downlink beamforming for energy-efficient heterogeneous ...€¦ · antennas, hetnet is more...

Downlink Beamforming for Energy-Efficient Heterogeneous Networkswith Massive MIMO and Small Cells

Nguyen, L., Tuan, H. D., Duong, Q., Dobre, O. A., & Poor, H. V. (2018). Downlink Beamforming for Energy-Efficient Heterogeneous Networks with Massive MIMO and Small Cells. IEEE Trans on WirelessCommunications, 17(5), 3386-3400. https://doi.org/10.1109/TWC.2018.2811472

Published in:IEEE Trans on Wireless Communications

Document Version:Peer reviewed version

Queen's University Belfast - Research Portal:Link to publication record in Queen's University Belfast Research Portal

Publisher rights© 2018 IEEE. This work is made available online in accordance with the publisher’s policies. Please refer to any applicable terms of use ofthe publisher.

General rightsCopyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associatedwith these rights.

Take down policyThe Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made toensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in theResearch Portal that you believe breaches copyright or violates any law, please contact [email protected].

Download date:29. Oct. 2020

https://doi.org/10.1109/TWC.2018.2811472

https://pure.qub.ac.uk/en/publications/downlink-beamforming-for-energyefficient-heterogeneous-networks-with-massive-mimo-and-small-cells(20a2b8ba-433c-486c-882b-851f10d67ccc).html

1

Downlink Beamforming for Energy-EfficientHeterogeneous Networks with Massive MIMO and

Small CellsLong D. Nguyen, Student Member, IEEE, Hoang D. Tuan, Trung Q. Duong, Senior Member, IEEE,

Octavia A. Dobre and H. Vincent Poor, Fellow, IEEE

Abstract—A heterogeneous network (HetNet) of a macro-cellbase station equipped with a large-scale antenna array (massiveMIMO) overlaying a number of small cell base stations (smallcells) can provide high quality of service (QoS) to multiple usersunder low transmit power budget. However, the circuit power foroperating such a network, which is proportional to the numberof transmit antennas, poses a problem in terms of its energyefficiency. This paper addresses the beamforming design at thebase stations to optimize the network energy efficiency under QoSconstraints and a transmit power budget. Beamforming tailoredfor weak, strong and medium cross-tier interference HetNetsis proposed. In contrast to the conventional transmit strategyfor power efficiency in meeting the users’ QoS requirements,which suggests the use of a few hundred antennas, it is foundout that the overall network energy efficiency quickly drops ifthis number exceeds 50. It is found that, for a given number ofantennas, HetNet is more energy-efficient than massive MIMOwhen considering overall energy consumption.

Index Terms—Heterogeneous networks, massive MIMO, smallcell, beamformer design, energy efficiency, optimization

I. INTRODUCTION

Massive MIMO [1], [2] and small cell networks [3] arepresently envisioned as two key technologies of the emerginggeneration of communication networks (5G) to support a1000-fold increase in network capacity. Since each of thesetechnologies alone is not expected to meet both the quality-of-service (QoS) and ubiquitous access requirements for 5G [4],combinations of the former overlaying the latter have attractedconsiderable research interest [5], [6]. In such heterogeneousnetworks (HetNets), the small cell base stations (SBSs) servestatic and low mobility users (SUEs) to explore their proximityto these users, while the massive MIMO base station (MBS)serves higher mobility users (MUEs) to explore its highcoverage area and favored channel characteristics. A main

This work was supported in part by the U.K. Royal Academy of Engineer-ing Research Fellowship under Grant RF1415\14\22 and U.K. Engineeringand Physical Sciences Research Council under Grant EP/P019374/1, inpart by the Australian Research Councils Discovery Projects under ProjectDP130104617, and in part by the U.S. National Science Foundation underGrants CNS-1456793 and ECCS-1343210

Long D. Nguyen and Trung Q. Duong are with Queen’s University, BelfastBT7 1NN, UK (e-mail:{lnguyen04,trung.q.duong}@qub.ac.uk)

Hoang D. Tuan is with the School of Electrical and Data Engineering,University of Technology Sydney, Sydney, NSW 2007, Australia (e-mail:[email protected])

Octavia A. Dobre is with Faculty of Engineering and Applied Science,Memorial University, NL A1B 3X5, Canada (email: [email protected])

H. Vincent Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544, USA (e-mail: [email protected]).

issue of HetNets is to manage both intra-tier interferencesand cross-tier interference between the MBS and SBSs [7],[8]. In [9], the SBSs were proposed to be turned-off if theycause an excessive interference to the MBS. Minimization ofthe downlink transmit power by multiflow-regularized-zero-forcing beamforming subject to users’ QoS constraints wasconsidered in [10], which involves a large-scale semi-definiteprogram of dramatically high computational complexity. Theso-called reserve time-division-duplexing of MBS operatingin downlink mode while the SBSs operate in uplink modeand vice versa was proposed in [6]. Being free of cross-tier interference, both MBS and SBSs when in downlink aresupposed to exploit the cross-tier channel state information(CSI) in suppressing their inter-link interference.

With the irreversible trend of network densification in 5Gand beyond [11], [12], a natural concern is its consumed power[13]. To meet the requirement of 1000-fold energy efficiencyfor new technologies [14], the energy efficiency (EE) in termsof the ratio between the information throughput and consumedpower has been introduced as a new figure of merit in assessingcommunication systems (see, e.g., [15]–[17] and referencestherein). While achieving lower transmit power in offeringbetter QoS with using more antennas, it should be realized thatboth MBS and SBSs then consume more circuit powers, whichare proportional to the number of their antennas. Since thelarge-scale analysis [6], [18], which is solely based on arbitrarylarge numbers of antennas and thus does not control theconsumed power, does not readily apply in the EE context, theproblem of determining the numbers of base stations, antennasand users in uplink to improve the EE was considered in [19].As surveyed in [17], so far the main tool for addressing the EEmaximization problems is Dinkelbach’s procedure of fractionalprogramming [20], even though the objective functions areno longer ratios of concave and convex functions; hence inthis case each Dinkelbach’s iteration invokes solution of adifficult nonconvex optimization problem, which is not easierthan the original optimization problem. Two separated energy-efficient beamforming problems were considered in [21]. Thefirst problem is energy-efficient MBS beamforming underconstrained interference to the SBSs’ users, while the secondproblem is the energy-efficient SBS beamforming ignoring theinterference from the MBSs. Each stationary point computedin each Dinkelbach’s iteration is not necessarily feasible. D.C.(difference of two convex functions) iterations [22] were em-ployed in [23] for computation of the nonconvex optimization

2

problem arisen in each Dinkelbach’s iteration for the SBSsunder constrained interference to the MBS-tier users. EachDinkelbach’s iteration in joint power allocation and remoteradio head (RRH)/high-power node (HPN) association forheterogeneous cloud radio access networks (H-CRAN) invokescomputation of a difficult mixed-integer optimization problem.Each Dinkelbach’s iteration in HetNets with fixed service rateconstraints [24] invokes computation of a very difficult mixed-combinatorial and nonconvex optimization problem, which isthen addressed by semi-definite relaxation.

This paper considers a HetNet of an MBS equipped with alarge antenna array overlaying multiple SBSs in serving bothMUEs and SUEs. The aim is beamforming design at both theMBS and SBSs to maximize the network EE under the users’QoS constraints. Such design problems under different beam-forming classes are formulated as maximizations of fractionalfunctions subject to convex constraints. Avoiding Dinkelbach’scomputationally inefficient iterations, path-following compu-tational procedures are developed; these invoke computationof a simple convex program to generate a better feasible pointand at least converge to a locally optimal solution. Simulationsunder different scenarios show that it is important to controlthe number of the MBS antennas maximizing the network EE,which is indeed in sharp contrast to the spectral efficiency (SE)orientated massive MIMO. Moreover, it is also shown in thepaper that indeed underlaid small cells are an effective toolfor substantially improving the network EE.

The paper is organized as follows. After the Introduction,Section II is devoted to the EE problem statement. Zero-forcing (ZF) MBS beamforming is addressed in Section III.Section IV considers other beamforming classes. A specialclass of the ZF MBS and SBS beamforming, with a differentsolution method, is treated in Section V. Simulation results arepresented in Section VI, which is followed by the Conclusions.Some fundamental inequalities used in the paper are providedin the Appendix.

Notation. Boldface uppercase and lowercase letters denotematrices and vectors, respectively. [x]+ , max{0, x} for ascalar x. The transpose and conjugate transpose of a matrixXXX are respectively represented by XXXT and XXXH . III and 000 standfor identity and zero matrices of appropriate dimensions. Tr(.)is the trace operator. ||xxx|| is the Euclidean norm of a vector xxxand ||XXX|| is the Frobenius norm of a matrix XXX . A complex-valued Gaussian random vector with mean xxx and covarianceRRRxxx is denoted by xxx ∼ CN (xxx,RRRxxx). For matrices XXX1, . . . ,XXXk

of appropriate dimension, denote by [XXX1; ...;XXXk] the matrix[XXXT

1 . . .XXXTk ]T .

