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On the Power Allocation for MIMO-NOMA
Systems with Layered Transmissions
Jinho Choi
Abstract—In this paper, we study optimal power allocation for
multiple input multiple output (MIMO) nonorthogonal multiple
access (NOMA) systems when a layered transmission scheme is
employed. An approach to maximize the sum rate of MIMO-
NOMA with layered transmissions is proposed once we show
that the sum rate is concave in allocated powers to multiple
layers of users. We also derive a closed-form expression for the
average sum rate when statistical channel state information (CSI)
is available at a transmitter, which allows us to allocate powers
to multiple layers for the maximization of the average sum rate.
We find lower- and upper-bounds on the average sum rate, from
which it is shown that the scaling property of MIMO-NOMA
with layered transmissions also holds as conventional MIMO does
(i.e., the average sum rate grows linearly with the number of
antennas).
Index Terms—non-orthogonal multiple access, power alloca-
tion, successive interference cancellation
I. INTRODUCTION
In order to improve the spectral efficiency in a cellular
system, nonorthogonal multiple access (NOMA) is recently
studied in [1]–[4]. In NOMA, the power domain is to be
exploited to improve the spectral efficiency when users have
different channel gains. For example, if there are one user
of high channel gain (for convenience, this user is referred
to as user 1) and the other user of low channel gain (this
user is referred to as user 2), a base station (BS) can transmit
two signals to both the two users simultaneously in the same
frequency band and time slot. The BS usually allocates a
high transmission power to user 2 and a low transmission
power to user 1. From this, at user 1, the signal to user 2
might be decodable. Thus, user 1 decodes the signal to user
2 first and then decodes his/her signal after subtracting the
decoded signal to user 2. At user 2, the signal to user 2 can
be decoded without significant interference from the signal to
user 1 as it might be weak. Clearly, in NOMA, by making
use of superposition coding (SC) [5] together with successive
interference cancellation (SIC) [6], it is possible to achieve
a higher spectral efficiency than that in orthogonal multiple
access (OMA).
NOMA has been extended to various systems and with mul-
tiple input multiple output (MIMO) systems. In [7], NOMA
is studied for downlink coordinated two-point systems. When
the BS is equipped with multiple antennas, beamforming can
be used for NOMA downlink transmissions as in [2], [8]. A
performance analysis is presented in [9] and a power allocation
problem for NOMA is studied in [10]. In [11], the ergodic
The author is with School of Information and Communications,
Gwangju Institute of Science and Technology (GIST), Korea (Email:
jchoi0114@gist.ac.kr).
capacity of MIMO-NOMA with two users is derived and
compared with that of OMA (time division multiple access
(TDMA) is used for OMA), which shows that NOMA can
provide a higher sum rate than OMA and the performance gap
grows with the channel gain difference between two users. In
addition, the power allocation between two users is carried
out for open-loop MIMO downlink transmissions (i.e., the BS
allocates the power to two users’ signals based on statistical
CSI).
While SC and SIC are employed in NOMA to deal with
inter-user interference, they are also used in single-user MIMO
systems [12]–[15]. Layered transmissions in MIMO based
on SC and SIC (often called horizontal Bell Labs layered
space-time (H-BLAST) schemes) are studied in [12], [14]. In
[16], the optimal power allocation for layered transmissions
is investigated. A salient feature of layered transmissions
is that the complexity to decode signals at a receiver is
low as sequence-by-sequence decoding with SIC is used. In
particular, MIMO detection, which can require a complexity
growing exponentially with the number of transmit anten-
nas for an optimal performance [17] [18], is not necessary
since a signal sequence in each layer is to be independently
detected/decoded. As a result, in layered transmissions, the
decoding complexity grows linearly with the number of layers
or transmit antennas.
We note that intra-user SC and SIC are employed for layered
transmissions in MIMO, while inter-user SC and SIC are
used in NOMA. Thus, in this paper, we consider layered
transmissions for MIMO-NOMA with intra-user and inter-user
SC and SIC in downlink transmissions. The main advantage of
(downlink) MIMO-NOMA with layered transmissions is that
the complexity of detection/decoding at users grows linearly
(not exponentially) with the number of transmit antennas or
layers, which would be crucial as the computing power and
energy of users’ devices are usually limited. In this paper,
we mainly focus on the power allocation for MIMO-NOMA
with layered transmissions when known instantaneous channel
state information (CSI) or statistical CSI is available at a BS.
The optimal power allocation is to maximize the sum rate by
optimally allocate powers to multiple layers under a maximum
transmission power constraint for each user. An approach
for the sum rate maximization is derived once we show
that the sum rate is concave in allocated powers to multiple
layers of users. Based on a derived closed-form expression
for the average sum rate, an optimal power allocation is
also considered when statistical CSI is available at the BS.
Throughout the paper, we confine ourselves to the case of two
users in the same frequency band and time slot in MIMO-
NOMA for tractable analysis although MIMO-NOMA with
2
layered transmissions is applicable to the case of more than
two users.
The main contributions of the paper are as follows: i)
an approach to perform the optimal power allocation for
MIMO-NOMA with layered transmissions with maximum
transmission power constraints is proposed; ii) a closed-form
expression for the average sum rate is derived for the optimal
power allocation with statistical CSI; iii) the scaling property
of MIMO-NOMA with layered transmissions is analytically
shown with upper- and lower-bounds on the average sum rate.
