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Full-duplex (FD) technology is currently under consideration for adoption in a range of legacy communications standards due to its attractive features. On the other hand, cellular networks are becoming increasingly heterogeneous as operators deploy a mix of macrocells and small cells. With growing tendency towards network densification, small cells are expected to play a key role in realizing the envisioned capacity objectives of emerging 5G cellular networks. From a practical perspective, small cells provide an ideal platform for deploying FD technology in cellular networks due to its lower transmit power, and lower cost for implementation compared with the macrocell counterpart. Motivated by these developments, in this paper, we analyze a two-tier heterogeneous cellular networks (HCNs) wherein the first tier comprises half-duplex (HD) macro base stations (BSs) and the second tier consists of FD small cells. Through a stochastic geometry approach, we characterize and derive the closed-form expressions for the outage probability and the rate coverage. Our analysis explicitly accounts for the spatial density, the SI cancellation capabilities, and the interference coordination based on enhanced inter-cell interference coordination (eICIC) techniques. Performance evaluation investigates the impact of different parameters on the outage probability and the rate coverage in various scenarios.
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IEEE ACCESS - SUBMITTED FOR PUBLICATION 1
Full-Duplex Small Cells for Next Generation
Heterogeneous Cellular Networks: A Case Study of
Outage and Rate Coverage Analysis
M. Omar Al-Kadri, Student Member, IEEE, Yansha Deng, Member, IEEE, Adnan Aijaz, Member, IEEE,
and Arumugam Nallanathan, Fellow, IEEE
Abstract—Full-duplex (FD) technology is currently under con-
sideration for adoption in a range of legacy communications
standards due to its attractive features. On the other hand,
cellular networks are becoming increasingly heterogeneous as
operators deploy a mix of macrocells and small cells. With
growing tendency towards network densification, small cells are
expected to play a key role in realizing the envisioned capacity
objectives of emerging 5G cellular networks. From a practical
perspective, small cells provide an ideal platform for deploying
FD technology in cellular networks due to its lower transmit
power, and lower cost for implementation compared with the
macrocell counterpart. Motivated by these developments, in this
paper, we analyze a two-tier heterogeneous cellular networks
(HCNs) wherein the first tier comprises half-duplex (HD) macro
base stations (BSs) and the second tier consists of FD small cells.
Through a stochastic geometry approach, we characterize and
derive the closed-form expressions for the outage probability and
the rate coverage. Our analysis explicitly accounts for the spatial
density, the SI cancellation capabilities, and the interference
coordination based on enhanced inter-cell interference coordina-
tion (eICIC) techniques. Performance evaluation investigates the
impact of different parameters on the outage probability and the
rate coverage in various scenarios.
Index Terms—full-duplex, enhanced inter-cell interference co-
ordination, heterogeneous cellular networks, outage probability,
rate coverage.
I. INTRODUCTION
IN half-duplex (HD) wireless communications systems, bi-
directional communications between a pair of nodes is
achieved with either frequency division duplexing (FDD) or
time division duplexing (TDD). The former technique employs
different frequency bands for the uplink (UL) and downlink
(DL), whereas, in the latter technique, a single channel is
shared in the time domain for both UL and DL. Such
techniques however are not suitable to fulfil the envisioned
requirements of next generation wireless systems [1]. His-
torically, simultaneous transmission and reception in wireless
communications was deemed infeasible in practice due to the
so called self-interference (SI), which is the interference gener-
ated by the transmitter of a radio on its own receiver. Recent
developments in SI cancellation techniques [2]–[6] have led
M. O. Al-Kadri, Y. Deng, and A. Nallanathan are with the Centre
for Telecommunications Research, King’s College London, London, UK
(email:{yansha.deng, arumugam.nallanathan, mhd_omar.alkadri}@kcl.ac.uk).
A. Aijaz was with the Centre for Telecommunications Research, King’s
College London, London, UK. He is now with the Telecommunications
Research Laboratory, Toshiba Research Europe Ltd., Bristol, UK (email:
adnan.a.aijaz@ieee.org).
to the practical realization of FD radios. The feasibility of
single-array FD transceivers has been presented in [7], [8].
FD technology has a number of attractive features e.g., it
can potentially double (theoretically) the ergodic capacity [9],
[10], reduce the feedback delay [11], decrease the end-to-end
delay [12], improve the network secrecy [13] and increase the
efficiency of network protocols (e.g., medium access control
[14]).
On the other hand, small cells are gaining increasing pop-
ularity in the next generation cellular systems. Small cells
provide an easy and cost-efficient deployment solution for
capacity and coverage improvements over the conventional
macro-centric networks [15], [16]. The low-powered nature of
small cells make them the ideal candidate for FD deployment
considering that the self-interference (SI) is more manageable
compared to the conventional high-power macro counterparts.
This inspires and motivates the investigation for the feasibility
and performance gains of FD small cells underlay heteroge-
neous cellular networks (HCNs).
An important issue for HCNs is the inter-cell interference,
which arises due to the dense unplanned deployments of small
cells, loud neighbors, and the closed subscriber group access.
To mitigate this interference, 3GPP has recently standardized
the enhanced Inter-Cell Interference Coordination (eICIC)
technique in Release 10 [17]. eICIC provides interference
cancellation techniques in time, frequency, and power control
domains. When the subframes of macro cells and small cells
are aligned, their control and data channel overlap with each
other. Therefore, eICIC mitigates the interference on the
control channel of small cells through Almost Blank Sub-
frames (ABS) at small cells. During ABS, the macro BSs only
transmits the reference signals, which allows small cell BSs
to schedule the associated users without interference from the
macrocells.
A. Related Work
Recent studies on modeling and analysis of HCNs heavily
rely on stochastic geometry framework [18]–[21]. Using these
tools, comprehensive modeling and analysis of legacy HCNs
has been carried out in [22]–[25]. In [26], the authors have pre-
sented the outage probability, the average ergodic rate, and the
minimum average user throughput for a downlink HD multi-
tier HCNs. They have concluded that neither the number of
BSs nor the tiers affect outage probability or average ergodic
IEEE ACCESS - SUBMITTED FOR PUBLICATION 2
rate in an interference-limited full-loaded HCNs with unbiased
cell association. These conclusions, however, may not hold
in environments which are prone to higher interference, like
HCNs comprising FD nodes.
