1
Non-Orthogonal Multiple Access in Large-Scale
Heterogeneous Networks
Yuanwei Liu, Member, IEEE, Zhijin Qin, Member, IEEE, Maged Elkashlan, Member, IEEE,
Arumugam Nallanathan, Fellow, IEEE, and Julie A. McCann, Member, IEEE,
Abstract—In this paper, the potential benefits of applying nonorthogonal multiple access (NOMA) technique in K-tier hybrid
heterogeneous networks (HetNets) is explored. A promising
new transmission framework is proposed, in which NOMA is
adopted in small cells and massive multiple-input multiple-output
(MIMO) is employed in macro cells. For maximizing the biased
average received power for mobile users, a NOMA and massive
MIMO based user association scheme is developed. To evaluate
the performance of the proposed framework, we first derive the
analytical expressions for the coverage probability of NOMA
enhanced small cells. We then examine the spectrum efficiency
of the whole network, by deriving exact analytical expressions
for NOMA enhanced small cells and a tractable lower bound for
massive MIMO enabled macro cells. Lastly, we investigate the
energy efficiency of the hybrid HetNets. Our results demonstrate
that: 1) The coverage probability of NOMA enhanced small
cells is affected to a large extent by the targeted transmit rates
and power sharing coefficients of two NOMA users; 2) Massive
MIMO enabled macro cells are capable of significantly enhancing
the spectrum efficiency by increasing the number of antennas;
3) The energy efficiency of the whole network can be greatly
improved by densely deploying NOMA enhanced small cell base
stations (BSs); and 4) The proposed NOMA enhanced HetNets
transmission scheme has superior performance compared to the
orthogonal multiple access (OMA) based HetNets.
Index Terms—HetNets, massive MIMO, NOMA, user association, stochastic geometry
I. I NTRODUCTION
The last decade has witnessed the escalating data explosion
on the Internet [2], which is brought by the emerging demanding applications such as high-definition videos, online games
and virtual reality. Also, the rapid development of internet
of things (IoT) requires for facilitating billions of devices to
communicate with each other [3]. Such requirements pose
new challenges for designing the fifth-generation (5G) networks. Driven by these challenges, non-orthogonal multiple
access (NOMA), a promising technology for 5G networks, has
attracted much attention for its potential ability to enhance
spectrum efficiency [4] and improving user access [5], [6].
The key idea of NOMA1 is to utilize a superposition coding
(SC) technology at the transmitter and successive interference
cancelation (SIC) technology at the receiver [7], and hence
multiple access can be realized in power domain via different
power levels for users in the same resource block. Some initial
research investigations have been made in this field [8]–[11].
The system-level performance of the downlink NOMA with
two users has been demonstrated in [8]. In [9], the performance
of a general NOMA transmission has been evaluated in which
one base station (BS) is able to communicate with several
spatial randomly deployed users. As a further advance, the
fairness issue of NOMA has been addressed in [10], by
examining appropriate power allocation policies among the
NOMA users. For multi-antenna NOMA systems, a twostage multicast beamforming downlink transmission scheme
has been proposed in [11], where the total transmitter power
was optimized using closed-form expressions.
Heterogeneous networks (HetNets) and massive multipleinput multiple-output (MIMO), as two “big three” technologies [12], are seen as the fundamental structure for the 5G
networks. The core idea of HetNets is to establish closer
BS-user links by densely overlaying small cells. By doing
so, promising benefits such as lower power consumption,
higher throughput and enhanced spectrum spatial reuse can
be experienced [13]. The massive MIMO regime enables tens
of hundreds/thousands antennas at a BS, and hence it is
capable of offering an unprecedented level of freedom to serve
multiple mobile users [14]. Aiming to fully take advantage
of both massive MIMO and HetNets, in [15], interference
coordination issues found in massive MIMO enabled HetNets
was addressed by utilizing the spatial blanking of macro
cells. In [16], the authors investigated a joint user association
and interference management optimization problem in massive
MIMO HetNets.
A. Motivation and Related Works
Part of this work has been presented in IEEE Global Communication
Conference (GLOBECOM), Dec. Washington D.C, USA, 2016 [1]. This
work was supported by the U.K Engineering and Physical Sciences Research
Council under Grant EP/N029720/1.
Y. Liu and A. Nallanathan are with the Department of Informatics,
King’s College London, London WC2R 2LS, U.K. (email: {yuanwei.liu,
arumugam.nallanathan}@kcl.ac.uk).
Z. Qin and J. McCann are with the Department of Computing,
Imperial College London, London SW7 2AZ, U.K. (email: {z.qin,
j.mccann}@imperial.ac.uk).
M. Elkashlan is with the School of Electronic Engineering and Computer
Science, Queen Mary University of London, London E1 4NS, U.K. (email:
maged.elkashlan@qmul.ac.uk).
Sparked by the aforementioned potential benefits, we therefore explore the potential performance enhancement brought
by NOMA for the hybrid HetNets. Stochastic geometry is
an effective mathematical tool for capturing the topological
randomness of networks. As such, it is capable of providing
tractable analytical results in terms of average network behaviors [17]. Some research contributions with utilizing stochastic
1 In this treatise, we use “NOMA” to refer to “power-domain NOMA” for
simplicity.
2
geometry approaches have been studied in the context of
Hetnets and NOMA [18]–[24]. For HetNets scenarios, based
on applying a flexible bias-allowed user association approach,
the performance of multi-tier downlink HetNets has been
examined in [18], where all BSs and users were assumed to
be equipped with a single antenna. As a further advance, the
coverage provability of the multi-antenna enabled HetNets has
been investigated in [19], using a simple selection bias based
cell selection policy. By utilizing massive MIMO enabled HetNets and a stochastic geometry model, the spectrum efficiency
of uplinks and downlinks were evaluated in [20] and [21],
respectively.
Regarding the literature of stochastic geometry based NOMA scenarios, an incentive user cooperation NOMA protocol
was proposed in [22] to tackle spectrum and energy issues, by
regarding near users as energy harvesting relays for improving
the reliability of far users. By utilizing signal alignment
technology, a new MIMO-NOMA design framework has been
proposed in a stochastic geometry based model [23]. Driven by
the security issues, two effective approaches—protection zone
and artificial noise has been utilized to enhance the physical
layer security for NOMA in large-scale networks in [24].
Very recently, the potential co-existence of two technologies,
NOMA and millimeter wave (mmWave) has been examined in
[25], in which the random beamforming technology is adopted.
Despite the ongoing research contributions having played a
vital role for fostering HetNets and NOMA technologies, to
the best of our knowledge, the impact of NOMA enhanced
hybrid HetNets design has not been researched. Also, there is
lack of complete systematic performance evaluation metrics,
i.e., coverage probability and energy efficiency. Different from
the conventional HetNets design [18], [20], NOMA enhanced
HetNets design poses three additional challenges: i) NOMA
technology brings additional co-channel interference from the
superposed signal of the connected BS; ii) NOMA technology
requires careful channel ordering design to carry out SIC
operations at the receiver; and iii) the user association policy
requires consideration of power sharing effects of NOMA.
Aiming at tackling the aforementioned issues, developing a
systematic mathematically tractable framework for intelligently investigating the effect of various types of interference on
network performance is desired.
B. Contributions and Organization
We propose a new hybrid HetNets framework with NOMA
enhanced small cells and massive MIMO aided macro cells.
We believe that the proposed structure design can contribute to
the design of a more promising 5G system due to the following
key advantages:
• High spectrum efficiency: With higher BS densities,
the NOMA enhanced BSs are capable of accessing the
served users closer, which increase the transmit signalto-interference-plus-noise ratio (SINR) by intelligently
tracking multi-category interference, such as inter/intratier interference and intra-BS interference.
• Low complexity: By applying NOMA in single-antenna
based small cells, the complex cluster based precod-
•
ing/detection design for MIMO-NOMA systems [26],
[27] can be avoided.
Fairness/throughput tradeoff: NOMA is capable of addressing fairness issues by allocating more power to weak
users [7], which is of great significance for HetNets when
investigating efficient resource allocation in sophisticated
large-scale multi-tier networks.