The rest of the paper is organized as follows. The sys-
tem model for MIMO-NOMA with layered transmissions is
presented in Section II. An approach for the optimal power
allocation to maximize the sum rate with known instantaneous
CSI is discussed in Section III. In Section IV, a closed-
form expression for the average sum rate is derived with
statistical CSI at the BS and the optimal power allocation to
maximize the average sum rate is studied. Simulation results
are presented in Section V. The paper is concluded with some
remarks in Section VI.
Notation: Matrices and vectors are denoted by upper- and
lower-case boldface letters, respectively. The superscripts T
and Hdenote the transpose and complex conjugate, respec-
tively. For a given vector, x,[x]mdenotes the mth element.
For a matrix X,[X]m,n represents the (m, n)th element. The
determinant of a square matrix Ais denoted by det(A).E[·]
and Var(·)denote the statistical expectation and variance,
respectively. CN(a,R)represents the distribution of circularly
symmetric complex Gaussian (CSCG) random vector with
mean vector aand covariance matrix R.
II. SYSTEM MODEL
In this section, we present a system model for MIMO-
NOMA [11] with a layered transmission scheme that is studied
in [13], [14].
A. MIMO-NOMA System
Suppose that a BS is to support two users in the same
frequency band and time slot using NOMA for downlink
transmissions. The BS is equipped with Nantennas and
each user is equipped with Mantennas. For convenience, we
assume that user 1 is sufficiently close to the BS, while user 2
is not close to the BS (e.g., a cell-edge user). Denote by Hk
the channel matrix from the BS to user k. The signal vector of
size M⇥1to user kis denoted by xk. It is assumed that xk
is a signal vector of a coded sequence. Thus, we may denote
a coded sequence by {xk(l)}, where xk(l)represents the lth
signal vector. For convenience, we omit the time index l.
At user k, the received signal vector is given by
yk=Hk(x1+x2)+nk,k2{1,2},(1)
where nk⇠CN(0,N
0I)is the background noise vector at
user k. Throughout the paper, we assume that E[xk]=0. In
general, we expect that the magnitudes of the elements of H1
are larger than those of H2due to large-scale fading. Thus,
the transmission power allocated to user 1 can be lower than
that to user 2 for a similar received power level and the power
difference is to be exploited in NOMA transmissions for a
higher spectral efficiency.
At user 1, since x2can have a higher power, it can be
decoded. Once x2is obtained, using SIC , the desired signal
can be decoded from
¯
y1=y1H1x2
=H1x1+n1.(2)
At user 2, since the power of x1might be lower than that of
x2,x2can be decoded in the presence of x1as interference.
In general, the BS would have more antennas than a user
has, i.e., N>M. In the case that N>M, we may assume
precoding at the BS with a precoding matrix of size N⇥M,
denoted by W. Then, the effective channel matrix becomes
HkW, which is an M⇥Mmatrix. Thus, throughout the paper,
we assume N=Mand Hkis a composite channel matrix.
Note that since we only focus on the power allocation in this
paper, we do not study precoding [4].
Although we focus on the case of two users in this paper, it
might be possible to consider the case of multiple users as in
[19] where a minorization-maximization algorithm (MMA) is
employed to approximate a non-convex sum rate maximization
problem together with precoding to maximize data rates.
Alternatively, a group of users can be divided into multiple
subgroups of users under a certain criterion (based on the
spatial correlation of channels and power differences between
users) that allows precoding to suppress or minimize the
interference between subgroups and exploits the gain from
NOMA within each subgroup of users. This issue will be
studied as a further research topic in the future.
B. Layered Transmissions
Suppose that the BS is to transmit Mcoded signals simul-
taneously to each user as in H-BLAST [12]. With the layered
transmission scheme, each user can carry out sequence-by-
sequence decoding with SIC for spatial multiplexing [12]–
[14]. Thus, the decoding complexity at users can be much
lower than that with non-layered transmission schemes (in par-
ticular, no joint MIMO detection is required with the layered
transmission scheme). However, unfortunately, this scheme
can reduce the achievable rate. To minimize the decrease of the
achievable rate by the layered transmission, we will consider
the optimal power allocation for MIMO-NOMA with layered
transmissions in Section III.
Denote by xm;kthe mth element of xk. It is assumed that
{xm;k(l)}is a coded sequence from the mth channel encoder.
That is, in each layer, the symbol sequence is assumed to be
independently encoded. Consider the QR factorization of Hk
as
Hk=QkRk,
where Qkis an M⇥Morthogonal matrix and Rkis an
M⇥Mupper triangular matrix. At user k, we can have
zk=QH
kyk
=Rk(x1+x2)+˜
nk,(3)
3
where ˜
nk=QH
knk. Note that since Qkis unitary, ˜
nk⇠
CN(0,N
0I). At layer m, the corresponding element of zkis
given by
zm;k=
M
X
q=m
rm,q;k(xm;1 +xm;2)+˜nm;k,(4)
where rm,q;krepresents the (m, q)th element of Rkand ˜nm;k
represents the mth element of ˜
nk. In this layered transmission
scheme, user 1 is to perform SIC with respect to the user 2’s
signals as well as his/her signals in the other layers.
At layer Mof user 1, we have
zM;1 =rM,M;1(xM;1 +xM;2 )+˜nM;1.(5)
Suppose that user 1 can decode {xM;2}and perform SIC to
decode {xM;1}. At layer M1, we have
zM1;1 =rM1,M1;1 (xM1;1 +xM1;2)
+rM1,M;1 (xM;1 +xM;2)+˜nM1;1.
Since {xM;1}and {xM;2 }are available, they can be canceled,
which results in the following signal:
zM1;1 rM1,M;1 (xM;1 +xM;2)
=rM1,M1;1 (xM1;1 +xM1;2)+˜nM1;1.(6)
In this layer, user 1 is to decode {xM1;2}, and perform SIC
to decode {xM1;1}.
