≥ q − ( ˜ p max , δ max ) , N5: ≤ l ≤ q − ( ˜ p max , δ max ) − q − ( , ) , N6: ≤ l + q − ( ˜ p , δ ) ≤ q − ( ˜ p max , δ max ) , ˜ C2: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Co-N3: q + ( λ ) + u ≥ q − ( ˜ p max ) , N7: ≤ u ≤ q − ( ˜ p max ) − q − ( ) , N8: ≤ u + q − ( ˜ p ) ≤ q − ( ˜ p max ) , ˜ C6: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Co-N4: q + ( ˜ p , ρ ) + v ≥ , N9: ≤ v ≤ q − ( ˜ p max ) − q − ( ) , N10: ≤ v + q − ( ˜ p ) ≤ q − ( ˜ p max ) , C4: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Co-N5: N ∑ n = ρ nm,k ≥ , ∀ m ∈ U k , ∀ k ∈ K , N12: N ∑ n = ρ nm,k ≤ , ∀ m ∈ U k , ∀ k ∈ K , C2–C4, ˜ C7where N1–N12 and C2–C4 and ˜ C7 are the constraints thatbuild normal sets and Co-N1–Co-N5 indicate the constraintsconstructing co-normal sets [13]–[15].IV. N
UMERICAL R ESULTS
In what follows, we evaluate the proposed optimizationsby using the transformed monotonic problem, P5 . Table. Ishows the considered chosen values for the parameters of thenetwork. To avoid numerical issues during simulations andsome simplifications, we assume that δ rn,k = δ ln + ,k , meaningthat the percentage of overlapping between left-hand-side ofchannel n + is the same as right-hand-side overlapping atchannel n . In this way, linear expressions w.r.t. δ rn,k and δ ln,k will be achieved in the interference terms defined in (3)–(5) which yields (√ δ rn,k + √ δ ln + ,k ) = δ rn,k . Poisson pointprocess (PPP) is leveraged to generate the users’ location in400 ×
400 m area where one BS is located at ( , ) m andthe second BS is placed at ( , ) m. To have a benchmark,we consider three scenarios, optimizing δ rn,k and δ ln,k for ∀ n ∈ N , ∀ k ∈ K where we name it as POD, the next scenario isoptimizing δ rn,k = δ and δ ln,k = δ for ∀ n ∈ N , ∀ k ∈ K where it isnamed as NPOD and the scenario where δ rn,k = δ ln,k = whichis NOMA-OFDM. The SE = SR W curves for different multipleaccess methods are shown in Fig. 2. Besides, the sum powervalues are shown for different schemes. We can see that the S E [ bp s / H z ] SP [ m W ] D-OMA, PODOFDMANOMA-OFDMD-OMA, NPOD
Fig. 2: Total SE and SP of the network for different multipleaccess schemes.
30 35 40 45 50 55 60
SE [bps/Hz] EE [ b it s / J ou l e / H z ] D-OMA, NPODD-OMA, PODOFDMANOMA-OFDM
Fig. 3: Total EE versus SE.power consumption for different multiple access methods isthe same except for OFDMA. Furthermore, the importanceof the sum power objective reduces in case of increasing ω , and hence, the power consumption increases as well.Moreover, when we optimize the percentage of overlappingamong channels individually denoted as POD scheme, higheramount of rate is achieved compared with the traditionalNOMA-OFDM case. Furthermore, we observe that the SE isan increasing function with respect to ω .It is observed that the D-OMA POD scheme outperformsother multiple access methods in Fig. 3, where the EE isplotted as a function of SE. Also, it is inferred that the D-OMAPOD scheme has higher SE for a fixed EE value, as a result itoutperforms other scenarios. The major contribution of usingthe D-OMA POD scheme is that a higher amount of rate can beachieved without expanding the current available bandwidth.Since the D-OMA POD scheme achieves higher SE withoutincreasing bandwidth, therefore, it would be interesting forTelecommunication operators. Because an operator leveragesa small amount of bandwidth in the frequency spectrum towork with and service their users. V. C ONCLUSION
In this study, we examined the performance of a D-OMAenabled network in the UL by proposing a multi-objectiveoptimization problem. The mathematical received signal in theuplink of a multi-AP network is identified and the achievablerate of a UE is extracted in terms of the ICI and partialICI signals depending on overlapping percentage betweensub-bands. To solve the proposed optimization problem, thebinary decision variables are relaxed and regulation termsare added to the objective function. Next, a method namedas Tchebyshev is leveraged to transform the multi-objectiveoptimization to a single-objective problem with the same con-straints. Then, the problem is reformulated into a monotonicoptimization framework by exploring the hidden monotonicityof the objective and constraints. The numerical results showthat the novel D-OMA method outperforms other traditionalmultiple access methods such as OFDMA, NOMA-OFDM,and OMA. R
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