Energy Efficiency Tradeoff in Interference Channels - IEEE Xplore

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Energy Efficiency Tradeoff in Interference Channels Guanding Yu, Lukai Xu, Daquan Feng, Zhaoyang Zhang, Geoffrey Ye Li, and Huazi Zhang

Abstract—Interference is one of the major obstacles to improving the performance in wireless communication systems. As the ever-growing data traffic is carried over extremely dense networks, how to deal with interference becomes even more relevant. In this paper, we investigate a network with N pairs of users transmitting on the same channel simultaneously from the energy efficiency (EE) perspective. For such an interference network, we aim to address two issues: what is the EE tradeoff between users and how to design energy-efficient resource allocation scheme? To answer these two questions, we formulate a non-concave multi-objective optimization problem (MOOP) to investigate the EE tradeoff, taking into account the minimum data rate requirement of each user. The weighted Tchebycheff method is utilized to solve the MOOP by converting it into a single-objective optimization problem, which is then solved by the Dinkelbach method and the concave-convex procedure (CCCP) method. Based on the above, a power control algorithm is developed for the interference network to achieve at least a local optimum. The proposed algorithm is compared with the orthogonal bandwidth sharing, where each user orthogonally shares the whole bandwidth without interfering each other. In this scenario, the weighted Tchebycheff and the Dinkelbach methods are also utilized to develop the optimal bandwidth allocation and power control algorithm. The performance of the proposed algorithms is verified by numerical results, which show that it is better to share the bandwidth orthogonally rather than nonorthogonally if the interference between each user pair is stronger than a given threshold. Index Terms—Interference channel, energy efficiency, bandwidth allocation and power control, multi-objective optimization

I. I NTRODUCTION The ever growing demand from mobile applications is driving the current infrastructure towards extremely dense networks where interference is becoming a major obstacle to further improving the performance of wireless communications. As a result, how to deal with interference has become one of the most important topics in recent years. Since late 1970s, the interference channel has been investigated from the perspective of information theory and lots of results on its capacity region have been derived [1]–[4]. It has been revealed that when interference is very strong, Manuscript received July 10, 2016; accepted August 7, 2016. This work was supported in part by the Zhejiang Provincial Natural Science Foundation under Grant LY14F010011, by the National Hi-Tech R&D Program of China under Grant 2014AA01A702, by the National Natural Science Foundation of China under Grant 61401388, and by the Fundamental Research Funds for the Central Universities under Grant 2016QNA5004. G. Yu, L. Xu, Z. Zhang, and H. Zhang are with the Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). D. Feng is with the College of Information Engineering, Shenzhen University, Shenzhen, China, 518060. E-mail: [email protected]. G. Y. Li is with the School of ECE, Georgia Institute of Technology, Atlanta, GA, USA. e-mail: [email protected].

the capacity of each user is as good as the interference-free scenario due to the fact that the interference signal is strong enough to be decoded and then cancelled [2]. For the general interference channel, the capacity region is still unknown and the best known achievable rate region so far is the HanKobayashi region [3] which, however, is extremely difficult to calculate. Moreover, a very complicated channel coding mechanism is required to achieve it. Therefore, though not optimal, it is still a common practice to treat interference as Gaussian noise in current wireless communication systems. In practice, dynamic radio resource management techniques have been utilized to mitigate inter-cell interference in cellular networks, such as dynamic power control [5], interference alignment [6], enhanced inter-cell interference cancellation [7], [8], etc. In ad hoc networks, e.g., wireless local area networks, wireless sensor networks, various distributed media access control mechanisms have been developed to manage inter-node interference. These techniques were generally designed with the major consideration of capacity enhancement. However, as wireless data traffic is expected to increase exponentially in the coming decade due to the proliferation of smart devices and mobile applications, the carbon footprint of wireless networks has become a critical problem both environmentally and economically. As a result, more stringent energy efficiency (EE) requirements are proposed for future green wireless communication systems [9], [10]. To this end, many green communication techniques have been developed recently. From the system point of view, adaptive base station sleep control [11] and shrinking the cell size [12], [13] are two effective ways to reduce the energy consumption of base stations. Besides, radio frequency (RF) power consumption can be further reduced by energy-efficient radio resource allocation, which has been widely investigated in various wireless systems, such as multiple-input multipleoutput (MIMO) [14]–[16], orthogonal frequency division multiple access (OFDMA) [17], [18], heterogeneous networks [13], [19]–[21], carrier aggregation [22], [23], etc. The interference issue has also been taken into account in energy-efficient design. In [24], a game-theory based distributed power control algorithm has been developed in OFDMA systems. In [18], a centralized energy-efficient resource allocation algorithm has been developed for multi-cell OFDMA systems, considering the multiple access interference. An energy-efficient opportunistic interference alignment has been proposed in [25], where the overall energy consumption is minimized while maintaining the minimum communication requirement. In [26], energy-efficient beamforming has been investigated for downlink multi-cell multiuser MIMO systems to maximize the weighted-sum EE of different cells. Although interference has been considered in these aforementioned

