Content uploaded by Tuan Hoang
Author content
All content in this area was uploaded by Tuan Hoang on Jun 12, 2020
Content may be subject to copyright.
1
Downlink Beamforming for Energy-Efficient
Heterogeneous Networks with Massive MIMO and
Small Cells
Long D. Nguyen, Hoang D. Tuan, Trung Q. Duong, Octavia A. Dobre and H. Vincent Poor
Abstract—A heterogeneous network (HetNet) of a macro-cell
base station equipped with a large-scale antenna array (massive
MIMO) overlaying a number of small cell base stations (small
cells) can provide high quality of service (QoS) to multiple users
under low transmit power budget. However, the circuit power for
operating such a network, which is proportional to the number
of transmit antennas, poses a problem in terms of its energy
efficiency. This paper addresses the beamforming design at the
base stations to optimize the network energy efficiency under QoS
constraints and a transmit power budget. Beamforming tailored
for weak, strong and medium cross-tier interference HetNets
is proposed. In contrast to the conventional transmit strategy
for power efficiency in meeting the users’ QoS requirements,
which suggests the use of a few hundred antennas, it is found
out that the overall network energy efficiency quickly drops if
this number exceeds 50. It is found that, for a given number of
antennas, HetNet is more energy-efficient than massive MIMO
when considering overall energy consumption.
Index Terms—Heterogeneous networks, massive MIMO, small
cell, beamformer design, energy efficiency, optimization
I. INTRODUCTION
Massive MIMO [1], [2] and small cell networks [3] are
presently envisioned as two key technologies of the emerging
generation of communication networks (5G) to support a
1000-fold increase in network capacity. Since each of these
technologies 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 attracted
considerable research interest [5], [6]. In such heterogeneous
networks (HetNets), the small cell base stations (SBSs) serve
static and low mobility users (SUEs) to explore their proximity
to these users, while the massive MIMO base station (MBS)
serves higher mobility users (MUEs) to explore its high
coverage area and favored channel characteristics. A main
issue of HetNets is to manage both intra-tier interferences
This work was supported in part by the Australian Research Councils
Discovery Projects under Project DP130104617, in part by the U.K. Royal
Academy of Engineering Research Fellowship under Grant RF1415/14/22
and U.K. Engineering and Physical Sciences Research Council under Grant
EP/P019374/1, and in part by the U.S. National Science Foundation under
Grants CNS-1456793 and ECCS-1343210
Long D. Nguyen and Trung Q. Duong are with Queen’s University, Belfast
BT7 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:
Tuan.Hoang@uts.edu.au)
Octavia A. Dobre is with Faculty of Engineering and Applied Science,
Memorial University, NL A1B 3X5, Canada (email: odobre@mun.ca)
H. Vincent Poor is with the Department of Electrical Engineering, Princeton
University, Princeton, NJ 08544, USA (e-mail: poor@princeton.edu).
and cross-tier interference between the MBS and SBSs [7],
[8]. In [9], the SBSs were proposed to be turned-off if they
cause an excessive interference to the MBS. Minimization of
the downlink transmit power by multiflow-regularized-zero-
forcing beamforming subject to users’ QoS constraints was
considered in [10], which involves a large-scale semi-definite
program of dramatically high computational complexity. The
so-called reserve time-division-duplexing of MBS operating
in downlink mode while the SBSs operate in uplink mode
and vice versa was proposed in [6]. Being free of cross-
tier interference, both MBS and SBSs when in downlink are
supposed to exploit the cross-tier channel state information
(CSI) in suppressing their inter-link interference.
With the irreversible trend of network densification in 5G
and beyond [11], [12], a natural concern is its consumed power
[13]. To meet the requirement of 1000-fold energy efficiency
for new technologies [14], the energy efficiency (EE) in terms
of the ratio between the information throughput and consumed
power has been introduced as a new figure of merit in assessing
communication systems (see, e.g., [15]–[17] and references
therein). While achieving lower transmit power in offering
better QoS with using more antennas, it should be realized that
both MBS and SBSs then consume more circuit powers, which
are proportional to the number of their antennas. Since the
large-scale analysis [6], [18], which is solely based on arbitrary
large numbers of antennas and thus does not control the
consumed power, does not readily apply in the EE context, the
problem of determining the numbers of base stations, antennas
and users in uplink to improve the EE was considered in [19].
As surveyed in [17], so far the main tool for addressing the EE
maximization problems is Dinkelbach’s procedure of fractional
programming [20], even though the objective functions are
no longer ratios of concave and convex functions; hence in
this case each Dinkelbach’s iteration invokes solution of a
difficult nonconvex optimization problem, which is not easier
than the original optimization problem. Two separated energy-
efficient beamforming problems were considered in [21]. The
first problem is energy-efficient MBS beamforming under
constrained interference to the SBSs’ users, while the second
problem is the energy-efficient SBS beamforming ignoring the
interference from the MBSs. Each stationary point computed
in 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
problem arisen in each Dinkelbach’s iteration for the SBSs
under constrained interference to the MBS-tier users. Each
2
Dinkelbach’s iteration in joint power allocation and remote
radio head (RRH)/high-power node (HPN) association for
heterogeneous cloud radio access networks (H-CRAN) invokes
computation of a difficult mixed-integer optimization problem.
Each Dinkelbach’s iteration in HetNets with fixed service rate
constraints [24] invokes computation of a very difficult mixed-
combinatorial and nonconvex optimization problem, which is
then addressed by semi-definite relaxation.
This paper considers a HetNet of an MBS equipped with a
large antenna array overlaying multiple SBSs in serving both
MUEs and SUEs. The aim is beamforming design at both the
MBS 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 fractional
functions subject to convex constraints. Avoiding Dinkelbach’s
computationally inefficient iterations, path-following compu-
tational procedures are developed; these invoke computation
of a simple convex program to generate a better feasible point
and at least converge to a locally optimal solution. Simulations
under different scenarios show that it is important to control
the 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 the
paper that indeed underlaid small cells are an effective tool
for 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 special
class of the ZF MBS and SBS beamforming, with a different
solution method, is treated in Section V. Simulation results are
presented in Section VI, which is followed by the Conclusions.
Some fundamental inequalities used in the paper are provided
in the Appendix.
Notation. Boldface uppercase and lowercase letters denote
matrices and vectors, respectively. [x]+,max{0, x}for a
scalar x. The transpose and conjugate transpose of a matrix
X
X
Xare respectively represented by X
X
XTand X
X
XH.I
I
Iand 0
0
0stand
for identity and zero matrices of appropriate dimensions. Tr(.)
is the trace operator. ||x
x
x|| is the Euclidean norm of a vector x
x
x
and ||X
X
X|| is the Frobenius norm of a matrix X
X
X. A complex-
valued Gaussian random vector with mean ¯
x
x
xand covariance
R
R
Rx
x
xis denoted by x
x
x∼ CN(¯
x
x
x,R
R
Rx
x
x). For matrices X
X
X1, . . . , X
X
Xk
of appropriate dimension, denote by [X
X
X1;...;X
X
Xk]the matrix
[X
X
XT
1. . . X
X
XT
k]T.
