1

Energy-Efficient Design of MIMO HeterogeneousNetworks with Wireless Backhaul

Howard H. Yang,Student Member, IEEE, Giovanni Geraci,Member, IEEE, andTony Q. S. Quek,Senior Member, IEEE

Abstract—As future networks aim to meet the ever-increasingrequirements of high data rate applications, dense and hetero-geneous networks (HetNets) will be deployed to provide bettercoverage and throughput. Besides the important implications forenergy consumption, the trend towards densification calls formore and more wireless links to forward a massive backhaultraffic into the core network. It is critically important to t akeinto account the presence of a wireless backhaul for the energy-efficient design of HetNets. In this paper, we provide a generalframework to analyze the energy efficiency of a two-tier MIMOheterogeneous network with wireless backhaul in the presenceof both uplink and downlink transmissions. We find that underspatial multiplexing the energy efficiency of a HetNet is sensitiveto the network load, and it should be taken into account whencontrolling the number of users served by each base station.We show that a two-tier HetNet with wireless backhaul canbe significantly more energy efficient than a one-tier cellularnetwork. However, this requires the bandwidth division betweenradio access links and wireless backhaul to be optimally designedaccording to the load conditions.

Index Terms—Green communications, wireless backhaul, het-erogeneous networks, stochastic geometry

I. I NTRODUCTION

In order to meet the exponentially growing mobile datademand, the next generation of wireless communication sys-tems targets a thousand-fold capacity improvement, and theprospective increase in energy consumption poses urgent en-vironmental and economic challenges [3], [4]. Green commu-nications have become an inevitable necessity, and much effortis being made both in industry and academia to develop newarchitectures that can reduce the energy per bit from currentlevels, thus ensuring the sustainability of future wireless net-works [5]–[9].

A. Background and Motivation

Since the current growth rate of wireless data exceeds bothspectral efficiency advances and availability of new wireless

Manuscript received September 15, 2015; revised January 24, 2016; ac-cepted March 17, 2016. The associate editor coordinating the review of thismanuscript and approving it for publication was Dr. Mohammad Reza Nakhai.

This work was supported in part by the A*STAR SERC under Grant1224104048, and the MOE ARF Tier 2 under Grant MOE2014-T2-2-002, and the SUTD-ZJU Research Collaboration under Grant SUTD-ZJU/RES/01/2014. This work was presented in part at 2016 IEEE InternationalConference on Acoustics, Speech and Signal Processing [1],and will bepresented in part to 2016 IEEE International Conference on Communications[2].

H. H. Yang, and T. Q. S. Quek are with the Singapore UniversityofTechnology and Design, Singapore (hao_yang@mymail.sutd.edu.sg,tonyquek@sutd.edu.sg). G. Geraci is with Bell Labs Nokia, Ireland (giovanni.geraci@nokia.com).

spectrum, a trend towards densification and heterogeneity isessential to respond adequately to the continued surge in mo-bile data traffic [10]–[12]. To this end, heterogeneous networks(HetNets) can provide higher coverage and throughput byoverlaying macro cells with a large number of small cellsand access points, thus offloading traffic and reducing thedistance between transmitter and receiver [13], [14]. Whensmall cells are densely deployed, forwarding a massive cellulartraffic to the backbone network becomes a key problem, and awireless backhaul is regarded as the only practical solutionfor outdoor scenarios where wired links are not available[15]–[19]. However, the power consumption incurred on thewireless backhaul links, together with the power consumedby the multitude of access points deployed, becomes a crucialissue, and an energy-efficient design is necessary to ensuretheviability of future wireless HetNets [20].

Various approaches have been investigated to improve theenergy efficiency of heterogeneous networks. Cell size, de-ployment density, and number of antennas were optimizedto minimize the power consumption of small cells [21],[22]. Cognitive sensing and sleep mode strategies were alsoproposed to turn off inactive access points and enhance theenergy efficiency [23], [24]. A further energy efficiency gainwas shown to be attainable by serving users that experiencebetter channel conditions, and by dynamically assigning usersto different tiers of the network [25], [26]. Although variousstudies have been conducted on the energy efficiency ofHetNets, the impact of a wireless backhaul has typicallybeen neglected. On the other hand, the power consumptionof backhauling operations at small cell access points (SAPs)might be comparable to the amount of power necessary tooperate macro base stations (MBSs) [27]–[29]. Moreover,since it is responsible to aggregate traffic from SAPs towardsMBSs, the backhaul may significantly affect the rates andtherefore the energy efficiency of the entire network. With apotential evolution towards dense infrastructures, wheremanysmall access points are expected to be used, it is of criticalimportance to take into account the presence of a wirelessbackhaul for the energy-efficient design of heterogeneousnetworks.

B. Approach and Main Outcomes

The main goal of this paper is to study the energy efficiencyof heterogeneous networks with wireless backhaul. We con-sider a two-tier HetNet which consists of MBSs and SAPs,where SAPs are connected to MBSs via a multiple-input-multiple-output (MIMO) wireless backhaul that uses a fraction

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of the total available bandwidth. We undertake an analyticalapproach to derive data rates and power consumption forthe entire network in the presence of both uplink (UL) anddownlink (DL) transmissions and spatial multiplexing. This isa practical scenario that has not yet been addressed. In thispaper, we model the spatial locations of MBSs, SAPs, anduser equipments (UEs) as independent homogeneous Poissonpoint processes (PPPs), and analyze the energy efficiency bycombining tools from stochastic geometry and random matrixtheory. Stochastic geometry is a powerful tool to analyzethe interference in large HetNets with a random topology[30], whereas random matrix theory enables a deterministicabstraction of the physical layer, for a fixed network topology[31]. Our analysis is general and encompasses all the keyfeatures of a heterogeneous network, i.e., interference, load,deployment strategy, and capability of the wireless infras-tructure components. With the developed framework, we canexplicitly characterize the power consumption of the HetNetdue to signal processing operations in macro cells, small cells,and wireless backhaul, as well as the rates and ultimately theenergy efficiency of the whole network. Our main contribu-tions are summarized below.

• We provide a general toolset to analyze the energyefficiency of a two-tier MIMO heterogeneous networkwith wireless backhaul. Our model accounts for bothUL and DL transmissions and spatial multiplexing, forthe bandwidth and power allocated between macro cells,small cells, and backhaul, and for the infrastructuredeployment strategy.

• We combine tools from stochastic geometry and randommatrix theory to derive the uplink and downlink ratesof macro cells, small cells, and wireless backhaul. Theresulting analysis is tractable and captures the effectsof multiantenna transmission, fading, shadowing, andrandom network topology.

• Using the developed framework, we find that the energyefficiency of a HetNet is sensitive to the load condi-tions of the network, thus establishing the importance ofscheduling the right number of UEs per base station whenspatial multiplexing is employed. Moreover, by com-paring the energy efficiency under different deploymentscenarios, we find that such property does not depend onthe infrastructure.

• We show that if the wireless backhaul is not allocatedsufficient resources, then the energy efficiency of a two-tier HetNet with wireless backhaul can be worse thanthat of a one-tier cellular network. However, the two-tierHetNet can achieve a significant energy efficiency gain ifthe backhaul bandwidth is optimally allocated accordingto the load conditions of the network.

The remainder of the paper is organized as follows. Thesystem model is introduced in Section II. In Section III, wedetail the power consumption of a heterogeneous network withwireless backhaul. In Section IV, we analyze the data rates andthe energy efficiency, and we provide simulations that confirmthe accuracy of our analysis. Numerical results are shown inSection V to give insights into the energy-efficient design of

MBS

SAP

UE

Wireless backhaul downlink

Macro/small cell downlink

Macro/small cell uplink

Wireless backhaul uplink

Fig. 1. Illustration of a two-tier heterogeneous network with wirelessbackhaul.

a HetNet with wireless backhaul. The paper is concluded inSection VI.