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works, they generally aim at maximizing the system-level EE, such as the overall system EE or the weighted-sum of per-cell EE. We consider an interference network with N pairs of users communicating simultaneously on the same channel. In such a system, we will address the following questions: what is the EE tradeoff between users1 and how to design energy-efficient resource allocation algorithm? In our previous works [19], [23], the EE tradeoffs in a time division duplex system and in a multi-RAT heterogeneous networks have been investigated, respectively. However, the interference issue has not been addressed in either works. The main contributions of this paper are summarized as follows. • Different from the existing works which focus on the overall system EE, we aim at achieving the EE tradeoff among different users in an N-user interference channel taking into account the minimum data rate requirement of each individual user. To accomplish this, a nonconvex multi-objective optimization problem (MOOP) is formulated. • To tackle this challenging non-convex MOOP, several novel mathematical tools have been introduced. Specifically, we first utilize the weighted Tchebycheff method to convert the problem into a single-objective optimization and then use the Dinkelbach method and the concaveconvex procedure (CCCP) to effectively solve it. Based on this, a power control algorithm is developed to achieve the EE tradeoff. The convergence of the algorithm is proved and its performance is verified by numerical simulation. • As a benchmark, we consider the scenario where each user orthogonally shares the whole bandwidth in a frequency division multiple access (FDMA) fashion without interfering each other. In this scenario, the weighted Tchebycheff method and the Dinkelbach method are also utilized to determine the optimal bandwidth allocation and power control scheme. Numerical results demonstrate that orthogonal resource sharing achieves a better EE when interference between each user pair is stronger than a given threshold. The rest of this paper is organized as follows. In Section II, we will introduce the system model and formulate the multiobjective optimization problem. In Section III, the algorithm to the EE optimization problem will be presented. The benchmark orthogonal resource sharing scenario will be discussed in Section IV. The simulation results are presented in Section V and the whole paper is concluded in Section VI. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this section, we will first introduce the system model and then formulate the EE optimization problem. A. System Model As shown in Fig. 1, we consider a network with N pairs of users communicating with each other, denoted 1 In

this paper, “user” is also used to represent “user pair” for brevity.

as N = {1, 2, · · · , N }. Let {Tx1 , Tx2 , · · · , TxN } and {Rx1 , Rx2 , · · · , RxN } denote the transmitter and the receiver, respectively, and W denote the overall bandwidth of the considered system. In this paper, we consider two scenarios for users to share the spectrum resource. •

•

Non-orthogonal sharing: all users may transmit on the whole bandwidth simultaneously with interference to each other. This is the situation of interference network, as shown in Fig. 1(a). Orthogonal sharing: each user orthogonally uses the spectrum resource, the bandwidth occupied by Txn is ∑i.e., N βn W where n=1 βn ≤ 1, as shown in Fig. 1(b). This scenario corresponds to the FDMA, with no interference among users.

The channel power gain between transmitters and receivers can be written as h = GΓd−τ , where G is the path loss constant, τ is the path loss exponent, Γ denotes the channel fading component, and d is the distance between the transmitter and the receiver. In the non-orthogonal sharing scenario, we denote hm,n as the channel power gain between Txm and Rxn , and in the orthogonal sharing scenario, we denote hn as the channel power gain between Txn and Rxn . We further assume that all channel state information (CSI) is known at a central controller, such as a base station in cellular systems or an access point in wireless LANs, where the resource allocation is performed. Each receiver first estimates the channel gain from its own transmitter, as well as the interference channel gain from other transmitters (in the non-orthogonal sharing scenario). Then, this information is gathered at the central controller, which is also responsible for sending the resource allocation results to each user after the algorithm is executed. In general, only a few users will share the same frequency channel in practical wireless systems. Therefore, the total amount of CSI feedback (N 2 ) would not be large. Even if full CSI is not able to obtain, our algorithm can still serve as a benchmark for other centralized algorithms with partial or imperfect CSI, as well as distributed algorithms without a central controller.

B. Problem Formulation As indicated before, the capacity region of interference channel is still an open challenge and the best known achievable region is the Han-Kobayashi region, which is very difficult to achieve. Therefore, in this paper, we treat interference as noise in the non-orthogonal sharing scenario. Although treating interference as noise is not optimal, it is the most common method in practical wireless systems. Assuming Gaussian signaling being used, the data rate of user pair n in the non-orthogonal sharing scenario in Fig. 1(a)



5[

7[

7[ h

hN

h

< <
ηi∗ . It is difficult to find the Pareto optimal EE directly from the optimization problem. Generally, a MOOP can only be solved by scalarizing the multiple objectives into a single one. There are various scalarization methods. Among them, the weighted Tchebycheff method can provide the complete Pareto optimal solutions with a low computational complexity and is the most effective one even if the objectives are non-convex. Therefore, we adopt it as the scalarization method in this paper. Before solving the MOOPs in (5) and (6), we need to introduce the concept of Utopia EE for each individual user, which is an essential element for the weighted Tchebycheff method. Definition 2 [27]: The Utopia EE for user n, ηu,n , is defined as the maximal EE this user could achieve, i.e., ηu,n = max ηn (p, β). (p,β)∈Θ



TABLE I: CCCP for the d.c. problem From the definition, the Utopia EE for a particular user is actually the maximum EE among all its Pareto optimal EEs. The optimization problems to find the Utopia EE in both nonorthogonal and orthogonal sharing scenarios can be formulated as NS ηu,n = max ηnNS (p), ∀n,

(7)

p∈P

Algorithm 1 CCCP for the d.c. problem 1: 2: 3: 4:

subject to the constraints in (5a) and (5b), and OS ηu,n = max ηnOS (p, β), ∀n,

Initialize the maximum tolerance ϵ. Set the iteration index k = 0 and the initial power vector ∀p(0) ∈ P. Do Update the power control by { ( )} p(k+1)=arg max fcave1 (p)−pT ∗∇fcave2 p(k) . p∈P

(8)

(p,β)∈Θ

5:

subject to the constraints in (6a), (6b), and (6c), respectively. Here, the bandwidth allocation in (7) is not needed since each user can use all bandwidth in the non-orthogonal sharing case. However, the bandwidth allocation should be considered in the orthogonal sharing case in (8). We shall also note that there is an Utopia EE for each user, which can be obtained by solving the problem in (7) or (8) independently. In other words, problem (7) or (8) should be solved N times with different objective functions but the same constraints to achieve the N Utopia EEs. Moreover, different from the problems in (5) and (6), the problems in (7) and (8) are single-objective optimization problems. In the next two sections, we will develop effective algorithms to find the Utopia EE and the Pareto optimal EE in the two scenarios, respectively.