At user 2, the SIC can be performed across layers assuming
that the signals from user 1 are interference.
III. OPTIMAL POWER ALLOCATION FOR LAYE R E D
TRANSMISSIONS WITH INSTANTANEOUS CSI
In this section, we consider an optimal power allocation
for MIMO-NOMA with layered transmissions with known
instantaneous CSI. Throughout this section, we assume block
fading channels where a coded sequence is transmitted within
a coherence time. While we assume that perfect CSI is
available at the BS in this section, we will consider the case
that statistical properties of the channels are available at the
BS and study the power allocation to maximize the average
sum rate in Section IV.
A. Optimal Power Allocation
For convenience, let
↵m=|rm,m;1|2and m=|rm,m;2 |2.(7)
The signal powers are denoted by Pm=E[|xm;1|2]and Qm=
E[|xm;2|2]. In addition, we assume that E[|˜nm;k|2]=N0=1
for normalization purposes.
Denote by ⌘m;kthe code rate of layer mfor user k. Then,
from (5), the code rate of layer Mfor user 2,⌘M;2, has to
satisfy the following inequality:
⌘M;2 log2✓1+ ↵MQM
↵MPM+1◆.(8)
Furthermore, at user 2, the following inequality is required:
⌘M;2 log2✓1+ MQM
MPM+1◆.(9)
This implies that
⌘M;2 min 8
<
:
log2⇣1+ ↵MQM
↵MPM+1 ⌘,
log2⇣1+ MQM
MPM+1 ⌘9
=
;
= log2✓1+ cMQM
cMPM+1◆,(10)
where cM=min{↵M,
M}. At user 1, the code rate of
the Mth layer signal of his/her signal needs to satisfy the
following inequality:
⌘M;1 log2(1 + ↵MPM).(11)
Thus, the (achievable) sum rate can be expressed as
R(P1,Q
1,...,P
M,Q
M)
=
M
X
m=1
log2✓1+ cmQm
cmPm+1◆+ log2(1 + ↵mPm).(12)
Theorem 1: R(P1,Q
1,...,P
M,Q
M)is concave in Pmand
Qm.
Proof: See Appendix A.
From Theorem 1, since the sum rate is concave, we may
consider the sum rate maximization problem as follows:
max{Pm},{Qm}R(P1,Q
1,...,P
M,Q
M)
subject to PM
m=1 Pm+QmPT,(13)
where PTis the total transmission power. Unfortunately, this
sum rate maximization may lead to an undesirable solution
in most cases. To see this, we can consider a special case of
M=1with ↵=↵1,=1, and c=min{↵1,
1}. In this
case, the sum rate maximization problem becomes
maxP,Q log2⇣1+ cQ
cP +1 ⌘+ log2(1 + ↵P)
subject to P+QPT.(14)
Since user 1 is closer to the BS than user 2, we usually have
↵>or c=. Thus, the sum rate becomes
R(P, Q) = log2(1+(P+Q))+log2(1+↵P)log2(1+P).
To maximize the sum rate, we need to allocate the maximum
power to P+Q, i.e., PT=P+Q, which results in
R(P, Q) = log2(1 + PT) + log2(1 + ↵P)log2(1 + P).
Since log2(1 + ↵P)log2(1 + P)is an increasing function
of Pfor given ↵>, we can see that the optimal power
for user 1 is P⇤=PT, while the optimal power for user 2 is
Q⇤=0. This shows that the sum rate maximization does not
lead to a fair power allocation between user 1 and user 2.
In order to be fair to user 2, it is necessary to impose the
power constraints separately. To this end, we may have the
total transmission power constraints to user 1 and user 2 as
follows: M
X
m=1
Pm¯
Pand
M
X
m=1
Qm¯
Q, (15)
where ¯
Pand ¯
Qare the maximum transmission powers to user
1 and user 2, respectively. Then, the sum rate maximization
problem can be formulated as
max{Pm},{Qm}R(P1,Q
1,...,P
M,Q
M)
subject to (15). (16)
4
B. Alternating Maximization
In this subsection, we consider an approach to find the
solution of the sum rate maximization problem in (16).
For convenience, let
P={p=[P1... P
M]T]|Pm0,X
m
Pm¯
P}
Q={q=[Q1... Q
M]T]|Qm0,X
m
Qm¯
Q}.(17)
In addition, let R(p,q)=R(P1,Q
1,...,P
M,Q
M). Then, to
solve (16), we can consider the alternating maximization (AM)
algorithm as follows:
q(t+1) = argmax
q2Q
R(p(t),q)(18)
p(t+1) = argmax
p2P
R(p,q(t+1)).(19)
where p(t)and q(t)are the power vectors after iteration tfor
user 1 and user 2, respectively.
Since Pand Qare convex sets and R(p,q)is concave, this
alternating maximization, which can be seen as a two-block
Gauss-Seidel method, can converge to the optimal solution for
any initial vectors p(0) 2Pand q(0) 2Q[20].
We now focus on the solution to the maximizations in (18)
and (19). The maximization in (18) can be carried out using
the water-filling theorem [5]. To see this, we need to consider
the terms of {Qm}in R(p,q)for given pas follows:
A(q;p)=X
m
log2(1 + cmPm+cmQm)
=X
m
log2((1 + cmPm)(1 + ¯cmQm))
=X
m
log2(1 + ¯cmQm)+Constant,(20)
where ¯cm=cm
1+cmPm. Thus, for given p, we have
Q⇤
m(p)=✓1
¯cm◆+
=✓1+cmPm
cm◆+
,(21)
where (x)+= max{0,x}and is a Lagrange multiplier that
can be decided to meet
¯
Q=X
m✓1+cmPm
cm◆+
.
