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sensors
Article
A Method of Implementing a 4 ×4 Correlation Matrix for
Evaluating the Uplink Channel Properties of MIMO
Over-the-Air Apparatus
Kazuhiro Honda
Citation: Honda, K. A Method of
Implementing a 4 ×4 Correlation
Matrix for Evaluating the Uplink
Channel Properties of MIMO
Over-the-Air Apparatus. Sensors 2021,
21, 6184. https://doi.org/10.3390/s
21186184
Academic Editor: Ángela María
Coves Soler
Received: 6 August 2021
Accepted: 13 September 2021
Published: 15 September 2021
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4.0/).
Graduate School of Engineering, Toyama University, 3190 Gofuku, Toyama 930-8555, Japan;
hondak@eng.u-toyama.ac.jp; Tel.: +81-76-445-6759
Abstract:
This paper presents a method of implementing a 4
×
4 correlation matrix for evaluating
the uplink channel properties of multiple-input multiple-output (MIMO) antennas using an over-
the-air measurement system. First, the implementation model used to determine the correlation
coefficients between the signals received at the base station (BS) antennas via the uplink channel
is described. Then, a methodology is introduced to achieve a 4
×
4 correlation matrix for a BS
MIMO antenna based on Jakes’ model by setting the initial phases of the secondary wave sources
in the two-dimensional channel model. The performance of the uplink channel for a four-element
MIMO terminal array antenna is evaluated using a two-dimensional bidirectional fading emulator.
The results show that the measured correlation coefficients between the signals received via the
uplink channel at the BS antennas using the proposed method are in good agreement with the
BS correlation characteristics calculated using Monte Carlo simulation and the theoretical formula,
thereby confirming the effectiveness of the proposed method.
Keywords:
uplink channel; 4
×
4 correlation matrix; initial phase; Jakes’ model; multiple-input
multiple-output (MIMO); over-the-air (OTA); bidirectional fading emulator; channel capacity
1. Introduction
Fifth-generation (5G) mobile communication systems, which will enable high speed,
low latency, and large capacity, are becoming commercially available globally. Thus far,
several new services that exploit these features, such as sports viewing [
1
] and autonomous
driving [
2
], have been considered. To transmit high-capacity data, such as video data from
a mobile terminal to a base station (BS), ultra-high-speed communication is required for the
uplink channel. In the 3rd Generation Partnership Project (3GPP), 5G supports downlink
and uplink peak rates of 20 and 10 Gbps, respectively [
3
]. Multiple-input multiple-output
(MIMO) systems are essential for achieving ultra-high-speed communication [
4
,
5
]. Hence,
evaluating the performance of MIMO terminals is necessary, not only for the downlink
channels, but also for the uplink channels.
A straightforward method for evaluating a MIMO terminal is field testing in an
actual scenario [
6
]. However, with field testing the repeatability and controllability of the
measured data cannot be observed, and, moreover, is a very time-consuming and labor-
intensive process. Hence, over-the-air (OTA) testing, which evaluates the performance of
MIMO mobile terminals by creating a realistic propagation environment in the laboratory,
is very important.
OTA measurement methods have been standardized by 3GPP and the Cellular Telecom-
munication and Internet Association (CTIA) [
7
,
8
]. In 3GPP, OTA test methodologies are
classified into three categories:
(1)
Reverberation chamber based methods,
(2)
Two-stage methods,
(3)
Multiple probe antenna based methods.
Sensors 2021,21, 6184. https://doi.org/10.3390/s21186184 https://www.mdpi.com/journal/sensors
Sensors 2021,21, 6184 2 of 20
Many papers in which OTA measurements have been done using these three methods
have been published. The evaluation target and issues focused on are highlighted in
Table 1
.
The characteristics of the three OTA measurement methods are summarized as follows.
Table 1. Novelty of the present work.