II. SYSTEM MODEL

A. Topology and Channel

We study a two-tier heterogeneous network which consistsof MBSs, SAPs, and UEs, as depicted in Figure 1. Thespatial locations of MBSs, SAPs, and UEs follow independentPPPsΦm, Φs, andΦu, with spatial densitiesλm, λs, andλu,respectively1. All MBSs, SAPs, and UEs are equipped withMm, Ms, and 1 antennas, respectively, each UE associateswith the base station that provides the largest average receivedpower, and each SAP associates with the closest MBS. Thelinks between MBSs and UEs, SAPs and UEs, and MBSsand SAPs are referred to asmacro cell links, small cell links,andbackhaul links, respectively. In light of its higher spectralefficiency [34], we consider spatial multiplexing where eachMBS and each SAP simultaneously serveKm andKs UEs,respectively. In practice, due to a finite number of antennas,MBSs and SAPs use traffic scheduling to limit the number ofUEs served toKm ≤ Mm andKs ≤ Ms [35]. Similarly, eachMBS limits toKb the number of SAPs served on the backhaul,with KbMs ≤ Mm. The MIMO dimensionality ratio for linearprocessing on macro cells, small cells, and backhaul is denotedby βm = Km

Mm, βs = Ks

Ms, andβb = KbMs

Mm, respectively. As

the MIMO dimensionality ratio at a base station reveals thenumber of active UEs within the cell coverage, for simplicityof notation, we refer to the MIMO dimensionality ratio as loadwhenever it does not cause ambiguity.

In this work, we consider a co-channel deployment of smallcells with the macro cell tier, i.e., macro cells and small cells

1Note that a PPP can serve as a good model not only for the opportunisticdeployment of small cell access points, but also for the planned deploymentof macro cell base stations, as verified by both empirical evidence [32] andtheoretical analysis [33].

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share the same frequency band for transmission2. In order toavoid severe interference which may degrade the performanceof the network, we assume that the access and backhaul linksshare the same pool of radio resources through orthogonaldivision, i.e., the total available bandwidth is divided into twoportions, where a fractionζb is used for the wireless backhaul,and the remaining(1− ζb) is shared by the radio access links(macro cells and small cells) [15], [17], [40], [41]. In orderto adapt the radio resources to the variation of the DL/ULtraffic demand, we assume that MBSs and SAPs operate in adynamic time division duplex (TDD) mode [42], [43], whereat every time slot, all MBSs and SAPs independently transmitin downlink with probabilitiesτm, τs, and τb on the macrocell, small cell, and backhaul, respectively, and they transmitin uplink for the remaining time3. We model the channelsbetween any pair of antennas in the network as independent,narrowband, and affected by three attenuation components,namely small-scale Rayleigh fading, shadowingSD and SB

for data link and backhaul link, respectively, and large-scalepath loss, whereα is the path loss exponent and the shadowing

satisfiesE[S2

α

D ] < ∞ andE[S2

α

B ] < ∞, and by thermal noisewith varianceσ2. We finally assume that all MBSs and SAPsuse a zero forcing (ZF) scheme for both transmission andreception, due to its practical simplicity [44]4.

B. Energy Efficiency

We consider the power consumption due to transmission andsignal processing operations performed on the entire network,therefore energy-efficiency tradeoffs will be such that savingsat the MBSs and SAPs are not counteracted by increasedconsumption at the UEs, and vice versa [6], [49]. We canidentify three main contributions to the power consumptionofthe heterogeneous network, namely the consumption on macrocells, small cells, and backhaul links. Consistent with previouswork [49]–[52], we account for the power consumption dueto transmission, encoding, decoding, and analog circuits.Adetailed model for the power consumption of the HetNet willbe given in Section III.

Let P [ Wm2 ] be the total power consumption per area, whichincludes the power consumed on all links. We denote byR[ bitm2 ] the sum rate per unit area of the network, i.e., the totalnumber of bits per second successfully transmitted per squaremeter. The energy efficiencyη = R

P is then defined as thenumber of bits successfully transmitted per joule of energyspent [49], [53]. For the sake of clarity, the main notationsused in this paper are summarized in Table I.

2Many frequency planning possibilities exist for MBSs and SAPs, where theoptimal solution is traffic load dependent. Though a non-co-channel allocationis justified for highly dense scenarios [36]–[38], in some cases a co-channeldeployment may be preferred from an operator’s perspective, since MBSs andSAPs can share the same spectrum thus improving the spectralutilization ratio[39].

3We note that different SAPs and MBSs may have different uplink/downlinkresource partitions for their associated UEs. Since the aggregate interference isaffected by the average value of such partitions, we assume fixed and uniformuplink/downlink partitions.

4Note that the results involving the machinery of random matrix theorycan be adjusted to account for different transmit precodersand receive filters,imperfect channel state information, and antenna correlation [45]–[48].

TABLE INOTATION SUMMARY

Notation Definition

P ; R; η Power per area; rate per area; energy efficiency

RDLm ; RDL

s ; RDL

bDownlink rate on macro cells, small cells, and back-haul

RULm ; RUL

s ; RUL

bUplink rate on macro cells, small cells, and backhaul

Pmt; Pst; Put Transmit power for MBSs, SAPs, and UEs

Pmb; Psb Backhaul transmit power for MBSs and SAPs

Pmc; Psc Analog circuit power consumption at macro cells andsmall cells

Pme; Pse; Pue Encoding power per bit on macro cells, small cells,and backhaul

Pmd; Psd; Pud Decoding power per bit on macro cells, small cells,and backhaul

Φm; Φs; Φu PPPs modeling locations of MBSs, SAPs, and UEs

λm; λs; λu Spatial densities of MBSs, SAPs, and UEs

Am; As Association probabilities for MBSs and SAPs

Mm; Ms Number of transmit antennas per MBS and SAP

Km; Ks; Kb UEs served per macro cell and small cell; SAPs perMBS on backhaul

βm; βs; βb Load on macro cells, small cells, and backhaul

τm; τs; τb Fraction of time in DL for macro cells, small cells,and backhaul

ζb; α Fraction of bandwidth for backhaul; path loss expo-nent

SD; SB Shadowing on radio access link and wireless back-haul

III. POWER CONSUMPTION

In this section, we model in detail the power consumptionof the heterogeneous network with wireless backhaul.

Since each UE associates with the base station, i.e., MBSor SAP, that provides the largest average received power, theprobability that a UE associates to a MBS or to a SAP can berespectively calculated as [54]

Am =λmP

2

α

mt

λmP2

α

mt + λsP2

α

st

(1)

and

As =λsP

2

α

st

λmP2

α

mt + λsP2

α

st

. (2)

In the remainder of the paper, we make use of the followingapproximation.

Assumption 1: We approximate the number of UEs, thenumber of SAPs associated to a MBS, and the number of UEsassociated to a SAP by constant valuesKm, Kb, and Ks,respectively, which are upper bounds imposed by practicalantenna limitations at MBSs and SAPs5.

The assumption above is motivated by the fact that the

5The number of base station antennas imposes a constraint on the maximumnumber of UEs scheduled for transmission. In fact, under linear precoding,the number of scheduled UEs should not exceed the number of antennas, inorder for the achievable rate not to be significantly degraded [55]–[57].

