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arXiv:1705.03325v1 [cs.IT] 9 May 2017
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 non-
orthogonal 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 associa-
tion, stochastic geometry
I. INTRO DUC TI O N
The last decade has witnessed the escalating data explosion
on the Internet [2], which is brought by the emerging demand-
ing 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) net-
works. 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].
Part of this work has been presented in IEEE Global Communication
Conference (GLOBECOM), Dec. Washington D.C, USA, 2016 [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).
The key idea of NOMA1is 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 two-
stage 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 multiple-
input multiple-output (MIMO), as two “big three” technolo-
gies [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
Sparked by the aforementioned potential benefits, we there-
fore 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 behav-
iors [17]. Some research contributions with utilizing stochastic
geometry approaches have been studied in the context of
1In this treatise, we use “NOMA” to refer to “power-domain NOMA” for
simplicity.
2
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 Het-
Nets 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 proto-
col was proposed in [22] to tackle spectrum and energy issues,
by regarding near users as energy harvesting relays for improv-
ing 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 intelli-
gently 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 signal-
to-interference-plus-noise ratio (SINR) by intelligently
tracking multi-category interference, such as inter/intra-
tier 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 ad-
dressing 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 beamform-
ing (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 ad-
dress 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 net-
work 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 ef-
ficiency, 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
Massive MIMO
User 1
SIC of User
m signal
User n signal
detection
User n
User m signal
detection
User m
NOMA
User 2 User N
……
Pico BS
Marco BS
Fig. 1. Illustration of NOMA and massive MIMO based hybrid HetNets.
II. NET WO RK MOD EL
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) Φ1and Φkand with density
λ1and λk, respectively. As it is common to overlay a high-
power 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 Mantennas,
each macro BS transmits signals to Nusers over the same
resource block (e.g., time/frequency/code). We assume that
M≫N > 1and linear ZFBF technique is applied at each
macro BS assigning equal power to Ndata 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,iPiL(dj,i)Bi,(1)
where Piis the transmit power of a i-th tier BS, an,i is the
power sharing coefficient for the near user, L(dj,i) = ηd−αi
j,i
is large-scale path loss, dj,i is the distance between the user
and a i-th tier BS, αiis the path loss exponent of the i-th
tier small cell, ηis the frequency dependent factor, and Biis
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
Pr,1=GMP1L(dℓ,1)/N, (2)
where P1is the transmit power of a macro BS, L(dℓ,1) =
ηd−α1
ℓ,1is the large-scale path loss, dℓ,1is the distance between
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,kmand do,knare the distances
from the k-th tier small cell BS to user mand user n,
respectively. Since it is not pre-determined that a typical user
is a near user nor 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 xdenotes 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 nto represent the typical user in
near user case. User nwill first decode the information of the
connected user m∗to the same BS with the following SINR
γkn→m∗=am,kPkgo,k L(do,kn)
an,kPkgo,k L(do,k ) + IM,k +IS,k +σ2,(3)
where am,k and an,k are the power sharing coefficients for
two users in the k-th layer, σ2is the additive white Gaussian
noise (AWGN) power, L(do,kn) = ηd−αi
o,knis the large-scale
path loss, IM,k =Pℓ∈Φ1
P1
Ngℓ,1L(dℓ,1)is the interference
from macro cells, IS,k =PK
i=2 Pj∈Φi\Bo,k Pigj,iL(dj,i)is
the interference from small cells, go,k and do,knrefer the
small-scale fading coefficients and distance between a typical
4
user and the BS in the k-th tier, gℓ,1and dℓ,1refer the small-
scale fading coefficients and distance between a typical user
and BS ℓin 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 jexcept 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,1following
Gamma distribution with parameters (N, 1).
