Power Efficient Resource Allocation for. Full-Duplex Radio Distributed Antenna Networks

Transcription

1 Power Efficient Resource Aocation for 1 Fu-Dupex Radio Distributed Antenna Networs Derric Wing Kwan Ng, Yongpeng Wu, and Robert Schober arxiv:submit/ [cs.it] 7 Feb 2015 Abstract In this paper, we study the resource aocation agorithm design for distributed antenna mutiuser networs with fu-dupex (FD radio base stations (BSs which enabe simutaneous upin and downin communications. The considered resource aocation agorithm design is formuated as an optimization probem taing into account the antenna circuit power consumption of the BSs and the quaity of service (QoS requirements of both upin and downin users. We minimize the tota networ power consumption by jointy optimizing the downin beamformer, the upin transmit power, and the antenna seection. To overcome the intractabiity of the resuting probem, we reformuate it as an optimization probem with decouped binary seection variabes and non-convex constraints. The reformuated probem faciitates the design of an iterative resource aocation agorithm which obtains an optima soution based on the generaized Bender s decomposition (GBD and serves as a benchmar scheme. Furthermore, to strie a baance between computationa compexity and system performance, a suboptima agorithm with poynomia time compexity is proposed. Simuation resuts iustrate that the proposed GBD based iterative agorithm converges to the goba optima soution and the suboptima agorithm achieves a cose-to-optima performance. Our resuts aso demonstrate the trade-off between power efficiency and the number of active transmit antennas when the circuit power consumption is taen into account. In particuar, activating an exceedingy arge number of antennas may not be a power efficient soution for reducing the tota system power consumption. In addition, our resuts revea that FD systems faciitate significant power savings compared to traditiona haf-dupex systems, despite the non-negigibe sef-interference. Index Terms Distributed antennas, fu-dupex radio, antenna seection, non-convex optimization, resource aocation. Derric Wing Kwan Ng, Yongpeng Wu, and Robert Schober are with the Institute for Digita Communications (IDC, Friedrich- Aexander-University Erangen-Nürnberg (FAU, Germany (emai:{wan, yongpeng.wu, schober}@nt.de. Derric Wing Kwan Ng and Robert Schober are aso with the University of British Coumbia, Vancouver, Canada.

2 2 I. INTRODUCTION The next generation wireess communication systems are required to support ubiquitous and high data rate communication appications with guaranteed quaity of service (QoS. These requirements transate into a tremendous demand for bandwidth and energy consumption. Mutipeinput mutipe-output (MIMO is a viabe soution for addressing these issues as it provides extra degrees of freedom in the spatia domain which faciitates a trade-off between mutipexing gain and diversity gain. Hence, a arge amount of wor has been devoted to MIMO communication over the past decades [1], [2]. However, the modest computationa capabiities of mobie devices imit the MIMO gains that can be achieved in practice. An attractive aternative for reaizing the performance gains offered by mutipe antennas is mutiuser MIMO, where a mutipeantenna transmitter serves mutipe singe-antenna receivers simutaneousy [3], [4]. In fact, the combination of mutiuser MIMO and distributed antennas is widey recognized as a promising technoogy for mitigating interference and extending service coverage [5] [7]. Specificay, distributed antennas introduce additiona capabiities for combating both path oss and shadowing by shortening the distances between the transmitters and the receivers. Nevertheess, if the number of antennas is very arge, the circuit power consumption of distributed antenna networs becomes non-negigibe compared to the power consumed for transmission. However, this probem has not been considered in most of the existing iterature [5] [7] on power efficient communication networ design. Furthermore, even with these powerfu MIMO techniques, spectrum scarcity is sti a major obstace in providing high speed upin and downin communications. Traditiona communication systems are designed for haf-dupex (HD transmission since this mode of operation faciitates ow-compexity transceiver design. In particuar, upin and downin communication are staticay separated in either time or frequency, e.g. via time division dupex or frequency division dupex, which eads to a oss in spectra efficiency. Even though different approaches have been proposed for improving the spectra efficiency of HD systems, e.g. dynamic upin-dowin scheduing/aocation in time division dupex communication systems [8], [9], the fundamenta spectra efficiency oss induced by the HD constraint remains unsoved. On the contrary, fu dupex (FD transmission aows downin and upin transmission to occur simutaneousy at the same frequency. In fact, FD radio has the potentia to doube the spectra efficiency of conventiona HD communication systems. However, in practice, the downin

3 3 transmission in FD systems creates sef-interference to the upin receive antennas which can be exceedingy arge compared to the received power of the usefu information signas. In fact, the huge difference in the power eves of the two signas saturates the dynamic range of the anaogto-digita converter (ADC essentiay preventing FD communication. Fortunatey, severa recent breathroughs in hardware (/signa processing agorithm design for suppressing sef-interference have been reported and FD radio prototypes have been successfuy presented [10] [14]. As a resut, FD radio has regained the attention of both industry [15] [18] and academia [19] [23]. In [19], the authors studied techniques for sef-interference suppression and canceation for FD mutipe-antenna reays. In [20], the outage probabiity of MIMO FD singe-user reaying systems was investigated. In [21], a resource aocation agorithm was proposed for maximization of the achievabe end-to-end system data rate of muticarrier mutiuser MIMO FD reaying systems. In [22], a suboptima beamformer design was considered to improve the spectra efficiency of a FD radio base station enabing simutaneous upin and downin communication. In [23], the concept of FD communication was extended to the case of massive MIMO where a FD radio reay is equipped with a arge number of antennas for suppressing the sef-interference and for enhancing the system throughput. However, the benefits of mutipe-antenna FD radio do not come for free. The rapidy escaating cost caused by the power consumption of the circuitries of arge antenna systems has ead to significant financia impications for service providers, which is often overooed in the iterature [10]-[23]. In fact, the systems in [10]-[23] are designed to serve pea service demands by activating a avaiabe antennas of the system, without considering the power consumption in the off-pea periods. However, the service oads vary across a wireess networ in practice, depending on the geographic ocation of the receivers and the time of day. Thus, we expect that extra power savings can be achieved by dynamicay switching off some of the antennas. Nevertheess, the optima number of active antennas has not been investigated from a system power efficiency point of view for FD radio communication, yet. In addition, there may be fewer degrees of freedom for sef-interference suppression at each FD radio base station in distributed antenna systems if the tota number of antennas in the networ is fixed. Thus, it is uncear whether the distributed antenna architecture eads to power savings for FD radio communication. Furthermore, unie for the orthogona transmission adopted in HD systems, the upin and downin transmit powers are couped in FD systems which mae the design of efficient resource aocation agorithms particuary chaenging.

