Joint Transmit Beamforming and Antenna Selection Design for MISO-NOMA Systems Based on Improper Gaussian Signaling

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Joint Transmit Beamforming and Antenna Selection
Design for MISO-NOMA Systems Based on
Improper Gaussian Signaling
Hao-Tse CHIU and Fumiaki MAEHARA
Graduate School of Fundamental Science and Engineering, Waseda University
3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email : fumiaki m@waseda.jp
Abstract—In this paper, we consider the joint antenna selection
(AS) and beamforming (BF) vector design problem for a down-
link two-user power-domain non-orthogonal multiple access (PD-
NOMA) system considering improper Gaussian signaling (IGS).
We formulate an optimization problem concerned about the
max-min rate between users with hardware constraints such as
the total power constraint (TPC), per-antenna power constraint
(PAPC), and the number of activated antennas along with
the conditions for successful successive interference cancellation
(SIC) that are proved in existing works. To deal with the
log-determinant objective function and binary optimization, we
propose a solution that combines successive concave quadratic
programming (SCQP) and the exact penalty approach to obtain
suboptimal results by iterations. As shown in simulation results,
IGS can prevail over its proper Gaussian signaling (PGS)
counterpart in terms of achievable sum rate (ASR) particularly
in high transmit signal-to-noise ratio (SNR) scenarios.
Index Terms—MISO-NOMA, beamforming design, antenna se-
lection, improper Gaussian signaling, max-min rate optimization.
I. INTRODUCTION
The fifth-generation wireless communication (5G) has been
widely studied and entered into the commercialized phase in
recent years. In general, 5G is framed into three scenarios
as ultra-reliable low-latency communications (URLLC), en-
hanced mobile broadband (eMBB), and massive machine type
communications (mMTC) [1]. Among them, eMBB highlights
the large capacity while URLLC supports ultra-low latency
and high connectivity. Nevertheless, due to the heterogeneity
and growing spectrum resource demand of 5G users, allow-
ing users to access the wireless network simultaneously has
become a challenging task. For typical orthogonal multiple
access (OMA) systems, it would be inefficient to allocate
resources within separate resource blocks to fulfill the varying
requirements of all users.
Conventionally, the wireless communication systems adopt
the proper Gaussian signaling (PGS) [2], under which trans-
mitted signals are uncorrelated with their complex conjugate or
equivalently, their real and imaginary parts have equal power.
As a general case of PGS, improper Gaussian signaling (IGS)
[3] relaxes the equal-power and the uncorrelated constraint. It
has been shown that by exploiting IGS, the system throughput
outperforms the PGS counterpart for both OMA and non-
orthogonal multiple access (NOMA) in interference-limited
network [4]–[7]. Compared with the covariance of PGS, IGS is
characterized by augmented covariance composed of pseudo-
covariance and covariance matrices. Practically, IGS can be
generated from proper signals via widely linear precoding
(WLP) (i.e., a pair of precoding matrices) at the transmitter,
which doubles the dimensions to be designed.
On the other hand, power-domain NOMA (PD-NOMA) [8],
[9] is one of the promising strategies to enable several users
to utilize the same resources concurrently while resources
are partitioned to serve OMA users. Coupled with successive
interference cancellation (SIC) at the receiver, PD-NOMA is
expected to improve spectral efficiency. In the past few years,
integrating the multiple-input multiple-output (MIMO) tech-
nique with NOMA has attracted great attention as well [10]–
[13]. To circumvent the power consumption, the hardware cost
of radio frequency (RF) chains and the computation load with
the increasing number of antennas, the antenna selection (AS)
technique can be applied to MIMO-NOMA as an effective
solution [12]. Additionally, MIMO-NOMA is also compatible
with beamforming (BF), which is widely adopted to alleviate
the interference between channels [13].
Distinct from the above literature, this paper inclines to
catch as many requirements as in practical applications. We
propose a max-min rate problem based on IGS with jointly
designing the AS and the BF vector under the total power
constraint (TPC), per-antenna power constraint (PAPC), as
well as the conditions for successful SIC. To the authors’
best knowledge, there is no existing work that considers the
joint design of the AS and the BF vector based on IGS.