II. PROBLEM STATEMENT FOR HETNETS

Consider a HetNet of an MBS of a large-scale NM antennaarray with NM up to several hundred and S SBSs, whichare referred as SBS 1, ...., SBS S. Each SBS s is equippedwith Ns antennas. The MBS serves M downlink MUEs, whileSBS s serves Ks downlink SUEs within its cell. All users areequipped with a single antenna. For convenience, denote byKM = {1, . . . ,M} the set of the MUEs and by {(s, `)|` ∈Ks , {1, . . . ,Ks}} the set of those SUEs that are served

TABLE I: Summation of used notations

Notation DescriptionNM number of MBS antennasNs number of SBS antennasM number of users served by MBS (MUEs)Ks number of users served by SBS s (SUEs)KM {1, . . . ,M}Ks {1, . . . ,Ks}k an MUE(s, `) an SUE served by SBS sIs set of MUEs interfered by SBS sNk set of SBS interfering to MUE k√βkhk channel from MBS to MUE k√βs,`χs,` channel from MBS to SUE (s, `)

hs,` channel from SBS s to SUE (s, `)fk MBS beamforming vector for MUE kfs,` SBS s’ beamforming vector for SUE (s, `)FM set of MBS beamforming vectorsFs set of SBS s’ beamforming vectorsFS {Fs, s = 1, . . . , S}FNk

{Fs, s ∈ Nk}F {FM ,FS} (set of all beamforming vectors)σmuik (FM ) inter-MUE interference to MUE kσs,`(Fs) inter-SUE interference to SUE (s, `)σmbis,` (FM ) MBS interference to SUE (s, `)

σsbik (FNk) SBS interference to MUE k

by SBS s. As Fig. 1 shows, in sharing the same spectrum atthe same time, the MBS interferes to all SUEs (s, `), whilethe SBSs interfere to those MUEs in their coverage range.Accordingly, Is with cardinality Is is defined as the set ofthose MUEs that are interfered by SBS s and Nk is the set ofthose SBSs that interfere to MUE k.

Fig. 1: Illustration to the HetNet. The dash lines denote the interference signals andinterference area for SBSs.

Similar to [1], [25] and [18], we will exploit the fol-lowing structure of the massive MIMO channel from theMBS to the MUEs:

√βkhk, and to the SUEs:

√βs,`χχχs,`,

where√βk and

√βs,` model the path loss and large-scale

fading from the MBS to MUE k and SUE (s, `), whilehk = (h1,k, ..., hNM ,k)T with hmk ∈ CN (0, 1) and χχχs,` =(χ1,s,`, . . . , χNM ,s,`)

T with χj,s,` ∈ CN (0, 1) represent thesmall-scale fading.

3

The complex baseband signal received by MUE k is

yk =√βk hhh

Hk fffkxk︸︷︷︸

desired signal

+∑

i∈KM\{k}

√βkhhh

Hk fff ixi︸︷︷︸

inter-MUE (co-tier) interference

+∑s∈Nk

Ks∑j=1

ηηηHs,kfffs,jxs,j︸︷︷︸SBS (cross-tier) interference

+nk, (1)

where fk ∈ CNM and xk are the beamforming vector andinformation from the MBS intended to MUE k, respectively,ηηηs,k is the channel vector from SBS s ∈ Nk to MUE k,fs,` ∈ CNs and xs,` are beamforming vector and informationfrom SBS s intended for SUE (s, `), and nk ∼ CN (0, σ2

k) isthe additive white Gaussian noise at MUE k.

The complex baseband signal received by SUE (s, `) is 1

ys,` = hhhHs,`fffs,`xs,`︸︷︷︸desired signal

+

M∑i=1

√βs,`χχχ

Hs,`fff ixi︸︷︷︸

MBS (cross-tier) interference

+∑

j′∈Ks\{`}

hhhHs,`fffs,j′xs,j′︸︷︷︸inter-SUE (co-tier) interference

+ns,`, (2)

where hs,` ∈ CNs is the channel vector and ns,` ∼CN (0, σ2

s,`) is the additive white Gaussian noise at SUE (s, `).Let

FM , [fk]k=1,...,M ∈ CNM×M ,Fs , [fs,`]`=1,...,Ks ∈ CNs×Ks

and

FS , {Fs, s = 1, . . . , S},FNk= {Fs, s ∈ Nk},

F , {FM ,FS}.

The network’s co-tier interferences are characterized by theinter-MUE and inter-SUE interference functions defined as

σmuik (FM ) , βk

∑i∈KM\{k}

|hhhHk fff i|2, k = 1, . . . ,M, (3)

and

σsuis,`(Fs) ,

∑j∈Ks\{`}

|hhhHs,`fffs,j |2, ` = 1, . . . ,Ks; s = 1, . . . , S,

(4)respectively. On the other hand, the network’s cross-tier inter-ferences are characterized by the MBS and SBSs interferencefunctions defined as

σmbis,` (FM ) , βs,`

M∑i=1

|χχχHs,`fff i|2 (5)

and

σsbik (FNk

) ,∑s∈Nk

Ks∑j=1

|ηηηHs,kfffs,j |2, k = 1, . . . ,M, (6)

1Note that in (2) we assume that the small cells are sufficiently far apartfrom each other so that the inter-small-cell interference can be ignored.This is also true for dense small cells where orthogonal frequency divisionmultiplexing is used to allow sufficiently far apart from each other cells touse the same carrier [26], [27].

respectively.The information throughputs at MUE k and SUE (s, `) (in

nats) are

rk(FM ,FNk) = ln

(1 +

βk|hhhHk fffk|2

σmuik (FM ) + σsbi

k (FNk) + σ2

k

)(7)

and

rs,`(Fs,FM ) = ln

(1 +

|hhhHs,`fffs,`|2

σsuis,`(Fs) + σmbi

s,` (FM ) + σ2s,`

).

(8)

The entire consumed power for the downlink transmission canbe expressed as

P (F) = Pmbs(FM ) +

S∑s=1

Ps(Fs), (9)

where

Pmbs(FM ) = α

M∑k=1

||fffk||2 +MPa + Pc (10)

is the power consumed by the MBS and

Ps(Fs) = αs

Ks∑`=1

||fffs,`||2 +NsPa,s + Pc,s (11)

is the power consumed by SBS s. There, α > 1 and αs > 1are the reciprocal of the drain efficiency of the amplifier ofthe MBS and SBS s, Pa and Pa,s represent the per-antennacircuit power of the MBS and SBS s, and Pc and Pc,s are thenon-transmission power of the MBS and SBS s, respectively.Accordingly, the total SBSs consumed power is

Psbs(FS) =

S∑s=1

Ps(Fs).

The EE maximization problem under QoS constraints andpower budget is formulated as

maxF

∑Mk=1 rk(FM ,FNk

) +∑Ss=1

∑Ks

`=1 rs,`(Fs,FM )

Pmbs(FM ) + Psbs(FS)

(12a)s.t. rk(FM ,FNk

) ≥ rk, k = 1, . . . ,M, (12b)rs,`(Fs,FM ) ≥ rs,` , ` = 1, . . . , Ns; s = 1, . . . , S,

(12c)M∑k=1

||fk||2 ≤ PmaxM , (12d)

Ks∑`=1

||fffs,`||2 ≤ Pmaxs , s = 1, ..., S, (12e)

where the constraints (12b)-(12c) set the QoS data rate require-ment at each MUE and SUE, and the constraints (12d)-(12e)keep the sum of the transmit power constraints at the MBSand SBSs under predefined budgets.

The paper follows the network centric techniques likeCloud-RAN and cooperative multipoint (CoMP) [28], where

4

the MBS and SBSs cooperate to solve the EE maximizationproblem (12) in a central manner. The conventional assumptionis that full channel state information (CSI) is available for theoptimization of problem (12).

The above problem is very complicated due the presenceof both intra-tier and cross-tier interferences, and the largedimension NM of the MBS beamforming vectors fk ∈ CNM .One can see from (1) and (2) that the MBS contributes a severeinterference to all MUEs and SUEs.

The next sections propose computational solutions for (12)by different classes of MBS and SBS beamforming.

III. ZERO-FORCING INTER-MUE INTERFERENCE BASEDBEAMFORMING (MZF)

For H = [h1 h2 ... hM ] ∈ CNM xM , which is very tall dueto NM >> M , there is the right inverse of the fat matrixHH = [hH1 ;hH2 ... ;hM ] ∈ CMxNM defined by

FFFM =[fff1 . . . fffM

]= HHH(HHHHHHH)−1, (13)

i.e.