II. PRO BL EM S TATEM EN T FO R HET NET S
Consider a HetNet of an MBS of a large-scale NMantenna
array with NMup to several hundred and SSBSs, which
are referred as SBS 1, ...., SBS S. Each SBS sis equipped
with Nsantennas. The MBS serves Mdownlink MUEs, while
SBS sserves Ksdownlink SUEs within its cell. All users are
equipped with a single antenna. For convenience, denote by
KM={1, . . . , M }the set of the MUEs and by {(s, `)|`∈
Ks,{1, . . . , Ks}} the set of those SUEs that are served
by SBS s. As Fig. 1 shows, in sharing the same spectrum at
the same time, the MBS interferes to all SUEs (s, `), while
TABLE I: Summation of used notations
Notation Description
NMnumber of MBS antennas
Nsnumber of SBS antennas
Mnumber of users served by MBS (MUEs)
Ksnumber of users served by SBS s(SUEs)
KM{1,...,M}
Ks{1,...,Ks}
kan MUE
(s, `)an SUE served by SBS s
Isset of MUEs interfered by SBS s
Nkset of SBS interfering to MUE k
√βkhkchannel from MBS to MUE k
pβs,`χs,` channel from MBS to SUE (s, `)
hs,` channel from SBS sto SUE (s, `)
fkMBS beamforming vector for MUE k
fs,` SBS s’ beamforming vector for SUE (s, `)
FMset of MBS beamforming vectors
Fsset of SBS s’ beamforming vectors
FS{Fs, s = 1,...,S}
FNk{Fs, s ∈ Nk}
F{FM,FS}(set of all beamforming vectors)
˜σmui
k(FM)inter-MUE interference to MUE k
˜σs,`(Fs)inter-SUE interference to SUE (s, `)
˜σmbi
s,` (FM)MBS interference to SUE (s, `)
˜σsbi
k(FNk)SBS interference to MUE k
the SBSs interfere to those MUEs in their coverage range.
Accordingly, Iswith cardinality Isis defined as the set of
those MUEs that are interfered by SBS sand Nkis the set of
those SBSs that interfere to MUE k.
Fig. 1: Illustration to the HetNet. The dash lines denote the interference signals and
interference area for SBSs.
Similar to [1], [25] and [18], we will exploit the fol-
lowing structure of the massive MIMO channel from the
MBS to the MUEs: √βkhk, and to the SUEs: pβs,`χ
χ
χs,`,
where √βkand pβs,` model the path loss and large-scale
fading from the MBS to MUE kand SUE (s, `), while
hk= (h1,k, ..., hNM,k )Twith hmk ∈ CN(0,1) and χ
χ
χs,` =
(χ1,s,`, . . . , χNM,s,` )Twith χj,s,` ∈ CN(0,1) represent the
small-scale fading.
The complex baseband signal received by MUE kis
yk=pβkh
h
hH
kf
f
fkxk
| {z }
desired signal
+X
i∈KM\{k}pβkh
h
hH
kf
f
fixi
| {z }
inter-MUE (co-tier) interference
+X
s∈Nk
Ks
X
j=1
η
η
ηH
s,kf
f
fs,j xs,j
| {z }
SBS (cross-tier) interference
+nk,(1)
3
where fk∈CNMand xkare the beamforming vector and
information from the MBS intended to MUE k, respectively,
η
η
ηs,k is the channel vector from SBS s∈ Nkto MUE k,
fs,` ∈CNsand xs,` are beamforming vector and information
from SBS sintended for SUE (s, `), and nk∼ CN(0, σ2
k)is
the additive white Gaussian noise at MUE k.
The complex baseband signal received by SUE (s, `)is 1
ys,` =h
h
hH
s,`f
f
fs,`xs,`
|{z }
desired signal
+
M
X
i=1 pβs,`χ
χ
χH
s,`f
f
fixi
| {z }
MBS (cross-tier) interference
+X
j0∈Ks\{`}
h
h
hH
s,`f
f
fs,j0xs,j0
| {z }
inter-SUE (co-tier) interference
+ns,`,(2)
where hs,` ∈CNsis 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 the
inter-MUE and inter-SUE interference functions defined as
eσmui
k(FM),βkX
i∈KM\{k}|h
h
hH
kf
f
fi|2, k = 1, . . . , M, (3)
and
eσsui
s,`(Fs),X
j∈Ks\{`}|h
h
hH
s,`f
f
fs,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 interference
functions defined as
eσmbi
s,` (FM),βs,`
M
X
i=1 |χ
χ
χH
s,`f
f
fi|2(5)
and
eσsbi
k(FNk),X
s∈Nk
Ks
X
j=1 |η
η
ηH
s,kf
f
fs,j |2, k = 1, . . . , M, (6)
respectively.
The information throughputs at MUE kand SUE (s, `)(in
nats) are
rk(FM,FNk) = ln 1 + βk|h
h
hH
kf
f
fk|2
eσmui
k(FM) + eσsbi
k(FNk) + σ2
k(7)
1Note that in (2) we assume that the small cells are sufficiently far apart
from each other so that the inter-small-cell interference can be ignored.
This is also true for dense small cells where orthogonal frequency division
multiplexing is used to allow sufficiently far apart from each other cells to
use the same carrier [26], [27].
and
rs,`(Fs,FM) = ln 1 + |h
h
hH
s,`f
f
fs,`|2
eσsui
s,`(Fs) + eσmbi
s,` (FM) + σ2
s,` !.
(8)
The entire consumed power for the downlink transmission can
be expressed as
P(F) = Pmbs(FM) +
S
X
s=1
Ps(Fs),(9)
where
Pmbs(FM) = α
M
X
k=1 ||f
f
fk||2+MPa+Pc(10)
is the power consumed by the MBS and
Ps(Fs) = αs
Ks
X
`=1 ||f
f
fs,`||2+NsPa,s +Pc,s (11)
is the power consumed by SBS s. There, α > 1and αs>1
are the reciprocal of the drain efficiency of the amplifier of
the MBS and SBS s,Paand Pa,s represent the per-antenna
circuit power of the MBS and SBS s, and Pcand Pc,s are the
non-transmission power of the MBS and SBS s, respectively.
Accordingly, the total SBSs consumed power is
Psbs(FS) =
S
X
s=1
Ps(Fs).
The EE maximization problem under QoS constraints and
power budget is formulated as
max
FPM
k=1 rk(FM,FNk) + PS
s=1 PKs
`=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
X
k=1 ||fk||2≤Pmax
M,(12d)
Ks
X
`=1 ||f
f
fs,`||2≤Pmax
s, 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 MBS
and SBSs under predefined budgets.
The paper follows the network centric techniques like
Cloud-RAN and cooperative multipoint (CoMP) [28], where
the MBS and SBSs cooperate to solve the EE maximization
problem (12) in a central manner. The conventional assumption
is that full channel state information (CSI) is available for the
optimization of problem (12).
The above problem is very complicated due the presence
of both intra-tier and cross-tier interferences, and the large
dimension NMof the MBS beamforming vectors fk∈CNM.
4
One can see from (1) and (2) that the MBS contributes a severe
interference to all MUEs and SUEs.
The next sections propose computational solutions for (12)
by different classes of MBS and SBS beamforming.
III. ZERO -FO RC IN G IN TE R-MUE INTERFERENCE BASED
BEAMFORMING (MZF)
For H= [h1h2... hM]∈CNMxM, which is very tall due
to NM>> M, there is the right inverse of the fat matrix
HH= [hH
1;hH
2... ;hM]∈CMxNMdefined by
¯
F
F
FM=hf
f
f1. . . f
f
fMi=H
H
H(H
H
HHH
H
H)−1,(13)
i.e.