FD-enabled wireless networks have been attracting growing
interest, recently [27]. In [28], the authors have derived the
expression for the throughput of hybrid duplex heterogeneous
networks composed of multi-tier networks, with access points
(APs) operating either in bi-directional FD mode or downlink
HD mode in each tier. The authors have concluded that having
tiers with hybrid duplex BSs degrades the performance, while
higher throughput was achieved when each tier operates in the
same duplex, either HD or FD rather than a mixture of both.
This motivates further research on two-tier HCNs with FD
small cells and HD macrocells, instead of considering hybrid
scenarios. In [29], the authors have derived the downlink rate
coverage probability of a user in a single-tier FD small cell
network with massive MIMO wireless backhauls. In [30],
the authors have introduced an FD-assisted cross-tier inter-
cell interference (ICI) mitigation scheme called fICIC, which
operates on small cells compared to the standardized eICIC
that operates on the macrocells. Such a change may lead
to modifications on the current backhaul affecting feasibility
of application. This motivates further investigation on the
application of eICIC on FD-enabled HCNs to avoid legacy
network modifications. In [31], the authors consider a hybrid
scenario where all BSs operate in FD mode. They derived
a closed-form expression for the critical value of the self-
interference attenuation power, which is required for the FD
users to outperform HD users. In [32], the authors have
considered a single tier mixed small cell network, where BSs
operate in either HD or FD, with all users operating in HD.
The effect of FD cells on the performance of the mixed system
was presented, however, inter-cell interference coordination
was not considered and only single tier was investigated.
B. Contributions and Outline
Different from the aforementioned studies, our objective in
this paper is to model and analyze an interference-coordinated
two-tier HCN with FD small cells. The key contributions of
this paper can be summarized as follows.
We formulate a tractable model for the interference
coordinated two-tier HCN with FD small cells, wherein
tier 1 comprises legacy HD macrocells and tier 2 consists
of FD small cells. By explicitly accounting for spatial
distribution of base stations, self-interference, transmit
power, cell association, uplink power control and ABS
factor, we provide signal-to-interference-plus-noise ratio
(SINR) expressions for users in the corresponding two
tiers. Specifically, the underlying model captures the DL
scenario for tier 1 and both UL and DL scenarios for tier
2, since tier 1 and tier 2 operate in HD and FD mode,
respectively.
Based on the system model for two-tier HCN with
FD small cells, we derive closed-form expressions for
outage probability of different tiers. The final expressions
explicitly account for interference coordination.
Fig. 1: Example cells of the system model, where macro BS
operates in HD mode, and small cells operate in FD mode.
We adopt the notion of rate coverage from [28], and
derive closed-form expressions for the corresponding two
tiers.
We conduct a comprehensive performance evaluation
through numerical as well as simulation studies. We
investigate the impact of various design parameters on
network performance in various scenarios.
The rest of the paper is organized as follows. Section II
provides the system model. In Section III, we analyze the
outage probability of two-tier HCNs with FD small cells. This
is followed by rate coverage analysis in section IV. Numerical
and simulation results are given in Section V. Finally, the paper
is concluded in Section VI.
II. SY ST EM MO DE L
We consider a two-tier HCNs, where tier 1 comprises macro
BSs operating in HD mode, and tier 2 consists of small cells
operating in FD mode, as illustrated in Fig. 1. Both tiers are
spatially distributed in R2following homogeneous Poisson
point processes (HPPPs) ΦS1and ΦS2, with intensities λS1and
λS2, respectively. All users operate in HD mode. The UL small
cell users are spatially located in R2following the HPPP ΦU2,
with intensity λu2. Assuming that the intensity of DL users is
high enough, and each user has data ready for transmission,
such that saturated traffic conditions hold. We also assume that
each small cell BS serves single active uplink user and single
downlink user per channel, and each macrocell BS serves
single active downlink user per channel. This assumption
is justified due to the conclusions in [28], that the highest
network performance is achieved when each tier in the network
operates in the same duplex, rather than having hybrid tiers.
We assume that the UL users share the macro DL frequency
to minimize the interference on the DL users, considering that
the density of small cells is usually significantly higher than
the density of macrocells. The full frequency reuse scenario
is assumed, such that all the cells use the same frequency
band. We assume that the channel coefficients are invariant
in each block and vary between different blocks. Moreover,
we assume that the channel hi,j between any pair of nodes
iand jis impaired by Rayleigh fading, and the path loss is
assumed to be inversely proportional to distance with the path
loss exponent α.
IEEE ACCESS - SUBMITTED FOR PUBLICATION 3
TABLE I: Frequently Used Notations
Notation Definition
ΦSxHPPP of base stations in tier x
ΦUxHPPP of users in tier x
λSxSpatial density of base stations in tier x
λuxSpatial density of users in tier x
SxBase station of tier x(1,2)
S
xAssociated BS of tier x(1,2)
uxUsers of tier x(1,2)
u0
xThe user at origin of tier x(1,2)
RSI Residual self-interference
PyxTransmission power of y(S, u)of tier x(1,2)
ha,b Small scale fading channel coefficient between aand b
Ra,b Distance between aand b
αa,b Pathloss exponent between aand b
IZ
yInterferences caused by y(S, u)in Z(U L, DL)
N0Additive Gaussian noise
ASxUsers association probability with BS of tier x(1,2)
We assume that the FD small cells are equipped with
a single antenna and achieve FD capability through the
techniques mentioned in [33], [34]. A node in FD mode
receives interference from its transmitted signal, and performs
SI interference cancellation. Since the amount of SI depends
on the transmission power at the receiver PS2, we define the
residual self-interference (RSI) power after performing the SI
cancellation as [28], [35], [36],
RSI =PS2HS I ,(1)
where HSI =|hS I |2is the RSI channel gain of the small cell
BS, and indicates the SI cancellation capability of that BS,
where hSI is the SI channel of the BS. Note that RSI = 0
denotes perfect cancellation capability.