6:

Update k = k + 1.

Until p(k) − p(k−1) < ϵ.

and

( fcave2 (p) = W log2

N ∑

) pm hm,n +

2 W σN

m=1,m̸=n

+ φ (pn + pe ) . Since both fcave1 (p) and fcave2 (p) are strictly concave functions on p and P is a convex set, the problem in (9) can be rewritten as the following d.c. structure max {fcave1 (p) − fcave2 (p)} .

(11)

p∈P

III. N ON - ORTHOGONAL S PECTRUM S HARING In this section, we consider the scenario that all users share the whole bandwidth, interfering with each other. We will develop a power control algorithm to achieve the Pareto optimal EE for the problem in (5) using the weighted Tchebycheff method. A. Solution of the Utopia EE First, we try to find a solution for the Utopia EE. We can see that the problem in (7) is a nonlinear fractional programming and the Dinkelbach algorithm [28] can be used to solve it, as shown in the following theorem, whose proof can be found in [28]. Theorem 1: Define { } (9) Un (φ) = max RnNS (p) − φPn (p) , ∀n, p∈P

( NS ) NS then ηu,n is achieved if and only if Un ηu,n = 0. Unfortunately, it is still difficult to obtain the optimum value since the object function in (9) is non-concave due to the interference in the denominator of RnNS (p). However, this problem has a difference-of-convex (d.c.) structure, which can be solved by the method elaborated as follows. We can rewrite the objective function in (9) as ∆ f (p) = RnNS (p) − φPn (p)= fcave1 (p) − fcave2 (p), (10)

where

( fcave1 (p) = W log2

N ∑ m=1

) pm hm,n +

2 W σN

,

A d.c. problem can be effectively solved by the primal-dual subdifferential method, namely, the d.c. algorithm (DCA) [29]. Moreover, since fcave2 (p) is differentiable, (11) can be further reduced to CCCP through the minorization-maximization method [30]. Based on the above analysis, we introduce the following theorem to solve the above problem, as proved in Appendix A. Theorem 2: The problem in (11) can be iteratively solved by the following concave programming { ( )} p(k+1) = arg max fcave1 (p) − pT ∗ ∇fcave2 p(k) , (12) p∈P

( ) ∆ where pT denotes the ]transpose of p and ∇fcave2 p(k) = [ (k) (k) (k) ∇1 , ∇2 , · · · , ∇N denotes the gradient of fcave2 (p) at p(k) , where (k) ∇i

=

  

(

W hi,n N ∑

m=1,m̸=n   φ,

(k) 2 pm hm,n +W σN

, ∀i ̸= n,

) ln 2

i = n.

We could see that the first part in (12), ( fcave1 ) (p), is concave, and the second part, −pT ∗ ∇fcave2 p(k) , is linear. Therefore, (12) is a standard concave optimization problem and thus can be efficiently solved by classic convex optimization methods. The algorithm of CCCP is summarized in Table I. We now discuss the convergence of the CCCP algorithm. As we have proved in (28) in Appendix A, the objective function in{ (12)}is monotonically increasing on the generated sequence p(k) unless p(k+1) = p(k) . Therefore, from [30], { } (k) p will converge to its stationary point p(∞) , which



TABLE II: The algorithm to find the Utopia EE for user n in the non-orthogonal scenario.

K

16 X

K16

Algorithm 2 The algorithm to find the Utopia EE for user n in the non-orthogonal scenario. 1: 2: 3: 4: 5:

K

16 X

K

16

Initialize the maximum tolerance ϵ. Set the iteration index j = 0 and initialize ∀p(0) ∈ P. Do ( ) Set φj = ηnNS p(j) . Calculate the optimal { power control } p(j+1)=arg max RnNS (p)−φj Pn (p)

K16

p∈P

6: 7:

by the CCCP algorithm in Table I. Update j = j + 1. ( ) ( ) Until RnNS p(j) − φj−1 Pn p(j) < ϵ.

K16

Fig. 2: The illustration of the Utopia EE and the Pareto optimal EE. also satisfies the Karush-Kuhn-Tucker conditions of problem (11). This indicates that by jointly applying the Dinkelbach method and the CCCP, at least a local optimal solution to the problem in (7) can be found. The detailed procedures of the proposed algorithm to find the Utopia EE for user n in the non-orthogonal sharing scenario are summarized in Table II. B. Solution of the Pareto Optimal EE NS After the Utopia EE for each user, ηu,n , ∀n, is obtained, we can scalarize the multi-objective optimization problem in (5) into a single-objective optimization problem using the weighted Tchebycheff method, which can be formulated as { ( NS )} NS ηwt = min max ϕn ηu,n − ηnNS (p) , (13) p∈P n∈N

where ϕ = (ϕ1 , ϕ2 , · · · , ϕN ) is the weight vector. From [27], all Pareto optimal EEs in problem (5) can be found by the weighted Tchebycheff method by changing the weight vector ϕ in problem (13). Therefore, to find the Pareto optimal solution to problem (5) is equivalent to solve the single-objective optimization problem (13) for a given ϕ. Moreover, the weighted Tchebycheff method can guarantee the complete Pareto optimal solutions, according to [27]. Fig. 2 illustrates an example in a two-user case. In the figure, the shadowing area shows the feasible EE region for the problem in (5), whose boundary (the solid curve) is the Pareto NS NS are the Utopia EE for users. As and ηu,2 optimal EE. ηu,1 illustrated in the figure, the Pareto optimal EE on the boundary can be achieved by different ϕn , n = 1, 2. The weight, ϕn , can be interpreted as the level of importance for each user. A larger weight can be assigned to the user that expects for a better EE, such as the user with lower battery power. In this case, its EE will be much closer to the Utopia EE, as illustrated in the figure. Now, we will propose a solution to solve the problem in NS (13) for the given ϕn and ηu,n , which can be rewritten as { ( )} NS ϕn ηu,n Pn (p) − RnNS (p) NS ηwt = min max . (14) p∈P n∈N Pn (p)