If the information of user m∗can be decoded successfully,
user nthen 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,kPkgo,k L(do,kn)
IM,k +IS,k +σ2.(4)
For the connected far user m∗served by the same BS, the
signal can be decoded by treating the message of user nas
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,kPkgo,k L(rk)
Ik,n +IM,k +IS,k +σ2,(5)
where Ik,n =an,kPkgo,kL(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 mto represent the typical user in far user
case. As such, for the connected near user n∗, it will first
decode the information of user mwith the following SINR
γkn∗→m=am,kPkgo,k L(rk)
an,kPkgo,kL(rk) + IM,k +IS,k +σ2.(6)
Once user mis decoded successfully, the interference from
a typical user mcan 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,kPkgo,k L(rk)
IM,k +IS,k +σ2.(7)
For user mthat connects to the k-th tier small cell, the
SINR can be expressed as
γkm=am,kPkgo,k L(do,km)
Ik,n∗+IM,k +IS,k +σ2,(8)
where Ik,n∗=an,k Pkgo,kL(do,km),L(do,km) = ηd−αk
o,km,
do,knis the distance between a typical user mand 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,1can be expressed as
γr,1=
P1
Nho,1L(do,1)
IM,1+IS,1+σ2,(9)
where IM,1=Pℓ∈Φ1\Bo,1
P1
Nhℓ,1L(dℓ,1)is the interference
from the macro cells, IS,1=PK
i=2 Pj∈ΦiPihj,iL(dj,i)is the
interference from the small cells; ho,1is the small-scale fading
coefficient between a typical user and the connected macro BS,
hℓ,1and dℓ,1refer to the small-scale fading coefficients and
distance between a typical user and the connected macro BS ℓ
except for the serving BS Bo,1in the macro cell, respectively,
hj,i and dj,i refer to the small-scale fading coefficients and
distance between a typical user and BS jin the i-th tier small
cell, respectively. Here, ho,1follows Gamma distribution with
parameters (M−N+ 1,1),hℓ,1follows Gamma distribution
with parameters (N , 1), and hj,i follows exponential distribu-
tion with unit mean.
III. COVE RAGE PRO BA BI L IT Y O F NON-ORTHOGONAL
MULTIP LE ACCES S BASE D SMAL L CELLS
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 aver-
age received power at the users. As such, based on (1) and (2),
the user association of macro cells and small cells are given
by the following. For simplicity, we denote ˜
Bik =Bi
Bk,˜αik =
αi
αk,˜α1k=α1
αk,˜αi1=αi
α1,˜
P1k=P1
Pk,˜
Pi1=Pi
P1, and ˜
Pik =Pi
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:
Ak=2πλkZ∞
0
rexp "−π
K
X
i=2
λi˜
Pik ˜
Bikδir2
˜αik
−πλ1 ˜
P1kGM
Nan,k Bk!δ1
r2
˜α1k
dr., (10)
and
A1=2πλ1Z∞
0
rexp
−π
K
X
i=2
λi an,i ˜
Pi1BiN
GM!δi
r2
˜αi1
−πλ1r2dr., (11)
respectively, where δ1=2
α1and δi=2
αi.
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
˜
Ak=λk
K
P
i=2
λi˜
Pik ˜
Bikδ+λ1˜
P1kGM
Nan,kBkδ,(12)
5
and
˜
A1=λ1
K
P
i=2
λian,i ˜
Pi1BiN
GMδ+λ1
,(13)
respectively, where δ=2
α.
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
fdo,k (x) =2πλkx
Ak
exp "−π
K
X
i=2
λi˜
Pik ˜
Bikδix2
˜αik
−πλ1 ˜
P1kGM
Nan,k Bk!δ1
x2
˜α1k.(14)
We then calculate the PDF of the distance between a typical
user and its connected macro BS. Based on (11), we obtain
fdo,1(x) = 2πλ1x
A1
exp
−π
K
X
i=2
λi an,i ˜
Pi1BiN
GM!δi
x2
˜αi1
−πλ1x2.(15)
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
LIS,k (s) = exp (−s
K
X
i=2
λi2πPiη(ωi,k (x0))2−αi
αi(1 −δi)×
2F11,1−δi; 2 −δi;−sPiη(ωi,k (x0))−αio,
(16)
where 2F1(·,·;·;·)is the is the Gauss hypergeometric function
[33, Eq. (9.142)], and ωi,k (x0) = ˜
Bik ˜
Pik
δi
2x
1
˜αik
0is the
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
LIM,k (s) = exp "−λ1πδ1
N
X
p=1 N
psP1
Nηp−sP1
Nηδ1−p
×B−sP1
Nη[ω1,k (x0)]−α1;p−δ1,1−N,(17)
where B(·;·,·)is the is the incomplete Beta function [33, Eq.