III = HHHHFFFM = [hhhH1 FFFM ; . . . ;hhhHMFFFM ]

= [hhhHi fff j ](i,j)∈KM×KM. (14)

which means that hhhHi fff i = 1 and hhhHi fff j = 0 for i 6= j. Usingthe normalized vectors fffk , fffk/‖fffk‖ , k = 1, ...,M , theMBS beamforming vector fk is sought in the class of

fk = pkfffk, k = 1, . . . ,M (15)

to cancel the inter-MUE interference σmuik (FM ) in (3):

hhhHk fff i = hhhHk fff i/||fff i|| = 0 for i 6= k.For p = (p1, ..., pM )T ∈ RM and

βk , βk|hhhHk fffk|2, (16)

the information throughput (in nats) for MUE k in (7) is

rk(pk,FNk) = ln(1 +

p2kβk

σ2k + σsbi

k (FNk)), (17)

with the SBS interference σsbik (FNk

) defined from (6), whilethe consumed power by the MBS transmission defined by (9)is now a quadratic form of p:

πmbs(p) = α

M∑k=1

p2k +MPa + Pc. (18)

The power constraint (12d) is nowM∑k=1

p2k ≤ Pmax

M , pk ≥ 0 , k = 1, ...,M. (19)

The information throughput for SUE (s, `) in (8) is re-expressed by

rs,`(Fs,p) = ln(1 +|hHs,`fffs,`|2

σmbis,` (p) + σsui

s,`(Fs) + σ2s,`

), (20)

where

σmbis,` (p) , βs,`

M∑i=1

p2i ||χχχHs,`fff i||2 (21)

is the MBS interference function (see (5)), and σsuis,`(Fs) is the

inter-SUE interference function defined from (4).Under the class of MZF, the EE maximization problem (12)

is now expressed by

maxp,FS

Φ(p,FS) ,(M∑k=1

ln(1 +p2kβk

σ2k + σsbi

k (FNk))

+

S∑s=1

Ks∑`=1

ln(1 +|hHs,`fffs,`|2


s,`(Fs) + σ2s,`

)

)/

(πmbs(p) + Psbs(FS))

s.t. (12e), (19), (22a)

ln

(1 +

p2kβk

σ2k + σsbi

k (FNk)

)≥ rk, k = 1, . . . ,M, (22b)

ln

(1 +

|hHs,`fffs,`|2


s,`(Fs) + σ2s,`

)≥ rs,`,

` = 1, . . . ,Ks; s = 1, ..., S. (22c)

The nonconvex constraint (22b) is seen equivalent to thefollowing second-order cone (SOC) constraint

pk

√βk ≥

√erk − 1

√σ2k + σsbi

k (FNk), k = 1, . . . ,M. (23)

As observed in [29], for fffs,` , e−.arg(hhhHs,`fffs,`)fffs,`, one

has |hhhHs,`fffs,`| = hhhHs,`fffs,` = <{hhhHs,`fffs,`} ≥ 0 in (20).Therefore, |hhhHs,`fffs,`|2 in (20) can be equivalently replaced by(<{hhhHs,`fffs,`})2 with <{hhhHs,`fffs,`} ≥ 0, ` = 1, . . . , Ns; s =1, . . . , S. Consequently, the nonconvex constraint (22c) is alsoequivalent to the SOC constraint

<{hHs,`fffs,`} ≥√

(ers,` − 1)√σmbis,` (p) + σsui

s,`(Fs) + σ2s,`,

` = 1, . . . ,Ks; s = 1, ..., S. (24)

Therefore, the EE maximization problem (22) is a nonconcavefunction maximization under convex constraints. Our focusnow is to handle its objective function. Let (p(n),F

(n)S ) be

a feasible point for (22) found from the (n − 1)th iteration.Using inequality (69) in the Appendix for

x = xk ,p2kβk

σ2k + σsbi

k (FNk), t , πmbs(p) + Psbs(FS)

andx = x

(n)k ,

(p(n)k )2βk

σ2k+σsbi

k (F(n)Nk

),

t = t(n) , πmbs(p(n)) + Psbs(F

(n)S ),

yields the following lower bounding approximation for the firstterm in the objective function in (22a):

ln

(1 +

p2kβk

σ2k + σsbi

k (FNk)

)P (p,FS)

≥

a(n)k − b(n)

k

σ2k + σsbi

k (FNk)

βkp2k

− c(n)k (πmbs(p) + Psbs(FS)) ≥

g(n)k (p,FS) (25)

5

over the trust region

2pk − p(n)k > 0, (26)

for

g(n)k (p,FS) , a

(n)k − b(n)

k

σ2k + σsbi

k (FNk)

βkp(n)k (2pk − p(n)

k )

−c(n)k (πmbs(p) + Psbs(FS)), (27)

where

0 < a(n)k , 2

ln(1 + x(n)k )

t(n)+

x(n)k

t(n)(x(n)k + 1)

,

0 < b(n)k ,

(x(n)k )2

t(n)(x(n)k + 1)

,

0 < c(n)k ,

ln(1 + x(n)k )

(t(n))2.

(28)

To address the second term in the objective function in (22a),by substituting

x = xs,` ,(<{hH

s,`fffs,`})2

σmbis,`(p)+σsui

s,`(Fs)+σ2s,`

,

t , πmbs(p) + Psbs(FS)

and

x = x(n)s,` ,

(<{hHs,`fff(n)s,` })2

σmbis,` (p(n)) + σsui

s,`(F(n)s ) + σ2

s,`

,

t = t(n) , πmbs(p(n)) + Psbs(F

(n)S ),

into (69) in the Appendix again and using the inequality(72) in the Appendix, we obtain its following lower boundingapproximation:

ln

(1 +

(<{hHs,`fffs,`})2


s,`(Fs) + σ2s,`

)/P (p,FS) ≥

a(n)s,` − b

(n)s,`


s,`(Fs) + σ2s,`

(<{hHs,`fffs,`})2

−c(n)s,` (πmbs(p) + Psbs(FS)) ≥ (29)

g(n)s,` (p,FS), (30)

for

g(n)s,` (p,FS) ,

a(n)s,` − b

(n)s,`


s,`(Fs) + σ2s,`

2<{hHs,`fff(n)s,` }<{hHs,`fffs,`} − (<{hHs,`fff

(n)s,` })2

−c(n)s,` (πmbs(p) + Psbs(FS)) (31)


2<{hhhHs,`ttts,`} ≥ <{hhhHs,`ttt(n)s,` }, ` = 1, . . . , Ns; s = 1, . . . , S,

(32)

where

0 < a(n)s,` , 2

ln(1 + x(n)s,` )

t(n)+

x(n)s,`

t(n)(x(n)s,` + 1)

,

0 < b(n)s,` ,

(x(n)s,` )2

t(n)(x(n)s,` + 1)

,

0 < c(n)s,` ,

ln(1 + x(n)s,` )

(t(n))2.

(33)

At the nth iteration, the following convex program is solvedto generate the next feasible point (p(n+1),F

(n+1)S ) for (22):

maxp,FS

Φ(n)(p,FS) , [

M∑k=1

g(n)k (p,FS) +

S∑s=1

Ks∑`=1

g(n)s,` (p,Fs)]

s.t. (12e), (19), (32), (23), (24), (26). (34)

It follows from (25) and (30) that

Φ(p,FS) ≥ Φ(n)(p,FS) ∀(p,FS) (35)

while it is trivial to check that

Φ(p(n),F(n)S ) = Φ(n)(p(n),F

(n)S ). (36)

As (p(n),F(n)S ) and (p(n+1),F

(n+1)S ) are a feasible point and

the optimal solution of the convex program (34), respectively,it also follows that

Φ(n)(p(n+1),F(n+1)S ) > Φ(n)(p(n),F

(n)S ), (37)

as far as(p(n+1),F

(n+1)S ) 6= (p(n),F

(n)S ), (38)

which together with (35) and (36) yield

Φ(p(n+1),F(n+1)S ) > Φ(p(n),F

(n)S ), (39)

showing that (p(n+1),F(n+1)S ) is a better feasible point for

(22) than (p(n),F(n)S ). Thus, in Algorithm 1, we propose a

path-following computational procedure for the EE maximiza-tion problem (22). An initial point (p(0),F

(0)S ) for (22) is

easily located because all the constraints in (22) are convex.For instance, it can be found from the following convexprogram:

minp,FS

πmbs(p)+Psbs(FS) s.t. (12e), (19), (23), (24). (40)

Algorithm 1 : Path-following algorithm for solving problem(22)

1: Initialization: Choose a feasible point (p(0),F(0)S ) for

(22). Set n := 0.2: Repeat3: Solve the problem (34) for its optimal solution

(p(n+1),F(n+1)S ).

4: Set n := n+ 1.5: Until convergence of the objective in (22).