The residual self-interfering channel gain HSI in (1) needs
to be characterized based on the applied SI cancellation
algorithm. Here, we consider the digital-domain cancellation,
where hSI can be presented as hSI =hSIcˆ
hSIcwhere
hSIcand ˆ
hSIcare the self-interfering channel and its estimate
as the self-interference is subtracted using the estimate [28],
[35]–[37], which allows HSI to be modeled as a constant
value, such that HSI =σ2
efor the estimation error variance
σ2
e[28], [35], [37]. Other SI cancellation algorithms, such as
analogue domain algorithms [38]–[40] or propagation domain
algorithms [36], [41], [42] will make the modeling of HSI
challenging. Therefore, in our analysis, we consider HSI to be
a constant value. Please note that the analysis can still be easily
extended to the case of random HSI within our framework.
For instance, once the probability density function (PDF) of
HSI is available for a certain SI cancellation algorithm, by
averaging the analytic results presented in this paper, over the
distribution of HSI , the results for the random HSI can be
derived.
We consider the maximum received power cell association
rule in the downlink transmission of HCNs, adopting the
flexible cell association without biasing [26]. In our case, the
association probability Afor the macrocells and the small cells
can be expressed by
AS1= 1 − AS2= 1 1 + λS1
λS2PS1
PS222!1
.(2)
and
AS2=PPr
S2> P r
S1= 1 + λS1
λS2PS1
PS221!1
,(3)
respectively, In (2) and (3), Pr
S1and Pr
S2are the received
power at the associating user from the macrocell and small
cell BSs, respectively. Moreover, α1and α2are the path loss
exponents of macrocells and small cells, respectively.
In this paper, we assume that the HCNs employs eICIC
technique for interference mitigation due to it’s wide usage
and popularity, with ABS transmission factor of ρdefined as
the ratio of ABS transmitted over the total transmitted frames.
A. Downlink SINR of Macrocell User
For a typical macrocell downlink user located at the origin
u0
1, associated with its serving macrocell BS S
1, the SINR is
expressed as
SI N RDL
u1=PS1|hS
1,u0
1|2RS
1,u0
1
α1
IUL
u2+IS2+IDL
S1+N0
,(4)
where
IUL
u2=X
u2ΦU2
Pu2|hu2,u0
1|2Ru2,u0
1
α2,
IS2=X
S2ΦS2
PS2|hS2,u0
1|2RS2,u0
1
α2,
IDL
S1=X
S1ΦS1\S
1
PS1|hS1,u0
1|2RS1,u0
1
α1.
given IUL
u2is the interference from small cell uplink users,
IS2is the interference from small cell BSs and IDL
S1is the
interference from other macrocell BSs.
In (4), Pu2is the transmit power of UL user associated
with small cell, hS
1,u0
1,hu2,u0
1,hS2,u0
1, and hS1,u0
1denote the
small scale fading channel coefficient for the channels of the
typical downlink user and its serving macrocell BS, small cell
users, small cell BSs and other non-associated macrocell BSs,
respectively. Moreover, RS
1,u0
1,Ru2,u0
1,RS2,u0
1, and RS1,u0
1
denote the distances between the typical downlink macrocell
user and its associated macrocell BS, small cell users, small
cell BSs, and other interfering macrocell BSs, respectively.
B. Downlink SINR of Small Cell User
For a typical small cell downlink user located at the origin
u0
2, associated with its serving small cell BS S
2, the SINR
expression is given by
SI N RDL
u2=PS2|hS
2,u0
2|2RS
2,u0
2
α2
IUL
u2+IS2+IDL
S1+N0
,(5)
during non ABS transmission, while SINR expression during
ABS transmission is given by
SI N RDL_AB S
u2=PS2|hS
2,u0
2|2RS
2,u0
2
α2
IUL
u2+IS2+N0
,(6)
IEEE ACCESS - SUBMITTED FOR PUBLICATION 4
where
IUL
u2=X
u2ΦU2
Pu2|hu2,u0
2|2Ru2,u0
2
α2,
IS2=X
S2ΦS2\S
2
PS2|hS2,u0
2|2RS2,u0
2
α2,
IDL
S1=X
S1ΦS1
PS1|hS1,u0
2|2RS1,u0
2
α1.
given IUL
u2is the interference from small cell uplink users,
IS2is the interference from small cell BSs and IDL
S1is the
interference from other macrocell BSs.
In (5) and (6), hS
2,u0
2,hu2,u0
2,hS2,u0
2, and hS1,u0
2denote
the small scale fading channel coefficient for the channels
of the downlink typical small cell user and its serving small
cell BS, small cell users, small cell BSs and macrocell BSs,
respectively. Further, RS
2,u0
2,Ru2,u0
1,RS2,u0
1, and RS1,u0
1
denote the distances between the typical small cell downlink
user and its associated small cell BS, small cell users, other
interfering small cell BSs, and macrocell BSs, respectively.
C. Uplink SINR of Small Cell BS
We assume that UL users utilize distance-proportional frac-
tional power control of the form Rα
x[43], where [0,1] is
the power control factor. Therefore, as users moves closer to
the associated BS, the transmit power required to maintain the
same received signal power decreases, which is a key issue for
battery-limited users.
For a typical small cell BS in the uplink located at the origin
S0
2, the SINR can be expressed as
SI N RU L
S2=Pu2|hu
2,S0
2|2Ru
2,S0
2
α2(1)
RSI +IU L
u2+IS2+IDL
S1+N0
,(7)
during non ABS transmission, while SINR expression during
ABS transmission is given by
SI N RU L_ABS
S2=Pu2|hu
2,S0
2|2Ru
2,S0
2
α2(1)
RSI +IU L
u2+IS2+N0
,(8)
where
IUL
u2=X
u2ΦU2
Pu2|hu2,S0
2|2Ru2,S0
2
α2,
IS2=X
S2ΦS2\S
2
PS2|hS2,S0
2|2RS2,S0
2
α2,
IDL
S1=X
S1ΦS1
PS1|hS1,S0
2|2RS1,S0
2
α1.
given IUL
u2denotes the interference from other small cell
uplink users, IS2is the interference from other small cell BSs
and IDL
S1is the interference from macrocell BSs.
When = 1, the numerator of (7) becomes Pu2|hu
2,S0
2|2,
with the pathloss completely inverted by the power control,
and when = 0, no channel inversion is performed and all
the nodes transmit using the same power.