Especially for PAPC, it is less concerned in the NOMA
optimization problems considering PGS, not to mention IGS.
In most existing works, only TPC is considered and we
cannot ensure whether the energy will allocate to some specific
antennas, which might cause some performance loss in some
extreme scenarios. Additionally, the joint AS and BF vector
design aim to combine the two advantages above, which are
both significant properties for future communication systems.
Although the problem is nonconvex, it is solved by iterative

algorithms to acquire suboptimal results.
The rest of the paper is organized as follows. The MISO-
NOMA system model, the mathematical analysis, and condi-
tions for successful SIC are provided in Section II. In Section
III, an optimization problem is formulated with the TPC
and the PAPC. The proposed solution is provided followed
by the problem. In Section IV, simulation results reveal the
advantages of the proposed method and the baselines. Finally,
concluding remarks are given in Section V.
Notations: x,X, and xrepresent column vectors, matrices,
and scalars, respectively. XTand XHdenote the transpose
and transpose-conjugate operation, respectively. The ith entry
of xand (i, j)th entry of Xare denoted by [x]iand [X]i,j ,
respectively. |x|,||x||2, and ||x||Fdenote the absolute value
of x, 2-norm, and Frobenius norm of x, respectively. <(·)
denotes the real part of a complex number. [X]2is XXH,
hX,Yi=tr(XHY), and |X|denotes the determinant. The
notation XYrepresents X−Yis positive semi-definite
(PSD) for Hermitian matrices Xand Y.x·yrepresents the
dot product of xand y.
II. SY ST EM MO DE L
A. System Model
In this paper, we consider a MISO downlink two-user
scenario as shown in Fig. 1. For the user grouping, we follow
[8], [12] and assume two users are randomly assigned to
the same group/cluster to perform NOMA. A base station
(BS) is equipped with an Nt-element antenna array and user
equipment (UE) is equipped with a single antenna each.
Firstly, at the BS, since PD-NOMA is adopted, the transmitted
signals are superimposed as
x=√ρ1x1+√ρ2x2(1)
where ρ1(ρ2)is the power allocation factor (PAF) for
UE1(UE2)and ρ1+ρ2= 1. The transmitted signal xi
is generated by the same pair of widely linear precoder
w1,w2∈CNtas [4]
xi=w1si+w2s∗
i(2)
where siis the desired signal for UEiand assumed to be
proper Gaussian signal with E[si]=0and E[|si|2]=1.
After performing WLP, we can generate improper signals from
proper ones.
The channel vector from the ith antenna at the BS to the
kth UE is denoted by hk∈C1×Nt, and its entries hk,i can
be modeled as independent and identically distributed (i.i.d.)
complex Gaussian random variables with variance σ2
hk. Data
at the BS are precoded by BF vectors and transmitted through
spatially independent flat Rayleigh fading channel. Then, the
received signal yiat UEican be express as
yi=hi(√ρ1x1+√ρ2x2) + ni(3)
where nidenotes the additive white Gaussian noise (AWGN)
of UEiwith variance σ2.
Fig. 1: Illustration of architecture of the MISO-NOMA system in this work.
Next, we can obtain the augmented equation of (3), which
is given by
yk
y∗
k=hk01×Nt
01×Nth∗
k2
X
l=1
√ρlw1w2
w∗
2w∗
1sl
s∗
l+nk
n∗
k
,ˆ
Hk
2
X
l=1
√ρlˆ
Wˆ
sl+ˆ
nk
(4)
Besides, the achievable rate expression at UEkbefore SIC is
computed by [14]
Rk=1
2log2
I2+ρk[ˆ
Hkˆ
W]2
X
m6=k
ρm[ˆ
Hkˆ
W]2+σ2I2

−1
(5)
For the special cases w2=0,xiin (2) becomes typical proper
Gaussian signal, so (5) yields
Rk=log2 1 + ρk|hkw1|2
Pm6=kρm|hkw1|2+σ2!(6)
which is the well-known rate expression for PGS cases.
