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Full duplex (FD) communication has emerged as an attractive solution for increasing the network throughput, by allowing downlink (DL) and uplink (UL) transmissions in the same spectrum. However, only employing FD base stations in heterogeneous cellular networks (HCNs) cause coverage reduction, due to the DL and UL interferences as well as the residual loop interference. We therefore propose HCNs with half duplex (HD) massive multiuser multiple-input multiple-output (MIMO) macrocell base stations (MBSs) to relax the coverage reduction, and FD small cell base stations (SBSs) to improve spectrum efficiency. A tractable framework of the proposed system is presented, which allows to derive exact and asymptotic expressions for the DL and the UL rate coverage probabilities, and the DL and the UL area spectral efficiencies (ASEs). Monte carlo simulations confirm the accuracy of the analytical results, and it is revealed that equipping massive number of antennas at MBSs enhances the DL rate coverage probability, whereas increasing FD SBSs increases the DL and the UL ASEs. The results also demonstrate that by tuning the UL fractional power control, a desirable performance in both UL and DL can be achieved.
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Massive Multiuser MIMO in Heterogeneous
Cellular Networks with Full Duplex Small
Cells
Sunila Akbar, Student Member, IEEE, Yansha Deng, Member, IEEE, Arumugam
Nallanathan, Fellow, IEEE, Maged Elkashlan, Member, IEEE, and George K.
Karagiannidi, Fellow, IEEE
Abstract
Full duplex (FD) communication has emerged as an attractive solution for increasing the network
throughput, by allowing downlink (DL) and uplink (UL) transmissions in the same spectrum. However,
only employing FD base stations in heterogeneous cellular networks (HCNs) cause coverage reduction,
due to the DL and UL interferences as well as the residual loop interference. We therefore propose
HCNs with half duplex (HD) massive multiuser multiple-input multiple-output (MIMO) macrocell base
stations (MBSs) to relax the coverage reduction, and FD small cell base stations (SBSs) to improve
spectrum efficiency. A tractable framework of the proposed system is presented, which allows to derive
exact and asymptotic expressions for the DL and the UL rate coverage probabilities, and the DL and
the UL area spectral efficiencies (ASEs). Monte carlo simulations confirm the accuracy of the analytical
results, and it is revealed that equipping massive number of antennas at MBSs enhances the DL rate
coverage probability, whereas increasing FD SBSs increases the DL and the UL ASEs. The results also
demonstrate that by tuning the UL fractional power control, a desirable performance in both UL and
DL can be achieved.
Index Terms
Heterogeneous cellular networks, massive multiuser MIMO, full duplex, spectral efficiency, stochas-
tic geometry.
S. Akbar, Y. Deng, and A. Nallanathan are with Center for Telecommunications Research, King’s College London, London,
UK (e-mail: {sunila.akbar, yansha.deng, arumugam.nallanathan}@kcl.ac.uk).
M. Elkashlan is with Queen Mary University of London, London E1 4NS, UK (e-mail: maged.elkashlan@qmul.ac.uk).
G. K. Karagiannidi is with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece (e-mail: geokarag@auth.gr).
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I. INTRODUCTION
The emerging fifth-generation (5G) wireless communication system targets higher data rates,
roughly 1000 times of the current fourth-generation (4G) system to support exponential increase
in wireless data transmissions [1]. In order to meet this target, heterogeneous cellular networks
(HCNs) are proposed to boost the network capacity through dense deployment of small cell base
stations (SBSs) [2], and multiuser multiple-input multiple-output (MIMO) with large number of
antennas at the base station (BS) enables fine-grained beamforming towards each mobile user
(MU), which brings ultra high throughput [3].
Recently, increasing research has been conducted on full-duplex (FD) communication, which
allows transmitting and receiving data simultaneously, within the same frequency band [4]. In
theory, FD data transmission is capable of doubling the spectral efficiency of half-duplex (HD)
system. However, FD has been previously regarded as hard to be realized in practice due to its
high residual self-interference (SI) problem. Fortunately, the recent advances on SI cancellation,
such as antenna separation schemes [5], beamforming-based techniques [6], and digital circuit
domain schemes [7], have demonstrated the feasibility of FD transmission for short to medium
range wireless communications. For instance, FD transmission can be realized at the access points
through shared or separated antenna configurations [8]. In terms of antenna usage, the efficiency
of the shared antenna configuration is higher than that of the separated one [9]. Besides, the shared
antenna configuration is a promising alternative for separated antenna configuration in short range
communications, where the transmit power is low and the antenna isolation requirement is less
rigorous compared with medium to long range communication [9]. As a promising candidate for
FD technology due to the low transmit power, the SBSs increases network capacity and coverage
[10]. As such, leveraging the use of FD technology at the SBSs and massive multiuser MIMO
at the macrocell base stations (MBSs) provide a potential solution to improve spectral efficiency
of HCNs.
A. Related Work
1) Multiuser MIMO in HCNs: [11] presented the coverage probability and area spectral
efficiency (ASE) for the downlink (DL) MU in HCNs with multiuser MIMO. It is shown that
for a given total number of transmit antennas, it is preferable to distribute the antennas across
large number of single-antenna BSs rather than small number of multi-antenna BSs. The work
in [10] was extended to [12], which studied the load balancing strategy, which maximizes the
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coverage probability. In [13], it is shown that massive multiuser MIMO BSs and small cells
BSs operating in time division duplexing (TDD) mode lead to high area throughput, which can
be further improved by installing more BS antennas or deploying more small cells. In [14], the
trade-off between the link reliability and the ASE of HCNs with multiuser MIMO was studied.
2) FD Communication in Cellular Networks: The performance gains brought by FD transmis-
sion in cellular networks have been studied in [15–22]. [15] concludes that making different tiers
operate in different duplex modes in heterogeneous networks enhances the network throughput.
In [16], the ASE was derived for small cell networks with FD, and the self-interference (SI) was
shown to be dominant compared to the aggregate interference. It is shown in [17] that small
cell in-band wireless backhaul has the potential to increase the throughput of massive multiuser
MIMO systems. The authors in [18] investigated the spectrum and energy efficiency of the
massive MIMO-enabled FD cellular networks. In [19], the rate coverage probability of a massive
multiuser MIMO-enabled wireless backhaul networks was evaluated, where each SBS can be
configured with either in-band or out-of-band FD backhaul mode. In [20], the authors studied
the joint in-band backhauling and interference mitigation problem in HCNs, which consists of a
massive multiuser MIMO MBS overlaid with self-backhauled small cells. Furthermore, the work
in [21] proposed in-band α-duplex scheme in multi-cell networks with FD operation in each cell,
which allows a partial overlap between DL and uplink (UL) frequency bands. The results in [21]
demonstrated that the overlap parameter, α, can be optimized to achieve maximum FD gain. In
[22], the cell association problem in multi-tier in-band FD networks was investigated. It is shown
that the proposed decoupled cell association, where MUs can be served by different BS in the
UL and DL transmission, outperforms the coupled cell association in which MUs associate to
the same BS in both DL and UL.
3) Spectral Efficiency and Link Reliability: The two important metrics to evaluate the perfor-
mance of HCNs, spectral efficiency and outage probability were evaluated in HCNs with wireless
power transfer in [23, 24]. The trade-off between the ASE and the link reliability was discussed
in wireless ad-hoc networks [25–27]. In these networks, increasing the density of transmitters
affects both link reliability and ASE, therefore the trade-off between them is essential to balance
both aspects. The trade-off between the ASE and the coverage probability has been studied in
massive multiuser MIMO HCNs [14] and a mixed multi-cell system composed of FD and HD
small cells [28].
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B. Motivation and Contributions
The aforementioned literature laid a solid foundation for the feasibility of FD communica-
tion in cellular networks. However, operating all the BSs in FD mode is likely to erode the
performance gain of the FD communication since simultaneous DL and UL operation on the
same band brings increased interference, and thus reduced coverage. Inspired by the work in
[15], where making different tiers operate in different duplex modes in heterogeneous networks
enhances the network throughput, we focus on HCNs, where only small cells operate in FD
mode, and the macrocells operate in HD mode. We consider FD deployment at the SBSs, due
to low transmit power and low mobility of the associated MUs [10]. However, we note that
increasing the FD SBSs increases the ASE of the network, they also increase the interference
due to the simultaneous DL and UL transmission on the same band which decreases coverage
[28]. In order to compensate this cost, a simple solution is to employ massive multiuser MIMO at
the BSs. However, due to the facts that: 1) a more powerful SI cancellation scheme is required to
make FD MIMO systems feasible, and 2) the residual interference at each receive chain increase
linearly with the number of antennas [29], we have considered massive multiuser MIMO only
at the MBSs. For the proposed HCNs, massive antennas at MBSs ensure coverage over large
areas, while SBSs act as capacity-drivers [13]. Furthermore, we employ distance-proportional
fractional power control in the UL, which provides coverage improvement to the cell-edge MUs
and efficient utilization of MUs’ battery [30, 31].