In (7), hu
2,S0
2,hu2,S0
2,hS2,S0
2, and hS1,S0
2denote the small
scale fading channel coefficient for the channels of small
cell uplink BS and its associated small cell uplink user,
other interfering small cell uplink users, other small cell BSs
and macrocell BSs, respectively. Moreover, Ru
2,S0
2,Ru2,S0
2,
RS2,S0
2, and RS1,S0
2denote the distances between the typical
small cell uplink BS and its associated small cell uplink user,
other interfering small cell uplink users, other small cell BSs
and macrocell BSs, respectively.
III. OUTAG E PROBABILITY ANALYS IS
In this section, we analyze the outage probability of two-tier
HCNs with FD small cells, which is a metric that represents
the average fraction of the cell area that is in outage at any
time. We define the outage probability Oas the probability
that the instantaneous SINR of a randomly located user is less
than a target SINR τ. Since the typical user is associated with
at most one tier, from the law of total probability, the outage
probability is given as
O=
K
X
k=1
OkAk,(9)
where Akis the per-tier association probability given in (3)
and (2), and Okis the outage probability of a typical user
associated with kth tier, and Kdenotes the number of tiers.
For a target SINR τkand a typical user SINRk(x)at a distance
xfrom its associated BS, the outage probability is given by
Ok=E[P[SI N Rk(x)< τk]] .(10)
Considering the chosen network model of HD macrocells
and FD small cells, the expression of the outage probability
becomes
O=ODL
1A1+ (ODL
2+OUL
2)A2,(13)
where ODL
1,ODL
2and OUL
2denote the outage probability of
macrocell downlink user, small cell downlink user, and small
cell uplink BS, respectively, and are derived in the following
section.
A. Outage Probability of Macrocell Downlink User
The probability density function (PDF) of the distance be-
tween the typical macrocell user and the associated macrocell
BS RS
1,u0
1[26], is given by
fRS
1,u0
1
(r) = 2πλS1
AS1
rexp (π
2
X
j=1
λj(PSj
PS1
)
2
αj1),
(14)
where AS1is given in (2).
Theorem 1. The outage probability ODL
1in HCNs com-
prised of HD macrocell and FD small cell, is defined as
the probability that the instantaneous SINR of a randomly
located macrocell downlink user is lower than a target τ1,
and expressed as
IEEE ACCESS - SUBMITTED FOR PUBLICATION 5
ODL
2=1 (2π(1 ρ)λS2
AS2Z
0
rexp nrα2PS2
1N0τ2π(η1r2)+(η2r2
α21)+(η3r2
α12)odr
+2πρλS2
AS2Z
0
rexp nrα2PS2
1N0τ2π(η1r2)+(η2r2
α21)odr),(11)
OUL
2=1 (2π(1 ρ)λS2Z
0
rexp nrα2(1)Pu2
1PS2σ2
eN0τ3π1r2) + (Γ2r2) + (Γ3r2
α12)odr
+ 2πρλS2Z
0
rexp nrα2(1)Pu2
1PS2σ2
eN0τ3π1r2) + (Γ2r2)odr),(12)
ODL
1= 1 (2πλS1
AS1Z
0
rexp nrα1PS1
1N0τ1
π1r2
α21) + (Ψ2r2
α21) + (Ψ3r2)odr),(15)
where
Ψ1=λu2Pu2
PS1222τ1
α222F1[1,12
α2
; 2 2
α2
;τ1],
Ψ2=λS2PS2
PS1222τ1
α222F1[1,12
α2
; 2 2
α2
;τ1],
Ψ3=λS1PS1
PS
1212τ1
α122F1[1,12
α1
; 2 2
α1
;τ1],
with 2F1[·]denote the Gauss hypergeometric function, and the
pathloss exponents αj>2.
Proof. See Appendix A.
B. Outage Probability of Small Cell Downlink User
The PDF of the distance between the typical small cell
downlink user and its associated BS RS
2,u0
2[26], is given
by
fRS
2,u0
2
(r) = 2πλS2
AS2
rexp nπ
2
X
j=1
λj(PSj/PS2)
2
αj2o.
(16)
Theorem 2. The outage probability ODL
2in HCNs com-
prised of HD macrocell and FD small cell, is defined as
the probability that the instantaneous SINR of a randomly
located small cell downlink user is lower than a target τ2,
during transmission of both ABS and non-ABS subframes, and
expressed as (11) at the top of this page, where
η1=λu2Pu2
PS2222τ2
α222F1[1,12
α2
; 2 2
α2
;τ2],
η2=λS2PS2
PS
2222τ2
α222F1[1,12
α2
; 2 2
α2
;τ2],
η3=λS1PS1
PS2212τ2
α122F1[1,12
α1
; 2 2
α1
;τ2],
for the pathloss exponents αj>2.
Proof. See Appendix B
C. Outage Probability of Small Cell Uplink BS
Since macrocells can only service one DL active user at a
time, the UL users can only be associated to the FD small
cells. Therefore, we assume that UL users are associated with
the small cells based on the nearest BS association rule, where
the PDF of the distance between the UL users and the small
cells RS2,uUL
2[26], is given as
fRS2,uUL
2
(r) = eλ2πr22πλ2r. (17)
Theorem 3. The outage probability OUL
2in HCNs comprised
of HD macrocell and FD small cell, is defined as the proba-
bility that the instantaneous SINR of a randomly located UL
small cell BS is lower than a target τ3during both ABS and
non-ABS subframes is given by (12) at the top of this page,
where
Γ1=λu2
2τ3
α222F1[1,12
α2; 2 2
α2τ3]
Γ2=λS2PS2
Pu2222τ3
α222F1[1,12
α2; 2 2
α2τ3]
Γ3=λS1PS1
Pu2212τ3
α122F1[1,12
α1; 2 2
α1;τ3],
for αj>2.
Proof. See Appendix C.