4

P = λm

[

τmPmt+(1−τm)KmPut+Pmf+PmaMm+PuaKm+τmKm (Pme+Pud)RDLm + (1−τm)Km (Pmd+Pue)R

ULm

]

+λs

[

τsPst+(1−τs)KsPut+Psf+PsaMs+PuaKs + τsKs (Pse+Pud)RDLs +(1− τs)Ks (Psd+Pue)R

ULs

]

+λm [τbPmb

+(1−τb)KbPsb + PmaMm+KbMsPsa+τbKbKs (Pme+Psd)RDLb +(1−τb)KbKs (Pmd+Pse)R

ULb

]

. (10)

number of UEsNm served by a MBS has distribution [54]

P(Nm = n) =3.53.5Γ(n+ 3.5)

(

λm

Amλu

)3.5

Γ(3.5)n! (1 + 3.5λm/λu)n+3.5 (3)

whereΓ(·) is the gamma function. LetKm be a limit on thenumber of users that can be served by a MBS, the probabilitythat a MBS serves less thanKm UEs is given by

P (Nm < Km) =

Km−1∑

n=0

3.53.5Γ(n+ 3.5)(

λm

Amλu

)3.5

Γ(3.5)n! (1 + 3.5λm/λu)n+3.5

≤

(

2λm

λu

)3.5 Km−1∑

n=0

Γ(n+ 3.5)

n!

3.53.5

Γ(3.5)(4)

which rapidly tends to zero asλu

λm

grows. This indicates thatin a practical network with a high density of UEs, i.e., whereλu ≫ λm, each MBS servesKm UEs with probability almostone. A similar approach can be used to show thatP(Ns <Ks) ≈ 0 andP(Nb < Kb) ≈ 0 whenλu ≫ λm andλs ≫ λm,respectively, and therefore each SAP servesKs UEs and eachMBS servesKb SAPs on the backhaul with probability almostone.

In the following, we use the power consumption modelintroduced in [49], which captures all the key contributions tothe power consumption of signal processing operations. Thismodel is flexible since the various power consumption valuescan be tuned according to different scenarios. We note thatthe results presented in this paper hold under more generalconditions and apply to different power consumption models[58], [59].

Under the previous assumption, and by using the model in[49], we can write the power consumption on each macro celllink as follows

Pm = τmPmt + (1− τm)KmPut + τmKm (Pme + Pud)RDLm

+ Pmc + (1− τm)Km (Pmd + Pue)RULm (5)

wherePmt andPut are the DL and UL transmit power fromthe MBS and theKm UEs, respectively,Pmc is the analogcircuit power consumption,Pme and Pmd are encoding anddecoding power per bit of information for MBS, whilePue andPud are encoding and decoding power per bit of informationfor UE, andRDL

m andRULm denote the DL and UL rates for

each MBS-UE pair. The analog circuit power can be modeledas [49]

Pmc = Pmf + PmaMm + PuaKm (6)

wherePmf is a fixed power accounting for control signals,baseband processor, local oscillator at MBS, cooling system,etc.,Pma is the power required to run each circuit component

attached to the MBS antennas, such as converter, mixer, andfilters,Pua is the power consumed by circuits to run a single-antenna UE. Under this model, the total power consumptionon the macro cell can be written as

Pm = τmPmt+(1−τm)KmPut+τmKm(Pme+Pud)RDLm

+Pmf+PmaMm+PuaKm+(1−τm)Km(Pmd+Pue)RULm .

(7)

Through a similar approach, the power consumption on eachsmall cell and backhaul link can be written as

Ps = τsPst + (1− τs)KsPut + Psf + τsKs (Pse + Pud)RDLs

+ PsaMs + PuaKs + (1− τs)Ks (Psd + Pue)RULs (8)

and

Pb = τbPmb + (1− τb)KbPsb + τbKbKs (Pme + Psd)RDLb

+ PmaMm +KbMsPsa + (1− τb)KbKs (Pmd + Pse)RULb ,(9)

respectively, the analog circuit power consumption in (9)accounts for power spent on out of band SAPs. In the aboveequations,Pst is the transmit power on a small cell,Pmb andPsb are the powers transmitted by MBSs and SAPs on thebackhaul, andPsf and Psa are the small-cell equivalents ofPmf andPma. Moreover,RDL

s andRULs denote the DL and

UL rates for each SAP-UE pair, andRDLb andRUL

b denote theDL and UL rates for each wireless backhaul link.

We can now write the total power consumption of theheterogeneous network with wireless backhaul.

Lemma 1: The power consumption per area in a heteroge-neous network with wireless backhaul is given by(10) shownon the top of this page.

Proof: Equation (10) follows from (7), (8), (9), and bynoting that under Assumption 1 the average power consump-tion per area can be expressed asP = Pmλm+Psλs+Pbλm.

IV. RATES AND ENERGY EFFICIENCY

In this section, we analyze the data rates and the energyefficiency of a HetNet with wireless backhaul, and we providesimulations that verify the accuracy of our analysis. Particu-larly, we combine tools from stochastic geometry and randommatrix theory to derive the uplink and downlink rates of macrocells, small cells, and wireless backhaul. The resulting analysisis tractable and captures the effects of multiantenna transmis-sion, fading, shadowing, and random network topology. Byusing the framework developed in this section, we will showthat the energy efficiency of a HetNet is sensitive to the loadconditions of the network, irrespective of the infrastructureused, and that a two-tier HetNet can achieve a significant

5

RDLm = (1− ζb)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνD

m

)

exp

(

−2π2λuP

δutE[S

δD]z

δ

α sin(

2πα

)

)

×exp

(

−τmamCα,Km(zPmt, t)

(

zPmt

Km

)δ

− τsasCα,Ks(zPmt, t)

(

zPst

Ks

)δ)

fLm(t)dtdz (11)

RULm = (1− ζb)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνU

m/t)

exp

{

−λuπE[SδD]

∫ ∞

0

1− e−Gmu

1 + z−1u1

δ /Put

du

−Γ (1 + δ) δπ2zδ

sin(δπ)

[

τmamPδmt

∏Km−1i=1 (i+ δ)

Γ(Km)Kδm

+τsasP

δst

∏Ks−1i=1 (i+ δ)

Γ(Ks)Kδs

]}

fLm(t)dtdz (15)

RDLs = (1 − ζb)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1−1

(1 + zPstt−1/Ks)∆s

)

exp

(

−2π2λuP

2

α

utE[S2

α

D ]z2

α

α sin(

2πα

)

)

× exp

(

−τsas Cα,Ks(zPst, t)

(

zPst

Ks

)δ

− τmam Cα,Km(zPst, t)

(

zPmt

Km

)δ)

fLs(t)dtdz (16)

RULs =(1− ζb)

∫ ∞

0

e−σ2z

z ln 2

[

1−

∫ ∞

0

fLs(t)dt

(1+zPut/t)∆s

]

exp

{

−λuπE[SδD]

∫ ∞

0

1− e−Gsz

1 + z−1u1

δ /Put

du

−Γ (1 + δ) δπ2zδ

sin(δπ)

[

τsasPδst

∏Ks−1i=1 (i+ δ)

Γ(Ks)Kδs

+τmamP

δmt

∏Km−1i=1 (i + δ)

Γ(Km)Kδm

]}

dz. (18)

energy efficiency gain over a one-tier network if the backhaulbandwidth is optimally allocated. Unless otherwise stated,the analytical expressions provided in this section are tightapproximations of the actual data rates. For a better readability,most proofs and mathematical derivations have been relegatedto the Appendix.