Method Target Issues Focused on
Reverberation chamber
based method
Mobile
terminal
Using a universal software radio peripheral [
9
]
2×2 MIMO, NLOS, 28 GHz [10]
Two-stage method Mobile
terminal
3D channel model, active antenna array [11]
Non-Stationary channel model [12]
Multiple probe antenna
based method
Mobile
terminal
Cluster environment [13]
Channel emulation technique [14]
Plane-wave field synthesis [15]
Bit error rate, IoT wireless device [16]
3D channel model, probe selection [17]
Base
station
Radiated testing of massive MIMO [18]
Probe selection algorithm [19]
In a reverberation chamber based method, the chamber refers to a metallic cavity or
cavities that can emulate an isotropic multipath environment that represents a reference
environment for systems designed to work during fading [
9
,
10
]. The Rayleigh propagation
environment in a chamber is well known as a good reference for mobile terminal systems
in urban environments. The advantages of this method are its simple structure, small size,
and low cost. However, it is difficult to control the multipath propagation characteristics,
such as the cross-polarization power ratio (XPR), spatial correlation, and cluster, owing to
the simplicity of the device.
In the two-stage method, the measurement is divided into two stages: first, measure-
ments of the MIMO antenna patterns that contain all the necessary information to evaluate
the antenna’s performance, such as radiation power, efficiency, and correlation, are made;
then, the measured antenna patterns are incorporated with the chosen MIMO OTA chan-
nel models for real-time emulation [
11
,
12
]. The advantage of this method is the reduced
number of instruments required, compared with the multiple probe antenna based method
introduced next. However, this method requires nonintrusive complex radiation pattern
measurements for accurate evaluation of the MIMO terminals.
In the multiple probe antenna based method, such as one with a fading emulator, a number
of test antenna probes located in the chamber reproduce the channel characteristics in a
controlled and repeatable manner to test MIMO performance [
13
–
17
]. The disadvantage
of this method is that it requires many instruments, such as a probe antenna, a power
divider, and a phase shifter. However, a fading emulator is capable of emulating realistic
and accurate multipath environments [20–24].
In OTA testing, investigations primarily focus on evaluating the performance of
mobile terminals in downlink channels [
9
–
17
]. For the uplink channel, OTA testing of a
massive MIMO BS has been conducted; however, MIMO terminal OTA testing has not yet
been reported [
18
,
19
]. In another paper, it was reported that there was a high correlation
coefficient between the received signals in the uplink channel for a BS antenna with a
separation of 5
λ
at 2 GHz [
25
]. Hence, the performance of the uplink channel from the
MIMO terminal must be evaluated considering the BS correlation conditions. Herein, the
correlation coefficient between the signals received at the BS antennas is referred to as the
BS correlation in the discussion that follows.
In one of my previous studies [
21
], in which the performance of the uplink channel
was examined, a control method for BS correlation of a 2
×
2 MIMO system based on
Jakes’ model [
26
] using a multiple probe antenna based method was proposed. In the 2
×
2
MIMO system, there are two BS antennas; therefore, the BS correlation can be controlled
by using the virtual point source separation to represent the spatial correlation of the BS
Sensors 2021,21, 6184 3 of 20
antennas. However, in a 4
×
4 MIMO system, there are six possible combinations of spatial
correlation, owing to the different relative separations between the BS antennas when the
four-element BS antennas are arranged in, for example, linear or circular arrays.
This paper presents a methodology to control the spatial correlation of the BS antennas
in a 4
×
4 MIMO system based on Jakes’ model with communication via the uplink
channel. First, the implementation model used to enable BS correlation with the BS antenna
arrangement is described. Then, a 4
×
4 correlation matrix of a BS MIMO antenna is
achieved by controlling the initial phases of the secondary wave sources depending on
the BS correlation. Finally, the validity of the proposed initial phase setting method used
to assess the 4
×
4 MIMO system was verified by evaluating the uplink channel using
bidirectional MIMO-OTA apparatus.
The remainder of this paper is organized as follows: in Section 2I present the propa-
gation model for the uplink channel; in Section 3the methodology of the proposed method
is described; Section 4shows the method for implementing a 4
×
4 correlation matrix; in
Section 5I discuss the experimental and analytical results; Section 6concludes the paper.
2. Uplink Channel Model
In the downlink channel, radio waves emitted from BS antennas are reflected and
diffracted by surrounding objects, such as buildings or trees, and they construct secondary
wave sources, that is, scatterers, around the MIMO terminal. Furthermore, radio waves
radiated from the secondary wave sources are combined to form a single coherent wave in
the antennas at the MIMO terminal via different routes. Then, the incoming radio waves
around the MIMO terminal generate a uniform distribution, and the correlation coefficient
between the signals received at the MIMO terminal antennas is low, regardless of the
distance between them [27].
