Joint Precoding and Power Control in Small-Cell Networks With Proportional-Rate MISO-BC Backhaul
Yanjie Dong, Md. Jahangir Hossain, Julian Cheng, Victor C. M. Leung
aa r X i v : . [ c s . I T ] A ug Joint Precoding and Power Control in Small-Cell Networks With Proportional-RateMISO-BC Backhaul
Yanjie Dong,
Student Member, IEEE ‡ , Md. Jahangir Hossain, Senior Member, IEEE † ,Julian Cheng, Senior Member, IEEE † , and Victor C. M. Leung, Fellow, IEEE ‡§‡
Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada † School of Engineering, The University of British Columbia, Kelowna, BC, Canada § College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, ChinaEmails: { ydong16, vleung } @ece.ubc.ca, { julian.cheng, jahangir.hossain } @ubc.ca Abstract —In the small-cell networks with multiple-input-single-output broadcasting (MISO-BC) backhauls, the jointdirty-paper coding and power control are investigated for theMISO-BC backhauls and access links in order to minimizethe system transmit power. Considering the proportional ratesof MISO-BC backhauls and flow-conservation constraints, theformulated optimization problem is non-convex. Moreover, theformulated problem couples the precoding vectors with thepower-control variables. In order to handle the non-convexoptimization problem and decouple the backhaul and access links,the structure of the formulated problem is investigated suchthat the optimal precoding vectors and optimal power-controlvariables are independently obtained. Moreover, the optimalprecoding vectors are obtained in closed-form expressions. Sim-ulation results are used to show the performance improvementover the benchmark scheme.
Index Terms —Power minimization, proportional rate, small-cell networks, wireless backhauling.
I. I
NTRODUCTION
The volume of mobile data experiences a high-pace growthduring the last decade under the development of mobileinternet. In order to support the ever-increasing data volume,the small-cell networks (SCNs) are proposed to offload themobile data from the traditional infrastructure to the small-cellbase stations (ScBSs). In the SCN, the low-power ScBSswill be ultra-densely deployed to improve the spectrum ef-ficiency of traditional infrastructure [1]. Moreover, the ScBSswill improve the energy efficiency by reducing the distancebetween the transmitters and receivers [1]. However, thesuccessful application of SCNs depends on the reliable andeconomical-friendly backhauls, which connect the ScBSs tothe core network. Therefore, the backhauling technology hasbeen considered as a key component for the fifth generation(5G) cellular networks [2]–[7].The backhauling technology can be classified into twocategories: wired backhauls [4], [5] and wireless backhauls[2], [3]. For example, the ScBSs can connect to the corenetwork via the wired backhauls, such as optical fibers or
This work was supported in part by the National Natural Science Foundationof China under Grant 61671088, in part by a UBC Four-Year DoctoralFellowship, in part by the Natural Science and Engineering Research Councilof Canada, and in part by the National Engineering Laboratory for Big DataSystem Computing Technology at Shenzhen University, China. digital subscriber lines [4], [5]. Though the wired backhaulscan provide high-speed connections, the capital expenditure ishigh. In some scenarios (e.g., temporary usage of backhaulsin the exhibitions and music concerts), the deployment ofwired backhauls is not necessary or not available. Therefore,the wireless backhauls are proposed as the promising alter-natives to the wired backhauls due to the low cost and fastdeployment [2], [3]. Moreover, the wireless backhauls alsoextend the coverage of cellular networks to the remote areaby providing the ScBSs with the plug-and-play backhauls.Based on the spectrum bands, the wireless backhauls arecategorized into millimeter-wave backhauls [2], [3], opticalwireless backhauls [8], radio-frequency (RF) backhauls [9]–[16] and hybrid backhauls [16], [17]. Several researchers haveinvestigated the characteristics of different spectrum bands forthe wireless backhauls [2], [3].Due to the low-cost RF devices, we investigate the re-source allocation in the SCNs with RF backhauls. In theSCNs with RF backhauls, the current resource allocationalgorithms mainly focus on the the admission control [9],power minimization [10], [11], system energy efficiency max-imization [12], [13], system security [14] and system capacitymaximization [15]–[17]. For example, Vu et al. proposed apower-control