IEEE ACCESS - SUBMITTED FOR PUBLICATION 6
2π(1 ρ)λS2
AS2Z
0
P PS2hS
2,u0
2RS
2,u0
2
α2
I> $!rexp
π
2
X
j=1
λj(Pj
PS2
)
2
αj2r
2
αj2
dr
+2πρλS2
AS2Z
0
P PS2hS
2,u0
2RS
2,u0
2
α2
I0> $!rexp
π
2
X
j=1
λj(Pj
PS2
)
2
αj2r
2
αj2
dr. (18)
IV. RATE COVERAGE ANALYS IS
In this section, we analyze the rate coverage of two-tier
HCNs with FD small cells. The rate coverage is defined in [44]
that the probability that a randomly chosen user can achieve
a target rate $, which is given by
Θ,P(R > $).(19)
Since the DL users can associate with either macro cells or
small cells in open-access mode, the overall rate coverage for
the chosen user in two-tier HCNs is given by
Θo=AS1P(RS1> $|AS1) + AS2P(RS2> $|AS2),(20)
where AS1and AS2denote the probability that a user
is associated with the macrocell or the small cell, and
P(RS1> $|AS1)and P(RS2> $|AS2)denote the rate cov-
erage conditioned on the association with the former and the
latter, respectively.
The rate achieved by a user associated with the tagged BS
in the xth-tier is given by
Rx=W
Nx
log2(1 + SI N Rx),(21)
where Wis the bandwidth of the frequency band, Nxis a
random variable which denotes the average number of users
associated with the tagged base station in the xth-tier, and
SI N Rxis the received signal-to-interference-plus-noise-ratio
from the serving base station for a user.
A. Rate Coverage for Macrocell Users in the Downlink
In Rayleigh fading environments, the rate coverage for a
macrocell DL user is given by
P(RS1> $|AS1)
=ENS1PSI N RDL
u1>2
$Nf
W1|AS1
=X
n0
P PS1hS
1,u0
1RS
1,u0
1
α1
I> $|AS1!
×P(NS2=n+ 1) ,(22)
where I=IUL
u2+IS2+IDL
S1+N0is the cumulative interference
from small cell UL users along with macrocell and small cell
BSs, and the additive Gaussian noise. $= 2
$NS1
W1.
According to [44], the distribution of the load associated
with the xth-tier is given by
P(Nx=n+ 1) =
3.53.5
n!
Γ (n+ 4.5)
Γ (3.5) λuAx
λxn3.5 + λuAx
λxn4.5
,(23)
with the mean load E[Nx] = 1 + 1.28λuAx
λx, where Γ (x) =
R
0tx1etdt is the Gamma function, and Axdenotes the
association probability of the xth-tier. Hence,
P PS1hS
1,u0
1RS
1,u0
1
α1
I> $|AS1!
=Z
0
P PS1hS
1,u0
1RS
1,u0
1
α1
I> $!fRS
1,u0
1
(r)
=2πλS1
AS1Z
0
P PS1hS
1,u0
1RS
1,u0
1
α1
I> $!r
×exp
π
2
X
j=1
λj(Pj
PS1
)
2
αj1r
2
αj1
dr. (24)
Using the derivation of outage probability for macrocell DL
users and (23), the final expression for DL rate coverage of
macrocell users can be obtained through (22).
B. Rate Coverage for Small Cell Users in the Downlink
Following the same derivation approach, the rate coverage
for DL small cell users is given by
P(RS2> $|AS2)
=ENS2ρPSI N RDL_AB S
u2>2
$Nf
W1|AS2
+ (1 ρ)PSI N RDL
u2>2
$Nf
W1|AS2
="(1 ρ)X
n0
P PS2hS
2,u0
2RS
2,u0
2
α2
I> $|AS2!
+ρX
n0
P PS2hS
2,u0
2RS
2,u0
2
α2
I0> $|AS2!#
×P(NS2=n+ 1) ,(26)
where I=IUL
u2+IS2+IDL
S1+N0denote the cumulative
interference from small cell UL users along with macrocell
and small cell BSs, receptively. I0=IUL
u2+IS2+N0is the
cumulative interference during ABS transmission, and $=
2
$NS1
W1.
IEEE ACCESS - SUBMITTED FOR PUBLICATION 7
2π(1 ρ)λS2Z
0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $!rexp
π
2
X
j=1
λj(Pj
PS2
)
2
αj2r
2
αj2
dr
+ 2πρλS2Z
0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I0+RSI > $!rexp
π
2
X
j=1
λj(Pj
PS2
)
2
αj2r
2
αj2
dr. (25)
Using the load distribution given in (23), (27) and (28), we
obtain (18) at the top of this page
(1 ρ)P PS2hS
2,u0
2RS
2,u0
2
α2
I> $|AS2!= (1 ρ)
×Z
0
P PS2hS
2,u0
2RS
2,u0
2
α2
I> $!fRS
2,u0
2
(r).
(27)
ρP PS2hS
2,u0
2RS
2,u0
2
α2
I0> $|AS2!=
ρZ
0
P PS2hS
2,u0
2RS
2,u0
2
α2
I0> $!fRS
2,u0
2
(r).
(28)
Finally, Using the derivation of outage probability for small
cell DL users and (23), the final expression for rate coverage
of small cell DL users can be obtained through (26).
C. Rate Coverage for Small cell BS in the Uplink
Similarly, the rate coverage for UL small cell BS is given
by
P(RS2> $|AS2)
=ENS2ρPSI N RU L_ABS
u2>2
$Nf
W1|AS2
+ (1 ρ)PSI N RU L
u2>2
$Nf
W1|AS2.
="(1 ρ)X
n0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $|AS2!
+ρX
n0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I0+RSI > $|AS2!#
×P(NS2=n+ 1) ,
(29)
where I=IUL
u2+IS2+IDL
S1+N0is the cumulative interference
from UL small cell users along with macrocell and small cell
BSs, and the Gaussian additive noise. I0=IUL
u2+IS2+N0
is the cumulative interference during ABS transmission, and
$= 2
$NS1
W1.
Using the load distribution given in (23) and both (30) and
(31) we obtain (25) at the top of this page
(1 ρ)P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $|AS2!=
(1 ρ)Z
0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $!fRS2,uUL
2
(r).
(30)
TABLE II: Parametric Values (unless otherwise specified)
Parameter Value
λxx(π×5002)1
PS1[dBm] 43 dBm (20 W)
PS,2[dBm] 23 dBm (200 mW)
Puyy[dBm] 23 dBm (200 mW)
W[Hz] 107
αkk4
τnn[dB] 0dB
RSI PS210LdB /10
LdB [dB] 38 dB
ρ0.3
0.2
ρP PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $|AS2!=
ρZ
0
P PS2hu0
2,S
2Ru0
2,S
2
α2(1)
I+RSI > $!fRS2,uUL
2
(r).
