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Energy Efficiency of Downlink TransmissionStrategies for Cloud Radio Access Networks

Binbin Dai, Student Member, IEEEand Wei Yu,Fellow, IEEE

Abstract—This paper studies the energy efficiency of the cloudradio access network (C-RAN), specifically focusing on two fun-damental and different downlink transmission strategies,namelythe data-sharing strategy and the compression strategy. Inthedata-sharing strategy, the backhaul links connecting the centralprocessor (CP) and the base-stations (BSs) are used to carryusermessages – each user’s messages are sent to multiple BSs; theBSslocally form the beamforming vectors then cooperatively transmitthe messages to the user. In the compression strategy, the usermessages are precoded centrally at the CP, which forwards acompressed version of the analog beamformed signals to the BSsfor cooperative transmission. This paper compares the energyefficiencies of the two strategies by formulating an optimizationproblem of minimizing the total network power consumptionsubject to user target rate constraints, where the total networkpower includes the BS transmission power, BS activation power,and load-dependent backhaul power. To tackle the discreteand nonconvex nature of the optimization problems, we utilizethe techniques of reweightedℓ1 minimization and successiveconvex approximation to devise provably convergent algorithms.Our main finding is that both the optimized data-sharing andcompression strategies in C-RAN achieve much higher energyefficiency as compared to the non-optimized coordinated multi-point transmission, but their comparative effectiveness in energysaving depends on the user target rate. At low user targetrate, data-sharing consumes less total power than compression,however, as the user target rate increases, the backhaul powerconsumption for data-sharing increases significantly leading tobetter energy efficiency of compression at the high user rateregime.

Index Terms—Cloud radio access network (C-RAN), data-sharing strategy, compression strategy, energy efficiency, powerminimization, base-station activation, base-station clustering,beamforming, backhaul power.

I. I NTRODUCTION

ULTRA-DENSE deployment of small cells and cooper-ative communications are recognized as two promising

technologies to meet the ever increasing demand of data trafficfor future wireless networks [1]. However, both technologiescome at the cost of increase in energy consumption because ofthe additional energy needed to support the increasing numberof base-station (BS) sites and the substantially increasedback-haul between the BSs for cooperation. The excessive energyconsumption of wireless networks not only has an ecologicalimpact in terms of carbon footprint but also has an economicalimpact on the operational cost to the mobile operators. Thus,the compelling call for improvement ofspectrum efficiency

Manuscript submitted on April 15, 2015; revised on September 15, 2015;accepted on December 11, 2015. This work was supported by HuaweiTechnologies, Canada, and Natural Sciences and Engineering Research Coun-cil (NSERC) of Canada. The authors are with The Edward S. Rogers Sr.Department of Electrical and Computer Engineering, University of Toronto,Toronto, ON M5S 3G4, Canada (e-mails:{bdai, weiyu}@comm.utoronto.ca).

in the fifth-generation (5G) wireless network needs to beaccompanied by a call for improvement ofenergy efficiencyto the same extent.

Cloud radio access network (C-RAN) is an emerging net-work architecture that shows significant promises in improv-ing both the spectrum efficiency and the energy efficiencyof current wireless networks [2]. In C-RAN, the BSs areconnected to a central processor (CP) through backhaul links.The benefits of the C-RAN architecture in energy saving areseveral-fold. First, under the C-RAN architecture, most ofthebaseband signal processing in traditional BSs can be migratedto the cloud computing center so that the conventional high-cost high-power BSs can be replaced by low-cost low-powerradio remote heads (RRHs). Second, the existence of CPalso allows for the joint precoding of user messages forinterference mitigation. With less interference generated, thetransmit power at the BSs can therefore be reduced. Third, ason average (and especially during non-peak time) a significantportion of network resources can be idle [3], the CP canperform joint resource allocation among the BSs to allocateresources on demand and put idle BSs into sleep mode forenergy saving [4].

The above-mentioned benefits of C-RAN in energy savingare concerned with the BS side. However, the additionalenergy consumption due to the increased backhaul betweenthe CP and the BSs also needs to be taken into account[5]. In this paper, we investigate the potential of C-RAN inimproving energy efficiency of the communication aspect ofthe network by considering the energy consumption due toBS activation, transmission, and backhaul provisioning. Thebackhaul energy consumption depends on the backhaul rate,which further depends on the interface between the CP andthe BSs. In this paper, we investigate two fundamental anddifferent transmission strategies for the downlink C-RAN.Inthe data-sharing strategy, the CP uses the backhaul links toshare user messages to a cluster of cooperating BSs. Thebackhaul cost of the data-sharing strategy depends on thenumber of BSs that the user messages need to be deliveredto: larger cluster size leads to larger cooperative gain, butalso higher backhaul rate. In an alternative strategy calledthe compression strategy, the CP performs joint precodingof the user messages centrally then forwards a compressedversion of the precoded signals to the BSs. The backhaulcost of the compression strategy depends on the resolutionof the compressed signals: higher-resolution leads to betterbeamformers, but also larger backhaul rate.

This paper aims to quantify the energy saving of C-RAN while accounting for both the BS and the backhaulenergy consumptions, and specifically to answer the question

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of between the data-sharing strategy and the compressionstrategy, which one is more energy efficient? The answerto this question is nontrivial as there are three factors thatcan lead to energy reduction: decrease in BS transmit power,turning-off of the BSs, and reduction in backhaul rate. Thesethree factors are interrelated. For example, it may be beneficialto keep more BSs active and to allocate higher backhaul ratein order to allow for better cooperation among the BSs sothat more interference can be mitigated. This leads to lesstransmit power consumption, but it can also lead to higherBS and backhaul power consumption. This paper intendsto capture such interplay using an optimization framework.Specifically, we propose a joint design of the BS transmitpower, BS activation and backhaul by minimizing network-wide power under given user rate constraints for both thedata-sharing strategy and the compression strategy. The result-ing optimization problems are nonconvex in nature and arehighly nontrivial to solve globally. This paper approximatesthe problems using reweightedℓ1 minimization technique [6]and successive convex approximation technique, and devisesefficient algorithms with convergence guarantee. We identifyoperating regimes where one strategy is superior to the other,and show that overall optimized C-RAN transmission canlead to more energy efficient network operation than the non-optimized coordinated multi-point (CoMP) transmission.

A. Related Work

The potential of C-RAN in improving the performance forfuture wireless networks has attracted considerable attentionsrecently. In the uplink, [7] and [8] show that with the capabilityof jointly decoding user messages in the CP, the throughputof traditional cellular networks can be significantly improved.A similar conclusion also has been drawn in the downlink.In particular, [9] proposes a joint design of beamforming andmultivariate compression to maximize the weighted sum ratefor the compression strategy, while [10] and [11] considerjoint beamforming and BS clustering design to maximize theweighted sum rate for data-sharing.

This paper focuses on the energy efficiency of C-RANin the downlink. Several metrics have been proposed in theliterature to measure the energy efficiency of a network. Forexample, the area power consumption metric (watts/unit area)is proposed in [12] to evaluate the energy efficiency of net-works of different cell site densities. Another widely adoptedmeasurement in energy efficiency is bits per joule metric,which has been studied in [13] under a simplified single-user-two-BSs model from an information-theoretical point of viewand also studied in [14]–[16] for orthogonal frequency divisionmultiple access (OFDMA) based cooperative networks fromthe practical system design point of view.