algorithm to minimize the transmit power ofthe RF backhauls subject to the communication quality-of-service (QoS) for the single-input-single-output broadcasting(SISO-BC) backhauls [10]. Since the proposed algorithmfocuses on the power control of backhauls, it induces asuboptimal performance when the radio resources of backhaulsand access links are considered. Using the SISO-BC backhauls,Zhang et al. investigated the joint backhaul bandwidth alloca-tion and access-link power control to maximize the systemenergy efficiency [12]. Since the multiple-input-single-outputbroadcasting (MISO-BC) channels have larger capacity regionthan the SISO-BC channels with the same pathloss, the linearprecoding schemes were investigated to the joint resourceallocation of backhauls and access links in the SCNs withMISO-BC backhauls [13]–[17].Compared with the linear precoding schemes, thedirty-paper coding scheme (DPCS) can improve the rate ofMISO-BC backhauls [18]. When the DPCS is used by theISO-BC backhauls, the impact of joint backhaul precodingand access-link power control has not been reported in thecurrent literature. Moreover, the current algorithms based onlinear precoding scheme in [13]–[17] cannot be used to designthe precoding vectors and power-control variables for theMISO-BC backhauls and access links, respectively. Besides,the fairness issue among the ScBSs is not considered in theSCNs with MISO-BC backhauls [13]–[17]. Therefore, we aremotivated to investigate the joint precoding and power control(JPPc) problem in the SCNs with proportional-rate MISO-BCbackhauls. Our contributions are summarized as follows. • In order to minimize the system transmit power, we for-mulate the JPPc problem when the MISO-BC backhaulsare used in the SCNs. We consider that the rates ofbackhauls satisfy a set of ratios in order to satisfy thefairness of ScBSs. • The formulated JPPc problem is non-convex. Moreover,the precoding vectors are coupled with the power-controlvariables. Hence, the JPPc problem is challenging to solvevia the standard solutions. • Based on the structure of JPPc problem, we obtain a setof optimal rate ratios for the MISO-BC backhaul. Basedon the optimal ratios, we can equivalently decouple theJPPc problem into two subproblems: access-link power-control subproblem and backhaul precoding subproblem.Therefore, the access-link power-control subproblem canbe solved via the convex optimization toolbox. Differentfrom [18], [19], we obtain the closed-form precodingvectors for the backhaul precoding subproblem when theencoding sequence of DPCS is given.Notation: W H denotes the hermitian of matrix W . C denotes the domain of complex values. The expectation of arandom variable is denoted as E {·} . The operator |·| and k·k F respectively denote the determinant and Frobenius norm of amatrix.The remaining of this paper is organized as follows. InSection II, the system model and problem formulation aredescribed. In Section III, the optimal algorithm is proposed toobtain the optimal power-control variables and the closed-formprecoding vectors. Numerical results are presented in SectionIV. Section V concludes this paper.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. Overall System Description
We consider the downlink transmission of SCN, which con-sists of an L -antenna gateway and M single-antenna ScBSs.The m -th ScBS is associated with N m single-antenna userequipments (UEs), m = 1 , , . . . , M . As shown in Fig. 1,the ScBSs communicate with the gateway and associated UEsvia the MISO-BC backhauls and RF access links, respectively.The MISO-BC backhauls and RF access links operate inthe orthogonal time slots in order to avoid the interference.Specifically, the gateway uses half portion of the frame forbackhauling transmission as shown in Fig. 2. The ScBSs use the other half portion of the frame for access-link transmissionas shown in Fig. 2.The ScBSs operate in the same channel. Therefore, theinter-cell and intra-cell interference exist among the ScBSs.We assume that the channel state information (CSI) ofMISO-BC backhauls and RF access links is perfectly knownat the gateway. Specifically, the CSI at receivers can beobtained via channel estimation of downlink pilots. TheCSI at transmitters can be obtained via uplink feedback infrequency-division duplex mode or channel reciprocity esti-mation in the time-division duplex mode [20]. Fig. 1: An illustration of the SCN with MISO-BC backhauls and RFaccess links.Fig. 2: An illustration of the backhaul link sharing scheme.