The main contribution of this work can be summarized as follows:
We model the K-tier HCNs with multiuser MIMO MBSs operating in HD mode and SBSs
operating in FD mode. We consider only the DL transmission for MBSs, and both DL and
UL transmissions for SBSs. We characterize the network interference generating from the
distributed FD SBSs and UL MUs for performance evaluation.
We derive analytical expressions for the DL rate coverage probability, the DL ASE of the
macrocells and small cells, the UL rate coverage probability, and the UL ASE of small cells
to evaluate the link reliability and spectral efficiency. To examine the impact of massive
multiuser MIMO antennas at the MBS, we derive easy to compute expressions for the
asymptotic DL and UL rate coverage probabilities, and asymptotic DL and UL ASEs as
the number of antennas at the MBSs grows large.
Numerical results demonstrate the effectiveness of massive multiuser MIMO at the MBSs
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TABLE I: Frequent Notations
Notation Definition Notation Definition
PMTransmit power of MBS PkTransmit power of kth tier SBS
ρkReceiver’s sensitivity at the kth tier SBS UL power control factor
αMPath loss exponent for macrocell αkPath loss exponent for kth tier small cell
βFrequency dependent constant value hRSI ,k
Residual self interfering channel
of a kth tier BS
NoNoise power Γ(.)Gamma function
B(.)[., .]Incomplete beta function 2F1[., .;.;.]Gauss hypergeometric function
and the FD SBSs in enhancing the rate coverage probabilities and the ASEs. Moreover,
we show that the distance-proportional fractional power control can be tuned to achieve
a desirable performance in both DL and UL, where decreasing the power control factor
degrades the UL rate coverage probability, but improves the DL rate coverage probability.
C. Paper Organization and Notations
The rest of the paper is organized as follows. In Section II, we discuss the system model of
HCNs with multiuser MIMO at the MBSs and FD operation at the SBSs. In Section III, we
present the cell association and derive the rate coverage probability and ASE both in the DL
and the UL. In Section IV, we evaluate DL and UL rate coverage probabilities and ASEs for
the massive multiuser MIMO regime. We present the performance comparison of the proposed
HCNs with FD SBSs with the conventional HCNs with HD SBSs in Section V. Finally, numerical
results are discussed in Section VI before the paper is concluded in Section VII.
The notations commonly used throughout the paper are presented in Table I.
II. SY ST EM MO DE L
A. Network Model
We consider K-tier HCNs, where the MBSs and the SBSs are spatially located in R2, following
homogeneous Poisson point process (HPPP), ΦbMand Φbkwith intensity λbMand λbk(k=
2,· · · , K), respectively. We consider massive multiuser MIMO at the MBSs, where each BS
is equipped with Nantennas, serving UMMUs (1< UMN), and operates in HD mode.
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MBS
SBS
DL/UL Signal Interference
Signals
Self Interference
Corresponding
Apx. Interference
Characterization
MU
Fig. 1: Example cells of the proposed HCNs with HD multiuser MIMO MBS and FD SBSs and
the interference characterizations.
Each SBS is equipped with single antenna, and is transmitting and receiving at the same time in
FD mode [32, 33]. All the MUs have single antenna and operate in HD mode. In this work, we
focus on the DL performance of the macrocell without pilot contamination, while the SBSs have
transmissions in the DL and UL simultaneously due to the FD operation. The performance of
the UL transmission of MBSs can be easily analyzed following our analytical framework, where
the scheduled macrocell MUs simultaneously transmit to their serving massive multiuser MBS
per resource block in the UL [34]. Linear receive filters are then used for UL signal detection.
In HCNs with FD small cells, the DL and UL small cell transmissions occur simultaneously
by reusing the spectrum of the DL macrocell transmissions due to the full frequency reuse in
HCNs [35]. The network is assumed to be fully-loaded, such that each MBS has UMactive
MUs [11, 12] 1, and each SBS serves one active DL MU and one active UL MU in each time
1Here, we limit ourselves to the ideal assumption with the fixed number of UMin each macrocell to ensure equivalent
performance at the macrocell MU in each macrocell as in [36, 37]. We note that in [38], the probability mass function of the
number of users served by a generic BS was derived by approximating the area of a Voronoi cell via a gamma-distributed random
variable. However, the result in [38] cannot be applied in this paper, since the Euclidean plane is not divided into Voronoi cells
based on the considered cell association methods. We highlight that it is an important work to study the case of the flexible
UMfollowing a certain distribution in less-dense scenarios.
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instant [22], as shown in Fig. 1. Accordingly, the intensity of the active DL MUs in HCNs is
λDL
u=UMλbM+
K
P
k=1
λbk, whereas the active UL MUs per tier are modeled by an independent
HPPP Φukwith intensity λUL
uk=λbk. The analysis will be performed at a typical MU, which is
assumed to be at the origin.
B. Channel Model
We model the channel path loss over the distance |x|as β|x|α, where βis the frequency
dependent constant value and αis the path loss exponent. The channels are modeled as inde-
pendent and identically distributed (i.i.d.) quasi-static Rayleigh fading. We assume TDD mode,
where channel reciprocity can be exploited and allows a BS to estimate its DL channels from
UL pilots sent by the MUs. Therefore, the resulting number of pilots scales linearly with the
number of MUs, and is independent of the number of antennas in contrast to frequency division
duplexing (FDD). We consider time division multiple access (TDMA), where several MUs share
the same channel in different time slots, thus the BS transmit power is independent of the density
of active MUs, and there is no intra-cell interference in each cell. In a snap of time, each MBS
can serve UMMUs and each FD SBS serves one DL and one UL MU per channel.
C. BS and MU Transmit Power Allocation
Each MBS and SBS transmit with fixed power, PMand Pk, respectively. To limit the UL
interference and reduce the overall power consumption of MUs, we employ distance-proportional
fractional power control [30] in the UL, where the MU at a distance dfrom the associated SBS
adjusts its transmit power with, Pu=ρkβdαj, to compensate large-scale fading. Note that,
01, is the power control factor, and ρkis the receiver sensitivity of the kth tier SBS.
D. Massive multiuser MIMO
Each MBS transmit UMdata streams using linear zero-forcing beamforming (ZFBF) with the
equal transmit power allocation [39], thus the uncorrelated intra-cell interference is suppressed.
We assume sophisticated channel estimation design with sufficient training information that
guarantees perfect CSI [40]. In the training phase, each MU sends a pre-assigned orthogonal
pilot sequence to the MBS, which is perfectly estimated by the MBS without pilot contamination,
therefore the perfect channel state information is available at the BSs and MUs. The non-
pilot contamination assumption is valid when the pre-assigned pilot sequences used in different
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macrocells are orthogonal to each other [41]. The maximum number of MUs per MBS depends
on the dimension of the UL pilot field. Accordingly, the number of channel vectors that can
be estimated and for which the DL precoder can be designed is determined. In our model, we
consider fixed number of MUs served by each MBS.
E. Self-Interference Cancellation for FD Small Cells
The SBS in FD mode receives self-interference from its transmitted signal, and performs SI
cancellation to combat it. Since, the amount of SI depends on the transmit power of the SBS,
we define residual SI power after SI cancellation as [42, 43]
PRSI (Pk) = Pk|hRSI ,k |2,(1)
where hRSI ,k is the residual self interfering channel of a kth tier BS, and hRSI ,k is character-
ized according to the cancellation algorithms. For instance, using digital-domain cancellation
algorithms, hRSI ,k can be modeled as hRSI ,k =hS,k ˆ
hS,k, where hS,k and ˆ
hS,k are the self-
interfering channel and its estimate channel, respectively [42, 44]. In [45], hRSI ,k is regarded as
a constant value with |hRSI ,k|2=σ2
efor the estimation error variance σ2
e. However, modeling
hRSI ,k is still challenging for other cancellation techniques, such as analog-domain schemes,
propagation-domain schemes, and combined schemes of different domains. The parametrization
of the self-IC capability in (1) can make the analysis more generic. Therefore, in our analysis,
we consider, a constant value for hRSI ,k 2, given as
|hRSI ,k |2= 10LdB,k/10 ,(2)
where LdB,k is the ratio between the residual self-interference after interference cancellation and
the transmit power at the kth tier BS as defined in [15].
F. Cell Association
To obtain the strongest received signal, we consider the maximum received power cell asso-
ciation rule in the DL transmission, where the DL MU connects to the BS, which provides the
2The analysis can be easily extended to the case of random hRSI ,k. For instance, once the probability density function
(PDF) of hRSI ,k is available for a certain self-IC algorithm, we can average the analytical results derived in the paper over the
distribution of hRSI ,k.
