Configuration of Base Station Antennas inMillimeter Wave MU-MIMO Systems
Lu Liu, Yafei Tian and Yifan Xue
School of Electronics and Information Engineering, Beihang University, Beijing, China
Email: {luliu, ytian, yfxue}@buaa.edu.cn,
Abstract—Millimeter wave (mmWave) communication is a sig-nificantly enabling technology in the fifth-generation (5G) cellularsystem to promote the capacity. In recent years, people havefocused on studying the propagation characteristics of mmWave.However, the impact of mmWave channel on the system capac-ity, especially in multi-user multi-input and multi-output (MU-MIMO) scenarios, has not been fully investigated. In this paper,we first study a low-complexity numerical calculation method ofthe sum-capacity in MU-MIMO downlink scenarios, and theninvestigate the effect of base station (BS) antenna configurationson the system capacity. The antenna spacing, deployment positionand array shape are respectively studied, and the cumulativedistribution functions (CDFs) of user correlations are analyzed.We find insights to achieve higher channel capacity and to guidethe deployment of BS antennas in practical cellular scenarios.
Index Terms—antenna array, array configuration, channelcapacity, millimeter-Wave, MU-MIMO.
I. INTRODUCTION
One of the primary technique in next generation cellular
system is milimeter wave (mmWave) communications, which
is the spectral frontier for wireless communication systems
nowadays [1]. The mmWave band can provide much wider
bandwidth comparing to today’s cellular networks, and can
promise much smaller base station (BS) antenna arrays due
to its short wavelength [2]. The propagation characteristic of
mmWave channel is different from the microwave channel.
Due to the larger penetration loss, less scattering and diffrac-
tion, mmWave channel is more sparse both in time domain
and space domain [3]–[5].
Channel capacity not only depends on the propagation
characteristics, but also depends on the antenna configurations.
The sub-channel correlation greatly circumscribes the capacity
of MIMO system [6]. One way to decrease the correlation is
to equip antennas with different polarizations and radiation
patterns. In [7], the author investigates MIMO systems where
antenna arrays are composed of different number of dipoles
in three axes (X,Y,Z) and then introduce diverse radiation
patterns to reduce correlation. The result demonstrated that
MlMO systems exploiting antenna pattern diversity allow for
performance improving. In [8], genetic algorithm (GA) are
utilized to design the antenna spacing of the linear array. Op-
timized results show that minute modification of the element
positions can improve the array pattern, especially for the
fewer elements case.
In cellular system, the space constraint of BS antenna
make it infeasible to deploy hundreds of antenna elements
in one horizontal or vertical dimension. In order to cope with
this limitation, people extend the line antenna array to two-
dimensional (2D) antenna arrays. The full-dimension MIMO
(FD-MIMO) system was proposed in [9], where 2D active
antennas are equipped on BSs. To accommodate the intro-
duction of 2D antenna arrays, the channel model is extended
to three dimensions (3D), where the azimuth and elevation
angles are both taken into account. The simulation results show
that the number or spacing of elements in two dimensions
can impact the throughput gain. Referring to the factors of
antennas, the geometry of antenna array has significant impact
on the eigenvalues of single-user MIMO (SU-MIMO) channel
[10], and thus influences channel capacity. In addition, there
are other studies on the formation, spacing and polarization of
antenna arrays [11]–[13].
In this paper, we study the relationship between array
factors and system performance in 3D mmWave multiple-
user MIMO (MU-MIMO) channels. We first study a low-
complexity numerical calculation method of the sum-capacity
in MU-MIMO downlink scenarios. Then combining MU-
MIMO with mmWave propagation, we analyze the impact of
BS antenna factors in detail, including the spacing between
antenna elements, the position of antennas, and the number
of elements on the horizontal or vertical dimensions. The
antenna spacing is changed with multiples of the wavelength
from 18λ to 32λ to find the effect of different array spacing
on the system performance. When BS antennas are deployed
indoors, we considered two positions, such as the ceiling
and the side wall, which are representative for the indoor
layout. Changing the number of elements on the horizontal
and vertical dimensions, we develop antenna arrays with kinds
of configurations and then find their difference on the capacity
as well as the user correlation coefficients.