One of the most common approaches to analyze a MIMO antenna system is to use
Monte Carlo simulation, where many scatterers are arranged on a circle, known as Clarke’s
model [
28
], to simulate a multipath fading channel in a mobile propagation environment;
this results in the evaluation of the MIMO terminal performance [24]. A multipath propa-
gation environment is simulated by the amplitude and phase of each wave radiated from
the scatterers, which is an aggregation of all the information in the radio waves emitted
by the BS antennas. A spatial fading emulator based on Clarke’s model is implemented
using Monte Carlo simulation; thus, controlling the amplitude and phase of the scatterers
is essential to create a realistic propagation environment.
However, for the uplink channel, the BS correlation is high regardless of the BS
antenna separation because the angular spread of the incoming wave for the BS antennas
is very narrow [
29
]. The BS correlation is known to be relatively high, even in the case of
array spacings of several wavelengths in a small cell environment [
25
]. Moreover, the BS
correlation is expected to be higher when the direction of the incident wave is parallel to
the BS antenna array. Hence, when the performance of the uplink channel from the MIMO
terminal is examined, the channel model must implement a high BS correlation that is
different from that of the downlink channel assumed to be independent and identically
distributed (i.i.d.).
Figure 1shows the propagation model for the uplink channel, wherein the radio
waves emitted from the MIMO terminal are reflected and diffracted over the full azimuth
by surrounding objects, such as buildings or trees, which are the relay points, that is,
probes, around the MIMO terminal. Furthermore, the radio waves radiating from the
probes combine to generate a single coherent wave at the BS antennas via different routes.
Sensors 2021,21, 6184 4 of 20
Sensors 2021, 21, x FOR PEER REVIEW 4 of 21
probes, around the MIMO terminal. Furthermore, the radio waves radiating from the
probes combine to generate a single coherent wave at the BS antennas via different routes.
Figure 1. Propagation model for the uplink channel.
The BS correlation for the uplink channel generally involves the characteristics of ra-
dio wave propagation, which is represented by multipath waves and the BS antenna char-
acteristics. In this paper, the sum of the two above-mentioned effects is attributed to the
characteristics of the multipath waves, which are represented by the amplitude and phase
of each wave radiating from the probes. Consequently, the area around the BS was not
modeled in Figure 1.
3. Method to Control the BS Correlation
Figure 2 illustrates two-dimensional channel models of the downlink and uplink
channels at the MIMO terminal [30]. In Figure 2, K secondary wave sources arranged on
a circle act as the radiating antennas (scatterers) and receiving antennas (probes) for the
downlink and uplink channels, respectively.
(a)
(b)
Figure 2. Two-dimensional channel model: (a) downlink; (b) uplink.
Base Station
Cluster
Uplink
Mobile Terminal
Probe
1m
ϕ
mK
ϕ
mi
ϕ
Base Station
Scatterer
#m#4
#2
#1
#4#n#2#1
DUT
v
φ
i
φ
#1
#i
#K
1m
ϕ
Km
ϕ
im
ϕ
Base Station
Probe
#m#4
#2
#1
#4#n#2#1
DUT
v
φ
i
φ
#1
#i
#K
Figure 1. Propagation model for the uplink channel.
The BS correlation for the uplink channel generally involves the characteristics of
radio wave propagation, which is represented by multipath waves and the BS antenna
characteristics. In this paper, the sum of the two above-mentioned effects is attributed to
the characteristics of the multipath waves, which are represented by the amplitude and
phase of each wave radiating from the probes. Consequently, the area around the BS was
not modeled in Figure 1.
3. Method to Control the BS Correlation
Figure 2illustrates two-dimensional channel models of the downlink and uplink
channels at the MIMO terminal [
30
]. In Figure 2,Ksecondary wave sources arranged on
a circle act as the radiating antennas (scatterers) and receiving antennas (probes) for the
downlink and uplink channels, respectively.
Sensors 2021, 21, x FOR PEER REVIEW 4 of 21
probes, around the MIMO terminal. Furthermore, the radio waves radiating from the
probes combine to generate a single coherent wave at the BS antennas via different routes.
Figure 1. Propagation model for the uplink channel.
The BS correlation for the uplink channel generally involves the characteristics of ra-
dio wave propagation, which is represented by multipath waves and the BS antenna char-
acteristics. In this paper, the sum of the two above-mentioned effects is attributed to the
characteristics of the multipath waves, which are represented by the amplitude and phase
of each wave radiating from the probes. Consequently, the area around the BS was not
modeled in Figure 1.