9
maximum long-term average received power [46]. The average received power at a typical DL
MU connected to the MBS m(mΦbM) is expressed as
Pr,M=Ga
PM
UM
βXm,uMαM,(3)
where the array gain Gaof zero-forcing beamforming (ZFBF) transmission is NUM+ 1 [47].
The average received power at a DL MU that is connected to the kth tier SBS bk(bkΦbk),
is expressed as
Pr,bk=PkβXbk,ukαk.(4)
We remind that for the UL transmission, the MUs can only associate with the FD SBSs.
Considering that the HD UL MU associated to the nearest BS can maximize the UL signal-
to-interference-plus-noise ratio (SINR) [22], we consider the nearest BS cell association in the
UL.
Based on the cell association model, the set of interfering MUs and FD SBSs may correlate.
However, to maintain model tractability, we assume that the set of interfering MUs is independent
of the set of interfering FD SBSs as in [16, 48].
G. SINR Models
1) DL SINR of a Macrocell MU: The SINR for a typical DL macrocell MU uM
0located at
the origin is given as
SINRDL
M=
PM
UMβgo,uM
0Xo,uM
0
αM
IM,uM
0+IS,uM
0+Ius
ul,uM
0
| {z }
IuM
0
+N0
,(5)
where go,uM
0Γ (NUM+ 1,1) is the small-scale fading channel power gain between the
typical DL macrocell MU and its serving MBS, and Xo,uM
0is the distance between the typical
DL macrocell MU and its serving MBS. In (5), IM,uM
0,IS,uM
0, and Ius
ul,uM
0are the interferences
from the other MBSs, the SBSs, and the UL small cell MUs given as
IM,uM
0=X
xΦM
b\o
PM
UM
hx,uM
0βXx,uM
0
αM,(6)
IS,uM
0=
K
X
j=2 X
yΦj
b
Pjhy,uM
oβXy,uM
0
αj,(7)
10
and
Ius
ul,uM
0=
K
X
j=2 X
zΦj
u
ρjβ|Rz,bz|αjhz,uM
0βXz,uM
0
αj,(8)
respectively. In (6), (7), and (8), hx,uM
0Γ (UM,1),hy,uM
0exp (1), and hz,uM
0exp (1)
denote the small-scale fading channel power gains from the MBSs to the typical DL macrocell
MU, from the SBSs to the typical DL macrocell MU, and from the UL small cell MUs to
the typical DL macrocell MU, respectively, and their corresponding distances are denoted as
Xx,uM
0,Xy,uM
0, and Xz,uM
0, respectively. In (8), ρjβ|Rz,bz|αjis the transmit power of the
UL MU at a distance of |Rz,bz|from its serving SBS, where ρjis the receiver sensitivity at the
SBS of the jth tier and is the power control factor.
2) DL SINR of a Small Cell MU: The SINR for a typical DL small cell MU uk
0located at
the origin can be written as
SINRDL
k=
Pkgo,uk
0βXo,uk
0
αk
IM,uk
0+IS,uk
0+Ius
ul,uk
0+N0
,(9)
where go,uk
0is the small-scale fading channel power gain between the typical DL small cell MU
and its serving SBS, and Xo,uk
0is the distance between the typical DL small cell MU and its
serving SBS. In (9), IM,uk
0,IS,uk
0, and Ius
ul,uk
0are the interference from the other MBSs, the SBSs,
and the UL small cell MUs, which are given as
IM,uk
0=X
xΦM
b
PM
UM
hx,uk
0βXx,uk
0
αM,(10)
IS,uk
0=
K
X
j=2 X
yΦj
b\o
Pjhy,uk
0βXy,uk
0
αj,(11)
and
Ius
ul,uk
0=
K
X
j=2 X
zΦj
u
ρjβ|Rz,bz|αjhz,uk
0βXz,uk
0
αj,(12)
respectively. In (10), hx,uk
0Γ (UM,1) denotes the small-scale fading channel power gain
between the MBSs and the typical DL small cell MU. The distances between the typical DL
small cell MU and the MBSs, the other small cell MU, and the UL small cell MU are denoted
as Xx,uk
0,Xy,uk
0, and Xz,uk
0, respectively.
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3) UL SINR of a Small Cell MU: The UL SINR for a typical SBS bk
0located at the origin
can be written as
SINRUL
k=
ρkgo,bk
0βXo,bk
0
αk(1)
PRSI (Pk) + IM,bk
0+IS,bk
0+Ius
ul,bk
0+N0
,(13)
where ρkis the receiver sensitivity of the serving SBS, go,bk
0exp (1) is the small-scale fading
channel power gain between the typical UL small cell MU and its serving SBS, Xo,bk
0is the
corresponding distance, and PRSI (Pk)is the residual SI power after performing cancellation
given in (1). In (13), IM,bk
0,IS,bk
0, and Ius
ul,bk
0are the interference from the MBSs, the other SBSs,
and the other UL small cell MUs given as
IM,bk
0=X
xΦM
b
PM
UM
hx,bk
0βXx,bk
0
αM,(14)
IS,bk
0=
K
X
j=2 X
yΦj
b\bk
0
Pjhy,bk
0βXy,bk
0
αj,(15)
and
Ius
ul,bk
0=
K
X
j=2 X
zΦj
u\o
ρjβ|Rz,bz|αjhz,bk
0βXz,bk
0
αj,(16)
respectively. In (14), hx,bk
0Γ (UM,1) denotes the small-scale fading channel power gain
between the typical UL small cell MU and the MBSs. The distances between the typical SBS
and the MBSs, the other SBSs, and the other UL small cell MUs are denoted as Xx,bk
0,Xy,bk
0,
and Xz,bk
0, respectively.
H. Interference Characterization
Characterizing the interference in proposed HCNs is the key challenge in evaluating the system
performance. The reason is the difficulty to obtain exact characteristics of the interference from
the UL small cell MUs to the DL macrocell MU, Ius
ul,uM
0in (8), the interference from the UL
small cell MUs to the DL small cell MU, Ius
ul,uk
0in (12), the interference from the MBSs in the
DL to the SBSs in the UL, IM,bk
0in (14), and the interference from the SBSs in the DL to the
SBSs in the UL, IS,bk
0in (15). We characterize the interferences as shown in Fig. 1 using similar
approximation as in [15]. For instance, to characterize Ius
ul,uM
0, we consider a DL macrocell MU
located at a, its serving MBS located at b, a FD SBS located at c, and its associated UL MU
at c+N(c), where N(c)is the relative location of small cell MU to its serving SBS at cin
12
the UL. Generally, the distance between DL macrocell MU aand FD SBS cis greater than
the distance between c+N(c)and c, i.e., kack  kN(c)k. Therefore, we assume that the
distance between macrocell MU at aand UL small cell MU at c+N(c)can be approximated
as the distance between a macrocell MU at aand the SBS at c. Likewise, we characterize the
interferences Ius
ul,uk
0,IM,bk
0, and IS,bk
0.
III. PERFORMANCE EVALUATION
Adjusting the number of antennas at MBSs with multiuser MIMO and the FD SBS densities
will affect both the link reliability and ASE of the HCNs. Accordingly, we analyze the perfor-
mance of the DL and the UL transmission of the HCNs in terms of rate coverage probability
and ASE. Since a typical MU can associate with at most one tier, the performance of each tier
as well as per tier association probability determine the overall performance of HCNs in the DL
and the UL as per the law of total probability. To facilitate the analysis, we first present the per
tier association probability.
A. DL Cell Association
The probability that a typical MU is associated with the MBS is given as in [46], where the
transmit powers of the MBSs and SBSs are given in (3) and (4), respectively.
ΛM=2πλbMZ
0
rexp nπλbMr2π
K
X
j=2
λbjPj
ΨPM2/αj
r2αM/αjodr,(17)
where
Ψ =NUM+ 1
UM
.(18)
Similarly, the probability that a typical MU is associated with the kth tier SBS is given as in
[46]
ΛDL
k=2πλbkZ
0
rexp nπ
K
X
j=2
λbj(Pjrαk/Pk)2jπλbMPMΨ
Pk2M
r2αkModr,(19)
where Ψis given in (18).
13
B. UL Cell Association
In the UL transmission, a typical MUs can only associate with the nearest FD SBS. The
probability that a typical MU is associated with the kth tier SBS is given as [46]
ΛUL
k= 2πλbkZ
0
rexpn
K
X
j=2
πλjr2αkjodr.(20)
C. DL Rate Coverage Probability
In this section, we derive the DL rate coverage probability of a typical MU in K-tier HCNs.