In this paper, we formulate the problem of minimizingthe total required power for downlink C-RAN in order toprovide a given set of quality-of-service (QoS) targets forthescheduled users. Such an optimization problem can also bethought of as minimizing watts per bit (or maximizing thebits per watt) at given user service rates. In this domain, mostof the previous works are restricted to the data-sharing strategy

[17]–[19]. Specifically, [17] proposes a joint BS selectionandbeamforming design algorithm to minimize the total powerconsumption in the downlink, while [18] generalizes to a jointdownlink and uplink total power minimization problem. Both[17] and [18] take advantage of the fact that if a BS is notselected to serve any user at the current time slot, it can beput into low-power sleep mode for energy saving purpose. Incontrast, [19] assumes fixed BS association but exploits thedelay tolerance of the users to improve the energy efficiencyin CoMP transmission. In delay-tolerant applications, BSscanaggregate the user messages and transmit them with high rateduring a short time frame while remain idle for the rest ofthe time slots under power-saving sleep mode. Fast deacti-vation/activation of hardware power-consuming componentsachieve significant energy reductions [4], [20].

In this paper, we adopt a similar energy saving perspectiveas in [17] but consider in addition the compression strategy.Compression strategy differs from data-sharing strategy in theway that the backhaul is utilized. We adopt the model proposedin [21] to model the power consumption of backhaul links asa linear function of backhaul rates. This is in contrast to [17],where the backhaul power is modeled as a step function withonly two levels of power consumption depending on whetherthe backhaul link is active or not.

In addition to the backhaul power, we also consider theBS power consumption by adopting the model proposed in[22], which approximates the power consumption of a BSas a piecewise linear function of transmit power. In suchmodel, BS sleep mode corresponds to a constant but lowerpower consumption with zero transmit power. BS active modecorresponds to a higher constant power plus a nonzero transmitpower. The overall framework of this paper is a joint optimiza-tion of BS transmit power, BS activation and backhaul rate forboth the compression and the data-sharing strategies.

From the optimization perspective, the total power con-sumption for the data-sharing strategy involves a sum ofweighted nonconvexℓ0-norms, which is highly nontrivialto optimize globally. Instead, we adopt the reweightedℓ1technique [6] to approximate the nonconvex total power intoa convex weighted sum of transmit power, where the weightsare iteratively updated in a way to reduce not only the numberof active BSs but also the backhaul rate. Such techniquehas also been applied to minimize the total backhaul rate in[23], and to optimize the tradeoff between the total transmitpower and total backhaul rate in [24]. On a related note,the discreteℓ0-norm can also be approximated using othertractable continuous functions such as Gaussian-like functionin [25] and exponential function in [26]. It has been reportedrecently in [27] that those approximation methods show sim-ilar effectiveness in inducing sparsity.

Further, the mathematical expression of the total powerconsumption for the compression strategy involves a differ-ence of two logarithmic functions, which is also nonconvex.We propose to approximate the first logarithmic functionusing the successive convex approximation technique, whichtransforms the objective function into a convex form. Theadopted reweightedℓ1 minimization technique and successiveconvex approximation technique in this paper are related to

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the majorization-minimization (MM) algorithm [28], whichdeals with an optimization problem with nonconvex objectivefunction by successively solving a sequence of optimizationproblems with approximate objective functions. This paperutilizes the known sufficient conditions of convergence fortheMM algorithm in the literature [29] to show the convergenceof the proposed algorithms for both the data-sharing and thecompression strategies.

Finally, we mention that the data-sharing and compressionstrategies considered in this paper are not the only possibilitiesfor the downlink of C-RAN. There is a potential to combinethese two strategies by sending directly the messages of onlythe strong users to the BSs and compressing the rest [30].Also, reverse compute-and-forward strategy that accountsforthe lattice nature of the transmitted message is also possible[31]. However, such strategy is difficult to optimize because ofthe need in choosing the right integer zero-forcing precodingcoefficients at the CP so that the effective noise, caused bythe non-integer penalty due to practical channels, at each useris minimized.

B. Main Contributions

This paper considers energy-efficient design of the data-sharing strategy and the compression strategy for downlinkC-RAN by formulating a problem of minimizing the totalnetwork power consumption subject to user rate constraints.The first contribution of this paper is the modeling of boththe BS power and the backhaul power consumption in thenetwork. The BS power consumption model includes a low-power sleep mode, while the backhaul power consumption ismodeled as a linear function of backhaul traffic rate.

For the data-sharing strategy, we propose a novel applicationof reweightedℓ1 minimization technique to approximate thenonconvex BS activation power and backhaul power. Suchapproximation technique reduces the nonconvex optimizationproblem to a conventional convex transmit power minimiza-tion problem, which can be solved efficiently using theuplink-downlink duality approach or through transformationas second-order cone programming (SOCP). Moreover, weadopt a reweighting function that enables us to connect thereweightedℓ1 minimization technique with the MM algorithm.This connection allows us to prove the convergence behaviorof the proposed algorithm for the data-sharing strategy.

For the compression strategy, in addition to the reweightedℓ1 approximation to the BS activation power as in the data-sharing strategy, we propose a successive convex approxima-tion to the backhaul power, which is in a nonconvex formas a difference of two logarithmic functions. The proposedsuccessive convex approximation technique and the reweightedℓ1 approximation technique can be combined together. Thecombined algorithm falls into the class of the MM algorithmsand has convergence guarantee.

Through simulations, we show that optimized data-sharingand compression strategies in C-RAN can bring much im-proved energy efficiency as compared to the non-optimizedCoMP transmission. However, the comparative energy savingof data-sharing versus compression depends on the user target

rates. The energy efficiency of the data-sharing strategy issuperior to that of the compression strategy in the low-rateregime. However, the backhaul power consumption of thedata-sharing strategy increases significantly with the user rate.Thus, in high user rate regime, the compression strategy maybe preferred from an energy saving perspective.

C. Paper Organization and Notations

The remainder of this paper is organized as follows. Sec-tion II introduces the system model and power consumptionmodel considered throughout this paper. Section III considersthe total power minimization under the data-sharing trans-mission strategy, while Section IV considers the compressionstrategy. Simulation results are presented in Section V andconclusions are drawn in Section VI.

Throughout this paper, lower-case letters (e.g.x) and lower-case bold letters (e.g.x) denote scalars and column vectorsrespectively. We useC to represent complex domain. Thetranspose, conjugate transpose andℓp-norm of a vector aredenoted as(·)T , (·)H and‖ · ‖p respectively. The expectationof a random variable is denoted asE [·]. Calligraphy letters areused to denote sets, while| · | stands for either the size of a setor the absolute value of a scalar, depending on the context.

II. SYSTEM AND POWER CONSUMPTION MODEL

In this section, we describe the overall system model andpower consumption model for the downlink C-RAN consid-ered throughout this paper.

A. System Model

Consider a downlink C-RAN withL BSs servingK users.All the BSs are connected to a CP via backhaul1 links andeach user receives a single independent data stream from theBSs. All the user messages are assumed to be available at theCP and are jointly processed before being forwarded to theBSs through the backhaul links. We assume that the CP hasaccess to global channel state information (CSI) but point outthat such assumption can be relaxed so that only the CSI fromthe neighboring BSs of each user is needed in the CP.

To simplify notations and ease analysis, we assume that theBSs and the users are equipped with a single antenna each,although the proposed algorithms in this paper can be easilygeneralized to the case of multi-antenna BSs as discussed laterin the paper. Letxl ∈ C denote the transmit signal at BSl,we can write the received signalyk ∈ C at userk as

yk = hHk x+ nk, k ∈ K = {1, 2, · · · ,K} (1)

wherex = [x1, x2, · · · , xL]T is the vector of transmit signals

across all theL BSs andhk ∈ CL×1 is the vector of channelgains from all theL BSs to userk. The received noisenk

is modeled as a complex Gaussian random variable with zero

1This paper refers the link between CP and BSs asbackhaul, whichis appropriate if the data-sharing strategy is used. However, in a C-RANarchitecture implementing the compression strategy wherethe BSs are simplyRRHs, the connection between RRH and CP can be referred to moreappropriately asfronthaul.

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mean and varianceσ2. Each user decodes its own messagesk ∈ C from the received signalyk.