4 4 In this paper, we address the above issues and study the resource aocation agorithm design for mutiuser distributed antenna communication networs. We minimize the tota networ power consumption whie taing into account the circuit power consumption of the distributed BS antennas and ensuring the QoS of both upin and downin users. In particuar, we propose an optima iterative resource aocation agorithm based on the generaized Bender s decomposition [24] [26]. Furthermore, we propose a suboptima resource aocation scheme with poynomia time computationa compexity based on the difference of convex functions (d.c. programming [27] which finds a oca optima soution for the considered optimization probem. A. Notation II. SYSTEM MODEL Matrices and vectors are represented by bodface capita and ower case etters, respectivey. A H, A T, Tr(A, and Ran(A represent the Hermitian transpose, the transpose, the trace, and the ran of matrix A, respectivey; A 0 and A 0 indicate that A is a positive definite and a positive semidefinite matrix, respectivey;i N is the N N identity matrix; C N M and H N denote the sets of a N M matrices and N N Hermitian matrices with compex entries, respectivey; diag(x 1,,x K denotes a diagona matrix with the diagona eements given by {x 1,,x K }; denotes the absoute vaue of a compex scaar; the circuary symmetric compex Gaussian distribution is denoted by CN(µ, C with mean vector µ and co-variance matrix C; stands for distributed as ; E{ } denotes statistica expectation; and x f denotes the gradient of a function f with respect to vector x. B. System Mode We consider a distributed antenna mutiuser communication networ. The system consists of a centra processor (CP, L FD radio base stations (BSs, and K mobie users, cf. Figure 1. Each FD radio BS is equipped with N T > 1 antennas for downin transmission and upin reception 1. The K users empoy singe-antenna HD mobie communication devices to ensure ow hardware compexity. In particuar, K U and K D users are schedued for simutaneous upin and downin transmission, respectivey, such that K U +K D = K. On the other hand, the CP is the core unit of the networ. In particuar, the FD radios are connected to the CP via bachau ins. In addition, 1 We assume that the antennas equipped at the FD BSs can transmit and receive simutaneousy which has been successfuy demonstrated in some FD radio prototypes [12].

5 Upin user 1 5 FD radio BS 1 Downin user 1 FD radio BS 2 Signa Co-channe Interference Bachau connection Centra processor (CP FD radio BS 3 Fig. 1. Mutiuser downin distributed antenna communication system mode with L = 3 fu dupex (FD radio base stations (BSs, K U = 1 upin user, and K D = 1 downin user. For the depicted case, the antennas equipped at FD radio BS 2 are switched to ide mode for reducing the tota power consumption in the networ. the CP has the fu channe state information of the entire networ and the data of a downin users for resource aocation. In this paper, we assume that the CP is a powerfu computing unit, e.g. a series of baseband units as in coud radio access networs (C-RAN, which computes the resource aocation poicy and broadcasts it to a FD radio BSs. Each FD radio BS receives the contro signas for resource aocation and the data of the K D downin users from the CP via a bachau in. Furthermore, the FD radio BSs transfer the received upin signas via bachau ins to the CP, where the information is decoded. In this paper, we assume that the bachau ins are impemented with optica fiber and have sufficienty arge capacity and ow atency to support rea time information exchange between the CP and the FD radio BSs. For studies on the impact of a imited bachau capacity on the performance of wireess systems, pease refer to [2], [28]. C. Channe Mode A frequency fat fading channe is assumed 2 in this paper. The received signas at downin user {1,...,K D } and the L FD radio BSs are given by K U y DL = h H D x+ Pj Ug j,d U j +n and (1 y UL = K U j=1 j=1 }{{} co channe interference P U j h U j d U j + }{{} H SIx +z, (2 sef interference 2 The frequency fat fading channe can be interpreted as one subcarrier of an orthogona frequency division mutipexing system.

6 6 respectivey, where x C N TL 1 denotes the joint transmit signa vector of the L FD radio BSs to thek D downin users. The downin channe between the L FD radio BSs and user is denoted by h D C N TL 1, and we use g j, C to represent the channe between upin user j and downin user. d U j and P U j are the transmit data and transmit power sent from upin user j to the L FD radio BSs, respectivey. h Uj C N TL 1 is the upin channe between upin user j and the L FD radio BSs. Due to simutaneous upin reception and downin transmission at the FD radio BSs, sef-interference from the downin impairs the upin signa reception. In practice, different interference mitigation techniques such as antenna canceation, baun canceation, and circuators [12], [13] have been proposed to aeviate the impairment caused by sef-interference. In order to isoate the resource aocation agorithm design from the specific impementation of sef-interference mitigation, we mode the residua sef-interference after interference canceation by matrix H SI C N TL N. Variabes h D, g j,, H SI, and h Uj capture the joint effect of path oss and mutipath fading. z CN(0,σ 2 zi N and n CN(0,σ 2 n represent the additive white Gaussian noise (AWGN at the L FD radio BSs and user, respectivey. In each scheduing time sot, K D independent signa streams are transmitted simutaneousy at the same frequency to the K D downin users. Specificay, a dedicated downin beamforming weight, w C, is aocated to downin user at the -th, {1,...,N TL}, antenna to faciitate downin information transmission. For the sae of presentation, we define a supervector w C NTL 1 for downin user as w = [ w 1 w2... ] wn TL T. (3 w represents the joint beamformer used by the N antennas shared by the FD radio BSs for serving downin user. Then, the information signa to downin user, x, can be expressed as x = w d D, (4 where d D C is the data symbo for downin user. Without oss of generaity, we assume that E{ d D 2 } = E{ d U j 2 } = 1, {1,...,K D },j {1,...,K U }. D. Networ Power Consumption Mode In our system mode, we incude the circuit power consumption of the system in the objective function in order to design a resource aocation agorithm which faciities power-efficient