A. Analysis

Under dynamic TDD [42], [43], transmissions are corruptedby DL interference from other MBSs and SAPs, and byUL interference from UEs that associated with other MBSsand SAPs. Specifically, the UL interference from UEs thatassociated with MBSs follow a homogeneous PPP with density(1 − τm)λmKm, and similarly, the UL interference fromUEs that associated with SAPs follow a homogeneous PPPwith density(1− τs)λsKs. By the composition theorem [60],we have the UL interfering UEs follow a PPP with densityλu = (1− τm)λmKm + (1 − τs)λsKs.

By noting that in practice, MBS can equip a large numberof antennas, we use random matrix theory tools to obtain theDL rate on a macro cell link.

Lemma 2: The downlink rate on a macro cell is given by(11), where δ = 2/α, am = λmπE[S

δD], as = λsπE[S

δD],

λu = (1− τm)λmKm + (1− τs)λsKs, while νDm, fLm(t), and

Cα,K(z, t) are given respectively as follows

νDm =Pmt (1− βm) (Gm)

α2

βmΓ(

1 + α2

) , (12)

fLm(t) = Gmδx

δ−1 exp(

−Gmxδ)

, x ≥ 0 (13)

Cα,K(z, t) =2

α

K∑

n=1

(

K

n

)[

B

(

1;K − n+2

α, n−

2

α

)

− B

(

(

1 +s

tK

)−1

;K − n+2

α, n−

2

α

)]

(14)

with Gm = am+as (Pst/Pmt)δ, andB(x; y, z) =

∫ x

0ty−1(1−

t)z−1dt the incomplete Beta function.Proof: See Appendix A.

We note that in the downlink, due to the maximum receivedpower association, interfering base station cannot be locatedcloser to the typical user than the tagged base station, i.e.,an exclusion region exists where the distance between a UEand the interfering base stations is bounded away from zero.However in the uplink, because of the PPP deployment as-sumption, an interfering base station can be located arbitrarilyclose to a typical MBS, i.e., the distance between a MBSand the interfering base stations can be arbitrarily small.Inthe following, we treat the latter as a composition of threeindependent PPPs with different spatial densities. We thenobtain the macro cell uplink rate as follows.

Lemma 3: The uplink rate on a macro cell is given by(15),with νUm = (1 − βm)MmPmt.

Proof: See Appendix B.Unlike the macro cell, due to the relatively small number of

antennas at the SAPs, random matrix theory tools cannot beemployed to calculate the rate on a small cell. We thereforeuse the effective channel distribution as follows.

Lemma 4: The downlink rate on a small cell is given by(16), where∆s = Ms −Ks + 1, and fLs

(t) is given as

fLs(t) = Gsδt

δ−1 exp(

−Gstδ)

, t ≥ 0 (17)

6

RDLb =

ζbMs

Ks

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνD

b

)

exp

(

−Γ (1 + δ) δπ2zδP δ

sb

sin(δπ)Γ(Ms)M δs

E[SδB](1− τb)λs

Ms−1∏

i=1

(i+ δ)

)

×

∫ ∞

0

exp

(

−τbabCα,KbMs(zPmb, t)

(

zPmb

KbMs

)δ)

fLb(t)dtdz (19)

RULb =

ζbMs

Ks

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνU

b/t)

exp

{

−τbabΓ(1+δ)δπ2P δ

mbzδ∏KbMs−1

i=1 (i+ δ)

sin(δπ) (MsKb)δΓ (MsKb)

}

× exp

{

− (1−τb) abKb

Ms∑

n=1

(

Ms

n

)∫ ∞

0

(

zu−1/δPsb/Ms

)n(1− e−abu)

(

1 + zu−1/δPsb/Ms

)Ms

du

}

fLb(t)dtdz (22)

RDLm,FDD = ξD(1− ζb)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνD

m

)

×exp

(

−τmamCα,Km(zPmt, t)

(

zPmt

Km

)δ

− τsasCα,Ks(zPmt, t)

(

zPst

Ks

)δ)

fLm(t)dtdz. (25)

RULm,FDD = (1− ζb)(1 − ξD)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνU

m/t)

× exp

(

−λuπE[SδD]

∫ ∞

0

1− e−Gmu

1 + z−1u1

δ /Put

du

)

fLU(t)dtdz (26)

with Gs = as + am (Pmt/Pst)δ.

Proof: See Appendix C.Following a similar approach as the one in Lemma 3, we

can obtain the uplink rate on a small cell.Lemma 5: The uplink rate on a small cell is given by(18)

on the top of previous page.Proof: The proof is similar to the one in Lemma 3 and

it is omitted.We now derive downlink and uplink rates on the wireless

backhaul of a heterogeneous network as follows.Lemma 6: The downlink rate on the wireless backhaul is

given by (19), where ab = λmπE[SδB], fLb

(t) and νDb aregiven as

fLb(t) = abδt

δ−1 exp(−abtδ), t > 0 (20)

νDb =Pmb(1− βb)a

δb

βbΓ(1 + 1/δ). (21)

Proof: See Appendix D.Lemma 7: The uplink rate on the wireless backhaul is given

by (22), whereνUb = (1− βb)MmPsb.Proof: See Appendix E.

Remark 1: In Lemma 6 and Lemma 7, we use ZF asthe precoding and receiving scheme, which is suboptimalcompared to the block diagonalization (BD) and can result inalower data rate achievable on the wireless backhaul [56], [57].However, the rate achievable under ZF is tractable, whereastothe best of the authors’ knowledge no closed form expressionisavailable for the rate achievable by BD. Furthermore, we showin Fig. 2 that the rate gap is limited, and that the rates underBD and ZF follow a similar trend. Therefore, our findings on

the energy efficiency tradeoffs remain valid irrespective of thescheme used.

By combining the previous results, we can now write thedata rate per area in a heterogeneous network with wirelessbackhaul.

Lemma 8: The sum rate per area in a heterogeneousnetwork with wireless backhaul is given by

R = B(

Kmλm +Ksλs

){

Am

[

τmRDLm + (1− τm)R

ULm

]

+As

[

τs min{

RDLs , RDL

b

}

+ (1− τs)min{

RULs , RUL

b

}]}

(23)

whereB is the total available bandwidth, andRDLm , RUL

m ,RDL

s , RULs , RDL

b , andRULb are given in(11), (15), (16), (18),

(19), and (22), respectively.Proof: See Appendix F.

We finally obtain the energy efficiency of a heterogeneousnetwork with wireless backhaul, defined as the number of bitssuccessfully transmitted per joule of energy spent.

Theorem 1: The energy efficiencyη of a heterogeneousnetwork with wireless backhaul is given by

η =B (Kmλm +Ksλs)

Pmλm + Psλs + Pbλm

(

Am

[

τmRDLm + (1− τm)R

ULm

]

+As

[

τs min{

RDLs , RDL

b

}

+ (1− τs)min{

RULs , RUL

b

}])

.

(24)

Proof: The result follows from Lemma 1 and Lemma 8and by noting that the energy efficiency is obtained as the ratiobetween the data rate per area and the power consumption perarea.