I
I
I=H
H
HH¯
F
F
FM= [h
h
hH
1¯
F
F
FM;. . . ;h
h
hH
M¯
F
F
FM]
= [h
h
hH
if
f
fj](i,j)∈KM×KM.(14)
which means that h
h
hH
if
f
fi= 1 and h
h
hH
if
f
fj= 0 for i6=j. Using
the normalized vectors ˜
f
f
fk,f
f
fk/kf
f
fkk, k = 1, ..., M , the
MBS beamforming vector fkis sought in the class of
fk=pk˜
f
f
fk, k = 1, . . . , M (15)
to cancel the inter-MUE interference ˜σmui
k(FM)in (3):
h
h
hH
k˜
f
f
fi=h
h
hH
k¯
f
f
fi/||¯
f
f
fi|| = 0 for i6=k.
For p= (p1, ..., pM)T∈RMand
¯
βk,βk|h
h
hH
k˜
f
f
fk|2,(16)
the information throughput (in nats) for MUE kin (7) is
rk(pk,FNk) = ln(1 + p2
k¯
βk
σ2
k+ ˜σsbi
k(FNk)),(17)
with the SBS interference ˜σsbi
k(FNk)defined from (6), while
the consumed power by the MBS transmission defined by (9)
is now a quadratic form of p:
πmbs(p) = α
M
X
k=1
p2
k+MPa+Pc.(18)
The power constraint (12d) is now
M
X
k=1
p2
k≤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 + |hH
s,`f
f
fs,`|2
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,`
),(20)
where
σmbi
s,` (p),βs,`
M
X
i=1
p2
i||χ
χ
χH
s,` ˜
f
f
fi||2(21)
is the MBS interference function (see (5)), and ˜σsui
s,`(Fs)is the
inter-SUE interference function defined from (4).
Under the class of MZF, the EE maximization problem (12)
is now expressed by
max
p,FS
Φ(p,FS),
M
X
k=1
ln(1 + p2
k¯
βk
σ2
k+ ˜σsbi
k(FNk))
+
S
X
s=1
Ks
X
`=1
ln(1 + |hH
s,`f
f
fs,`|2
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,`
)!/
(πmbs(p) + Psbs (FS))
s.t.(12e),(19),(22a)
ln 1 + p2
k¯
βk
σ2
k+ ˜σsbi
k(FNk)≥¯rk, k = 1, . . . , M, (22b)
ln 1 + |hH
s,`f
f
fs,`|2
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,` !≥¯rs,`,
`= 1, . . . , Ks;s= 1, ..., S. (22c)
The nonconvex constraint (22b) is seen equivalent to the
following second-order cone (SOC) constraint
pkq¯
βk≥√e¯rk−1qσ2
k+ ˜σsbi
k(FNk), k = 1, . . . , M. (23)
As observed in [29], for ¯
f
f
fs,` ,e−.arg(h
h
hH
s,`f
f
fs,`)f
f
fs,`, one
has |h
h
hH
s,`f
f
fs,`|=h
h
hH
s,` ¯
f
f
fs,` =<{h
h
hH
s,` ¯
f
f
fs,`} ≥ 0in (20).
Therefore, |h
h
hH
s,`f
f
fs,`|2in (20) can be equivalently replaced by
(<{h
h
hH
s,`f
f
fs,`})2with <{h
h
hH
s,`f
f
fs,`} ≥ 0, ` = 1, . . . , Ns;s=
1, . . . , S. Consequently, the nonconvex constraint (22c) is also
equivalent to the SOC constraint
<{hH
s,`f
f
fs,`} ≥p(e¯rs,` −1)qσmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,`,
`= 1, . . . , Ks;s= 1, ..., S. (24)
Therefore, the EE maximization problem (22) is a nonconcave
function maximization under convex constraints. Our focus
now 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,p2
k¯
βk
σ2
k+ ˜σsbi
k(FNk), t ,πmbs (p) + Psbs(FS)
and
¯x=x(n)
k,(p(n)
k)2¯
βk
σ2
k+˜σsbi
k(F(n)
Nk),
¯
t=t(n),πmbs(p(n)) + Psbs (F(n)
S),
yields the following lower bounding approximation for the first
term in the objective function in (22a):
ln 1 + p2
k¯
βk
σ2
k+ ˜σsbi
k(FNk)
P(p,FS)≥
a(n)
k−b(n)
k
σ2
k+ ˜σsbi
k(FNk)
¯
βkp2
k−c(n)
k(πmbs(p) + Psbs (FS)) ≥
g(n)
k(p,FS)(25)
over the trust region
2pk−p(n)
k>0,(26)
5
for
g(n)
k(p,FS),a(n)
k−b(n)
k
σ2
k+ ˜σsbi
k(FNk)
¯
βkp(n)
k(2pk−p(n)
k)
−c(n)
k(πmbs(p) + Psbs (FS)),(27)
where
0< a(n)
k,2ln(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,`f
f
fs,`})2
σmbi
s,`(p)+ ˜σsui
s,`(Fs)+σ2
s,`
,
t,πmbs(p) + Psbs (FS)
and
¯x=x(n)
s,` ,(<{hH
s,`f
f
f(n)
s,` })2
σmbi
s,` (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 bounding
approximation:
ln 1 + (<{hH
s,`f
f
fs,`})2
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,` !/P (p,FS)≥
a(n)
s,` −b(n)
s,`
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,`
(<{hH
s,`f
f
fs,`})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,`
σmbi
s,` (p) + ˜σsui
s,`(Fs) + σ2
s,`
2<{hH
s,`f
f
f(n)
s,` }<{hH
s,`f
f
fs,`} − (<{hH
s,`f
f
f(n)
s,` })2
−c(n)
s,` (πmbs(p) + Psbs (FS)) (31)
over the trust region
2<{h
h
hH
s,`t
t
ts,`} ≥ <{h
h
hH
s,`t
t
t(n)
s,` }, ` = 1, . . . , Ns;s= 1, . . . , S,
(32)
where
0< a(n)
s,` ,2ln(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 solved
to generate the next feasible point (p(n+1),F(n+1)
S)for (22):
max
p,FS
Φ(n)(p,FS),[
M
X
k=1
g(n)
k(p,FS) +
S
X
s=1
Ks
X
`=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 convex
program:
min
p,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: Repeat
3: 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.
IV. OTHER SCHEMES
The MZF as given by (15) cancels only the inter-MUE
interference, under which the MBS interference ˜σmbi(FM)is
not controlled. In this section we consider other classes of
MBS and SBSs beamforming to enhance both cross-tier and
co-tier interferences in optimizing the EE of the system.
6
A. Zero forcing co-tier interference based beamforming
(MZF+SZF)
In this scheme, referred as MZF+SZF with SZF used as an
abbreviation to represent ”zero-forcing inter-SUE interference
based beamforming”, the MBS beamforming vector fkis
sought in the class of MZF, while the SBS beamforming vector
fs,` 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 sto its SUEs except that
to SUE (s, `). To nullify the inter-SUE interference in (4), the
following condition must be fulfilled:
HH
s,`f
f
fs,` = 0
0
0∈CKs−1, ` = 1, . . . , Ks,(42)
requiring
Ns> Ks.
Such fs,` is parametrized as
fs,` =G
G
Gs,`t
t
ts,`,(43)
where G
G
Gs,` ∈CNs×(Ns−Ks+1) is an orthogonal basis for the
null space of HH
s,` and t
t
ts,` ∈CNs−Ks+1. Consequently, the
information throughput at SUE (s, `)in (8) is
r(2)
s,` (ts,`) = ln 1 + |¯
hH
s,`t
t
ts,`|2/(σmbi
s,` (p) + σ2
s,`)(44)
with σmbi
s,` (p)defined in (21) and ¯
hs,` ,GH
s,`hs,` .