Similar to the method solving the Utopia EE, the above problem is a fractional optimization and can be solved by the Dinkelbach method [28]. Similar to Theorem 1, we have the following theorem, which is proved in [31]. Theorem 3: Define { ( NS ) } V (α) = min max ϕn ηu,n Pn (p)−RnNS (p) −αPn (p) , (15) p∈P n∈N

( NS ) NS then ηwt is achieved if and only if V ηwt = 0. Similar to the problem in (9), (15) is nonconvex, however, it can be solved by the Lagrange method { and( the CCCP method jointly. Define ) } NS l = max ϕn ηu,n Pn (p) − RnNS (p) − αPn (p) , then n∈N from the parametric algorithm, the problem in (15) is equivalent to the following problem min l,

(16)

p∈P,l

subject to ( ) NS l ≥ ϕn ηu,n Pn (p) − RnNS (p) − αPn (p), ∀n.

(16a)

The Lagrangian function for the above problem can be written as L (p, l, λ, µ, ν) N { ( NS ) } ∑ =l+ λn ϕn ηu,n Pn (p) − RnNS (p) −αPn (p)−l + +

N ∑

n=1

µn (pn − Pmax ) { ( ( R ) min νn 2 W −1

n=1 N ∑

n=1

N ∑

2 pm hm,n +W σN m=1,m̸=n

)

} −pn hn,n ,

where λ, µ, ν are Lagrange multiplier vectors. Further, the Lagrangian dual problem will be max

min L (p, l, λ, µ, ν) .

λ≽0,µ≽0,ν≽0 p≽0,l

(17)

Then, the above Lagrangian dual problem can be solved by the following iterative algorithms.



1) Inner Loop: The inner loop is to solve min L (p, l) for a p≽0,l

given (λ, µ, ν), which is a non-convex optimization problem and difficult to solve. However, we can decompose L (p, l) into L (p, l) = Lvex1 (p, l) − Lvex2 (p) ,

(18)

where Lvex1 (p, l) { ( N )} N ∑ ∑ 2 =l− λn ϕn W log2 pm hm,n + W σN n=1

m=1

N { } ∑ NS Pn (p)−αPn (p)−l + µn (pn −Pmax ) + λn ϕn ηu,n n=1 n=1 { ) } ( ( R ) N N ∑ ∑ min 2 + νn 2 W −1 −pn hn,n , pm hm,n +W σN N ∑

n=1

) ) (∑ ( R min N 2 p h + W σ , ς1 , where ϑ = 2 W − 1 m m,n N m=1,m̸=n ς2 , and ς3 are the positive step sizes. From the above analysis, the problem in (17) has been solved by jointly performing the CCCP method and the Lagrangian dual method. Accordingly, the weighted Tchebycheff optimization problem in (13) can be solved by a proposed three-step algorithm as follows. The first step is the Dinkelbach method, which updates α in (15) until V (α) = 0, the second step is updating the Lagrangian multiplier according to (21), and the third step is the CCCP method in (20). We should note that the three steps are nested. The details of the proposed algorithm are presented in Table III. TABLE III: The algorithm to find the Pareto optimal EE in the non-orthogonal scenario.

m=1,m̸=n

and

  N  ∑ Lvex2 (p) = − λn ϕn W log2  n=1

 N  ∑ 2  pm hm,n +W σN . 

m=1,m̸=n

Therefore, min L (p, l) is the minimization of a d.c. function p≽0,l

p∈P,l

1: 2: 3: 4:

under a convex constraint set, as min {Lvex1 (p, l) − Lvex2 (p)} .

Algorithm 3 The algorithm to find the Pareto optimal EE in the non-orthogonal scenario.

(19)

5: 6:

Similarly, we can utilize the CCCP method to solve the above problem, as shown in the following theorem, which can be proved in a similar way as Theorem 2. Theorem 4: The problem in (19) can be iteratively solved by the following sequential convex programming { ( )} p(k+1)=arg min Lvex1 (p, l) − pT ∗ ∇Lvex2 p(k) , (20) p∈P,l

] ( ) ∆ [ (k) (k) (k) denotes the where Lvex2 p(k) = ∇1 , ∇2 , · · · , ∇N gradient of Lvex2 (p) at p(k) , where           N   ∑ hi,n (k) ) ∇i =− λn ϕn W ( .   N ∑   (k) n=1,n̸=i   2  pm hm,n + W σN ln 2    m=1,m̸=n

We could see that the first part of (20), Lvex1((p, l), ) is convex, and the second part of (20), −pT ∗ ∇Lvex2 p(k) , is linear. Therefore, (20) is a convex optimization problem and thus can be efficiently solved by classic convex optimization methods. 2) Outer Loop: Since the Lagrangian dual function is differentiable, the subgradient method can be used to find (λ, µ, ν) for given (p, l). The subgradient update equations are given by λn = [λn + { ( NS ) }]+ ς1 ϕn ηu,n , ∀n, Pn (p) − RnNS (p) − αPn (p) − l + µn = [µn + ς2 (pn − Pmax )] , ∀n, + νn = [νn + ς3 (ϑ − pn hn,n )] , ∀n, (21)

7: 8:

Initialize the maximum tolerance ϵ1 , ϵ2 . Set the iteration index k = 0 and initialize ∀p(0) ∈ P. NS Calculate ηu,n , ∀n by Algorithm 2. Do { } NS NS ϕn (ηu,n Pn (p(k) )−Rn (p(k) )) Calculate αk = max . Pn (p(k) ) n∈N Initialize λ, µ, ν, j = 0, l(0) = 0. Do Using the CCCP algorithm to find (p∗ , l∗ ) = arg min {Lvex1 (p, l)−Lvex2 (p)}. p∈P,l

9:

Update λ, µ, ν using (21), j = j + 1, p(j) = p∗ , l =l .

