Joint Power Optimization for Device-to-Device Communication in Cellular Networks with Interference Control

Transcription

1 Joint Power Optimization for evice-to-evice ommunication in ellular Networks with nterference ontrol Ali Ramezani-Kebra, Student Member, EEE, Min ong, Senior Member, EEE, Ben Liang, Senior Member, EEE, Gar Boudreau, Senior Member, EEE, and S. Hossein Seedmehdi Abstract For device-to-device 2 communication underlaid in a cellular network with uplink resource sharing, both cellular and 2 pairs ma cause significant inter-cell interference at a neighboring base station BS. n this work, under optimal BS receive beamforming, we jointl optimize the power of a cellular user U and a 2 pair for their sum rate imization, while satisfing minimum SNR requirements and worst-case limit in multiple neighboring cells. We solve this non-conve joint optimization problem in two steps. First, the necessar and sufficient condition for the 2 admissibilit under given constraints is obtained. Net, we consider joint power control of the U and 2 transmitters. We propose a power control algorithm to imize the sum rate. epending on the severit of that 2 and U ma cause, we categorize the feasible solution region into five cases, each of which ma further include several scenarios based on minimum SNR requirements. The proposed algorithm is optimal when to a single neighboring cell is considered. For multiple neighboring cells, we provide an upper bound on the performance loss b the proposed algorithm and conditions for its optimalit. We further etend our consideration to the scenario of multiple Us and 2 pairs, and formulate the joint power control and U-2 matching problem. We show how our proposed solution for one U and one 2 pair can be utilized to solve this general joint optimization problem. Simulation demonstrates the effectiveness of our power control algorithm and the nearl optimal performance of the proposed approach in the setting of multiple Us and 2 pairs. nde Terms evice-to-device communication, inter-cell interference, power control, receive beamforming.. NTROUTON The increasing popularit of mobile devices with data hungr applications has resulted in a fast growing demand for high-rate data access eperiences. t is anticipated that it will be challenging for the currentl deploed Long Term Evolution LTE and LTE-Advanced sstems to satisf such demand in the future. New technologies to further improve This work was funded in part b Ericsson anada, b the Natural Sciences and Engineering Research ouncil NSER of anada under ollaborative Research and evelopment Grant RPJ and iscover Grants. A. Ramezani-Kebra and B. Liang are with the epartment of Electrical and omputer Engineering, Universit of Toronto, Toronto, Ontario M5S 3G4, anada aramezani@ece.utoronto.ca; liang@ece.utoronto.ca. M. ong is with the epartment of Electrical, omputer and Software Engineering, Universit of Ontario nstitute of Technolog, Oshawa, Ontario LH 7K4, anada min.dong@uoit.ca. G. Boudreau and S. H. Seedmehdi are with Ericsson anada, Ottawa, Ontario, anada gar.boudreau@ericsson.com; hossein.seedmehdi@ericsson.com. A preliminar version of this work was presented in []. the capacit and spectrum efficienc are required for future cellular sstem evolution. One promising solution is deviceto-device 2 communication, in which nearb users can setup a direct communication link to transmit data to each other without going through the base station BS, b reusing cellular spectrum resources [2] [5]. Such resource reuse b both cellular users Us and 2 pairs can offload cellular traffic and improve radio resource utilization. As a result, it is shown that 2 communication can increase the overall network spectrum efficienc [2]. For 2 communication underlaid in a cellular sstem to reuse spectrum resource assigned to Us, uplink resource sharing is in general preferred for several reasons [6], [7]. ownlink spectrum reuse for 2 communication would require a 2 pair to have a new transmit chain, which is more costl than uplink spectrum reuse, where onl a new receive chain is required at the 2 receiver such as in LTE sstems. n addition, uplink traffic is often lighter than downlink traffic, with uplink resources more likel being available for 2 communication. Furthermore, it is easier to manage the interference incurred at the BS [4]. When a 2 pair reuses the channel resource of a U, the generate intra-cell interference to each other. Furthermore, both 2 and U transmissions cause inter-cell interference to neighboring cells. Thus, the use of 2 communication and corresponding resource allocation need to ensure satisfactor qualit-of-service QoS for both 2 pairs and Us, as well as to maintain a satisfactor limit to neighboring cells. For a 2 underlaid cellular network, interference management to 2s and Us in the same cell has been investigated in various aspects in the literature [6] [22]. The works in [6] [5] focus on interference management to meet minimum QoS requirements for both 2s and Us. Maimizing the sum rate of 2s and Us while meeting the minimum QoS requirements through power control, resource allocation, or association techniques among users is considered in [6] [9]. The problem of joint 2s and Us association and power control to imize the sum rate has been considered in [20], [2]. espite the above results, the due to 2 communication has not been investigated in the eisting literature. For a practical sstem, the caused b both 2s and Us in a neighboring cell should be carefull controlled to not eceed a certain level. n addition, due to the challenges involved in the problem, eisting power allocation schemes for interference mitigation proposed in the literature are tpicall

2 2 heuristics whose performance gap from the optimal cannot be guaranteed. n this work, we consider 2 communication underlaid in a cellular sstem for uplink resource sharing. We assume all users are equipped with a single antenna, while the BS is equipped with multiple antennas. First, we focus on one 2 pair sharing the uplink resource assigned to one U. We aim at jointl optimizing the power control at the U and 2 transmitters, under optimal BS receive beamforming, to imize the sum rate of the 2 and U while satisfing minimum SNR requirements and obeing worst-case to multiple neighboring cells. The formulated joint optimization problem is non-conve. We propose a two-step approach to find a solution: We determine the admissibilit of the 2 pair under the power, SNR, and constraints, through a feasibilit test. The optimal beam vector for the U is provided, as well as the necessar and sufficient condition for the 2 admissibilit. Assuming the 2 pair is admissible, we propose an approimate power control algorithm to imize the sum rate. We obtain the power solution of the U and 2 in closed form, b analzing the feasible solution region of the problem and the characteristics of the solution in the feasible region. We show that, depending on the severit of the that 2 and U each ma cause to the neighboring cell, the shape of the feasible solution region can be categorized into five cases, each of which ma further include several scenarios depending on the minimum SNR requirements. With a total of siteen unique scenarios, we derive the joint power solution in closed form for each scenario. The proposed algorithm is optimal when to a single neighboring cell is considered. For to multiple neighboring cells, we provide an upper bound on the performance loss b the proposed algorithm and conditions for its optimalit. Net, we etend our consideration to the scenario of multiple Us and 2 pairs, and formulate the joint power control and U-2 matching problem. The joint optimization problem is a mied integer programming problem that is difficult to solve. nstead, as a suboptimal approach, we show how our provided solution for one U and one 2 pair can be utilized to find a solution b breaking down the joint optimization problem into a joint power optimization problem and a U- 2 matching problem. Simulation shows that substantial performance gain is achieved b the proposed power control algorithm over two alternative approaches for a single U and 2 pair. Furthermore, for multiple Us and 2 pairs, simulation shows that our proposed approach provides close to optimal performance. A. Related Work To limit the intra-cell interference due to resource sharing b Us and 2s, different approaches have been proposed to meet minimum QoS requirements. As one of the earliest works, [8] has proposed a simple power control scheme for the 2 pair to constrain the SNR degradation of the U, with limited interference coordination available between 2 and U. n [3], the interference link condition between 2 and U is obtained through the 2 pairs received power measurement during uplink transmission, and an interferenceaware resource allocation scheme for 2 pairs has been proposed. To limit the interference at the 2 receiver, [7] has proposed an interference limited cell area, where a 2 pair and multiple Us cannot coeist for channel reuse. For uplink resource sharing, [4] has proposed to scale the power of a 2 transmitter according to the pathloss between the 2 transmitter and the BS to satisf the U SNR requirement. Without consideration, the sum rate imization of 2 and U under their respective minimum QoS requirements has been studied in the literature. n [6], optimal timefrequenc resource allocation and power control for sum rate imization of a 2 pair and a U has been studied, under rate limitation due to modulation and coding, and the U s minimum QoS requirement. For a single-antenna sstem, optimal power allocation for sum rate imization of a 2 pair and a U has been obtained in [7]. n contrast, we consider a multi-antenna BS and constraints when optimizing the U and 2 powers, which is more general and technicall more challenging. With onl the statistics of the interfering link from a U to a 2 pair being available at the BS, [8] have proposed a probabilistic access control for the 2 pair to imize the epected sum rate for uplink resource sharing. A low compleit 2-U association scheme has been proposed in [9] to imize the sum rate of 2 pairs and Us under power and QoS constraints. The gaming approach has also been considered for 2 resource sharing [22], [23]. The problem of joint association and power control for 2 pairs has been studied in [22] using a pricing-based game theoretical approach to satisf the SNR requirements of 2 pairs and Us. A nontransferable coalition formation game has been considered in [23] to solve the energ-efficient resource sharing problem for mobile 2 multimedia communication. To the best of our knowledge, neither to neighboring cells nor BS receive beamforming has been considered in the literature studing the joint power optimization of U and 2 for sum rate imization under their respective minimum SNR requirements. B. Organization and Notations The rest of this paper is organized as follows. n Section, the sstem model is described and the sum rate imization problem is formulated. n Section, a closed-form epression of the optimal beam vector is derived, and the necessar and sufficient condition for admissibilit of the 2 pair is obtained. The power control solution at the U and 2 transmitter is obtained through the stud of five cases of conditions in Section V. n Section V, we etend our consideration to multiple Us and 2 pairs, and present our approach for the joint power control and U-2 matching problem b utilizing the previous solution. Numerical results are presented in Section V, and conclusions are drawn in Section V.