For Θ
Θ
Θ`,ks
,G
G
GH
s,`h
h
hs,ks∈CNs−Ks+1 and TNk
,[Ts]s∈Nk,
the SBSs interference to MUE kin (6) is
σsbi
k(TNk),eσsbi
k([Gs,`ts,` ]s∈Nk,`=1,...,Ks)
=X
s∈Nk
Ks
X
`=1 |Θ
Θ
ΘH
`,kst
t
ts,`|2.(45)
Now, recalling the definition (21) for the MBS interference to
the SUEs, the EE maximization problem is formulated as
max
p,T
M
X
k=1
ln(1 + p2
k¯
βk
σ2
k+σsbi
k(TNk))
π(p,T)
+
S
X
s=1
Ks
X
`=1
ln(1 + (<{¯
hH
s,`t
t
ts,`})2
σmbi
s,` (p) + σ2
s,`
)
π(p,T)(46a)
s.t.(19),
pkq¯
βk≥√e¯rk−1qσ2
k+σsbi
k(TNk), k = 1, . . . , M,
(46b)
<{¯
hH
s,`t
t
ts,`} ≥ p(e¯rs,` −1)qσmbi
s,` (p) + σ2
s,`,
`= 1, . . . , Ks;s= 1, ..., S, (46c)
Ks
X
`=1 ||t
t
ts,`||2≤Pmax
s, s = 1, ..., S. (46d)
Initialized from a feasible point (p(0),T(0) ), which is found
from the convex program
min
p,Tπ(p,T)s.t. (19),(46b),(46c),(46d)(47)
at the nth iteration the following convex program is solved to
generate the next iterative point (p(n+1),T(n+1))for (46):
max
p,T[
M
X
k=1
g(n)
k(p,T) +
S
X
s=1
Ks
X
`=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
σ2
k+σsbi
k(TNk)
¯
βkp(n)
k(2pk−p(n)
k)
−c(n)
kπ(p,T),(49)
with a(n)
k,b(n)
kand c(n)
kdefined from (28) for
x(n)
k,(p(n)
k)2¯
βk
σ2
k+σsbi
k(T(n)
Nk), t(n),π(p(n),T(n)),
and
f(n)
s,` (p,T),
a(n)
s,` −b(n)
s,`
σmbi
s,` (p) + σ2
s,`
2<{¯
hH
s,`t
t
t(n)
s,` }<{¯
hH
s,`t
t
ts,`} − (<{¯
hH
s,`t
t
t(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,` ,(<{¯
hH
s,`t
t
t(n)
s,` })2/σmbi
s,` (p(n))), t(n),π(p(n),T(n)).
Similar to Proposition 1, it can be easily shown that the
computational 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 solution
satisfying the KKT conditions.
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 an
abbreviation to represent ”zero-forcing inter-MUE and MBS
interferences based beamforming”, the MBS beamforming
vector fkis designed to force both inter-MUE interference
and MBS interference to zero, while the SBS beamforming
vector fs,` is parametrized by (43) in forcing the inter-SUE
interference to zero.
Define the interfering channels from the MBS to the SUEs
χ
χ
χs= [χ
χ
χs,`]`=1,...,Ks∈CNM×Ks,
χ
χ
χ,[χ
χ
χs]s=1,...,S ∈CNM×PS
s=1 Ks
and
H
H
Hmbs ,[H
H
H χ
χ
χ]∈CNM×(M+PS
s=1 Ks),
which is still a very tall as the total number M+PS
s=1 Ksof
users is still small compared to the number NMof the MBS’s
antennas. Then the right inverse of the fat matrix H
H
HH
mbs is
defined as
¯
F
F
Fmbs =h¯
g1. . . ¯
gM. . . ¯
gM+PS
s=1 Ksi
,H
H
Hmbs(H
H
HH
mbsH
H
Hmbs)−1(51)
∈CNM×(M+PS
s=1 Ks),
7
i.e., hHH
mbs ¯
g1. . . ¯
gMHH
mbs h¯
gM+1 . . . ¯
gM+PS
s=1 Ksii
=HH
mbs ¯
F
F
Fmbs
=I∈R(M+PS
s=1 Ks)×(M+PS
s=1 Ks).
Particularly,
H
H
HH
mbs [¯
g1. . . ¯
gM] = I
O∈C(M+PS
s=1 Ks)×M.
Using the normalized vectors
˜
fk,¯
gk/||¯
gk||, k = 1, ..., M, (52)
the MBS beamforming vector fkis sough in the class of (15) to
nullify the inter-MUE interference and the MBS interference
to the SUEs. Under the definition (16) for ¯
βkwith ˜
fkdefined
in (52) and the definition (43) for parametrizing beamforming
vectors fs,` of SZF, the EE maximization problem (12) is now
formulated as
max
p,T
M
X
k=1
ln 1 + p2
k¯
βk/(σsbi
k(TNk) + σ2
k)
π(p,T)
+
S
X
s=1
Ks
X
`=1
ln 1+(<{¯
h
h
hH
s,`t
t
ts,`})2/σ2
s,`
π(p,T)(53a)
s.t. (19),(46b),(46d),(53b)
<{¯
hH
s,`t
t
ts,`} ≥ p(e¯rs,` −1)σs,`,(53c)
`= 1, . . . , Ks;s= 1, ..., S.
Initialized from a feasible point (p(0),T(0) ), which is found
from the convex program
min
p,Tπ(p,T)s.t. (19),(46b),(46d),(53c)(54)
at the nth iteration the following convex program is solved to
generate the next iterative feasible point (p(n+1),T(n+1))for
(53):
max
p,T[
M
X
k=1
g(n)
k(p,T) +
S
X
s=1
Ks
X
`=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,`
σsui
s,`(Ts) + σ2
s,`
2<{¯
h
h
hH
s,`t
t
t(n)
s,` }<{¯
h
h
hH
s,`t
t
ts,`} − (<{¯
h
h
hH
s,`t
t
t(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,` ,(<{¯
h
h
hH
s,`t
t
t(n)
s,` })2
σsui
s,`(T(n)
s) + σ2
s,`
, t(n),π(p(n),T(n)).
Similar to Proposition 1, it can be easily shown that the
computational procedure that invokes the convex program
(55) to generate the next iterative point, is path-following for
(53), which at least converges to its locally optimal solution
satisfying the KKT conditions.