7 7 communication. Thus, we mode the power dissipation in the system as the sum of one static term and four dynamic terms as foows [24]: U TP (w,s,pj U = P 0 + s P Active + (1 s P Ide } {{ } Antenna power consumption K D K U +η ε D w 2 + ε U ζ j Pj U, (5 j=1 } {{ } Ampifer power consumption wherep 0 is the aggregated static power consumption of the CP, a FD radio BSs, and a bachau ins. s {0,1} is a binary seection variabe. In particuar, s = 1 and s = 0 indicate that the -th antenna in the FD communication system is in active mode and ide mode, respectivey, s wi be optimized to minimize the tota networ power consumption in the next section. P Active > 0 is the signa processing power that is consumed if an antenna is active. P Active incudes the power dissipations of the transmit fiter, mixer, frequency synthesizer, digita-toanaog converter, etc. In this paper, an FD radio antenna is considered active if it serves at east one user in the system. P Ide > 0 is the required power consumption of an antenna in ide mode, i.e., if it is not serving any user, and P Active > P Ide hods in genera. K D N w 2 is the tota power radiated by the L FD radio BSs for downin transmission. ε D 1 and ε U 1 are constants which account for the inefficiency of the power ampifier 3 adopted for downin and upin transmission, respectivey. In other words, ε KD N D w 2 and ε KU U j=1 PU j are the tota power consumptions of the power ampifiers for downin and upin transmission, respectivey. η 0 and ζ j 0 in the ast two terms of (5 are constant weights which can be chosen by the system designer to prioritize the importance of the tota downin transmit power and the transmit power of individua upin users j {1,...,K U }, respectivey. III. PROBLEM FORMULATION In this section, we first introduce the QoS metrics for the considered FD radio communication networ. Then, we formuate the resource aocation agorithm design as a non-convex optimization probem. 3 We assume Cass A power ampifiers with inear characteristic are impemented in the transceivers. In practice, the maximum power efficiency of Cass A ampifiers is 25%.

8 8 A. Achievabe Data Rate The achievabe data rate (bit/s/hz between the L FD radio BSs and downin user {1,...,K D } is given by C = og 2 (1+Γ DL, where ΓDL = h H D w 2 K D h H D w t 2 + (6 K U j=1 PU j g j, 2 +σn 2 is the receive signa-to-interference-pus-noise ratio (SINR at downin user. t On the other hand, we assume that the CP empoys a inear receiver for decoding of the received upin information. Therefore, the achievabe data rate between the L FD radio BSs and upin user j is given by ( Cj UL = og 2 1+Γ UL j I j = v H j ( K D, Γ UL j = PU j vh j h U j 2 σ 2 z v j 2 +I j, (7 H SI w w H HH SI v j + K U r j P U r vh j h U r 2, (8 where v j C N TL 1 is the receive beamforming vector for decoding of the information for upin user j. In this paper, maximum ratio combining (MRC is adopted, i.e., the receive beamformer for upin userj is chosen asv j = N s R h Uj to maximize the signa strength of the received ( signa, where R diag 0,,0,1,0,,0, {1,...,LN }{{}}{{} T }, is a diagona matrix. It is ( 1 LN T nown that MRC achieves a good system performance, especiay if a arge number of antennas is empoyed, and has been widey adopted in the iterature [3], [4]. Remar 1: We note that zero-forcing beamforming (ZFBF or minimum mean square error beamforming (MMSE-BF are not considered for upin signa detection since they do not faciitate an efficient resource aocation agorithm design for the considered networ. Using MRC, the upin SINR of user j is given by ( Pj U Tr h Uj h H N U j m=1 Γ UL j = I j = ( K D Tr N n=1 s ms n R m h Uj h H U j R H n σ 2 z Tr ( N s h Uj h H U j R +I j, where (9 H SI w w H H H SI K U + Pr U Tr (h Ur h HUr r j s m s n R m h Uj h H U j R H n s m s n R m h Uj h H U j R H n (10. (11

9 9 B. Optimization Probem Formuation The system objective is to minimize the tota networ power consumption whie providing QoS for reiabe communication to both upin and downin users simutaneousy. We obtain the optima resource aocation agorithm poicy by soving the foowing optimization probem: minimize U TP (w,s,p U w,s,pj U j s.t. C1: h H D w 2 DL K D h H D w t 2 + Γreq, {1,...,K D }, K U j=1 PU j g j, 2 +σn 2 t C2: Γ UL j C3: K D Γ UL req j, j {1,...,K U }, w 2 s Pmax DL, {1,...,N }, C4: 0 Pj U Pmax U j, j {1,...,K U }, C5: s {0,1}, {1,...,N }. (12 Γ DL req and Γ UL req j in constraints C1 and C2 denote the minimum receive SINR required by downin user and upin user j for successfu information decoding, respectivey. In C3, we constrain the maximum radiated power of the -th antenna in the system to P DL max to satisfy the maximum power spectra mas imit. C4 imits the maximum transmit power and ensures the non-negativity of the transmit power of upin user j. C5 constrains the optimization variabes which contro the active and ide states of the antennas in the system to be binary. Remar 2: In this paper, energy/power saving is achieved by optimizing not ony the upin and downin transmit powers, but aso by optimizing the states of the antennas in the networ. Thereby, it is expected that switching the antennas on and off adaptivey according to the channe conditions is an effective strategy for reducing the networ power consumption when the QoS requirements are not stringent or the number of users is ow. IV. RESOURCE ALLOCATION ALGORITHM DESIGN The optimization probem in (12 is a mixed non-convex and combinatoria optimization probem. The combinatoria nature is due to the binary seection variabes in C5. Aso, variabe s is couped with both downin beamforming vector w and upin power aocation variabe Pj U in constraint C2. Furthermore, constraint C1 is non-convex with respect to w. In the foowing, we first transform the optimization probem into an equivaent form and obtain the

10 10 goba optima soution by using the generaized Bender s decomposition. Then, we propose a suboptima poynomia time agorithm which is inspired by the difference of convex functions program. A. Probem Reformuation In this section, we reformuate the considered optimization probem in (12 using the definitions W = w w H, H D = h D h H D, and H Uj = h Uj h H U j. This eads to K D K U minimize P 0 + s P Active + (1 s P Ide +η ε D Tr(W +ε U ζ j P W H N,s,Pj U j U,qm,n s.t. C1: Tr(H D W Γ DL req C2: P U j Γ UL req j Tr K D K U Tr(H D W j + Pj U g j, 2 +σn 2,, t (H Uj ( σz 2 Tr q, H Uj R +Tr + r j Tr C3: K D (H Ur j=1 q m,n R m H Uj R H n ( K D H SI W H H SI Pr U q m,n R m H Uj R H n, j {1,...,K U }, Tr(W R s P DL max, {1,...,N }, C4, C5, q m,n R m H Uj R H n C6: W 0,, C7: Ran(W 1,, C8: 0 q m,n s m, m,n {1,...,N }, C9: q m,n s n, m,n, C10: q m,n s n +s m 1, m,n. (13 Constraints C6, C7, and W H N TL,, are imposed to guarantee that W = w w H hods after optimization. q m,n is an auxiiary continuous optimization variabe which is introduced to hande the product of two binary variabes s n s m in constraint C2, cf. (9 (11. In particuar, because of constraints C8 C10, q m,n wi have a binary vaue if s is binary. We note that constraint C2 is sti non-convex due to the product terms q m,n P U t and q m,n W which is an obstace for the design of a computationay efficient resource aocation agorithm. In order to circumvent this difficuty, we adopt the big-m formuation [29], [30] to decompose the product terms. First, we introduce auxiiary variabes P U j,m,n = PU j q m,n and j=1 W m,n = W q m,n.