7

RDLs,FDD = ξD(1− ζb)

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2

(

1−1

(1 + zPstt−1/Ks)∆s

)

× exp

(

−τsas Cα,Ks(zPst, t)

(

zPst

Ks

)δ

− τmam Cα,Km(zPst, t)

(

zPmt

Km

)δ)

fLs(t)dtdz. (28)

RULs,FDD = (1− ζb)(1 − ξD)

∫ ∞

0

e−σ2z

z ln 2

[

1−

∫ ∞

0

fLU(t)dt

(1+zPut/t)∆s

]

exp

(

−λuπE[SδD]

∫ ∞

0

1− e−Gsu

1 + z−1u1

δ /Put

du

)

dz (29)

RDLb,FDD =

ξBζbMs

Ks ln 2

∫ ∞

0

∫ ∞

0

(

1− e−zνD

b

)

zeσ2zexp

(

−τbabCα,KbMs(zPmb, t)

(

zPmb

KbMs

)δ)

fLb(t)dtdz (30)

RULb,FDD =

(1− ξB) ζbMs

Ks

∫ ∞

0

e−σ2z

z ln 2

∫ ∞

0

(

1− e−zνU

b/t)

fLU(t)dt

× exp

{

− (1−τb) abKb

Ms∑

n=1

(

Ms

n

)∫ ∞

0

(

zu−1/δPsb/Ms

)n(1− e−abu)

(

1 + zu−1/δPsb/Ms

)Ms

du

}

dz (31)

Equation (24) quantifies how all the key features of a hetero-geneous network, i.e., interference, deployment strategy, andcapability of the wireless infrastructure components, affect theenergy efficiency when a wireless backhaul is used to forwardtraffic into the core network. Several numerical results basedon (24) will be shown in Section V to give more practicalinsights into the energy-efficient design of a heterogeneousnetwork with wireless backhaul. In Section IV-C, we providesimulations to validate the analysis presented in this section.

B. Rate Analysis of FDD network

Our results are general and hold under both time divisionduplex (TDD) and frequency division duplex (FDD). In fact,TDD and FDD are equivalent in that they all divide up thespectrum orthogonally [61]. In this section, we show theextension of our framework to the case of FDD, where aportion ξD of the radio access spectrum is assigned to thedownlink, and the remaining fraction(1 − ξD) is assignedto the uplink. Similarly, on the wireless backhaul, a fractionξB is reserved for the downlink, and the remaining fraction(1 − ξB) is reserved for the uplink. In addition, in orderto increase the spectral efficiency, we consider a decoupledUL/DL association, where the downlink UEs are associatedwith the base station that provides the largest received power,and the uplink UEs are associated with the closest MBSs orSAPs. As such, the rate of DL macro cell, UL macro cell, DLsmall cell, UL small cell, and the wireless backhaul can bewritten as follows.

Lemma 9: Under FDD, the downlink rate on a macro cellis given by(25) in the previous page.

Proof: See Appendix G.Lemma 10: Under FDD, the uplink rate on a macro cell

is given by(26) in the previous page, wherefLU(t) is given

as

fLU(t) = δGtδ−1 exp

(

−Gtδ)

,

G = (λs + λm)πE[SδD]. (27)

Proof: The proof is similar to the one in Lemma 3 andit is omitted.

Lemma 11: Under FDD, the downlink rate on a small cellis given by(28).

Proof: The proof is similar to the one in Lemma 4 andit is omitted.

Lemma 12: Under FDD, the uplink rate on a small cell isgiven by(29).

Proof: The proof is similar to the one in Lemma 5 andit is omitted.

Lemma 13: Under FDD, the downlink rate on the wirelessbackhaul is given by(30).

Proof: The proof is similar to the one in Lemma 6 andit is omitted.

Lemma 14: Under FDD, the uplink rate on the wirelessbackhaul is given by(31).

Proof: The proof is similar to the one in Lemma 7 andit is omitted.

C. Validation

We now show simulation results that confirm the accuracyof the analysis provided in this section. In our simulations,all cells operate under dynamic TDD, the locations of MBSs,SAPs, and UEs are generated as PPPs, and the typical UEis located at the origin. We use the following values for thenumber of antennas and the transmit power:Mm = 100,Ms =4, Pmt = 47.8dBm, andPst = 23.7dBm.

In Fig. 2 we compare the rate on the wireless backhaul underblock diagonalization (BD) and zero forcing (ZF), respectively,with different numbers of SAPs. Although ZF achieves a lowerrate than BD, the rate gap is limited as the antenna numbergrows, and the rates under BD and ZF follow a similar trend.Therefore, the conclusions drawn in this paper on the energyefficiency tradeoffs remain valid irrespective of the schemeused.

Fig. 3 compares the simulated macro cell downlink rateto the analytical result obtained in Lemma 2 with differentantenna numbers at the MBS. The downlink rate is plotted

8

20 30 40 50 60 70 80 90 1005

10

15

20

25

30

35

40

45

50

PSfrag replacements

MBS antenna number,Mm

Bac

khau

ldow

nlin

ksu

mra

te

BD: Kb = 2ZF: Kb = 2

BD: Kb = 4ZF: Kb = 4

Fig. 2. Downlink rate on the wireless backhaul, by using block diagonalizationand zero forcing, with different numbers of SAPs.

30 35 40 45 501

1.5

2

2.5

PSfrag replacements

MBS transmit power,Pmt [dBm]

Mac

roce

lld

own

link

rate

,RDL

m

Sim.: Mm = 20, Km = 5Ana.: Mm = 20, Km = 5Sim.: Mm = 100, Km = 25Ana.: Mm = 100, Km = 25

Fig. 3. Comparison of the simulations and numerical resultsfor macro celldownlink rate.

versus the transmit power at the MBSs. The figure showsthat analytical results and simulations fairly well match,thusconfirming the accuracy of Lemma 2.

V. NUMERICAL RESULTS

In this section, we provide numerical results to show howthe energy efficiency is affected by various network parametersand to give insights into the optimal design of a heterogeneousnetwork with wireless backhaul. As an example, we considertwo different deployment scenarios, namely (i) a dense deploy-ment of low-power SAPs with a small number of antennas,here denoted asfemto cells, and (ii) a less dense deploymentof larger and more powerful SAPs, here denoted aspico cells,and we refer tolight load and heavy loadconditions as theones of a network withβm = βs = βb = 0.25 and0.9 ≤ βm,βs, βb < 1, respectively. We consider a network operating at2GHz, we set the path loss exponent toα = 3.8 to modelan urban scenario, the shadowingSB and SD are set to be

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12x 10

5

PSfrag replacements

Fraction of bandwidth for backhaul,ζb

En

erg

yE

ffici

ency

,η[b

its/J

]

Femto cells (light load)Pico cells (light load)Femto cells (heavy load)Pico cells (heavy load)

Fig. 4. Energy efficiency of a heterogeneous network that uses pico cellsand femto cells, respectively, versus fraction of bandwidth ζb allocated to thebackhaul, under different load conditions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PSfrag replacements

Load on the backhaul,βb

Op

timal

frac

tion

of

ban

dw

idth

,ζ∗ bKs = 3Ks = 2Ks = 1

Fig. 5. Optimal fraction of bandwidth to be allocated to the backhaul versusload on the backhaul, for various values of the number of UEs per SAP,Ks.

lognormal distributed asSB = 10XB

10 andSD = 10XD

10 , whereXB ∼ N(0, σ2

B) and XD ∼ N(0, σ2D), with σB = 3dB

and σD = 6dB, respectively [62]. In addition, we set thebackhaul transmit power equal to the radio access power,i.e., Pmb = Pmt, Psb = Pst. All other system and powerconsumption parameters are set as follows:Pmt = 47.8dBm,for pico cell SAPsPst = 30dBm, for femto cell SAPPst =23.7dBm, Put = 17dBm, Pma = 1W, for pico cell SAPsPsa = 0.8W, for femto cell SAPPsa = 0.8W, Pua = 0.1W[32]; Pmf = 225W, for pico cell SAPsPsf = 7.3W, forfemto cell SAPsPsf = 5.2W [58], [59]; Pme = 0.1W/Gb,Pmd = 0.8W/Gb, Pse = 0.2W/Gb, Psd = 1.6W/Gb,Pue = 0.3W/Gb, Pmd = 2.4W/Gb [49].