Different from most existing stochastic geometry based
single cell research contributions in terms of NOMA [9], [22]–
[25], we consider multi-cell multi-tier scenarios in this treatise,
which is more challenging. In this framework, we consider
a downlink K-tier HetNets, where macro BSs are equipped
with large antenna arrays with linear zero-forcing beamforming (ZFBF) capability to serve multiple single-antenna users
simultaneously, and small cells BSs equipped with single
antenna each to serve two single-antenna users simultaneously
with NOMA transmission. Based on the proposed design, the
primary theoretical contributions are summarized as follows:
1) We develop a flexible biased association policy to address the impact of NOMA and massive MIMO on the
maximum biased received power. Utilizing this policy,
we first derive the exact analytical expressions for the
coverage probability of a typical user associating with
the NOMA enhanced small cells for the most general
case. Additionally, we derive closed-form expressions in
terms of coverage probability for the interference-limited
case that each tier has the same path loss.
2) We derive the exact analytical expressions of the NOMA
enhanced small cells in terms of spectrum efficiency.
Regarding the massive MIMO enabled macro cells, we
provide a tractable analytical lower bound for the most
general case and closed-form expressions for the case
that each tier has the same path loss. Our analytical
results illustrate that the spectrum efficiency can be
greatly enhanced by increasing the scale of large antenna
arrays.
3) We finally derive the energy efficiency of the whole
network by applying a popular power consumption
model [28]. Our results reveal that NOMA enhanced
small cells achieve higher energy efficiency than macro
cells. It is also shown that increasing antenna numbers
at the macro cell BSs has the opposite effect on energy
efficiency.
4) We show that the NOMA enhanced small cell design
has superior performance over conventional orthogonal
multiple access (OMA) based small cells in terms of
coverage probability, spectrum efficiency and energy efficiency, which demonstrates the benefits of the proposed
framework.
The rest of the paper is organized as follows. In Section
II, the network model for NOMA enhanced hybrid HetNets
is introduced. In Section III, new analytical expressions for
the coverage probability of the NOMA enhanced small cells
are derived. Then spectrum efficiency and energy efficiency
are investigated in Section IV and Section V, respectively.
Numerical results are presented in Section VI, which is
followed by the conclusions in Section VII.
3
User n signal
detection
Massive MIMO
SIC of User
m signal
User m signal
detection
User 1
Pico BS
User n
User m
Marco BS
NOMA
……
User 2
Fig. 1.
User N
Illustration of NOMA and massive MIMO based hybrid HetNets.
II. N ETWORK M ODEL
where Pi is the transmit power of a i-th tier BS, an,i is the
−αi
power sharing coefficient for the near user, L (dj,i ) = ηdj,i
is large-scale path loss, dj,i is the distance between the user
and a i-th tier BS, αi is the path loss exponent of the i-th
tier small cell, η is the frequency dependent factor, and Bi is
the identical bias factor which are useful for offloading data
traffic in HetNets.
2) Average received power in massive MIMO aided macro
cells: In macro cells, as the macro BS is equipped with
multiple antennas, macro cell users experience large array
gains. By adopting the ZFBF transmission scheme, the array
gain obtained at macro users is GM = M − N + 1 [29], [31].
As a result, the average power received at users connecting to
macro BS ℓ (where ℓ ∈ ΦM ) is given by
A. Network Description
Focusing on downlink transmission scenarios, we consider a
K-tier HetNets model, where the first tier represents the macro
cells and the other tiers represent the small cells, such as pico
cells and femto cells. The positions of macro BSs and all the
k-th tier (k ∈ {2, · · · , K}) BSs are modeled as homogeneous
poisson point processes (HPPPs) Φ1 and Φk and with density
λ1 and λk , respectively. As it is common to overlay a highpower macro cell with successively denser and lower power
small cells, we apply massive MIMO technologies to macro
cells and NOMA to small cells in this work. As shown in
Fig. 1, in macro cells, BSs are equipped with M antennas,
each macro BS transmits signals to N users over the same
resource block (e.g., time/frequency/code). We assume that
M ≫ N > 1 and linear ZFBF technique is applied at each
macro BS assigning equal power to N data streams [29].
Perfect downlink channel state information (CSI) are assumed
at the BSs. In small cells, each BS is equipped with single
antenna. Such structure consideration is to avoid sophisticated
MIMO-NOMA precoding/detection in small cells. All users
are considered to be equipped with single antenna each. We
adopt user pairing in each tier of small cells to implement
NOMA to lower the system complexity [22]. It is worth
pointing out that in long term evolution advanced (LTE-A),
NOMA also implements a form of two-user case [30].
B. NOMA and Massive MIMO Based User Association
In this work, a user is allowed to access the BS of any tier,
which provides the best coverage. We consider flexible user
association based on the maximum average received power of
each tier.
1) Average received power in NOMA enhanced small cells:
Different from the convectional user association in OMA,
NOMA exploits the power sparsity for multiple access by
allocating different powers to different users. Due to the
random spatial topology of the stochastic geometry model,
the space information of users are not pre-determined. The
user association policy for the NOMA enhanced small cells
assumes that a near user is chosen as the typical one first. As
such, at the i-th tier small cell, the averaged power received at
users connecting to the i-th tier BS j (where j ∈ Φi ) is given
by:
Pr,i = an,i Pi L (dj,i ) Bi ,
(1)
Pr,1 = GM P1 L (dℓ,1 ) /N,
(2)
where P1 is the transmit power of a macro BS, L (dℓ,1 ) =
−α1
is the large-scale path loss, dℓ,1 is the distance between
ηdℓ,1
the user and a macro BS.
C. Channel Model
1) NOMA enhanced small cell transmission: In small cells,
without loss of generality, we consider that each small cell
BS is associated with one user in the previous round of user
association process. Applying the NOMA protocol, we aim to
squeeze a typical user into a same small cell to improve the
spectral efficiency. For simplicity, we assume that the distances
between the associated users and the connected small cell BSs
are the same, which can be arbitrary values and are denoted
as rk , future work will relax this assumption. The distance
between a typical user and the connected small cell BS is
random. Due to the fact that the path loss is more stable and
dominant compared to the instantaneous small-scale fading
[32], we assume that the SIC operation always happens at the
near user. We denote that do,km and do,kn are the distances
from the k-th tier small cell BS to user m and user n,
respectively. Since it is not pre-determined that a typical user
is a near user n or a far user m, we have the following near
user case and far user case.
Near user case: When a typical user has smaller distance
to the BS than the connected user (x ≤ rk , here x denotes the
distance between the typical user and the BS), then we have
do,km = rk . Here we use m∗ to represent the user which has
been already connected to the BS in the last round of user
association process, we use n to represent the typical user in
near user case. User n will first decode the information of the
connected user m∗ to the same BS with the following SINR
γkn→m∗ =
am,k Pk go,k L (do,kn )
,
an,k Pk go,k L (do,k ) + IM,k + IS,k + σ 2
(3)
where am,k and an,k are the power sharing coefficients for
two users in the k-th layer, σ 2 is the additive white Gaussian
−αi
is the large-scale
noise (AWGN) power, L (do,kn ) = ηdo,k
n
∑
P1
L (dℓ,1 ) is the interference
path loss, IM,k =
ℓ∈Φ1 ∑
N gℓ,1∑
K
from macro cells, IS,k =
i=2
j∈Φi \Bo,k Pi gj,i L (dj,i ) is
the interference from small cells, go,k and do,kn refer the
small-scale fading coefficients and distance between a typical
4
user and the BS in the k-th tier, gℓ,1 and dℓ,1 refer the smallscale fading coefficients and distance between a typical user
and BS ℓ in the macro cell, respectively, gj,i and dj,i refers
to the small-scale fading coefficients and distance between a
typical user and its connected BS j except the serving BS
Bo,k in the i-th tier small cell, respectively. Here, go,k and gj,i
follow exponential distributions with unit mean. gℓ,1 following
Gamma distribution with parameters (N, 1).
If the information of user m∗ can be decoded successfully,
user n then decodes its own message. As such, the SINR at
a typical user n, which connects with the k-th tier small cell,
can be expressed as
γkn =
an,k Pk go,k L (do,kn )
.
IM,k + IS,k + σ 2
(4)
For the connected far user m∗ served by the same BS, the
signal can be decoded by treating the message of user n as
interference. Therefore, the SINR that for the connected user
m∗ to the same BS in the k-th tier small cell can be expressed
as
γkm∗ =
am,k Pk go,k L (rk )
,
Ik,n + IM,k + IS,k + σ 2
(5)
where Ik,n = an,k Pk go,k L (rk ), and L (rk ) = ηrk −αk .