3 3 h g i i Fig. : 2 communication in uplink cellular communication. Notation: We use to denote the Euclidean norm of a vector. stands for an N N identit matri.. SYSTEM MOEL AN PROBLEM FORMULATON A. Sstem Model We stud the underlaing 2 communication in a cellular sstem, where 2 devices reuse the spectrum resource alread assigned to the Us for their uplink communication. We assume orthogonal spectrum resource allocation among the Us within a cell. Thus, these Us do not interfere with each other. When a 2 pair communicates using the channel assigned to a U, the 2 pair and the U cause intra-cell interference to each other. We first focus on a single U and a single 2 pair attempting to reuse the U s channel as shown in Fig.. We assume that all users are equipped with a single antenna and the BS is equipped with N antennas. The BS centrall coordinates the transmission from the U and the 2 pair. n Section V, we etend our consideration to the scenario of multiple Us and 2 pairs. Let P and P denote the transmit power of the 2 pair and the U, respectivel. The SNR at the 2 receiver, denoted b γ, is given b γ = P h 2 σ 2 +P g 2 whereh is the channel between the 2 pair,g is the interference channel between the U and the 2 receiver, and σ 2 is the noise variance at the 2 receiver. The uplink received SNR at the BS from the U is given b γ = P w H h 2 σ 2 +P w H g 2 2 where h N is the channel between the U and the BS, g N is the interference channel between the 2 transmitter and the BS, w is the receive beam vector at the BS with unit norm, i.e., w 2 =, and σ 2 is the noise variance at the BS. Both 2 and U transmissions cause in a neighboring cell. n this work, we focus on for uplink transmission at neighboring BSs. However, our approach can also be applied to considering in the downlink transmission scenario. Let b denote the number of neighboring cells. Let f,i N The noise term in SNR epressions, i.e., σ 2 and σ 2, can be treated as the receiver noise plus inter-cell interference power. and f,i N denote the channels from the U and the 2 transmitter to neighboring BS i, respectivel, for i =,,b. Since the beam vector at neighboring BS i is tpicall unknown to the U and 2 pair, we consider the worst-case, denoted b P,i, given b P,i = P f,i 2 +P f,i 2. 3 Note that P is an upper bound of the actual, because for a neighboring BS with the beam vector w, the received signal is w H f f. We assume perfect knowledge of the communication channels and intra-cell interfering channels, i.e., h, h, g c, and g. 2 For the channels, note that onl channel power gains are needed to obtain P,i. The can be measured at neighboring BSs and shared with the BS of the desired cell through the backhaul. B. Problem Formulation Let P and P denote the imum transmit power at the U and 2 transmitters, respectivel. Our goal is to imize the sum rate of the 2 pair and the U b optimizing the transmit powers {P,P } and the receive beam vector w, while satisfing the worst-case and minimum SNR requirements under per-node power constraints. The problem is formulated as follows: P: P,P,w log+γ +log+γ subject to γ γ, 4 γ γ, 5 P P, P P, 6 P,i Ĩ, i =,,b 7 where γ and γ are the minimum SNR requirements of the U and 2 pair, respectivel, and Ĩ is the worst-case threshold in neighboring cells. The optimization problem P is non-conve, due to the non-conve objective function. Solving P requires two steps. First, we need to determine whether the 2 pair can be admitted to reuse the U s channel. Second, if the 2 pair can be admitted, we optimize the powers and beam vector to imize the sum rate in P. The first problem can be cast as a feasibilit test as shown in the net section. For the second problem, we will derive the optimal power solution {P o,po } when b =. For b >, we will propose an approimate power control algorithm b appling the results obtained for b =.. AMSSBLTY OF 2 Given the power constraints, SNR requirements, and threshold, the admissibilit of the 2 pair can be determined b evaluating the feasibilit of the problem in P given b Find {P,P,w} 8 subject to 4, 5, 6, 7. 2 Note that the additional signaling overhead due to 2 communication is mainl on the feedback of channels {h,g } to the BS. These two channels are tpicall necessar for establishing 2 communication.

4 4 A necessar condition for P being feasible is that U SNR constraint 4 under the optimal w should be met for some {P,P }. For an given {P,P }, we obtain the optimal beam vector w o that imizes γ. This is a receive beamforming problem given b P w H H w w: w 2 = w H Λ w where H = h h H and Λ = σ 2 + P g g H. The imization problem 9 is a generalized eigenvalue problem, and the optimal beam vector is given b w o = Λ h Λ h. 0 The SNR of the U in 2 under the optimal w o is given b 9 w γ = P h H Λ h which is a function of {P,P }. From the definition of Λ, and appling the matri inversion lemma [24], we derive Λ as Λ = σ 2 P g g H σ 2 +P g 2. Substituting the epression of Λ into and after some algebraic manipulation, the SNR constraint 4 under the optimal beamforming can be re-epressed as P h 2 ρ 2 σ 2 γ 2 + σ 2 P g 2 where ρ = hh g h g is the correlation coefficient of the channels h and g, and ρ. For P to be feasible, there should eist at least one power solution pair {P,P } such that constraints 5-7 and 2 hold. For notation simplicit, in the following, we denote = P and = P. We now stud the SNR constraints 5 and 2 and state the following lemma. Lemma : For γ = γ and γ = γ, the power solution {, } is unique and is given b = ξ 2 K, = ξ 2 K βk 3 σ2 K 3 3 where α = σ2 γ h, β = γ 2 h, K 2 = ρ 2, K 2 = σ 2 g, K 2 3 = g 2, K 4 = βαk3 + σ 2 K K 2, K 5 = 4 K βk 2 αk 3 +σ 2, and ξ = β αk 3 +σ 2 K K 2 + K 2 4 +K 5. Proof: See Appendi A. B Lemma and combining constraints 6 and 7, the necessar and sufficient condition for the 2 pair to be admissible is given as follows. Necessar and sufficient condition: The 2 pair is admissible if {, } in 3 satisfies 0 < P, 4 0 < P 5 c,i +,i, i =,,b 6 f where c,i =,i 2 f, and,i =,i 2. Ĩ Ĩ Note that {, } is the minimum power level required to satisf the minimum SNR requirements. Thus, for an feasible {,}, we have and. onstraints 4 and 5 ensure the imum power at the 2 and U are enough to meet their respective SNR requirements. onstraint 6 ensures the constraints can be satisfied. V. POWER ONTROL FOR 2 AN U Assuming the 2 pair is admissible, we now solve the sum rate imization problem P. With the optimal w o given in 0, we need to solve P with respect to {P,P }. ue to the non-conve objective, finding an optimal solution is challenging. nstead, we propose the following approimation to obtain the power solution. Note that the constraints in 7 can be equivalentl written as c,i +,i, i =,,b. 7 We replace these constraints with a single constraint and ensure that it satisfies the requirements in all neighboring cells. We denote this constraint b c +, where c and are determined as follows. efine c, i c,i,, i,i, c min,ip i,i. Note that if there eists i such that c,i P, then from 7, i c,i, and we have c = c,. Otherwise, if c,i P < for, and ỹ min,ip i c,i all i, then the intersection of c + = and = P is given b,p. Similarl, if there eists j such that,j P, then =,. Otherwise, the intersection of c + = and = P is given b P,ỹ. Based on this, we determine c and in four possible cases: 3 f i,j, such that c,i P and,j P : c = c, and =,. 2 f i, such that c,i P, and j,,j P < : c = c, and = c,ỹ P. 3 f i, c,i P <, and j, such that,j P : c = c2, P and c 2 =,. 4 f i,j, c,i P < and,j P < : c = P P P ỹ and c P 2 = ỹ P P ỹ. Note that when b =, the approimation becomes accurate to represent the constraint. Substituting the epression of γ under the optimalw o at the left hand side of 2 in P, and approimating multiple constraints in 7 with a single constraint, we modif P into the following problem P2: logr, 8, subject to K l γ, 9 K 2 + a σ 2 +K 3 γ, 20 P, 2 c Note that other values of c and ma also be chosen; however, our method strives to create a new feasible region that is close to the original one, so that the loss due to approimation is small.