C. Adaptively suppressed co-interference based beamforming
(AZMI+SZF)
Denote by S1={1, ..., S1}the set of those SBSs that
are located sufficiently near to the MBS, and thus, their
SUEs are under the strong MBS interference, while denote by
S2={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 as
an abbreviation to represent ”adaptively zero-forcing MBS in-
terference based beamforming”, the SBS beamforming vector
fs,` is parametrized by (43) to force the inter-SUE interference
to zero. On the other hand, the MBS beamforming vector fk
is designed based on (15) with ˜
fkdefined in (52) with
¯
g1. . . ¯
gM. . . ¯
gM+PS1
s1=1 Ks1,
H
H
Hmbs,1(H
H
HH
mbs,1H
H
Hmbs,1)−1∈CNM×(M+PS1
s1=1 Ks1)(57)
and
χ
χ
χs1= [χ
χ
χs1,`]`=1,...,Ks1∈CNM×Ks1,
χ
χ
χ1,[χ
χ
χs1]s1=1,...,S1∈CNM×PS1
s1=1 Ks1,
H
H
Hmbs,1,[H
H
H χ
χ
χ1]∈CNM×(M+PS1
s1=1 Ks1),
to nullify the inter-MUE interference and the strong MBS
interference to SUEs (s1, `), s1∈ S1. The EE maximization
problem (12) becomes
max
p,T
M
X
k=1
ln(1 + p2
k¯
βk
σ2
k+σsbi
k(TNk))
π(p,T)
+
S
X
s=1
Ks
X
`=1
ln(1 + (<{¯
h
h
hH
s,`t
t
ts,`})2
σmbi
s,` (p) + σ2
s,`
)
π(p,T)(58a)
s.t.(19),(46d),(58b)
pkq¯
βk≥√e¯rk−1qσ2
k+σsbi
k(TNk), k = 1, . . . , M,
(58c)
<{¯
h
h
hH
s,`t
t
ts,`} ≥ σs,` pers,` −1,
`= 1, . . . , Ks;s= 1, ..., S1,(58d)
<{¯
hH
s,`t
t
ts,`} ≥ p(e¯rs,` −1)qσmbi
s,` (p) + σ2
s,`,
`= 1, . . . , Ks;s= 1, ..., S2.(58e)
Initialized from a feasible point, which is found from the
convex program
min
p,Tπ(p,T)s.t. (19),(46d),(58c),(58d),(58e),(59)
8
at the nth iteration, the following convex program is solved
to generate a feasible point (p(n+1),T(n+1) )for (58):
max
p,T
M
X
k=1
g(n)
k(p,T) +
S
X
s=1
Ks
X
`=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)
kand c(n)
k
from (28), for
x(n)
k,(p(n)
k)2¯
βk
σ2
k+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,` ,(<{¯
hH
s,`t
t
t(n)
s,` })2
σmbi
s,` (p(n)) + σ2
s,`
, t(n),π(p(n),T(n)).
Similar to Proposition 1, it can be easily shown that the
computational procedure that invokes the convex program
(60) to generate the next iterative point, is path-following for
(58), which at least converges to its locally optimal solution
satisfying the KKT conditions.
V. ENERGY-EFFIC IE NT Z ERO-FORCING HETNET
BEAMFORMING (EE ZF)
To show the advantage of HetNets over massive MIMO in
terms of the EE, in this section we address the EE maximiza-
tion problems in the class of zero-forcing beamforming at both
MBS and SBSs, i.e. the the BMS beamforming vector fkis
sought in the class of (15) with ˜
fkdefined from (51) and (52)
to cancel both inter-MUE and MBS interferences while the
SBS beamforming vector fs,` is also sought to cancel both
inter-SUE and SBS interferences as detailed below.
With Hs∈CNs×Isdefined from (41) and
Hsbs,s ,[ [hs,`]`=1,...,KsHs]∈CNs×(Ks+Is)
the right inverse of HH
sbs,sis
[¯
fs,1. . . ¯
fs,Ks. . . ¯
fs,Ks+Is],Hsbs,s(HH
sbs,sHsbs,s )−1.
Using the normalized vectors ˜
fs,` =¯
fs,`/||¯
fs,`||,`=
1, . . . , Ks, to cancel both inter-SUE and SBS interferences
the SUE beamformers fs,` is sought in the form
fs,` =ps,`˜
fs,`, ` = 1, . . . , Ns;s= 1, . . . , S. (61)
For ¯
βkdefined from (16) and ¯
βs,` ,|hH
s,`˜
fs,`|2, while ps,
(ps,`)s=1,...,S ;`=1,...,Ns, the EE maximization problem (12) is
thus
max
p,pSPM
k=1 ln 1 + ¯
βkp2
k/σ2
k
πmbs(p) + πsbs (pS)
+PS
s=1 PKs
`=1 ln 1 + ¯
βs,`p2
s,`/σ2
s,`
πmbs(p) + πsbs (pS)(62a)
s.t.(19),
ln 1 + ¯
βkp2
k/σ2
k≥rk, k = 1, ..., M, (62b)
ln 1 + ¯
βs,`p2
s,`/σ2
s,`≥rs,` ,
`= 1, . . . , Ks;s= 1, . . . , S, (62c)
Ks
X
`=1
p2
s,` ≤Pmax
s, s = 1, .., S, (62d)
where
πsbs(pS),
S
X
s=1
[αs
Ks
X
`=1
p2
s,` +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 each
Dinkelbach’s iteration admits a closed-form solution; thus,
Dinkelbach’s algorithm is very computationally efficient.
First, it follows from (62b) and (62c) that
p2
k≥¯pk:= σ2
k(e¯rk−1)/¯
βk,
p2
s,` ≥¯ps,` := σ2
s,`(e¯rs,` −1)/¯
βs,`.
By making the variable change
p2
k= ˜pk+ ¯pk, p2
s,` = ˜ps,` + ¯ps,`,
it is straightforward to solve (62) by applying Dinkelbach’s
algorithm, which seeks τ > 0such that the optimal solution
of the following optimization problem is zero:
max
˜
p,˜
pS
M
X
k=1
ln ak+¯
βk˜pk/σ2
k
+
S
X
s=1
Ks
X
`=1
ln as,` +¯
βs,` ˜ps,`/σ2
s,`
−τ(˜πmbs(˜
p) + ˜πsbs(˜
pS)) (63a)
s.t.
M
X
k=1
˜pk≤¯
Pmax
M,˜pk≥0, k = 1, ..., M, (63b)
Ks
X
`=1
˜ps,` ≤¯
Pmax
s, s = 1, .., S, (63c)
where ak= 1 + ¯
βk¯pk/σ2
k,¯
PMcir =αPM
k=1 ¯pk+MPa+Pc,
¯
Pmax
M=Pmax
M−PM
k=1 ¯pk,˜πmbs(˜
p),αPM
k=1 ˜pk+¯
PMcir, and
as,` = 1 + ¯
βs,` ¯ps,`/σ2
s,`,¯
Ps,cir =αsPKs
`=1 ¯ps,` +Ps,cir,¯
Pmax
s=
Pmax
s−PKs
`=1 ¯ps,`,˜πsbs (˜
pS),PS
s=1[αsPKs
`=1 ˜ps,` +¯
Ps,cir].
Problem (63) admits the optimal solution in the closed-form:
˜p∗
k=1
(τα +λM)−akσ2
k
¯
βk+
, k = 1, . . . , M,
˜p∗
s,` ="1
(ταs+λs)−as,`σ2
s,`
¯
βs,` #+
,
`= 1, . . . , Ks;s= 1, . . . , S,
(64)
9
where λM= 0 when
M
X
k=1 1
τα −akσ2
k
¯
βk+
≤¯
Pmax
M.
Otherwise, λM>0is located through the bisection method
such that
M
X
k=1 1
(τα +λM)−akσ2
k
¯
βk+
=¯
Pmax
M.(65)
Analogously, λs= 0 when
Ks
X
`=1 "1
ταs−as,`σ2
s,`
¯
βs,` #+
≤¯
Pmax
s.
Otherwise, λs>0is located through the bisection method
such that
Ks
X
`=1 "1
(ταs+λs)−as,`σ2
s,`
¯
βs,` #+
=¯
Pmax
s.(66)
The above proposed Dinkelbach’s computational procedure
for (62) is summarized in Algorithm 2.