Far user case: When a typical user has a larger distance to
the BS than the connected user(x > rk ), we have do,kn = rk .
Here we use n∗ to represent the user which has been already
connected to the BS in the last round of user association
process, we use m to represent the typical user in far user
case. As such, for the connected near user n∗ , it will first
decode the information of user m with the following SINR
γkn∗→m =
am,k Pk go,k L (rk )
.
an,k Pk go,k L (rk ) + IM,k + IS,k + σ 2
(6)
Once user m is decoded successfully, the interference from
a typical user m can be canceled, by applying SIC technique.
Therefore, the SINR at the connected user n∗ to the same BS
in the k-th tier small cell is given by
γkn∗ =
an,k Pk go,k L (rk )
.
IM,k + IS,k + σ 2
γ km
(8)
−αk
where Ik,n∗ = an,k Pk go,k L (do,km ), L (do,km ) = ηdo,k
,
m
do,kn is the distance between a typical user m and the
connected BS in the k-th tier.
2) Massive MIMO aided macro cell transmission: Without
loss of generality, we assume that a typical user is located at
the origin of an infinite two-dimension plane. Based on (1)
and (2), the SINR at a typical user that connects with a macro
BS at a random distance do,1 can be expressed as
γr,1 =
P1
N ho,1 L (do,1 )
IM,1 + IS,1 + σ
,
2
III. C OVERAGE P ROBABILITY OF N ON - ORTHOGONAL
M ULTIPLE ACCESS BASED S MALL C ELLS
In this section, we focus our attention on analyzing the
coverage probability of a typical user associated to the NOMA
enhanced small cells, which is different from the conventional
OMA based small cells due to the channel ordering of two
users.
A. User Association Probability and Distance Distributions
As described in Section II-B, the user association of the
proposed framework is based on maximizing the biased average received power at the users. As such, based on (1) and (2),
the user association of macro cells and small cells are given
Bi
, α̃ik =
by the following. For simplicity, we denote B̃ik = B
k
αi
P1
Pi
α1
αi
Pi
αk , α̃1k = αk , α̃i1 = α1 , P̃1k = Pk , P̃i1 = P1 , and P̃ik = Pk
in the following parts of this work.
Lemma 1. The user association probability that a typical user
connects to the NOMA enhanced small cell BSs in the k-th tier
and to the macro BSs can be calculated as:
[
∫ ∞
K
) δi 2
(
∑
Ak =2πλk
r exp −π
λi P̃ik B̃ik r α̃ik
0
−πλ1
(7)
For user m that connects to the k-th tier small cell, the
SINR can be expressed as
am,k Pk go,k L (do,km )
=
,
Ik,n∗ + IM,k + IS,k + σ 2
∑
where IM,1 = ℓ∈Φ1 \Bo,1 PN1 hℓ,1 L (dℓ,1 ) is the interference
∑K ∑
from the macro cells, IS,1 = i=2 j∈Φi Pi hj,i L (dj,i ) is the
interference from the small cells; ho,1 is the small-scale fading
coefficient between a typical user and the connected macro BS,
hℓ,1 and dℓ,1 refer to the small-scale fading coefficients and
distance between a typical user and the connected macro BS ℓ
except for the serving BS Bo,1 in the macro cell, respectively,
hj,i and dj,i refer to the small-scale fading coefficients and
distance between a typical user and BS j in the i-th tier small
cell, respectively. Here, ho,1 follows Gamma distribution with
parameters (M − N + 1, 1), hℓ,1 follows Gamma distribution
with parameters (N, 1), and hj,i follows exponential distribution with unit mean.
P̃1k GM
N an,k Bk
) δ1
r
2
α̃1k
and
A1 =2πλ1
∫
∞
0
−πλ1 r
2
]
r exp −π
dr.,
2
α1
K
∑
i=2
λi
(
dr.,
an,i P̃i1 Bi N
GM
(10)
)δi
2
r α̃i1
(11)
2
αi .
respectively, where δ1 =
and δi =
Proof: Using a similar method to Lemma 1 of [18], (10)
and (11) can be easily obtained.
Corollary 1. For the special case that each tier has the same
path loss exponent, i.e., α1 = αk = α, the user association
probability of the NOMA enhanced small cells in the k-th tier
and the macro cells can be expressed in closed form as
Ãk =
(9)
i=2
(
K
∑
i=2
λk
(
)δ
)δ ,
P̃1k GM
λi P̃ik B̃ik + λ1 N an,k Bk
(
(12)
5
and
Ã1 =
λ1
K
∑
i=2
respectively, where δ =
λi
(
2
α.
an,i P̃i1 Bi N
GM
)δ
,
(13)
+ λ1
Remark 1. The derived results in (12) and (13) demonstrate
that by increasing the number of antennas at the macro
cell BSs, the user association probability of the macro cells
increases and the user association probability of the small
cells decreases. This is due to the large array gains brought
by the macro cells to the users served. It is also worth noting
that increasing the power sharing coefficient, an , results in
a higher association probability of small cells. As an → 1,
the user association becomes the same as in the conventional
OMA based approach.
We consider the probability density function (PDF) of the
distance between a typical user and its connected small cell
BS in the k-th tier. Based on (10), we obtain
[
K
(
)δi 2
∑
2πλk x
λi P̃ik B̃ik x α̃ik
exp −π
fdo,k (x) =
Ak
i=2
(
) δ1
2
P̃1k GM
(14)
−πλ1
x α̃1k .
N an,k Bk
We then calculate the PDF of the distance between a typical
user and its connected macro BS. Based on (11), we obtain
(
) δi
K
∑
2
an,i P̃i1 Bi N
2πλ1 x
λi
x α̃i1
exp −π
fdo,1 (x) =
A1
GM
i=2
]
(15)
−πλ1 x2 .
B. The Laplace Transform of Interference
The next step is to derive the Laplace transform of a typical
user. We denote that Ik = IS,k + IM,k is the total interference
to the typical user in the k-th tier. The laplace transform of Ik
is LIk (s) = LIS,k (s) LIM,k (s). We first calculate the Laplace
transform of interference from the small cell BS to a typical
user LIS,k (s) in the following Lemma.
Lemma 2. The Laplace transform of interference from the
small cell BSs to a typical user can be expressed as
{
K
2−αi
∑
λi 2πPi η(ωi,k (x0 ))
×
LIS,k (s) = exp −s
αi (1 − δi )
i=2
(
)}
−αi
,
2 F1 1, 1 − δi ; 2 − δi ; −sPi η(ωi,k (x0 ))
(16)
where 2 F1 (·, ·; ·; ·) is the is the Gauss hypergeometric function
) δ2i 1
(
α̃
x0 ik is the
[33, Eq. (9.142)], and ωi,k (x0 ) = B̃ik P̃ik
nearest distance allowed between the typical user and its
connected small cell BS in the k-th tier.
Proof: See Appendix A.
Then we calculate the laplace transform of interference from
the macro cell to a typical user LIM,k (s) in the following
Lemma.
Lemma 3. The Laplace transform of interference from the
macro cell BSs to a typical user can be expressed as
[
)p (
)δ1 −p
N ( )(
∑
N
P1
P1
−s η
s η
LIM,k (s) = exp −λ1 πδ1
N
N
p
p=1
(
)]
P1
−α
×B −s η[ω1,k (x0 )] 1 ; p − δ1 , 1 − N
,
(17)
N
where B (·; ·, ·) is the is the incomplete Beta function [33, Eq.
) δ21 1
(
P̃1k GM
x α̃1k is the nearest
(8.319)], and ω1,k (x0 ) = an,k Bk N
distance allowed between a typical user and its connected BS
in the macro cell.
Proof: See Appendix B.
C. Coverage Probability
The coverage probability is defined as that a typical user
can successfully transmit signals with a targeted data rate Rt .
According to the distances, two cases are considered in the
following.
Near user case: For the near user case, x0 < rk , successful
decoding will happen when the following two conditions hold:
1) The typical user can decode the message of the connected user served by the same BS.
2) After the SIC process, the typical user can decode its
own message.
As such, the coverage probability of the typical user on the
condition of the distance x0 in the k-th tier is:
Pcov,k (τc , τt , x0 )|x0 ≤rk = Pr {γkn→m∗ > τc , γkn > τt } ,
(18)
where τt = 2Rt − 1 and τc = 2Rc − 1. Here Rc is the targeted
data rate of the connected user served by the same BS.