In (19), for given q, we need to maximize R(p,q). Using
the method of Lagrange Multipliers, we have the following
unconstrained problem:
max
p
R(p,q)⌫X
m
p,(22)
where ⌫is a Lagrange multiplier. Taking the first order
derivative with respect to Pm, we have
B(Pm)=⌫,(23)
where
B(Pm)= ↵m
1+↵mPm
+cm
1+cmQm+cmPm
cm
1+cmPm
.
It is easy to show that B(Pm)is a decreasing function. Thus,
for a given ⌫, we can find a unique Pmthat satisfies (23),
which is denoted by P⇤
m. Note that B(0) = ↵mcm+
cm
1+cmQm. Thus, for ⌫>B
m(0),P⇤
m=0. The value of ⌫can
be decided to hold PmP⇤
m=¯
Pusing the bisection method
based on the fact that PmPmincreases as ⌫decreases.
The computational complexity of the AM algorithm de-
pends on the number of iterations and the complexity of the
bisection method that is used to solve (18) and (19) (note that
in the water-filling theorem to solve (18), we also need to use
the bisection method to decide in (21)). The computational
complexity of the bisection method is O(log2✏), where ✏
is the tolerance. From simulations, we observe that a few
iterations (less than 5) are required for the AM algorithm to
converge. For a fixed torelance and number of iterations, the
complexity becomes linear in M, because for given and ⌫,
the complexity to decide powers are proportional to M. Thus,
the overall complexity is not high.
It is noteworthy that another advantage of layered trans-
missions over non-layered transmissions is a lower amount
of CSI feedback for the power allocation. As shown above,
for the power allocation, the BS needs to know the squared
magnitudes of the diagonal elements of {Rk}, i.e., {↵m}
and {m}, not the whole channel matrices, {Hk}. Thus, the
number of the parameters for CSI feedback is Mfrom each
user, not M2.
IV. OPTIMAL POWER ALLOCATION TO MAXIMIZE
AVERAGE SUM RATE
In this section, we consider the power allocation to maxi-
mize the average sum rate with maximum transmission power
constraints for MIMO-NOMA with layered transmissions once
we derive a closed-form expression for the average sum rate.
We also derive bounds on the average sum rate.
A. Derivation of Average Sum Rate and Power Optimization
From (12), the average sum rate for given pand qis given
by
¯
R(p,q)=E[(p,q)]
=
M
X
m=1
Elog2✓1+ cmQm
cmPm+1◆
+E[log2(1 + ↵mPm)] .(24)
Throughout this section, we consider the following assumption
to find the average sum rate.
A) We assume that the elements of Hkare iid and
[Hk]n,m ⇠CN(0,
2
k).
For convenience, define the normalized signal-to-noise ratio
(SNR) for user 1 as Z=↵m
2
1
or Z=m
2
2
for user 2. Then,
under A), we have the following probability density function
(pdf) of Z[21] [22]:
fm(z)= 1
(Mm)!zMmez,z0.(25)
Let
Cm;1(Pm)=E[log2(1 + ↵mPm)] ,(26)
5
which is the ergodic capacity of layer mof user 1 for given
Pm. Under A), from [23], we have
Cm;1(Pm)=Z1
0
log2(1 + 2
1Pmz)fm(z)dz
=e
1
2
1Pm
ln 2
Mm
X
q=0
Eq+1 ✓1
2
1Pm◆,(27)
where Eq(x)=R1
1exyyqdy. For convenience, we define
⌧n(x)= e1
x
ln 2
n1
X
q=0
Eq+1 ✓1
x◆
=e1
x
ln 2
n
X
q=1
Eq✓1
x◆.(28)
Thus, the average sum rate at user 1 with layered transmissions
becomes
C1(p)=
M
X
m=1
Cm;1(Pm)
=
M
X
m=1
⌧Mm+1(2
1Pm).(29)
This has been derived in [16].
The average sum rate of user 2 is a bit involved. Let
C2(p,q)=
M
X
m=1
Cm;2(Pm,Q
m),(30)
where C2,m(Pm,Q
m)is the ergodic capacity of layer mof
user 2 for given Pmand Qm, which is given by
C2,m(Pm,Q
m)=Elog2✓1+ cmQm
cmPm+1◆
=m(Pm+Qm)m(Pm).(31)
Here,
m(x)=E[log2(1 + cmx))] .(32)
Theorem 2: Under A), we have
m(x)=
Mm
X
p=0
gp✓2
1
2
2
,M m+1
◆⌧Mm+p+1 (x),
(33)
where =2
12
2
2
1+2
2
and
gp(t;n)=✓p+n1
p◆tn+tp
(1 + t)n+p.(34)
Proof: See Appendix B.
Finally, from (29), (30), and (31), we can have a closed-
form expression for the average sum rate as follows:
¯
R(p,q)=C1(p)+C2(p,q)
=
M
X
m=1 ✓⌧Mm+1(2
1Pm)
+m(Pm+Qm)m(Pm)◆.(35)
Using the closed-form expression for the average sum rate
in (35), we can consider the following maximization of the
average sum rate:
max
p2P,q2Q
¯
R(p,q).(36)
According to Theorem 1, R(p,q)is concave in pand q.
Since ¯
R(p,q)is the average of R(p,q), it is also concave.