Algorithm 2 : Dinkelbach’s algorithm for solving problem
(62)
1: Initialization: Solve (63) for initial τ > 0. If its optimal
value is greater than zero, set τ=τand reset τ←2τ
and solve (63) again. Otherwise (its optimal value is lower
than zero) set ¯τ=τ. End up by having τand ¯τsuch that
the optimal value of (63) is positive for τ=τand is
negative for τ= ¯τ. The optimal τfor zero optimal value
of (63) lies on [τ, ¯τ];
2: Bisection Method
3: Repeat
4: Solve (63) for τ= (τ+ ¯τ)/2. If its optimal value is
positive, then reset τ←τ. Otherwise (its optimal value
is negative), reset ¯τ←τ.
5: Until ¯τ−τ≤(tolerance) to have the optimal value of
(63) equal to zero.
VI. NUMERICAL SIMULATIONS
In this section, we evaluate the performance of the proposed
algorithms by numerical simulations. Consider a circular cell
HetNet with radius 1km, where the MBS is at the center
and S= 6 underlaid SBSs are distributed either equally
at the cell edge or nearly the MBS, or half of which are
equally distributed at the cell edge with another half distributed
nearly the MBS as depicted in Fig. 2a, Fig. 2b or Fig. 2c,
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 the
interference area of each SBS is 50 m×3 = 150m. The MBS
is equipped with NM>32 antennas and each SBS is equipped
with Ns= 4 antennas. The other simulation parameters are
provided in Table I, which follow the prior works [10], [31].
The channel vector hs,` from SBS sin (2) is still genereated
by √qs,` ˜
hs,` with √qs,` and ˜
hs,` = (˜
h1,s,`, ...., ˜
hNs,s,`)T,
˜
hi,s,` ∈ CN(0,1) representing the path loss and large-scale
fading and the small-scale fading, respectively. There are
M= 16 MUEs with 10 MUEs uniformly distributed outside
the SBSs’ coverage and each of the remaining 6MUEs
located in the interference area of one of SBSs. Thus, the
SBS’s interference to the MUEs is the same under these three
settings. Each SBS serves two SUEs. The QoS requirement
for all users is 0.4bps/Hz or 4Mbps [32, Table I].
TABLE II: Simulation Setup
Parameter Assumption
Carrier frequency / Bandwidth 2 GHz / 10 MHz
MBS transmission power 46 dBm
SBS transmission power 30 dBm
Path loss from MBS to user 148.1 + 37.6log10 R[dB], Rin km
Path loss from SBSs to user 127 + 30log10 R[dB], Rin km
Shadowing standard deviation 8 dB
Noise power density -174 dBm/Hz
The power amplifiers parameter α0= 1/0.388,αs= 1/0.052
The circuit power per antenna Pa,0= 189 mW, Pa,s = 5.6 mW
The non-transmission power Pc,0= 40 dBm, Pc,s = 20 dBm
−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 SC1
SC2
SC6
SC5
(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
SC6
SC5
(b) A strongly coupled HetNet
10
−1000 −800 −600 −400 −200 0 200 400 600 800 1000
−1000
−800
−600
−400
−200
0
200
400
600
800
1000
MBS
SC2
SC3
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 proposed
path-following computational procedures for solving each par-
ticular EE maximization problem in the scenario of weakly
coupled HetNets. which is also observed in other scenarios.
The EE objective is iteratively increase and converges rapidly
within several iterations.
0 2 4 6 8 10 12 14
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Number of iterations
EE (bits/Joule/Hz)
MZF
MZF+SZF
ZMI+SZF
Fig. 3: The convergence vs. iteration number under NM= 64 and QoS ¯rk= ¯r(s,`)≡
4Mbps.
Note that MZF and MZF+SZF use the same class (13), (15)
of zero forcing inter-MUE interference MBS beamforming
vector fkfor the EE maximization problems (22) and (46).
They achieve the same EE performance but MZF+SZF is
obviously more computationally efficient; as such only the
curve 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 interference
enhancement means in HetNets.
40 50 60 70 80 90 100
2.5
3
3.5
4
4.5
Number of MBS antennas
EE (bits/Joule/Hz)
MZF+SZF
MZF+CoZF
ZMI+SZF
ZMI+CoZF
EE ZF
Fig. 4: The EE performance vs. the number of MBS antennas in weakly coupled HetNets.
QoS ¯rk= ¯r(s,`)≡4Mbps.
Fig. 4 depicts the EE performance vs. the number NMof
the MBS antennas. Under the weakly coupled scenario, the
MBS interference to the SUEs is weak, leaving its cancelation
unnecessary. This explains why MZF, which ignores this in-
terference in the EE maximization problem (22), outperforms
ZMI+SZF, which nullifies it in the EE maximization problem
(53).
The performance gap between MZF and ZMI+SZF is
narrower as NMincreases, making the MBS interference
stronger. However, there is a huge gap at both NM= 40 and
NM= 60, under which either MZF or ZMI+SZF achieves
their maximum EE.
It has been shown in [33] that the following conventional
SBS 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(¯
HH
s¯
Hs)−1and ¯
Hs= [hs,j ]j∈Ks∈CNs×Ks, which
stacks all channels from SBS sto its SUEs. Figures 4 to 7 also
plot the EE by these conventional class of zero-forcing SBS
beamforming under different classes of MBS beamforming,
which also demonstrates that this conventional zero-forcing
SBS beamforming is not energy-efficient for HetNets and is
clearly outperformed by the SBS beamforming with beam-
forming vectors parametrized by (43).
11
40 50 60 70 80 90 100
2
4
6
8
10
12
14
Number of MBS antennas
Transmit power at MBS (W)
ZMI+SZF
EE 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
Number of MBS antennas
Transmit power at SBSs (W)
ZMI+SZF
EE 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, achieve
the same EE. Both schemes particularly force the inter-SUE
interference to zero. SBS interference is compensated by MBS
beamforming without hurting the network’s EE in ZMI+SZF
but is nullified by SBS beamforming in EE ZF. Fig. 5 shows
that, as expected, the former requires less SBSs’ transmission
power to optimize the EE than the latter. Furthermore, it also
reveals that when the number NMof the MBS antennas is
less than 60, the network’s EE is optimised by requiring less
transmission power, i.e. the spectral efficiency compensates
well the massive MIMO’s circuit power. However, when the
number NMis more than 60, the MBS interference becomes
the main factor to hurt the network’s sum throughput in the
numerator of the EE objective. SBSs also need more power
to compensate this MBS interference to maintain the QoS
requirements.
B. Strongly coupled HetNet
40 50 60 70 80 90 100
2.8
3
3.2
3.4
3.6
3.8
Number of MBS antennas
EE (bits/Joule/Hz)
MZF+SZF
MZF+CoZF
ZMI+SZF
ZMI+CoZF
EE ZF
Fig. 6: The EE performance vs. the number of MBS antennas in strongly coupled
HetNets. QoS ¯rk= ¯r(s,`)≡4Mbps.
In this scenario, the MBS interference to the SUEs is very
strong as all SUEs are located sufficiently near to the MBS.
This suggests that the MBS needs to control its interference
to make the overall network energy-efficient. However, as
Fig. 6 shows, the EE performance achieved by MZF for
NM= 40, which ignores this interference while enhancing
inter-SUE interference, is almost the same as that achieved by
ZMI+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 compensate
such MBS interference. However, as NMbecomes larger than
40 and the MBS interference becomes too severe, the former
cannot compensate the latter and the MZF’s performance dete-
riorates. The zero-forcing the strong MBS interference comes
into fruition, making ZMI+SZF easily outperform MZF.