The DL rate coverage probability of a random MU in the K-tier HCNs is given by
CDL(RDL) = ΛMCM(RDL) +
K
X
k=2
ΛDL
kCDL
k(RDL),(21)
where ΛMand ΛDL
kare given in (17) and (19), respectively, CM(RDL)is the DL rate coverage
probability between a typical MU and its serving MBS, and CDL
k(RDL)is the DL rate coverage
probability between a typical MU and its serving SBS.
In (21), the DL rate coverage probability between a typical MU and its serving MBS is defined
as
CM(RDL) =E
Xo,uM
0
Pr hSINRDL
MXo,uM
0γDLXo,uM
0i,(22)
where SINRDL
Mis given in (5) and γDL is given as
γDL = eRDL 1.(23)
In (23), RDL is the DL rate threshold. Similarly, the DL rate coverage probability of typical MU
at a distance Xo,uk
0from its associated SBS in the kth tier is defined as
CDL
k(RDL) =E
Xo,uk
0
Pr hSINRDL
kXo,uk
0γDLXo,uk
0i,(24)
where SINRDL
kand γDL are given in (9) and (23), respectively.
Theorem 1. The DL rate coverage probability of a typical MU associated with the MBS is
derived as
CM(RDL) = 2πλbM
ΛMZ
0
x
NUM
X
n=0
(xαM)n
n!(1)nXn!
n
Q
l=1
ml(l!)ml
κ(x)
n
Y
l=1 ψ(l)(xαM)mldx, (25)
14
where
κ(x) = exp (γDLUMqN0
PMβζγDLUMq
PMβπλbMx2π
K
X
j=2
λbkPj
ΨPM2
αjx
2αM
αj),(26)
Pis over all n-tuples of non-negative integers (m1, ..., mn)that satisfy the constraint
n
P
l=1
l . ml=
n,Ψis given in (18),ΛMis given by (17),γDL is given in (23), and ζ(.),ψ(1)(.), and ψ(l)(.)
are given as
ζ(s) =2πλbM
UM
X
ν=1 UM
νPM
UM
βν
sνsPM
UMβν+2
αM
αM
BsPM
UMβxαMν2
αM
,1UM+
K
X
j=2
2πλbj(sPjβ
DM
j(x)2αj
αj22F1αj2
αj
,1; 2 2
αj
;sPjβ(DM
j(x))αj+Z
0Zr2
0
1
1+(jβ(1))1uαj/2rαj
π
K
X
j=2
λbje
π
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr),
(27)
ψ(1)(q) = γDLUMN0
PMβ2πλbMUMγDL x2αM
αM22F1αM2
αM
, UM+ 1; 2 2
αM
;γDLqxαM
K
X
j=2
2πλbj(
γDLUMPj
PM
DM
j(x)2αj
αj22F1αj2
αj
,2; 2 2
αj
;γDLUMPjq
PM
(DM
j(x))αjZ
0Zr2
0PMβuαj/2rαj
γDLUMρ
1 + PMβuαj/2rαj
γDLUMρ2
q2π
K
X
j=2
λbje
π
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr),
(28)
and
ψ(l)(q) =2πλbM
(UM+l1)!
(UM1)! (γDL)
2
αM(q)l+2
αM
αM
B(γDLqxαM)l2
αM
,1UMl+
K
X
j=2
2πλbj(l!
γDLUMPj
PM2
αM(q)l+2
αM
αM
B
γDLUMPjq
PM(DM
j(x))αMl2
αM
,lZ
0Zr2
0PMβuαj/2rαj
γDLUMρ
l
Y
i=2
(i)PMβuαj/2rαj
γDLUMρ+q(l+1) π
K
X
j=2
λbje
π
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr).
(29)
In (27),(28) and (29),λIΦUL
bj
(r),γDL, and λIΦUL
bj
(r)are given in (A.8),(23), and (A.8) respec-
tively, and DM
j(x)is the distance between the closest interferring BS of the jth tier and the
typical MMU given as
DM
j(x) = Pj
ΨPM1
αjx
αM
αj,(30)
where Ψis given in (18).
Proof. See Appendix A.
15
Theorem 2. The DL rate coverage probability of a typical MU associated with the kth tier SBS
is derived as
CDL
k(RDL)
=2πλbk
ΛDL
kZ
0
xexp (γDLxαkN0
PkβΞ(x)π
K
X
j=2
λbjPjxαk
Pk2
αjπλbMPMΨ
Pk2
αMx
2αk
αM)dx,
(31)
where Ψ,ΛDL
k,γDL, and Ξ(x)are given in (18),(19), and (23), and (32), respectively. In (31),
Ξ(.)is given as
Ξ(x) =2πλbM
UM
X
ν=1 UM
νγDLxαkPM
PkUMνγDLxαkPM
PkUMν+2
αM
αM
BγDLxαkPM
PkUM(Dk
M(x))αMν2
αM
,1UM
+
K
X
j=2
2πλbj(γDLxαkPj
PkDk
j(x)2αj
(αj2) 2F1αj2
αj
,1; 2 2
αj
;γDLPj
Pk
(Dk
j(x))αkαj+Z
0Zr2
0
1
1+(γDLxαk
Pkβρjβ(1))1uαj/2rαjπ
K
X
j=2
λbje
π
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr),
(32)
where λIΦUL
bj
(r)and γDL are given in (A.8) and (23), respectively. In (32),Dk
M(x)is the distance
between the closest interfering MBS and the typical small cell MU, and Dk
j(x)is the distance
between the closest interfering BS in the jth tier and the typical small cell MU, given as
Dk
M(x) = ΨPM
Pk1
αMx
αk
αM,(33)
and
Dk
j(x) = Pj
Pk1
αjx
αk
αj.(34)
In (33),Ψis given in (18).
Proof. The proof follows analogous steps to Theorem 1.
D. DL Area Spectral Efficiency
The DL ASE measures the capacity of HCNs in the DL defined by [11, 49, 50]. In this section,
we define the DL ASE of the proposed model as
ASEDL =λbMUMCM(RDL) ln(1 + γDL)
| {z }
ASEDL
MB S
+
K
X
k=2
λbkCDL
k(RDL) ln(1 + γDL)
| {z }
ASEDL
SBS
,(35)
16
where CM(RDL),CDL
k(RDL), and γDL are given in (25), (31), and (23), respectively.
In the following, we present the UL performance metrics which reflect the effect of the self-IC,
the density of SBSs, the transmit power of SBSs, and the power control on the UL performance
in the HCNs. We characterize the UL performance in terms of the UL rate coverage probability
and the UL ASE.
E. UL Rate Coverage Probability
In this section, we derive the UL rate coverage probability using
CUL(RUL ) =
K
X
k=2
ΛUL
kCUL
k(RUL),(36)
where ΛUL
kis given in (20), and CUL
kis the UL rate coverage probability between a typical MU
and its serving SBS defined as
CUL
k(RUL) =E
Xo,bk
0
Pr hSINRUL
kXo,bk
0γULXo,bk
0i,(37)
where SINRUL
kis given in (13) and γUL is given as
γUL = eRUL 1,(38)
and RUL is the UL rate threshold.
Theorem 3. The UL rate coverage probability of a typical MU associated with the kth tier SBS
is derived as
CUL
k(RUL) = 2πλbk
ΛUL
kZ
0
xexp nγULxαk(N0+Pk|hRS I ,k|2)
(ρkβdαj)βΥ(x)
K
X
j=2
πλbj(Pj/Pk)2j
x2αkjodx, (39)
where ΛUL
k,γUL, and |hRS I ,k|2are given in (20),(38) and (2), respectively. In (39),Υ(.)is given
as
Υ(x) =2πλbM
UM
X
ν=1 UM
νγUL xαkPM
ρkxαkβUMαMΓν2
αMΓUMν+2
αM
Γ (UM)+
K
X
j=2
2πλbj(γULPjxαk
ρkxαkβ
x2αj
(αj2)2F11,12
αj
; 2 2
αj
;γULPj
ρkxαkβx(αkαj)+Z
0Zr2
0
1
1 + γULx(1)αk
ρkρjβ1uαj/2rαj
π
K
X
j=2
λbje
π
K
P
j=2
λbjudu rdr).
(40)
Proof. The proof follows analogous steps to Theorem 1.
17
F. UL Area Spectral Efficiency
In this section, we derive the UL ASE in the K-tier HCNs. The UL ASE measures the capacity
of HCNs in the UL, given as
ASEUL =
K
X
k=2
λbkCUL
k(RUL) ln(1 + γUL),(41)
where CUL
k(RUL)and γUL are given in (39) and (38), respectively.
IV. ASYMPTOTIC PERFORMANCE EVALUATION: MASSI VE M ULTIUSER MIMO REGIME
In this section, we analyze the asymptotic performance of K-tier HCNs in which MBSs
are equipped with massive multiuser MIMO antennas. The large number of antennas focusses
energy into ever smaller regions of space to bring huge improvements in throughput and energy
efficiency. We refer to the massive multiuser MIMO regime as the case where 1< UMN.