( ) ( ) Until p(j) , l(j) − p(j−1) , l(j−1) < ϵ1 . Update k = k + 1, p(k) = p∗ . Until ( NS ( )) ( ) ( ) ϕn ηu,n Pn p(k) −RnNS p(k) −αk−1 Pn p(k) < ϵ2 . (j)

10: 11: 12:

∗

The following corollary shows that the convergence of the proposed algorithm can be guaranteed, which is proved in Appendix B. Corollary 1: The convergence of Algorithm 3 can be guaranteed. We shall note that although the convergence can be guaranteed, the algorithm only converges to a local optimum. However, according to [29], it often converges to the global optimum for a suitable starting point. We now discuss the computational complexity of the proposed algorithm. The computational complexity of an optimization problem mainly includes two parts. The first part is the required number of iterations until convergence and the second part is the computational complexity in each iteration. Since both Lvex1 (p, l) and Lvex2 (p) are convex piecewiselinear, the CCCP algorithm will converge at least linearly to the stationary point according to [34]. The Dinkelbach method and the subgradient method also converge at least linearly according [31] and [33], respectively. On the other hand, in



each iteration, the optimization problem in (20) is convex and we can use the interior point method to solve it. From [35], the computational complexity of the interior point method is O(N 3 ). Therefore, we conclude that the proposed algorithm has a polynomial time complexity, which is desirable for practical implementation. IV. O RTHOGONAL S PECTRUM S HARING In this section, we will develop a joint bandwidth and power allocation algorithm to achieve the Pareto optimal EE in the orthogonal sharing scenario, using the weighted Tchebycheff method.

TABLE IV: The algorithm to find the Pareto optimal EE in the orthogonal sharing scenario. Algorithm 4 The algorithm to find the Pareto optimal EE in the orthogonal sharing scenario. 1: 2: 3: 4: 5: 6:

A. Solution of the Utopia EE

(p,β)∈Θn∈N

First, to find the Utopia EE in (8), we have to optimize both bandwidth and power allocation. However, for a given power allocation, pn , the EE increases with the bandwidth, βn . Therefore, we can separate the problem into bandwidth allocation and power allocation. For the bandwidth allocation problem, we have the following lemma to achieve the Utopia EE for user n, which is proved in Appendix C. Lemma 1: The optimal bandwidth allocation to achieve the Utopia EE for user n can be written as { zi , ∑ i ̸= n, ∗ βi = zi , i = n, 1− i̸=n

where zi , ∀i ̸= n is the solution of RiOS (Pmax , zi ) = Rmin . Here, we assume that all users can achieve the minimum data rata requirement with the available bandwidth if they transmit at the maximum power, which can be guaranteed by some admission control procedures. According to Lemma 1, the optimal power allocation can be reformulated as the following optimization problem ( ) βn∗ W log2 1 + β ∗pnWhσn2 n N OS = max ηu,n , ∀n, (22) pn pn + pe subject to 0 ≤ pn ≤ Pmax , RnOS

(p, β) ≥ Rmin .

Initialize the maximum tolerance ϵ. Set the iteration index k = 0 and initialize (p(0) , β (0) ) ∈ Θ. OS Calculate ηu,n , ∀n by Lemma 1. Do { } OS OS ϕn (ηu,n Pn (p(k) )−Rn (p(k) ,β(k) )) Calculate θk = max . Pn (p(k) ) n∈N ( ) Find p(k+1){, β (k+1) ( OS ) } = min max ϕn ηu,n Pn (p)−RnOS (p, β) −θk Pn (p) .

(22a) (22b)

Since the objective in (22) is a single variable function, the OS optimal ηu,n can be easily achieved by standard optimization tools, such as Newton’s method. B. Solution of the Pareto Optimal EE After the utopia EE is obtained for each user, we can also utilize the weighted Tchebycheff method to solve the problem in (6), which can be reformulated as )} { ( ROS (p,β) OS OS − nPn (p) ηwt = min max ϕn ηu,n (p,β)∈Θ n∈N { } (23) OS OS ϕn (ηu,n Pn (p)−Rn (p,β)) ∆ = min max . Pn (p) (p,β)∈Θ n∈N

This is a standard convex generalized fractional programming (GFP) since the numerator and the denominator of (23) is

7: 8:

Update k = k + 1. Until ( OS ( (k)) OS ( (k) (k) )) ( ) ϕn ηu,n Pn p −Rn p , β −θk−1 Pn p(k) gth , and vice versa. It is extremely difficult to analytically determine gth since two Pareto optimal regions need to be compared. Even in the case that each user has the same weight, i.e., ϕn = 1/N, ∀n,



TABLE V: Simulation parameters

the following equation should be solved f (g) = f0 , where

(25)

( W /N log2 1 +

f0 = max ηnOS (p) = max {pOS n }

f (g)=max ηnNS (p)=max {pNS n }

)

pOS n + pe

{pOS n }

and

pOS n h 2 W σN /N

( W log2 1 +

pNS n h 2 (N −1)pNS n gh+W σN

, )

pNS n + pe

{pNS n }

.