B. Signal Models1) Signal Models in Backhauls:
At the gateway, theinformation-bearing signal for the m -th ScBS is defined as w m ∈ C L × . Hence, the transmit signal at the gateway isgiven as w = M X m =1 w m . (1)The overall transmit covariance matrix is obtained as W = P Mm =1 W m , where W m , E { w m w H m } is the transmit co-variance matrix for the m -th ScBS.Let h m ∈ C L × denote the channel-coefficient vectorbetween the gateway and the m -th ScBS (the m -th backhaulink). The Rayleigh fading is considered; therefore, eachentry of the m -th channel-coefficient vector h m follows acircularly-symmetric-complex-Gaussian (CSCG) distribution C N (cid:0) , Ω − m (cid:1) . Here, the value Ω m denotes the pathloss of the m -th link.The received signal at the m -th ScBS is denoted as y m = h H m w + z m (2)where z m is the additive white Gaussian noise with the CSCGdistribution as C N (cid:0) , σ (cid:1) .Based on the DPCS, the gateway communicates withthe ScBSs via the MISO-RF backhauls. In the DPCS, theinformation-bearing signals are sequentially encoded. Let π , { π , π , . . . , π M } denote the encoding order of ScBSs. Here,the term π m denotes the index of information-bearing signalsencoded at the ( M + 1 − m ) -th order. Therefore, the channelcapacity of backhauls is denoted as [18] C B ( π , { h m , W m } ∀ m )= ( R B π m ≤ s log Ψ π m Θ π m , s.t. M X m =1 Tr ( W m ) ≤ P max0 ) (3)where Ψ π m , h H π m m X k =1 W π k h π m + σ (4)and Θ π m , h H π m m − X k =1 W π k h π m + σ (5)with Θ π = σ . Here, P max0 is the maximum transmit powerof gateway.
2) Signal Models in Access Links:
After decoding theinformation-bearing signals, the m -th ScBS broadcasts theinformation-bearing signals to the associated UEs. The re-ceived signal of the n -th UE at the m -th ScBS (the ( m, n ) -thUE) is denoted as y m,n = g m,n √ v m,n x m,n + g m,n X i = n √ v m,i x m,i + X j = m N j X i =1 g j,n √ v j,i x j,i + z m,n (6)where g m,n , x m,n and v m,n are, respectively, the channelcoefficient, information-bearing signal and transmit power forthe ( m, n ) -th UE. Here, g m,n follows a CSCG distribution C N (cid:0) , ω − m,n (cid:1) with ω m,n as the pathloss of the ( m, n ) -th UEto the m -th ScBS. The term z m,n is the AWGN at the ( m, n ) -thUE with CSCG distribution C N (cid:0) , σ (cid:1) .Based on (6), the received SINR at the ( m, n ) -th UE isdenoted as SINR A m,n = v m,n | g m,n | I INTRA m,n + I INTER m,n + σ (7)where I INTRA m,n and I INTER m,n are, respectively, defined as I INTRA m,n , X i = n v m,i | g m,n | (8) and I INTER m,n , X j = m N j X i =1 v j,i | g j,n | . (9) C. Problem Formulation
Since the m -th backhaul link is shared by the N m UEs ofthe m -th ScBS, the flow-conservation constraint of the m -thbackhaul is denoted as R B m ≥ N m X n =1 R A m,n (10)where R A m,n = log (cid:0) SINR A m,n (cid:1) .Our objective is to minimize the system transmit powerof the gateway and ScBSs via the joint design of precod-ing covariance matrices { W m } ∀ m and proportional ratios { φ m } ∀ m of MISO-BC backhauls, the power-control variables { v m,n } ∀ m,n of access links. Let Y denote the set of opti-mization variables as Y , { W m , φ m , v m,n } ∀ m,n , where φ m is the ratio of rate that is shared by the m -th backhaul linkwith P Mm =1 φ m = 1 . In order to minimize the system transmitpower, we formulated the JPPc problem as min Y M X m =1 Tr ( W m ) + M X m =1 N m X n =1 v m,n (11a)s.t. M X m =1 Tr ( W m ) ≤ P max0 (11b) R B : . . . : R B M = φ : . . . : φ M (11c) R B m ≥ N m X n =1 R A m,n , ∀ m (11d) N m X n =1 v m,n ≤ P max m , ∀ m (11e) R A m,n ≥ R REQ m,n , ∀ m, n (11f)where P max m and R REQ m,n are respectively the maximum transmitpower of the m -th ScBS and the communication QoS require-ment of the ( m, n ) -th UE.III. C ENTRALIZED O PTIMAL S OLUTION