In Fig. 4, we compare the energy efficiency of heteroge-neous networks that use pico cells and femto cells, respec-tively, under various load conditions and for different portionsof the bandwidth allocated to the wireless backhaul. The figure

9

25 30 35 40 45 50 552

3

4

5

6

7

8

9

10

11

12x 10

5

PSfrag replacements

En

erg

yef

ficie

ncy

,η[b

its/J

]

Femto cells (light load)

Femto cells (heavy load)Pico cells (light load)

Pico cells (heavy load)

MBS transmit power,Pmt [dBm]

Fig. 6. Energy efficiency of a heterogeneous network that uses pico cellsand femto cells, respectively, versus power allocated to the backhaul, underdifferent load conditions.

shows that femto cell and pico cell deployments exhibit similarperformance in terms of energy efficiency. Moreover, Fig. 4shows that the energy efficiency of the network is highlysensitive to the portion of bandwidth allocated to the backhaul,and that there is an optimal value ofζb which maximizes theenergy efficiency of the HetNet. The optimal value ofζb is notaffected by the network infrastructure, i.e., it is the sameforpico cells and femto cells. However, the optimalζb increasesas the load on the network increases. In fact, when more UEsassociate with each SAPs, more data need to be forwardedfrom MBSs to SAPs through the wireless backhaul in orderto meet the rate demand. In summary, the figure shows thatirrespective of the deployment strategy, an optimal backhaulbandwidth allocation that depends on the network load can behighly beneficial to the energy efficiency of a heterogeneousnetwork.

In Fig. 5, we plot the optimal valueζ∗b for the fractionof bandwidth to be allocated to the backhaul as a functionof the load on the backhaulβb. We consider femto celldeployment for three different values of the number of UEs perSAP,Ks. Consistently with Fig. 4, this figure shows that theoptimal fraction of bandwidthζ∗b to be allocated to the wirelessbackhaul increases asβb or Ks increase, since the load on thewireless backhaul becomes heavier and more resources areneeded to meet the data rate demand.

In Fig. 6, we plot the energy efficiency of the HetNetas a function of the MBS transmit power under differentdeployment strategies and load conditions. The figure showsthat the energy efficiency is sensitive to the MBS transmitpower, and that there is an optimal value for the transmitpower, given by a tradeoff between the data rate that thewireless backhaul can support and the power consumptionincurred. Under spatial multiplexing, the data rate of thenetwork is affected by the number of scheduled UEs per basestation antenna, which we denote as the network load. As aconsequence, the network load affects the data rate, and inturn affects the energy efficiency.

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9x 10

5

PSfrag replacements

Number of SAPs per MBS,Kb

En

erg

yef

ficie

ncy

,η[b

its/J

]

OptimalProportionalFixedOne-Tier

Fig. 7. Energy efficiency versus number of SAPs per MBS under variousbandwidth allocation schemes.

In Fig. 7, we plot the energy efficiency of the network versusthe number of SAPs per MBS. We consider four scenarios: (i)optimal bandwidth allocation, where the fraction of bandwidthζb for the backhaul is chosen as the one that maximizesthe overall energy efficiency; (ii) proportional bandwidthal-location, where the fraction of bandwidth allocated to thebackhaul is equal to the fraction of load on the backhaul,i.e., ζb = KbKs

Km+KbKs

[41]; (iii) fixed bandwidth allocation,where the bandwidth is equally divided between macro-and-small-cell links and wireless backhaul, i.e.,ζb = 0.5; and(iv) one-tier cellular network, where no SAPs or wirelessbackhaul are used at all, and all the bandwidth is allocatedto the macro cell link, i.e.,ζb = 0. Fig. 7 shows that in atwo-tier heterogeneous network there is an optimal number ofSAPs associated to each MBS via the wireless backhaul thatmaximizes the energy efficiency. Such number is given by atradeoff between the data rate that the SAPs can provide tothe UEs and the total power consumption. This figure alsoindicates that if the wireless backhaul is not supported well,a two-tier HetNet with wireless backhaul can be worse thana single tier cellular network in terms of energy efficiency.However, when the backhaul bandwidth is optimally allocated,the HetNet can achieve a significant energy efficiency gainover a one tier deployment.

VI. CONCLUDING REMARKS

In this work, we undertook an analytical study for theenergy-efficient design of heterogeneous networks with awireless backhaul. We used a general model that accountsfor uplink and downlink transmissions, spatial multiplexing,and resource allocation between radio access links and back-haul. Our results revealed that, irrespective of the deploymentstrategy, it is critical to control the network load in orderto maintain a high energy efficiency. Moreover, a two-tierheterogeneous network with wireless backhaul can achieve asignificant energy efficiency gain over a one-tier deployment,as long as the bandwidth division between radio access linksand wireless backhaul is optimally designed.

10

The framework provided in this paper allows to explicitlycharacterize the power consumption of the HetNet due to thesignal processing operations in macro cells, small cells, andwireless backhaul, as well as the data rates and ultimately theenergy efficiency of the whole network. More generally, ourwork helps to understand how all the key features of a hetero-geneous network, i.e., interference, load, deployment strategy,and capability of the wireless infrastructure components,affectthe energy efficiency when a wireless backhaul is used toforward traffic into the core network.

This paper considered the current state-of-the-art co-channeldeployment of small cells with the macro cell tier. In thenear future, an orthogonal, ultra-dense deployment of smallcells could be used to further boost the network capacityby targeting static users. Investigating up to what extentthe wireless backhaul capability can support such ultra-densetopology, and designing idle-mode mechanisms for an energy-efficient and sustainable ultra-dense deployment are regardedas concrete directions for future work.

APPENDIX

A. Proof of Lemma 2

The channel matrix between a MBS to itsKm asso-ciated UEs can be written asH = L

1

2H, where L =diag{L−1

1 , ..., L−1Km

}, with Li = rαi /Si being the path lossfrom the MBS to itsi-th UE, whereri is the correspondingdistance andSi denotes the shadowing,H = [h1, ...,hKm

]T istheKm×Mm small scale fading matrix, withhi ∼ CN (0, I).The ZF precoder is then given byW = ξH∗(HH

∗)−1, whereξ2 = 1/tr[(H∗

H)−1] normalizes the transmit power [62]. Inthe following, we use the notationΦU as ΦD to denote thesubsets ofΦ that transmit in uplink and downlink, respectively,we further denoteUx as the set of UEs that are associatedwith access pointx, and denotex as the transmitter thatlocates closest to the origin. Since the locations of MBSs andSAPs follow a stationary PPP, we can apply the Slivnyark’stheorem [60], which implies that it is sufficent to evaluate thesignal-to-interference-plus-noise ratio (SINR) of a typical UEat the origin. As such, by noticing that under dynamic TDD,every wireless link experiences interference from the downlinktransmitting MBSs and SAPs, and from the uplink transmittingUEs, the downlink SINR between a typical UE at the originand its serving MBS can be written as

γDLm =

Pmt|h∗xm,owxm,o|2L

−1xm,o

Imuoc + Iu + σ2

(28)

where hxm,o is the small scale fading,wxm,o is the ZFprecoding vector,Lxm,o denotes the corresponding path loss,while Imu

oc is the aggregate interference from other cells tothe MBS UE, andIu denotes the interference from UEs,respectively given as follows