11 11 Then, we impose the foowing additiona constraints: C11: PU j,m,n P U max j q m,n, j,m,n, C12: PU j,m,n P U j, j,m,n, (14a C13: PU j,m,n P U j (1 q m,n P U max j, j,m,n, C14: PU j,m,n 0, (14b C15: C17: W m,n W m,n I NP DL max q m,n,,m,n, C16: W (1 q m,n I NP DL max,,m,n, C18: W m,n W,,m,n, (14c W m,n 0,,m,n. (14d In particuar, constraints C11-C18 invove ony continuous optimization variabes, i.e., P j, P j,m,n, q m,n, and W, which faciitates the design of an efficient resource aocation agorithm. Subsequenty, we substitute P U j,m,n = PU j q m,n and which yieds C2: + 1 Γ UL req j Tr (H Uj ( σz 2 Tr q, H Uj R +Tr Tr r j (H Ur W m,n P U j,m,n R mh Uj R H n ( K D = W q m,n into the couped variabes in C2 P U r,m,nr m H Uj R H n H SI Wm,n H H SIR m H Uj R H n, j {1,...,K U }. (15 The big-m formuation inearizes the terms q m,n P U r and q m,n W such that constraint C2 is an affine function with respect to the new optimization variabes P U j,m,n and W m,n. We note that constraints C2 and C2 are equivaent when constraints C5 and C11 C18 are satisfied. As a resut, the considered optimization probem (13 can be transformed into the foowing equivaent probem: minimize W H N, W m,n,s,q m,n, P j U, P j,m,n U P 0 + s.t. C1, C2, C3, C4, C6, C8 C18, K D K U s P Active + (1 s P Ide +η ε D Tr(W + ε U ζ j Pj U C5: s {0,1},, C7: Ran(W 1,, (16 and we can focus on the design of an agorithm for soving the optimization probem in (16. Now, the remaining non-convexity of optimization probem (16 is due to constraints C5 and C7. Remar 3: We note that the upin-downin duaity approach in [31], [32] cannot be appied to our probem for the foowing two reasons. First, the upin and downin transmit power j=1

12 12 variabes are couped in constraints C1 and C2. Second, the upin and downin transmit powers of each transceiver are constrained. B. Optima Iterative Resource Aocation Agorithm Now, we adopt the generaized Bender s decomposition (GBD to hande the constraints invoving binary optimization variabes [24] [26], i.e., C3, C8, C9, and C10. In particuar, we decompose the probem in (16 into two sub-probems: (a a prima probem which is a non-convex optimization probem invoving continuous optimization variabes{w, W m,n,p U j, P U j,m,n,q m,n }; (b a master probem which is a mixed integer inear program (MILP. Specificay, the prima probem is soved for given s which yieds an upper bound for the optima vaue of (16. In contrast, the soution of the master probem provides a ower bound for the optima vaue of (16. Subsequenty, we sove the prima and master probems iterativey unti the soutions converge. In the foowing, we first propose agorithms for soving the prima and master probems in the i-th iteration, respectivey. Then, we describe the iterative procedure between the master probem and the prima probem. 1 Soution of the prima probem in the i-th iteration: For given and fixed input parameters s = s (i obtained from the master probem in the i-th iteration, we minimize the objective function with respect to variabes {W, minimize W H N, W m,n,q m,n, P j U, P j,m,n U W m,n,p U j, P U j,m,n,q m,n} in the prima probem: K D K U P 0 + s P Active + (1 s P Ide +η ε D Tr(W +ε U ζ j Pj U s.t. C1, C2, C3, C4, C16 C18. (17 We note that constraint C5 in (16 wi be handed by the master probem since it invoves ony the binary optimization variabe s. Now, the ony obstace in soving (17 is the combinatoria ran constraint in C7 and we adopt the SDP reaxation approach to hande this non-convexity. In particuar, we reax constraint C7: Ran(W 1 by removing it from the probem formuation, such that the considered probem in (17 becomes a convex SDP and can be soved efficienty by numerica methods designed for convex programming such as interior point methods [33]. If the soution W of the reaxed version of (17 is a ran-one matrix for a downin users, then the probem in (17 and its reaxed version share the same optima soution and the same optima objective vaue. j=1

13 13 Now, we study the tightness of the adopted SDP reaxation. The SDP reaxed version of (17 is jointy convex with respect to the optimization variabes and satisfies Sater s constraint quaification. Thus, strong duaity hods and soving the dua probem is equivaent to soving (17. To obtain the dua probem, we define the Lagrangian of the reaxed version of (17 as ( ( L Θ,Φ = U TP W,s,Pj U +f 1 (Θ,Φ+f 2 (Θ,Φ, where (18 U TP (W,s,P U j = P 0 + s P Active + (1 s P Ide +η K D K U f 1 (Θ,Φ = Tr(Z W K D [ + α Tr(H D W ( K U Tr + ψ j j=1 ( K D +Tr K U + j=1 K U + j=1 K D + K D + Γ DL req t ( H Uj N m=1 j=1 K D K U β j,m,n PU j,m,n + λ j (Pj U Pmax U j j=1 K U ] Tr(H D W j + Pj U g K U j, 2 +σz 2 χ j Pj U j=1 N n=1 P U j,m,n R mh Uj R H n Γ UL req j H SI Wm,n H H SIR m H Uj R H n K U µ j,m,n ( P j,m,n U PU max j q m,n + K D + N τ j,m,n ( P j,m,n U PU j { Tr { Tr K D f 2 (Θ,Φ = + D C15,m,n ( Wm,n ϕ m,n (q m,n s n + Here,Θ = {W,s,P U j + Tr r j j=1 ς m,n q m,n K U ε D Tr(W +ε U ζ j Pj U (19 j=1 ( +σz 2 Tr q, H Uj R (H Ur j=1 P r,m,nr U m H Uj R H n ξ j,m,n (Pj U (1 q m,n Pmax U j P j,m,n U ( Wm,n } I NPmax DL q m,n +D C16,m,n W D C17,m,n ( W (1 q m,n I NP DL max W m,n ρ ( Tr(W R s P DL max + } D C18,m,n Wm,n, and (20 κ m,n (q m,n s m ω m,n (s n +s m 1 q m,n. (21, W m,n, P U j,m,n,q m,n} andφ = {α,ψ j,ρ,{λ j,χ j },Z,{ς m,n,κ m,n },ϕ m,n, ω m,n, µ j,m,n τ j,m,n,ξ j,m,n,β j,m,n,d C15,m,n,D C16,m,n,D C17,m,n,D C18,m,n } are the coections of