(8.319)], and ω1,k (x0) = ˜
P1kGM
an,kBkN
δ1
2x1
˜α1kis the nearest
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 con-
nected 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 x0in the k-th tier is:
Pcov,k (τc, τt, x0)|x0≤rk= Pr {γkn→m∗> τc, γkn> τt},
(18)
where τt= 2Rt−1and τc= 2Rc−1. Here Rcis 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 −τcan,k ≥0holds, the conditional
coverage probability of a typical user for the near user case
is expressed in closed-form as
Pcov,k (τc, τt, x0)|x0≤rk= exp −ε∗(τc, τt)xαk
0σ2
Pkη
−λ1δ1π˜
P1kε∗(τc, τt)/Nδ1x
2
˜α1k
0Qn
1,t (τc, τt)
−
K
X
i=2
λiδiπ˜
Bik2
αi−1˜
Pik2
αix
2
˜αik
0
1−δi
Qn
i,t (τc, τt)
.(19)
Otherwise, Pcov,k (τc, τt, x0)|x0≤rk= 0. Here, εn
t=
τt
an,k ,εf
c=τc
am,k−τcan,k ,ε∗(τc, τt) = max εf
c, εn
t,
Qn
i,t (τc, τt) = ε∗(τc, τt)2F11,1−δi; 2 −δi;−ε∗(τc,τt)
˜
Bik ,
6
and Qn
1,t (τc, τt) =
N
P
p=1 N
p(−1)δ1−p×
B−ε∗(τc,τt)an,kBk
GM;p−δ1,1−N.
Proof. Substituting (3) and (4) into (18), we obtain
Pcov,k (τc, τt, x0)|x0≤rk= Pr go,knPkη
xαi
0(Ik+σ2)> ε∗(τc, τt)
=e−ε∗(τc,τt)xαk
0σ2
PkηEIke−ε∗(τc,τt)xαk
0y
Pkη
=e−ε∗(τc,τt)xαk
0σ2
PkηLIkε∗(τc, τt)
Pkηxαk
0.(20)
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−τtan,k ≥0holds, the coverage probability
of a typical user for the far user case is expressed in closed-
form as
Pcov,k (τt, x0)|x0>rk= exp (−εf
txαk
0σ2
Pkη
−λ1δ1π˜
P1kεf
t/Nδ1x
2
˜α1k
0Qf
1,t (τt)
−
K
X
i=2
λiδiπ˜
Bik2
αi−1˜
Pik2
αix
2
˜αik
0
1−δi
Qf
i,t (τt)
.(21)
Otherwise, Pcov,k (τt, x0)|x0>rk= 0. Here
εf
t=τt
am,k−τtan,k , and Qf
1,t (τt) =
N
P
p=1 N
p(−1)δ1−pB−εf
tan,kBk
GM;p−δ1,1−N
Qf
i,t (τt) = εf
t2F11,1−δi; 2 −δi;−εf
t
˜
Bik .
Proof. Based on (8), we have
Pcov,k (τt, x0)|x0>rk= Pr (go,km>εf
txαi
0Ik+σ2
Pkη).
(22)
Following the similar procedure to obtain (19), with inter-
changing ε∗(τc, τt)with εf
t, 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 asso-
ciated to the k-th tier small cells is expressed as
Pcov,k (τc, τt) = Zrk
0
Pcov,k (τc, τt, x0)|x0≤rkfdo,k (x0)dx0
+Z∞
rk
Pcov,k (τt, x0)|x0>rkfdo,k (x0)dx0,(23)
where Pcov,k (τc, τt, x0)|x0≤rkis given in (19),
Pcov,k (τt, x0)|x0>rkis given in (21), and fdo,k (x0)is
given in (14).