A. SINR Models
1) DL SINR of a Macrocell MU: With massive multiuser MIMO at the MBSs, the SINR for
a typical DL MMU defined in (5) can be simplified to
SINRDL
MmM=
PMβXo,uM
0
αM
IMmM,uM
0+IS,uM
0+Ius
ul,uM
0
|{z }
IuM
omM
+N0
,(42)
where the massive multiuser MIMO gain, NUM+ 1, and the impact of equal power allocation
per backhaul stream (i.e., the denominator of MBS’s transmit power PM
UM) have already been incor-
porated in (45). In (42), IMmM,uM
0=PxΦM
b\o
PM
UMβhx,uM
oXx,uM
0
αM(a)
PxΦM
b\oPMβXx,uM
0
αM,
the approximation in (a) results due to the fact that with the large number of UM, i.e., (1 <
UMN), the small scale channel fading vanish by the channel hardening effect as in [19]. In
(42), IS,uM
0and Ius
ul,uM
0are given in (7) and (8), respectively.
2) DL SINR of a Small Cell MU: For the massive multiuser MIMO, the DL SINR for a
typical small cell MU defined in (9) can be given as
SINRDL
kmM=
Pkgo,uk
0βXo,uk
0
αk
IMmM,uk
0+IS,uk
0+Ius
ul,uk
0+N0
,(43)
where IMmM,uk
0PxΦM
b\oPMβXx,uk
0
αM, i.e., there is no short-term fading factor due to
channel hardening effect. In (43), IS,uk
0and Ius
ul,uk
0are given in (11) and (12), respectively.
18
3) UL SINR of a Small Cell MU: For the massive multiuser MIMO case, the UL SINR for
a typical SBS given in (13) can be written as
SINRUL
kmM=
ρkgo,bk
0βXo,bk
0
αk(1)
PRSI (Pk) + IMmM,bk
0+IS,bk
0+Ius
ul,bk
0+N0
,(44)
where IMmM,bk
0PxΦM
b\oPMβXx,bk
0
αM, i.e., there is no short-term fading factor due to
channel hardening effect. In (44), PRSI (Pk),IS,bk
0, and Ius
ul,bk
0are given in (1), (11) and (12),
respectively.
B. Asymptotic DL Rate Coverage Probability
In this analysis, we use the following formula for the DL rate coverage probability of the
macrocell with massive multiuser MIMO [51]
CM(RDL)mM=E
Xo,uM
0
ESINRDL
MhPr hSINRDL
MXo,uM
0γDL
MmMXo,uM
0i,(45)
where
γDL
MmM=γDL
Ψ,(46)
and γDL and Ψare given in (23) and (18), respectively. The DL and UL rate coverage probabilities
definitions for the small cells in the massive multiuser MIMO case will be the same as in the
multiuser MIMO case, which are defined as (24), and (37), respectively.
We present the asymptotic DL rate coverage probability of a typical MU associated with the
MBS and the kth tier SBS in Theorem 4and Theorem 5, respectively.
Theorem 4. For the massive multiuser MIMO regime, the DL rate coverage probability of a
typical MU associated with the MBS is derived as
CM(RDL)mM=2πλbM
ΛMZ
0
x1
21
πZ
0
Imhexp χ1(x, w) πλbMχ2(x, w)
K
X
j=2
2πλbj
{χ3(x, w) + χ4(x, w)} − πλbMx2π
K
X
j=2
λbkPj
ΨPM2/αj
r2αM/αjidw
wdx,
(47)
19
where
χ1(x, w) =jw PMβ
γDL
MmMxαMNo!,(48)
χ2(x, w) =
Γ12
αM+2
αMΓu2
αM,jwPMβ
xαM
(jwPMβ)2
αM
x2,(49)
χ3(x, w) = Pjαj/2(jw)β(xαM
ΨPM)2j1
αj2!2F11,12
αj
; 2 2
αj
;jwΨPM
xαM,(50)
χ4(x, w) = Z
0Zr2
0
1
1+(jwρjβ(1))1uαj/2rαjπ
K
X
j=2
λbjeπ
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr,
(51)
and ΛM,γDL
MmM,Ψ, and λIΦUL
bj
(r)are given in (17),(46),(18), and (A.8), respectively.
Proof. See Appendix B.
Theorem 5. For the massive multiuser MIMO regime, the DL rate coverage probability of a
typical MU associated with the kth tier SBS is derived as
CDL
k(RDL)mM=2πλbk
ΛDL
kZ
0
x"1
21
πZ
0
Imexp njwNoπλbM$1(x, w)
K
X
j=2
2πλbj
{$2(x, w) + $3(x, w)} − π
K
X
j=2
λbjPjxαk
Pk2
αjπλbMPMΨ
Pk2
αMx
2αk
αMo
1 + jwPkβ
γDLxαk1#dw
wdx, (52)
where
$1(x, w) =
Γ12
αM+2
αMΓu2
αM,jwPMβ
(DM
k(x))αM
(jwPMβ)2
αM
(DM
k(x))2,(53)
$2(x, w) = (Pj
Pk)2j(jw)βPk(xαk)2j1
αj2!2F11,12
αj
; 2 2
αj
;jwβPj
xαk(Dk
j(x))αkαj,
(54)
$3(x, w) = Z
0Zr2
0
1
1+(jwρjβ(1))1uαj/2rαjπ
K
X
j=2
λbjeπ
K
P
j=2
λbjuλIΦUL
bj
(r)du rdr,
(55)
and ΛDL
k,γDL,Ψ,DM
k(x), and λIΦUL
bj
(r)are given in (19),(23),(18),(33), and (A.8), respectively.
20
Proof. The proof follows analogous steps to Theorem 4.
C. Asymptotic UL Rate Coverage Probability
We present the asymptotic UL rate coverage probability of a typical MU associated with the
kth tier SBS in Theorem 6.
Theorem 6. For the massive multiuser MIMO regime, the UL rate coverage probability of a
typical MU associated with the kth tier SBS is derived as
CUL
k(RUL)mM=2πλbk
ΛUL
kZ
0
x"1
21
πZ
0
Imexp njwNoπλbMϑ1(x, w)
K
X
j=2
2πλbj{ϑ2(x, w)
+ϑ3(x, w)} −
K
X
j=2
πλbjx
2αk
αjPj
Pk2
αjo1 + jwρkxαkββ
γULxαk1#dw
wdx,
(56)
where
ϑ1(x, w) =(jwPMβ)2
αMΓ12
αM,(57)
ϑ2(x, w) =(jwPjβ)x2αj
(αj2)2F11,12
αj
; 2 2
αj
;jwβPj
xαkx(αkαj),(58)
ϑ3(x, w) = Z
0Zr2
0
1
1 + jwρjβ11uαj/2rαjπ
K
X
j=2
λbjeπ
K
P
j=2
λbjudu rdr, (59)
and ΛUL
kand γUL are given in (20) and (38), respectively.
Proof. The proof follows analogous steps to Theorem 4.
V. PERFORMANCE COMPARISON WITH THE CONVENTIONAL HD HCNS
In order to compare the performance of the proposed HCNs with FD SBSs with that of the
conventional HD HCNs with HD SBSs, we define the total ASE of a random MU in HCNs as
ASE =ASEDL +ASEUL,(60)
where ASEDL and ASEUL are given in (35) and (41), respectively. Furthermore, we define the
ASE of FD small cell MUs as
ASESBS =ASEDL
SBS +ASEUL,(61)
where ASEDL
SBS and ASEUL are given in (35) and (41), respectively.
21
TABLE II: Parameter Values unless specified
Parameter Value Parameter Value Parameter Value Parameter Value
λbM5×105µ20 PM40 dBm P233 dBm
αM3.5α24ρ240 dBm 0.9
RDL 0.5 RUL 0.5N0100 dBm PkhLI 0
6 6.5 7 7.5 8 8.5 9 9.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
N
Exact Simu.
Asymp. Simu.
Exact Analysis
Asymp. Analysis
Rate Coverage Probability
DL Macrocell
DL Small cell
UL Small cell
(a) Rate coverage probability versus small N(multiuser MIMO).
100 200 300 400 500 600 700 800 900 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
N
Exact Simu.
Asymp. Simu.
Exact Analysis
Asymp. Analysis
Rate Coverage Probability
DL Macrocell
DL Small cell
UL Small cell
(b) Rate coverage probability versus large N(massive multiuser
MIMO).
Fig. 2: Rate coverage probability versus the number of MBS antennas with UM= 5.