In this paper, we investigate two fundamental but differenttransmission strategies, the data-sharing strategy and the com-pression strategy, for the downlink C-RAN for delivering themessagesk to userk via the transmit signalx from the BSs. Inparticular, we compare the potential of these two strategies inimproving the energy efficiency. Before discussing the detailsof the two strategies, we first describe the power consumptionmodel adopted in this paper.

B. Power Consumption Model

Traditional cellular network transmission strategy designtypically only considers transmit power at each BS, whichis written as

Pl,tx = E[|xl|

2]≤ Pl, l ∈ L = {1, 2, · · · , L} , (2)

where Pl is the transmit power budget available at BSl.However, a full characterization of power consumption at a BSshould also consider the efficiency of the power amplifier andother power-consuming components such as baseband unit,cooling system, etc. In addition, the power consumption ofbackhaul links connecting the BSs to the CP also needs tobe taken into account for the specific C-RAN architectureconsidered in this paper. In the following, we describe thepower consumption model adopted in this paper for the BSsand the backhaul links respectively.

1) Base-Station Power Consumption:The characteristic ofpower-consuming components in a BS depends on the BSdesign. We adopt the following unified power consumptionmodel proposed in [22], which is applicable for different typesof BSs. This model approximates the BS power consumptionas a piecewise linear function of the transmit powerPl,tx:

PBSl =

{ηlPl,tx + Pl,active, if 0 < Pl,tx ≤ Pl

Pl,sleep, if Pl,tx = 0, l ∈ L

(3)where ηl > 0 is a constant reflecting the power amplifierefficiency, feeder loss and other loss factors due to powersupply and cooling for BSl, Pl,tx is the transmit powerdefined in (2) andPl,active is the minimum power requiredto support BSl with non-zero transmit power. If BSl hasnothing to transmit, it can be put into sleep mode with lowpower consumptionPl,sleep. Typically, Pl,sleep < Pl,active sothat it is beneficial to turn BSs into sleep mode, wheneverpossible, for energy-saving purpose.

2) Backhaul Power Consumption:In C-RAN, the BSs areconnected to the CP with the backhaul links. The power con-sumption due to backhaul links varies with different backhaultechnologies. In this paper, we model the backhaul as a setof communication channels, each with capacityCl and powerdissipationPBH

l,max, and write the backhaul power consumptionas

PBHl =

RBHl

Cl

PBHl,max = ρlR

BHl , l ∈ L (4)

whereρl = PBHl,max/Cl is a constant scaling factor andRBH

l isthe backhaul traffic between BSl and the CP. This model hasbeen used in [21] for microwave backhaul links and can also

be generalized to other backhaul technologies, such as passiveoptical network, fiber-based Ethernet, etc., as mentioned in[32]. Note that [17] also considers the sleep mode capabilityfor backhaul links. We point out that such considerationcan be unified withPl,active and Pl,sleep in the BS powerconsumption model (3).

3) Total Power Consumption:Based on the above BSpower consumption model (3) and backhaul power consump-tion model (4), we can write the total power consumptionPtotal for C-RAN as

Ptotal =∑

l∈L

(PBSl + PBH

l

)

=∑

l∈L

(

ηlPl,tx + 1{Pl,tx} (Pl,active − Pl,sleep)

+ Pl,sleep + ρlRBHl

)

=∑

l∈L

(

ηlPl,tx + 1{Pl,tx}Pl,∆ + ρlRBHl

)

+∑

l∈L

Pl,sleep

︸ ︷︷ ︸

constant

(5)

where1{·} is the indicator function defined as

1{x} =

{1, if x > 00, otherwise

, (6)

andPl,∆ = Pl,active − Pl,sleep is the difference between theminimum active BS power consumption and the sleep modeBS power consumption.

As we can see from (5), there are three possibilities inimproving the energy efficiency of C-RAN: reducing thetransmit power, putting BSs into sleep mode, and decreasingthe backhaul traffic. However, these three aspects cannot berealized simultaneously: deactivating more BSs means reducedcapability for interference mitigation among the active BSs,which leads to higher transmit power in order to maintainthe QoS for the users; higher backhaul rate can allow formore user information being shared among the BSs so thatthe BSs can better cooperate to mitigate interference, thuslesstransmit power may be needed. A joint design is necessary inorder to balance the roles of transmit power, BS activationand backhaul traffic rate in achieving energy efficiency. Inthe following, we describe the general problem formulationconsidered in this paper for such joint design used in both thedata-sharing and the compression strategies.

C. Energy Efficiency Maximization

This paper aims to understand the energy efficiency fordownlink C-RAN, which can be defined as the ratio of theachievable sum rate and the sum power consumption, i.e.∑

kRk

Ptotalwhere Rk is the data rate for userk determined

by the specific transmission strategy andPtotal is the totalconsumed power defined in (5). Towards this end, this papertakes the similar approach as in [19] to fix the service rates of

5

scheduled users and consider the minimization of total powerconsumption:

minimize Ptotal (7)

subject to Rk ≥ rk, ∀k ∈ K

E[|xl|

2]≤ Pl, ∀l ∈ L

whererk is the fixed target rate for userk. The solution tothe above problem gives us the energy efficiency

∑kRk

Ptotalof the

system at the operating point(r1, r2, · · · , rK). To maximizeenergy efficiency, we need to further search over all operatingpoints. For the rest of the paper, we study and comparethe minimum required total power for different transmissionstrategies under the same operating point(r1, r2, · · · , rK) inthe downlink of C-RAN. Note that problem (7) implicitlyassumes fixed user scheduling. There also exists a possibilityof doing joint user scheduling and power minimization by con-sidering a problem of minimizing the total power consumptionacrossmultiple time slotssubject to a minimum target foreach user’saveragerate. Such problem is considerably morecomplicated.

D. Data-Sharing versus Compression

Data-sharing and compression are two fundamentally dif-ferent transmission strategies for the downlink of C-RAN fordelivering data to the users. These two strategies correspondto alternative functional splits in C-RAN. In the data-sharingtransmission strategy, the CP routes each scheduled user’sin-tended message to a cluster of BSs through the backhaul links;the cluster of BSs then cooperatively serve that user throughjoint beamforming. In contrast, in the compression strategy, theprecoding operation is implemented centrally at the CP, whichthen forwards a compressed version of the analog beamformedsignal to the BSs through the backhaul/fronthaul links. TheBSs then simply transmit the compressed beamforming signalsto the users [9], [30], [33].

The data-sharing strategy differs from the compressionstrategy in backhaul utilization. In data-sharing, the backhaulrate is a function of the user message rate and the BS clustersize, while in the compression strategy the backhaul cost isdetermined by the compression resolution. Intuitively, astheuser target rate and BS cluster size increase, the backhaulrate for the data-sharing strategy would increase significantly,leading to high energy consumption. However, in the low userrate regime where the BS cluster size is small, data-sharingcanbe more efficient than compression as the latter suffers fromquantization noise. Therefore, there exists a tradeoff betweendata-sharing and compression in terms of backhaul rate andenergy efficiency at different user target rate operating points.In the following two sections, we describe in details the data-sharing strategy and the compression strategy, and proposecorresponding algorithms to find the minimum required totalpower for each strategy.

Throughout this paper, we primarily account for the energyconsumption due tocommunicationsin either the backhaul orthe transmission front-end at the BSs, rather than the energyconsumption due tocomputing. There is significant additionalenergy saving due to migrating signal processing from the BSs

Central Processors1, s2

s1s1 s2s2

RBH1 = R1 RBH

2 =

R1 +R2

RBH3 = R2

x1 = w11s1 x2 = w21s1+w22s2

x3 = w32s2

BS 1 BS 2 BS 3

User1 User2

Fig. 1. Downlink C-RAN with data-sharing transmission strategy. In thisillustrative example, the CP transmits user1’s messages1 to BSs1 and2, anduser2’s messages2 to BSs2 and3. The BS cluster(1, 2) then cooperativelyserves user1 and the BS cluster(2, 3) cooperatively serves user2 throughjoint beamforming.

to the cloud computer center in the C-RAN architecture. Werefer the readers to [34].