3. Method to Control the BS Correlation
Figure 2 illustrates two-dimensional channel models of the downlink and uplink
channels at the MIMO terminal [30]. In Figure 2, K secondary wave sources arranged on
a circle act as the radiating antennas (scatterers) and receiving antennas (probes) for the
downlink and uplink channels, respectively.
(a)
(b)
Figure 2. Two-dimensional channel model: (a) downlink; (b) uplink.
Base Station
Cluster
Uplink
Mobile Terminal
Probe
1m
ϕ
mK
ϕ
mi
ϕ
Base Station
Scatterer
#m#4
#2
#1
#4#n#2#1
DUT
v
φ
i
φ
#1
#i
#K
1m
ϕ
Km
ϕ
im
ϕ
Base Station
Probe
#m#4
#2
#1
#4#n#2#1
DUT
v
φ
i
φ
#1
#i
#K
Figure 2. Two-dimensional channel model: (a) downlink; (b) uplink.
Sensors 2021,21, 6184 5 of 20
For the downlink channel, illustrated in Figure 2a, radio waves emitted from BS
antennas are received by the antennas in the device under test (DUT) via the scatterers
surrounding the DUT. In this paper, non-line-of-sight (NLOS) scenarios are examined; thus,
Clarke’s model is used, which uniformly inputs radio waves from the full azimuth.
In the fading emulator or Monte Carlo simulation, radio waves radiated from the
scatterers that surround the DUT antennas, which are controlled phases of the signal used
to emulate the Rayleigh fading channel, are summed around the DUT antennas, and the
desired radio environment is generated. Furthermore, when the amplitudes of the signals
radiated from the scatterers are controlled according to the power spectrum of the incident
wave, a cluster propagation environment can be represented.
The phase-shift of the carrier wave from the i-th scatterer according to the m-th BS
antenna, Pmi (t), is calculated as follows:
Pmi (t)=2πfDtcos(φi−φv)+ϕmi, (1)
where
φi
is the azimuth angle of the i-th scatterer, and
φv
is the direction of movement
of the DUT antenna.
fD
is the maximum Doppler frequency in Hertz, which is given by
fD=v/λ
, where
v
is the speed of the DUT antennas and
λ
is the wavelength of the carrier
wave.
ϕmi
, which is set by a random number, is the initial phase of the i-th scatterer with
respect to the m-th BS antenna, and this can control the correlation coefficient between the
channels [7].
The correlation coefficient between the signals received via the downlink by the DUT
antennas is determined by the radiation pattern around the antennas. In the case of the
Jakes’ model, the spatial correlation between the antennas can be expressed as
ρ=J02πd
λ, (2)
where
J0()
is a zeroth-order Bessel function of the first kind, and dis the distance between
the isotropic antennas. As indicated in Equation (2), the correlation coefficient between
the signals received at the DUT antennas depends on the geometrical phase between
the antennas regardless of the initial phase of the scatterers which can be represented by
another channel from the BS antenna.
Similarly, in the fading emulator, the correlation coefficient between the received
signals, considering the phase difference generated by the geometrical relationship between
the DUT antennas and each scatterer, is equivalent to the spatial correlation calculated by
Equation (2). Hence, the geometrical relationships between the DUT antennas and the
scatterers change with the position of the DUT antennas at the fading emulator, and this is
very important for controlling the correlation coefficient.
In the uplink illustrated in Figure 2b, channel reciprocity theory is used to describe
that radio waves emitted from the DUT antennas received by the BS antennas via the
probes surrounding the DUT. As mentioned in Section 2, in this paper, the initial phase
of the probes represents the BS correlation characteristics for the whole uplink channel
including the path and the BS antenna properties.
The incoming signal at the BS is calculated by combining the signals from all the
probes that are superimposed on the received signal and the phase-shift values of the
probes. Therefore, the combined signal varies depending on the initial phases of the
probes, even though the set of received signals from the probes are the same since the
terminal antenna in a fading emulator does not move, which results in the generation of an
incoming signal to any BS antenna. Based on the above-mentioned principle, the setting of
the initial phases of the probes is very important to embody a realistic BS correlation for
the uplink channel.