C. Mixed-coupled HetNets
In this scenario, the MBS interference to the SUEs is strong
only for the half, which is located sufficiently near to the
MBS, and is weak for the other half, which is located far way
from the MBS. From the previous results, it is expected that
AZMI+SZF, which forces only the strong MBS interference
to zero and ignores the weak MBS interference in the EE max-
imization problem (58), will be efficient. Fig. 7 confirms this
intuition. Interestingly, ZMI+SZF and AZMI+SZF achieve
their best EE at NM= 50, where their performance gap is
clearly visualized.
In summary, the best beamforming strategy is to ignore the
interference when it is weak, enhance it when it is medium-
strong and cancel it when it is strong. The weak interference
does not only make obtaining CSI difficult, but is not needed
for optimization either.
12
40 50 60 70 80 90 100
2
2.5
3
3.5
4
Number of MBS antennas
EE (bits/Joule/Hz)
MZF+SZF
MZF+CoZF
ZMI+SZF
ZMI+CoZF
AZMI+SZF
AZMI+CoZF
EE ZF
Fig. 7: The EE performance vs. the number of MBS antennas in mixed-coupled HetNets.
QoS ¯rk= ¯r(s,`)≡4Mbps.
D. HetNet EE vs. massive MIMO EE
To have an appropriate setting for EE ZF in the EE opti-
mization problem (62), the number Nsof each SBS antennas
is set to 6. The effectiveness of HetNets is demonstrated by
comparing its EE performance with that achieved by a massive
MIMO with a MBS equipped with NM+PS
s=1 Nsantennas
to serve M+PS
s=1 Ksusers in the two following schemes:
•Optimal power allocation for zero-forcing beamforming
referred as to MBS PA:
max
pPM+PS
s=1 Ks
k=1 ln 1 + ¯
βkp2
k/σ2
k
πmbs(p)(67a)
s.t.
M+PS
s=1 Ks
X
k=1
p2
k≤Pmax
M,(67b)
ln 1 + ¯
βkp2
k/σ2
k≥rk,(67c)
k= 1,...,(M+
S
X
s=1
Ks),
which is solved by the same Dinkelbach’s type algorithm
proposed in Section V.
•Equal power allocation for zero-forcing
beamforming referred as to MBS EPA with the
EE [
M+PS
s=1 Ks
X
k=1
ln 1 + ¯
βkp2
e/σ2
k]/πmbs,EPA, where
pe=qPmax
M/(M+PS
s=1 Ks)is the equal power
allocation to all UE and πmbs,EPA =αP max
M+PMcir.
In simulations, the proposed Dinkelbach’s type algorithm
converges within 10 iterations to the optimal solutions in all
solved problems.
Fig. 8 shows the significant benefit of using HetNets instead
of massive MIMO for the system EE. It can be seen that the
EE in massive MIMO is sensitive to the number of users,
which are near to the BS (near users). More near users lead
to a better EE in massive MIMO. There are many near users
in the massive MIMO corresponding to the strongly coupled
HetNets. The number of near users in the massive MIMO
corresponding to the mixed-coupled HetNets is more than that
in the massive MIMO corresponding to the weakly couples
HetNets. On the other hand, the EE in HetNets is dependent
on both number of near MUEs and degree of the MBS
interference to SUEs. For this reason, the EE is achieved best
in the weakly HetNets, second best in the strongly coupled
HetNets, and last in the mixed-coupled HetNets. Comparing
to the mix-coupled HetNets, the MBS interference is stronger
but the number of near MUEs is more so the former still
achieve a better EE than the latter.
70 80 90 100 110 120 130 140
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Total number of antennas
EE (bits/Joule/Hz)
EE ZF
MassiveMIMO_PA
MassiveMIMO_EPA
(a) Weakly coupled HetNet
70 80 90 100 110 120 130 140
0.5
1
1.5
2
2.5
3
3.5
4
Total number of antennas
EE (bits/Joule/Hz)
EE ZF
MassiveMIMO_PA
MassiveMIMO_EPA
(b) Strongly coupled HetNet
70 80 90 100 110 120 130 140
0.5
1
1.5
2
2.5
3
3.5
4
Total number of antennas
EE (bits/Joule/Hz)
EE ZF
MassiveMIMO_PA
MassiveMIMO_EPA
(c) Mixed-coupled HetNet
Fig. 8: The EE performance vs. the number NMof MBS antennas from 40 to 100
antennas and fixed antennas per SBS with 6SBSs. The throughput threshold per UE is
4Mbps.
13
VII. CONCLUSIONS
We have considered various classes of beamforming in
HetNets to optimize their energy-efficiency, which is expressed
as the ratio of the sum throughput and consumed power. These
problems have been formulated as maximizations of highly
difficult fractional functions, subject to nonconvex constraints
for user QoS satisfaction, and were solved by the proposed
path-following algorithms. Numerical examples have shown
the efficiency of these algorithms. More importantly, they have
shown that in contrast to maximizing the spectral efficiency,
which suggests using as many antennas as possible, the EE
drops very quickly when this number exceeds 50, which is
quite small in the massive MIMO context. HetHets exhibit
superior performance in terms of EE when compared with
massive MIMO, for a given number of antennas.
APPENDIX:FU NDA ME NTAL INEQUAL ITIES
We exploit the fact that the function f(x, t) = ln(1+1/x)
tis
convex in x > 0, t > 0which can be proved by examining its
Hessian. The following inequality for all x > 0,¯x > 0,t > 0
and ¯
t > 0then holds true [34]:
ln(1 + 1/x)
t≥f(¯x, ¯
t) + h∇f(¯x, ¯
t),(x, t)−(¯x, ¯
t)i
= 2ln(1 + 1/¯x)
¯
t+1
¯
t(¯x+ 1)
−x
(¯x+ 1)¯x¯
t−ln(1 + 1/¯x)
¯
t2t, (68)
where ∇is the gradient operation.
By replacing 1/x →xand 1/¯x→¯xin (68), we have
ln(1 + x)
t≥a−b
x−ct, (69)
where
a= 2ln(1+¯x)
¯
t+¯x
¯
t(¯x+1) >0, b =¯x2
¯
t(¯x+1) >0,
c=ln(1+¯x)
¯
t2>0.
By replacing |x|2→xand |¯x|2→¯xin (69) we have
ln(1 + |x|2)
t≥¯a−¯
b
|x|2−¯ct
≥¯a−¯
b
2<{x¯x∗}−|¯x|2−¯ct (70)
over the trust region
2<{x¯x∗}−|¯x|2>0,(71)
where
¯a= 2 ln(1+|¯x|2)
¯
t+|¯x|2
¯
t(|¯x|2+1) >0,
¯
b=|¯x|4
¯
t(|¯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 with
very 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, “Massive
MIMO 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, and
massive 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 use
of excess antennas: Massive MIMO, small cells, and TDD,” Bell Labs
Technical 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 system
analysis,” IEEE J. Sel. Areas. Commun., vol. 33, no. 6, pp. 1187–1200,
Jun. 2015.
[7] A. Ghosh et al, “Heterogeneous cellular networks: from theory to
practice,” IEEE Commun. Mag., vol. 50, no. 6, pp. 54–64, Jun. 2012.