III. D ATA -SHARING STRATEGY

In this section, we study the minimum total power requiredfor the data-sharing strategy in order to support the givenscheduled users at guaranteed service rates.

A. Problem Formulation

Consider the data-sharing transmission strategy for thedownlink of C-RAN as illustrated in Fig. 1, where the eachuser’s message is shared among a cluster of serving BSs. Letwlk ∈ C be the beamforming coefficient for BSl to serveuserk. If BS l is not part of userk’s serving cluster,wlk isset to be zero. The transmit signalxl at BS l can be writtenas xl =

∑

k∈K wlksk. We model the user messagessk ’sas independent and identically distributed complex Gaussianrandom variables with zero mean and unit variance. Thetransmit powerPl,tx formulated in (2) can be written as

Pl,tx =∑

k∈K

|wlk|2, l ∈ L. (8)

Substitutingxl =∑

k∈K wlksk into (1), the received signalyk at userk is

yk = hHk wksk +

∑

j 6=k

hHk wjsj + nk, k ∈ K, (9)

where wk = [w1k, w2k, · · · , wLk]T is the network beam-

former for user k. Based on (9), the received signal-to-interference-plus-noise ratio (SINR) at userk can be expressedas

SINRk =

∣∣h

Hk wk

∣∣2

∑

j 6=k

∣∣hH

k wj

∣∣2+ σ2

, k ∈ K (10)

6

and the achievable rate for userk is then

Rk = log2

(

1 +SINRk

Γm

)

, k ∈ K, (11)

whereΓm stands for the signal-to-noise ratio (SNR) gap dueto practical modulation scheme.

For the data-sharing strategy, if userk is served by BSl, then the CP needs to send userk’s messagesk, alongwith the beamforming coefficientwlk, to BS l through thebackhaul link. In this paper, we assume that the channels areslow varying and ignore the backhaul required for sharing CSIand beamformers, and only consider the backhaul capacityconsumption due to data-sharing. Hence, the backhaul ratefor BS l, RBH

l , is the accumulated data rates of those usersserved by BSl, which can be formulated as

RBHl =

∑

k∈K

1{|wlk|2}Rk, l ∈ L, (12)

where1{|wlk|2} is the indicator function defined in (6) and

indicates whether or not BSl serves userk.Substituting (8) and (12) into (5), the total power mini-

mization problem (7) can be formulated for the data-sharingstrategy as

minimize{wlk}

∑

l∈L

(

ηl∑

k∈K

|wlk|2+ 1{

∑k∈K

|wlk|2}Pl,∆

+ ρl∑

k∈K

1{|wlk|2}Rk

)

(13a)

subject to Rk = log2

(

1 +SINRk

Γm

)

≥ rk, k ∈ K (13b)∑

k∈K

|wlk|2 ≤ Pl, l ∈ L. (13c)

Note that the∑

l∈L Pl,sleep term in (5) is a constant and hasbeen dropped in the objective function (13a). It is easy to seethat the minimum rate constraint (13b) is met with equalityat the optimal point. Hence, problem (13) can be equivalentlyformulated as

minimize{wlk}

∑

l∈L

(

ηl∑

k∈K

|wlk|2+ 1{

∑k∈K

|wlk|2}Pl,∆

+ ρl∑

k∈K

1{|wlk|2}rk

)

(14a)

subject to SINRk ≥ γk, k ∈ K (14b)∑

k∈K

|wlk|2≤ Pl, l ∈ L (14c)

where the variableRk in (13a) is replaced by the targetrate rk in (14a) and γk = Γm (2rk − 1) in (14b). Thenew SINR constraint (14b) is also met with equality at theoptimality. However, we keep (14b) as an inequality constraint,so that it can be reformulated as a convex second-order cone(SOC) constraint [35]. Note that problem (14) is equivalentto problem (13) in the sense that they have the same optimalsolutions and the same feasibility region.

Note that the above optimization is over the beamformingcoefficients and also implicitly over the BS cluster for each

user. The overall optimization problem (14) aims to choosethe optimal cluster of serving BSs for each scheduled userfor minimizing the total power consumption while satisfyingthe user QoS constraints. Due to the indicator functions in theobjective function (14a), problem (14) is nonconvex (discrete),so finding its global optimum solution is challenging. In thefollowing, we propose to approximate the nonconvex indicatorfunction using reweighted convexℓ1-norm and show thatwith a particular reweighting function the proposed algorithmalways converges2.

B. Proposed Algorithm

We make an observation that the indicator function definedin (6) is equivalent to theℓ0-norm of a scalar. Theℓ0-normof a vector is defined as the number of nonzero entries in thevector, so it reduces to the indicator function in the scalarcase.In compressive sensing literature [6], nonconvexℓ0-norm min-imization problem can be approximated as convex reweightedℓ1 minimization problem. We take advantage of this techniqueand propose to approximate the indicator functions in theobjective function (14a) as

1{∑

k∈K|wlk|

2} =

∥∥∥∥∥

∑

k∈K

|wlk|2

∥∥∥∥∥0

≈ µl

∑

k∈K

|wlk|2 (15)

1{|wlk|2} =

∥∥∥|wlk|

2∥∥∥0≈ νlk |wlk|

2 (16)

with weightsµl andνlk iteratively updated according to

µl = f

(∑

k∈K

|wlk|2 , τ1

)

=c1

∑

k∈K |wlk|2 + τ1

,

νlk = f(

|wlk|2 , τ2

)

=c2

|wlk|2 + τ2

(17)

where{wlk} is the beamformer from the previous iteration,τ1 > 0 and τ2 > 0 are some constant regularization factors,andc1, c2 are constants.

Note that in the above iterative updates ofµl and νlk, theBSs with small transmit power,

∑

k∈K |wlk|2, or small trans-

mit power to userk, |wlk|2, at current iteration are given larger

weightsµl or νlk in the next iteration. This further decreases∑

k∈K |wlk|2 or |wlk|

2 in the next iteration, and eventuallyforces BSl toward sleep mode (i.e.,

∑

k∈K |wlk|2= 0) or to

be removed from userk’s serving cluster (i.e.,|wlk|2= 0). The

weight µl has the effect of putting appropriate BSs to sleepmode, whileνlk has the effect of determining the BS clustersize for userk, which in turn affects the backhaul capacityconsumption of userk.

The resulting optimization problem after theℓ1-norm ap-proximation is formulated as follows:

minimize{wlk}

∑

l∈L

∑

k∈K

αlk |wlk|2 (18)

subject to (14b), (14c)

2In fact, it converges to the stationary point solution of an approximationto problem (14).

7

Algorithm 1 Total Power Minimization for Data-SharingStrategy

Initialization : Set the initial values for{µl, νlk} according to(17) with the {wlk} chosen as a feasible point of problem(14);Repeat:

1) Fix {µl, νlk}, find the optimal{wlk} by solving problem(18) using the uplink-downlink duality approach [36] orby transforming it into an SOCP problem [35];

2) Update{µl, νlk} according to (17).

Until convergence

where αlk = ηl + µlPl,∆ + ρlνlkrk. Problem (18) is aweighted sum transmit power minimization problem, whichcan be solved efficiently through the uplink-downlink dualityapproach [36] or by transforming it into an SOCP problem[35]. We now summarize the proposed algorithm to solve thetotal power minimization problem (14) for the data-sharingstrategy in Algorithm 1.