5 5 where R, = a + σ 2 +K + K l, 3 K P A E B P A B F with a = h 2 and l = h 2 /σ 2. Let A denote the feasible solution region of the problem P2. Note that b modifing P to P2, we shrink the original feasible region of P to A. This is done b replacing the feasible region boundaries formed b the intersections of the multiple constraints in 7 with a single boundar described b 22. The following lemma gives the locations of the optimal power pair in A. Lemma 2: The optimal power solution pair o, o is at the vertical, horizontal, or tilted boundar of A, given b = P, = P, or c + =, respectivel. Proof: See Appendi B. Note that, depending on the sstem parameter setting, the shape of the feasible regiona varies. The boundaries ofa ma or ma not include the tilted boundar segment c + =. n the following, we consider both cases and obtain the optimal solution to P2 for each case. f the boundaries of A do not include c + =, the optimal solution satisfies 22 with strict inequalit. B Lemma 2, it follows that at least one of o and o equals its imum value P or P, and the optimal o, o is at either the vertical or horizontal boundar line of A. n this case, the optimal power pair is given in the following proposition. Proposition : f the boundaries of the feasible region A do not include c + =, then the optimal power pair o, o for P2 is at one end point of the vertical or horizontal boundar line segment of A. Proof: See Appendi. f the boundaries of A include c + =, then at optimalit, o, o is on the horizontal, vertical, or tilted boundar line of A. The following proposition provides the solution to P2 in this case. Proposition 2: f the boundaries of the feasible region A include c + =, then the optimal power pair o, o is given in one of the two cases: An end point of the horizontal, vertical, or tilted boundar line segment of A ; or 2 an interior point of tilted boundar line segment of A, whose -coordinate is one of the roots of the following quartic equation e 4 4 +e 3 3 +e 2 2 +e +e 0 = 0 24 where {e i } 4 i=0 are given as in..5 in Appendi. The optimal U power is o = o /c. Proof: See Appendi. Note that the roots of a quartic equation have closed-form epressions. Furthermore, we do not need to compute all the roots of 24, since not all of them are ina. n the following, we classif different scenarios leading to different tpes of the boundaries for A. We obtain simple inequalities to check the conditions under which each scenario applies. For each scenario, we discuss the corresponding optimal power solution o, o. P P a Scenario A B c Scenario 3 B G e Scenario 5 P P P P P b Scenario 2 B G d Scenario 4 f Scenario 6 Fig. 2: Moderate from U and 2. A. Moderate from U and 2 We first consider the case where the boundar line c + = intersects both the horizontal and vertical boundar lines. This means that c P + P, and the - intercept and -intercept of c + = are greater than P and P, respectivel. These result in the following conditions P F P H P P c c P P c, 25 P. 26 Note that given the definitions of c and, conditions 25 and 26 mean that the caused b the U or 2 transmitter alone, at each imum power, is less than the threshold, while combined the caused b both of them is greater than the threshold. n other words, both the U and the 2 transmitter cause relativel moderate to the neighboring cell. n this case, depending on SNR requirements 9 and 20, there are several different shapes of the feasible regiona, as shown in Figs. 2a 2f, wherea is the shaded area. The curve

6 6 Q G Q G Q G P A E F F P P T P a Scenario 7 b Scenario 8 a Scenario 3 b Scenario 4 Q P A E B P A M P H H P T P P c Scenario 9 d Scenario 0 c Scenario 5 d Scenario 6 P A B P B G Fig. 4: a Strong from U and 2; b-d Weak. e Scenario F T f Scenario 2 Fig. 3: a-c Strong from U and moderate from 2; d-f Moderate from U and strong from 2. and line passing through point correspond to constraints 9 and 20 with equalit. The feasible region A s in Figs. 2a 2f correspond to si scenarios, depending on whether the line and curve passing through point intersect the vertical, horizontal, or tilted boundar. n the following, we derive the optimal power control solution in each of these si scenarios. Scenario : The feasible solution regiona is depicted in Fig. 2a as the shaded area. Points A, B,, E, and are the intersections of two lines or a line and curve. Both curve -E and line -A intersect with the horizontal boundar line = P. n this scenario, we have E B. t occurs under the following condition: K 2 K γ lp c P. 27 B Lemma 2 and Proposition, the optimal power pair o, o can be one of points A and E. Therefore, the set F T of candidate power pairs is given b { P A. = βσ 2 +K 3P,P, K,P } K γ lp 2 Scenario 2: As shown in Fig. 2b, in this scenario, the curve -F and line -A intersect the tilted boundar line and horizontal boundar line, respectivel. Note that F is the - coordinate of point F, which is the intersection of the curve -F and the tilted boundar line B-F. We can find F b setting constraints 9 and 22 with equalit. This results in a quadratic equation given b K 2 θ+k 2 αc = 0 29 whereθ = K K 2 αc. The uniqueness of the feasible solution of 29 is stated in the following lemma. Lemma 3: For A as shown in Fig. 2b, the feasible solution of the quadratic equation 29 is unique and is given b F = θ + θ 2 4 K K 2 αc K Proof: See Appendi E. Note that in Scenario 2, we have A B F P. The conditions for this scenario to happen are as follows: βσ 2 +K 3 P c P c P, 3 F P. 32 B Proposition 2, the candidate pairs for o, o are points A, B, F, and an interior point of line B-F. For the last case