By defining w,{(w1,w2)}and if we focus on UE2first,
(5) can be further rewritten as a function of w, which provides
additional degree of freedom (DoF) to improve the system
performance
R2=1
2
1
ln2ln 
I2+ [√ρ2ˆ
H2ˆ
W]2[√ρ1ˆ
H2ˆ
W]2+σ2I2−1
=1
2
1
ln2ln I2+ [X2]2[X1]2+σ2I2−1
,1
2
1
ln2r2(w)
(7)
B. Conditions for Successful SIC
In two-user PD-NOMA systems, the information of the
weaker user (i.e., with lower channel gain) would be decoded

first by SIC at the receiver of the stronger user. Without loss
of generality, we assume that UE1is the strong user and UE2
is the weak user. For PGS cases (i.e.,w2=0), to successfully
perform SIC at strong user, the SINR at user 1 for decoding
s2should be no less than that at UE2, which is called implicit
SINR constraint in [15] and equivalently expressed as
|h1w1|2≥ |h2w1|2(8)
Moreover, as for IGS cases, to ensure UE1can successfully
decode s2, the rate of UE2should be also bounded by the
observed rate at UE1for decoding s2,ˆ
R2[16]
R2≤ˆ
R2,1
2log2



I2+ [√ρ2ˆ
H1ˆ
W]2[√ρ1ˆ
H1ˆ
W]2+σ2I2−1



(9)
which also implies R2=min{R2,ˆ
R2}. To simplify the
expression, [16] derives a lemma of the channel condition that
can ensure the achievable rate of user 2 can achieve its channel
capacity as well as ensure the successful SIC at the strong user,
which is similar to (8) and given by
hH
1h1hH
2h2(10)
However, one cannot randomly generate the channel pairs
that always meet the lemma. By broadening the gap of channel
gains and lower the number of transmit and received antennas
can increase the probability of achieving the condition.
To sum up, for PGS cases, we would introduce an extra
constraint (8) to ensure successful SIC while we assign the
simulation parameters to fulfill (10) with high probability for
IGS cases, which is also stated to be easily achieved in practice
[16].
III. PROB LE M FOR MU LATI ON A ND PROPOSED SOLUTION
We formulate an optimization problem of the AS vector
and the BF vector to maximize the minimum achievable rate
of users. In Section III-A, an initial joint optimization problem
is described. In Section III-B, a proposed solution to obtain
suboptimal results is given.
A. Problem Statement
Assume the channel state information (CSI) is perfectly
known at the BS. The max-min rate optimization problem is
formulated as
max
w,bmin {R1, R2}(11a)
s.t.||w1||2+||w2||2≤Psum (11b)
||[w1]m||2+||[w2]m||2≤Ppapc ·bm(11c)
bm∈ {0,1}, m = 1,2,··· , Nt(11d)
Nt
X
m=1
bm=L(11e)
The variables in (11) include the TPC Psum; the PAPC
Ppapc; the AS indicator bm, which equals to 1 if the mth
transmitted antenna is selected and bm= 0, otherwise. (11b)-
(11c) indicate the hardware power constraint to avoid power
amplifier (PA) non-linearity for the BS and even each antenna
at the BS. (11d)-(11e) limit the number of simultaneously
activated antenna to be L.
B. Proposed Solution
We consider the system to be IGS, which is a more
general scheme than PGS. Different from the PGS counterpart
(6), which is a function of a single beamforming vector,
the objective function (11a) involves a pair of beamforming
vectors w1and w2. Even in MISO cases, (11a) is a log-
determinant function and thus more complicated than the
logarithmic fraction function as (6).