Note that a similar problem as to (14) is considered in ourprevious work [24], where we formulate the problem as atradeoff between the BS transmit power and the backhaulcapacity. This paper considers a more realistic BS powerconsumption model with sleep mode capability, and alsoaccounts for backhaul power consumption. The consideredproblem (14) in this paper can also be thought of as providinga tradeoff between the per-BS power consumption and theper-BS backhaul capacity consumption, where the tradeoffconstantρl is specifically chosen according to the backhaulpower consumption model (4).

C. Convergence Analysis

Algorithm 1 relies on the reweighting heuristic (17) todeactivate BSs and reduce the BS cluster size for energy savingpurpose. To establish the convergence proof for Algorithm 1under arbitrary reweighting function is challenging, however,we show in the following that if the reweighting function ischosen as

f (x, τ) =1

(x+ τ) ln (1 + τ−1), (19)

i.e. the constants in (17) are chosen asc1 = 1

ln(1+τ−1

1 ), c2 =

1

ln(1+τ−1

2 ), Algorithm 1 can be seen as a special case of the

MM algorithms [28] and is guaranteed to converge.Theorem 3.1:Starting with any initial point, the sequence{

w(n)lk

}∞

n=1generated by Algorithm 1 with the reweighting

function chosen as (19) is guaranteed to converge.Proof: See Appendix A.

Finally, we point out that the choice of the reweightingfunction (19) is not unique. There exist other reweightingfunctions that may work well in different problem setups[6]. Recently, [27] has experimented with other approximationfunctions to theℓ0-norm, e.g. exponential function and arc-tangent function, in addition to the logarithmic function (32)used in this paper, and observed similar effectiveness of thesefunctions in inducing sparsity.

D. Complexity Analysis

Algorithm 1 is an iterative procedure between updating theweights {µl, νlk} and solving the weighted transmit powerminimization problem (18). The problem (18) can be formu-lated as an SOCP and solved using the interior-point method,e.g. using the convex optimization solver [37]. The total num-ber of variables in problem (18) isLK and the total numberof SOC constraints is(L+K). The complexity order forsolving such a problem through interior-point method is givenas O

(

(L+K) (LK)3)

[38]. Assuming that Algorithm 1requires a total number ofT1 weight updates, the overall com-plexity order for Algorithm 1 is thenO

(

T1 (L+K) (LK)3)

.Note that in the above complexity order,K is the number of

scheduled users, which is comparable to the number of activeBSs in the network. In addition, instead of considering all theL BSs in the entire network, we can set the nearestLc < LBSs around each scheduled user as its candidate serving BScluster. This further reduces the computational complexity forAlgorithm 1 with negligible performance loss.

E. Generalization to the Multi-Antenna System

Algorithm 1 can be readily generalized to the case withmultiple transmit antennas at each BS. In such case, oneonly needs to replace the beamforming coefficientwlk withthe beamforming vectorwlk ∈ CNl×1 from BS l to userk,whereNl is the number of antennas at BSl. The rest of theoptimization parameters are straightforward extensions basedon wlk [24].

Algorithm 1 can also be applied to the case with multiplereceive antennas at each user but with fixed receive beam-former. In this case, the only change is to replace the channelgain vectorhk with the effective channel gainhk = Hkuk,whereHk ∈ CL×Mk anduk ∈ CMk×1 are the channel matrixseen by userk and the receive beamformer at userk, Mk

is the number receive antennas at userk. However, the jointdesign of transmit beamformer and receive beamformer forthe multiple receive antennas case is more complicated. Onepossible way is to iteratively design the transmit beamformerassuming fixed receive beamformer and update the receiver asthe optimal minimum mean square error (MMSE) beamformer.

IV. COMPRESSIONSTRATEGY

In this section, we aim to minimize the total power con-sumption for downlink C-RAN under the compression strat-egy.

A. Problem Formulation

Consider the compression transmission strategy for down-link C-RAN as illustrated in Fig. 2. Letxl =

∑

k∈K wlkskdenote the beamformed signal formed in the CP for BSl.The CP compressesxl into xl and sendsxl to BS l. In thispaper, we assume that eachxl is compressed independently3

and model the compression procedure as the following forwardtest channel:

xl = xl + el, l ∈ L, (20)

3Correlated compression is also possible and has been considered in [9].

8

Central Processorxl = wl1s1 + wl2s2

l = 1, 2, 3

x1 x2 x3RBH1 RBH

2RBH

3

x1 = x1 + e1 x2 = x2 + e2 x3 = x3 + e3

BS 1 BS 2 BS 3

User1 User2

Fig. 2. Downlink C-RAN with compression transmission strategy. In thisillustrative example, the CP centrally precodes both users’ messagess1 ands2 to xl, l = 1, 2, 3, and forwards the compressed precoded signalsxl, l =1, 2, 3, to each of the BSs. Each BS transmits the compressed beamformedsignal received from the CP to both users.

whereel ∈ C is the quantization noise independent ofxl andis assumed to be Gaussian distributed with zero mean andvarianceq2l . Substituting (20) to (2), the transmit power at BSl under the compression strategy can be written as

Pl,tx =∑

k∈K

|wlk|2 + q2l , l ∈ L. (21)

Comparing (21) with (8), we can see that different from thedata-sharing strategy, the BS transmit power in the compres-sion strategy involves a quantization noise power in additionto the beamforming power.

Substituting (20) into (1), the received signalyk at userkunder the compression strategy can be written as

yk = hHk wksk +

∑

j 6=k

hHk wjsj + h

Hk e+ nk, k ∈ K, (22)

wheree = [e1, e2, · · · , eL]T is the quantization noise vector

transmitted from all theL BSs. As we can see, besides theinter-user interference and background noise, each user nowalso receives an additional quantization noise termh

Hk e from

the BSs. The user received SINR is expressed as

SINRk =

∣∣h

Hk wk

∣∣2

∑

j 6=k

∣∣hH

k wj

∣∣2+∑

l∈L |hlkql|2 + σ2

, k ∈ K.

(23)The backhaul capacity consumption for the compression

strategy is related to the level of quantization noiseq2l : lowerquantization noise requires higher backhaul rate. Under theforward test channel (20), the achievable compression rateisthe mutual information betweenx and x, according to rate-

distortion theory [39], given aslog2(

1 +∑

k∈K|wlk|

2

q2l

)

, where∑

k∈K |wlk|2 is the power of the signal to be compressed,

i.e. xl. However, practical quantizer may be far from the

theoretically ideal quantizer. Similar to [33], we introduce anotion of gap to rate-distortion limit, denote asΓq > 1, toaccount for the loss due to practical quantizer and formulatethe backhaul capacity consumption for BSl as

RBHl = log2

(

1 +Γq

∑

k∈K |wlk|2

q2l

)

, l ∈ L. (24)

Substituting (21) and (24) into (5), the total power mini-mization problem for the compression strategy is formulatedas follows

min{wlk,ql}

∑

l∈L

(

ηl

(∑

k∈K

|wlk|2 + q2l

)

+ 1{∑

k∈K|wlk|

2+q2l }Pl,∆

+ρl log2

(

1 +Γq

∑

k∈K |wlk|2

q2l

))

(25a)

s. t. SINRk ≥ γk, k ∈ K (25b)∑

k∈K

|wlk|2+ q2l ≤ Pl, l ∈ L (25c)

where the SINR in (25b) is defined in (23). Due to the indicatorfunction as well as the backhaul rate expression in (25a), theoptimization problem (25) is nonconvex. In the following, wedescribe the techniques to approximate (25a) in a convex form.