Since the JPPc problem (11) contains the non-convexproportional-rate constraints in (11c) and the non-convex flow-conservation constraints in (11d), the optimization problemis challenging to handle with standard optimization tools.Therefore, we are motivated to investigate the structure of JPPcproblem (11) such that an optimal solution is obtained.Based on the structure of JPPc problem (11), we pro-pose a two-stage optimization framework. In the optimizationframework, the JPPc problem (11) is decoupled into access-link power-control subproblem and backhaul precoding sub-problem. After several algebraic manipulations, we obtain theoptimal power-control variables via the off-the-shelf toolbox,e.g., CVX [21]. Solving the backhaul precoding subproblem,we obtain the optimal closed-form precoding vectors for thegateway. . Optimal Proportional Ratios for MISO-BC Backhauls
Before analyzing the optimal proportional ratios, we firstintroduce a proposition as follows.
Proposition Ignoring the proportional-rate constraints in(11c), the constraints in (11d) and (11f) are active when theproblem (11) is optimally solved.
Proof:
See Appendix A.When the constraints (11d) and (11f) are active, the requiredrate for the m -th backhaul is obtained as R B m = R REQ m , N m X n =1 R REQ m,n (12)with m = 1 , , . . . , M .Substituting (12) into (11c), we obtain the optimal propor-tional ratios { φ ∗ m } ∀ m as φ ∗ m = R REQ m,nN m P n =1 R REQ m,n (13)otherwise, more transmit power is required to the MISO-BCbackhauls.Based on Proposition 1 and performing several algebraicmanipulations, we equivalently decouple the JPPc problem(11) into access-link power-control subproblem and backhaulprecoding subproblem as min { v m,n } ∀ m,n M X m =1 N m X n =1 v m,n (14a)s.t. N m X n =1 v m,n ≤ P max m , ∀ m (14b)SINR m,n = Γ REQ m,n , ∀ m, n (14c)and min { W m } ∀ m M X m =1 Tr ( W m ) (15a)s.t. M X m =1 Tr ( W m ) ≤ P max0 (15b) R B m = R REQ m , ∀ m (15c)where Γ REQ m,n = exp (cid:0) R REQ m,n (cid:1) − . B. Optimal Power control for Access Links
In order to solve the access-link power-control subproblem(14), we obtain a set of equivalent constraints to (14c) as v m,n | g m,n | Γ REQ m,n = X i = n v m,i | g m,n | + X j = m N j X i =1 v j,i | g j,n | + σ . (16) Substituting (16) into (14), we obtain a convex version ofaccess-link power-control subproblem as min { v m,n } ∀ m,n M X m =1 N m X n =1 v m,n (17a)s.t. N m X n =1 v m,n ≤ P max m and (16) , ∀ m. (17b)Since the optimization problem (17) is convex, the opti-mal power-control variables { v m,n } ∀ m,n can be obtained viasecond-order cone programming or semidefinite programming. C. Optimal Precoding Vectors for MISO-BC Backhauls
We propose a new method to obtain the closed-form optimalsolution. Based on the uplink-downlink duality, we obtain a setof equivalent convex constraints to (15c).The received signal in the dual uplink channel of (2) isobtained as y = M X m =1 h m √ w m x m + z (18)where x m , w m and z are, respectively, dual signal of the m -thScBS, the power of dual signal and additive white Gaussiannoise at the gateway. Here, the AWGN at the gateway followsCSCG distribution with mean zero and covariance matrix σ I .Since the downlink channel capacity with the encoding order π in (2) is equal to