Imuoc =

∑

xm∈ΦDm \xm

Pmtgxm,o

KmLxm,o+∑

xs∈ΦDs

Pstgxs,o

KsLxs,o(29)

and

Iu =∑

xu∈ΦUu

Put|hxu,o|2

Lxu,o(30)

whereasgxm,o and gxs,o represent the effective small-scalefading from the interfering MBSxm and SAPxs to the origin,respectively, given by [63]

gxm,o =∑

u∈Uxm

Km|h∗xm,owxm,u|

2 ∼ Γ (Km, 1) (31)

andgxs,o =

∑

u∈Uxs

Ks|h∗xs,owxs,u|

2 ∼ Γ (Ks, 1) . (32)

By conditioning on the interference, whenKm,Mm → ∞with βm = Km/Mm < 1, the SINR under ZF precodingconverges to [45]

γDLm → γDL

m =PmtMm

(Imuoc + Iu + σ2)

∑Km

j=1 e−1j

, a.s. (33)

whereei is the solution of the fixed point equation

L−1xm,ui

ei= 1+

J

Mm, i = 1, 2, ...,Km (34)

with J =∑Km

j=1 L−1xm,uj

e−1j . By summing (34) overi we

obtain

J = Km +Km

MmJ. (35)

Solving the equation above results inJ = KmMm/(Mm −Km), and by substituting the value ofJ into (34) we canhave

1

ei=

Mm

Mm −Km· Lxm,ui

(36)

which substituted into (33) yields

γDLm =

(1− βm)MmPmt

(Imuoc + Iu + σ2)

∑Km

j=1 Lxm,uj

. (37)

Notice that{Lxm,uj}Km

j=1 is an independent independent andidentically distributed (i.i.d.) sequence with finite firstmoment,given by

E[

Lxm,uj

]

= Γ

(

1 +1

δ

)

G−1m < ∞,

by applying the strong law of large numbers (SLLN) to (37),we have

γDLm →

(1− βm)G1/δm Pmt

βmΓ(

1 + 1δ

)

(Imuoc + Iu + σ2)

, a.s. (38)

As such, using the continuous mapping theorem and thelemma in [64], we can compute the ergodic rate as

E[

log2(

1 + γDLm

)]

=1

ln 2E

[

ln

(

1 +νDm

Imuoc + Iu + σ2

)]

=

∫ ∞

0

e−σ2z

z ln 2

(

1− e−νD

mz)

E[

e−zIu]

E

[

e−zImu

oc

]

dz. (39)

Due to the composition of independent PPPs and the dis-placement theorem [30], the interferenceIu follows a homo-geneous PPP with spatial densityλu = (1− τm)λmKm +

11

(1− τs)λsKs, and the corresponding Laplace transform isgiven as [60]

E[

e−zIu]

= exp

(

−2π2λuE[S

2

α

D ]P2

α

utz2

α

α sin(

2πα

)

)

. (40)

As for the Laplace transform ofImuoc , the conditional Laplace

transform onLxm,o can be computed as

E

[

e−zImu

oc |Lxm,o= t]

=exp

(

−τmamCα,Km(zPmt, t)

(

zPmt

Km

)δ

− τsasCα,Ks(zPmt, t)

(

zPst

Ks

)δ)

.

(41)

Notice thatLxm,o has its distribution given by (13), and therateRDL

m given as

RDLm = (1− ζb)E

[

log2(

1 + γDLm

)]

, (42)

substituting (40) and (41) into (39), and deconditionLxm,o

with respect to (13) we have the corresponding result.

B. Proof of Lemma 3

Let us consider a UE transmitting in uplink to a typical MBSlocated at the origin, which employs a ZF receive filterr

∗o,xu

=

h∗o,xu

(∑

u∈Uoho,uh

∗o,u)

−1 [62], the SINR is then given by

γULm =

PutL−1o,xu

|r∗o,xuho,xu

|2

(Imbsoc + Iu + σ2) ‖ro,xu

‖2(43)

whereImbsoc denotes the interference from other cells received

at the MBS. By conditioning on the interference, whenKm,Mm → ∞ with βm = Km/Mm < 1, the SINR aboveconverges to [45]

γULm → γUL

m =PutMm(1− βm)L

−1o,xu

Imbsoc + Iu + σ2

, a.s. (44)

By using the continuous mapping theorem [64], the uplinkergodic rate can be calculated as

E[

log2(

1 + γULm

)]

=1

ln 2E

[

ln

(

1 +νUmL

−1o,xu

Imbsoc + Iu + σ2

)]

=

∫ ∞

0

∫ ∞

0

(

1− e−zνU

m/t)

zeσ2z ln 2E[

e−zIu]

E

[

e−zImbs

oc

]

fLm(t)dzdt.

(45)

The Laplace transform ofImbsoc can be computed as

E

[

e−zImbs

oc

]

= exp

(

−Γ(1+δ) δπ2zδ

sin(δπ)

[

τmamPδmt

∏Km−1i=1 (i+δ)

Γ(Km)Kδm

+τsasP

δst

∏Ks−1i=1 (i+δ)

Γ(Ks)Kδs

])

. (46)

On the other hand, to consider the uplink interference fromUEs, we use the result in [65] where the path loss from

MBS UEs and SAP UEs are modeled as two independentinhomogeneous PPP with intensity measure being

Λ(m)mu (dx) = δamx

δ−1[

1− exp(

−Gmxδ)]

, (47)

Λ(m)su (dx) = δasx

δ−1[

1− exp(

−Gmxδ)]

. (48)

The Laplace transform of the UE interference can then becalculated as

E[e−zIu ] = exp

(

−(1− τm)Km

∫ ∞

0

Λ(m)mu (dx)

1 + z−1x/Put

− (1− τs)Ks

∫ ∞

0

Λ(m)su (dx)

1 + z−1x/Put

)

= exp

(

−λuπE[SδD]

∫ ∞

0

1− e−Gmu

1 + z−1u1

δ /Put

du

)

(49)

As such, noticing that

RULm = (1− ζb)E

[

log2(

1 + γULm

)]

(50)

the result follows by substituting (46) and (49) into (45).

C. Proof of Lemma 4

With a similar approach in the proof of Lemma 2, it issufficient to consider a typical UE that locates at the originand associates with an SAP, the corresponding downlink SINRcan then be written as

γDLs =

Pst|h∗xs,o

wxs,o|2L−1

xs,o

Isuoc + Iu + σ2(51)

where wxs,o is the ZF precoding vector, andIsuoc is theaggregate interference from other cells to the SAP UE, givenas follows

Isuoc =∑

xs∈ΦDs \xs

Pstgxs,o

KsLxs,o+

∑

xm∈ΦDm

Pmtgxm,o

KmLxm,o. (52)

By noting that|Ksh∗xs,o

wxs,o|2 ∼ Γ(∆s, 1) with ∆s = Ms −

Ks + 1 [14], we can write the rate of an SAP UE as [64]

E[

log2(

1 + γDLs

)]

=

∫ ∞

0

∫ ∞

0

e−σ2z

z ln 2E

[

e−zIsu

oc

]

E[

e−zIu]

×

[

1−1

(1+zPstt−1/Ks)∆s

]

fLs(t)dzdt. (53)

On one hand,E[

e−zIu]

is given in (40), on the other, bydenotingLxs,o = t, the conditional Laplace transform ofIsuoccan be derived as

E

[

e−zIsu

oc |Lxs,o = t]

= exp

(

−τsasCα,Ks(zPst, t)

(

zPst

Ks

)δ

− τmamCα,Km(zPst, t)

(

zPmt

Km

)δ)

. (54)

As Lxs,o follows a distribution as (17), andRDLs is given as

RDLs = (1− ζb)E

[

log2(

1 + γDLs

)]

(55)

by substituting (40) and (54) into (53), and deconditioningLxs,o, we have the desired result.