B. Proposed Algorithm

The difficulties in solving problem (25) lie in both theindicator function and the nonconvex backhaul rate expressionin the objective function (25a). For the indicator function, wecan utilize the similar technique used in the previous sectionto approximate it using reweightedℓ1-norm:

1{∑

k∈K|wlk|

2+q2l }

=

∥∥∥∥∥

∑

k∈K

|wlk|2+ q2l

∥∥∥∥∥0

≈ βl

(∑

k∈K

|wlk|2+ q2l

)

(26)

where βl is iteratively updated according to the followingreweighting function

βl = f

(∑

k∈K

|wlk|2+ q2l , τ3

)

=c3

∑

k∈K |wlk|2+ q2l + τ3

(27)where{wlk, ql} come from the previous iteration,τ3 > 0 issome constant regularization factor, andc3 is a constant.

For the backhaul rate (24), we can express it as a differenceof two logarithmic functions:log2

(

q2l + Γq

∑

k∈K |wlk|2)

−

2ρl log2 ql. Although the second term−2ρl log2 ql is convexin ql, the first term is still nonconvex. To deal with thenonconvexity of the backhaul rate, we propose to successivelyapproximate the first logarithmic function using the followinginequality

log2

(

q2l + Γq

∑

k∈K

|wlk|2

)

≤ log2 λl +q2l + Γq

∑

k∈K |wlk|2

λl ln 2−

1

ln 2(28)

9

Algorithm 2 Total Power Minimization for CompressionStrategy

Initialization : Set the initial values for{βl} and{λl} accord-ing to (27) and (29) respectively with{wlk, ql} chosen as afeasible point of problem (25);Repeat:

1) Fix {βl, λl}, find the optimal{wlk, ql} by solving theconvex optimization problem (30);

2) Update{βl} and {λl} according to (27) and (29) re-spectively.

Until convergence

due to the concavity oflog2(x). The above inequality achievesequality if and only if

λl = q2l + Γq

∑

k∈K

|wlk|2. (29)

The right-hand side of (28) is a convex quadratic function in{wlk, ql} for fixed λl. This fact motivates us to successively

solve the problem (25) withlog2(

q2l + Γq

∑

k∈K |wlk|2)

re-placed by the right-hand side of (28) for fixedλl, then toiteratively updateλl according to (29).

Combining the above describedℓ1-norm reweighting andsuccessive convex approximation techniques, we get the re-sulting optimization problem under fixedβl andλl as

minimize{wlk,ql}

∑

l∈L

∑

k∈K

φl |wlk|2 +

∑

l∈L

(ψlq

2l − 2ρl log2 ql

)

+∑

l∈L

ρl

(

log2 λl −1

ln 2

)

︸ ︷︷ ︸

constant

(30)

subject to (25b), (25c)

whereφl = ηl+βlPl,∆+ρlΓq

λl ln 2 andψl = ηl+βlPl,∆+ ρl

λl ln 2 .Similar to the SINR constraint in (14b), the constraint (25b)can also be equivalently reformulated as an SOC constraint.Thus, problem (30) is a convex optimization problem andcan be solved efficiently using standard convex optimizationsolver, e.g. [37], with polynomial complexity.

We summarize the proposed algorithm for solving problem(25) in Algorithm 2, which admits guaranteed convergenceproperty as stated in the following theorem.

Theorem 4.1:Starting with any initial point, the sequence{

w(n)lk , q

(n)l

}∞

n=1generated by Algorithm 2 with the reweight-

ing function in (27) chosen as (19) is guaranteed to converge.Proof: See Appdendix B

Algorithm 2 shows a similar computational complexity asAlgorithm 1 but with additionalL quantization noise variablesto be optimized in each iteration. Assuming that Algorithm 2converges inT2 iterations, its complexity order is then givenasO

(

T2 (L+K) (LK + L)3)

.

C. Generalization to the Multi-Antenna System

Algorithm 2 can be readily applied to the scenario wheremultiple transmit antennas are available at the BSs assuming

TABLE ISIMULATION PARAMETERS.

Cellular HexagonalLayout 7-cell wrapped-around

Channel bandwidth 10 MHzDistance between cells 0.8 kmNumber of RRHs/cell 4

Number of antennas/(RRH, user) (1, 1)Maximum transmit power for RRHPl 20 WattsActive mode power for RRHPl,active 84 WattsSleep mode power for RRHPl,sleep 56 Watts

Slope of transmit powerηl 2.8Backhaul link capacityCl 100 Mbps

Maximum backhaul powerPBHl,max

50 WattsAntenna gain 15 dBi

Background noise −169 dBm/HzPath loss from RRH to user 128.1 + 37.6 log10(d)

Log-normal shadowing 8 dBRayleigh small scale fading 0 dB

SNR gapΓm 0 dBGap to rate-distortion limitΓq 4.3 dB

Reweighting parameters (τ1, τ2, τ3) (10−5, 10−8, 10−5)

km

km 0

0.5

1

1.5

−0.5

−1

−1.50 0.5 1 1.5−0.5−1−1.5

Fig. 3. A cellular topology with7 cells and4 RRHs each cell, where eachdot represents a RRH.

that the CP performs independent compression for each an-tenna. Joint compression among the antennas may improve theperformance but results in a different optimization problem.Similar to Algorithm 1, generalization of Algorithm 2 to mul-tiple receive antennas at the user side is also straightforward ifthe receive beamformer is assumed to be fixed, however, jointdesign of transmit and receive beamformer is by no meanstrivial and requires additional efforts.

V. NUMERICAL EVALUATION OF ENERGY EFFICIENCY

In this section, we evaluate the energy efficiency of the data-sharing and compression strategies for downlink C-RAN usingthe proposed algorithms for a7-cell network with wrapped-around topology. Each cell here refers to a geographic areawith 4 RRHs as shown in Fig. 3. Equivalently, the networkconsists of28 BSs. The out-of-cell interference combinedwith background noise is set as−150 dBm/Hz. The gapto rate-distortion limit is set asΓq = 4.3 dB corresponding

10

Iteration

Nu

mb

ero

fA

ctiv

eB

Ss

Data-Sharing (Alg.1) w/1 user/cell

Data-Sharing (Alg.1) w/2 users/cell

Data-Sharing (Alg.1) w/3 users/cell

Compression (Alg.2) w/1 user/cell

Compression (Alg.2) w/2 users/cell

Compression (Alg.2) w/3 users/cell

10 20 30 40 50 60 705

10

15

20

25

30

35

Fig. 4. Trajectories of number of active BSs underrk = 20 Mbps targetrate for each user.

Iteration

Ob

ject

ive

Val

ue

of

Pro

ble

m(3

1)/(3

5)

Data-Sharing (Alg.1) w/1 user/cell

Data-Sharing (Alg.1) w/2 users/cell

Data-Sharing (Alg.1) w/3 users/cell

Compression (Alg.2) w/1 user/cell

Compression (Alg.2) w/2 users/cell

Compression (Alg.2) w/3 users/cell

10 20 30 40 50 60 70 800

400

800

1200

1600

2000

Fig. 5. Convergence behavior of proposed algorithms underrk = 20 Mbpstarget rate for each user.

to the uncoded fixed rate uniform scalar quantizer [40]. Forsimplicity, the SNR gap is set to beΓm = 0 dB. Theparameters in the BS power consumption model (3) are takenfrom [22] while the parameters for the backhaul power model(4) are from [21]. All the parameters related to the simulationsare listed in Table I.

We first evaluate the effectiveness of the proposedℓ1-normreweighting technique in turning off BSs. We plot the numberof active BSs remained in each iteration for both data-sharing(Algorithm 1) and compression (Algorithm 2) strategies inFig. 4. The user target rate is set to be20 Mbps for every userand different number of scheduled users are simulated. Insteadof considering all the28 BSs in the entire network as potentialserving BSs for each user, we set the initial BS cluster for each

user as the strongestLc = 14 BSs to reduce the amount ofCSI acquisition for the CP. From Fig. 4 we can see that allthe BSs are active at the first iteration, however, the numberof active BSs decreases as the iteration goes on. Intuitively,more users are served, more BSs need to remain active. Thiscan be verified from Fig. 4. We also observe from Fig. 4 thatAlgorithm 1 for data-sharing exhibits faster convergence speedthan Algorithm 2 for compression, where the former convergeswithin 30 iterations while the later requires50 iterations toconverge in the worst case.