the dual uplink channel capacity with theinverse decoding order as [18], [22] C D ( π , { h m , w m } ∀ m )= ( R D π m ≤ log (cid:12)(cid:12) Ψ π m (cid:12)(cid:12)(cid:12)(cid:12) Θ π m (cid:12)(cid:12) , s.t. M X m =1 w m ≤ P max0 ) (19)where Ψ π m and Θ π m are, respectively, defined as Ψ π m , M X k = m h π k w π k h H π k + σ I (20)and Θ π m , M X k = m +1 h π k w π k h H π k + σ I (21)with Θ π M = σ I .Moreover, the injection between w π m and W π m is definedas [22] W π m = Θ − π m u π m Θ π m w π m u H π m Θ − π m (22)where the vector u π m is obtained via singular-value decom-position as Θ − π m h π m Θ − π m = u π m λ π m with u H π m u π m = 1 andeigen-value λ π m .Based on the duality of uplink and downlink channels, weobtain R B π m = R D π m = log (cid:12)(cid:12) Ψ π m (cid:12)(cid:12) − log (cid:12)(cid:12) Θ π m (cid:12)(cid:12) , ∀ m (23)with P Mm =1 w π m = P Mm =1 Tr ( W π m ) .ropping the power constraint in (15b) and replacing thebackhaul rate R B π m in (15c) with (23), the optimization prob-lem (15) is transformed as min { w πm } ∀ m M X m =1 w π m (24a)s.t. log (cid:12)(cid:12) Ψ π m (cid:12)(cid:12) − log (cid:12)(cid:12) Θ π m (cid:12)(cid:12) = R REQ π m , ∀ m. (24b)The optimization problem (24) is equivalent to problem (15)if and only if the optimal value of (24) is less than or equalto P max0 . Based on (19)–(21), we observe that the data rate ofthe m -th backhaul increases with w π m . Moreover, the data rateof the k -th backhaul decreases with the power of dual signal w π m when k < m . Therefore, setting the constraints in (24b)active, we obtain the optimal power of dual signal (cid:8) w ∗ π m (cid:9) ∀ m as w π m = exp (cid:0) R REQ π m (cid:1) − h H π m Θ − π m h π m . (25)Substituting (25) into (22) and performing some algebraicmanipulations, we obtain the closed-form optimal downlinkprecoding vectors { w π m } ∀ m as w π m = exp (cid:0) R REQ π m (cid:1) − h H π m Θ − π m h π m Θ − π m h π m (cid:13)(cid:13)(cid:13) Θ − π m h π m Θ − π m (cid:13)(cid:13)(cid:13) F . (26)After obtaining the CSI of backhauls and access links, thegateway can obtain the optimal precoding vectors in two steps:1) calculating the set of optimal power of dual signals fordual uplink channel based on (25); and 2) calculating theoptimal precoding vectors for downlink channel based on (26).Based on (13), (17) and (26), we can obtain the optimalproportional ratios, access-link power-control variables andbackhaul precoding vectors.IV. S IMULATION R ESULTS
We use numerical results to verify our proposed algo-rithm. In order to illustrate the performance improvement,we also include a benchmark scheme where the backhaulsuse zero-forcing beamforming. The pathloss equations for thebackhauls and access links are, respectively, given as [23] Ω m = 32 . ( f c ) + 31 . ( D m ) dB (27)and ω m,n = 17 . . ( f c ) + 38 . ( d m,n ) dB (28)where D m and d m,n are the distances of m -th backhaul andthe ( m, n ) -th access link, respectively. The power of AWGNis − dBm . The gateway is equipped with eight antennas.There are four ScBSs, and each ScBS is associated with oneUEs. The maximum transmit powers of the gateway and ScBSsare dBm and dBm. This value is obtained when power spectrum density of AWGN is − dBm/Hz, the noise figures of gateway and UE are dB, the noise figure ofScBS is dB, and the bandwidth of system is KHz.
We define the system outage event as either the backhaulsor the access links are in outage. When the system outageevent happens, the gateway and ScBSs will not transmit anyinformation.