Similar to [30, Prop. 1] we have the following result.Proposition 1: At least, Algorithm 1 converges to a locally

optimal solution of (22) satisfying the Karush-Kuhn-Tucker(KKT) conditions of optimality.

6

IV. OTHER SCHEMES

The MZF as given by (15) cancels only the inter-MUEinterference, under which the MBS interference σmbi(FM ) isnot controlled. In this section we consider other classes ofMBS and SBSs beamforming to enhance both cross-tier andco-tier interferences in optimizing the EE of the system.

A. Zero forcing co-tier interference based beamforming(MZF+SZF)

In this scheme, referred as MZF+SZF with SZF used as anabbreviation to represent ”zero-forcing inter-SUE interferencebased beamforming”, the MBS beamforming vector fk issought in the class of MZF, while the SBS beamforming vectorfs,` is designed to force the inter-SUE interference to zero.For each SUE (s, `) define the interfering channel

Hs,` , [hs,j ]j∈Ks\{`} ∈ CNs×(Ks−1), (41)

which stacks all channels from SBS s to its SUEs except thatto SUE (s, `). To nullify the inter-SUE interference in (4), thefollowing condition must be fulfilled:

HHs,`fffs,` = 000 ∈ CKs−1, ` = 1, . . . ,Ks, (42)

requiring

Ns > Ks.

Such fs,` is parametrized as

fs,` = GGGs,`ttts,`, (43)

where GGGs,` ∈ CNs×(Ns−Ks+1) is an orthogonal basis for thenull space of HH

s,` and ttts,` ∈ CNs−Ks+1. Consequently, theinformation throughput at SUE (s, `) in (8) is

r(2)s,` (ts,`) = ln

(1 + |hHs,`ttts,`|2/(σmbi

s,` (p) + σ2s,`))

(44)

with σmbis,` (p) defined in (21) and hs,` , GH

s,`hs,`.For ΘΘΘ`,ks , GGGHs,`hhhs,ks ∈ CNs−Ks+1 and TNk

, [Ts]s∈Nk,

the SBSs interference to MUE k in (6) is

σsbik (TNk

) , σsbik ([Gs,`ts,`]s∈Nk,`=1,...,Ks

)

=∑s∈Nk

Ks∑`=1

|ΘΘΘH`,ksttts,`|

2. (45)

Now, recalling the definition (21) for the MBS interference tothe SUEs, the EE maximization problem is formulated as

maxp,T

M∑k=1

ln(1 +p2kβk

σ2k + σsbi

k (TNk))

π(p,T)

+

S∑s=1

Ks∑`=1

ln(1 +(<{hHs,`ttts,`})2

σmbis,` (p) + σ2

s,`

)

π(p,T)(46a)

s.t. (19),

pk

√βk ≥

√erk − 1

√σ2k + σsbi

k (TNk), k = 1, . . . ,M,

(46b)

<{hHs,`ttts,`} ≥√

(ers,` − 1)√σmbis,` (p) + σ2

s,`,

` = 1, . . . ,Ks; s = 1, ..., S, (46c)Ks∑`=1

||ttts,`||2 ≤ Pmaxs , s = 1, ..., S. (46d)

Initialized from a feasible point (p(0),T(0)), which is foundfrom the convex program

minp,T

π(p,T) s.t. (19), (46b), (46c), (46d) (47)

at the nth iteration the following convex program is solved togenerate the next iterative point (p(n+1),T(n+1)) for (46):

maxp,T

[

M∑k=1

g(n)k (p,T) +

S∑s=1

Ks∑`=1

f(n)s,` (p,T)]

s.t. (19), (26), (32), (46d), (46b), (46c),

(48)

where

g(n)k (p,T) , a

(n)k − b(n)

k

σ2k + σsbi

k (TNk)

βkp(n)k (2pk − p(n)

k )

−c(n)k π(p,T), (49)

with a(n)k , b(n)

k and c(n)k defined from (28) for

x(n)k ,

(p(n)k )2βk

σ2k + σsbi

k (T(n)Nk

), t(n) , π(p(n),T(n)),

and

f(n)s,` (p,T) ,

a(n)s,` − b

(n)s,`

σmbis,` (p) + σ2

s,`

2<{hHs,`ttt(n)s,` }<{hHs,`ttts,`} − (<{hHs,`ttt

(n)s,` })2

−c(n)s,` π(p,T) (50)

with a(n)s,` , b(n)

s,` and c(n)s,` defined by (33) for

x(n)s,` , (<{hHs,`ttt

(n)s,` })

2/σmbis,` (p(n))), t(n) , π(p(n),T(n)).

Similar to Proposition 1, it can be easily shown that thecomputational procedure that invokes the convex program(48) to generate the next iterative point, is path-following for(46), which at least converges to its locally optimal solutionsatisfying the KKT conditions.

7

B. Zero-forcing inter-MUE and MBS and inter-SUE interfer-ence beamforming (ZMI+SZF)

In this scheme, referred as ZMI+SZF with ZMI used as anabbreviation to represent ”zero-forcing inter-MUE and MBSinterferences based beamforming”, the MBS beamformingvector fk is designed to force both inter-MUE interferenceand MBS interference to zero, while the SBS beamformingvector fs,` is parametrized by (43) in forcing the inter-SUEinterference to zero.

Define the interfering channels from the MBS to the SUEs

χχχs = [χχχs,`]`=1,...,Ks∈ CNM×Ks ,

χχχ , [χχχs]s=1,...,S ∈ CNM×∑S

s=1Ks

andHHHmbs , [HHH χχχ] ∈ CNM×(M+

∑Ss=1Ks),

which is still a very tall as the total number M +∑Ss=1Ks of

users is still small compared to the number NM of the MBS’santennas. Then the right inverse of the fat matrix HHHH

mbs isdefined as

FFFmbs =[g1 . . . gM . . . gM+

∑Ss=1Ks

], HHHmbs(HHH

HmbsHHHmbs)

−1 (51)

∈ CNM×(M+∑S

s=1Ks),

i.e., [HH

mbs

[g1 . . . gM

]HH

mbs

[gM+1 . . . gM+

∑Ss=1Ks

]]= HH

mbsFFFmbs

= I ∈ R(M+∑S

s=1Ks)×(M+∑S

s=1Ks).

Particularly,

HHHHmbs [g1 . . . gM ] =

[IO

]∈ C(M+

∑Ss=1Ks)×M .

Using the normalized vectors

fk , gk/||gk||, k = 1, ...,M, (52)

the MBS beamforming vector fk is sough in the class of (15) tonullify the inter-MUE interference and the MBS interferenceto the SUEs. Under the definition (16) for βk with fk definedin (52) and the definition (43) for parametrizing beamformingvectors fs,` of SZF, the EE maximization problem (12) is nowformulated as

maxp,T

M∑k=1

ln(1 + p2

kβk/(σsbik (TNk

) + σ2k))

π(p,T)

+

S∑s=1

Ks∑`=1

ln(

1 + (<{hhhHs,`ttts,`})2/σ2s,`

)π(p,T)

(53a)

s.t. (19), (46b), (46d), (53b)


(ers,` − 1)σs,`, (53c)

` = 1, . . . ,Ks; s = 1, ..., S.

Initialized from a feasible point (p(0),T(0)), which is foundfrom the convex program

minp,T

π(p,T) s.t. (19), (46b), (46d), (53c) (54)

at the nth iteration the following convex program is solved togenerate the next iterative feasible point (p(n+1),T(n+1)) for(53):

maxp,T

[

M∑k=1

g(n)k (p,T) +

S∑s=1

Ks∑`=1

f(n)s,` (p,T)]

s.t. (19), (26), (32), (46b), (46d), (53c),

(55)

where g(n)k (p,T) is defined in (49), and

f(n)s,` (p,T) ,

a(n)s,` − b

(n)s,`

σsuis,`(Ts) + σ2

s,`

2<{hhhHs,`ttt(n)s,` }<{hhh

Hs,`ttts,`} − (<{hhhHs,`ttt

(n)s,` })2

−c(n)s,` π(p,T), (56)

with a(n)s,` , b(n)

s,` and c(n)s,` defined by (33) for

x(n)s,` ,

(<{hhhHs,`ttt(n)s,` })2

σsuis,`(T

(n)s ) + σ2

s,`

, t(n) , π(p(n),T(n)).