Proof. Based on (19) and (21), considering the distant distri-
butions 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 ˜α1k= ˜αik = 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=αand σ2= 0, the coverage
probability of a typical user can be expressed in closed-form
as follows:
˜
Pcov,k (τc, τt) =
bk1−e−π(bk+cn
1(τc,τt)+cn
2(τc,τt))r2
k
bk+cn
1(τc, τt) + cn
2(τc, τt)
+bke−π(bk+cf
1(τt)+cf
2(τt))r2
k
bk+cf
1(τt) + cf
2(τt),(24)
where bk=
K
P
i=2
λi˜
Pik ˜
Bikδ+λ1˜
P1kGM
Nan,kBkδ
,
cn
1(τc, τt) = λ1δ1˜
P1kε∗(τc,τt)
Nδ˜
Qn
1,t (τc, τt),
cn
2(τc, τt) =
K
P
i=2
λiδi(˜
Bik)2
α−1(˜
Pik)2
α
1−δi
˜
Qn
i,t (τc, τt),
cf
1(τt) = λ1δ1˜
P1kεf
t
Nδ1˜
Qf
1,t (τt), and
cf
2(τt) =
K
P
i=2
λiδi(˜
Bik)2
α−1(˜
Pik)2
α
1−δ˜
Qf
i,t (τt). Here,
˜
Qn
1,t (τc, τt),˜
Qn
i,t (τc, τt),˜
Qf
1,t (τt), and ˜
Qf
i,t (τt)are based on
interchanging the same path loss exponents, i.e. α1=αk=α,
for each tier from Qn
1,t (τc, τt), Qn
i,t (τc, τt), Qf
1,t (τt), and
Qf
i,t (τt).
Proof. If α1=αk=αhols, (10) can be rewritten as
˜
Ak=λ1
bk
,(25)
Then we have
˜
fdo,k (x) = 2πbkxexp −πbkx2.(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
2This is a common assumption in stochastic geometry based large-scale
networks [18], [34].
7
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 −τtan,k <0, will lead to the coverage
probability always being zero.
IV. SPE CTRU M EFFICI ENC Y
To evaluate the spectrum efficiency of the proposed NOMA
enhanced hybrid HetNets framework, we calculate the spec-
trum 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]. 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:
τn
k=2πλk
Akln 2 "Z
am,k
an,k
0
¯
Fγkm∗(z)
1 + zdz +Z∞
0
¯
Fγkn(z)
1 + zdz#,
(27)
where ¯
Fγkm∗(z)and ¯
Fγkn(z)are given by
¯
Fγkm∗(z) = Zrk
0
xexp −σ2zrkαk
(am,k −an,kz)Pkη
−Θzrkαk
(am,k −an,kz)Pkη+ Λ (x)dx, (28)
and
¯
Fγkn(z) = Zrk
0
xexp Λ (x)−σ2zxαk
an,kPkη−Θzxαk
an,kPkηdx.
(29)
Here Λ (x) = −π
K
P
i=2
λi˜
Pik ˜
Bikδix2
˜αik −
πλ1˜
P1kGM
Nan,kBkδ1x2
˜α1kand Θ (s)is given by
Θ (s) = λ1πδ1
N
X
p=1 N
psP1
Nηp−sP1
Nηδ1−p
×B−sP1
Nη[ω1,k (x)]−α1;p−δ1,1−N
+s
K
X
i=2
λi2πPiη(ωi,k (x))2−αi
αi(1 −δi)
×2F11,1−δi; 2 −δi;−sPiη(ωi,k (x))−αii.(30)
Proof. See Appendix C.
Lemma 7. The achievable ergodic rate of the k-th tier small
cell for the far user case can be expressed as follows:
τf
k=2πλk
Akln 2 "Z∞
0
¯
Fγkn∗(z)
1 + zdz +Z
am,k
an,k
0
¯
Fγkm(z)
1 + zdz#,
(31)
where ¯
Fγkm(z)and ¯
Fγkn∗(z)are given by
¯
Fγkm(z) = Z∞
rk
xexp −σ2zxαk
Pkη(am,k −an,kz)
−Θzxαk
Pkη(am,k −an,kz)+ Λ (x)dx, (32)
and
¯
Fγkn∗(z) = Z∞
rk
xexp Λ (x)−σ2zrkαk
Pkηan,k
−Θzrkαk
Pkηan,k dx.
(33)
Proof. The proof procedure is similar to the approach of ob-
taining (27), which is detailed introduced in Appendix C.
Theorem 2. Conditioned on the HPPPs, the achievable er-
godic rate of the small cells can be expressed as follows:
τk=τn
k+τf
k,(34)
where τn
kand τf
kare 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
complexity. However, the exact analytical results require high
order derivatives of the Laplace transform with the aid of Faa
Di Bruno’s formula [35]. When the number of antennas goes
large, it becomes mathematically intractable to calculate the
derivatives due to the unacceptable complexity. In order to
evaluate spectrum efficiency for the whole system, we provide
a tractable lower bound of throughput for macro cells in the
following theorem.