Since analytically solving the above equation would be involved, we will compare the EE between the orthogonal and the non-orthogonal sharing methods numerically. D. Partial Orthogonal Sharing From the above discussion, whether orthogonal sharing is better than non-orthogonal sharing depends on the strength of mutual interference. Intuitively, to further improve the performance, the partial orthogonal sharing method can be employed [32]. That is, a part of the bandwidth is orthogonally shared while the other part is non-orthogonally shared. Moreover, the non-orthogonal part can be further split into several blocks, each is shared non-orthogonally by a set of users. However, this scenario is extremely complicated. In an N-user system, the overall bandwidth should be split into 2N − 1 blocks (see Fig. 3 for the example of a 3-user system). Furthermore, the transmit power of each user can be optimally allocated to different bandwidth blocks. We can easily find some situations where partial orthogonal sharing is better than both orthogonal and non-orthogonal sharing. However, there is no simple and insightful result in partial orthogonal sharing case. Due to page limit, we leave this issue as our future work.

wholebandwidth

user1

user2

user3

user1,2

user1,3

user2,3

user1,2,3

orthogonallyshared

nonͲorthogonallyshared

Fig. 3: Partial orthogonal sharing in a 3-user case.

V. N UMERICAL R ESULTS In this section, the proposed energy-efficient algorithms will be validated by numerical simulations. We consider a system with N pairs of users where N varies from 2 to 8. The distance between the i-th user pair is denoted as di , which varies from 100 - 300 m. The channel gains between users are with independent and identically distributed (i.i.d) Rayleigh fading. The path loss exponent is set to 4 and a log-normal shadow fading with 8 dB standard deviation is considered. We assume that each user has the same maximum transmit power, Pmax , and the same data rate requirement, Rmin . The fixed circuit power consumption is assumed to be 24 dBm.

Parameter Distance between user pairs Noise spectral density Bandwidth Path loss model Shadowing standard deviation Minimum required data rate, Rmin Number of user pairs, N Maximum transmit power, Pmax Fixed circuit power consumption, pe Interference-to-signal ratio, g

Value 100 - 300 m -174 dBm/Hz 10 MHz 148+40log10(d[km]) 3 dB 1 Mbps 2-8 24 dBm 24 dBm 0.1 - 0.6

In the non-orthogonal sharing (NS) scenario, we assume that for each user the average interference power gain to all other users are identical, i.e., hn,m = ghn,n , ∀m ̸= n, where g is the interference-to-signal ratio and h stands for the average channel power gain. 2 We will compare the performances between the orthogonal sharing (OS) and the NS scenarios with different g. Other simulation parameters are listed in Table V unless otherwise stated. We first present the Utopia EE and the Pareto optimal EE in a four-user case, where each user pair has a distance of 150 m, 150 m, 200 m, and 200 m, respectively. Fig. 4(a) and Fig.4 (b) depict the performance in the OS and the NS scenarios with ϕ = (0.25, 0.25, 0.25, 0.25) and ϕ = (0.1, 0.2, 0.3, 0.4), respectively. From the figures, user 3 and user 4 achieve higher Utopia EE and Pareto optimal EE than the other two due to their shorter communication distances. From Fig. 4(a), the gap between the Utopia EE and the Pareto optimal EE for each user in both NS and OS scenarios is almost the same because of the equivalent weight value. However, in Fig. 4(b), the Pareto optimal EEs of user 3 and user 4 will increase due to their relatively higher weight values, i.e., ϕ3 and ϕ4 , compared with those of other users. On the other hand, the gap between the Utopia EE and the Pareto optimal EE decreases from user 1 to user 4, resulting from the min-max operation in the weighted Tchebycheff method. In Fig. 5, we plot the Utopia EE and the Pareto optimal EE with different weights in the two-user case with nonorthogonal sharing. We fix the weight of user pair 2 as ϕ2 = 1. Fig. 5(a) corresponds to the symmetric user case where each user pair has the same distance of 200 m. In this case, the Utopia EE is the same for both users. However, the Pareto optimal EE for user 1 increases with its weight, ϕ1 . Consequently, the Pareto optimal EE for user 2 will decrease. Both users can achieve an identical Pareto optimal EE when ϕ1 = ϕ2 = 1 and user 1 has a higher Pareto optimal EE than user 2 when ϕ1 > 1. We also plot the gap between the Utopia EE and the Pareto optimal EE for each user. Comparing the two curves, we observe that the ratio between the gaps is almost equal to ϕ1 : ϕ2 , which demonstrates the effectiveness of the proposed weighted Tchebycheff method. In Fig. 5(b), 2 We shall note that the assumption of same transmit power and identical channel gain is only for numerical simulations, and our algorithms can work well with arbitrary transmit power and channel gain.



60

100 90 80

ηNS 2

50

ηNS u,1 ηNS u,2

40

60

EE/(Mb/J)

EE (Mb/J)

70

ηNS 1

Utopia EE OS: Pareto optimal EE OS: gap NS: Pareto optimal EE NS: gap

50 40

gap1 gap2

30

20

30 10

20 10 0

0 0.5

1

2

3

1

1.5

4

2 φ1

2.5

3

3.5

user index

(a) Symmetric user case with d1 = d2 = 200 m. (a) ϕ = (0.25, 0.25, 0.25, 0.25). 160 ηNS 1

100

80

EE (Mb/J)

70

140

EE/(Mb/J)

90

Utopia EE OS: Pareto optimal EE OS: gap NS: Pareto optimal EE NS: gap

60 50

ηNS 2

120

ηNS u,1

100

gap1

ηNS u,2 gap2

80 60

40 40

30 20

20

0 0.5

10 0

1

2

3

4

user index

(b) ϕ = (0.1, 0.2, 0.3, 0.4).

Fig. 4: The EE for different users in both NS and OS scenarios. Rmin = 1 Mbps, N = 4, d1 = d2 = 200 m, d3 = d4 = 150 m.

the results in the asymmetric user case are presented where two user pairs have different distances, i.e., d1 = 100 m, d2 = 200 m. In this case, the Utopia EEs are different for the two users. However, the ratio between the gaps is still approximately equal to ϕ1 : ϕ2 . We have obtained similar results in the orthogonal sharing scenario. However, the figures are not presented due to page limits. In Fig. 6 (a) and Fig. 6(b), we plot the Pareto optimal EE for the two-user case with different communication distances in the OS and the NS scenarios, respectively. In both figures, d1 and d2 denote the communication distances of the first and the second user pair, respectively. For convenience, we neglect the minimum data rate requirement of each user pair by setting Rmin = 0. From the figures, the EE tradeoff between two users can be clearly observed. We compare the EE tradeoff curves in the OS and the NS scenarios in Fig. 7 with different interference-to-signal ratios, g. Fig. 7(a) corresponds to the symmetric user case where two

1

1.5

2 φ1

2.5

3

3.5

(b) Asymmetric user case with d1 = 100 m, d2 = 200 m.