C. Adaptively suppressed co-interference based beamforming(AZMI+SZF)

Denote by S1 = {1, ..., S1} the set of those SBSs thatare located sufficiently near to the MBS, and thus, theirSUEs are under the strong MBS interference, while denote byS2 = {S1+1, ..., S} the set of those SBSs located far to MBS,and thus, their SUEs are under the weak MBS interference.In this scheme, referred to as AZMI+SZF with AZMI used asan abbreviation to represent ”adaptively zero-forcing MBS in-terference based beamforming”, the SBS beamforming vectorfs,` is parametrized by (43) to force the inter-SUE interferenceto zero. On the other hand, the MBS beamforming vector fkis designed based on (15) with fk defined in (52) with[

g1 . . . gM . . . gM+

∑S1s1=1Ks1

],

HHHmbs,1(HHHHmbs,1HHHmbs,1)

−1 ∈ CNM×(M+∑S1

s1=1Ks1) (57)

and

χχχs1 = [χχχs1,`]`=1,...,Ks1∈ CNM×Ks1 ,

χχχ1 , [χχχs1 ]s1=1,...,S1 ∈ CNM×∑S1

s1=1Ks1 ,

HHHmbs,1 , [HHH χχχ1] ∈ CNM×(M+∑S1

s1=1Ks1),

8

to nullify the inter-MUE interference and the strong MBSinterference to SUEs (s1, `), s1 ∈ S1. The EE maximizationproblem (12) becomes

maxp,T

M∑k=1

ln(1 +p2kβk

σ2k + σsbi

k (TNk))

π(p,T)

+

S∑s=1

Ks∑`=1

ln(1 +(<{hhhHs,`ttts,`})2

σmbis,` (p) + σ2

s,`

)

π(p,T)(58a)

s.t. (19), (46d), (58b)

pk

√βk ≥

√erk − 1

√σ2k + σsbi

k (TNk), k = 1, . . . ,M,

(58c)

<{hhhHs,`ttts,`} ≥ σs,`√ers,` − 1,

` = 1, . . . ,Ks; s = 1, ..., S1, (58d)


(ers,` − 1)√σmbis,` (p) + σ2

s,`,

` = 1, . . . ,Ks; s = 1, ..., S2. (58e)

Initialized from a feasible point, which is found from theconvex program

minp,T

π(p,T) s.t. (19), (46d), (58c), (58d), (58e), (59)

at the nth iteration, the following convex program is solvedto generate a feasible point (p(n+1),T(n+1)) for (58):

maxp,T

M∑k=1

g(n)k (p,T) +

S∑s=1

Ks∑`=1

f(n)s,` (p,T) (60a)

s.t. (19), (26), (32), (46d), (58c), (58d), (58e), (60b)

where g(n)k (p,T) is defined in (49), with a(n)

k , b(n)k and c(n)

k

from (28), for

x(n)k ,

(p(n)k )2βk

σ2k + σsbi

k (T(n)Nk

), t(n) , π(p(n),T(n)),

and f (n)s,` (p,T) is defined from (50), with a(n)

s,` , b(n)s,` and c(n)

s,`

from (33), for

x(n)s,` ,

(<{hHs,`ttt(n)s,` })2

σmbis,` (p(n)) + σ2

s,`

, t(n) , π(p(n),T(n)).


V. ENERGY-EFFICIENT ZERO-FORCING HETNETBEAMFORMING (EE ZF)

To show the advantage of HetNets over massive MIMO interms of the EE, in this section we address the EE maximiza-tion problems in the class of zero-forcing beamforming at bothMBS and SBSs, i.e. the the BMS beamforming vector fk issought in the class of (15) with fk defined from (51) and (52)to cancel both inter-MUE and MBS interferences while the

SBS beamforming vector fs,` is also sought to cancel bothinter-SUE and SBS interferences as detailed below.

With Hs ∈ CNs×Is defined from (41) and

Hsbs,s , [ [hs,`]`=1,...,KsHs] ∈ CNs×(Ks+Is)

the right inverse of HHsbs,s is

[fs,1 . . . fs,Ks. . . fs,Ks+Is ] , Hsbs,s(H

Hsbs,sHsbs,s)

−1.

Using the normalized vectors fs,` = fs,`/||fs,`||, ` =1, . . . ,Ks, to cancel both inter-SUE and SBS interferencesthe SUE beamformers fs,` is sought in the form

fs,` = ps,`fs,`, ` = 1, . . . , Ns; s = 1, . . . , S. (61)

For βk defined from (16) and βs,` , |hHs,`fs,`|2, while ps ,(ps,`)s=1,...,S;`=1,...,Ns

, the EE maximization problem (12) isthus

maxp,pS

∑Mk=1 ln

(1 + βkp

2k/σ

2k

)πmbs(p) + πsbs(pS)

+

∑Ss=1

∑Ks

`=1 ln(

1 + βs,`p2s,`/σ

2s,`

)πmbs(p) + πsbs(pS)

(62a)

s.t. (19),

ln(1 + βkp

2k/σ

2k

)≥ rk , k = 1, ...,M, (62b)

ln(1 + βs,`p

2s,`/σ

2s,`

)≥ rs,`,

` = 1, . . . ,Ks; s = 1, . . . , S, (62c)Ks∑`=1

p2s,` ≤ Pmax

s , s = 1, .., S, (62d)

where

πsbs(pS) ,S∑s=1

[αs

Ks∑`=1

p2s,` +NsPa,s + Pc,s].

One can see that the objective in (62a) is the ratio of con-cave and convex functions, for which Dinkelbach’s algorithm[20] is applicable. In what follows, we will show that eachDinkelbach’s iteration admits a closed-form solution; thus,Dinkelbach’s algorithm is very computationally efficient.

First, it follows from (62b) and (62c) that

p2k ≥ pk := σ2

k(erk − 1)/βk,p2s,` ≥ ps,` := σ2

s,`(ers,` − 1)/βs,`.

By making the variable change

p2k = pk + pk, p

2s,` = ps,` + ps,`,

it is straightforward to solve (62) by applying Dinkelbach’salgorithm, which seeks τ > 0 such that the optimal solution

9

of the following optimization problem is zero:

maxp,pS

M∑k=1

ln(ak + βkpk/σ

2k

)+

S∑s=1

Ks∑`=1

ln(as,` + βs,`ps,`/σ

2s,`

)− τ(πmbs(p) + πsbs(pS)) (63a)

s.t.M∑k=1

pk ≤ PmaxM , pk ≥ 0 , k = 1, ...,M, (63b)

Ks∑`=1

ps,` ≤ Pmaxs , s = 1, .., S, (63c)

where ak = 1 + βkpk/σ2k, PMcir = α

∑Mk=1 pk + MPa + Pc,

PmaxM = Pmax

M −∑Mk=1 pk, πmbs(p) , α

∑Mk=1 pk + PMcir, and

as,` = 1+ βs,`ps,`/σ2s,`, Ps,cir = αs

∑Ks

`=1 ps,`+Ps,cir, Pmaxs =

Pmaxs −

∑Ks

`=1 ps,`, πsbs(pS) ,∑Ss=1[αs

∑Ks

`=1 ps,` + Ps,cir].Problem (63) admits the optimal solution in the closed-form:

p∗k =

[1

(τα+ λM )− akσ

2k

βk

]+

, k = 1, . . . ,M,

p∗s,` =

[1

(ταs + λs)−as,`σ

2s,`

βs,`

]+

,

` = 1, . . . ,Ks; s = 1, . . . , S,

(64)

where λM = 0 whenM∑k=1

[1

τα− akσ

2k

βk

]+

≤ PmaxM .

Otherwise, λM > 0 is located through the bisection methodsuch that

M∑k=1

[1

(τα+ λM )− akσ

2k

βk

]+

= PmaxM . (65)

Analogously, λs = 0 when

Ks∑`=1

[1

ταs−as,`σ

2s,`

βs,`

]+

≤ Pmaxs .

Otherwise, λs > 0 is located through the bisection methodsuch that

Ks∑`=1

[1

(ταs + λs)−as,`σ

2s,`

βs,`

]+

= Pmaxs . (66)

The above proposed Dinkelbach’s computational procedurefor (62) is summarized in Algorithm 2.

VI. NUMERICAL SIMULATIONS

In this section, we evaluate the performance of the proposedalgorithms by numerical simulations. Consider a circular cellHetNet with radius 1 km, where the MBS is at the centerand S = 6 underlaid SBSs are distributed either equallyat the cell edge or nearly the MBS, or half of which areequally distributed at the cell edge with another half distributednearly the MBS as depicted in Fig. 2a, Fig. 2b or Fig. 2c,

Algorithm 2 : Dinkelbach’s algorithm for solving problem(62)

1: Initialization: Solve (63) for initial τ > 0. If its optimalvalue is greater than zero, set τ = τ and reset τ ← 2τand solve (63) again. Otherwise (its optimal value is lowerthan zero) set τ = τ . End up by having τ and τ such thatthe optimal value of (63) is positive for τ = τ and isnegative for τ = τ . The optimal τ for zero optimal valueof (63) lies on [τ , τ ];

2: Bisection Method3: Repeat4: Solve (63) for τ = (τ + τ)/2. If its optimal value is

positive, then reset τ ← τ . Otherwise (its optimal valueis negative), reset τ ← τ .