14 14 prima and dua variabes, respectivey; α 0,ψ j 0,ρ 0,{λ j,χ j } 0,Z 0,ς m,n 0,κ m,n 0,ϕ m,n 0,ω m,n 0,µ j,m,n 0, τ j,m,n 0,ξ j,m,n 0,β j,m,n 0,D C15,m,n 0,D C16,m,n 0,D C17,m,n 0, and D C18,m,n 0, are the scaar/matrix dua variabes for constraints C1 C4, C8 C18, respectivey. Function U TP (W,s,Pj U in (19 is the objective function of the SDP reaxed version of probem (17; f 1 (Θ,Φ in (20 is a function invoving the constraints that do not depend on the binary optimization variabes; f 2 (Θ,Φ in (21 is a function invoving the constraints incuding s (i. These functions are introduced here for notationa simpicity and wi be expoited for faciitating the presentation of the soutions for both the prima probem and the master probem. For a given s, the dua probem of the SDP reaxed optimization probem in (17 is given by ( maximize minimizel Θ,Φ. (22 Φ Θ We define Θ (i = {W,s,P U j, m,n W, P U j,m,n,q m,n } and Φ(i = {Φ } as the optima prima soution and the optima dua soution of the SDP reaxed probem in (17 in the i-th iteration. Now, we introduce the foowing theorem regarding the tightness of the adopted SDP reaxation. Theorem 1: Assuming the channe vectors of the downin users, h D, {1,...,K D }, can be modeed as statisticay independent random variabes, then the soution of the SDP reaxed version of (17 is ran-one, i.e., Ran(W = 1,, with probabiity one. Thus, the optima downin beamformer for user, i.e., w, is the principa eigenvector of W. Proof: Pease refer to Appendix A for a proof of Theorem 1. On the other hand, we formuate an 1 -minimization probem for the case when (17 is infeasibe for given binary variabes s (i. The 1 -minimization probem is given as: minimize W H N, W m,n,q m,n, P j U, P j,m,n U,νC3,ν m,n C8,νC9 m,n,νc10 m,n ν C3 + s.t. C1, C2, C4, C6, C11 C18, C3: Tr(W R s (ip DL max +ν C3, {1,...,N }, ν C8 m,n + ν C9 m,n +ν C10 m,n (23 C8: 0 q m,n s m (i+ν C8 m,n, m,n {1,...,N TL}, C9: q m,n s n (i+ν C9 m,n, m,n, C10: ν C10 m,n +q m,n s n (i+s m (i 1, m,n, C19: ν C3,ν C8 m,n,ν C9 m,n,ν C10 m,n 0,,m,n,.

15 15 Equation (23 is an SDP probem and can be soved by interior point methods with poynomia time computationa compexity. We note that the objective function in (23 is the sum of the constraint vioations with respect to the probem in (17. Besides, the corresponding dua variabes and the optima prima variabes wi be used as the input to the master probem for the next iteration [25]. We adopt a simiar notation as in (13 to denote the prima and dua variabes in (23. In particuar, the prima and dua soutions for the 1 -minimization probem in (23 are denoted as Θ = {W,s,P U j, W m,n, P U j,m,n,q m,n } and Φ = {α,ψ j,ρ,{λ j,χ j },Z,ς m,n,κ m,n,ϕ m,n,ω m,n, µ j,m,n, τ j,m,n,ξ j,m,n,β j,m,n,d C15,m,n,D C16,m,n, D C17,m,n,D C18,m,n }, respectivey. The prima and dua variabes wi be expoited as inputs for the constraints of the master probem. 2 Soution of the master probem in the i-th iteration: For notationa simpicity, we define F and I as the sets of a iteration indices at which the prima probem is feasibe and infeasibe, respectivey. Then, we formuate the master probem which utiizes the soutions of (13 and (23. The master probem in the i-th iteration is given as foows: minimize,s (24a s.t. C5, (24b ξ(φ(t,s,t {1,...,i} F, (24c 0 ξ(φ(t,s,t {1,...,i} I, (24d where s and are optimization variabes for the master probem and ξ(φ(t,s = minimize U TP (W,s,Pj U +f 1 (Θ,Φ(t+f 2 (Θ,Φ(t, (25 W H N, W m,n,q m,n, P j U, P j,m,n U ξ(φ(t,s = minimize W H N, W m,n,q m,n, P j U, P j,m,n U f 1 (Θ,Φ(t+f 2 (Θ,Φ(t. (26 Equations (25 and (26 are two different minimization probems defining the constraint set of the master probem in (24. In particuar, ξ(φ(t,s,t {1,...,i} F in (24c and 0 ξ(φ(t,s,t {1,...,i} I in (24d denote the sets of hyperpanes spanned by the optimaity cut and the feasibiity cut from the first to the i-th iteration, respectivey. The two different types of hyperpanes reduce the search region for the goba optima soution. Moreover, both ξ(φ(t,s and ξ(φ(t,s are aso functions of s which is the optimization variabe of the outer minimization in (24.