Accordingly, successive convex quadratic programming
(SCQP) [17] is adopted to solve problem (11) by iterations in
this work. Let w(κ)be a feasible point for (11) resulted from
the (κ−1)th iteration. By utilizing the inequality (15) in the
Appendix A, we can obtain a lower-bound concave quadratic
function of r2(w)as
r2(w)≥r(κ)
2(w)
,a(κ)+ 2<{hA(κ),X2i}
− hB(κ),[X2]2+ [X1]2+σ2I2i
(12)
where
0> a(κ),r(κ)
2(w(κ))−tr([X(κ)
2]H([X(κ)
1]2+σ2I2)−1[X(κ)
2])
A(κ),([X(κ)
1]2+σ2I2)−1[X(κ)
2]
0B(κ),([X(κ)
1]2+σ2I2)−1−([X(κ)
1]2+σ2I2+ [X(κ)
2]2)−1
Similarly, r1(w)can also be approximated by (15) and we
can successively approximate r1(w)and r2(w)by the right-
hand side of (12). Therefore, for the κth iteration, we solve
the following convex optimization problem (13) to generate
the next feasible point w(κ+1)
max
wmin nr(κ)
1, r(κ)
2o(13a)
s.t.(11b) −(11e) (13b)
As for the AS constraints, we adopt another algorithm
based on the exact penalty approach [18]. We collect the AS
indicators together as band introduce a new slack variable t.
Then the problem (13) can be rewritten as
max
w,bI,bII t−µ|L−(bI)TbI I |(14a)
s.t. t ≤r(κ)
1(14b)
t≤r(κ)
2(14c)
||[w1,1]m||2+||[w2,1]m||2≤Ppapc ·bI
m(14d)
||[w1,1]m||2+||[w2,1]m||2≤Ppapc ·bII
m(14e)
XbI
m=XbII
m=L(14f)
0≤bI
m, bII
m≤1(14g)
(11b)
where the binary constraint (11d) is replaced by PbI
mbII
m=L
and (14g). The proof is expounded in [18]. By solving (14),

we can acquire at least one suboptimal solution set {w,b}
and the whole procedures are shown in Algorithm 1.
Algorithm 1 Proposed Exact Penalty Approach-based Succes-
sive Convex Quadratic Programming (SCQP)
Input: ρ1, ρ2, L, h1,h2, Psum, Ppapc , κmax, µ, β , 1, 2.
Output: w.
1: Initialization: Set κ= 0 and a feasible set w(0) ,bI,0
that satisfies (11b), (14f), and (14g).
2: repeat
3: Solve (14) for the next feasible pointw,bII,κ.
4: If (bII ,κ)TbI I,κ =Lor (bI I,κ)TbI I,κ ≥(bI ,κ)TbI ,κ +
β, keep µthe same, otherwise increase µ(e.g.,µ←2µ).
βis a proper positive value (e.g.,L/20).
5: {w(κ+1),bI ,κ}←{w,bII,κ }
6: κ←κ+ 1
7: until ||w(κ+1) −w(κ)||F< 1,|(bII ,κ)TbI I,κ −L|< 2,
or the given maximum iteration number κmax is reached,
where 1, 2denotes the convergence threshold.
IV. SIMULATION RESULT
In this section, we show simulation results to verify the
performance of the proposed problem. The algorithm in Sec-
tion III is implemented with CVX [19], a package used to
solve convex programs. In this work, we set the initial PAFs
ρ= [0.4 0.6],Nt= 16 and L= 5,16. We have 100
independent channel realizations for each case and channel
coefficients are assigned to be zero mean and variance (0dB,-
10dB) for (σh2
1,σh2
2), respectively. For the power constraints,
we set Psum = 1 and Ppapc = 0.3. Penalty scalars are
µ= 0.005, µ1=µ2= 0.0005 (µ1and µ2are parameters
for Appendix B), and the number of maximum iteration for
Algorithm 1 is κmax = 100. For performance comparisons, we
take the conventional PGS zero-forcing beamforming (ZFBF)
as one of the baseline, named as NOMA-ZFBF-PGS, which
generates beamforming vectors for each user and can be
directly computed by the psuedo-inverse of the channel matrix,
i.e.,pPsum/2[(hT
1·b)†/||hT
1·b||; (hT
2·b)†/||hT
2·b||]T.
Beside NOMA-ZFBF-PGS, the following cases are evaluated
for comparisons:
•NOMA-maxmin-IGS: proposed problem (14).
•NOMA-maxmin-PGS: results of (16).
•NOMA-max2-PGS: results of (16) without (16l).
•OMA-MASR-PGS: proposed problem for OMA with
the objective function (11a) being maxmize R1+R2.