7 7 to happen, o should be within the range cp < o < F, which means the roots of 24 should satisf this range constraint. Let S 2 be the set of roots that meet the above range constraint. The corresponding set of points on the interior of { } line B-F is given b A 2 =, c2 /c : S2. Now, we have the set of candidate pairs for o, o as { P A.2 =, F, F /c, } /,P,A 2 33 βσ 2 +K 3P,P c P where the first three pairs are the coordinates for points A, F, and B, respectivel. 3 Scenario 3: As illustrated in Fig. 2c, in this scenario, the curve - and line -A intersect the horizontal and vertical boundar lines, respectivel. The entire tilted boundar B- is in the feasible region. n this scenario, we have A B and. The conditions for this scenario to occur are given b α and 3. K +K 2 /P P, 34 c B Proposition 2, o, o could be either points A, B,,, or if o, o lies on the interior of line B-, o should be within the range B < o <, i.e., cp < o < P. Let S 3 denote the set of roots 24 satisfing the above range constraint. B Proposition 2, the set of candidate pairs on the { interior} of line B- is given b A 3 = o, o /c : S 3. Thus, in Scenario 3, the set of candidate pairs is given b { P A.3 = βσ 2 +K 3 P,P, /,P, K },A +K 2 /P 3 c P P,α P, P /c, 35 where the first four pairs are the coordinates of points A, B,, and, respectivel. 4 Scenario 4: As shown in Fig. 2d, in this scenario, both curve -F and line -G intersect the tilted boundar line B-. The condition for this scenario to happen is as follows: c P G F P 36 where G can be obtained b setting constraints 20 and 22 with equalit, given b G = σ2 β +βk 3/c. 37 +βk 3 /c B Proposition 2, to find the optimal power pair, we need to consider points G and F. The set of candidate power pairs { on the interior of line } G-F is given b A 4 = o, o /c : S4 where S4 denotes the set of roots 24 which meet G < o < F. Thus, the set of candidate pairs is given as follows: P A.4 = { G, G /c, F, F /c,a 4 }. 38 TABLE : Moderate from U and 2 under conditions 25 and 26 ondition Set of candidates for the optimal powers 27 P o = P A. in 28 3 and 32 P o = P A.2 in 33 3 and 34 P o = P A.3 in P o = P A.4 in and 39 P o = P A.5 in 40 4 P o = P A.6 in 42 5 Scenario 5: As shown in Fig. 2e, in this scenario, line -G intersects tilted boundar line G-, while curve - intersects the vertical boundar line = P. n this scenario, we have B G P and. The conditions under which this scenario happens are given b c P and 34. G P, 39 Based on Proposition 2, o, o could be either points G,,, or if o, o is on the interior of line G-, o should be within the range G < o < P. Let S 5 denote the set of roots of 24 within the above range of o. The set of candidate points on the interior of line G- { } is A 5 = o, o /c : S5. Hence, the set of candidate pairs for o, o in this scenario is given b { P A.5 = G, c 2 G c P,α, P, P /c K +K 2 /P,,A 5 } Scenario 6: As depicted in Fig. 2f, this scenario happens when both curve - and line -H intersect the vertical boundar line = P. n this scenario, we have H. The condition for this scenario to happen is given b P βσ2 βk 3 P. 4 c n this scenario, the boundaries of A do not include the tilted boundar line. B Proposition, it is sufficient to consider onl points H and to find the optimal power solution. The set of candidates for the optimal powers is thus given b P A.6 = { P,α P, P K +K 2 /P, } βσ2. 42 βk 3 We summarize in Table the candidate solution sets for the above si scenarios for the case where conditions 25 and 26 hold. B. Strong from U and Moderate from 2 onsider the case in which the tilted line c + = onl intersects the vertical line = P, but does not intersect the horizontal line = P. This means that the -intercept and

8 8 -intercept of c + = are greater than P and less than P, respectivel. The resulting conditions are given b P c, 43 and 26. Note that the condition 43 means the caused b the U under its imum power is greater than the threshold. Along with 26, these conditions correspond to the case where the caused b the U is strong, while the caused b the 2 transmitter is moderate. n this case, b SNR requirements 9 and 20, the necessar and sufficient condition to set up 2 communication is given b 4 and 6. The feasible region A can have three different shapes as shown in Figs. 3a 3c, depending on whether the line and curve passing through point intersect the vertical or tilted boundar line. Accordingl, we derive the optimal power control solution in these three scenarios as follows. Scenario 7: As shown in Fig. 3a, in this scenario, both curve -F and line -G intersect the tilted boundar line. The condition under which this scenario happens is given b 0 G F P 44 where F and G are given in 30 and 37, respectivel. Let P B. denote the set of candidate pairs in this scenario. Note that the feasible region A has the same shape as that in Scenario 4 in Fig. 2d. Thus, these two scenarios have the same set of candidate pairs, i.e., P B. = P A.4. 2 Scenario 8: As depicted in Fig. 3b, in this scenario, line -G and curve - intersect the tilted boundar line G- and vertical boundar line, respectivel. This scenario occurs when G and, which results in the following conditions: 0 G P, 45 and 34. From Fig. 3b, we see that the shape of A is the same as that of Scenario 5 in Fig. 2e. Thus, these two scenarios have the same set of candidate pairs. Let P B.2 denote the set of candidate power pairs in Scenario 8. We have P B.2 = P A.5. 3 Scenario 9: As illustrated in Fig. 3c, this scenario happens when both curve - and line -H intersect the vertical boundar line. B similar discussion as the above, the condition for this scenario to happen is the same as that in Scenario 6 in Fig. 2f, which is given b 4. As a result, the set of candidate pairs, denoted b P B.3, is the same as that in Scenario 6, i.e., P B.3 = P A.6. The candidate solution sets for these three scenarios for the case with conditions 26 and 43 are summarized in Table.. Moderate from U and Strong from 2 Now consider the case in which line c + = onl intersects the horizontal line = P, but does not intersect the vertical line = P. Similar to the discussion in Section V-B, this means that the -intercept and -intercept TABLE : Strong from U and Moderate from 2 under conditions 26 and 43 ondition Set of candidates for the optimal powers 44 P o = P B. = P A.4 34 and 45 P o = P B.2 = P A.5 4 P o = P B.3 = P A.6 of c + = are greater than P and less than P, respectivel, which are equivalent to the following conditions: P c, 46 2 and 25. These conditions correspond to the case where the caused b the U is moderate, while the caused b the 2 transmitter is strong. Similar to the previous cases, depending on the SNR requirements 9 and 20, the feasible region of A can have three different shapes as shown in Figs. 3d 3f. The optimal power pair for each of these three scenarios are discussed below. Scenario 0: The feasible region is shown in Fig. 3d. n this scenario, both curve -E and line -A intersect the horizontal boundar line. The shape of the feasible regiona is the same as that of Scenario, with the condition to occur given b 27. Thus, the set of candidate pairs, denoted b P., is given b P. = P A.. 2 Scenario : As shown in Fig. 3e, this scenario happens when curve -F and line -A intersect the tilted boundar line B-F and horizontal boundar line, respectivel. n this scenario, we have A B F T, which results in the following conditions: c P F c, 47 and 3. Similarl, the shape of A is the same as that of Scenario 2 in Fig. 2b. As a result, the set of candidate pairs, denoted b P., is given b P. = P A.2. 3 Scenario 2: As depicted in Fig. 3f, in this scenario, both curve -F and line -G intersect the tilted boundar line. With a similar approach, we can derive the condition for this scenario to happen as follows: c P G F. 48 Again, A in this scenario has the same shape as that in Scenario 4, and thus, the set of candidate pairs, denoted b P.3, is given b P.3 = P A.4. The candidate solution sets for these three scenarios for the case with conditions 25 and 46 are summarized in Table.. Strong from U and 2 When the tilted line c + = does not intersect either the vertical or horizontal line, we have the feasible region A as shown in Fig. 4a. n this case the -intercept and - intercept of c + = are less than P and P,