Since the relationship between each factor and the system
capacity is complicated, it is intractable to derive a rigorous
mathematical formula. Hence, Monte Carlo simulations are
employed to observe cumulative distribution function (CDF)
of channel correlation coefficients and the sum-capacity. After
comprehensive study, we will obtain some general rules to
guide the deployment and configuration of the BS antenna
array in practical 5G systems.
The rest of this paper is organized as follows. In Section II,
we introduce channel models for two BS deployment position-
s. Then we formulate the calculation of channel capacity in
Section III. In section IV, we investigate the impact of various
antenna factors on the system performance in detail. Finally,
Section V concludes the paper.
II. CHANNEL MODEL
We consider a downlink MU-MIMO transmission scenario,
where multiple users are randomly distributed in an indoor
environment. Due to severe path losses, the mmWave environ-
ment is well characterized by a clustered channel model, i.e.,
the Saleh-Valenzuela model [14]. The channel matrix between
the BS and one user is defined as
H =
√NtNr
NclNray
Ncl∑i=1
Nray∑l=1
αilar(φril, θ
ril)a
Ht (φt
il, θtil). (1)
In (1), Nt = Nth ×Ntv is the number of transmitter antennas
on the BS, and Nr = Nrh × Nrv is the number of receiver
antennas on the user end. Nxh and Nxv represent the number
of antenna elements on the horizontal and vertical dimensions
respectively, where x ∈ {t, r} denotes the transmitter or
receiver. When Nxh = 1 or Nxv = 1, the array is linear,
otherwise it is planar. Ncl and Nray denote the number of
clusters and the number of rays in each cluster. Generally,
all of the clusters are consumed to be uniformly distributed,
while the rays in one cluster follow Laplace distribution in
their own angel spread. αil represents the gain of the l-th ray
in the i-th cluster. We suppose that it is i.i.d. and follows the
distribution CN (0, σ2α,i) where σ2
α,i is the average power of
the i-th cluster.ax(φ
xil, θ
xil) is the array response vector in which φx
il is the
azimuth angle and θxil is the elevation angle. The angles with
superscript t denote AoDs and that with superscript r denote
AoAs. The array response vector can be formulated as
ax(φxil, θ
xil) =
1√Nx
[1, e−jvxil , . . . , e−j(Nxh−1)vx
il ]T
⊗[1, e−juxil , . . . , e−j(Nxv−1)ux
il ]T ,
(2)
where uxil and vxil are the phase difference between adjacent el-
ements on the vertical and horizontal dimensions. The symbol
⊗ denotes Kronecker product. While establishing the channel
model, we found that the array response vector ax(φxil, θ
xil) is
different in two kinds of antenna deployments, which will be
shown in detail in the next two subsections.
A. The Array Deployed on the Side WallIf the antenna array is on the side wall, we introduce the
system schematic as shown in Fig. 1 and take the link between
the BS and one user as an example.In Fig. 1, φ is the angle between the ray and the positive
x-axis while θ is the angle between the ray and the x-y
plane. Then the phase difference between adjacent elements
on vertical and horizontal dimensions are
uxil =
2πdxvλ
sin θxil, (3)
vxil =2πdxhλ
cos θxil cosφxil, (4)
where dv and dh are antenna spacing on the vertical and
horizontal dimensions, and λ is the signal wavelength.
X
Y
Z
d1
d2
V
H
User
Fig. 1: The system model with the array on the side wall.
B. The Array Deployed on the Ceiling
If the antenna array is on the ceiling, we establish the
coordinate system as shown in Fig. 2. The definitions of angles
keep the same, and the two kinds of phase difference are
respectively
uxil =
2πdxvλ
cos θxil cosφxil, (5)
vxil =2πdxhλ
cos θxil sinφxil. (6)
X
Y
Z
d1
d2
V
H
User
Fig. 2: The system model with the array on the ceiling.