Sensors 2021,21, 6184 6 of 20
Figure 3shows the implementation model used to achieve the BS correlation. In
Figure 3, the virtual point sources illustrated by the green circles are positioned to generate
a geometrical phase difference that realizes the desired BS correlation. Therefore, it is
different from the actual BS antenna arrangement and the channel model used to evaluate
a MIMO antenna using the fading emulator or Monte Carlo simulation.
Sensors 2021, 21, x FOR PEER REVIEW 6 of 21
geometrical phase difference that realizes the desired BS correlation. Therefore, it is dif-
ferent from the actual BS antenna arrangement and the channel model used to evaluate a
MIMO antenna using the fading emulator or Monte Carlo simulation.
Figure 3. Implementation model for BS correlation.
The received signals of each BS antenna can be observed using different initial phase
matrices depending on the desired BS correlation m
Φ. When the reference source is the
virtual point source of BS #1 placed at the center of the implementation model, the virtual
point source of BS #m is arranged at a distance d1m from that of BS #1. Then, the distance
d1m is decided depending on the desired BS correlation using Jakes’ model. Therefore, the
geometrical phase difference between the virtual point source of BS #m and probe #i mi
α
is given by
1
2cos
m
mi i
d
π
α
φ
λ
=. (3)
The initial phase matrix corresponding to BS #m m
Φ is the sum of the initial phase
matrix for BS #1, which is generated by a random number, and the geometrical phase
difference matrix using Equation (3), and it is defined as
[]
1
11 1 21 2 1
mm
mmKmK
ϕα ϕα ϕ α
=+
=+ + +
ΦΦΑ
. (4)
In Jakes’ model, spatial correlation is calculated as a function of the antenna separa-
tion. However, in this paper, it was necessary to determine the distance between the vir-
tual point sources d according to the desired BS correlation. In [21], a third-order approx-
imate formula for the relationship between the spatial correlation and antenna separation
was derived via the least-squares method. However, the accuracy of the estimated for-
mula is poor when the spatial correlation is close to 1. The spatial correlation is expected
to be close to 1 in a four-element BS antenna depending on the direction of the incident
wave; therefore, a high-accuracy 4 × 4 correlation matrix cannot be realized using the con-
ventional method.
In this paper, the geometrical phase difference required to complete the desired BS
correlation is determined via the bisection method using Jakes’ model, as depicted in Fig-
ure 4. First, the distance between the virtual point sources is set to the range from 0 to 0.4
λ, as denoted by the blue arrows in Figure 4, so that the spatial correlation satisfies 0–1.
Then, the spatial correlation at d = 0.2 λ, which is the midpoint of the blue arrow, is calcu-
lated using a zeroth-order Bessel function of the first kind, as indicated in Equation (2). If
i
φ
1m
d
1
2cos
m
mi i
d
π
α
φ
λ
=
Virtual Point
Source Probe
1m
ϕ
im
ϕ
Km
ϕ
x
y
#1 #m
Figure 3. Implementation model for BS correlation.
The received signals of each BS antenna can be observed using different initial phase
matrices depending on the desired BS correlation
Φm
. When the reference source is the
virtual point source of BS #1 placed at the center of the implementation model, the virtual
point source of BS #mis arranged at a distance d
1m
from that of BS #1. Then, the distance
d
1m
is decided depending on the desired BS correlation using Jakes’ model. Therefore, the
geometrical phase difference between the virtual point source of BS #mand probe #i
αmi
is
given by
αmi =2πd1m
λcos φi. (3)
The initial phase matrix corresponding to BS #m
Φm
is the sum of the initial phase
matrix for BS #1, which is generated by a random number, and the geometrical phase
difference matrix using Equation (3), and it is defined as
Φm=Φ1+Am
=ϕ11 +αm1ϕ21 +αm2· · · ϕK1+αmK (4)
In Jakes’ model, spatial correlation is calculated as a function of the antenna separation.
However, in this paper, it was necessary to determine the distance between the virtual
point sources daccording to the desired BS correlation. In [21], a third-order approximate
formula for the relationship between the spatial correlation and antenna separation was
derived via the least-squares method. However, the accuracy of the estimated formula
is poor when the spatial correlation is close to 1. The spatial correlation is expected to
be close to 1 in a four-element BS antenna depending on the direction of the incident
wave; therefore, a high-accuracy 4
×
4 correlation matrix cannot be realized using the
conventional method.