16 TABLE I OPTIMAL ITERATIVE RESOURCE ALLOCATION ALGORITHM BASED ON GBD 16 Agorithm Generaized Bender s Decomposition 1: Initiaize the maximum number of iterations L max, UB(0 =, LB(0 =, and a sma constant ϑ 0 2: Set iteration index i = 1 and start with s (i = 1,, 3: repeat {Loop} 4: Sove (17 by SDP reaxation for a given set of s (i 5: if (17 is feasibe then 6: Obtain an intermediate resource aocation poicy Θ (i = {W,s,Pj U m,n U, W, P j,m,n,q m,n}, the corresponding Lagrange mutipier set Φ(i, and an intermediate objective vaue f 0 7: The upper bound vaue is updated with UB(i = min{ub(i 1,f 0}. If UB(i = f 0, we set the current optima 8: ese poicy Θ current = Θ(i 9: Sove the 1-minimization probem in (23 and obtain an intermediate resource aocation poicy Θ(i = 10: end if {W,s,Pj U m,n U, W, P j,m,n,q m,n} and the corresponding Lagrange mutipier set Φ(i 11: Sove the master probem in (24 for s, save s (i+1 = s, and obtain the i-th ower bound, i.e., LB(i 12: if LB(i UB(i ϑ then 13: Goba optima = true, return {W,s,P U j, 14: ese 15: i = i+1 16: end if 17: unti i = L max m,n W, P j,m,n,q U m,n} = {Θ current} Now, we introduce the foowing proposition for the soution of the two minimization probems in (25 and (26. Proposition 1: The soutions of (25 and (26 for index t {1,...,i} are the soutions of (17 and (23 in the t-th iteration, respectivey. Proof: Pease refer to Appendix B for a proof of Proposition 1. The master probem in (24 is transformed to an MILP by appying Proposition 1 to sove (25 and (26. Hence, the master probem can be soved by using standard numerica sovers for MILPs such as Mose [34] and Gurobi [35]. We note that an additiona constraint is imposed to the master probem in each additiona iteration, thus the objective vaue of (24, i.e., the ower bound of (16, is a monotonicay non-decreasing function with respect to the number of iterations. 3 Overa agorithm: The proposed iterative resource aocation agorithm is summarized in Tabe I and is impemented by a repeated oop. For the initiation, we first set the iteration index i to one and the binary variabes s (i to one, e.g. s (1 = 1,. In the i-th iteration, we sove the probem in (17 via SDP reaxation. Two different types of Lagrange mutipiers are

17 17 defined depending on the feasibiity of the prima probem. If the probem is feasibe for a given s (i (ines 6, 7, then we obtain an intermediate resource aocation poicy Θ(i, an intermediate objective vauef 0, and the corresponding Lagrange mutipier set Φ(i. In particuar, Φ(i is used to generate an optimaity cut in the master probem. Aso, the optima resource aocation poicy and the performance upper boundub(i are updated if the computed objective vaue is the owest across a the iterations. On the contrary, if the prima probem is infeasibe for a given s (i (ine 9, then we sove the 1 -minimization probem in (23 and obtain an intermediate resource aocation poicy Θ(i and the corresponding Lagrange mutipier set Φ(i. This information wi be used to generate an infeasibiity cut in the master probem. We note that the upper bound is obtained ony from the feasibe prima probem. Subsequenty, we sove the master probem based on Θ(t and Θ(i, t {1,...,i}, via a standard MILP numerica sover. Due to wea duaity [26], the optima vaue of the origina optimization probem in (17 is bounded beow by the objective vaue of the master probem in each iteration. The agorithm stops when the difference between the i-th ower bound and the i-th upper bound is smaer than a predefined threshod ϑ 0 (ines We note that when the master and the prima probems can be soved in each iteration, the proposed agorithm is guaranteed to converge to the optima soution [25, Theorem 6.3.4]. C. Suboptima Resource Aocation Agorithm Design The optima iterative resource aocation agorithm proposed in the ast section has a nonpoynomia time computationa compexity due to the MILP master probem 4. In this section, we propose a suboptima resource aocation agorithm which has a poynomia time computationa compexity. The starting point for the design of the proposed suboptima resource aocation agorithm is the reformuated optimization probem in (13. 1 Probem reformuation via difference of convex functions programming: The major obstace in soving (13 are the binary constraints. Hence, we rewrite constraint C5 in its equivaent form: C5a: 0 s 1, {1...,L} and C5b: s s 2 0. (27 Now, optimization variabes s in C5a are continuous vaues between zero and one whie constraint C5b is the difference of two convex functions. By using the SDP reaxation approach 4 The optima agorithm serves mainy as a performance benchmar for the proposed suboptima agorithm.

18 TABLE II SUBOPTIMAL ITERATIVE RESOURCE ALLOCATION ALGORITHM 18 Agorithm Successive Convex Approximation 1: Initiaize the maximum number of iterations L max, penaty factor φ 0, iteration index i = 0, and s (i 2: repeat {Loop} 3: Sove (31 for a given s (i+1 and obtain the intermediate resource aocation poicy {W,s,Pj U m,n U, W, P j,m,n,q m,n} 4: Set s (i+1 = s and i = i+1 5: unti Convergence or i = L max as in the optima resource aocation agorithm, we can reformuate the optimization probem as ( minimize U TP W,s,P W H N, W,b,PU j, P j U (28 j,m,n U,qm,n s.t. Θ D, C5b, whered denotes the convex feasibe soution set spanned by constraints C1, C2, C3, C4, C5a, C6, and C8 C18. The ony non-convexity in (28 is due to constraint C5b which is a reverse convex function [27]. Now, we introduce the foowing Theorem for handing the constraint. Theorem 2: For a arge constant vaue φ 1, (28 is equivaent 5 to the foowing probem: ( ( minimize U TP W,s,P U W H N, W,b,PU j, P j +φ s j,m,n U,qm,n s.t. Θ D. In particuar, φ acts as a arge penaty factor for penaizing the objective function for any s that is not equa to 0 or 1. Proof: Pease refer to Appendix C for a proof of Theorem 2. The probem in (29 is in the canonica form of difference of convex (d.c. functions programming. Specificay, g(s = N s2 is a concave function and we minimize d.c. functions over a convex constraint set. As a resut, we can appy successive convex approximation [36] to obtain a oca optima soution of (29. 2 Suboptima iterative agorithm: Since g(s is a differentiabe convex function, inequaity s 2 (29 g(s g(s (i + s g(s (i (s s (i, {1,...,N }, (30 aways hods for any feasibe point s (i, where the right hand side of (30 is an affine function [33] and represents a goba underestimator of g(s. 5 Here, equivaence means that both probems share the same optima objective vaue and the same optima resource aocation poicy.