Thus, we can use the AM algorithm to find the solution of
(36). That is,
q(t+1) = argmax
q2Q
¯
R(p(t),q)(37)
p(t+1) = argmax
p2P
¯
R(p,q(t+1)).(38)
Unlike the case of the maximization with known CSI, we do
not have a closed-form solution of each maximization. Thus,
we could resort to any gradient ascent algorithm [24] that
requires the first order derivatives of a given objective function.
In the following result, we derive the first order derivatives of
¯
R(p,q).
Theorem 3: Let gp,m =gp⇣2
1
2
2
,M m+1
⌘for nota-
tional convenience. Then, we have
d¯
R(p,q)
dQm
=dm(Pm,Q
m)
dQm
=⌫m(Pm+Qm),(39)
where
⌫m(z)= e1
z
z2ln 2
⇥
Mm
X
p=0
gp,m
Mm+p
X
q=0
¯
Eq✓1
z◆.(40)
Here,
¯
Eq(x)=Eq(x)Eq+1 (x).
In addition, we have
d¯
R(p,q)
dPm
=
Mm
X
q=0 ✓e
1
2
1Pm
2
1P2
mln 2 ¯
Eq✓1
2
1Pm◆
+⌫m(Pm+Qm)⌫m(Pm)◆.(41)
Proof: See Appendix C.
B. Bounds
In [11], the power allocation for MIMO-NOMA with non-
layered transmissions is studied with the ergodic capacity.
Although it is not clearly mentioned in [11], the derived
ergodic capacity is an upper-bound. In particular, since
E[min{X, Y }]min{E[X],E[Y]},(42)
the result in [11, Eq. (8)] becomes an upper-bound. Using (42),
we can also find an upper-bound on the average sum rate of
6
MIMO-NOMA with non-layered transmissions. In particular,
an upper-bound on C2(p,q)can be found as follows:
C2(p,q)=
M
X
m=1
Elog2✓1+ cmQm
cmPm+1◆
=
M
X
m=1
E"log2 1+min(↵mQm
↵mPm+1 ,
mQm
mPm+1 )!#
M
X
m=1
E[log2(1 + m(Pm+Qm))]
E[log2(1 + mPm))]
=
M
X
m=1
⌧Mm+1 2
2(Pm+Qm)
⌧Mm+1 2
2Pm,(43)
where the inequality is due to (42) and the assumption that
2
1>
2
2. Thus, an upper-bound on ¯
R(p,q)is given by
¯
R(p,q)
M
X
m=1 ✓⌧Mm+1 2
1Pm
+⌧Mm+1 2
2(Pm+Qm)⌧Mm+1 2
2Pm◆.(44)
We can have a lower-bound on C2,m(Pm,Q
m)as follows.
Theorem 4: Under A), we have
C2,m(Pm,Q
m)˜⌧Mm+1 (Pm,Q
m),(45)
where
˜⌧n(x, y )=⌧n(x+y)⌧n(x).(46)
Proof: See Appendix D.
From (46), a lower-bound on ¯
R(p,q)becomes
¯
R(p,q)
M
X
m=1 ✓⌧Mm+1(2
1Pm)
+⌧Mm+1((Pm+Qm)) ⌧Mm+1 (Pm)◆.(47)
From (44) and (47), we can see that the bounds can be tight
if =2
1
2
1+2
2
2
2⇡2
2. This is the case that 2
12
2.
Using the upper- and lower-bounds on ¯
R(p,q), we can
show that the scaling property of MIMO-NOMA is identical
to that of MIMO [25], [26]. For convenience, we consider the
equal power allocation, i.e., Pm=¯
P
Mand Qm=¯
Q
M. For
C1(p), we can consider the following approximation [17]:
exEq(x)⇡1
x+q.
Using this, from (29), we have
C1⇡1
ln 2
M
X
m=1
Mm+1
X
q=1
1
M
2
1¯
P+q
⇡1
ln 2
M
X
m=1
ln ✓1+(Mm+ 1)2
1¯
P
M◆
=1
ln 2
M
X
m=1
ln ✓1+m2
1¯
P
M◆
⇡1
ln 2 ZM
0
ln ✓1+x2
1¯
P
M◆dx
=M✓✓1+ 1
2
1¯
P◆log2(1 + 2
1¯
P)1
ln 2 ◆.(48)
This shows that C1=O(M), which is the scaling property of
MIMO with layered transmissions and equal power allocation.
Similarly, we can show that C2=O(M)with both the upper-
and lower-bounds in (44) and (47), respectively. Thus, we have
¯
R(p,q)=O(M)with equal power allocation, which shows
that MIMO-NOMA with layered transmissions has the average
sum rate that grows linearly with M.
V. S IMULATION RESULTS
In this section, we present simulation results. We assume the
channel model in Assumption A) with 2
1=1and 2
2=¯!. In
general, we have ¯!1as the distance between the BS and
user 2 is longer than that between the BS and user 1.
A. Power Allocation with Known CSI
In this subsection, we consider the optimal power alloca-
tion for MIMO-NOMA with layered transmissions when the
instantaneous CSI (i.e., {↵m}and {m}) is available at the
BS. For comparison purposes, we consider the achievable rate
of non-layered transmissions, which is given by
R(C1,C2)
=min⇢log2det(I+(I+H1C1HH
1)1H1C2HH
1)
log2det(I+(I+H2C1HH
2)1H2C2HH
2)
+ log2det(I+H1C1HH
1),(49)
where Ck=E[xkxH
k]. For non-layered transmissions, we
only consider the case that C1=¯
P
MIand C2=¯
Q
MI, i.e.,
equal power allocation. As mentioned earlier, if non-layered
transmissions are used, users need to perform joint detection1
for Msymbols, which might be computationally expensive.