We then evaluate the convergence behavior of the proposedalgorithms in Fig. 5. As in Fig. 4, different number ofscheduled users are tested and each user’s target rate is settobe20 Mbps. As we can see, the objective values monotonicallydecrease and converge for both the data-sharing (Algorithm1)and the compression (Algorithm 2) strategies. Similar to Fig. 4,we also observe faster convergence speed for data-sharing thanfor compression in Fig. 5. This is possibly due to the fact thatcompression strategy involves more variables to be optimized.

We now compare the performance of the data-sharingstrategy and the compression strategy in terms of power savingin Fig. 6. In addition, we consider two reference schemes. Inthe first scheme, each user is only served by its strongest BSthat is not already associated with another user. This schemeis termed as “Single BS Association”, for which the transmitpower for each user can be minimized using the strategy in[41]. In the second scheme, each user’s message is sharedamong the4 RRHs in its own cell and is cooperatively servedby the4 RRHs using the coordinated beamforming strategy of[36]. Such scheme is termed as “Per-Cell CoMP”. The “SingleBS Association” and “Per-Cell CoMP” are two extreme casesin terms of number of active BSs: the former only hasK(number of scheduled users) active BSs while in the latter allthe 28 BSs in the entire network remain active.Each point inFig. 6 is averaged over100 channel realizations.

As we can see from Fig. 6, “Per-Cell CoMP” consumes themost power since all the BSs are active in this scheme. “SingleBS Association” consumes the least power, similar to the data-sharing strategy, but only at low user rate regime because theminimum number of BSs are selected to serve the users in thisscheme. However, as the user rate or the number of scheduledusers increases, “Single BS Association” becomes infeasiblevery quickly. For instance, in Fig. 6(b) where there are2 usersper cell, “Single BS Association” can only support each userwith 10 Mbps service rate, while in Fig. 6(c) the case of 3users per cell, “Single BS Association” is not feasible evenat10 Mbps per user.

It is worth noting that “Single BS Association” consumesthe same amount of power as data-sharing in the low usertarget rate regime in Fig. 6, as the latter essentially reducesto single BS association at low user rates. However, there isstill significant advantage in migrating signal processingto thecloud in a C-RAN as compared to the conventional cellulararchitecture in term of computation power saving, which isnot included in the model in this paper. It is also worth notingthat the optimized data-sharing and compression strategiesoutperform the non-optimized per-Cell CoMP significantly inFig. 6, highlighting the importance of optimization approaches

11

User Target Rate (Mbps)

Tota

lP

ower

Co

nsu

mp

tion

(kW

atts

) CompressionData-SharingPer-Cell CoMPSingle BS Association

10 20 30 40 50 60 701.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(a) 1 user per cell

User Target Rate (Mbps)To

tal

Pow

erC

on

sum

ptio

n(k

Wat

ts)

CompressionData-SharingPer-Cell CoMPSingle BS Association

10 20 30 40 50 60 701.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(b) 2 users per cell

User Target Rate (Mbps)

Tota

lP

ower

Co

nsu

mp

tion

(kW

atts

) CompressionData-SharingPer-Cell CoMP

10 20 30 40 50 60 701.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(c) 3 users per cell

Fig. 6. Total power consumption comparison between different schemes. Missing points indicate that the correspondingscheme is infeasible. For instance,“Single BS Association” is feasible only for the first three points in (a), and is only feasible for the very first point in (b), and is infeasible in (c). For “Per-CellCoMP” scheme in (c), it is only feasible at10 Mbps target rate.

proposed in this paper. Other optimized CoMP schemes withlarger cooperation cluster may consume less BS transmissionpower, however, the overall power consumptions can be stillvery high if all the BSs remain active.

We also observe from Fig. 6 that neither the data-sharing northe compression strategy dominates the other over the entireuser target rate regime. For example in Fig. 6(b), althoughdata-sharing consumes less power than compression when theuser target rate is below30 Mbps, its power consumptionincreases dramatically with the user rate and eventually crossesover the total power consumed by compression after40 Mbpstarget rate. Similar trend can be observed from Fig. 6(a) and6(c). This trend is parallel to the observation made in [33],in which these two downlink strategies are compared fromthe utility maximization perspective with limited backhaulconstraint. It is observed in [33] that with low backhaulrate, data-sharing produces higher utility than compressionwhile with high backhaul rate, compression outperforms data-sharing.

To investigate further, we plot the individual power con-sumption of BSs and backhaul links for both the data-sharingand the compression strategies in Fig. 7 for the case of2users per cell. As seen from Fig. 7, although the BS powerconsumptions for data-sharing and compression are similarineach case of the user target rate, the backhaul power consump-tions are significantly different and are the determining factorin the choice of strategies. As the user target rate increases,the backhaul power consumption for data-sharing increasessignificantly and crosses over the compression strategy ataround30 Mbps. This is because in data-sharing, each user’smessage needs to be delivered to each one of its servingBSs through backhaul links. So, the backhaul rate directlydepends on both the user target rate and the BS cluster size.Note that as user target rate increases, the size of serving BScluster also increases. The two factors together contribute toa much higher backhaul rate. In contrast, the backhaul rate ofcompression strategy depends on the logarithm of the signal-

User Target Rate (Mbps)

Pow

er(k

Wat

ts)

Data-Sharing BS Power

Data-Sharing Backhaul Power

Compression BS Power

Compression Backhaul Power

10 20 30 40 50 60 70

1

2

3

4

5

6

7

0

Fig. 7. Power consumptions of BSs and backhaul links with2 users eachcell.

to-quantization-noise ratio, which only increases gradually asuser target rate increases. Also, note that in the low userrate regime, data-sharing consumes less backhaul power thancompression in Fig. 7. This is because it is more efficient toshare data directly than to compress when only a few BSs areinvolved.

In Fig. 8, we compare the percentage of active BSs inthe data-sharing strategy versus in the compression strategy.Similar to Fig. 6, each point in Fig. 8 is averaged over100channel realizations. As we can see, with higher user targetrate and more users to be served, more BSs need to remainactive for transmission. Also, from Fig. 8, it is observed thatthe compression strategy tends to turn off more BSs than thedata-sharing strategy.

12

User Target Rate (Mbps)

Fra

ctio

no

fA

ctiv

eB

Ss

CompressionData Sharing

10 20 30 40 50 60 700.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(a) 1 user per cell

User Target Rate (Mbps)F

ract

ion

of

Act

ive

BS

s

CompressionData Sharing

10 20 30 40 50 60 700.4

0.5

0.6

0.7

0.8

0.9

1.0

(b) 2 users per cell

User Target Rate (Mbps)

Fra

ctio

no

fA

ctiv

eB

Ss

CompressionData Sharing

10 20 30 40 50 60 70

0.7

0.8

0.9

0.65

0.75

0.85

0.95

1.0

(c) 3 users per cell

Fig. 8. Comparison of average fraction of active BSs. Note that the “Per-Cell CoMP” scheme considered in Fig. 6 corresponds to100% of active BSs, whilethe percentage of active BSs for “Single BS Association” scheme depends on the number of users in each cell, which are25%, 50% and75% respectivelyfor 1, 2 and3 users per cell, although only those few points corresponding to Fig. 6 are feasible.

VI. CONCLUSION

This paper compares the energy efficiency between the data-sharing strategy and the compression strategy in downlink C-RAN. We formulate the problem as that of minimizing thetotal network power consumption subject to user target rateconstraints, with both the BS power consumption and thebackhaul power consumption taken into account. By taking ad-vantage of theℓ1-norm reweighting technique and successiveconvex approximation technique, we transform the nonconvexoptimization problems into convex form and devise efficientalgorithms with provable convergence guarantees.