9 9 TABLE : Moderate from U and Strong from 2 under conditions 25 and 46 ondition Set of candidates for the optimal powers 27 P o = P. = P A. 3 and 47 P o = P.2 = P A.2 48 P o = P.3 = P A.4 respectivel, which are equivalent to conditions 43 and 46. n this scenario, the necessar and sufficient condition to have an admissible 2 pair is reduced to 6. enote this case b Scenario 3. n this scenario, we have 0 G F. 49 Since A in both this scenario and Scenario 4 have the same shape, the set of candidate pairs, denoted b P., is given b P. = P A.4. E. Weak n all above cases, the line c + = intersects at least one of the horizontal and vertical boundar lines i.e., = P and = P. Now we consider the case where there is no intersection betweenc + = and either of the horizontal and vertical boundar lines. This case happens when c P + P <. 50 Note that condition 50 occurs when both the U and the 2 transmitter cause weak to the neighboring cell, e.g., the both are located near the center of the cell. Shown in Figs. 4b 4d, there are three possible shapes for the feasible region A in this case, depending the SNR requirements 9 and 20. The optimal power pairs for these three scenarios are discussed as follows. Scenario 4: As shown in Fig. 4b, this scenario happens when curve -E and line -A intersect the horizontal boundar line. The condition for this scenario to happen is E P, i.e., K K 2 P. 5 γ lp Since the shape of A is the same as that in Scenario, the set of candidates for the optimal powers, denoted b P E., is given b P E. = P A.. 2 Scenario 5: As illustrated in Fig. 4c, in this scenario, line -A and curve - intersect the horizontal and vertical boundar lines, respectivel. n this scenario, we have A P and P. Solving for A and, we have the following conditions: α βσ 2 +K 3 P P, 52 P. 53 K +K 2 /P TABLE V: Weak under condition 50 ondition Set of candidates for the optimal powers 5 P o = P E. = P A. 52 and 53 P o = P E.2 in P o = P E.3 = P A.6 Algorithm Approimate power control algorithm nput: α, β, a, l, K, K 2, K 3, {c,i,,i} b i=, γ, γ, P, P Output: P, o P, o and w o : heck the feasibilit condition : etermine c,, a, a 2, b, b 2, F, and G. 3: if 25 and 26 hold then 4: ompute candidate solution set in Table. 5: else if 26 and 43 hold then 6: ompute candidate solution set in Table. 7: else if 25 and 46 hold then 8: ompute candidate solution set in Table. 9: else if 43 and 46 hold then 0: ompute candidate solution set P o = P. in Section V-. : else if 50 holds then 2: ompute candidate solution set in Table V. 3: end if 4: Enumerate among candidate solution set P o to find the optimum solution. 5: Obtain the optimum beam vector 0 using P o and P. o B Proposition, o, o could be at either of points A,, or M, and the set of candidate pairs is given b { P E.2 = βσ 2 +K 3 P,P, P,P, P,α K } +K 2 /P Scenario 6: The feasible region is shown in Fig. 4d. This scenario happens when both line -H and curve - intersect the vertical boundar line. The condition for this scenario to occur is H P, i.e., P βσ2 βk 3 P. 55 Since the shape of A is the same as that in Scenario 6, the set of candidate pairs, denoted b P E.3, is given b P E.3 = P A.6. The candidate solution sets for these three scenarios for the case with condition 50 are represented in Table V. Finall, we summarize the steps to solve the optimization problem P in Algorithm. We note that the optimal solution in an scenario can be obtained in closed form. F. Performance Bound Analsis Note that Algorithm is ver efficient in obtaining the power solution pair {P o,po }, as the candidate pairs are all given in closed-form. However, it onl provides the optimal solution for b =. For b >, since A is a subset of the original feasible region of P, Algorithm is suboptimal. n the following, we provide an upper bound on the performance loss of the proposed algorithm and provide the conditions for its optimalit. Let A o denote the original feasible region of P, and we have A A o. An eample for b = 2 is shown in

10 0 P L,U L A A A L U3 U2 P Fig. 5: Approimating the true feasible region. Fig. 5, where A and A o are given b A and A A2, respectivel. Let points U and U 2 denote the intersections of the boundar of A o with the line and curve corresponding to minimum SNR requirements 20 and 9, respectivel. t follows that U U2 and U2 U. Similarl, let points L and L 2 denote the intersections of the boundar of A with the same line and curve, and we have L L2 and L2 L. We denote the set of all corner points of A b L, e.g., L = {L,L 2,L 3 } in Fig. 5. Note that L 4. We define a new point U 3 = U, U2, U, U2 as shown in Fig. 5 for b = 2. For simplicit, we use R Z to denote logr Z, Z in 8 for point Z in A. Also, let R A and R opt denote the sum rate achieved b Algorithm and the imum sum rate under an optimal solution of P, respectivel. We have the following results on the performance of Algorithm. Proposition 3: For b =, R opt = R A. For b >, the performance loss of Algorithm is bounded b R opt R A {R U,R U2,R U3 } l L {R l}. 56 Furthermore, R opt = R A if one of the following conditions holds: constraints in 7 results in a single tilted boundar line for A o. 2 L = L2 or equivalentl U = U2. 3 L = L2 or equivalentl U = U2. Proof: See Appendi F. Note that in Proposition 3, ondition means A = A o. For onditions 2 or 3, the line and curve associated with 20 and 9 both intersect either the vertical or horizontal boundar of A o. V. EXTENSON TO MULTPLE US AN 2 PARS So far, we have provided a power control solution for one U and one 2 pair. We now etend our consideration to the scenario of multiple Us and 2 pairs. onsider a multichannel communication sstem e.g., OFMA with N orthogonal subchannels in each cell. We assume a full loaded network with N Us, and there are N 2 pairs with N N. Without loss of generalit, we assume U j uses subchannel j for j = {,,N }. Each 2 pair reuses at most one subchannel, and the subchannel of each U can be reused b at most one 2 pair. efine indicator k,j {0,} such that k,j = if 2 pair k reuses U j s subchannel and k,j = 0 otherwise. Let P = [P,,,P,N,P,,,P,N ] T, = [,,,,N,, N,N ] T, andw = [w T,,wT N ] T. The objective is to imize the overall sum rate of all 2 pairs and Us b optimizing the transmit power vector P, the indicator vector, and the receive beam vector w, while satisfing the worst-case and minimum SNR requirements under per-node power constraints. The formulated problem is given b 4 P3: log+γ,j + k,j log+γ,k subject to P,w, k j P,j w H j h,j 2 σ 2 + k,j P,k w H j g,k 2 γ, j P,k h,k 2 σ 2,k + k,jp,j g j,k 2 γ, k P,j P, P,k P, j,k P,i,j Ĩ, j,i =,,b k,j, k,j, j,k k j k,j {0,}, j,k where denotes the set of admissible 2 pairs. 2 pair k is admissible if it can reuse at least one subchannel from. Note that problem P3 is a mied integer programming problem and is challenging to solve. nstead, we consider a suboptimal approach b utilizing our proposed Algorithm as follows: etermine the admissibilit of an 2 pair k to reuse U j s subchannel, for k =,,N, j =,,N. 2 For all k andj, if 2 pairk is admissible to use U j s subchannel, we jointl optimize their transmit powers to imize their sum rate, which is given b problem P with the solution provided b Algorithm. 3 We solve the U-2 matching problem to optimall assign each admissible 2 pair to a U. n particular, we define a bipartite graph between Us and 2 pairs. Each edge between a 2 pair and a U indicates that the pairing of the 2 pair and the U is feasible. The weight of the edge is given b the sum rate or rate gain of the 2 pair and the U, under the approimate power control solution provided b Algorithm. This U-2 matching problem is an assignment problem, whose optimal solution can be achieved b using the well-known Hungarian algorithm [26]. 4 For overall sstem optimization to imize the sum rate, one would need to jointl optimize the subchannel assignment for multiple Us and 2 pairs, and power allocation for them. This problem is known in the literature to be NP-hard [25]. n addition, re-allocating resources to all Us and 2 pairs whenever a new 2 pair join or leave the sstem will increase the signaling overhead significantl. nstead, our problem formulation aims for practical design, where the addition of a 2 pair does not disturb the eisting subchannel assignments for Us.