19 19 As a resut, for any given vaue of s (i, we sove the foowing optimization probem, ( minimize U TP W,s,P W H N, W,b,s,Pj U, P j U j,m,n U,qm,n ( +φ s 2 2 s (i (s s (i s.t. Θ D, (31 which eads to an upper bound of (29. Then, to tighten the obtained upper bound, we empoy an iterative agorithm which is summarized in Tabe II. First, we initiaize the vaue of s (i for iteration index i = 0. Then, in each iteration, we sove (31 for given vaues of s (i, cf. ine 3, and update s (i+1 (s (i with the intermediate soution s, cf. ine 4. The proposed iterative method generates a sequence of feasibe soutions s (i+1 with respect to (29 by soving the convex upper bound probem (31 successivey. As shown in [36], the proposed suboptima iterative agorithm converges to a oca optima soution 6 of (29 with poynomia time computationa compexity. In fact, the proposed suboptima agorithm benefits from the convexity of (31 and different numerica methods can be used to efficienty sove (31. In particuar, when the prima-dua path-foowing interior-point method is used with a proper choice of erne(/barrier function, cf. [37], [38], the computationa compexity of the proposed suboptima agorithm is O(L max (N 2 n((n 2 /ǫ with respect to N for a given soution accuracy ǫ > 0 [39], where O( stands for the big-o notation. The computationa compexity is significanty reduced compared to the computationa compexity of an exhaustive search which is given by O(2 N TL (N n(n /ǫ with respect to N, i.e., cf. Figure 3. Remar 4: The proposed agorithm requires s (i to be a feasibe point for the initiaization, i.e., for i = 0. This point can be easiy obtained since the constraints in (29 span a convex set. V. SIMULATION RESULTS In this section, we evauate the system performance of the proposed resource aocation designs via simuations. There are L = 3 FD radio BSs in the system, which are paced at the corner points of an equiatera triange. The inter-site distance between any two FD radio BSs is 250 meters. The upin and downin users are uniformy distributed inside a disc with radius 500 meters centered at the centroid of the triange. We set the constant weights for the downin and upin power consumption as η = ζ j = 1, j {1,...,K U }. The penaty termφfor the proposed 6 By foowing a simiar approach as in the proof of Theorem 1, it can be shown that Ran(W = 1 hods despite the adopted SDP reaxation.

20 TABLE III SYSTEM PARAMETERS 20 Carrier center frequency and path oss exponent 1.9 GHz and 3.6 Mutipath fading distribution and tota noise variance, σ 2 z Minimum required SINR for upin user j, Γ UL req j Power ampifier power efficiency and antenna power consumption in ide mode, P Ide Max. transmit power for downin and upin, P DL max and P U max j Rayeigh fading and 62 dbm 10 db 1/ε D = 1/ε U = 0.2 and 0 dbm 48 dbm and 23 dbm suboptima agorithm is set to 10P DL max. Aso, P 0 = 0 is adopted in a simuation resuts 7. Uness specified otherwise, we assume 50 db of sef-interference canceation 8 at the FD radio BSs and the circuit power consumption per antenna is P Active = 30 dbm. The antenna gains for the BSs and the users are 10 dbi and 0 dbi, respectivey, and there are N T = 20 antennas equipped in each FD BS resuting in N = 60 antennas in the networ. Furthermore, a downin users require identica minimum SINRs, i.e., Γ DL req = Γ DL req,. The performance of the proposed agorithms is compared with the performances of the foowing four baseine systems designed for pea system oad when a the avaiabe antennas are activated. In particuar, we minimize the tota system power consumption of a four baseine systems using a simiar approach as for the schemes proposed in this paper but set s = 1, {1,...,N }. The baseine systems are configured as foows. Baseine 1: a FD distributed antenna system (FD-DAS; Baseine 2: a HD distributed antenna system (HD-DAS; Baseine 3: a FD system with co-ocated antennas (FD- CAS; Baseine4: a HD system with co-ocated antennas (HD-CAS. For the HD communication systems, we adopt static time division dupex such that upin and downin communication occur in non-overapping equa-ength time intervas. In other words, both sef-interference and the upin-to-downin co-channe interference are avoided. For a fair performance comparison between HD and FD systems, we setog 2 (1+Γ UL req j = 1/2og 2 (1+Γ UL HD andog 2 (1+Γ DL req j = 1/2og 2 (1+Γ DL HD req j such that the minimum required SINRs for the upin users, Γ UL HD, and downin users, Γ DL HD req j, become Γ UL HD req j = (1 + Γ UL req j 2 1 and Γ DL HD req j = (1 + Γ DL req 2 1, respectivey, to account for the penaty due to the oss in spectra efficiency of the HD protoco. Aso, the power consumption of downin and upin transmission in the objective function of the HD systems is reduced by a factor of two as at a given time either upin or downin transmission is performed. For the CAS, we assume that there is ony one BS ocated at the center of the system, which is equipped with the same number of antennas as a FD BSs in the req j req j 7 We note that the vaue of P 0 does not affect the resource aocation agorithm design. 8 We assume a baun anaog circuit is impemented in the FD radio BSs which can cance 50 db of sef-interference [11]. The residua sef-interference is handed by the beamforming matrix W via the proposed optimization framewor.

21 Average tota system power consumption (dbm Objective vaue of equation (29 (dbm Upper bound, SINR = 24 db 40 Lower bound, SINR = 24 db Upper bound, SINR = 21 db Lower bound, SINR = 21 db Number of iterations SINR = 24 db Optima vaue, SINR = 24 db SINR = 21 db Optima vaue, SINR = 21 db Computationa compexity Proposed suboptima agorithm Brute force search Computationa compexity reduction Number of iteartions Number of BS antennas (N T L Fig. 2. Convergence of the proposed iterative agorithms. Fig. 3. Computationa compexity versus the tota number of transmit antennas in the system, N TL. distributed setting combined, i.e., N. Furthermore, for a baseine systems, we remove the maximum transmit power constraints imposed for the downin and upin transmissions, i.e., constraints C3 and C4. The ey parameters adopted in the simuations are provided in Tabe III. A. Convergence and Computationa Compexity of the Proposed Iterative Agorithms Figure 2 iustrates the convergence of the proposed optima and suboptima agorithms for different minimum required SINRs for downin users, Γ DL req. There are K D = 4 downin users and K U = 2 upin users in the system. It can be seen from the upper haf of Figure 2 that the proposed optima agorithm in Tabe I converges to the optima soution in ess than 350 iterations, i.e., the upper bound vaue meets the ower bound vaue. On the other hand, from the ower haf of Figure 2, we observe that the suboptima agorithm converges to a oca optima vaue after ess than 20 iterations. In the seque, we show the performance of the suboptima iterative agorithm for 20 iterations. Figure 3 compares the computationa compexity of the brute force approach with that of the proposed suboptima agorithm 9 for 20 iterations and soution accuracy ǫ = 0.1. The system setting is identica to the scenario in Figure 2 and the resuts are computed based on the big- O compexity anaysis in Section IV. As can be observed, the proposed suboptima resource aocation agorithm requires a significanty ower computationa compexity compared to the brute force approach, especiay for arge numbers of antennas. 9 The proposed optima agorithm may have the same computationa compexity as the brute force approach in the worst case scenario athough this sedom happens in practice.