5: Until τ − τ ≤ ε (tolerance) to have the optimal value of(63) equal to zero.

respectively. These scenarios correspond to weakly coupled,strongly coupled and mixed-coupled HetNets, respectively.The radius of each small cell is 50 m and the radius of theinterference area of each SBS is 50 m×3 = 150m. The MBSis equipped with NM > 32 antennas and each SBS is equippedwith Ns = 4 antennas. The other simulation parameters areprovided in Table I, which follow the prior works [10], [31].The channel vector hs,` from SBS s in (2) is still genereatedby √qs,`hs,` with √qs,` and hs,` = (h1,s,`, ...., hNs,s,`)

T ,hi,s,` ∈ CN (0, 1) representing the path loss and large-scalefading and the small-scale fading, respectively. There areM = 16 MUEs with 10 MUEs uniformly distributed outsidethe SBSs’ coverage and each of the remaining 6 MUEslocated in the interference area of one of SBSs. Thus, theSBS’s interference to the MUEs is the same under these threesettings. Each SBS serves two SUEs. The QoS requirementfor all users is 0.4 bps/Hz or 4 Mbps [32, Table I].

TABLE II: Simulation Setup

Parameter AssumptionCarrier frequency / Bandwidth 2 GHz / 10 MHzMBS transmission power 46 dBmSBS transmission power 30 dBmPath loss from MBS to user 148.1 + 37.6log10R [dB], R in kmPath loss from SBSs to user 127 + 30log10R [dB], R in kmShadowing standard deviation 8 dBNoise power density -174 dBm/HzThe power amplifiers parameter α0 = 1/0.388, αs = 1/0.052The circuit power per antenna Pa,0 = 189 mW, Pa,s = 5.6 mWThe non-transmission power Pc,0 = 40 dBm, Pc,s = 20 dBm

10

−1000 −800 −600 −400 −200 0 200 400 600 800 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

MBS

SC3

SC4SC1

SC2

SC6SC5

(a) A weakly coupled HetNet

−1000 −800 −600 −400 −200 0 200 400 600 800 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

MBS

SC3

SC4

SC2

SC1

SC6SC5

(b) A strongly coupled HetNet

−1000 −800 −600 −400 −200 0 200 400 600 800 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

MBS

SC2SC3

SC5

SC1

SC6

SC4

(c) A mixed-coupled HetNet

Fig. 2: Three examples of HetNets with different locations of SBSs.

A. Weakly coupled HetNet

Fig. 3 provides the typical convergence of the proposedpath-following computational procedures for solving each par-ticular EE maximization problem in the scenario of weaklycoupled HetNets. which is also observed in other scenarios.

The EE objective is iteratively increase and converges rapidlywithin several iterations.

0 2 4 6 8 10 12 140.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of iterations

EE

(bi

ts/J

oule

/Hz)

MZFMZF+SZFZMI+SZF

Fig. 3: The convergence vs. iteration number under NM = 64 and QoS rk = r(s,`) ≡4 Mbps.

Note that MZF and MZF+SZF use the same class (13), (15)of zero forcing inter-MUE interference MBS beamformingvector fk for the EE maximization problems (22) and (46).They achieve the same EE performance but MZF+SZF isobviously more computationally efficient; as such only thecurve of MZF+SZF’s EE is provided in the next simulations.This observation implies that:(i) The SBSs’ interference to the MUEs can be easily com-pensated in HetNets without hurting the network’s EE, and(ii) The zero-forcing inter-SUE interference SBS beamform-ing vector fs,` as parametrized by (43) provides interferenceenhancement means in HetNets.

40 50 60 70 80 90 1002.5

3

3.5

4

4.5

Number of MBS antennas

EE

(bi

ts/J

oule

/Hz)

MZF+SZFMZF+CoZFZMI+SZFZMI+CoZFEE ZF

Fig. 4: The EE performance vs. the number of MBS antennas in weakly coupled HetNets.QoS rk = r(s,`) ≡ 4 Mbps.

Fig. 4 depicts the EE performance vs. the number NM ofthe MBS antennas. Under the weakly coupled scenario, theMBS interference to the SUEs is weak, leaving its cancelationunnecessary. This explains why MZF, which ignores this in-terference in the EE maximization problem (22), outperformsZMI+SZF, which nullifies it in the EE maximization problem(53).

The performance gap between MZF and ZMI+SZF isnarrower as NM increases, making the MBS interference

11

stronger. However, there is a huge gap at both NM = 40 andNM = 60, under which either MZF or ZMI+SZF achievestheir maximum EE.

It has been shown in [33] that the following conventionalSBS beamforming vector fs,` to force the inter-SUE inter-ference to zero is not quite energy-efficient for multi-small-cell networks: fs,` = ps,`fs,`/||fs,`|| with [fs,1 . . . fs,`] =Hs(H

Hs Hs)

−1 and Hs = [hs,j ]j∈Ks∈ CNs×Ks , which

stacks all channels from SBS s to its SUEs. Figures 4 to 7 alsoplot the EE by these conventional class of zero-forcing SBSbeamforming under different classes of MBS beamforming,which also demonstrates that this conventional zero-forcingSBS beamforming is not energy-efficient for HetNets and isclearly outperformed by the SBS beamforming with beam-forming vectors parametrized by (43).

40 50 60 70 80 90 1002

4

6

8

10

12

14


Tra

nsm

it po

wer

at M

BS

(W)

ZMI+SZFEE ZF

(a) MBS transmission

40 50 60 70 80 90 100

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7


Tra

nsm

it po

wer

at S

BSs

(W

)

ZMI+SZFEE ZF

(b) SBSs transmission

Fig. 5: The transmit power at MBS and SBSs in weakly coupled HetNet.

On the other hand, observe that ZMI+SZF and EE ZF,which use the same class of MBS beamforming, achievethe same EE. Both schemes particularly force the inter-SUEinterference to zero. SBS interference is compensated by MBSbeamforming without hurting the network’s EE in ZMI+SZFbut is nullified by SBS beamforming in EE ZF. Fig. 5 showsthat, as expected, the former requires less SBSs’ transmissionpower to optimize the EE than the latter. Furthermore, it alsoreveals that when the number NM of the MBS antennas isless than 60, the network’s EE is optimised by requiring lesstransmission power, i.e. the spectral efficiency compensates

well the massive MIMO’s circuit power. However, when thenumber NM is more than 60, the MBS interference becomesthe main factor to hurt the network’s sum throughput in thenumerator of the EE objective. SBSs also need more powerto compensate this MBS interference to maintain the QoSrequirements.

B. Strongly coupled HetNet

40 50 60 70 80 90 1002.8

3

3.2

3.4

3.6

3.8

Number of MBS antennasE

E (

bits

/Jou

le/H

z)

MZF+SZFMZF+CoZFZMI+SZFZMI+CoZFEE ZF

Fig. 6: The EE performance vs. the number of MBS antennas in strongly coupledHetNets. QoS rk = r(s,`) ≡ 4 Mbps.

In this scenario, the MBS interference to the SUEs is verystrong as all SUEs are located sufficiently near to the MBS.This suggests that the MBS needs to control its interferenceto make the overall network energy-efficient. However, asFig. 6 shows, the EE performance achieved by MZF forNM = 40, which ignores this interference while enhancinginter-SUE interference, is almost the same as that achieved byZMI+SZF, which forces both the MBS interference and inter-SUE interference to zero. This can be explained as the inter-SUE interference enhancement in MZF could still compensatesuch MBS interference. However, as NM becomes larger than40 and the MBS interference becomes too severe, the formercannot compensate the latter and the MZF’s performance dete-riorates. The zero-forcing the strong MBS interference comesinto fruition, making ZMI+SZF easily outperform MZF.

C. Mixed-coupled HetNets

In this scenario, the MBS interference to the SUEs is strongonly for the half, which is located sufficiently near to theMBS, and is weak for the other half, which is located far wayfrom the MBS. From the previous results, it is expected thatAZMI+SZF, which forces only the strong MBS interferenceto zero and ignores the weak MBS interference in the EE max-imization problem (58), will be efficient. Fig. 7 confirms thisintuition. Interestingly, ZMI+SZF and AZMI+SZF achievetheir best EE at NM = 50, where their performance gap isclearly visualized.

In summary, the best beamforming strategy is to ignore theinterference when it is weak, enhance it when it is medium-strong and cancel it when it is strong. The weak interference

12

does not only make obtaining CSI difficult, but is not neededfor optimization either.

40 50 60 70 80 90 1002

2.5

3

3.5

4


EE

(bi

ts/J

oule

/Hz)

MZF+SZFMZF+CoZFZMI+SZFZMI+CoZFAZMI+SZFAZMI+CoZFEE ZF

Fig. 7: The EE performance vs. the number of MBS antennas in mixed-coupled HetNets.QoS rk = r(s,`) ≡ 4 Mbps.