Based on (18), for the near user case, we can obtain
the expressions for the conditional coverage probability of a
typical user in the following Lemma.
Lemma 4. If am,k − τc an,k ≥ 0 holds, the conditional
coverage probability of a typical user for the near user case
is expressed in closed-form as
{ ∗
k 2
ε (τc , τt ) xα
0 σ
Pcov,k (τc , τt , x0 )|x0 ≤rk = exp −
Pk η
) δ1 2
(
α̃
x0 1k Qn1,t (τc , τt )
− λ1 δ1 π P̃1k ε∗ (τc , τt ) /N
) α2 −1 ( ) α2 2
(
α̃ik
i
i
K
x0
P̃ik
∑ λi δi π B̃ik
−
Qni,t (τc , τt ) . (19)
1 − δi
i=2
Otherwise, Pcov,k (τc , τt , x0 )|x0 ≤rk = 0. Here, εnt =
}
{
τt
τc
f
= am,k −τ
, ε∗ (τc , τt ) = max εfc , εnt ,
an,k , εc
c an,k
)
(
∗
c ,τt )
Qni,t (τc , τt ) = ε∗ (τc , τt ) 2 F1 1, 1 − δi ; 2 − δi ; − ε (τ
,
B̃
ik
6
N ( )
∑
δ1 −p
N
×
and Qn1,t (τc , τt ) =
p (−1)
p=1
)
( ∗
ε (τc ,τt )an,k Bk
; p − δ1 , 1 − N .
B −
GM
Proof: Substituting (3) and (4) into (18), we obtain
}
{
go,kn Pk η
∗
>
ε
(τ
,
τ
)
Pcov,k (τc , τt , x0 )|x0 ≤rk = Pr
c t
i
2
xα
0 (Ik + σ )
{
αk 2
αk }
∗
∗
ε (τc ,τt )x0 σ
ε (τc ,τt )x0 y
−
−
Pk η
Pk η
=e
E Ik e
( ∗
)
α
ε∗ (τc ,τt )x0 k σ 2
ε (τc , τt ) αk
−
Pk η
L Ik
x0
.
(20)
=e
Pk η
Then by plugging (16) and (17) into (20), we obtain the
conditional coverage probability for the near user case in (19).
The proof is complete.
Far user case: For the far user case, x0 > rk , successful
decoding will happen if the typical user can decode its own
message by treating the connected user served by the same BS
as noise. The conditional coverage probability of a typical user
for the far user case is calculated in the following Lemma.
Lemma 5. If am,k −τt an,k ≥ 0 holds, the coverage probability
of a typical user for the far user case is expressed in closedform as
{
ε f x αk σ 2
Pcov,k (τt , x0 )|x0 >rk = exp − t 0
Pk η
) δ1 2
(
α̃
x0 1k Qf1,t (τt )
− λ1 δ1 π P̃1k εft /N
) α2 −1 ( ) α2 2
(
α̃
i
i
K λ δ π B̃
P̃ik
x0 ik
∑
ik
i i
f
−
Qi,t (τt ) . (21)
1 − δi
i=2
Otherwise,
Pcov,k (τt , x0 )|x0 >rk
=
0.
Here
f
τt
=
εft
(τ
)
=
,
and
Q
1,t t
am,k −τt an,k
( f
)
N ( )
∑
ε a
Bk
δ1 −p
N
B − t Gn,k
; p − δ1 , 1 − N
p (−1)
M
p=1
)
(
εf
Qfi,t (τt ) = εft 2 F1 1, 1 − δi ; 2 − δi ; − B̃t .
ik
Proof: Based on (8), we have
{
(
)}
i
εft xα
Ik + σ 2
0
.
Pcov,k (τt , x0 )|x0 >rk = Pr go,km >
Pk η
(22)
Following the similar procedure to obtain (19), with interchanging ε∗ (τc , τt ) with εft , we obtain the desired results in
(21). The proof is complete.
Based on Lemma 4 and Lemma 5, we can calculate
the coverage probability of a typical user in the following
Theorem.
Theorem 1. The coverage probability of a typical user associated to the k-th tier small cells is expressed as
∫ rk
Pcov,k (τc , τt , x0 )|x0 ≤rk fdo,k (x0 ) dx0
Pcov,k (τc , τt ) =
0
∫ ∞
Pcov,k (τt , x0 )|x0 >rk fdo,k (x0 ) dx0 ,
(23)
+
rk
where
Pcov,k (τc , τt , x0 )|x0 ≤rk
is
given
in
(19),
Pcov,k (τt , x0 )|x0 >rk is given in (21), and fdo,k (x0 ) is
given in (14).
Proof: Based on (19) and (21), considering the distant
distributions of a typical user associated to the k-th user small
cells, we can easily obtain the desired results in (23). The proof
is complete.
Although (23) has provided the exact analytical expression
for the coverage probability of a typical user, it is difficult to
directly obtain insights from this expression. Driven by this,
we provide one special case that considers each tier with the
same path loss exponents. As such, we have α̃1k = α̃ik = 1.
In addition, we consider the interference limited case, where
the thermal noise can be neglected2 . Then based on (23), we
can obtain the closed-form coverage probability of a typical
user in the following Corollary.
Corollary 2. With α1 = αk = α and σ 2 = 0, the coverage
probability of a typical user can be expressed in closed-form
as follows:
(
)
n
n
2
bk 1 − e−π(bk +c1 (τc ,τt )+c2 (τc ,τt ))rk
P̃cov,k (τc , τt ) =
bk + cn1 (τc , τt ) + cn2 (τc , τt )
f
f
2
bk e−π(bk +c1 (τt )+c2 (τt ))rk
+
,
(24)
bk + cf1 (τt ) + cf2 (τt )
(
)δ
(
)δ
K
∑
GM
+ λ1 NP̃a1kn,k
λi P̃ik B̃ik
where bk
=
,
Bk
i=2
(
)
δ
∗
cn1 (τc , τt )
=
λ1 δ1 P̃1k ε N(τc ,τt ) Q̃n1,t (τc , τt ),
2
2 −1
K λ δ B̃
α
∑
(P̃ik ) α n
i i ( ik )
Q̃i,t (τc , τt ),
cn2 (τc , τt )
=
1−δi
i=2
) δ1
(
P̃ εft
and
Q̃f1,t (τt ),
=
λ1 δ1 1k
cf1 (τt )
N
2
2 −1
K λ δ B̃
α
∑
(P̃ik ) α f
i i ( ik )
Q̃i,t (τt ). Here,
cf2 (τt )
=
1−δ
i=2
Q̃n1,t (τc , τt ) , Q̃ni,t (τc , τt ) , Q̃f1,t (τt ), and Q̃fi,t (τt ) are based on
interchanging the same path loss exponents, i.e. α1 = αk = α,
for each tier from Qn1,t (τc , τt ) , Qni,t (τc , τt ) , Qf1,t (τt ), and
Qfi,t (τt ).
Proof: If α1 = αk = α hols, (10) can be rewritten as
Ãk =
λ1
,
bk
(25)
Then we have
)
(
f˜do,k (x) = 2πbk x exp −πbk x2 .
(26)
Then by plugging (26) into (23) and after some mathematical
manipulations, we can obtain the desired results in (24).
Remark 2. The derived results in (24) demonstrate that the
coverage probability of a typical user is determined by both
the target rate of itself and the target rate of the connected
user served by the same BS. Additionally, inappropriate power
allocation such as, am,k −τt an,k < 0, will lead to the coverage
probability always being zero.
2 This is a common assumption in stochastic geometry based large-scale
networks [18], [34].
7
IV. S PECTRUM E FFICIENCY
To evaluate the spectrum efficiency of the proposed NOMA
enhanced hybrid HetNets framework, we calculate the spectrum efficiency of each tier in this section.
A. Ergodic Rate of NOMA enhanced Small Cells
Rather than calculating the coverage probability of the case
with fixed targeted rate, the achievable ergodic rate for NOMA
enhanced small cells is opportunistically determined by the
channel conditions of users. It is easy to verify that if the
far user can decode the message of itself, the near user
can definitely decode the message of far user since it has
better channel conditions [9], [35]. Recall that the distance
order between the connected BS and the two users are not
predetermined, as such, we calculate the achievable ergodic
rate of small cells both for the near user case and far user
case in the following Lemmas.