(31)
Finally, Using the derivation of outage probability for UL
small cell BS and (23), the final expression for rate coverage
of UL small cell BS can be obtained through (29).
V. NUMERICAL AND SIMULATION RESULTS
In this section, we evaluate the performance of two-tier
HCNs with FD small cells. Specifically, we investigate how
different parameters affect network performance in terms of
the outage probability and the rate coverage. The simulation
methodology comprises independent realization of PPP dis-
tributions for the BSs of two tiers, followed by realization of
user distribution and the association process. After that, outage
probability and rate coverage are calculated based on the
cumulative interference. The parameters used for the analysis
and simulation are stated in Table II. Monte Carlo simulations
have been conducted to obtain the results, averaged over 10000
iterations, which are then compared with numerical evaluation
of the derived expressions.
Fig. 2 plots the outage probability of a typical DL user
associated with macrocell BS, small cell BS, and random type
of BS in the DL, as a function of small cell BSs density λ2.
We observe that the outage probability of macrocell DL user
increases with increasing the small cell BS density. This results
from the increase in aggregate interference from the small
cell BSs, as shown in (4). Additionally, the outage probability
of macrocell DL user decreases with increasing the transmit
power at the macrocell BS, which is due to the increase in
SINR at the typical downlink user associated with macrocell
IEEE ACCESS - SUBMITTED FOR PUBLICATION 8
12345678910
Small Cell BS Density λ2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Downlink Outage Probability
Small cell DL PS1 =40 dBm
Small cell DL PS1 =43 dBm
Macrocell DL PS1 =40 dBm
Macrocell DL PS1 =43 dBm
Random DL PS1 =40 dBm
Random DL PS1 =43 dBm
Simulation
× ( 500 )-1
2
Fig. 2: Outage probability of macrocell and small cell down-
link as a function of small cell density λ2.
12345678910
Small Cell BS Density λ2×-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Outage Probability
Macrocell DL
Small cell DL
Small cell UL, =0.2
Small cell UL, =0.7
Random User, =0.2
Random User, =0.7
Simulation
( 500 )
2
Fig. 3: Outage probability as a function of small cell density
λ2.
BS, as shown in (4). Interestingly, for the typical downlink user
associated with small cell BS, the outage probability decreases
with increasing the small cell BS density. This is because
densification of tier 2 reduces the inter-link distances between
the typical downlink user and the associated small cell BS,
as shown in (5). In addition, the outage probability of typical
small cell DL user increase with increasing the transmit power
at the macrocell BS, which is due to the increase in aggregate
interference caused by macrocell BSs, as shown in (5). Finally,
outage probability of a random DL user, which is defined as
ODL
1A1+ODL
2A2, increases with both the increase of small
cell density, and the decrease of transmit power of macrocell
BS. This is because ODL
1in the expression is lower than ODL
2,
therefore the expression reflects such tendency. Note that the
simulation results closely follow the analytical results, and
therefore, validate the analytical modeling.
12345678910
Small Cell BS Density λ2 ×( 500 )
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Rate Coverage Probability
UL PS2 =23 dBm, =0.7
UL PS2 =23 dBm, =0.2
UL PS2 =26 dBm, =0.7
UL PS2 =26 dBm, =0.2
DL PS2 =23 dBm
DL PS2 =26 dBm
Simulation
2-1
Fig. 4: Rate coverage as a function of small cell density λ2.
Fig. 3 plots the outage probability of macrocell DL user,
small cell DL user, small cell UL BS, and a randomly located
user versus the density of small cell BSs. In this figure,
we focus on the impact of small cell BSs density on the
outage probability of a randomly located user. Interestingly,
we observe that the outage probability of a randomly located
user is not significantly affected by the increase in the small
cell BS density. It suffers from slight increase that results from
aggregate interference from the small cell BSs, as shown in
(13). We also evaluate the impact of uplink power control
factor, on outage. As shown by the results, a higher value of
results in a higher outage probability, for small cell user in the
uplink, due to reduced uplink transmit power as a consequence
of more aggressive power control. The simulation results also
closely follow the analytical results.
Fig. 4 plots rate coverage for a random DL user and UL
small cell BS versus the small cell BSs density. We note that
the rate coverage of a random DL user decrease as the density
of small cell BSs increases. This is because of increase in
aggregate interference caused by small cell BSs, as seen in
(4). Similarly, the rate coverage of a random DL user decreases
with increasing the transmission power of small cell BSs. On
the contrary, rate coverage of an UL small cell BS increase
with increase of small cell BSs density. This is due to the
fact that densification reduces the inter-link distance between
a user and it’s associated BS, which can be verified by (7).
Similarly, the rate coverage of an UL small cell BS increases
with increasing the transmission power of small cell BSs due
to higher SINR of small cell UL BS as can be verified by (7).
We also evaluate the impact of uplink power control factor,
on rate coverage. As shown by the results, a higher value of
results in a lower rate coverage probability, for small cell
user in the uplink, due to reduced uplink transmit power as a
consequence of more aggressive power control.
Fig. 5 plots the rate coverage of UL small cell BSs as a
function of the ABS factor ρ. We note that the rate coverage of
UL small cell BSs increases as ρincreases. This is because of
the aggregate interference caused by macrocell BSs, which can
IEEE ACCESS - SUBMITTED FOR PUBLICATION 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ABS factor ρ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
UL Rate Coverage Probability
λ2 = λ1
λ2 = 2λ1
λ2 = 5λ1
λ2 = 10λ1
Simulation
Fig. 5: UL rate coverage probability as a function of ABS
factor ρ.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ABS factor ρ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
DL Rate Coverage Probability
λ2 = λ1
λ2 = 2λ1
λ2 = 5λ1
λ2 = 10λ1
Simulation
Fig. 6: DL rate coverage probability as a function of ABS
factor ρ.
be seen in (7) and (8). Similarly, in Fig. 6, the rate coverage
of random DL users increases with increasing ρfor the same
reason, which can be seen in (5) and (6). Fig. 7 shows that the
outage probability of a random user decreases as ρincreases.
This is due to the fact that interference originated by macrocell
BSs decreases with increasing ρ, as seen in (4) and (6).