VI. NUMERICAL RE SU LTS
In this section, we investigate the system performance in the DL and the UL in terms of the
rate coverage probability and the ASE of HCNs with multiuser MIMO antennas at the MBSs
and FD operation at the SBSs. We compare the performance of HCNs with multiuser MIMO
at MBSs and FD at SBSs with that of massive multiuser MIMO at MBSs and FD at SBSs. We
plot the DL rate coverage probability, the DL ASE, the UL rate coverage probability, and the
UL ASE using (21), (35), (36), and (41), respectively. We validate the accuracy of the derived
expressions for a two-tier HCNs with network radius An=π(1000)2km2consisting of HD
macrocells with density λbMand FD small cells with density λb2, via Monte Carlo simulations.
The interference approximations in Section II-H are not made in the simulation. The simulation
is repeated and averaged over 10,000 iterations. The results presented in the figures of this
22
section validate the accuracy of our approach to characterize the interferences and show that the
assumptions made have a minor effect on the accuracy of the proposed analytical model. Unless
specified, the parameter values used in this section are listed in Table II.
A. Impact of number of multiuser MIMO/massive multiuser MIMO antennas at the MBS on the
DL and UL Rate Coverage Probability
Fig. 5(a) and Fig. 5(b) compare the DL and the UL rate coverage probability with the multiuser
MIMO at the MBS to that with massive multiuser MIMO at the MBS as a function of the number
of antennas at the MBS. We see that the asymptotic rate coverage probability of small cell MU
closely matches the exact rate coverage probability in DL and UL both for small and large N.
This observation can be attributed to the fact that changing Nat the interfering MBS does not
change the distributions of short term fading factors hx,uk
0in (10) and hx,bk
0in (14), for the exact
case, which are ignored in the asymptotic case due to channel hardening effect. As expected, the
DL rate coverage probabilities of the macrocell MU and the small cell MU in massive multiuser
MIMO case is higher than those in multiuser MIMO case due to the large antenna array gain.
However, the UL rate coverage probability of MU remains constant with increasing Nfor both
small and large Ndue to that: 1) the UL MU can only associate with the SBSs, and 2) the
interferences from NMBS antennas do not add coherently such that for the same total transmit
power, the interference level from a MBS to an UL MU is the same, regardless of the number
of Nunder i.i.d. Rayleigh fading channels.
B. Impact of number of SBSs density on the DL and UL rate coverage probability
Fig. 3 compares the DL and the UL rate coverage probability with massive multiuser MIMO at
the MBS as a function of the ratio between the SBSs density to the MBSs density (µ=λb2bM).
The increase in λb2improves the DL rate coverage probabilities of macrocell MU and small
cell MU. This is according to the the fact that increasing λb2decreases the distance between
the typical small cell MU and the serving SBS. Thus, the MUs transmit with less power due to
distance-proportional fractional power control, which in turn reduces the UL interference for the
macrocell MU and the small cell MU. However, increasing λb2decreases the UL rate coverage
probability due to the increased interference from larger number of SBSs.
23
10 15 20 25 30 35 40 45 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Rate Coverage Probability, N=128,
UM=2
µ
Simu. DL HCNs
Simu. DL Small cell
Simu. DL Macrocell
Simu. UL Small cell
Asymp. Analysis
Fig. 3: Rate coverage probability versus the
ratio between SBSs density to MBSs density.
N
100 200 300 400 500 600 700 800 900 1000
×10
-4
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Area Spectral Efficiency (nats/sec/Hz.m )
2
FD HCNs
HD Small Cells
FD Small Cells
HD HCNs
Asymp. Analysis
Fig. 4: ASE versus the number of MBS anten-
nas with UM= 5.
C. Performance comparison of the proposed HCNs with the conventional HCNs
Fig. 4 compares the ASE of the proposed HCNs with FD SBSs to that of the conventional
HCNs with HD SBSs. We plot the ASE of a random MU in the proposed HCNs using (60),
and that in the conventional HCNs using ASEDL in (35) with no UL interference from the MUs,
i.e., Ius
ul,uM
0= 0 in (5) and Ius
ul,uk
0= 0 in (9). We plot the ASE of a small cell MU of the
proposed HCNs using (61), and that of conventional HCNs using ASEDL
SBS in (35) with no UL
interference from the MUs, i.e., Ius
ul,uk
0= 0 in (9). The ASE of the proposed HCNs is observed
to be higher than that of the conventional HCNs. This suggests the ASE improvement brought
by simultaneous transmission in DL and UL due to FD SBSs which dominates the resulting
additional interferences. With the increase in the number of antennas at the MBSs, the ASE of
the HCNs increases due to the increase in the rate coverage probability with larger Nas shown
in Fig. 2b. Moreover, similar trends are observed for the small cell tier with improved ASE than
that of the HCNs.
D. Impact of SBS density with different number of MBS antennas on the DL Performance
Fig. 5(a) and Fig. 5(b) examine the trade-off between the DL ASE and the DL rate coverage
probability versus the ratio between density of SBSs to density of MBS (µ=λb2bM)and
the number of MBS antennas both for multiuser MIMO and massive multiuser MIMO at the
24
0.5 0.52 0.54 0.56 0.58 0.6 0.62
0
1
2
3
4
5
6
7
8x 10−4
Rate Coverage Probability
Simu. N=4,
DL HCNs
DL Small Cell
µ=10
µ=50
DL Macrocell
µ=50 µ=10
µ=10
µ=50
Exact Analysis
UM=2
Simu. N=6,
UM=2
Area Spectral Efficiency (nats/sec/Hz.m
2
)
(a) ASE versus Rate Coverage Probability, small N(multiuser
MIMO).
0.55 0.6 0.65 0.7 0.75 0.8 0.85
1
2
3
4
5
6
7
8
9
10 x 10−4
Rate Coverage Probability
Area Spectral Efficiency (nats/sec/Hz.m
DL HCNs
DL Small Cell
µ=10
µ=50
µ=50
µ=10
µ=10
DL Macrocell
µ=50
Simu. N=64,
Asymp. Analysis
UM=5
Simu. N=128,
UM=5
2
)
(b) ASE versus Rate Coverage Probability, large N(massive
multiuser MIMO).
Fig. 5: The tradeoff between the ASE and the rate coverage probability for various number of
MBS antennas with PkhLI = 0.
MBSs. In Fig. 5(a) and Fig. 5(b), we consider = 0 and the transmit power at the MU Puis
taken as 23 dBm. Clearly, the DL ASE and rate coverage probability with massive multiuser
MIMO at the MBS are higher as compared to those with multiuser MIMO at the MBSs due
to sharp beamforming. The massive number of antennas at MBSs brings the higher DL rate
coverage probability and ASE. The DL ASE and the rate coverage probability of macrocell
MU decreases with increasing the SBSs density due to the increased interference from SBSs.
However, increasing the SBSs density λbkincreases the DL ASE, but decreases the DL rate
coverage probability of small cell MU. With the increase in the SBSs density, the number of DL
small cell transmissions and the aggregate interference from small cells increase, which results
in a trade-off between DL ASE and rate coverage probability for the small cell MUs. We have
shown that the UL interference can be reduced by employing UL power control in the UL,
which improves the DL rate coverage probability in Fig. 3.
E. Impact of SBS density with different MBS and SBS transmit powers on the DL and UL
Performance
Fig. 6 plots the DL and UL ASE and rate coverage probability as a function of the transmit
powers at the MBSs and SBSs. In Fig. 6, we consider = 0 and the transmit power at the MU
25
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
Rate Coverage Probability
M=40 dBm, P =30 dBm
P
µ=10
µ=50
µ=50
µ=10
µ=10
µ=50
µ=50
DL HCNs
DL Small Cell
DL Macrocell
UL Small Cell
2
M=40 dBm, P =37 dBm
P2
M=46 dBm, P =30 dBm
P2
Asymp. Analysis
µ=10
Area Spectral Efficiency (nats/sec/Hz.m
2
)
Fig. 6: The tradeoff between the ASE and the rate coverage probability for various MBS and
SBS transmit powers with N= 128 and UM= 5.
Puis taken as 23 dBm. Increasing the MBS transmit power increases the DL ASE and the rate
coverage probability of all tiers, which is due to the increase of SINRDL
Min (5), and the reduced
distance between the typical small cell MU and the associated SBS. Moreover, we observe the
decrease in the UL ASE and the UL rate coverage probability with the increase in PMand
Pk, which is due to the increased cross-tier and co-tier interferences as can be seen from (13).
Furthermore, we observe that the increase in the SBS density increases the UL rate coverage
probability in contrast to the decreased DL rate coverage probability for small cell MU as shown
in Fig. 5a and Fig. 5b, which is due to the decreased distance between the UL small cell MU
and the serving SBS. It can thus be concluded that the SBS density and the BS transmit power
of each tier can be tuned to achieve joint DL and UL performance gains with FD SBSs.