Theorem 3. The lower bound of the achievable ergodic rate
of the macro cells can be expressed as follows:
τ1,L = log2 1 + P1GMη
NR∞
0(Q1(x) + σ2)xα1fdo,1(x)dx!,
(35)
where fdo,1(x)is given in (15),Q1(x) =
2P1ηπλ1
α1−2x2−α1+PK
i=2 2πλiPiη
αi−2[ωi,1(x)]2−αi, and
ωi,1(x) = an,i ˜
Pi1BiN
GM
δi
2x1
˜αi1is denoted as the nearest
distance allowed between the i-th tier small cell BS and the
typical user that is associated with the macro cell.
8
Proof. See Appendix C.
Corollary 3. If α1=αk=αholds, the lower bound of
the achievable ergordic rate of the macro cell is given by in
closed-form as
˜τ1,L = log2 1 + P1GMη/N
ψ(πb1)−1+σ2Γα
2+ 1(πb1)−α
2!,
(36)
where ψ=2P1ηπλ1
α−2+
K
P
i=2 2πλiPiη
α−2an,i ˜
Pi1BiN
GMδ−1
and
b1=
K
P
i=2
λian,i ˜
Pi1BiN
GMδ+λ1.
Proof. When α1=αk=α, (11) can be rewritten as
˜
A1=λ1
b1
,(37)
Then we have
˜
fdo,1(x) = 2πb1xexp −πb1x2.(38)
By substituting the (38) into (35), we can obtain
˜τ1,L = log2
1 + P1GMη/N
R∞
0˜
Q1(x) + σ2xα˜
fdo,1(x)dx
,
(39)
where ˜
Q1(x) = 2P1ηπλ1
α−2x2−α+
K
P
i=2 2πλiPiη
α−2an,i ˜
Pi1BiN
GMδ−1x2−α+σ2. Then with the aid
of [33, Eq. (3.326.2)], we obtain the desired closed-form
expression as (36). The proof is complete.
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.
C. Spectrum Efficiency of the Proposed Hybrid Hetnets
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
τSE,L=A1Nτ1,L +XK
k=2 Akτk,(40)
where Nτ1and τkare the lower bound spectrum efficiency of
macro cells and the exact spectrum efficiency of the k-th tier
small cells. Here, Akand A1are obtained from (10) and (11),
τkand τ1,L are obtained from (34) and (35), respectively.
V. ENE RGY EFFI CI E NC Y
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.
A. Power Consumption Model
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
Pi,total =Pi,static +Pi
εi
,(41)
where Pi,static is the static hardware power consumption of
small cell BSs in the i-th tier, and εiis 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 +
3
X
a=1 Na∆a,0+Na−1M∆a,1+P1
ε1
,
(42)
where P1,static is the static hardware power consumption of
macro cell BSs, ε1is the efficiency factor for the power
amplifier of macro cell BSs, and ∆a,0and ∆a,1are 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)
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
Θk
EE =τk
Pk,total
,(44)
where τkis obtained from (34).
Based on (42) and (43), the energy efficiency of macro cell
is expressed as
Θ1
EE =Nτ1,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:
ΘHetnets
EE =A1Θ1
EE +XK
k=2 AkΘk
EE,(46)
where Akand A1are obtained from (10) and (11),Θk
EE and
Θ1
EE are obtained from (44) and (45).
9
TABLE I
TABL E O F PAR AM ET ER S
Monte Carlo simulations repeated 105times
The radius of the plane 104m
Carrier frequency 1GHz
The BS density of macro cells λ1=5002×π−1
Pass loss exponent α1= 3.5,αk= 4
The noise figure Nf= 10 dB
The noise power σ2=−90 dBm
Static hardware power consumption P1,total = 4 W, Pi,total = 2 W
Power amplifier efficiency factor ε1=εi= 0.4
Precoding power consumption ∆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
VI. NU M ER ICA L RESU LTS
In this section, numerical results are presented to facilitate
the performance evaluations of NOMA enhanced hybrid K-
tier HetNets. The noise power is σ2=−170 + 10 ×
log10 (BW ) + Nf. The power sharing coefficients of NOMA
for each tier are same as am,k =amand an,k =anfor
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 param-
eters 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 in-
creasing 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
3The power consumption parameters applied in this treatise are based on
an established massive MIMO model proposed in [28], [36].