Fig. 5: EE versus ϕ1 in the two-user case with non-orthogonal sharing. N = 2, ϕ2 = 1, Rmin = 1 Mbps, g = 0.1.

users have the same communication distance, d1 = d2 = 200 m, while Fig. 7(b) corresponds to the asymmetric user case where two users have different communication distances, d1 = 100 m, d2 = 200 m. From both figures, the EE decreases with g as interference will deteriorate both the spectral and energy efficiencies. Moreover, when g is small, which means that interference is weak, non-orthogonal sharing has a better performance than orthogonal sharing. However, when the interference is larger than a threshold, orthogonal sharing will be preferred. To further investigate the effect of interference on the EE performance, we compare the EE between the OS and the NS scenarios for different g in Fig. 8. A symmetric user case where each user pair has the same distance of 200 m and the same weight value of ϕi = 1/N, ∀i, is shown in Fig. 8(a). From the figure, the same Pareto optimal EE can be achieved by each user, which decreases with the number of users. Moreover, as interference increases, the EE in the non-orthogonal sharing scenario decreases, which indicates that orthogonal sharing can achieve a better EE than non-



60

160 d1=100 m, d2=100 m d1=100 m, d2=200 m

140

50

d1=200 m, d2=200 m 120

d1=100 m, d2=300 m d1=300 m, d2=300 m

40 η2 (Mb/J)

100 ηOS (Mb/J) 2

OS NS, g=0.1 NS, g=0.3 NS, g=0.5 NS, g=0.7 NS, g=0.9

80

30

60

20 40

10

20 0

0

20

40

60

80

100

120

140

160

0

ηOS (Mb/J) 1

0

(a) Orthogonal sharing scenario.

20

30 η1 (Mb/J)

40

50

60

d1=100 m, d2=100 m

OS NS, g=0.1 NS, g=0.3 NS, g=0.5 NS, g=0.7

d1=100 m, d2=200 m

140

d1=200 m, d2=200 m 120

60

(a) Symmetric user case with d1 = d2 = 200 m.

160

50

d1=100 m, d2=300 m d1=300 m, d2=300 m

40 η2 (Mb/J)

100 ηNS (Mb/J) 2

10

80 60

30

20 40 20 0

10

0

20

40

60

80

100

120

140

160

ηNS (Mb/J) 1

(b) Non-orthogonal sharing scenario. g = 0.1.

Fig. 6: The EE tradeoff in the two-user case with different communication distances. Rmin = 0 Mbps.

orthogonal sharing for strong interference. In Fig. 8(b), we show the Pareto optimal EE for different communication distances in a symmetric four-user case where each user pair has the same weight and the same communication distance d varying from 100 m to 300 m. From the figure, the EE decreases as the distance increases. We can also see that orthogonal sharing is better than non-orthogonal sharing when interference is strong. VI. C ONCLUSION In this paper, we have investigated the N-user interference channel from the EE perspective. To find the EE tradeoff between users and design the energy-efficient resource allocation algorithm in such network, a multi-objective optimization problem is formulated, taking into account the minimum data rate requirement of each user. To solve it, we first utilize the weighted Tchebycheff method to convert it into a singleobjective optimization problem and then apply the Dinkelbach method and the CCCP method to develop a local optimal power control algorithm. A benchmark scenario where each

0

0

20

40

60

80 η1 (Mb/J)

100

120

140

160

(b) Asymmetric user case with d1 = 100 m, d2 = 200 m.

Fig. 7: The EE tradeoff in the two-user case with different g. Rmin = 0 Mbps.

user orthogonally shares the bandwidth has also been investigated. The weighted Tchebycheff method and the Dinkelbach method have been used to develop the optimal bandwidth allocation and power control for this scenario. Numerical results have been presented to demonstrate the effectiveness of the proposed algorithms, which show that the orthogonal bandwidth sharing scenario can achieve a better EE than the interference scenario when the interference between each user pair is stronger than a given threshold. Therefore, our findings provide a valuable reference for designing an opportunistic energy-efficient resource sharing scheme in interference networks. An adaptive mode switching between the two modes can be designed to further improve the performance, e.g., when interference is larger than a given threshold, orthogonal sharing should be used; otherwise, non-orthogonal sharing should be used. Moreover, we have assumed that each user is equipped with a single antenna in this paper. However, the developed algorithms can be easily extended into MIMO systems after some modifications, which is omitted here due to page limit. Another future direction is the design of distributed algorithm



50

increases monotonically in each iteration, i.e.,

ηOS,N = 2

( ) (a) ( ) f p(k+1) ≥ F p(k+1) , p(k) (b) ( ) (c) ( ) ≥ F p(k) , p(k) = f p(k) ,

ηOS,N = 4

45

ηOS,N = 8 40

NS

η

35 EE (Mb/J)

,N = 2

ηNS,N = 4 ηNS,N = 8

where (a) and (c) follow from (26) while (b) follows from (27). The basic idea of CCCP is to iteratively linearise the convex part of the d.c. objective function, −fcave2 (p). According to [30], we can use Taylor approximation to construct the minorization function F . Since −fcave2 (p) is convex and differentiable in its feasible region, we have the following inequality