22 Average tota system power consumption (dbm Optima agorithm Suboptima agorithm Baseine 1: FD DAS Baseine 2: HD DAS Baseine 3: FD CAS Baseine 4: HD CAS Performance gain Minimum required SINR of downin users (db Fig. 4. Average tota system power consumption (dbm versus the minimum required SINRs for the downin users, Γ DL req, for different systems. The doube-sided arrows indicate the performance gain of the proposed FD system compared to traditiona HD communication systems. B. Average Tota System Power Consumption In Figure 4, we study the average tota system power consumption versus the minimum required SINRs of the downin users, Γ DL req. There are K D = 4 downin users and K U = 2 upin users in the system. It can be observed that the average tota system power consumption increases graduay with Γ DL req. In fact, as the QoS requirements of the downin users become more stringent, a higher downin transmit power is needed to fufi the requirement. At the same time, the sef-interference power increases with the downin transmit power. Thus, the FD radio BSs have to utiize more degrees of freedom for sef-interference suppression, and as a consequence, ess degrees of freedom are avaiabe for reducing the tota system power consumption. On the other hand, the proposed suboptima iterative resource aocation agorithm offers practicay the same performance as the optima agorithm for the considered scenario. As can be observed, the two proposed agorithms faciitate significant power savings compared to a baseine system architectures (which activate aways a avaiabe antennas, especiay for ow to moderate system oads, i.e., Γ DL req 21 db. Indeed, activating a antennas may not be beneficia for the tota system power consumption when the oad of the system is reativey sma, since in this case, the power consumption caused by an extra antenna circuit outweighs the power reduction for information transmission offered by the extra activated antenna. Nevertheess, the performance gap between the two proposed agorithms and baseine system 1 diminishes as

23 65 23 Average tota system power consumption (dbm Performance gain HD systems Proposed agorithms FD systems Optima agorithm Suboptima agorithm Baseine 1: FD DAS Baseine 2: HD DAS Baseine 3: FD CAS Baseine 4: HD CAS Number of downin users (K D Fig. 5. Average tota system power consumption (dbm versus the number of downin users for Γ DL req = 21 db and different systems. The doube-sided arrows indicate the performance gain of the proposed FD protoco compared to the HD protoco. the minimum required SINRs for the downin users increase. In particuar, the BSs are forced to transmit with high power to satisfy the more stringent QoS requirements when the number of activated antennas is sma. As a resut, the two proposed agorithms have to activate more antennas, cf. aso Figure 6, for improving the power efficiency of the system which yieds a simiar resource aocation as baseine system 1. Additionay, the two proposed agorithms outperform HD baseine systems 2 and 4 by a considerabe margin. As can be seen, in the HD systems, an exceedingy arge system power consumption is required to meet the more stringent minimum required downin SINRs to compensate for the spectra efficiency oss inherent to the HD protoco. Furthermore, the distributed antennas depoyed in the proposed systems provide spatia diversity across the networ which shortens the distance between transmitters and receivers. This accounts for the power saving enabed by the two proposed agorithms compared to baseine CASs 3 and 4. Figure 5 depicts the average tota system power consumption versus the number of downin users for a minimum required downin SINR of Γ DL req = 21 db. There are K U = 2 upin users in the system. It is observed that the average tota system power consumption increases with the number of downin users. As more downin users request communication services from the system, more QoS constraints are imposed on the optimization probem in (12 which reduces the size of the feasibe soution set and thus resuts in a higher tota system power consumption. In addition, the two proposed resource aocation agorithms outperform a baseine schemes

24 Average number of activated antennas (N T L Optima agorithm, K D = 4 Suboptima agorithm, K D = 4 Optima agorithm, K D = 3 Suboptima agorithm, K D = 3 Optima agorithm, K D = 2 Suboptima agorithm, K D = 2 Optima agorithm, K D = 1 Suboptima agorithm, K D = 1 K D = 4 K D = 3 K D = 2 Average number of activated antennas (N T L Baseine systems 24 Optima agorithm, Γ DL = 21 db req Suboptima agorithm, Γ DL = 21 db req Optima agorithm, Γ DL = 18 db req Suboptima agorithm, Γ DL = 18 db req Optima agorithm, Γ DL = 12 db req Suboptima agorithm, Γ DL = 12 db req Baseine systems K D = 1 10 Proposed agorithms Minimum required SINR of downin users (db Circuit power consumption per active antenna (dbm Fig. 6. Γ req. Average number of activated antennas versus Fig. 7. Average number of activated antennas versus circuit power consumption per active antenna. due to the adopted optimization framewor and the distributed antenna architecture. C. Average Number of Activated Antennas In Figure 6, we study the average number of activated antennas versus the minimum required downin SINR, Γ DL req, for different numbers of downin users. It can be observed that the average number of activated antennas increases with increasing minimum required SINR for the downin users. Athough activating an extra antenna for signa transmission and reception consumes extra power in the circuit, i.e.,p Active P Ide > 0, a arger number of activated antennas increases the degrees of freedom of the system which is beneficia if the QoS constraints are stringent. Specificay, with more antennas, the direction of beamforming matrix W can be more accuratey steered towards downin user which substantiay reduces the necessary downin transmit power to achieve a certain QoS. Moreover, the reduced downin transmit power aso decreases the sef-interference which in turn reduces the required upin transmit power. In fact, for a sma number of activated antennas, the FD radio BSs are required to transmit with exceedingy high power if Γ DL req is arge. As a resut, the FD radio BSs prefer to activate more antennas to improve the power efficiency of information transmission, when the cost of activating extra antennas is ess than the associated potentia transmit power saving. On the other hand, it can be observed that the proposed schemes activate more antennas when more downin users are in the system. In fact, the downin co-channe interference increases with the number of downin users. Furthermore, the co-channe interference cannot be suppressed by simpy increasing the downin transmit power for a downin users. Thus, extra spatia degrees of freedom are beneficia for decreasing the system power consumption. In Figure 7, we show the average number of activated antennas versus the circuit power