III. CAPACITY ANALYSIS
In the analysis of data rate, different precoding approaches
could lead to different results. Considering that we mainly
concentrate on the relationship between antenna factors and
the system performance, to avoid the influence of specific
precoding method, we directly analyze the channel sum-
capacity.
It is well known that dirty paper coding (DPC) can achieve
the capacity region of MIMO broadcast channel (BC) [15],
but the implementation of DPC has very high complexity.
According to the duality between MIMO-BC and MIMO
multiple-access channel (MAC), the capacity region of a
MIMO-BC is the same as the capacity region of its dual
MIMO-MAC [16]. Considering a K-user Gaussian MIMO-
BC, the capacity region is represented as
Cunion(P,HH)
�⋃
∑Ki=1 Pi�P
CMAC(P1, . . . , PK ;HH)
=⋃
∑Ki=1 Tr(Pi)�P
{(R1, . . . , RK) :
∑i∈S
Ri
� log∣∣∣I+∑
i∈SHH
i PiHi
∣∣∣, ∀S ⊆ {1, . . . ,K}}. (7)
In (7), H is the channel matrix between the BS and
users where H = [H1, . . . ,HK ]T
. Hi is the channel matrix
of the i-th user. We assume H is constant and is known
perfectly at the transmitter and at all receivers. Pi is individual
power constraint for each transmitter in the dual MAC, and∑Ki=1 Pi ≤ P is the sum power constraint. The positive
semidefinite matrix Pi = ViVHi represents every set of MAC
covariance matrix, where Vi denotes the precoding matrix for
the i-th user equipment (UEi). Ri is the achievable rate of
UEi.
The calculation of capacity region involves a series of
optimization problem of the sum-rate in any possible user set
S . If we are interested in the sum-capacity instead of the whole
capacity region, the problem will be much simplified. Consider
a BS equipped with Nt = Nth × Ntv antennas and UEs
equipped with single antenna. Then, the matrix Hi degrades
to a vector hi. To find the sum-capacity of this MU-MIMO
channel, (7) can be converted as the following optimization
problem
maxP
|I+HHPH|s.t. Tr(P) ≤ P, (8)
where P = diag(P1, . . . , PK) and H = [h1, . . . ,hK ]T
. Note
that the sum-capacity optimization problem of MU-MIMO
is different with the capacity optimization problem of SU-
MIMO. In (8), if P is the covariance matrix of a single
user precoding matrix, the optimal result will be obtained by
singular value decomposition of HH and water filling power
allocation. But for the MU-MIMO case, P is a diagonal matrix
and only involves transmit power of each user. It has no rule
on how to allocate the powers according to the channel matrix
H, which is the combination of channel vectors of different
users.
The problem (8) is convex, which can be solved by standard
convex optimization algorithm, like interior-point method [17].
However, the computational complexity of the interior-point
method is still high for this specific problem. Furthermore, we
cannot find the condition when the sum-capacity is achieved.
Observing the optimization problem (8), we can recognize
that the objective function is a quadratic form whose maximum
value will increase along with the increasing of sum transmit
power. Therefore, the optimal result should be obtained when
the sum power reaches the constraint. In other words, the
optimization problem (8) is the same as
maxP
|I+HHPH|s.t. Tr(P) = P. (9)
We propose a gradient based method to solve problem (9).
Define L = |I+HHPH|, then the gradient over P is
∂L
∂P= |HHPH| (H(I+HHPH)−1HH
)T. (10)
The diagonal element of ∂L∂P is the derivative of L with respect
to the diagonal elements of P.
Theorem 1. In a K-user dual MIMO-MAC channel, whereeach user is equipped with single antenna, the maximum of itssum-capacity is achieved when the transmit powers of someusers are zero and the gradients over the transmit powers ofother users are equal.