200 220 240 260 280 300 320 340 360 380 400
Distance of ScBS to Gateway (meters) S ys t e m T r an s m i t P o w e r C on s u m p t i on ( m W ) PROP, m,nREQ = uniform(35, 45)PROP, m,nREQ = uniform(30, 40)ZFBF, m,nREQ = uniform(35, 45)ZFBF, m,nREQ = uniform(30, 40)
Fig. 3: The variation of system power consumption over the distanceof backhauls.
200 220 240 260 280 300 320 340 360 380 400
Distance of ScBS to Gateway (meters) S ys t e m O u t age P r obab ili t y PROP, m,nREQ = uniform(35, 45)PROP, m,nREQ = uniform(30, 40)ZFBF, m,nREQ = uniform(35, 45)ZFBF, m,nREQ = uniform(30, 40)
Fig. 4: The variation of system outage over the distance of backhauls.
Figure 3 shows the variation of the system transmit powerover the distance of backhauls. We observe that the systemtransmit power increases with the distance of backhauls. Afterreaching the peak value, the system transmit power starts todecrease. For example, the system transmit power decreasesafter the distance of backhauls is greater that meters. Thesetwo observations can be explained as follows. As the distanceof backhauls increases, the required power of the backhaulsto deliver a certain amount of information increases. Since thegateway has a limitation on the maximum transmit power, thesystem outage probability increases as shown in Fig. 4. Whena certain threshold is surpassed, the SCN has a high probabilityto be in an outage event. In other words, the SCN will be silentwith a high probability. Therefore, the system transmit powerdecreases.Figure 4 illustrates the variation of the system outageprobability over the distance of backhauls. We observe thatthe outage probability of the SCN also increases when thedistances of fronthaul links decrease. This observation can bexplained as follows. When the distances of fronthaul linksdecrease, the outage probability of access links increases dueto the increasing interference among the access links.Based on Fig. 3 and Fig. 4, we observe that our proposedscheme outperforms the ZFBF scheme which is used [16].When the required SINR of UEs are uniformly drawn fromthe ranges (35 , and (30 , , our proposed scheme canrespectively reduce at most . % and . % of systemtransmit power when compared with the ZFBF scheme. Whenthe required SINR of UEs are uniformly drawn from (35 , ,our proposed scheme can reduce the outage probability by . % compared with the ZFBF scheme. Moreover, ourproposed scheme can reduce the outage probability by . %compared with the ZFBF scheme when the required SINR ofUEs are uniformly drawn from (30 , .V. C ONCLUSIONS
We investigated the structure of the formulated JPPc prob-lem in order to optimally minimize the system transmit powerwith proportional-rate constraints for MISO-BC backhauls.Based on the structure of the JPPc problem, we obtained theoptimal ratios for the backhauls such that we can separate theprecoding vectors and power-control variables into two sub-problems: access-link power-control subproblem and backhaulprecoding subproblem. The optimal power-control variablescan be obtained via the standard convex optimization tool-box. Leveraging the information-theoretical uplink-downlinkduality, we obtain the closed-form expression of the precodingvectors. Simulation results are used to show the performanceimprovement over the benchmark scheme.A
PPENDIX AP ROOF OF P ROPOSITION { v m,n } ∀ m,n do not guarantee that constraints in (11f)are inactive. Without loss of generality, we assume that the ( m, n ) -th constraint in (11f) is inactive. We can obtain alower transmit power ˜ v m,n ≤ v m,n such that the ( m, n ) -thconstraint in (11f) is active. Using the transmit power ˜ v m,n ,we observe that the remaining constraints in (11f) are stillsatisfied. Besides, we have M X m =1 N m X n =1 v m,n ≥ M X m =1 N m X n =1 v m,n + ˜ v m,n − v m,n . (29)Based on (29), the power-control variables { v m,n } ∀ m,n is not optimal. Therefore, we obtain the contradiction. Weconclude that the optimal power-control variables { v m,n } ∀ m,n guarantee that constraints in (11f) are active.When the constraint in (11c) are ignored, we can use thesimilar arguments to prove that the constraints in (11d) areactive. R EFERENCES[1] J. Liu, M. Sheng, and J. Li, “Improving network capacity scaling law inultra-dense small cell networks,”
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