11 Sum rate bit/channel use Rate gain bit/channel use Regime 2 2 Regime Proposed algorithm N =8 BL heuristic N =8 0 U-priorit heuristic N = 8 Proposed algorithm N =4 8 BL heuristic N =4 U-priorit heuristic N = 4 Proposed algorithm N =2 6 BL heuristic N =2 U-priorit heuristic N = P /σ 2 db a Sum rate Regime Regime 2 Proposed algorithm N =8 BL heuristic N =8 U-priorit heuristic N = 8 Proposed algorithm N =4 BL heuristic N =4 U-priorit heuristic N = 4 Proposed algorithm N =2 BL heuristic N =2 U-priorit heuristic N = P /σ 2 db b Rate gain Fig. 6: The sum rate and rate gain versus P /σ 2 d /d 0 = 0.2. Sum rate bit/channel use Rate gain bit/channel use Regime Regime 2 2 Proposed algorithm d = 0.d0 0 BL heuristic d = 0.d0 U-priorit heuristic d = 0.d0 Proposed algorithm d = 0.2d0 8 BL heuristic d = 0.2d0 U-priorit heuristic d = 0.2d P /σ 2 db Regime a Sum rate Regime 2 Proposed algorithm d = 0.d0 BL heuristic d = 0.d0 U-priorit heuristic d = 0.d0 Proposed algorithm d = 0.2d0 BL heuristic d = 0.2d0 U-priorit heuristic d = 0.2d P /σ 2 db b Rate gain Fig. 7: The sum rate and rate gain versus P /σ 2 N = 4. Remark : Note that the suboptimalit of the above approach lies onl in the approimation of multiple constraints b a single constraint. t follows that this approach is optimal for P3 if one of the conditions in Proposition 3 is satisfied. We can further reduce the compleit of the U-2 assignment problem in Step 3 above b proposing two suboptimal U-2 matching schemes. nstead of the reward rate or gain, we define the cost on an edge between 2 pair k and U j in the bipartite graph as follows: The intra-cell interference channel gain between U j and 2 receiver k, i.e., g j,k termed Suboptimal U- 2 matching A. The weight of 2 transmit power in the constraint 22, i.e., for 2 pair k on subchannel j termed Suboptimal U-2 matching B. We will show through simulation that our proposed approach provides performance ver close to that of the jointl optimal solution for P3. V. NUMERAL RESULTS n our simulation, we consider that the cell of interest contains one U and one 2 pair. Assume that the BS is located at coordinates 0,0, and that the U, the 2 transmitter, and the 2 receiver are located at 0,0.5d 0, 0, 0.75d 0 d /2, and 0, 0.75d 0 +d /2, respectivel. Unless otherwise mentioned, we consider one neighboring cell with its BS located at 2d 0,0. Let d, d g, and d g denote the distances between the U and the BS, the U and the 2 receiver, and the 2 transmitter and the BS, respectivel, and let d f and d f denote the distance between the U and the neighboring BS, and the 2 transmitter and the neighboring BS, respectivel. We set d = 0.5d 0, d g =.25d 0 d /2, d g = 0.75d 0 + d /2, d f = 2.066d 0, and d f = d /d 0 2 d 0. For all the links, we assume the path loss eponent is set to 4. The channel coefficients are assumed to be Gaussian with zero-mean and variance d/d 0 4. We set σ 2 = σ 2 =, γ = γ = 3 db, P = P = P, and Ĩ = N 0 where 0 is the threshold reference, such that Ĩ is a function of the number of antennas at the BS. We set 0 /σ 2 = 3 db. We use 5000 channel realizations to evaluate the average performance. We evaluate the rate gain obtained b adding 2 communication. t is the difference of the imum sum rate of P provided b Algorithm and that when there is no 2 pair in the cell. Furthermore, for performance comparison, we consider two baseline algorithms: boost-and-limit BL heuristic, where the unique power solution, in 3 is boosted proportionall with a common factor ζ such that either the imum power constraint 2 or the limit 22 is met with equalit, i.e., further boosting the powers would violate at least one constraint. This BL algorithm is

12 2 Sum rate bit/channel use Proposed algorithm d = 0.d0 BL heuristic d = 0.d0 U-priorit heuristic d = 0.d0 2 Proposed algorithm d = 0.2d0 BL heuristic d = 0.2d0 U-priorit heuristic d = 0.2d d/d 0 a Sum rate Sum rate bit/channel use Proposed algorithm N =8 BL heuristic N =8 0 U-priorit heuristic N = 8 Proposed algorithm N =4 8 BL heuristic N =4 U-priorit heuristic N = 4 Proposed algorithm N =2 6 BL heuristic N =2 U-priorit heuristic N = /σ 2 db a Sum rate 4 0 Rate gain bit/channel use Proposed algorithm d = 0.d0 BL heuristic d = 0.d0 U-priorit heuristic d = 0.d0 2 Proposed algorithm d = 0.2d0 BL heuristic d = 0.2d0 U-priorit heuristic d = 0.2d d/d 0 b Rate gain Fig. 8: The sum rate and rate gain versus d/d0 N = 4,P /σ 2 = 0 db. Rate gain bit/channel use /σ 2 db b Rate gain Proposed algorithm N = 8 BL heuristic N = 8 U-priorit heuristic N = 8 Proposed algorithm N = 4 BL heuristic N = 4 U-priorit heuristic N = 4 Proposed algorithm N = 2 BL heuristic N = 2 U-priorit heuristic N = 2 Fig. 9: The sum rate and rate gain versus 0 /σ 2 d /d 0 = 0.2,P /σ 2 = 20 db. also promising in a practical point of view. Note that the scheduling BS can easil compute, and then boost the power of the U and 2 pair until either the imum power is achieved or a neighboring cell alerts regarding the level. This can be achieved b simple binar feedback through the backhaul. Hence, there is no need for an channel echange among the cells, which reduces signaling overhead substantiall. 2 U-priorit heuristic, aiming at imizing SNR γ of the U. t selects the imum feasible U power with the minimum feasible 2 power, satisfing constraints Note that in this U-priorit heuristic, we prioritize the U in terms of choosing a specific end point of the feasible region. A. Single Neighboring ell We first evaluate how the performance changes with the imum transmit power. The sum rate and rate gain versus the normalized imum power, P /σ 2, under Algorithm, the BL heuristic, and the U-priorit heuristic, are shown in Figs. 6a and 6b, respectivel, for N = 2,4,8. n Fig. 6a, for the increment of the sum rate and rate gain over P /σ 2 under Algorithm, we observe two regimes: i Regime, where the sum rate is an increasing function of P. n this regime, the is relativel weak, which is similar to the case in Section V-E. n this case, as shown in Scenarios 4 6, the feasible region is not affected b the constraint, and the candidates for the optimal power pair in Table V are directl functions of P. As a result, the sum rate increases linearl with P in this regime. ii Regime 2, where the sum rate converges. n this regime, the is relativel strong, which is similar to the case in Section V-. n this case, as shown in Scenario 3, the feasible region is not changed b P, and the candidates for the optimal power pair are functions of Ĩ. Hence, the sum rate is controlled b the fied threshold. We observe in Fig. 6b that the rate gain is increasing in Regime and decreasing in Regime 2. To see wh this happens, notice that, the in the 2 mode is caused b both the U and the 2 transmitter, while in the non-2 mode, it is caused b the U onl. Hence, in non-2 mode, the U can use a higher power for transmission, and the corresponding rate gain b the 2 mode is reduced. Furthermore, comparing these three algorithms, we see that the optimal solution b the proposed algorithm provides significant sum rate improvement over the BL and U-priorit heuristics in both regimes for all values of N. Note that the BL heuristic outperforms the Upriorit heuristic. n order to stud the effect of the 2 distance on the performance, the sum rate and rate gain versus P /σ 2 for d /d 0 = 0. and 0.2 are shown in Figs. 7a and 7b, respectivel. We set N = 4. Note that for the proposed