Fig. 8 plots the relation between small cell UL rate cov-
erage probability and the residual SI cancellation RSI =
PS2.10LdB/10 , where LdB is the ratio of RSI after interfer-
ence cancellation is applied to the transmission power at the
receiver. We observe that outage probability of a randomly
located user is initially high, especially when SI cancellation
capability is low (LdB <15 ), then it decreases with increas-
ing LdB, until it nearly stabilise beyond (LdB >37 ). This
is because the high SI cancellation capabilities improve the
performance of FD links as can be seen in (7). Additionally, we
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ABS factor ρ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Outage Probability
λ2 = λ1
λ2 = 2λ1
λ2 = 5λ1
λ2 = 10λ1
Simulation
Fig. 7: Outage probability in relation to ABS transmission
factor ρ.
-50-40-30-20-100
LdB
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Outage Probability
λ2 = λ1
λ2 = 2λ1
λ2 = 5 λ 1
λ2 = 10λ1
Fig. 8: Outage probability as a function of the SI cancellation
capability LdB.
observe that the outage probability in high small cell densities
is more sensitive to LdB variations. This is due to increased
FD links in higher small cell densities since only the small
cell BSs operate in FD mode.
In Fig. 9, we plot the relation between small cell UL rate
coverage probability and SI cancellation capability LdB. Since
only the small cell BSs operate in FD mode, SI only applies
to those BSs. We note that the rate coverage increases with
the increase of LdB. This is because higher SI cancellation
improves the performance of FD links, as can be seen in (7)
and (8). Moreover, increasing the density of small cell BSs
increases the rate coverage. This is due to the fact that more
FD links exist in higher small cell densities.
VI. CONCLUSIONS
Unprecedented technological developments like network
densification and FD communications will be crucial in shap-
IEEE ACCESS - SUBMITTED FOR PUBLICATION 10
-50-40-30-20-100
LdB
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
UL Rate Coverage Probability
λ2 = λ1
λ2 = 2λ1
λ2 = 5λ1
λ2 = 10λ1
Fig. 9: UL rate coverage probability as a function of the self-
interference cancellation capability LdB.
ing 5G radio access networks for achieving the envisioned
capacity objectives. Realizing the FD capability at small cells
is particularly attractive due to simplicity, superior SI cancel-
lation (compared to macrocells), and widespread deployment.
In this paper, we have investigated the performance of two-
tier interference-coordinated HCNs with FD small cells. We
have derived closed-form expressions for outage probability
and rate coverage in two-tier HCNs with FD small cells
explicitly accounting for interference coordination between
macro and small cells. Performance evaluation investigates the
impact of different network parameters on both outage and rate
coverage probabilities. The results demonstrate that the outage
probability and the rate coverage improves with higher ABS
factor and better underlying SI cancellation capabilities of FD
small cells.
APPENDIX
A. Proof of Theorem 1.
From (10), the outage probability ODL
1is given as
ODL
1=EPSI N RDL
u1< τ1
=1 Z
0
PSI N RDL
u1> τ1fRS
1,u0
1
(r)dr
=1 2πλS1
AS1Z
0
PSI N RDL
u1> τ1r
×exp nπ
2
X
j=1
λj(PSj
PS1
)
ζ1
rζ1odr, (32)
where ζ1=2
αj1.
By setting Q1=IUL
u2+IS2+IDL
S1+N0, We rewrite
SI N RDL
u1as hS
1,u1
PS1
1rα1Q1. Therefore,
PSI N RDL
u1> τ1=PhS1,u1> PS1
1rα1τ1Q1
= exp rα1PS1
1N0τ1LIUL
u2(rα1PS1
1τ1)
× LIS2(rα1PS1
1τ1)LIDL
u1(rα1PS1
1τ1).(33)
Starting with the Laplace transform of the interference
originated from small cell UL users on the macrocell DL user,
presented in (33), we have
LIUL
u2(rα1PS1
1τ1) = EIUL
u2exp rα1PS1
1τ1IUL
u2
=EΦU2hexp nrα1Pu2
PS1
τ1X
u2ΦU2
hu2,u0
1Ru2,u0
1
α2oi
(a)
=
exp n2πλu2Z
0
1− LhS1,u1rα1Pu2
PS1
τ1xα2xdxo
= exp n2πλS1Z
011
1 + rα1PS1
PS
1
τ1xα1xdxo
= exp n2πλu2Z
0
x
1 + rα1Pu2
PS1τ11xα2
dxo.(34)
where (a) is provided in [45]. Note that the integration limits
in (34) are from 0 to since the small cell UL users can
be at any distance from the DL macrocell users. Now, with a
change of variables
v1= (rα1Pu2
PS1τ1)22x2,We express
LIUL
u2(rα1PS1
1τ1) = exp nπλu2Pu2
PS122
×Z1(τ1, α2)r2
α21o,(35)
where
Z1(τ1, α2) = τ22
1Z
x1
1
1 + vα2/2
1
dv1
=2τ1
α222F1[1,122; 2 22;τ1].(36)
In (36), 2F1[·]denote the Gauss hypergeometric function
and x1= (11)22. The expression holds for α2>2.
Similarly, we can derive the Laplace transform for the
interference from small cells BSs expressed in (33), as
LIS2(rα1PS1
1τ1) = exp nπλS2(PS2/PS1)22
×2τ1
α222F1[1,122; 2 22;τ1]o,(37)
for α2>2. We finally derive the Laplace transform for the
interference originated from macrocell BSs expressed in (33)
using similar approach, as
LIDL
S1(rα1PS1
1τ1) = exp nπλS1PS1/PS
121
×2τ1
α122F1[1,121,τ1]o,(38)
for α1>2. Now, plugging (35), (37) and (38) into
P[SI N RDL
u1> τ1]we obtain
IEEE ACCESS - SUBMITTED FOR PUBLICATION 11
P[SI N RDL
u1> τ1] = exp nrα1PS1
1N0τ1
πΨ1r2
α21+Ψ2r2
α21+Ψ3r2o.(39)
given
Ψ1=λu2Pu2
PS1222τ1
α222F1[1,122; 2 22;τ1],
Ψ2=λS2PS2
PS1222τ1
α222F1[1,122; 2 22;τ1],
Ψ3=λS1PS1
PS
1212τ1
α122F1[1,121; 2 22;τ1],
where αj>2. Therefore, the final expression for a randomly
located macrocell DL user is given by (15).