In Fig. 1, we show the performances of MIMO-NOMA with
non-layered and layered transmissions with a fixed ¯
P=6dB
and M=6for different total transmission powers of user
2. For the case of non-layered transmissions, the equal power
allocation is considered as mentioned earlier. For the case of
1For joint detection, we can use the maximum likelihood (ML) detector.
Unfortunately, the computational complexity of the ML detection grows
exponentially with M[18]. On the other hand, the complexity for signal
detection based on the QR factorization is linear in M. Since the complexity
of the QR factorization is O(M3), the proposed approach has a much lower
computational complexity in the signal detection than the non-layered scheme
for a large M.
7
layered transmissions, the optimal and equal power allocations
are considered. While the non-layered transmission scheme
can have a higher sum rate than the layered transmission
scheme as shown in Fig. 1 (a), the optimal power allocation
can provide a higher sum rate than the equal power allocation
in the layered transmission scheme. In Fig. 1 (b), it is
interesting to see that the rates of user 2 in the non-layered and
layered transmission schemes are similar when ¯
Qis low, while
the gap increases with ¯
Q. This shows that the performance
loss due to nulling at the users by the QR factorization
for layered transmission becomes significant when the total
allocated power, ¯
Q, increases and it is not compensated by the
optimal power allocation. To avoid this, we need to consider
joint detection/decoding when ¯
Qis high at the expense of high
computational complexity for joint detection.
6 8 10 12 14 16 18 20
10
12
14
16
18
20
22
24
¯
Q(dB)
Sum Rates
Nonlayered (Equal Power)
Layered (Opt. Power)
Layered (Equal Power)
(a)
6 8 10 12 14 16 18 20
0
2
4
6
8
10
12
¯
Q(dB)
Rate of Each User
User 1: Nonlayered (Equal Power)
User 1: Layered (Opt. Power)
User 1: Layered (Equal Power)
User 2: Nonlayered (Equal Power)
User 2: Layered (Opt. Power)
User 2: Layered (Equal Power)
(b)
Fig. 1. Achievable rates for different total transmission powers of user 2, ¯
Q
when M=6,¯
P=6dB, and ¯!=6dB: (a) sum rates; (b) achievable
rates of each user.
Fig. 2 shows the sum rates for different channel gains of
user 2, ¯!, when M=6,¯
P=6dB, and ¯
Q=6dB. The
optimal power allocation has about 1 bps/Hz more sum rate
than the equal power allocation in the layered transmission
scheme for all values of ¯!=2
2.
−10 −9−8−7−6−5−4−3−2−1 0
9
10
11
12
13
14
15
¯ω(dB)
Sum Rates
Nonlayered (Equal Power)
Layered (Opt. Power)
Layered (Equal Power)
Fig. 2. Sum rates for different channel gains of user 2 when M=6,¯
P=6
dB, and ¯
Q=6dB.
In Fig. 3, assuming that the total transmission power, ¯
P+¯
Q,
is fixed and set to 6 dB, we obtain the sum rate for different
fractions of ¯
P. As ¯
Papproaches the total transmission power
or ¯
P
¯
P+¯
Qapproaches 1, the sum rate increases as expected.
Interestingly, we can see that the difference between the sum
rates of the layered and non-layered transmission schemes is
smaller as ¯
P
¯
P+¯
Qdecreases. That is, if the total transmission
power to user 2 is sufficiently larger than that to user 1, the sum
rate of the layered transmission scheme can be close to that of
the non-layered transmission scheme. Note that although the
sum rate increases with ¯
P, we may need to scarify the sum
rate to pursue the fairness by increasing ¯
Q(i.e., allocating
more power to user 2). In this case, the layered transmission
scheme could be a good choice as it reduces the complexity
of decoding at users and it provides a sum rate close to that
with non-layered transmissions.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4
5
6
7
8
9
10
11
¯
P/(
¯
P+
¯
Q)
Sum Rates
Nonlayered (Equal Power)
Layered (Opt. Power)
Layered (Equal Power)
Fig. 3. Sum rates for different fractions of ¯
Pwhen M=6,¯
P+¯
Q=6dB,
and ¯!=3dB.
Fig. 4 shows the sum rates for different numbers of an-
tennas, M, when ¯
P=¯
Q=6dB and ¯!=6dB. An
8
interesting observation is that the scaling property in MIMO
is also valid in MIMO-NOMA (with both layered and non-
layered transmission schemes) with instantaneous CSI. Note
that the gap between the sum rates of nonlayered and layered
transmissions grows with Mas shown in Fig. 4. In layered
transmissions, due to nulling, the performance is degraded,
although a linear complexity of detection/decoding in Mis
achieved. We can also claim that this performance degradation
grows linearly with Mdue to the scaling property in both
MIMO and MIMO-NOMA.
1 2 3 4 5 6 7 8 9 10
2
4
6
8
10
12
14
16
18
20
22
M
Sum Rates
Nonlayered (Equal Power)
Layered (Opt. Power)
Layered (Equal Power)
Fig. 4. Sum rates for different numbers of antennas, M, when ¯
P=¯
Q=6
dB and ¯!=6dB.
B. Power Allocation with Statistical CSI
In this subsection, we consider the power allocation to max-
imize the average sum rate for MIMO-NOMA with layered
transmissions.