(Detailed in Appendix C)
Fig. 2a and Fig. 2b reveal the comparison of ASR after
perfect SIC. It is found that NOMA-maxmin-IGS always
outperform others except for NOMA-ZFBF-PGS especially at
a high transmit signal-to-noise ratio (SNR). It demonstrates
that by applying IGS, the ASR does have the potential to
improve with the design of extra dimensions, no matter which
objective function is used for PGS cases. Although ZFBF
exhibits supremacy against others, there is a crucial point that
for ZFBF, we only limit the total transmit power of BF vectors,
so the PAPC is relaxed. Therefore, in some cases, we cannot
ensure PAPC always holds, which is a minor disadvantage
compared with designed BF vectors.
5 10 15 20 25
0
1
2
3
4
5
6
7
8
ASR
ASR comparison
(a) Nt= 16, L = 5
5 10 15 20 25
0
2
4
6
8
10
12
ASR
ASR comparison
(b) Nt= 16, L = 16
Fig. 2: ASR v.s. transmit SNR for all cases
V. CONCLUSION
In this paper, we formulate a joint optimization problem
for downlink two-user MISO-NOMA concerned about the
AS and the BF vector design with the TPC and the PAPC.
Moreover, IGS, the more general scheme than PGS, is high-
lighted in the system analysis. Under IGS, the max-min rate
optimization, whose objective function is a log-determinant
function, is too sophisticated to be solved. Consequently, to
obtain a suboptimal solution, we combine SCQP and the exact
penalty approach to cope with the problem. Simulation results
demonstrate that in terms of ASR, the proposed NOMA-
maxmin-IGS prevails other PGS cases exclusive of NOMA-
ZFBF without PAPC.
APPENDIX A
FUN DAM EN TAL INEQUAL IT Y USE D IN SCQP
To deal with the objective function of IGS cases, the
following inequality holds true for PSD matrices X,Y,¯
X,¯
Y
with dimension 2×2[17]

ln I2+ [X]2Y−1≥ln I2+ [ ¯
X]2¯
Y−1−tr([ ¯
X]2¯
Y−1)
+ 2<trh¯
Y−1¯
X,Xi
− h ¯
Y−1−(¯
Y+ [ ¯
X]2)−1,[X]2+Yi
(15)
APPENDIX B
PROP ER GAU SS IA N SIGNALING CONSIDERED IN THIS
WORK
By substituting w2=0and (6) into (11). Defining W=
w1wH
1and Hk=hH
khk. We can formulate an optimization
problem based on PGS and exact penalty approach [18]:
max
WI,WII ,bI,bI I tr(WIH2) + tr(WII H2)
−µ1|P2
sum −tr(WIWII )| − µ2|L−(bI)TbI I |
(16a)
s.t.tr(WI)≤Psum (16b)
tr(WII )≤Psum (16c)
WI,WII 0(16d)
[WI]m,m ≤PpapcbI
m, m = 1,2,·· · , Nt
(16e)
[WII ]m,m ≤Ppapc bII
m, m = 1,2,·· · , Nt
(16f)
XbI
m=XbII
m=L(16g)
XbI
mbII
m=L(16h)
0≤bI
m, bII
m≤1(16i)
tr(WIH1)≥tr(WIH2)(16j)
tr(WII H1)≥tr(WI I H2)(16k)
ρ1tr(WH1)≥ρ2tr(WH2)(16l)
Note that in PGS cases, we simplify the objective function into
maximizing R2instead of the origin max-min function. This
simplification stems from the fact that (11a) and maxW,bR2
would be equivalent if (16l) holds. In the simulation, we also
display the difference between the existing of the inequality
for more comprehensive comparison.
APPENDIX C
OMA CA SE S CONSIDERED IN THIS WOR K
For conventional two-user OMA systems, SINRs can be
expressed as SINRi=tr(WHi)/σ2, i = 1,2and Ri=
0.5log2(1 + SINRi). The corresponding MASR optimization
problem is given by
maximize
W,bpk1k2(17a)
s.t. ki≤1 + SINRi, i = 1,2(17b)
(11b)−(11e)
which can be solved after some mathematical rearrangements
of the objective function and constraints.
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