30 25 20 15 10 5 0.1

0.2

0.3

0.4

0.5

0.6

g

(a) Symmetric user case with d1 = d2 = 200 m. ϕi = 1/N, ∀i. 70 ηOS,d = 100 m ηOS,d = 200 m 60

ηOS,d = 300 m ηNS,d = 100 m ηNS,d = 200 m

EE (Mb/J)

50

ηNS,d = 300 m

−fcave2 (p) ≥ −fcave2 (q) (29) T − (p − q) ∗ ∇fcave2 (q) , ∀p, q ∈ P. By adding fcave1 (p) into both sides of the inequality, we have f (p) = fcave1 (p) − fcave2 (p) (30) T ≥ fcave1 (p) − fcave2 (q) − (p − q) ∗ ∇fcave2 (q) . Define the minorization function as ∆

F (p, q) = fcave1 (p) − fcave2 (q) T − (p − q) ∗ ∇fcave2 (q) .

40

30

20

10 0.1

(28)

0.2

0.3

0.4

0.5

0.6

g

(b) Symmetric user case with different communication distances. N = 4, ϕ = (0.25, 0.25, 0.25, 0.25).

Fig. 8: The EE comparison between the orthogonal and nonorthogonal sharing scenarios. Rmin = 1 Mbps.

with partial or imperfect CSI, which bears practical importance and deserves further investigation.

A PPENDIX A P ROOF OF T HEOREM 2 According to [30], we can construct a minorization function F for f (p), such that { f (p) ≥ F (p, q) , ∀p, q ∈ P, (26) f (p) = F (p, p) , ∀p ∈ P. The above indicates that F is a lower bound of f , and F = f when q = p. Then, the minorization algorithm corresponding with F will update p iteratively as )} { ( (27) p(k+1) = arg max F p, p(k) . p∈P

{ ( )} The iteration stops when p(k) = arg max F p, p(k) , or p∈P

(k)

p − p(k+1) ≤ ϵ in practice. We can also prove that f (p)

(31)

Then, based on (27), the problem in (11) can be sequentially solved by the following concave programming { } ( (k) ) f (p) − f p cave1 cave2 ( )T ( ) p(k+1) = arg max p∈P − p − p(k) ∗ ∇fcave2 p(k) (32) { ( (k) )} ∆ T = arg max fcave1 (p) − p ∗ ∇fcave2 p . p∈P

A PPENDIX B P ROOF OF C OROLLARY 1 We will prove that all the three steps in Algorithm 3 are convergent. The first step is the Dinkelbach method, whose convergence has been proved in [31]. Moreover, as analyzed in Section III-A, the convergence of the CCCP method in the third step has also been proved in [30]. We now prove that the second step, the subgradient method, is also convergent. Let (λ1 , µ1 , ν1 ) and (λ2 , µ2 , ν2 ) be the Lagrangian multipliers in the q-th and the (q + 1)-th subgradient update process, respectively. Then, we have λ1 ≽ λ2 , µ1 ≽ µ{2 , ν1 ≽ ν2 , according to} (21). (0) Furthermore, denote p(0) (λ, and { (∞) } µ, ν) , l (λ, µ, ν) (∞) p (λ, µ, ν) , l (λ, µ, ν) as the initial point and the stationary point in the inner loop, i.e., the CCCP algorithm, for given (λ, µ, ν), respectively, and ∆ L∗ (λ, µ, ν) = min L (p, l, λ, µ, ν) as the minimum value p∈P,l

by the CCCP algorithm. Then, we have L∗ (λ2 , µ(2 , ν2 ) ) = L( p(∞) (λ2 , µ2 , ν2 ) , l(∞) (λ2 , µ2 , ν2 ) , λ2 , µ2 , ν2) ≥ L (p(∞) (λ2 , µ2 , ν2 ) , l(∞) (λ2 , µ2 , ν2 ) , λ1 , µ1 , ν1 ) ≥ L p(∞) (λ1 , µ1 , ν1 ) , l(∞) (λ1 , µ1 , ν1 ) , λ1 , µ1 , ν1 = L∗ (λ1 , µ1 , ν1 ) . (33)



In the above, the first “≥” comes from the fact that L is a monotonic{function over (λ, µ, ν) and the second } “≥” is because that p(∞) (λ1 , µ1 , ν1 ) , l(∞) (λ1 , µ1 , ν1 ) achieves the( minimum of L. Note that in some cases, the local)minimum L p(∞) (λ1 , (µ1 , ν1 ) , l(∞) (λ1 , µ1 , ν1 ) , λ1 , µ1 , ν1 might be ) larger than L p(∞)(λ2 , µ2 , ν2 ) ,l(∞)(λ2 , µ2 , ν2 ) , λ1 , µ1 , ν1 . In this situation, we can re-do the CCCP method in the q-th subgradient by setting ( (0) update process, ) the (0) initial point p (λ , µ , ν ) , l (λ , µ , ν ) as 1 1 1 1 1 1 ( (∞) ) p (λ2 , µ2 , ν2 ) , l(∞) (λ2 , µ2 , ν2 ) . Then, the second “≥” can be guaranteed by the CCCP algorithm, as proved in Appedix A. Now we can easily prove that the subgradient method is convergent according to (33), which ends the proof. A PPENDIX C P ROOF OF L EMMA 1 OS We can rewrite ηu,n as

( OS ηu,n = max a (1 − x) log2 1+ x

where x =

i=N ∑

b (1 − x)

βi (0 ≤ x ≤ 1) , a =

i=1,i̸=n

)

W pn +pe ,

, ∀n and b =

(34) pn h n 2 W σN

.

It is easy to check that the objective function in (34) decreases with x. Therefore, (34) is equivalent to N ∑

min (p,β)∈Θ

βi ,

(35)

i=1,i̸=n

subject to 0 ≤ pi ≤ Pmax , ∀i ̸= n, RiOS 0