Proof. Assume that P has random initial values, P(0) =
diag(P(0)1 , P
(0)2 , . . . , P
(0)K ), where Tr(P(0)) = P . Then, we
can get
diag
(∂L
∂P
)∣∣∣∣P=P(0)
=
[∂L
∂P1,∂L
∂P2, . . . ,
∂L
∂PK
]∣∣∣∣P=P(0)
.
(11)
Note that the partial derivative over one transmit power ∂L∂Pi
actually depends on itself and all other transmit powers.
When P(0) is not the optimal one to get the maximum of
sum-rate, there exists m and n leading to
∂L
∂Pm�= ∂L
∂Pn
∣∣∣∣P=P(0)
, (12)
where m,n ∈ {1, 2, . . . ,K}. It means that the data rate grows
with different speed over Pm and Pn. If ∂L∂Pm
> ∂L∂Pn
, L grows
more rapidly over Pm than over Pn. In other words, adding
a small amount of power ε to Pm will be more effective to
increase data rate than adding ε to Pn. Considering that ∂L∂Pm
and ∂L∂Pn
are continuous, when ε is small enough, we can
always get higher data rate by setting
P(1) = diag[P
(0)1 , . . . , P (0)
m + ε, . . . , P (0)n − ε, . . . , P
(0)K
].
(13)
In this way, the sum-rate can keep increasing and the sum-
power constraint is always satisfied.
Since the transmit power cannot be less than 0, when
one variable Pi approaches 0, it will stay at 0 and other
variables keep to change. Finally, the gradients over all non-
zero transmit powers will be equal, and the achieved sum-
rate is the sum-capacity of the given K-user MIMO-MAC
channel.
The proposed optimization algorithm follows the proof. At
first, set initial values for P, for example Pi = PK for all
i. Then calculate the gradient matrix ∂L∂P using (10). Find
the transmit power Pm with the maximal gradient component∂L∂Pm
, and the transmit power Pn with the minimal gradient
component ∂L∂Pn
. Update P using (13). If one transmit power
Pi approaches 0, find the second minimal gradient component
and keep updating the transmit power matrix. The algorithm
terminates when the gradients over all non-zero transmit
powers are equal (within a given precision). In our statistics,
the proposed algorithm can be about 100 times faster than the
interior-point method used in CVX toolbox, while simulation
results of the two methods are almost the same.
IV. FACTORS AFFECTING CAPACITY
In this section, we strive to find factors of antenna arrays that
affecting MU-MIMO system capacity in mmWave propagation
environments. We simulate in a typical indoor office scenario
described in 3GPP TR38.900 R14 [5], where the length is
120 m, the width is 50 m and the height is 3 m. There are 16
users uniformly distributed at the height of 1 m, and the BS
antennas are 3 m high. The channel parameters are given by
Ncl = 8, Nray = 10 and σ2α,i = 1. The large scale pathloss is
calculated according to [5] in the indoor-office scenario with
28 GHz center frequency .
About the parameters of angles, in the cluster level, the
angle spread of AoDs in line-of-sight (LOS) is 15◦ in the
azimuth and is 7.5◦ in the elevation, in which central AoDs
of each cluster in azimuth and elevation follow the uniform
distribution. The central AoAs of each cluster in azimuth
and elevation follow the uniform distribution varying from 0◦
to 360◦. In the ray level, the AoDs and AoAs in azimuth
and elevation of rays follow the Laplacian distribution. The
angle spread of rays in each cluster is 5◦. Other simulation
parameters are listed in Table I and II for various scenarios.
A. The Antenna Spacing and Deployment Position
If we enlarge or reduce the spacing between antenna ele-
ments such as dh and dv , the phase difference in (3)-(6) will
change. Then the array response and the channel matrix will
also change according to (2) and (1). Therefore, the correlation
of users and the channel capacity may be influenced. We want
to find the relationship between the antenna spacing and the
system performance.