13 3 Sum rate bit/channel use Rate gain bit/channel use Optimal power control and optimal U-2 matching Approimate power control and optimal U-2 matching BL heuristic power control and optimal U-2 matching 20 U-priorit heuristic power control and optimal U-2 matching Approimate power control and suboptimal U-2 matching A Approimate power control and suboptimal U-2 matching B P /σ 2 db a Sum rate Optimal power control and optimal U-2 matching Approimate power control and optimal U-2 matching BL heuristic power control and optimal U-2 matching U-priorit heuristic power control and optimal U-2 matching Approimate power control and suboptimal U-2 matching A Approimate power control and suboptimal U-2 matching B P /σ 2 db b Rate gain Fig. 0: The sum rate and rate gain versus P /σ 2 N = 4,d /d 0 = 0.,b = 6. algorithm, the sum rate and rate gain improve significantl as d /d 0 decreases. However, the performance of the Upriorit heuristic is not sensitive to d /d 0. This is due to the objective of the sum rate of both the 2 and U in Algorithm, resulting in significant rate improvement when the 2 channel is ver strong, i.e., the 2 distance is small. Let d denote the 2-BS distance, which is defined as the distance between the middle point of the 2 pair and the BS. The 2 transmitter and receiver are placed at 0, d d /2 and 0, d + d /2, respectivel. Then we have d g = 0.5d 0 + d d /2, d g = d + d /2 and d f = 2 2 +d g /d 0 2 d 0. Figs. 8a and 8b show the sum rate and rate gain versus d/d 0, respectivel. We set N = 4, P /σ 2 = 0 db, and d /d 0 = 0. and 0.2. ncreasing d increases the distance between the U and the 2 pair, resulting in reduced intra-cell interference at the BS and at the 2 receiver. Furthermore, the distance between the 2 transmitter and the neighboring BS increases, which leads to reduction. As epected, both the sum rate and rate gain improve as d/d 0 increases. Furthermore, we observe from Fig. 8b that the gap of the rate gain between the optimal solution b Algorithm and the U-priorit heuristic is significant; and it increases as the 2-BS distance increases. We now stud the effect of the threshold reference 0 on the performance. The sum rate and the rate gain versus 0 /σ 2 are shown in Figs. 9a and 9b, respectivel, for d /d 0 = 0.2 and P /σ 2 = 20 db. We observe that both the sum rate and rate gain improve when 0 increases. For small 0 values, the sum rate is an increasing function of 0 /σ 2 since the is relativel strong the case in Section V-, and the candidates for the optimal power pair are functions of 0 /σ 2. As 0 increases, the constraint becomes inactive as the case in Section V-E and the sum rate converges due to the fied P. B. Multiple Neighboring ells and Multiple Users We now consider multiple neighboring cells and stud the performance of Algorithm. Besides multiple neighboring cells, we further consider multiple Us and 2 pairs in the cell of interest as discussed in Section V. We set the number of neighboring cells as b = 6. We consider 3 Us and 3 2 pairs that are randoml dropped in the cell of interest. We compare the following schemes: optimal power control and optimal U-2 matching, where the optimal power control is obtained b ehaustive search; 2 proposed approach in Section V, i.e., the approimate power control Algorithm and optimal U-2 matching; 3 BL heuristic power control and optimal U-2 matching; 4 U-priorit heuristic power control and optimal U-2 matching; 5 approimate power control and suboptimal U-2 matching A given in Section V; 6 approimate power control and suboptimal U-2 matching B given in Section V. n Figs. 0a and 0b, the sum rate and rate gain versus P /σ 2 under different power control methods and matching schemes are shown. We set N = 4 and d /d 0 = 0.. We observe that the performance of our proposed approach is close to that of the optimal power control with optimal U- 2 matching. n particular, in the region where the sum rate is an increasing function of P, the performance b both power control schemes overlap. This demonstrates the merit of our proposed approimate power control algorithm to provide a simple closed-form solution. n addition, it can be seen that the approimate power control algorithm with an of the three U-2 matching schemes outperforms the BL and U-priorit heuristics with optimal U-2 matching. Furthermore, for the approimate power control algorithm, the gap between optimal U-2 matching and suboptimal U- 2 matching A is small. This is because when U-2 matching A is used to define the bipartite graph, the intracell interference a U causes to the matched 2 receiver is small. This results in a high 2 rate. Note that unlike the optimal U-2 matching solution, where the optimal powers for pairing each 2 pair and a U are needed to determine the best matching, for the suboptimal U-2 matching A scheme, onl the channel power of the intra-cell interference channel needs to be known at the BS. As a result, the computational compleit of suboptimum U- 2 matching A is drasticall reduced. V. ONLUSON n this paper, we have studied power control to imize the sum rate of a U and a 2 pair, subject to minimum

14 4 SNR requirements, power constraints, and worst-case constraints to neighboring cells. With optimal BS receive beamforming for uplink transmission, we have proposed an efficient approimate power control algorithm to obtain the powers of the U and 2 transmitters in closed form. epending on the conditions from the U and the 2 pair, we have divided the problem into five cases, each including several different scenarios due to minimum SNR requirements. The proposed algorithm is optimal when the to a single neighboring cell is considered. For multiple neighboring cells, we have given a performance bound on our proposed algorithm, and further provided conditions for which our approimation becomes optimal. We have further considered the general scenario of multiple Us and 2 pairs, and have shown how our previousl proposed solution can be utilized to find a solution to the joint power control and U-2 matching problem. Simulation demonstrates that substantial performance gain can be achieved b our proposed power control algorithm over two alternative approaches. t also shows our proposed approach provides close to optimal performance for the scenario of multiple Us and 2 pairs, despite its low compleit. Finall, we note that studing the sum rate imization problem in a scenario with imperfect S is an interesting topic for future work. APPENX A PROOF OF LEMMA Proof: onsidering 2 with equalit, we rewrite it as = η = K α A. +K 2 / where α = σ2 γ h, K 2 = ρ 2, and K 2 = σ 2 g. Taking the 2 first and second derivatives, we have dη d = αk K 2 K 2 +K 2 2 > 0, +K 2 / d 2 η d 2 = 2αK K 2 K K 3 +K 2 3 0, +K 2 / since K and K 2 > 0, i.e., η is a concave strictl increasing function. Note that constraint 5 is characterized b a line on the power plane, i.e., = β. σ 2 +K3 Solving the intersection of this line and the curve A., we obtain 3. APPENX B PROOF OF LEMMA 2 Proof: Given an power pair, in the interior of A, there eists ζ >, such that ζ,ζ A. n the following, we show that Rζ,ζ > R,. Substituting ζ,ζ into 23, we have a +Φζ Rζ,ζ = + σ 2 /ζ +K B. 3 where Φζ = ζ K K 2/ζ+ l. a t is straightforward to show that + σ 2 /ζ+k3 > + a for ζ >. n order to complete the proof, it is +K3 σ 2 sufficient to show that Φζ is an increasing function of ζ for a given,. Taking the first derivative, we have dφζ dζ = l K 2 K +ϕ ζ+k 2 /ζ 2 > 0 B.2 where ϕ = ζ 2 K +K 2 /ζ K 2 /ζ + K since K and ϕ > 0. Therefore, we have Rζ,ζ > R,, i.e., the optimal solution pair o, o is not in the interior of A. APPENX PROOF OF PROPOSTON Proof: Substituting = P and = P into 23, we define h = R,P and g = RP,. To show that the imum of g for ã b is obtained when o is at either ã or b, it is sufficient to show that g is a strictl monotonic function, or g is a strictl conve function. n the following, we show that g is such a function. n both cases, since P2 is a imization problem, o is an end point of the domain determined ba. A similar proof is also provided for h. The function g can be written as g = + α α + 2+ α 3 where α = ap /K 3, α 2 = σ 2 /K 3, and α 3 = l K P K 2+P. Taking the first derivative of g, we have dg d = α α 2 α 3 +µ α where µ = α 3 α α α 2 α 3. Since α 2 > 0, α 3 > 0, and 0, either dg d > 0, i.e., g is a strictl increasing function or dg d = 0 ma have a valid solution onl if µ < 0. Supposing µ < 0 and taking the second derivative, we have d 2 g d = 2α α2α3 2 α 2+ > 0, since µ < 0 implies α 3 α 2 α 3 > 0. n other words, g is a conve function. Similarl, h can be written as h = + β β 2 K K 2/+ where β = a σ 2 +K3P + and β 2 = lp. Taking the first derivative of h, we have dh d = ĥ+ω +K 2 2 where ĥ = β + β2 K 2 + 2β K 2 + β2 K and ω = β K2 2 +β β 2 K2 2 β 2 K K 2. Since K, K 2 > 0,β > 0,β 2 > 0, and 0, eitherh is a strictl increasing function or dh d = 0 ma have a valid solution onl if ω < 0. Supposing ω < 0 and taking the second derivative of h, we have d2 h d = 2βK2 +β 2 K K 2 2ω 2 +K 2 > 0, 3 i.e., h is a conve function. APPENX PROOF OF PROPOSTON 2 Proof: B Proposition, if o = P or o = P, then o, o is an end point of the vertical or horizontal boundar line segment of A, respectivel. f constraint