Lemma 6. The achievable ergodic rate of the k-th tier small
cell for the near user case can be expressed as follows:
τkn
2πλk
=
Ak ln 2
[∫
am,k
an,k
0
F̄γkm∗ (z)
dz +
1+z
∫
∞
0
]
F̄γkn (z)
dz ,
1+z
(27)
where F̄γkm∗ (z) and F̄γkn (z) are given by
F̄γkm∗ (z) =
rk
σ 2 zrk αk
x exp −
(am,k − an,k z) Pk η
0
(
)
]
zrk αk
−Θ
+ Λ (x) dx, (28)
(am,k − an,k z) Pk η
∫
[
where F̄γkm (z) and F̄γkn∗ (z) are given by
[
∫ ∞
σ 2 zxαk
F̄γkm (z) =
x exp −
Pk η (am,k − an,k z)
rk
)
]
(
zxαk
+ Λ (x) dx, (32)
−Θ
Pk η (am,k − an,k z)
and
F̄γkn∗ (z) =
∫
∞
rk
σ 2 zrk αk
x exp Λ (x) −
−Θ
Pk ηan,k
[
(
)]
zrk αk
dx.
Pk ηan,k
(33)
Proof: The proof procedure is similar to the approach of
obtaining (27), which is detailed introduced in Appendix C.
Theorem 2. Conditioned on the HPPPs, the achievable ergodic rate of the small cells can be expressed as follows:
τk = τkn + τkf ,
(34)
where τkn and τkf are obtained from (27) and (31).
Note that the derived results in (34) is a double integral
form, since even for some special cases, it is challenging to
obtain closed form solutions. However, the derived expression
is still much more efficient and also more accurate compared
to using Monte Carlo simulations, which highly depends on
the repeated iterations of random sampling.
B. Ergodic Rate of Macro Cells
In massive MIMO aided macro cells, the achievable ergodic
rate can be significantly improved due to multiple-antenna
array gains, but with more power consumption and high
and
[
(
)]
∫ rk
complexity. However, the exact analytical results require high
σ 2 zxαk
zxαk
x exp Λ (x) −
F̄γkn (z) =
−Θ
dx. order derivatives of the Laplace transform with the aid of Faa
an,k Pk η
an,k Pk η
0
(29) Di Bruno’s formula [36]. When the number of antennas goes
large, it becomes mathematically intractable to calculate the
) δi 2
(
K
∑
α̃ik
− derivatives due to the unacceptable complexity. In order to
λi P̃ik B̃ik x
Here
Λ (x)
=
−π
evaluate spectrum efficiency for the whole system, we provide
i=2
(
) δ1 2
P̃1k GM
a tractable lower bound of throughput for macro cells in the
α̃1k
πλ1 N an,k Bk
x
and Θ (s) is given by
following theorem.
)p (
)δ1 −p
N ( )(
∑
P1
P1
N
Theorem 3. The lower bound of the achievable ergodic rate
−s η
s η
Θ (s) = λ1 πδ1
p
N
N
of the macro cells can be expressed as follows:
p=1
(
)
)
(
P1
−α
P1 GM η
× B −s η[ω1,k (x)] 1 ; p − δ1 , 1 − N
,
τ1,L = log2 1 + ∫ ∞
N
N 0 (Q1 (x) + σ 2 )xα1 fdo,1 (x) dx
K
2−αi
∑
λi 2πPi η(ωi,k (x))
(35)
+s
α
(1
−
δ
)
i
i
i=2
where fdo,1 (x) is given ( in )(15), Q1 (x)
=
(
)]
−α
× 2 F1 1, 1 − δi ; 2 − δi ; −sPi η(ωi,k (x)) i .
(30) 2P1 ηπλ1 x2−α1 + ∑K 2πλi Pi η [ωi,1 (x)]2−αi , and
i=2
α1 −2
αi −2
) δ2i 1
(
Proof: See Appendix C.
an,i P̃i1 Bi N
x α̃i1 is denoted as the nearest
ωi,1 (x) =
GM
Lemma 7. The achievable ergodic rate of the k-th tier small distance allowed between the i-th tier small cell BS and the
cell for the far user case can be expressed as follows:
typical user that is associated with the macro cell.
]
[∫
∫ aam,k
∞
Proof: See Appendix C.
(z)
F̄γkn∗ (z)
n,k F̄γk
2πλk
m
dz +
dz ,
τkf =
Ak ln 2 0
1+z
1+z
Corollary 3. If α1 = αk = α holds, the lower bound of
0
(31) the achievable ergordic rate of the macro cell is given by in
8
closed-form as
(
A. Power Consumption Model
1+
τ̃1,L = log2
P1 GM η/N
)
(
−α
−1
ψ(πb1 ) + σ 2 Γ α2 + 1 (πb1 ) 2
)
,
(36)
)(
)δ−1
K (
∑
an,i P̃i1 Bi N
2πλi Pi η
1 ηπλ1
and
+
where ψ = 2Pα−2
α−2
GM
i=2
(
)δ
K
∑
a P̃ B N
b1 =
λi n,i Gi1M i
+ λ1 .
i=2
Proof: When α1 = αk = α, (11) can be rewritten as
Ã1 =
λ1
,
b1
(38)
By substituting the (38) into (35), we can obtain
P1 GM η/N
,
)
τ̃1,L = log2 1 + ∫ ∞ (
2 xα f˜
Q̃
(x)
+
σ
(x)
dx
1
d
o,1
0
(39)
2P1 ηπλ1 2−α
+
α−2 x
)δ−1
an,i P̃i1 Bi N
x2−α
GM
where Q̃1 (x) =
)(
K (
∑
+ σ 2 . Then with the aid
of [33, Eq. (3.326.2)], we obtain the desired closed-form
expression as (36). The proof is complete.
Pi
,
εi
(41)
where Pi,static is the static hardware power consumption of
small cell BSs in the i-th tier, and εi is the efficiency factor
for the power amplifier of small cell BSs in the i-th tier.
The power consumption of macro cell BSs is given by
P1,total = P1,static +
(
)
f˜do,1 (x) = 2πb1 x exp −πb1 x2 .
2πλi Pi η
α−2
Pi,total = Pi,static +
(37)
Then we have
i=2
To calculate the energy efficiency, we first need to model
the power consumption parameter of both small cell BSs and
macro cell BSs. The power consumption of small cell BSs is
given by
3
∑
( a
) P1
,
N ∆a,0 + N a−1 M ∆a,1 +
ε1
a=1
(42)
where P1,static is the static hardware power consumption of
macro cell BSs, ε1 is the efficiency factor for the power
amplifier of macro cell BSs, and ∆a,0 and ∆a,1 are the
practical parameters which are depended on the chains of
transceivers, precoding, coding/decoding, etc3 .
B. Energy Efficiency of NOMA enhanced Small Cells and
Macro Cells
The energy efficiency is defined as
ΘEE =
Total data rate
.
Total energy consumption
(43)
Remark 3. The derived results in (36) demonstrate that the
achievable ergordic rate of the macro cell can be enhanced
by increasing the number of antennas at the macro cell BSs.
This is because the users in the macro cells can experience
larger array gains.
Therefore, based on (43) and the power consumption model for
small cells that we have provided in (41), the energy efficiency
of the k-th tier of NOMA enhanced small cells is expressed
as
τk
,
(44)
ΘkEE =
Pk,total
C. Spectrum Efficiency of the Proposed Hybrid Hetnets
where τk is obtained from (34).
Based on (42) and (43), the energy efficiency of macro cell
is expressed as
Based on the analysis of last two subsections, a tractable
lower bound of spectrum efficiency can be given in the
following Proposition.
Proposition 1. The spectrum efficiency of the proposed hybrid
Hetnets is
∑K
τSE,L = A1 N τ1,L +
Ak τk ,
(40)
k=2
where N τ1 and τk are the lower bound spectrum efficiency of
macro cells and the exact spectrum efficiency of the k-th tier
small cells. Here, Ak and A1 are obtained from (10) and (11),
τk and τ1,L are obtained from (34) and (35), respectively.
V. E NERGY E FFICIENCY
In this section, we proceed to investigate the performance of
the proposed hybrid HetNets framework from the perspective
of energy efficiency, due to the fact that energy efficiency is
an important performance metric for 5G systems.
Θ1EE =
N τ1,L
,
P1,total
(45)
where τ1,L is obtained from (35).