In Fig. 5 (a), we present the average sum rates (ergodic
capacities) of MIMO-NOMA with layered and non-layered
transmissions and MIMO-OMA with non-layered transmis-
sions under equal power allocation when ¯
P=6,¯
Q= 10 dB,
and ¯!=9dB. For MIMO-OMA, we consider TDMA where
each user has half of a unit time slot. While MIMO-NOMA
with layered transmissions has a lower sum rate than MIMO-
NOMA with non-layered transmissions, it can have a higher
sum rate than MIMO-OMA with non-layered transmissions.
This shows that MIMO-NOMA with layered transmissions
is a good choice that can enjoy a trade-off between the
complexity (in terms of the complexity of the user’s receiver)
and performance (in comparison with the sum rate of MIMO-
OMA).
Fig. 5 (b) shows the analytical results and simulation results
of the average achievable rate of each user. It is shown that
the derived analytical results agree with simulation results.
In Fig. 6, we show the average sum rates of MIMO-NOMA
with layered transmissions under optimal and equal power
allocations for different total transmission powers of user 2.
We note that if ¯
Qis sufficiently large, the performance of equal
power allocation approaches that of optimal power allocation.
This behavior is similar to that of the water filling theorem [5].
2 4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
M
Sum Rate
Ergodic Capacity of NOMA (Nonlayered)
Ergodic Capacity of OMA (Nonlayered)
Ergodic Capacity of NOMA (Layered)
(a)
2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
25
30
M
Achievable Rate of Each User
user 1 (theory)
user 1 (sim.)
user 2 (theory)
user 2 (sim.)
(b)
Fig. 5. Average sum rates of different MIMO transmission schemes for
various numbers of antennas, M, when ¯
P=6,¯
Q=10dB, and ¯!=9dB:
(a) sum rates; (b) achievable rates of each user (MIMO-NOMA with layered
transmissions).
In the water filling theorem, the optimal power allocation to
maximize the sum rate approaches the equal power allocation
as the total transmission power increases. Thus, we can see that
the equal power allocation can be used in MIMO-NOMA with
layered transmissions for sufficiently high total transmission
powers.
In Fig. 7, the impact of ¯!on the average sum rate is
considered when M=6,¯
P=6dB, and ¯
Q=6dB. There is
about 1 bps/Hz gap in terms of average sum rate between the
performances of the equal and optimal power allocations. This
gap decreases with ¯!. From this, we can see that the optimal
power allocation would be more important as ¯!is smaller or
the distance between the BS and user 2 increases (provided
that the distance between the BS and user 1 is fixed).
Fig. 8 shows the average sum rate for different numbers of
antennas, M, when ¯
P=¯
Q=6dB and ¯!=6dB. As
discussed in Subsection IV-B, MIMO-NOMA has the scaling
property that the average sum rate grows linearly with M. In
Fig. 8, we can confirm this scaling property. We also note
9
6 8 10 12 14 16 18 20
10
12
14
16
18
20
22
24
26
28
¯
Q(dB)
Sum Rates
Sim. (Equal Power)
Theory (Equal Power)
Sim. (Opt. Power)
Theory (Opt. Power)
Fig. 6. Average sum rates for different total transmission powers of user 2,
¯
Qwhen M=6,¯
P=6dB, and ¯!=6dB.
−10 −9−8−7−6−5−4−3−2−1 0
10.5
11
11.5
12
12.5
13
13.5
¯ω(dB)
Sum Rates
Sim. (Equal Power)
Theory (Equal Power)
Sim. (Opt. Power)
Theory (Opt. Power)
Fig. 7. Average sum rates for different channel gains of user 2 when M=6,
¯
P=6dB, and ¯
Q=6dB.
that the gap between the average sum rates of the equal and
optimal power allocations grows with M. This shows that
as the number of layers, M, increases, the optimal power
allocation is more important.
VI. CONCLUSIONS
We studied the optimal power allocation to maximize the
sum rate of MIMO-NOMA with layered transmissions when
each user has a total transmission power constraint. Once we
showed that the sum rate is concave in allocated powers to
multiple layers of users, we proposed an approach for the
optimal power allocation based on the AM algorithm for the
cases of known instantaneous CSI and statistical CSI at the
BS. To perform the optimal power allocation with statistical
CSI, we derived a closed-form expression for the average sum
rate as well as upper- and lower-bounds. From the upper- and
lower-bounds on the average sum rate, we demonstrated that
the scaling property holds (i.e., it was shown that the average
sum rate grows linearly with the number of antennas).
1 2 3 4 5 6 7 8 9 10
2
4
6
8
10
12
14
16
18
20
M
Sum Rates
Sim. (Equal Power)
Theory (Equal Power)
Sim. (Opt. Power)
Theory (Opt. Power)
Fig. 8. Average sum rates for different numbers of antennas, M, when ¯
P=
¯
Q=6dB and ¯!=6dB.
While we have focused on the power allocation to maximize
the sum rate with two users in this paper, there are also other
issues in MIMO-NOMA. An extension to more than two users
is an important issue and a generalization with beamforming
would be practically interesting. Those issues will be studied
as further research topics in the future.
APPENDIX A
PROOF O F THEOREM 1
From (12), it can be shown that
R(P1,Q
1,...,P
M,Q
M)= 1
ln 2 X
m
m(Pm,Q
m),(50)
where
m(Pm,Q
m)=ln(1+cm(Pm+Qm))
+ln(1+↵mPm)ln(1 + cmPm).(51)
To prove Theorem 1, it is sufficient to show that m(Pm,Q
m)
is concave. To this end, we need to find the Hessian matrix of
m(Pm,Q
m). It can be shown that
@2 m
@P2
m
=@2 m
@Q2
m
=@2 m
@Pm@Qm
=c2
m
(1 + cm(Pm+Qm))2.