D. HetNet EE vs. massive MIMO EE

To have an appropriate setting for EE ZF in the EE opti-mization problem (62), the number Ns of each SBS antennasis set to 6. The effectiveness of HetNets is demonstrated bycomparing its EE performance with that achieved by a massiveMIMO with a MBS equipped with NM +

∑Ss=1Ns antennas

to serve M +∑Ss=1Ks users in the two following schemes:

• Optimal power allocation for zero-forcing beamformingreferred as to MBS PA:

maxp

∑M+∑S

s=1Ks

k=1 ln(1 + βkp

2k/σ

2k

)πmbs(p)

(67a)

s.t.

M+∑S

s=1Ks∑k=1

p2k ≤ Pmax

M , (67b)

ln(1 + βkp

2k/σ

2k

)≥ rk, (67c)

k = 1, . . . , (M +

S∑s=1

Ks),

which is solved by the same Dinkelbach’s type algorithmproposed in Section V.

• Equal power allocation for zero-forcingbeamforming referred as to MBS EPA with the

EE [

M+∑S

s=1Ks∑k=1

ln(1 + βkp

2e/σ

2k

)]/πmbs,EPA, where

pe =√PmaxM /(M +

∑Ss=1Ks) is the equal power

allocation to all UE and πmbs,EPA = αPmaxM + PMcir.

In simulations, the proposed Dinkelbach’s type algorithmconverges within 10 iterations to the optimal solutions in allsolved problems.

Fig. 8 shows the significant benefit of using HetNets insteadof massive MIMO for the system EE. It can be seen that theEE in massive MIMO is sensitive to the number of users,which are near to the BS (near users). More near users leadto a better EE in massive MIMO. There are many near usersin the massive MIMO corresponding to the strongly coupled

HetNets. The number of near users in the massive MIMOcorresponding to the mixed-coupled HetNets is more than thatin the massive MIMO corresponding to the weakly couplesHetNets. On the other hand, the EE in HetNets is dependenton both number of near MUEs and degree of the MBSinterference to SUEs. For this reason, the EE is achieved bestin the weakly HetNets, second best in the strongly coupledHetNets, and last in the mixed-coupled HetNets. Comparingto the mix-coupled HetNets, the MBS interference is strongerbut the number of near MUEs is more so the former stillachieve a better EE than the latter.

70 80 90 100 110 120 130 1400.5

1

1.5

2

2.5

3

3.5

4

4.5

Total number of antennas

EE

(bi

ts/J

oule

/Hz)

EE ZFMassiveMIMO_PAMassiveMIMO_EPA

(a) Weakly coupled HetNet

70 80 90 100 110 120 130 1400.5

1

1.5

2

2.5

3

3.5

4


EE

(bi

ts/J

oule

/Hz)


(b) Strongly coupled HetNet

70 80 90 100 110 120 130 1400.5

1

1.5

2

2.5

3

3.5

4


EE

(bi

ts/J

oule

/Hz)


(c) Mixed-coupled HetNet

Fig. 8: The EE performance vs. the number NM of MBS antennas from 40 to 100antennas and fixed antennas per SBS with 6 SBSs. The throughput threshold per UE is4 Mbps.

13

VII. CONCLUSIONS

We have considered various classes of beamforming inHetNets to optimize their energy-efficiency, which is expressedas the ratio of the sum throughput and consumed power. Theseproblems have been formulated as maximizations of highlydifficult fractional functions, subject to nonconvex constraintsfor user QoS satisfaction, and were solved by the proposedpath-following algorithms. Numerical examples have shownthe efficiency of these algorithms. More importantly, they haveshown that in contrast to maximizing the spectral efficiency,which suggests using as many antennas as possible, the EEdrops very quickly when this number exceeds 50, which isquite small in the massive MIMO context. HetHets exhibitsuperior performance in terms of EE when compared withmassive MIMO, for a given number of antennas.

APPENDIX: FUNDAMENTAL INEQUALITIES

We exploit the fact that the function f(x, t) = ln(1+1/x)t is

convex in x > 0, t > 0 which can be proved by examining itsHessian. The following inequality for all x > 0, x > 0, t > 0and t > 0 then holds true [34]:

ln(1 + 1/x)

t≥ f(x, t) + 〈∇f(x, t), (x, t)− (x, t)〉

= 2ln(1 + 1/x)

t+

1

t(x+ 1)

− x

(x+ 1)xt− ln(1 + 1/x)

t2t, (68)

where ∇ is the gradient operation.By replacing 1/x→ x and 1/x→ x in (68), we have

ln(1 + x)

t≥ a− b

x− ct, (69)

where

a = 2 ln(1+x)t + x

t(x+1) > 0, b = x2

t(x+1) > 0,

c = ln(1+x)t2 > 0.

By replacing |x|2 → x and |x|2 → x in (69) we have

ln(1 + |x|2)

t≥ a− b

|x|2− ct

≥ a− b

2<{xx∗} − |x|2− ct (70)


2<{xx∗} − |x|2 > 0, (71)

wherea = 2 ln(1+|x|2)

t + |x|2t(|x|2+1) > 0,

b = |x|4t(|x|2+1) > 0,

c = ln(1+|x|2)t2 > 0.

Finally, we also have the following inequality

x2

t≥ 2

xx

t− x2

t2t ∀ x > 0, x > 0, t > 0, t > 0. (72)

REFERENCES

[1] F. Rusek et al., “Scaling up MIMO: Opportunities and challenges withvery large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60,Jan. 2013.

[2] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “MassiveMIMO for next generation wireless systems,” IEEE Commun. Mag.,vol. 52, no. 2, pp. 186–195, Feb. 2014.

[3] J. Hoydis, M. Kobayashi, and M. Debbah, “Green small-cell networks,”IEEE Veh. Technol. Mag., vol. 6, no. 1, pp. 37–43, Jan. 2011.

[4] V. Jungnickel et al, “The role of small cells, coordinated multipoint, andmassive MIMO in 5G,” IEEE Commun. Mag., vol. 52, no. 5, pp. 44–51,May 2014.

[5] J. Hoydis, K. Hosseini, S. t. Brink, and M. Debbah, “Making smart useof excess antennas: Massive MIMO, small cells, and TDD,” Bell LabsTechnical J., vol. 18, no. 2, pp. 5–21, Apr. 2013.

[6] L. Sanguinetti, A. L. Moustakas, and M. Debbah, “Interference manage-ment in 5G reverse TDD HetNets with wireless backhaul: a large systemanalysis,” IEEE J. Sel. Areas. Commun., vol. 33, no. 6, pp. 1187–1200,Jun. 2015.

[7] A. Ghosh et al, “Heterogeneous cellular networks: from theory topractice,” IEEE Commun. Mag., vol. 50, no. 6, pp. 54–64, Jun. 2012.

[8] H. S. Dhillon, M. Kountouris, and J. G. Andrews, “Downlink MIMOHetNets: Modeling, ordering results and performance analysis,” IEEETrans. Wireless Commun., vol. 12, no. 10, pp. 5208–5222, Oct. 2013.

[9] A. Adhikary, H. S. Dhillon, and G. Caire, “Massive-MIMO MeetsHetNet: Interference coordination through spatial blanking,” IEEE J.Sel. Areas. Commun., vol. 33, no. 6, pp. 1171–1186, Jun. 2015.

[10] E. Bjornson, M. Kountouris, and M. Debbah, “Massive MIMO and smallcells: improving energy efficiency by optimal soft-cell coordination,” inICT, May 2013, pp. 1–5.

[11] N. Bhushan et al, “Network densification: the dominant theme forwireless evolution into 5G,” IEEE Commun. Mag., vol. 52, no. 2, pp.82–89, Feb. 2014.

[12] J. G. Andrews, X. Zhang, G. D. Durgin, and A. K. Gupta, “Are weapproaching the fundamental limits of wireless network densification?”IEEE Commun. Mag., vol. 54, no. 10, pp. 184–190, Oct. 2016.

[13] A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok, “The globalfootprint of mobile communications: The ecological and economicperspective,” IEEE Commun. Mag., vol. 49, no. 8, pp. 55–62, Aug. 2011.

[14] GreenTouch Green Meter Research Study, “Reducing the net energyconsumption in communications networks by up to 90% by 2020(version 1.0),” https://s3-us-west-2.amazonaws.com/belllabs-microsite-greentouch/, Jun. 2013.

[15] C. Isheden, Z. Chong, E. Jorswieck, and G. Fettweis, “Framework forlink level energy efficiency optimization with informed transmitter,”IEEE Trans. Wireless Commun., vol. 11, no. 8, pp. 2946–2957, Aug.2012.

[16] R. Cavalcante, S. Stanczak, M. Schubert, A. Eisenlatter, and U. Turke,“Toward energy-efficienct 5G wireless communications technologies,”IEEE Signal Process. Mag., vol. 13, no. 11, pp. 24–34, Nov. 2014.