C. Energy Efficiency of the Proposed Hybrid Hetnets
According to the derived results of energy efficiency of
NOMA enhanced small cells and macro cells, we can express
the energy efficiency in the following Proposition.
Proposition 2. The energy efficiency of the proposed hybrid
Hetnets is as follows:
∑K
Ak ΘkEE ,
(46)
ΘHetnets
= A1 Θ1EE +
EE
k=2
where Ak and A1 are obtained from (10) and (11), ΘkEE and
Θ1EE are obtained from (44) and (45).
9
TABLE I
TABLE OF PARAMETERS
105 times
104 m
1 GHz
(
)−1
λ1 = 5002 × π
α1 = 3.5, αk = 4
Nf = 10 dB
σ 2 = −90 dBm
P1,total = 4 W, Pi,total = 2 W
ε1 = εi = 0.4
∆1,0 = 4.8, ∆2,0 = 0
∆3,0 = 2.08 × 10−8
∆1,1 = 1, ∆2,1 = 9.5 × 10−8
∆3,1 = 6.25 × 10−8
Marco cells
Pico cells
User association probability
Monte Carlo simulations repeated
The radius of the plane
Carrier frequency
The BS density of macro cells
Pass loss exponent
The noise figure
The noise power
Static hardware power consumption
Power amplifier efficiency factor
Precoding power consumption
——
——
——
0.7
Femto cells
0.6
Simulation
B2 =10
0.5
0.4
0.3
0.2
50
B2 =20
100
150
200
250
300
350
400
450
500
M
In this section, numerical results are presented to facilitate
the performance evaluations of NOMA enhanced hybrid Ktier HetNets. The noise power is σ 2 = −170 + 10 ×
log10 (BW ) + Nf . The power sharing coefficients of NOMA
for each tier are same as am,k = am and an,k = an for
simplicity. BPCU is short for bit per channel use. Monte Carlo
simulations marked as ‘◦’ are provided to verify the accuracy
of our analysis. Table I summarizes the the simulation parameters used in this section.
A. User Association Probability and Coverage Probability
Fig. 2 shows the effect of the number of antennas equipped
at each macro BS, M , and the bias factor on the user
association probability, where the tiers of HetNets are set to
be K = 3, including macro cells and two tiers of small cells.
The analytical curves representing small cells and macro cells
are from (10) and (11), respectively. One can observe that as
the number of antennas at each macro BS increases, more
users are likely to associate to macro cells. This is because
the massive MIMO aided macro cells are capable of providing
larger array gain, which in turn enhances the average received
power for the connected users. This observation is consistent
with Remark 1 in Section III. Another observation is that increasing the bias factor can encourage more users to connect to
the small cells, which is an efficient way to extend the coverage
of small cells or control the load balance among each tier of
HetNets. Fig. 3 plots the coverage probability of a typical user
associated to the k-tier NOMA enhanced small cells versus
the bias factor. The solid curves representing the analytical
results of NOMA are from (23). One can observe that the
coverage probability decreases as the bias factor increases,
which means that the unbiased user association outperforms
the biased one, i.e., when B2 = 1, the scenario becomes
unbiased user association. This is because by invoking biased
user association, users cannot be always associated to the BS
which provides the highest received power. But the biased user
association is capable of offering more flexibility for users as
well as the whole network, especially for the case that cells
are fully over load. We also demonstrate that NOMA has
3 The power consumption parameters applied in this treatise are based on
an established massive MIMO model proposed in [28], [37].
Fig. 2. User association probability versus antenna number with different
bias factor, with K = 3, N = 15, P1 = 40 dBm, P2 = 30 dBm and
P3 = 20 dBm, rk = 50 m, am = 0.6, an = 0.4, λ2 = λ3 = 20 × λ1 , and
B3 = 20 × B2 .
0.8
Analytical NOMA
Simulation NOMA
OMA
a m =0.8, a n =0.2
0.7
Coverage probability
VI. N UMERICAL R ESULTS
0.6
a m =0.7, a n =0.3
0.5
0.4
a m =0.6, a n =0.4
0.3
0.2
5
10
15
20
25
30
B2
Fig. 3. Coverage probability comparison of NOMA and OMA based small
cells. K = 2, M = 200, N = 15, λ2 = 20 × λ1 , Rt = Rc = 1 BPCU,
rk = 10 m P1 = 40 dBm, and P2 = 20 dBm.
superior behavior over OMA scheme4 . Actually, the OMA
based HetNets scheme has been analytically investigated in
the previous research contributions such as [18], the OMA
benchmark adopted in this treatise is generated by numerical
approach. It is worth pointing out that power sharing between
two NOMA users has a significant effect on coverage probability, and optimizing the power sharing coefficients can further
enlarge the performance gap over OMA based schemes [27],
which is out of the scope of this paper.
Fig. 4 plots the coverage probability of a typical user
associated to the k-tier NOMA enhanced small cells versus
both Rt and Rc . We observe that there is a cross between
these two plotted surfaces, which means that there exists an
optimal power sharing allocation scheme for the given targeted
rate. In contrast, for fixed power sharing coefficients, e.g.,
am = 0.9, an = 0.1, there also exists optimal targeted rates of
two users for coverage probability. This figure also illustrates
4 The OMA benchmark adopted in this treatise is that by dividing the two
users in equal time/frequency slots.
10
18
m
n
a =0.6, a =0.4
m
n
Coverage probability
1
0.8
0.6
0.4
6
0.2
Spectrum efficiency (bit/s/Hz)
16
a =0.9, a =0.1
4
0
5
4
3
2
2
1
0
Rt (BPCU)
Marco cells P 1 =30 dBm
Small cells P 1 =30 dBm
12
HetNets P 1 =30 dBm
10
Marco cells P 1 =40 dBm
Small cells P 1 =40 dBm
HetNets
8
HetNets P 1 =40 dBm
Simulation
6
Small cells
4
2
0
Rc (BPCU)
5
0
10
15
20
25
30
B2
Fig. 4. Successful probability of typical user versus targeted rates of Rt and
Rc , with K = 2, M = 200, N = 15, λ2 = 20 × λ1 , rk = 15 m, B2 = 5,
P1 = 40 dBm, and P2 = 20 dBm.
Fig. 6. Spectrum efficiency of the proposed framework. rk = 50 m, am =
0.6, an = 0.4, K = 2, M = 50, N = 5, P2 = 20 dBm, and λ2 =
100 × λ1 .
3
Energy efficiency (bits/Hz/Joule)
3.5
Spectrum efficiency (bit/s/Hz)
Macro cells
14
3
NOMA
2.5
2
1.5
Analytical NOMA, P 2 = 20 dBm
1
Analytical NOMA, P 2 =30 dBm
Simulation
0.5
OMA
OMA,P2 =30 dBm
OMA,P2 =20 dBm
0
5
10
2.5
2
NOMA small cells
1.5
Macro cells M=200
NOMA small cells M=200
HetNets M=200
Macro cells M=50
NOMA small cells M=50
HetNets M=50
OMA small cells M=200
OMA small cells M=50
HetNets
OMA small cells
1
Macro cells
0.5
0
15
20
25
30
5
10
15
20
25
30
B2
B2
Fig. 5. Spectrum efficiency comparison of NOMA and OMA based small
cells. K = 2, M = 200, N = 15, rk = 50 m, am = 0.6, an = 0.4,
λ2 = 20 × λ1 , and P1 = 40 dBm.
Fig. 7. Energy efficiency of the proposed framework. K = 2, rk = 10
m, am = 0.6, an = 0.4, N = 15, P1 = 30 dBm, P2 = 20 dBm, and
λ2 = 20 × λ1 .
that for inappropriate power and targeted rate selection, the
coverage probability is always zero, which also verifies our
obtained insights in Remark 2.
Fig. 6 plots the spectrum efficiency of the proposed hybrid
HetNets versus bias factor, B2 , with different transmit powers,
P1 . The curves representing the spectrum efficiency of small
cells, macro cells and HetNets are from (40). We can observe
that macro cells can achieve higher spectrum efficiency compared to small cells. This is attributed to the fact that macro
BSs are able to serve multiple users simultaneously offering
promising array gains to each user, which has been analytically
demonstrated in Remark 3. It is also shown that the spectrum
efficiency of macro cells improves as the bias factor increases.