Thus, the Hessian matrix is negative semi-definite, because for
any vector [x1x2]T, we have
[x1x2]"@2 m
@P2
m
@2 m
@Pm@Qm
@2 m
@Pm@Qm
@2 m
@Q2
m#x1
x2
=c2
m
(1 + cm(Pm+Qm))2(x1+x2)20,(52)
which implies that m(Pm,Q
m)is concave in Pmand Qm.
10
APPENDIX B
PROOF O F THEOREM 2
Let F↵m(x)and Fm(x)denote the cumulative distribution
functions (cdfs) of ↵mand m, respectively. Since cm=
min{↵m,
m}, the cdf of cbecomes [27]
Fcm(x)=1Pr(↵m>x)Pr(m>x)
=1(1 Fm(x/2
1))(1 Fm(x/2
2))
=Fm(x/2
1)+Fm(x/2
2)
Fm(x/2
1)Fm(x/2
2),(53)
where Fm(x)is the cdf of Zand given by
Fm(x)=(Mm+1,x)
(Mm)! .(54)
Here, (x)is the Gamma function and (n, x)is the lower
incomplete Gamma function, which is given by
(n, x)=Zx
0
tn1etdt.
Thus, we have
fcm(x)= 1
2
1✓1Fm✓x
2
2◆◆fm✓x
2
1◆
+1
2
2✓1Fm✓x
2
1◆◆fm✓x
2
2◆.(55)
Then, it can be shown that
m(x)=Z1
0
log2(1 + 2
1xt)fm(t)dt
| {z }
=a)
Z1
0
log@(1 + 2
1xt)Fm✓2
1
2
2
t◆fm(t)dt
| {z }
=b)
+Z1
0
log2(1 + 2
2xt)fm(t)dt
| {z }
=c)
Z1
0
log2(1 + 2
2xt)Fm✓2
2
2
1
t◆fm(t)dt
| {z }
=d)
.(56)
The first and third terms are
a) = ⌧Mm+1 2
1x
c) = ⌧Mm+1 2
2x.(57)
Let !=2
1
2
2
and ¯!=2
2
2
1
. If nis a nonnegative integer, the
lower incomplete Gamma function becomes
(n, t)=(n1)! 1et
n1
X
p=0
tp
p!!.(58)
Using this, after some manipulations, we can find the second
and forth terms as follows:
b) = ⌧Mm+1(2
1x)
Mm
X
p=0
˜gp(!;Mm+ 1)⌧p+Mm+1 ✓2
1x
1+!◆(59)
d) = ⌧Mm+1(2
2x)
Mm
X
p=0
˜gp(¯!;Mm+ 1)⌧p+Mm+1 ✓2
2x
1+¯!◆,(60)
where
˜gp(t;n)=✓p+n1
p◆tp
(1 + t)n+p.(61)
Since ¯!=1
!, we can show that
2
1
1+!=2
2
1+¯!=2
12
2
2
1+2
2
=.
Thus, we have
m(x) = a) b) + c) d)
=
Mm
X
p=0
(˜gp(!;Mm+ 1) + ˜gp(¯!;Mm+ 1)) ⇥
⌧p+Mm+1 ✓2
12
2x
2
1+2
2◆.(62)
Finally, noting that ˜gp(!;n)+˜gp(¯!;n)becomes gp(!;n)in
(34) and from (62), we have (33). This completes the proof.
APPENDIX C
PROOF O F THEOREM 3
The first order derivative of ⌧n(x)is obtained in [16] as
d⌧n(x)
dx =e1
x
x2ln 2
n1
X
q=0 ✓Eq✓1
x◆Eq+1 ✓1
x◆◆
=e1
x
x2ln 2
n1
X
q=0
¯
Eq✓1
x◆.(63)
The first order derivatives in (39) and (41) can be obtained
by using (63). Since the derivations are straightforward and
tedious, we omit them.
APPENDIX D
PROOF O F THEOREM 4
To find a lower-bound on C2,m(Pm,Q
m), we rewrite (31)
as
C2,m(Pm,Q
m)=
Mm
X
p=0
gp(!, M m+ 1)
⇥˜⌧Mm+1+p(Pm,Q
m).(64)
From (26), (27), (28), and (46), we can show that
˜⌧n(x, y )=⌧n(x+y)⌧n(x)
=Elog2✓1+ ↵M+1ny
1+↵M+1nx◆.(65)
11
Noting that log2⇣1+ ay
1+ax ⌘is an increasing function of a
for given x, y > 0, according to [28], it follows
Elog2✓1+ ↵my
1+↵mx◆Elog2✓1+ ↵m+1y
1+↵m+1x◆,
(66)
because ↵mis stochastically larger than ↵m+1 due to
Pr(↵m+1 x)=Fm+1 ✓x
2
1◆>Pr(↵mx)=Fm✓x
2
1◆,
where Fm(x)is defined in (54). Consequently, we have
˜⌧n(x, y )˜⌧n+1(x, y).(67)
Applying (67) to (64), we have
C2,m(Pm,Q
m)
Mm
X
p=0
gp(!, M m+ 1)
⇥˜⌧Mm+1 (Pm,Q
m).(68)
In (68), we can show that
Mm
X
p=0
gp(!, M m+ 1)
=
Mm
X
p=0 ✓Mm+p
p◆!Mm+1 +!p
(1 + !)Mm+1+p
=
Mm
X
p=0 ✓Mm+p
p◆
⇥⇥zMm+1(1 z)p+(1z)Mm+1zp⇤
=1,(69)
where z=!
1+!and the last equality is due to [29]. Finally,
substituting (69) into (68), we have (45), which completes the
proof.
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