When the BS is at the side wall with 50 m width, the layout
of indoor office scenario is shown in Fig. 3. The antenna
array of the BS is square with Ntv = 8, Nth = 8, totally 64
elements, while the antenna of the user is single. We change
the BS antenna spacing by multiple of wavelength and study
its influence on sum data rate. Set the simulation parameters
as those in Table I. The sum-capacity is calculated with the
proposed gradient algorithm, and is shown in Fig. 4. As a
comparison, the sum data rate achieved by the zero-forcing
(ZF) precoding method is also provided. The simulation results
demonstrate that as the antenna spacing increases, the data rate
continues to rise and eventually reaches a plateau. Therefore,
in practical systems, when the antenna array is placed on the
side wall, we can greatly improve the system capacity through
increasing the antenna spacing appropriately.
When the BS is at center of the ceiling, we set antenna
arrays as those in the side wall scenario and apply the same
method to study the effect of antenna spacing on the sum data
rate. Fig. 5 displays the simulation results with the parameters
in Table I. The trend of lines is similar to that in Fig. 4, but the
achieved data rate is higher. When the antenna spacing is less
than 4λ, the data rate increases more rapidly. In this scenario,
when the antenna spacing is small, even though we enlarge
it a little, the data rate can remarkably get higher. Therefore,
widening antenna spacing will be more effective for the BS
antennas on the ceiling than those on the side wall. In Fig. 4
and Fig. 5, we can also see that, there is a large gap between
the sum-capacity and the achieved sum-rate of ZF precoding
when the BS antenna array is on the side wall, while the gap
greatly shrinks when the BS antenna array is deployed on the
ceiling.
TABLE I: Simulation Parameters
BS array Nth = 8, Ntv = 8Receiver array Nrh = 1, Nrv = 1
Antenna spacing 18λ, 1
4λ, 1
2λ, λ, 2λ, 4λ, 8λ, 16λ, 32λ
50m
Nth
Ntv
BS
User
1m
3m
120m
Fig. 3: Layout of the indoor office scenario.
0 5 10 15 20 25 300
10
20
30
40
50
60
Antenna Spacing (λ)
Dat
a R
ate
(bps
/Hz)
Sum−CapacityZF−Precoding
Fig. 4: Data rate with different antenna spacing when the BS
antenna array is on the side wall.
B. The Configuration of the Antenna Array
The BS antennas we considered are all 2D planar arrays
which can be divided into horizontal and vertical dimensions.
0 5 10 15 20 25 300
10
20
30
40
50
60
70
80
Antenna Spacing (λ)
Dat
a R
ate
(bps
/Hz)
Sum−CapacityZF−Precoding
Fig. 5: Data rate with different antenna spacing when the BS
antenna array is on the ceiling.
In this part, we change the number of antenna elements
on these two dimensions to form different kinds of array
configurations. For example, in the above subsections, the BS
antenna arrays are square, and each array has 8 elements on
both dimensions. However, what will happen on the system
performance if the number of elements on vertical and hori-
zontal dimensions are not equal?
Firstly, take the BS array on the side wall as an example.
When Nth > Ntv , we call it “fat array”. Otherwise, if
Nth < Ntv , we call it “thin array”. Each array has 64 elements.
Other parameters are listed in Table II. In Fig. 6, the sum-
capacities are shown for seven kinds of array configurations. In
general, when the array is on the side wall and the total number
of elements is constant, the more elements are put on the
horizontal dimension, the higher capacity that can be achieved.
Additionally, the gap of the capacity curves of fat arrays is
wider than that of thin arrays. Therefore, it will be more
effective to increase the elements on horizontal dimension.
Moreover, we plot the CDF of correlation among users in Fig.
7 when the antenna spacing is 4λ. It can be seen that when the
array is fat, the correlation among users is smaller than that of
the thin array, so the inter-user interference is weaker and the
capacity is higher. Hence, if the antenna array is arranged on
the side wall, we should increase elements on the horizontal
dimension as much as possible.