50 100 150 200 250 300 350 400 450 500
M
0.2
0.3
0.4
0.5
0.6
0.7
User association probability
Marco cells
Pico cells
Femto cells
Simulation
B2=20
B2=10
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.
5 10 15 20 25 30
B2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Coverage probability
Analytical NOMA
Simulation NOMA
OMA
am=0.8, an=0.2
am=0.7, an=0.3
am=0.6, an=0.4
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 mP1= 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 probabil-
ity, 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 Rtand 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
4The OMA benchmark adopted in this treatise is that by dividing the two
users in equal time/frequency slots.
10
Rt (BPCU)
Rc (BPCU)
6
4
0
5
0.2
2
4
0.4
3
Coverage probability
0.6
21
0.8
0
0
1
am=0.9, an=0.1
am=0.6, an=0.4
Fig. 4. Successful probability of typical user versus targeted rates of Rtand
Rc, with K= 2,M= 200,N= 15,λ2= 20 ×λ1,rk= 15 m, B2= 5,
P1= 40 dBm, and P2= 20 dBm.
that for inappropriate power and targeted rate selection, the
coverage probability is always zero, which also verifies our
obtained insights in Remark 2.
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.
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 com-
pared 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.
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
5 10 15 20 25 30
B2
0
0.5
1
1.5
2
2.5
3
3.5
Spectrum efficiency (bit/s/Hz)
Analytical NOMA, P2= 20 dBm
Analytical NOMA, P2=30 dBm
Simulation
OMA,P2=30 dBm
OMA,P2=20 dBm
NOMA
OMA
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.
5 10 15 20 25 30
B2
0
2
4
6
8
10
12
14
16
18
Spectrum efficiency (bit/s/Hz)
Marco cells P1=30 dBm
Small cells P1=30 dBm
HetNets P1=30 dBm
Marco cells P1=40 dBm
Small cells P1=40 dBm
HetNets P1=40 dBm
Simulation
Macro cells
HetNets
Small cells
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.
5 10 15 20 25 30
B2
0
0.5
1
1.5
2
2.5
3
Energy efficiency (bits/Hz/Joule)
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
Macro cells
OMA small cells
NOMA small cells
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.
are as follows: 1) The energy efficiency of the macro cells
11
decrease as the number of antenna increases. Although enlarg-
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. CO NC LUS ION S
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 perfor-
mance. 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 direc-
tion is to optimize power sharing coefficients among NOMA
users to further enhance the performance of the proposed
framework.
APP EN D IX A : PROO F O F LEM MA 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
(a)
=EΦi
XK
i=2 Y
j∈Φi\Bo,k
Egj,i he−sPigj,iηd−αi
j,i i
(b)
=exp −
K
X
i=2
λi2πZ∞
ωi,k(x0)1−Egj,i he−gj,i sPiη
rαiirdr!
= exp −
K
X
i=2
λi2πZ∞
ωi,k(x0)1− Lgj,i sPiηr−αirdr!
(c)
=exp −
K
X
i=2
λi2πZ∞
ωi,k(x0)1−1 + sPiηr−αi−1rdr!,
(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.
APP EN D IX B: PROO F OF LEM MA 3
Based on (3), the Laplace transform of the interference from
macro cell BSs can be expressed as follows:
LIM,k (s) = EIM,k "exp −sX
ℓ∈Φ1
P1
Ngℓ,1L(dℓ,1)!#
=EΦ1"Y
ℓ∈Φ1
Egℓ,1exp −sP1
Ngℓ,1ηd−α1
ℓ,1#
(a)
=exp −λ12πZ∞
ωi,1(x)1−Egℓ,1e−sP1gℓ,1η
Nr α1rdr!,
(B.1)
where (a) is obtained with the aid of invoking generating-
function of PPP. Recall that the gℓ,1follows Gamma
distribution with parameter (N , 1). With the aid of
Laplace transform for the Gamma distribution, we ob-
tain Egℓ,1exp −sP1
Ngℓ,1ηr−α1 =Lgℓ,1sP1
Nηr−α1=
1 + sP1
Nηr−α1−N. As such, we can rewrite (B.1) as
LIM,k (s) =
exp −λ12πZ∞
ω1,k(x) 1−1 + sP1η
Nrα1−N!rdr!