[8] H. S. Dhillon, M. Kountouris, and J. G. Andrews, “Downlink MIMO
HetNets: Modeling, ordering results and performance analysis,” IEEE
Trans. Wireless Commun., vol. 12, no. 10, pp. 5208–5222, Oct. 2013.
[9] A. Adhikary, H. S. Dhillon, and G. Caire, “Massive-MIMO Meets
HetNet: 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 small
cells: improving energy efficiency by optimal soft-cell coordination,” in
ICT, May 2013, pp. 1–5.
[11] N. Bhushan et al, “Network densification: the dominant theme for
wireless 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 we
approaching 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 global
footprint of mobile communications: The ecological and economic
perspective,” IEEE Commun. Mag., vol. 49, no. 8, pp. 55–62, Aug. 2011.
[14] GreenTouch Green Meter Research Study, “Reducing the net energy
consumption 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 for
link 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 dense
networks for maximal energy efficiency: Small cells meet massive
MIMO,” IEEE J. Sel. Areas. Commun., vol. 34, no. 4, pp. 832–847,
Apr. 2016.
[20] W. Dinkelbach, “On nonlinear fractional programming,” Management
Science, 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 power
allocation in wireless networks by local D.C. programming,” IEEE
Trans. 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 heterogeneous
networks,” IEEE J. Sel. Areas. Commun., vol. 33, no. 10, pp. 2104–2117,
Oct 2015.
14
[25] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Massive MU-MIMO
downlink tdd systems with linear precoding and downlink pilots,” in
Proc. IEEE Allerton Conf., Oct. 2013, pp. 293–298.
[26] N. Saquib, E. Hossain, L. B. Le, and D. I. Kim, “Interference man-
agement in OFDMA femtocell networks: issues and approaches,” IEEE
Wireless Commun., vol. 19, no. 3, pp. 86–95, Jun. 2012.
[27] A. Abdelnasser, E. Hossain, and D. I. Kim, “Clustering and resource
allocation 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 conic
optimization 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 algorithm
for 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, “Interference
alignment 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. Areas
Commun., vol. 32, no. 6, pp. 1065–1082, Jun. 2014.
[33] L. D. Nguyen, H. D. Tuan, and T. Q. Duong, “Energy-efficient signalling
in 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.
He received the B.S. degree in electronics and
telecommunication engineering and the M.S. degree
in telecommunication engineering from the Ho Chi
Minh City University of Technology, Vietnam, in
2013 and 2015, respectively. He is currently pur-
suing the Ph.D. degree with Queen’s University
Belfast. His research interests include convex op-
timization techniques, heterogeneous network, relay
networks, and massive MIMO.
Hoang Duong Tuan received the Diploma (Hons.)
and Ph.D. degrees in applied mathematics from
Odessa State University, Ukraine, in 1987 and 1991,
respectively. He spent nine academic years in Japan
as an Assistant Professor in the Department of
Electronic-Mechanical Engineering, Nagoya Univer-
sity, from 1994 to 1999, and then as an Associate
Professor in the Department of Electrical and Com-
puter Engineering, Toyota Technological Institute,
Nagoya, from 1999 to 2003. He was a Professor with
the School of Electrical Engineering and Telecom-
munications, University of New South Wales, from 2003 to 2011. He is
currently a Professor with the School of Electrical and Data Engineering
and a core member of Global Big Data Technologies Centre, University
of Technology Sydney. He has been involved in research with the areas
of optimization, control, signal processing, wireless communication, and
biomedical engineering for more than 20 years.
Trung Q. Duong (S’05, M’12, SM’13) received his
Ph.D. degree in Telecommunications Systems from
Blekinge Institute of Technology (BTH), Sweden in
2012. Since 2013, he has joined Queen’s Univer-
sity Belfast, UK as a Lecturer (Assistant Profes-
sor). His current research interests include small-cell
networks, physical layer security, energy-harvesting
communications, cognitive relay networks. He is
the author or co-author of more than 260 technical
papers published in scientific journals (142 articles)
and presented at international conferences (121 pa-
pers).
Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONS
ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-
CATIONS, I ET COMMUNICATIONS, and a Senior Editor for IE EE COMMU-
NICATIONS LETTE RS. He was awarded the Best Paper Award at the IEEE
Vehicular Technology Conference (VTC-Spring) in 2013, IEEE International
Conference on Communications (ICC) 2014, and IEEE Global Communi-
cations Conference (GLOBECOM) 2016. He is the recipient of prestigious
Royal Academy of Engineering Research Fellowship (2016-2021).
Octavia A. Dobre (M’05, SM’07) received the Dipl.
Ing. and Ph.D. degrees from Politehnica Univer-
sity of Bucharest (formerly Polytechnic Institute of
Bucharest), Romania, in 1991 and 2000, respec-
tively. She was a Royal Society Scholar in 2000
and a Fulbright Scholar in 2001. Between 2002
and 2005, she was with Politehnica University of
Bucharest and New Jersey Institute of Technology,
USA. In 2005, she joined Memorial University,
Canada, where she is currently a Professor and
Research Chair. She was a Visiting Professor with
Universit de Bretagne Occidentale, France, and Massachusetts Institute of
Technology, USA, in 2013. Her research interests include 5G enabling
technologies, blind signal identification and parameter estimation techniques,
cognitive radio systems, network coding, as well as optical and underwater
communications among others. Dr. Dobre serves as the Editor-in-Chief of the
IEEE COMMUNICATIONS LETTERS, as well as an Editor of the IEEE
SYSTEMS and IEEE COMMUNICATIONS SURVEYS AND TUTORIALS.
She was an Editor and a Senior Editor of the IEEE COMMUNICATIONS
LETTERS, an Editor of the IEEE TRANSACTIONS ON WIRELESS COM-
MUNICATIONS and a Guest Editor of other prestigious journals. She served
as General Chair, Tutorial Co-Chair, and Technical Co-Chair at numerous
conferences. She is the Chair of the IEEE ComSoc Signal Processing and
Communications Electronics Technical Committee, as well as a Member-
at-Large of the Administrative Committee of the IEEE Instrumentation and
Measurement Society. Dr. Dobre is a Fellow of the Engineering Institute of
Canada.
H. Vincent Poor (S’72, M’77, SM’82, F’87) re-
ceived the Ph.D. degree in EECS from Princeton
University in 1977. From 1977 until 1990, he was
on the faculty of the University of Illinois at Urbana-
Champaign. Since 1990 he has been on the faculty
at Princeton, where he is currently the Michael
Henry Strater University Professor of Electrical En-
gineering. During 2006 to 2016, he served as Dean
of Princetons School of Engineering and Applied
Science. He has also held visiting appointments at
several other universities, including most recently at
Berkeley and Cambridge. His research interests are in the areas of information
theory and signal processing, and their applications in wireless networks,
energy systems and related fields. Among his publications in these areas is
the recent book Information Theoretic Security and Privacy of Information
Systems (Cambridge University Press, 2017).
Dr. Poor is a member of the National Academy of Engineering and the
National Academy of Sciences, and is a foreign member of the Chinese
Academy of Sciences, the Royal Society, and other national and international
academies. He received the Marconi and Armstrong Awards of the IEEE
Communications Society in 2007 and 2009, respectively. Recent recognition
of his work includes the 2017 IEEE Alexander Graham Bell Medal, Honorary
Professorships at Peking University and Tsinghua University, both conferred
in 2017, and a D.Sc. honoris causa from Syracuse University also awarded
in 2017.