The main conclusions of this paper are that C-RAN sig-nificantly improves the range of feasible user data rates ina wireless cellular network, and that both data-sharing andcompression strategies bring much improved energy efficiencyto downlink C-RAN as compared to non-optimized CoMP.Moreover, between the data-sharing strategy and the com-pression strategy, either may be preferred depending on thedifferent target rate regimes: at low user target rate, data-sharing consumes less power, while at high user target ratecompression is preferred since the backhaul rate for data-sharing increases significantly as user rate increases.

APPENDIX APROOF FORTHEOREM 3.1

The idea is to show that Algorithm 1 converges to thestationary point solution of the following problem:

min{wlk}

∑

l∈L

(

ηl∑

k∈K

|wlk|2+ Pl,∆

ln(

1 + τ−11

∑

k∈K |wlk|2)

ln(1 + τ−1

1

)

+ ρl∑

k∈K

ln(

1 + τ−12 |wlk|

2)

ln(1 + τ−1

2

) rk

)

(31)

s. t. (14b), (14c)

First, we note that problem (31) differs from the originalproblem (14) in that theℓ0-norms in the objective function

(14a) are approximated by the logarithmic functions in (31).This approximation stems from the relation that withx ≥ 0,

1{x} = ‖x‖0 = limτ→0

ln(1 + xτ−1

)

ln (1 + τ−1). (32)

Now, due to the concavity oflnx, the inequalitylnx ≤lnx0 + x−1

0 x − 1 holds for anyx > 0, x0 > 0 and achievesequality if and only ifx = x0. Hence, we have

(31)≤∑

l∈L

(

ηl∑

k∈K

|wlk|2

+ Pl,∆

lnxl + x−1l

(

1 + τ−11

∑

k∈K |wlk|2)

− 1

ln(1 + τ−1

1

)

+ ρl∑

k∈K

ln ylk + y−1lk

(

1 + τ−12 |wlk|

2)

− 1

ln(1 + τ−1

2

) rk

)

,

∀xl > 0, ylk > 0 (33)

with equality if and only if

xl = 1 + τ−11

∑

k∈K

|wlk|2, ylk = 1 + τ−1

2 |wlk|2. (34)

Although problem (31) is nonconvex due to the logarithmicobjective function, the function on the right-hand side of (33)is a convex quadratic function in{wlk}. Based on this fact,we can develop an MM algorithm to solve problem (31) bysolving a sequence of convex optimization problems with theobjective function in (31) replaced by the convex function in(33) and iteratively updating the parametersxl, ylk accordingto (34). Comparing such MM algorithm with Algorithm 1, itis easy to see that Algorithm 1 reduces to the MM algorithmif the reweighting function in (17) is chosen as (19).

It is known in the literature [29] that an MM algorithmis guaranteed to converge to the stationary point solutionsofthe original problem if the approximate function satisfies thefollowing conditions: 1) it is continuous, 2) it is a tight upper

13

bound of the original objective function and 3) it has the samefirst-order derivative as the original objective function at thepoint where the upper bound is tight. It is easy to check thatthe function in (33) satisfies all these sufficient conditions.Thus, Algorithm 1, which is equivalent to an MM algorithm,must converge.

APPENDIX BPROOF FORTHEOREM 4.1

Similar to the proof for Theorem 3.1, the idea is show thatAlgorithm 2 converges to the stationary point solution of thefollowing optimization problem:

minimize{wlk,ql}

∑

l∈L

(

ηl

(∑

k∈K

|wlk|2+ q2l

)

+ Pl,∆

ln(

1 + τ−13

(∑

k∈K |wlk|2+ q2l

))

ln(1 + τ−1

3

)

+ ρl log2

(

1 +Γq

∑

k∈K |wlk|2

q2l

))

(35)

subject to (25b), (25c) ,

which is a logarithmic approximation to the original problem(25).

Problem (35) is nonconvex due to the logarithmic functionsin its objective function. However, we can develop an MMalgorithm to solve (35) by solving a sequence of convexoptimization problems with the objective function in (35)replaced by its upper bound shown below

∑

l∈L

(

ηl

(∑

k∈K

|wlk|2+ q2l

)

− 2ρl log2 ql

+ Pl,∆

ln zl + z−1l

(

1 + τ−13

(∑

k∈K |wlk|2+ q2l

))

− 1

ln(1 + τ−1

3

)

+ ρl

(

log2 λl +q2l + Γq

∑

k∈K |wlk|2

λl ln 2−

1

ln 2

))

(36)

and iteratively updating the parameterszl as zl = 1 +

τ−13

(∑

k∈K |wlk|2 + q2l

)

and λl as (29). It is easy to seethat such MM algorithm is equivalent to Algorithm 2 withthe reweighting function in (27) chosen as (19). We can alsoeasily verify that the majorizing function in the right-hand sideof (36) satisfies all the sufficient conditions in [29] for theconvergence guarantee of MM algorithm. Hence, Algorithm 2must converge.

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Binbin Dai (S’12) received the B.E. degree inInformation Science and Engineering from Chien-Shiung Wu Honors College, Southeast University,Nanjing, China, in 2011 and M.A.Sc degree in Elec-trical and Computer Engineering from University ofToronto, Toronto, Ontario, Canada, in 2014. He iscurrently working towards the Ph.D degree with theDepartment of Electrical and Computer Engineering,University of Toronto, Toronto, Ontario, Canada.His research interests include optimization, wirelesscommunications and signal processing.

Wei Yu (S’97-M’02-SM’08-F’14) received theB.A.Sc. degree in Computer Engineering and Math-ematics from the University of Waterloo, Waterloo,Ontario, Canada in 1997 and M.S. and Ph.D. degreesin Electrical Engineering from Stanford University,Stanford, CA, in 1998 and 2002, respectively. Since2002, he has been with the Electrical and Com-puter Engineering Department at the University ofToronto, Toronto, Ontario, Canada, where he is nowProfessor and holds a Canada Research Chair (Tier1) in Information Theory and Wireless Communica-

tions. His main research interests include information theory, optimization,wireless communications and broadband access networks.

Prof. Wei Yu currently serves on the IEEE Information TheorySocietyBoard of Governors (2015-17). He is an IEEE Communications SocietyDistinguished Lecturer (2015-16). He served as an Associate Editor for IEEETRANSACTIONS ON INFORMATION THEORY (2010-2013), as an Editor forIEEE TRANSACTIONS ONCOMMUNICATIONS (2009-2011), as an Editor forIEEE TRANSACTIONS ONWIRELESSCOMMUNICATIONS (2004-2007), andas a Guest Editor for a number of special issues for the IEEE JOURNAL

ON SELECTED AREAS IN COMMUNICATIONS and the EURASIP JOURNAL

ON APPLIED SIGNAL PROCESSING. He was a Technical Program co-chairof the IEEE Communication Theory Workshop in 2014, and a TechnicalProgram Committee co-chair of the Communication Theory Symposium atthe IEEE International Conference on Communications (ICC)in 2012. Hewas a member of the Signal Processing for Communications andNetworkingTechnical Committee of the IEEE Signal Processing Society (2008-2013).Prof. Wei Yu received a Steacie Memorial Fellowship in 2015,an IEEECommunications Society Best Tutorial Paper Award in 2015, an IEEE ICCBest Paper Award in 2013, an IEEE Signal Processing Society Best PaperAward in 2008, the McCharles Prize for Early Career ResearchDistinction in2008, the Early Career Teaching Award from the Faculty of Applied Scienceand Engineering, University of Toronto in 2007, and an EarlyResearcherAward from Ontario in 2006. He was named a Highly Cited Researcher byThomson Reuters in 2014.