Then consider that the BS array is at the center of the
ceiling. When Nth > Ntv , we call it “wide array”. Otherwise,
if Nth < Ntv , we call it “long array”. We take the same
method to get simulation results of capacity and correlation
coefficients which are shown in Fig. 8 and Fig. 9. As can
be seen from Fig. 8, the difference of capacity is most
recognizable when the antenna spacing is 12λ. Thus, in Fig.
9 we assume the antenna spacing as 12λ to see the CDF of
channel correlation coefficients. We can see that the wide
arrays have advantages over the long arrays, so it is more
effective to add elements on the horizontal dimension (parallel
to the long side wall). But if the long array configuration has
to be used, putting all elements on a vertical line (parallel to
the short side wall) can achieve the highest capacity.
TABLE II: Simulation Parameters
BS array
Nth Ntv
64 132 216 48 84 162 321 64
Receiver array Nrh = 1, Nrv = 1
Antenna spacing Side 12λ, λ, 2λ, 4λ, 8λ, 16λ
Ceiling 18λ, 1
4λ, 1
2λ, λ, 2λ, 4λ
0 2 4 6 8 10 12 14 16Antenna Spacing ( )
20
25
30
35
40
45
50
55
Cap
acity
(bps
/Hz)
Nth=64 Ntv=1 (Fat)
Nth=32 Ntv=2 (Fat)
Nth=16 Ntv=4 (Fat)
Nth=8 Ntv=8 (Square)
Nth=4 Ntv=16 (Thin)
Nth=2 Ntv=32 (Thin)
Nth=1 Ntv=64 (Thin)
Fig. 6: Capacity of BS antennas with various configurations
on the side wall.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Correlation Coefficient
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CD
F
Nth=64 Ntv=1 (Fat)
Nth=32 Ntv=2 (Fat)
Nth=16 Ntv=4 (Fat)
Nth=8 Ntv=8 (Square)
Nth=4 Ntv=16 (Thin)
Nth=2 Ntv=32 (Thin)
Nth=1 Ntv=64 (Thin)
Fig. 7: CDF of users correlation for BS arrays on the side
wall with various configurations and antenna spacing = 4λ.
0 0.5 1 1.5 2 2.5 3 3.5 4Antenna Spacing ( )
40
45
50
55
60
65
70
75
Cap
acity
(bps
/Hz)
Nth=64 Ntv=1 (Wide)
Nth=32 Ntv=2 (Wide)
Nth=16 Ntv=4 (Wide)
Nth=8 Ntv=8 (Square)
Nth=4 Ntv=16 (Long)
Nth=2 Ntv=32 (Long)
Nth=1 Ntv=64 (Long)
Fig. 8: Capacity of BS antennas with various configurations
on the ceiling.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Correlation Coefficient
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CD
F
Nth=64 Ntv=1 (Wide)
Nth=32 Ntv=2 (Wide)
Nth=16 Ntv=4 (Wide)
Nth=8 Ntv=8 (Square)
Nth=4 Ntv=16 (Long)
Nth=2 Ntv=32 (Long)
Nth=1 Ntv=64 (Long)
Fig. 9: CDF of users correlation for BS arrays on the ceiling
with various configurations and antenna spacing = 12λ.
V. CONCLUSION
In this paper, we study how the array factors of BS
antennas affect the system performance in millimeter-wave
MU-MIMO systems. We first study the property of optimal
power allocation when the sum-capacity is achieved, and then
propose a low-complexity calculation method for the sum-
capacity. Then we simulate the system performance under
different kinds of deployments and configurations of the BS
array in an indoor office scenario. In an appropriate range,
with the antenna spacing increasing, the channel capacity will
gradually increase. With a fixed height, deploying the BS
antennas on the ceiling can achieve higher capacity. If the
total number of elements in the array is constant, allocating
more number of elements on the horizontal dimension is more
effective. The channel correlation properties among users are
also demonstrated. In a given propagation environment, the
guidance of BS antenna configuration is trying every aspect
to reduce the channel correlation among users.
ACKNOWLEDGEMENTS
This work was supported by the National Natural Science
Foundation of China under Grant 61371077.
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