The reason is again that when more low SINR macro cell users
are associated to small cells, the spectrum efficiency of macro
cells can be enhanced.
B. Spectrum Efficiency
Fig. 5 plots the spectrum efficiency of small cells with
NOMA and OMA versus bias factor, B2 , with different
transmit powers of small cell BSs, P2 . The curves representing
the performance of NOMA enhanced small cells are from (34).
The performance of conventional OMA based small cells is
illustrated as a benchmark to demonstrate the effectiveness
of our proposed framework. We observe that the spectrum
efficiency of small cells decreases as the bias factor increases.
This behavior can be explained as follows: larger bias factor
associates more macro users with low SINR to small cells,
which in turn degrades the spectrum efficiency of small cells. It
is also worth noting that the performance of NOMA enhanced
small cells outperforms the conventional OMA based small
cells, which in turn enhances the spectrum efficiency of the
whole HetNets.
C. Energy Efficiency
Fig. 7 plots the energy efficiency of the proposed hybrid
HetNets versus the bias factor, B2 , with different numbers of
transmit antenna of macro cell BSs, M . Several observations
are as follows: 1) The energy efficiency of the macro cells
decrease as the number of antenna increases. Although enlarg-
11
ing the number of antenna at the macro BSs offers a larger
array gains, which in turn enhances the spectrum efficiency.
Such operations also bring significant power consumption
from the baseband signal processing of massive MIMO, which
results in decreased energy efficiency. 2) Another observation
is that NOMA enhanced small cells can achieve higher energy
efficiency than the massive MIMO aided macro cells. It means
that from the perspective of energy consumption, densely
deploying BSs in NOMA enhanced small cell is a more
effective approach. 3) It is also worth noting that the number
of antennas at the macro cell BSs almost has no effect on
the energy efficiency of the NOMA enhanced small cells. 4)
It also demonstrates that NOMA enhanced small cells has
superior performance than conventional OMA based small
cells in terms of energy efficiency. Such observations above
demonstrate the benefits of the proposed NOMA enhanced
hybrid HetNets and provide insightful guidelines for designing
the practical large scale networks.
VII. C ONCLUSIONS
In this paper, a novel hybrid HetNets framework has been
designed. A flexible NONA and massive MIMO based user
association policy was considered. Stochastic geometry was
employed to model the networks and evaluate its performance.
Analytical expressions for the coverage probability of NOMA enhanced small cells were derived. It was analytically
demonstrated that the inappropriate power allocation among
two users will result ‘always ZERO’ coverage probability.
Moreover, analytical results for the spectrum efficiency and
energy efficiency of the whole network was obtained. It was
interesting to observe that the number of antenna at the macro
BSs has weak effects on the energy efficiency of NOMA
enhanced small cells. It has been demonstrated that NOMA
enhanced small cells were able to coexist well with the current
HetNets structure and were capable of achieving superior
performance compared to OMA based small cells. Note that
applying NOMA scheme also brings hardware complexity
and processing delay to the existing HetNets structure, which
should be taken into considerations. A promising future direction is to optimize power sharing coefficients among NOMA
users to further enhance the performance of the proposed
framework.
A PPENDIX A: P ROOF OF L EMMA 2
(c)
(
= exp −
K
∑
λi 2π
i=2
∫
∞
ωi,k (x0 )
(
(
1 − 1 + sPi ηr
)
−αi −1
)
)
rdr ,
(A.1)
where (a) is resulted from applying Campbell’s theorem, (b)
is obtained by using the generating-function of PPP, and (c)
is obtained by gj,i follows exponential distribution with unit
mean. By applying [33, Eq. (3.194.2)], we can obtain the
Laplace transform in an more elegant form in (16). The proof
is complete.
A PPENDIX B: P ROOF OF L EMMA 3
Based on (3), the Laplace transform of the interference from
macro cell BSs can be expressed as follows:
[
(
)]
∑ P1
LIM,k (s) = EIM,k exp −s
gℓ,1 L (dℓ,1 )
N
ℓ∈Φ1
[
)]]
[
(
∏
P1
−α1
Egℓ,1 exp −s gℓ,1 ηdℓ,1
= EΦ1
N
ℓ∈Φ
)
( 1 ∫
(
[ sP g η ])
∞
1 ℓ,1
(a)
−
= exp −λ1 2π
rdr ,
1 − Egℓ,1 e N rα1
ωi,1 (x)
(B.1)
where (a) is obtained with the aid of invoking generatingfunction of PPP. Recall that the gℓ,1 follows Gamma distribution with parameter (N, 1). With the aid of
Laplace transform
for the Gamma
distribution,
we ) ob[
(
(
)]
tain Egℓ,1 exp −s PN1 gℓ,1 ηr−α1
= Lgℓ,1 s PN1 ηr−α1 =
)−N
(
. As such, we can rewrite (B.1) as
1 + s PN1 ηr−α1
LIM,k (s) =
(
∫
exp −λ1 2π
(a)
∞
ω1,k (x)
= exp
−2πλ1
(
)
(
)−N )
sP1 η
1− 1+
rdr
N r α1
)p ∫
n ( )(
∑
n
sηP1
p=1
p
N
∞
ω1,k (x0 )
r
(
1+
)δ N ( )
(b)
sηP1 1 ∑ N
δ −p
= exp −πλ1 δ1
(−1) 1
p
N
p=1
∫ −ω1,k (x)−α sηP1 /N p−δ1 −1 ]
t
dt ,
×
N
(1 − t)
0
[
(
−α1 p+1
sηP1
rα N
)N dr
(B.2)
Based on (3), the Laplace transform of the interference from
small cell BSs can be expressed as follows:
[
]
LIS,k (s) = EIS,k e−sIS,k
[
]
∑
∏
−α
K
(a)
i
= EΦ i
Egj,i e−sPi gj,i ηdj,i
where (a) is obtained by applying binomial expression and
after some mathematical manipulations, and (b) is obtained
by using t = −sηr−α1 P1 /N . Based on [33, Eq. (8.391)],
we can obtain the Laplace transform of IM,k as given in (17).
The proof is complete.
= exp −
For the near user case in small cells, the achievable ergodic
rate in the k-th tier can be expressed as
i=2
(b)
(
(
= exp −
K
∑
i=2
K
∑
i=2
j∈Φi \Bo,k
λi 2π
λi 2π
∫
∞
ωi,k (x0 )
∫
∞
ωi,k (x0 )
(
(
[ gj,i sPi η ])
rdr
1 − Egj,i e− rαi
))
(
1 − Lgj,i sPi ηr−αi rdr
)
)
A PPENDIX C: P ROOF OF L EMMA 6
τkn = E {log2 (1 + γkm∗ ) + log2 (1 + γkn )}
12
1
=
ln 2
∫
∞
0
F̄γkm∗ (z)
1
dz +
1+z
ln 2
∫
∞
0
F̄γkn (z)
dz. (C.1)
1+z
We need to obtain the expressions for F̄γkn (z) first. Based on
(4), we can obtain
[
]
∫ rk
an,k Pk go,k ηx−αk
Pr
F̄γkn (z) =
>
z
fdo,k (x) dx
IM,k + IS,k + σ 2
0
(
)
)
(
∫ rk
zxαk
σ 2 zxαk
L Ik
fdo,k (x) dx,
exp −
=
an,k Pk η
an,k Pk η
0
(C.2)
By combining (17) and (16), we can obtain the Laplace
transform of Ik∗ as LIk∗ (s) = exp (−Θ (s)), where Θ (s) is
given in (30). By plugging (14) and LIk∗ (s) into (C.2), we
obtain the complete cumulative distribution function (CCDF)
of γkn in (29). In the following, we turn to our attention
to derive the CCDF of γkm∗ . Based on (5), we can obtain
F̄γkm∗ (z) as
∫ rk
fdo,k (x)×
F̄γkm∗ (z) =
0
[
) ]
(
IM,k + IS,k + σ 2 z
Pr (am,k − an,k z) go,k >
dx.