F. Impact of SI cancellation capability with different SBS transmit power on the DL and UL
Performance
Fig. 7 examines the impact of the SI cancellation capability LdB on the DL and UL rate
coverage probabilities. As expected, increasing LdB decreases the UL rate coverage probability
of the small cell MU. Moreover, increasing the SBS transmit power decreases the UL rate
coverage probability of the small cell MU, due to the increased self interference. However,
26
-120 -115 -110 -105 -100 -95 -90 -85 -80 -75 -70
Rate Coverage Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
LdB
=30 dBm
P
2=37 dBm
P
2
Simu.
Simu.
Asymp. Analysis
DL HCNs
UL Small Cell
Fig. 7: Rate coverage probability versus SI can-
cellation capability for various SBSs transmit
powers with N= 128 and UM= 5.
ρ
-60 -50 -40 -30 -20 -10 0
Rate Coverage Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
=0.9
Simu. with UL-PC,
ε
ε
Simu. without UL-PC
=1
Asymp. Analysis
Simu. with UL-PC,
DL Macrocell
UL Small cell
DL Small cell
Fig. 8: Rate coverage probability versus SBSs
receivers sensitivity for various SBSs power
control factors with N= 128 and UM= 5.
increasing the SBS transmit power increases the DL rate coverage probability of a random MU,
due to the increase of SINRDL
kin (44).
G. Impact of receiver sensitivity at the SBS with different power control factors
Fig. 8 plots the DL and the UL rate coverage probability versus the receiver’s sensitivity at
SBSs ρ2for various power control factors . Increasing ρ2increases the UL rate coverage
probability, and degrades the DL rate coverage probability. This is due to the reason that
decreasing the the SBS receiver sensitivity (i.e., an increase in ρ2) increases the transmit power
required at each MU to perform channel inversion towards serving SBS, which in turn increases
the useful signal power at the its associated SBS and the interference at the other BSs and
MUs. Similarly, higher power control factor improves the UL performance, but degrades the
DL performance. These results demonstrate that ρ2and can be optimized for joint DL and
UL performance gain. We also compare the DL and UL performance of HCNs with UL power
control to that without UL power control when the MUs transmit power is Pu= 23 dBm. The
UL rate coverage probability in HCNs without UL power control is shown to be very small due
to the increased inter-cell interference from the UL MUs.
27
VII. CONCLUSION
In this paper, we have presented a tractable model for massive multiuser MIMO-enabled HCNs
with FD small cells. Relying on stochastic geometry, we have derived the analytical expressions
for DL rate coverage probability and ASE for macrocell and small cells, and UL rate coverage
probability and ASE for small cells. We have also presented the asymptotic expressions as the
number of antennas at MBS goes to infinity. Numerical results demonstrated the benefits brought
by massive multiuser MIMO in achieving high rate coverage probability and the benefits brought
by of FD SBSs in achieving high ASE. It is shown that the SBSs density and the number of
antennas at the MBSs can be used as design parameters to target optimal DL ASE and DL
rate coverage probability. The results also demonstrate that, to achieve similar performance in
the DL and the UL, UL power control should be employed. With the advancements of massive
multiuser MIMO and SI cancellation in FD, the proposed HCNs will prove to be a promising
candidate for 5G systems.
APPENDIX A
PROOF OF THEOREM 1
From (22), the rate coverage probability of the macrocell tier is given by deconditioning over
Xo,uM
0as
CM(RDL) = Z
0
Pr SINRDL
M||Xo,uM
0|=xγDLXo,uM
0f|Xo,M|(x) dx(A.1)
where f|Xo,M|(x)is the PDF of the distance between a typical MU and its serving MBS given
by [46] as follows
f|Xo,M|(x) =2πλbM
ΛM
xexp πλbMx2π
K
X
j=2
λbjPj
ΨPM2/αj
x2αM/αj,(A.2)
Substituting SINRDL
M(x) from (5) into (A.1) and simplifying we obtain
Pr PM
UMβgo,u0MxαM
IuM
0+N0
> τ!(a)
=Z
0
eτUMxαM(γDL +N0)
PMβ
NUM
X
n=0 τUMxαM(γDL+N0)
PMβn
n!dPr(IuM
0γDL),
(b)
=
NUM
X
n=0
(xαM)n
n!(1)!
dneτUMxαMN0
PMβLIuM
0τUMq
PMβ
dqnq=xαM
,(A.3)
28
where (a) follows from go,uM
0Γ (NUM+ 1,1) and (b) follows from some some mathematical
manipulations. In (A.3), LIuM
0
is the Laplace transform of the PDF of IuM
0given as
LIuM
0
(s) = LIM,uM
0
(s)LIS,uM
0
(s)LIuS
ul,uM
0
(s),(A.4)
where LIM,uM
0
(s),LIS,uM
0
(s), and LIuS
ul,uM
0
(s)are the Laplace transform of the PDF of IM,uM
0,
IS,uM
0, and IuS
ul,uM
0, respectively. In (A.4), LIM,uM
0
(s)is derived as
LIM,uM
0
(s) =Eexp sX
mΦM
b\0
PM
UM
hm,uM
0β|Xm,uM
0|αM
(a)
= exp 2πλbM
UM
X
ν=1 UM
νZ
xPM
UMβνsν(rαM)ν
1 + sPM
UMβrαMUMrdr,(A.5)
where (a) is obtained by using generating functional of PPP [52], hm,uM
0Γ (UM,1) , and using
Binomial expansion. Likewise, LIS,uM
0
(s)is evaluated as
LIS,uM
0
(s) = exp
K
X
j=2
2πλbjZ
DM
j(x)sPjβrαj
1 + sPjβrαjrdr,(A.6)
where DM
j(x)is the distance between a typical MU and the closest interfering BS in the jth tier
given in (30). In (A.4), LIuS
ul,uM
0
(s)is evaluated as
LIus
ul,uM
0
(s)(a)
= exp
K
X
j=2
2πλbjZ
0
(1 exp(πλbj
AUL
j
r2))ERjβ1Rαjrαj
1 + jβ1Rαjrαjrdr,
(A.7)
where (a) follows from the probability generating functional of a PPP and the fact that the UL
interference field is a non-homogeneous PPP with distance dependent density function given as
λIΦUL
bj
(r) = λbj(1 exp(πλbj
AUL
j
r2)) (A.8)
where (AUL
j=λbj/
K
P
i=2
λbi)is the repulsion parameter as in [53]. In (A.8), the integral has a
lower limit of zero as the nearest UL MU of FD SBS can be arbitrarily close to the typical
macrocell MU. Using the PDF of serving link distances given in (C.1), we derive LIus
ul,uM
0
(s).
Plugging (A.5), (A.6) and (A.7) into (A.4), after some manipulations, LIuM
0
(s)is derived as
LIuM
0
(s) = eζ(s),(A.9)
where ζ(s)is given by (27). Substituting (A.9) into (A.3), simplifying using the Faa di Bruno’s
formula, and finally plugging into (A.1), we obtain (25).
29
APPENDIX B
PROOF OF THEOREM 2
The rate coverage probability of a typical MU associated with the kth tier SBS is evaluated
following the similar steps as of Theorem 1 with the PDF of the distance between a typical DL
MU and its serving SBS is given by [46]
f|Xo,k|(x)|DL =2πλbkrexp π
K
X
j=2
λbj(Pjrαk/Pk)2jπλbMPMΨ
Pk2M
r2αkM,(B.1)
where Ψis given in (18).
APPENDIX C
PROOF OF THEOREM 3
The UL rate coverage probability of a typical MU associated with the kth tier SBS is evaluated
following the similar steps as of Theorem 1 with the PDF of the distance between a typical UL
MU and its serving SBS is given by [46]
f|Xo,k|(x)|U L = 2π
K
X
j=2
λbjxexp nπ
K
X
j=2
λbjx2o.(C.1)
APPENDIX D
PROOF OF THEOREM 4
Based on (42), the DL rate coverage probability of the macrocell with massive multiuser
MIMO at the BSs, can be given as
CM(RDL)mM =
Z
0
FIuM
0mMPMβ
γDL
MmM|Xo,uM
0|αMN0f|Xo,M|(x) dx, (D.1)
where we resort to apply the Gil-Pelaez inversion theorem [54] and the CDF of the interference
FIuM
0mM
(.)can be derived as
FIuM
0mM
(x) =1
21
π
Z
0
Im"LIMmM,uM
0
(jw)LIS,uM
0
(jw)LIus
ul,uM
0
(jw)
exp jwPMβ
γDL
MmMxαMN0#dw
w,(D.2)
where Im(.)represents the imaginary part of the argument. In (D.2), the Laplace transform of
IMmM,uk
0can be derived as under
LIMmM,uM
0
(jw) (a)
= exp 2πλbM
Z
x1exp{−(jw)PMβrαM}rdr,(D.3)
30
where (a) follows from the probability generating functional of PPP. Solving the integral in
(D.3) we derive LIMmM,uk
0
(jw). In (D.2), the Laplace transforms of IS,uM
0and Ius
ul,uM
0can be
evaluated as (A.6) and (A.7), respectively. Finally substituting LIMmM,uM
0(jw),LIS,uM
0
(jw),
and LIus
ul,uM
0
(jw) into (D.2), and plugging (D.2) into (D.1), we obtain Theorem 4.