15 5 22 is active at optimalit, the optimal power is the solution of the following optimization problem: + l, a σ 2 +K 3 subject to c + =. + K K 2 + Substituting = /c into the objective function above, we have R, where R = + a a K 4 + b b 2 K K 2+. Since R is continuous and has a first-order derivative, the optimum o is either the-coordinate of an end point of the tilted line segment ofa or obtained b solving d R/d = 0, which results in the quartic equation e 4 4 +e 3 3 +e 2 2 +e +e 0 = 0 where e 0 = aa K 2 2 b + a 2 b K K 2 a 2 b 2K 2 2 e = 2aa b 2 K 2 2 +aa K 2 b + 2aa K K 2 b. +aa b K 2 +2a 2 b 2K 2 K +2a a 2 b 2 K2 2 +aa K 2 +2a a 2 b K K 2.2 e 2 = aa b 2 K 2 3K 4+aa +b K +a 2 b K K 2 a a 2 4a a 2 b 2 K 2 K +a 2 b 2 K 2 2a a 2 a 2 b 2 K.3 e 3 = 2aa b 2 K +2a a 2 b 2 K 2a 2 b 2 K 2 K a a 2 e 4 = a2 a a 2 b 2 K,.4.5 with a = σ 2 +K 3 /c, a 2 = K3 /c, b = l/c, and b 2 = l /c. APPENX E PROOF OF LEMMA 3 Proof: We show that the alternative solution of the quadratic equation is not feasible, i.e., ˆ F = θ θ 2 4 K K 2 αc 2 K < 0. E. n order to reject ˆ F, it is sufficient to show that αc <. Substituting P into 2 and considering ρ 2, we have P h 2 σ 2 > γ. E.2 Rearranging the inequalit E.2, we have αc <c P <, where the first and second inequalit hold due to the definition of α and condition of moderate from U 25, respectivel. APPENX F PROOF OF PROPOSTON 3 Proof: efining R L = l L {R l }, we have R A R L. F. A new feasible region A = A o A3 is formed b replacing the corner of A o with a rectangular one including U 3. Since A o A, we have R opt R U = {R U,R U2,R U3 } F.2 where the right-hand side of F.2 is the optimal sum rate within A b Proposition. ombining F. and F.2, we have the upper bound on the sum-rate loss of of Algorithm given b 56. Based on ondition, we have A = A o, and thus the solution of Algorithm is optimal. For onditions 2, it means the line and curve associated with 20 and 9, respectivel, both intersect the vertical boundar of A o, i.e., L = U and L 2 = U 2. n this case, b definition of U 3, we have U 3 = U. B F. and F.2, we have{r U,R U2 } R A R opt {R U,R U2 }. Thus, R A = R opt. For ondition 3, the line and curve both intersect the horizontal boundar of A o with L = U and L 2 = U 2. Following a similar argument, we have U 3 = U 2, and R A = R opt. REFERENES [] A. Ramezani-Kebra, M. ong, B. Liang, G. Boudreau, and S. H. Seedmehdi, Optimal power allocation in device-to-device communication with SMO uplink beamforming, in Proc. EEE SPAW, Stockholm, Sweden, June 205, pp [2] K. oppler, M. Rinne,. Wijting,. Ribeiro, and K. Hugl, eviceto-device communication as an underla to LTE-Advanced networks, EEE ommun. Mag., vol. 47, pp , ec [3] G. Fodor, E. ahlman, G. Mildh, S. Parkvall, N. Reider, G. Miklós, and Z. Turáni, esign aspects of network assisted device-to-device communications, EEE ommun. 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Quart., vol. 2, pp , Mar Min ong S 00-M 05-SM 09 received the B.Eng. degree from Tsinghua Universit, Beijing, hina, in 998, and the Ph.. degree in electrical and computer engineering with minor in applied mathematics from ornell Universit, thaca, NY, in From 2004 to 2008, she was with orporate Research and evelopment, Qualcomm nc., San iego, A. n 2008, she joined the epartment of Electrical, omputer and Software Engineering at Universit of Ontario nstitute of Technolog, Ontario, anada, where she is currentl an Associate Professor. She also holds a status-onl Associate Professor appointment with the epartment of Electrical and omputer Engineering, Universit of Toronto since Her research interests are in the areas of statistical signal processing for communication networks, cooperative communications and networking techniques, and stochastic network optimization in dnamic networks and sstems. r. ong received the the 2004 EEE Signal Processing Societ Best Paper Award, the Best Paper Award at EEE in 202, and the Earl Researcher Award from Ontario Ministr of Research and nnovation in 202. She is the co-author of the Best Student Paper Award of Signal Processing for ommunications and Networks in EEE ASSP 6. She served as an Associate Editor for the EEE TRANSATONS ON SGNAL PROESSNG , and as an Associate Editor for the EEE SGNAL PROESSNG LETTERS She was a smposium lead co-chair of the ommunications and Networks to Enable the Smart Grid Smposium at the EEE nternational onference on Smart Grid ommunications SmartGridomm in 204. She has been an elected member of EEE Signal Processing Societ Signal Processing for ommunications and Networking SP-OM Technical ommittee since 203. Ben Liang S 94-M 0-SM 06 received honorssimultaneous B.Sc. valedictorian and M.Sc. degrees in Electrical Engineering from Poltechnic Universit in Brookln, New York, in 997 and the Ph.. degree in Electrical Engineering with a minor in omputer Science from ornell Universit in thaca, New York, in 200. n the academic ear, he was a visiting lecturer and postdoctoral research associate with ornell Universit. He joined the epartment of Electrical and omputer Engineering at the Universit of Toronto in 2002, where he is now a Professor. His current research interests are in networked sstems and mobile communications. He has served as an editor for the EEE Transactions on ommunications since 204, and he was an editor for the EEE Transactions on Wireless ommunications from 2008 to 203 and an associate editor for Wile Securit and ommunication Networks from 2007 to 206. He regularl serves on the organizational and technical committees of a number of conferences. He is a senior member of EEE and a member of AM and Tau Beta Pi. Ali Ramezani-Kebra S 3 received the B.Sc. degree from the Universit of Tehran, Tehran, ran, and the M.A.Sc degree from Queen s Universit, Kingston, anada, respectivel. Then he joined the Wireless omputing Lab WL of the epartment of Electrical and omputer Engineering, the Universit of Toronto, Toronto, anada, where he is currentl a Ph.. candidate. His research interests cover the broad area of nternet of Things, with special emphasis on 2/M2M/V2V sstems, optimization, learning algorithms, and resource allocation. Mr. Ramezani-Kebra was the recipient of the V. L. Henderson and M. Bassett Research Fellowship, the EEE Kingston Section M.Sc. Research Ecellence Award, and Ontario Graduate Scholarship OGS. Gar Boudreau M 84-SM received a B.A.Sc. in Electrical Engineering from the Universit of Ottawa in 983, an M.A.Sc. in Electrical Engineering from Queens Universit in 984 and a Ph.. in electrical engineering from arleton Universit in 989. From 984 to 989 he was emploed as a communications sstems engineer with anadian Astronautics Limited and from 990 to 993 he worked as a satellite sstems engineer for MPR Teltech Ltd. For the period spanning 993 to 2009 he was emploed b Nortel Networks in a variet of wireless sstems and management roles within the MA and LTE basestation product groups. n 200 he joined Ericsson anada where he is currentl emploed in the LTE sstems architecture group. His interests include digital and wireless communications as well as digital signal processing.

17 Hossein Seedmehdi received a Master s degree from the National Universit of Singapore in 2008 and a Ph degree from the Universit of Toronto in 204 both in Electrical and omputer Engineering. He is currentl affiliated with Ericsson anada where he is working on the 5thgeneration 5G of wireless technologies including the commercialization of massive MMO. He has numerous papers in the area of wireless sstems in addition to more than 0 patents. His research interests include nformation Theor for Wireless ommunications, Signal Processing for MMO channels, Algorithms for Radio Access Networks, and Future Radio Technologies. 7