12

D. Proof of Lemma 6

We consider the signal received at thek-th antenna of atypical SAP located at the origin, which is served by a MBSthrough the wireless backhaul and has the SINR given as

γDLb,k =

Pmb

∣

∣

∣h∗b,kvb,k

∣

∣

∣

2

L−1xb,o

σ2 + Im + Is(56)

where vb,k denotes the ZF precoder, whileIm and Is arethe aggregated interference from downlink transmitting MBSsand uplink transmitting SAPs in the wireless backhaul, respec-tively, with expression given as follows

Im =∑

xm∈ΦDm \xm

Pmbgxm,o

KbMsLxm,o(57)

and

Is =∑

xs∈ΦUs

Psbgxs,o

MsLxs,o(58)

where gxm,o ∼ Γ(KbMs, 1) and gxs,o ∼ Γ(Ms, 1) are theeffective small scale fading from the interfering MBSxm andSAP xs to the origin.

By conditioning on the interferenceIm + Is, and by usinga similar approach as the one in the proof of Lemma 2, whenKb,Mm → ∞ with βb = KbMs/Mm < 1, the SINR in (56)satisfies [45]

γDLb,k → γDL

b,k =Pmb(1− βb)a

δb

βbΓ(

1 + 1δ

)

(σ2 + Im + Is), a.s. (59)

By using the continuous mapping theorem and the lemma in[64], we have the following holds

E[

log2(

1 + γDLb,k

)]

=

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνD

b

)

× E[

e−zIs]

E[

e−zIm]

dz. (60)

With the effective channel distribution available, we can com-pute the Laplace transform ofIs as

E[e−zIs ]

= exp

(

−Γ (1 + δ) δπ2zδP δ

sb

sin(δπ)Γ(Ms)M δs

E[SδB](1− τb)λs

Ms−1∏

i=1

(i + δ)

)

.

(61)

On the other hand, conditioning onLxb,o = t, we have theconditional Laplace transform given as

E[

e−zIm |Lxb,o = t]

= exp

(

−τbabCα,KbMs(zPmb, t)

(

zPmb

KbMs

)δ)

. (62)

Notice thatLxb,o has its distribution as (20), and the downlinkrate achievable on the wireless backhaul given as

RDLb =

ζbMs

KsE[

log2(

1 + γDLb,k

)]

(63)

Lemma 6 then follows from substituting (61) and (62) into(60), and deconditioning with respect to (20).

E. Proof of Lemma 7

Let us consider an SAP transmitting in uplink to a typicalMBS located at the origin, which employs a ZF receive filter,the received SINR from thekth antenna is then given by

γULb,k =

PsbL−1o,xb

|r∗b,khb,k|2

(Im + Is + σ2) ‖rb,k‖2(64)

By conditioning on the interferenceIm+Is, whenKb,Mm →∞ with βb = KbMs/Mm < 1, the SINRγUL

b,k satisfies [45]

γULb,k → γDL

b,k =PsbMm(1− βb)L

−1xb,o

σ2 + Im + Is, a.s. (65)

By using the continuous mapping theorem and the lemma in[64], we have the following holds

E[

log2(

1 + γULb,k

)]

=

∫ ∞

0

e−σ2z

z ln 2

(

1− e−zνU

b

)

× E[

e−zIs]

E[

e−zIm]

dz. (66)

On one hand, the Laplace transform ofIm can be calculatedas

E[

e−zIm]

= exp

{

−τbabΓ(1+δ)δπ2P δ

mbzδ∏KbMs−1

i=1 (i+ δ)

sin(δπ) (MsKb)δΓ (MsKb)

}

.

(67)

On the other, to consider the uplink interference from SAPs,we use the result in [65] where the path loss from theinterfering SAPs are modeled as an inhomogeneous PPP withintensity measure being

Λ(m)s (dx) = δabx

δ−1[

1− exp(

−abxδ)]

. (68)

As such, the Laplace transform ofIs in the uplink can becomputed as

E[

e−zIs]

= exp (− (1− τb)Kb

×

∫ ∞

0

[

1−1

(1 + zPsbx−1/Ms)Ms

]

Λ(m)s (dx)

)

.

(69)

Since the uplink rate achievable on the wireless backhaul isgiven as

RULb =

ζbMs

KsE[

log2(

1 + γULb,k

)]

(70)

the result then follows by substituting (67) and (69) into (66).

F. Proof of Lemma 8

The average rate for a typical UE located at the origin isgiven by

R = AmRm +AsRs (71)

whereRm andRs are the data rates when the UE associatesto a MBS and a SAP, respectively, given by

Rm = τmRDLm + (1− τm)R

ULm (72)

13

and

Rs = τs min{

RDLs , RDL

b

}

+(1−τs)min{

RULs , RUL

b

}

. (73)

As each MBS and each SAP serveKm and Ks UEs,respectively, the total density of active UEs is given byKmλm+Ksλs. LetB be the available bandwidth, the sum rateper area is obtained asR = (Kmλm +Ksλs)BR. Lemma 8then follows from Lemmas 2 to 7 and by the continuousmapping theorem.

G. Proof of Lemma 9

We consider a typical UE that locates at the origin, noticethat under FDD, the wireless link experiences interferencefrom the downlink transmitting MBSs and SAPs. As such,the SINR can be written as

γDLm,FDD =

Pmt|h∗xm,owxm,o|2L

−1xm,o

Imuoc + σ2

. (74)

By conditioning on the interference, whenKm,Mm → ∞with βm = Km/Mm < 1, the SINR under ZF precodingconverges to [45]

γDLm,FDD → γDL

m,FDD =νDm

Imuoc + σ2

, a.s. (75)

by using the continuous mapping theorem, and the lemma in[64], we have the following

E[

log2(

1 + γDLm,FDD

)]

=

∫ ∞

0

(

1− e−zνD

m

)

zeσ2z ln 2E

[

e−zImu

oc

]

dz

(76)

The result then follows by noticing thatRDLm,FDD = ξD(1 −

ζb)E[log2(1 + γDLm,FDD)], and by substituting (41) into (76).

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30 35 40 45 501

1.5

2

2.5

MBSTRANSMITPOWERPMT

MA

CR

OC

ELL

DO

WN

LIN

KR

AT

E

SimulationsSimulationsMm20KmAnalysisMm20Km5SimulationsMm100Km25AnalysisMm100Km25

20 30 40 50 60 70 80 90 1005

10

15

20

25

30

35

40

45

50

MBSANTENNANUMBER

BA

CK

HU

ALD

OW

NLI

NK

RA

TE

BLOCK4ZEROFORCIN4BLOCK2ZEROFORCIN2

MBS

SAP

UE

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9x 10

5

NUMBEROFSAPSKB

EN

ER

GY

EF

FIC

IEN

CY

OPTIMALPROPORTIONFIXEDONETIER

25 30 35 40 45 50 552

3

4

5

6

7

8

9

10

11

12x 10

5

POWERONBAKCHAUL

EN

ER

GY

EF

FIC

IEN

CY

LIGHTLOADFEMTOCELLIGHTLOADPICOCELLHHHEAVYLOADFEMTOCELLHEAVYLOADPICOCELL

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

BETAB

ZE

TA

OP

TIM

AL

KSEQU3KSEQU2KSEQU1

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12x 10

5

FRACTIONOFBANDWITH

EN

ER

GY

EF

FIC

IEN

CY

FEMTOCELLLIGHTLOADLDDPICOCELLLIGHTLOADFEMTOCELLHEAVYLOPICOCELLHEAVYLOAD