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... The downlink coverage performances of SU-MIMO and MU-MIMO transmission schemes have been widely studied in the literature [29], [31], [32], [33], [34], [35], [53], [30] utilizing stochastic geometry tools. Locations of the BSs are generally modeled using Poisson Point Process (PPP) and users are assumed to be connected to the nearest BS [29]. ...
... Therefore, the exact coverage probability of MIMO requires the calculation of the n-th derivative of Laplace transform of aggregate interference [31]. In an eort to calculate the n-th derivative of Laplace transform of interference, various approaches have been investigated such as Taylor expansion [32], Faa di Bruno's formula [33,34,30] and Toeplitz matrix representation [31], [35]. ...
... derived coverage probability of downlink MIMO cellular networks in[34] is also not in closed form and contains an integral operation. Although the methods proposed in[31],[33],[35] &[30] lead to closed-form solutions, the derived expressions are complicated and include special functions. ...
Thesis
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Cellular wireless communications systems transformed the way people communicated by combining communication and mobility. Since the early 1970s, the mobile industry has been driving the development and evolution of cellular technology. The rst generation only offered voice communication, whereas the second generation offered both voice and data services. Furthermore, as cellular wireless technology advanced, 3G enabled video conferencing and other services. With rising demand, 4G emerged, ensuring us with ultra-broadband internet access. 5G will be able to provide us with services that we have never had before. Cellular wireless networks are typically modeled by placing base stations on a regular hexagonal grid and mobile users either randomly or deterministically distributed. These models have been widely used, but they have the disadvantage of being both highly idealized and not very tractable, so complex system-level simulations are used to assess coverage probability and rate. Models that are more tractable have long been desired. In this dissertation, we develop novel methods based on stochastic geometry to analyze the coverage and average spectral e ciency of downlink cellular wireless networks. In the rst part of the dissertation, we consider to analyze the performance of SISO (Single Input Single Output) transmission scheme. Regarding this analysis, both the positions of base stations (BS) and mobile users are modeled as points of an independent Poisson Point Process' (PPP). We consider to use the max C/I (Carrier to Interference Ratio) scheduling method, to select the user with the best SIR (Signal to Interference Ratio) value among the users that are connected to the same base station. Since the base stations are randomly placed and users are linked to their closest base stations, the C/I ratios of multiple users become conditionally dependent. For such a realistic deployment scenario, we utilize stochastic geometry tools and copula functions to take into account this correlation and analyze the coverage probability and spectral e ciency. Our simulation results demonstrate that the commonly used independence assumption of the user C/I values overestimates both actual coverage probability and spectral e ciency of the system. In the second part, we analyze the coverage performance of multi-user multipleinput multiple-output (MU-MIMO) downlink cellular networks. We model the locations of the base stations and mobile users using Poisson Point Processes and employ maximum ratio transmission precoding at the base station. Our analysis builds upon Prony's method to approximately calculate the coverage probability using the sum of exponentials which facilitates the computation of Laplace transform of both intra-cell and inter-cell interference. We demonstrate the accuracy of derived approximate coverage probability expressions through numerical simulations for different MU-MIMO con gurations. In the third part, we analyze the downlink coverage performance of general user, arbitrarily located in the 2-D region, edge user and worst user sit on the boundary between two neighbor cells and vertex where three neighbor cells meet respectively, considering the Coordinated Multipoint (CoMP) transmission scheme by using stochastic geometry approach. In addition, we analyze the performance of two users, in terms of average spectral e ciency in the CoMP transmission scheme where the users are connected to the closest and second closest BS's, by using stochastic geometry approach. The correlation of SIR values of two users are again modeled by using copula functions. The max C/I scheduling method is employed to select the user in this scheme.
... aggregate interference [4]. In an effort to calculate the n-th derivative of Laplace transform of interference, various approaches have been investigated such as Taylor expansion [5], Faa di Bruno's formula [6], [7], [10] and Toeplitz matrix representation [4], [8]. ...
... The work in [5] focuses on multi-antenna ad-hoc networks and presents an outage probability expression containing integral and nested sum expressions. The derived coverage probability of downlink MIMO cellular networks in [7] is also not in closed form and contains an integral operation. Although the methods proposed in [4], [6], [8] & [10] lead to closedform solutions, the derived expressions are complicated and include special functions. ...
... Np n=1 a n exp(b n θr α ij (I i1 + I i2 )N ) (7) where N p denotes the number of exponential functions used in the approximation. Therefore, we can express the average of coverage probability conditioned on distance, I i1 and I i2 as: ...
... Wireless backhauling through full-duplex communication has the potential to surmount the limitations [9]. Although none of the technologies individually can fulfill the expectations of next-generation wireless networks [6,10], a combination of the aforementioned technologies has recently attracted the interest of researchers [10][11][12]. In addition, a major part of research has concentrated on how to reduce power consumption of a next-generation network. ...
... Here, MBS equipped with m-MIMO is used to backhaul data to SBSs and to serve the users simultaneously. Besides, SBSs in the proposed model are equipped with fullduplex hardware [11]. Hence, SBS can receive data from MBS and transmit data to users in the same spectrum. ...
... It is assumed that the MBS and SBSs are operating in half-duplex mode and full-duplex mode, respectively. In-band full-duplex communication is undertaken where portion of bandwidth is allocated for access link between the MBS and user and (1 − ) for backhaul link (from MBS to SBS) and access link (from SBS to user) [11,25]. Imperfect CSI is assumed to be available from MBS to user link due to the mobility of users. ...
Article
Full-text available
In this paper, a massive MIMO enabled backhaul model for two-tier heterogeneous networks with in-band full-duplex transmission and imperfect CSI is provided. Because of a massive number of antennas at the macro base station and densification of small cells, circuit power consumption increases in the system. It is requisite to study energy efficiency of such backhaul networks. This paper aims at maximizing EE for optimal user association, spectrum allocation, and power allocation while accounting for QoS and backhaul constraints. A joint optimization problem is formulated as a non-convex mixed integer non-linear problem which provides an un-affordable computational complexity. To crack the problem efficiently, it is progressively divided into sub-problems until each one is convex and is solved separately to obtain the optimal solution. Furthermore, complexity of the proposed algorithm is evaluated and Jain’s fairness index is measured. Simulation results of the proposed distributive algorithm are shown and compared with that of the existing algorithm (Max SINR) to manifest the strength of the proposed algorithm. The impact of self-interference cancellation factor and channel estimation error on EE is clearly reflected by the numerical results. In addition, it is observed that using the proposed algorithm results in better EE than increasing the number of antennas.
... Therefore, the exact coverage probability of MIMO requires the calculation of the n-th derivative of Laplace transform of [4]. In an effort to calculate the n-th derivative of Laplace transform of interference, various approaches have been investigated such as Taylor expansion [5], Faa di Bruno's formula [6], [7], [10] and Toeplitz matrix representation [4], [8]. ...
... The work in [5] focuses on multi-antenna ad-hoc networks and presents an outage probability expression containing integral and nested sum expressions. The derived coverage probability of downlink MIMO cellular networks in [7] is also not in closed form and contains an integral operation. Although the methods proposed in [4], [6], [8] & [10] lead to closedform solutions, the derived expressions are complicated and include special functions. ...
Article
Full-text available
In this letter, we analyze the coverage performance of multi-user multiple-input multiple-output (MU-MIMO) downlink cellular networks. We model the locations of the base stations and mobile users using Poisson Point Processes and employ maximum ratio transmission precoding at the base station. Our analysis builds upon Prony’s method to approximately calculate the coverage probability using the sum of exponentials which facilitates the computation of Laplace transform of both intra-cell and inter-cell interference. We demonstrate the accuracy of derived approximate coverage probability expressions through numerical simulations for different MU-MIMO configurations.
... This smart system entails all the possible smart devices, gadgets and sensors, all in all actuating a smart and automatized infrastructure. This infrastructure therefore highlights the utilization of the D2D communication network [9]- [11], the massive MIMO [12], small cell access points (SCA) [13], the IoT [14] with the network cloud [15], [16], all in all forming a part of the said 5G/B5G cellular framework [17], [18]. ...
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