Pk ηrk −αk
(C.3)
We first calculate the first part of (D.3) as
∑ P
1
E
hℓ,1 L (dℓ,1 ) do,1 = x
N
ℓ∈Φ1 \Bo,1
∫
(a) P1
= ηE {hℓ,1 } λ1
r−α1 dr
N
∫ ∞ R
(b)
= 2πP1 ηλ1
r1−α1 dr
x
2P1 ηπλ1 2−α1
=
,
x
α1 − 2
(D.4)
where (a) is obtained by applying Campbell’s theorem, and
(b) is obtained since the expectation of hℓ,1 is N . Then we
turn to our attention to the second part of (D.3), with using
the similar approach, we obtain
}
{∑
K ∑
E
Pi hj,i L (dj,i ) do,1 = x
i=2
=
K (
∑
i=2
j∈Φi
)
2πλi Pi η
2−αi
[ωi,1 (x)]
.
αi − 2
(D.5)
By substituting (D.4) and (D.5) into (D.2), we obtain the
desired results in (35). The proof is complete.
a
, it is easy to observe that
Note that for the case z ≥ am,k
n,k
a
F̄γkm∗ (z) = 0. For the case z ≤ am,k
, following the similar
n,k
procedure of deriving (29), we can obtain the ergodic rate of
the existing user for the near user case as (28). The proof is
complete.
A PPENDIX D: P ROOF OF T HEOREM 3
With the aid of Jensen’s inequality, we can obtain the lower
bound of the achievable ergodic rate of the macro cells as
(
( {
})−1 )
−1
E {log2 (1 + γr,1 )} ≥ τ1,L = log2 1 + E (γr,1 )
(D.1)
By invoking the law of large numbers, we have ho,1 ≈ GM .
Then based on (9), τ1,L can be approximated as follows:
{
}
{(
)
}
N
−1
E IM,1 + IS,1 + σ 2 xα1
E (γr,1 )
≈
P G η
∫ ∞ 1 M
(
)
N
=
E { IM,1 + IS,1 | do,1 = x} + σ 2
P1 GM η 0
(D.2)
× xα1 fdo,1 (x) dx.
We turn to our attention to the expectation, denoting Q1 (x) =
E { IM,1 + IS,1 | do,1 = x}, with the aid of Campbell’s Theorem, we obtain
∑ P
1
hℓ,1 L (dℓ,1 ) do,1 = x
Q1 (x) = E
N
ℓ∈Φ1 \Bo,1
{∑
}
K ∑
+E
Pi hj,i L (dj,i ) do,1 = x
i=2
j∈Φi
K
2P1 ηπλ1 2−α1 ∑
2πλi
x
+
=
α1 − 2
i=2
(
)
Pi η
2−αi
,
[ωi,1 (x)]
αi − 2
(D.3)
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Yuanwei Liu (S’13, M’16) received his Ph.D. degree in electrical engineering from the Queen Mary
University of London, UK in 2016. Before that, he
received his M.S. degree and B.S. degree from the
Beijing University of Posts and Telecommunications
in 2014 and 2011, respectively. He is currently the
Post-Doctoral Research Fellow with the Department
of Informatics, King’s College London, U.K.
His research interests include 5G Wireless Networks, Internet of Things (IoT), Stochastic Geometry and Matching Theory. He received the Exemplary
Reviewer Certificate of the IEEE W IRELESS C OMMUNICATION L ETTER
in 2015 and IEEE T RANSACTIONS ON C OMMUNICATIONS in 2017. He
currently serves as an Editor of IEEE C OMMUNICATIONS L ETTERS and
IEEE ACCESS. He has served as a TPC member for many IEEE conferences
such as GLOBECOM and VTC.
Zhijin Qin (S’13, M’16) received her double B.Sc
degrees from Beijing University of Posts and Telecommunications, Beijing, China and Queen Mary
University of London, London, UK, and the Ph.D.
degree in electronic engineering from Queen Mary
University of London, London, U.K., in 2012 and
2016, respectively. She is currently the research
associate with Department of Computing, Imperial
College London, London, U.K. She was awarded
the best paper at Wireless Technology Symposium,
April 2012, London, U.K., when she was an undergraduate student.
Her research interests include low power wide area network in IoT,
fog networking, compressive spectrum sensing in cognitive radio, and nonorthogonal multiple access in 5G network. She currently serves as an Editor of
IEEE ACCESS. She has served as a TPC member for many IEEE conferences
such as ICC’16 and VTC’15 and VTC’14.
Maged Elkashlan (M’06) received the Ph.D. degree in Electrical Engineering from the University
of British Columbia, Canada, 2006. From 2006 to
2007, he was with the Laboratory for Advanced
Networking at University of British Columbia. From
2007 to 2011, he was with the Wireless and Networking Technologies Laboratory at Commonwealth
Scientific and Industrial Research Organization (CSIRO), Australia. During this time, he held an
adjunct appointment at University of Technology
Sydney, Australia. In 2011, he joined the School
of Electronic Engineering and Computer Science at Queen Mary University
of London, UK, as an Assistant Professor. He also holds visiting faculty
appointments at the University of New South Wales, Australia, and Beijing
University of Posts and Telecommunications, China. His research interests fall
into the broad areas of communication theory, wireless communications, and
statistical signal processing for distributed data processing and heterogeneous
networks.
Dr. Elkashlan currently serves as an Editor of IEEE T RANSACTIONS
ON W IRELESS C OMMUNICATIONS , IEEE T RANSACTIONS ON V EHICULAR
T ECHNOLOGY, and IEEE C OMMUNICATIONS L ETTERS. He also serves as
Lead Guest Editor for the special issue on “Green Media: The Future of
Wireless Multimedia Networks” of the IEEE W IRELESS C OMMUNICATIONS
M AGAZINE, Lead Guest Editor for the special issue on “Millimeter Wave
Communications for 5G” of the IEEE C OMMUNICATIONS M AGAZINE, Guest
Editor for the special issue on “Energy Harvesting Communications” of the
IEEE C OMMUNICATIONS M AGAZINE, and Guest Editor for the special issue
on “Location Awareness for Radios and Networks” of the IEEE J OURNAL ON
S ELECTED A REAS IN C OMMUNICATIONS. He received the Best Paper Award
at the IEEE International Conference on Communications (ICC) in 2014,
the International Conference on Communications and Networking in China
(CHINACOM) in 2014, and the IEEE Vehicular Technology Conference
(VTC-Spring) in 2013. He received the Exemplary Reviewer Certificate of
the IEEE Communications Letters in 2012.
14
Arumugam Nallanathan (S’97-M’00-SM’05-F’17)
is Professor of Wireless Communications in the
Department of Informatics at Kings College London
(University of London). He served as the Head
of Graduate Studies in the School of Natural and
Mathematical Sciences at Kings College London,
2011/12. He was an Assistant Professor in the Department of Electrical and Computer Engineering,
National University of Singapore from August 2000
to December 2007. His research interests include
5G Wireless Networks, Internet of Things (IoT) and
Molecular Communications. He published more than 300 technical papers
in scientific journals and international conferences. He is a co-recipient of
the Best Paper Award presented at the IEEE International Conference on
Communications 2016 (ICC2016) and IEEE International Conference on
Ultra-Wideband 2007 (ICUWB 2007). He is an IEEE Distinguished Lecturer.
He has been selected as a Thomson Reuters Highly Cited Researcher in 2016.
He is an Editor for IEEE Transactions on Communications and IEEE Transactions on Vehicular Technology. He was an Editor for IEEE Transactions
on Wireless Communications (2006-2011), IEEE Wireless Communications
Letters and IEEE Signal Processing Letters. He served as the Chair for the
Signal Processing and Communication Electronics Technical Committee of
IEEE Communications Society and Technical Program Chair and member of
Technical Program Committees in numerous IEEE conferences. He received
the IEEE Communications Society SPCE outstanding service award 2012 and
IEEE Communications Society RCC outstanding service award 2014.
Julie
McCann
directs
the
Adaptive
Emergent/Embedded/Ephemeral
Systems
Engineering (AE3SE) research group which
explores how wireless sensing-based actuator
systems interact with physical world and how this
knowledge can inform protocols, edge-analytics
and delay tolerant control systems. She currently
co-directs the Intel Collaborative Research Institute
in Urban IoT systems and the cross Imperial
Smart Connected Futures Centre who explore
IoT, Cyber-physical systems, crowdsourcing and
related topics. She is an active programme committee member for many of
the adaptive computing, sensor network and communications journals and
conferences and has also co-chaired conferences such as IEEE SECON,
ACM Self-Adaptive and Self-Organising Systems etc. She has invited to
give many keynotes, distinguished seminars and has lead and participated
in academic and industrial panel discussions and has been called to advise
government on aspects concerning IoT, procurement and security. .