B. Proof of Theorem 2.
From (10), the outage probability ODL
2, considering the
adapted eICIC mechanism is given by
ODL
2=(1 ρ)EhPSI N RDL
u2< τ2i
+ρEhPSI N RDL_AB S
u2< τ2i,(40)
where ρis the ABS transmission ratio.
Starting by the first term of (40), we have
(1 ρ)EPSI N RDL
u2> τ2
= 1 (1 ρ)Z
0
PSI N RDL
u2> τ2fRS
2,u0
2
(r)dr
= 1 2π(1 ρ)λS2
AS2Z
0
PSI N RDL
u2> τ2r
×exp nπ
2
X
j=1
λj(Pj
PS2
)
ζ2
rζ2odr,
(41)
where ζ2=2
αj2.
By setting Q2=IUL
u2+IS2+IDL
S1+N0, we get
PSI N RDL
u2> τ2=PhS2,u2> PS2
1rα2τ2Q2
=Z
0
exp{−rα2PS2
1τ2q}fQ2(q)dq
=EQ2exp{−rα2PS2
1τ2q}
= exp rα2PS2
1N0τ2LIUL
u2(rα2PS2
1τ2)
× LIS2(r2PS2
1τ2)LIDL
S1(rα2PS2
1τ2).(42)
Following the approach presented in Appendix A, we can
obtain the Laplace transforms in (42), starting with the Laplace
transform of the interference originated from UL to DL small
cell users as follows
LIUL
u2(rα2PS2
1τ2) = exp nπr2λu2Pu2
PS222
Y1(τ2, α2)o
(43)
given
Y1(τ2, α2) = τ22
2Z
τ2(22)
1
1 + yα2/2
1
dy1
=2τ2
α222F1[1,12
α2
; 2 2
α2
;τ2],(44)
where α2>2, and y1= (rα2PS2
PS
2
τ2)22r2.
Similarly, the second Laplace transform in (42) of the
interference originated from small cell BS on DL small cell
user is given as
LIS2(rα2PS2
1τ2) = exp nπr2λS2PS2
PS
222
Y1(τ2, α2)o
(45)
The Laplace transform of the interference originated from
macrocell BS on DL small cell user is given as
LIDL
S1(rα2PS2
1τ2) = exp nπλS1PS1
PS221
×Y2(τ2, α1)r2
α12o,(46)
given
Y2(τ2, α1) = τ21
2Z
τ2(21)
1
1 + yα1/2
2
dy2
=2τ2
α122F1[1,12
α1
; 2 2
α1
;τ2],(47)
where α1>2, and y2= (rα2PS1
PS2τ2)21r2.
Plugging (43), (45) and (46) into PSI N RDL
u2> τ2we
obtain
PhSI N RDL
u2> τ2i= exp nrα2PS2
1N0τ2π(η1r2)
+ (η2r2
α21)+(η3r2
α12)o,(48)
where
η1=λu2Pu2
PS222
Y1(τ2, α2)
η2=λS2PS2
PS
222
Y1(τ2, α2)
η3=λS1PS1
PS221
Y2(τ2, α1).
Similarly, the analysis of the second term in (40) of the
outage probability during ABS subframes transmission is
given as follows.
First, we consider the SINR expressed in (6) for the ABS
subframes transmission. By setting Q
2=IUL
u2+IS2+N0, we
have
PSI N RDL_AB S
u2> τ2=PhS2,u2> PS2
1rα2τ2Q
2
=Z
0
exp{−rα2PS2
1τ2q}fQ
2(q)dq
=EQ
2exp{−rα2PS2
1τ2q}
= exp rα2PS2
1N0τ2LIUL
u2(rα2PS2
1τ2)
× LIS2(rα2PS2
1τ2).(49)
IEEE ACCESS - SUBMITTED FOR PUBLICATION 12
Since we have previously derived LIUL
u2(rα2PS2
1τ2)
and LIS2(rα2PS2
1τ2), we can obtain the probability
PSI N RDL_AB S
u2> τ2as
PSI N RDL_AB S
u2> τ2= exp nrα2PS2
1N0τ2
π(η1r2)+(η2r2
α21)o.(50)
Therefore, the final expression for the outage probability for
a randomly located DL small cell user, considering eICIC is
given in (11).
C. Proof of Theorem 3.
From (10), The outage probability OUL
2can be obtained by
OUL
2=(1 ρ)EhPSI N RU L
S2< τ3i
+ρEhPSI N RU L_ABS
S2< τ3i(51)
Starting by the first term of (51), we have
(1 ρ)EPSI N RU L
S2> τ2
= 1 (1 ρ)Z
0
PSI N RU L
S2> τ2fRS2,uUL
2
(r)dr
= 1 2π(1 ρ)λS2Z
0
PSI N RU L
S2> τ2r
×exp πλ2r2dr,(52)
Following the same steps used in previous derivations,
taking into account the power control factor , we obtain the
final expression of the UL small cell outage probability as
given in (12).
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The paper studies the suppression of cross-tier inter-cell interference (ICI) generated by a macro base station (MBS) to pico user equipments (PUEs) in heterogeneous networks (HetNets). Different from existing ICI avoidance schemes such as enhanced ICI cancellation (eICIC) and coordinated beamforming, which generally operate at the MBS, we propose a full duplex (FD) assisted ICI cancellation (fICIC) scheme, which can operate at each pico BS (PBS) individually and is transparent to the MBS. The basic idea of the fICIC is to apply FD technique at the PBS such that the PBS can send the desired signals and forward the listened cross-tier ICI simultaneously to PUEs. We first consider the narrowband single-user case, where the MBS serves a single macro UE and each PBS serves a single PUE. We obtain the closed-form solution of the optimal fICIC scheme, and analyze its asymptotical performance in ICI-dominated scenario. We then investigate the general narrowband multi-user case, where both MBS and PBSs serve multiple UEs. We devise a low-complexity algorithm to optimize the fICIC aimed at maximizing the downlink sum rate of the PUEs subject to user fairness constraint. Finally, the generalization of the fICIC to wideband systems is investigated. Simulations validate the analytical results and demonstrate the advantages of the fICIC on mitigating cross-tier ICI.