(a)
=exp
−2πλ1
n
X
p=1 n
psηP1
NpZ∞
ω1,k(x0)
r−α1p+1
1 + sηP1
rαNNdr
(b)
=exp "−πλ1δ1sηP1
Nδ1N
X
p=1 N
p(−1)δ1−p
×Z−ω1,k(x)−αsηP1/N
0
tp−δ1−1
(1 −t)Ndt#,(B.2)
where (a)is obtained by applying binomial expression and
after some mathematical manipulations, and (b)is obtained
by using t=−sηr−α1P1/N. Based on [33, Eq. (8.391)],
we can obtain the Laplace transform of IM ,k as given in (17).
The proof is complete.
APP EN D IX C: PROO F OF LEM MA 6
For the near user case in small cells, the achievable ergodic
rate in the k-th tier can be expressed as
τn
k=E{log2(1 + γkm∗) + log2(1 + γkn)}
12
=1
ln 2 Z∞
0
¯
Fγkm∗(z)
1 + zdz +1
ln 2 Z∞
0
¯
Fγkn(z)
1 + zdz. (C.1)
We need to obtain the expressions for ¯
Fγkn(z)first. Based on
(4), we can obtain
¯
Fγkn(z) = Zrk
0
Pr an,kPkgo,k ηx−αk
IM,k +IS,k +σ2> zfdo,k (x)dx
=Zrk
0
exp −σ2zxαk
an,kPkηLIkzxαk
an,kPkηfdo,k (x)dx,
(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 γknin (29). In the following, we turn to our attention
to derive the CCDF of γkm∗. Based on (5), we can obtain
¯
Fγkm∗(z)as
¯
Fγkm∗(z) = Zrk
0
fdo,k (x)×
Pr "(am,k −an,kz)go,k >IM,k +IS,k +σ2z
Pkηrk−αk#dx.
(C.3)
Note that for the case z≥am,k
an,k , it is easy to observe that
¯
Fγkm∗(z) = 0. For the case z≤am,k
an,k , following the similar
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.
APP EN D IX D : PROO F O F THE O RE M 3
With the aid of Jensen’s inequality, we can obtain the lower
bound of the achievable ergodic rate of the macro cells as
E{log2(1 + γr,1)} ≥ τ1,L = log21 + En(γr,1)−1o−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:
En(γr,1)−1o≈N
P1GMηEIM,1+IS,1+σ2xα1
=N
P1GMηZ∞
0E{IM,1+IS,1|do,1=x}+σ2
×xα1fdo,1(x)dx. (D.2)
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 Theo-
rem, we obtain
Q1(x) = E
X
ℓ∈Φ1\Bo,1
P1
Nhℓ,1L(dℓ,1)
do,1=x
+EXK
i=2 Xj∈Φi
Pihj,iL(dj,i)
do,1=x
=2P1ηπλ1
α1−2x2−α1+
K
X
i=2
2πλiPiη
αi−2[ωi,1(x)]2−αi,
(D.3)
We first calculate the first part of (D.3) as
E
X
ℓ∈Φ1\Bo,1
P1
Nhℓ,1L(dℓ,1)
do,1=x
(a)
=P1
NηE{hℓ,1}λ1ZR
r−α1dr
(b)
=2πP1ηλ1Z∞
x
r1−α1dr
=2P1ηπλ1
α1−2x2−α1,(D.4)
where (a)is obtained by applying Campbell’s theorem, and
(b)is obtained since the expectation of hℓ,1is N. Then we
turn to our attention to the second part of (D.3), with using
the similar approach, we obtain
EXK
i=2 Xj∈Φi
Pihj,iL(dj,i)
do,1=x
=
K
X
i=2 2πλiPiη
αi−2[ωi,1(x)]2−αi.(D.5)
By substituting (D.4) and (D.5) into (D.2), we obtain the
desired results in (35). The proof is complete.
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