[17] A. Zappone, L. Sanguinetti, G. Bacci, E. Jorswieck, and M. Debbah,“Energy-efficient power control: A look at 5G wireless technologies,”IEEE Trans. Signal Process., vol. 64, no. 7, pp. 1668–1683, Jul. 2016.

[18] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral effi-ciency of very large multiuser MIMO systems,” IEEE Trans. Commun.,vol. 61, no. 4, pp. 1436–1449, Apr. 2013.

[19] E. Bjornson, L. Sanguinetti, and M. Kountouris, “Deploying densenetworks for maximal energy efficiency: Small cells meet massiveMIMO,” IEEE J. Sel. Areas. Commun., vol. 34, no. 4, pp. 832–847,Apr. 2016.

[20] W. Dinkelbach, “On nonlinear fractional programming,” ManagementScience, vol. 13, no. 7, pp. 492–498, Jul. 1967.

[21] S. He, Y. Huang, H. Wang, S. Jin, and L. Yang, “Leakage-aware energy-efficient beamforming for heterogeneous multicell multiuser systems,”IEEE J. Sel. Areas. Commun., vol. 32, no. 6, pp. 1268–1281, Jun. 2014.

[22] H. H. Kha, H. D. Tuan, and H. H. Nguyen, “Fast global optimal powerallocation in wireless networks by local D.C. programming,” IEEETrans. Wireless Commun., vol. 11, no. 2, pp. 510–515, Feb. 2012.

[23] R. Ramamonjison and V. K. Bhargava, “Energy efficiency maximiza-tion framework in cognitive downlink two-tier networks,” IEEE Trans.Wireless Commun., vol. 14, no. 3, pp. 1468–1479, Mar. 2015.

[24] J. Tang, D. K. C. So, E. Alsusa, K. A. Hamdi, and A. Shojaeifard,“Resource allocation for energy efficiency optimization in heterogeneousnetworks,” IEEE J. Sel. Areas. Commun., vol. 33, no. 10, pp. 2104–2117,Oct 2015.

[25] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Massive MU-MIMOdownlink tdd systems with linear precoding and downlink pilots,” inProc. IEEE Allerton Conf., Oct. 2013, pp. 293–298.

14

[26] N. Saquib, E. Hossain, L. B. Le, and D. I. Kim, “Interference man-agement in OFDMA femtocell networks: issues and approaches,” IEEEWireless Commun., vol. 19, no. 3, pp. 86–95, Jun. 2012.

[27] A. Abdelnasser, E. Hossain, and D. I. Kim, “Clustering and resourceallocation for dense femtocells in a two-tier cellular ofdma network,”IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1628–1641, Mar.2014.

[28] D. Gesbert, S. Hanly, H. Huang, S. S. Shitz, O. Simeone, and W. Yu,“Multi-cell MIMO cooperative networks: A new look at interference,”IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp. 1380–1408, Dec. 2010.

[29] A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precoding via conicoptimization for fixed MIMO receivers,” IEEE Trans. Signal Process.,vol. 54, no. 1, pp. 161–176, Jan. 2006.

[30] B. R. Marks and G. P. Wright, “A general inner approximation algorithmfor nonconvex mathematical programs,” Operation Research, vol. 26,no. 4, pp. 681–683, Apr. 1978.

[31] G. Liu, M. Sheng, X. Wang, W. Jiao, Y. Li, and J. Li, “Interferencealignment for partially connected downlink MIMO heterogeneous net-works,” IEEE Trans. Commun., vol. 63, no. 2, pp. 551–564, Feb 2015.

[32] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K.Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. AreasCommun., vol. 32, no. 6, pp. 1065–1082, Jun. 2014.

[33] L. D. Nguyen, H. D. Tuan, and T. Q. Duong, “Energy-efficient signallingin QoS constrained heterogeneous networks,” IEEE Access, vol. 4, pp.7958–7966, 2016.

[34] H. Tuy, Convex Analysis and Global Optimization (second edition).Springer, 2016.

Long D. Nguyen was born in Dongnai, Vietnam. Hereceived the B.S. degree in electronics and telecom-munication engineering and the M.S. degree intelecommunication engineering from Ho Chi MinhCity University of Technology (HCMUT), Vietnam,in 2013 and 2015, respectively. He is currentlywith Queens University Belfast, as a PhD student.His research interests include convex optimizationtechniques, heterogeneous network, relay networks,and massive MIMO.

Hoang Duong Tuan received the Diploma (Hons.)and Ph.D. degrees in applied mathematics fromOdessa State University, Ukraine, in 1987 and 1991,respectively. He spent nine academic years in Japanas an Assistant Professor in the Department ofElectronic-Mechanical Engineering, Nagoya Univer-sity, from 1994 to 1999, and then as an AssociateProfessor in the Department of Electrical and Com-puter Engineering, Toyota Technological Institute,Nagoya, from 1999 to 2003. He was a Professor withthe School of Electrical Engineering and Telecom-

munications, University of New South Wales, from 2003 to 2011. He iscurrently a Professor with the School of Electrical and Data Engineeringand a core member of Global Big Data Technologies Centre, Universityof Technology Sydney. He has been involved in research with the areasof optimization, control, signal processing, wireless communication, andbiomedical engineering for more than 20 years.

Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Swedenin 2012. Currently, he is with Queen’s UniversityBelfast (UK), where he was a Lecturer (AssistantProfessor) from 2013 to 2017 and a Reader (As-sociate Professor) from 2018. His current researchinterests include Internet of Things (IoT), wirelesscommunications, molecular communications, andsignal processing. He is the author or co-author of290 technical papers published in scientific journals

(165 articles) and presented at international conferences (125 papers).Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IET COMMUNICATIONS, and a Lead Senior Editor for IEEECOMMUNICATIONS LETTERS. He was awarded the Best Paper Award atthe IEEE Vehicular Technology Conference (VTC-Spring) in 2013, IEEEInternational Conference on Communications (ICC) 2014, IEEE GlobalCommunications Conference (GLOBECOM) 2016, and IEEE Digital SignalProcessing Conference (DSP) 2017. He is the recipient of prestigious RoyalAcademy of Engineering Research Fellowship (2016-2021) and has won aprestigious Newton Prize 2017.

Octavia A. Dobre (M’05-SM’07) received the Dipl.Ing. and Ph.D. degrees from Politehnica Univer-sity of Bucharest (formerly Polytechnic Institute ofBucharest), Romania, in 1991 and 2000, respec-tively. She was a Royal Society Scholar in 2000and a Fulbright Scholar in 2001. Between 2002and 2005, she was with Politehnica University ofBucharest and New Jersey Institute of Technology,USA. In 2005, she joined Memorial University,Canada, where she is currently a Professor andResearch Chair. She was a Visiting Professor with

Universit de Bretagne Occidentale, France, and Massachusetts Institute ofTechnology, USA, in 2013. Her research interests include 5G enablingtechnologies, blind signal identification and parameter estimation techniques,cognitive radio systems, network coding, as well as optical and underwatercommunications among others. Dr. Dobre serves as the Editor-in-Chief of theIEEE COMMUNICATIONS LETTERS, as well as an Editor of the IEEESYSTEMS and IEEE COMMUNICATIONS SURVEYS AND TUTORIALS.She was an Editor and a Senior Editor of the IEEE COMMUNICATIONSLETTERS, an Editor of the IEEE TRANSACTIONS ON WIRELESS COM-MUNICATIONS and a Guest Editor of other prestigious journals. She servedas General Chair, Tutorial Co-Chair, and Technical Co-Chair at numerousconferences. She is the Chair of the IEEE ComSoc Signal Processing andCommunications Electronics Technical Committee, as well as a Member-at-Large of the Administrative Committee of the IEEE Instrumentation andMeasurement Society. Dr. Dobre is a Fellow of the Engineering Institute ofCanada.

H. Vincent Poor (S’72, M’77, SM’82, F’87) re-ceived the Ph.D. degree in EECS from PrincetonUniversity in 1977. From 1977 until 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the facultyat Princeton, where he is currently the MichaelHenry Strater University Professor of Electrical En-gineering. During 2006 to 2016, he served as Deanof Princetons School of Engineering and AppliedScience. He has also held visiting appointments atseveral other universities, including most recently at

Berkeley and Cambridge. His research interests are in the areas of informationtheory and signal processing, and their applications in wireless networks,energy systems and related fields. Among his publications in these areas isthe recent book Information Theoretic Security and Privacy of InformationSystems (Cambridge University Press, 2017).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and is a foreign member of the ChineseAcademy of Sciences, the Royal Society, and other national and internationalacademies. He received the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognitionof his work includes the 2017 IEEE Alexander Graham Bell Medal, HonoraryProfessorships at Peking University and Tsinghua University, both conferredin 2017, and a D.Sc. honoris causa from Syracuse University also awardedin 2017.

downlink beamforming for energy-efficient heterogeneous ...€¦ · antennas, hetnet is more...

Documents