In this paper, the geometrical phase difference required to complete the desired BS
correlation is determined via the bisection method using Jakes’ model, as depicted in
Figure 4
. First, the distance between the virtual point sources is set to the range from 0
to 0.4
λ
, as denoted by the blue arrows in Figure 4, so that the spatial correlation satisfies
0–1. Then, the spatial correlation at d= 0.2
λ
, which is the midpoint of the blue arrow, is
calculated using a zeroth-order Bessel function of the first kind, as indicated in
Equation (2)
.
If the absolute value of the difference between the calculated spatial correlation and the
desired BS correlation,
|∆ρ|
, is less than the threshold value, which is a sufficiently small
Sensors 2021,21, 6184 7 of 20
value, the distance between the virtual point sources is set. When it is greater than
the threshold value and the calculated spatial correlation is greater than the desired BS
correlation, the range of the distance between the virtual point sources is modified to the
upper half of the original range. In the other case, the range is changed to the lower half.
By repeating this operation, the distance between the virtual point sources that enables
the desired BS correlation can be determined with high accuracy. Notably, the distance
between the virtual point sources is set to 0 when the desired BS correlation is 1.
Sensors 2021, 21, x FOR PEER REVIEW 7 of 21
the absolute value of the difference between the calculated spatial correlation and the de-
sired BS correlation,
ρ
Δ, is less than the threshold value, which is a sufficiently small
value, the distance between the virtual point sources is set. When it is greater than the
threshold value and the calculated spatial correlation is greater than the desired BS corre-
lation, the range of the distance between the virtual point sources is modified to the upper
half of the original range. In the other case, the range is changed to the lower half. By
repeating this operation, the distance between the virtual point sources that enables the
desired BS correlation can be determined with high accuracy. Notably, the distance be-
tween the virtual point sources is set to 0 when the desired BS correlation is 1.
Figure 4. Virtual point source separation vs. desired BS correlation.
4. Method for Implementing a 4 × 4 Correlation Matrix
4.1. Corellation Characteristcs of the BS
Radio waves that arrive at the BS from the MIMO mobile terminal have a narrow
angular spread and are incident from various directions depending on the position of the
MIMO terminal [29]; thus, the BS correlation matrix is determined by the arrangement of
the BS antennas, the direction of the incident wave, and the angular spread of the incident
waves. In a 4 × 4 MIMO system, the realized 4 × 4 correlation matrix has six correlation
values (
ρ
12,
ρ
13,
ρ
14,
ρ
23,
ρ
24, and
ρ
34). These correlation values are converted into the six
distances between the virtual point sources, as explained in Section 3, and it is necessary
to properly arrange the four virtual point sources to achieve the six BS correlations.
For generality, Figure 5 shows a model in which four BS antennas are arranged arbi-
trarily. In the figure, the wavelength numbers denote the distances between the antennas.
The four BS antennas are isotropic antennas. When the angular spread of the incident
wave is sufficiently narrow, the spatial correlation (absolute value of the complex corre-
lation) between the antennas can be calculated as follows [31]:
22 2 2
2
4sin
2
exp cos 2
s
s
d
d
j
πφσ
π
ρφ
λλ
=−
, (5)
where d is the distance between the BS antennas,
φ
s is the angle of the incident wave rela-
tive to the line that connects the individual BS array, and σ is the angular spread of the
incident wave. In this paper, the 4 × 4 correlation matrix was calculated assuming σ = 1.5°
[29].
Figure 4. Virtual point source separation vs. desired BS correlation.
4. Method for Implementing a 4 ×4 Correlation Matrix
4.1. Corellation Characteristcs of the BS
Radio waves that arrive at the BS from the MIMO mobile terminal have a narrow
angular spread and are incident from various directions depending on the position of the
MIMO terminal [
29
]; thus, the BS correlation matrix is determined by the arrangement of
the BS antennas, the direction of the incident wave, and the angular spread of the incident
waves. In a 4
×
4 MIMO system, the realized 4
×
4 correlation matrix has six correlation
values (
ρ12
,
ρ13
,
ρ14
,
ρ23
,
ρ24
, and
ρ34
). These correlation values are converted into the six
distances between the virtual point sources, as explained in Section 3, and it is necessary to
properly arrange the four virtual point sources to achieve the six BS correlations.
For generality, Figure 5shows a model in which four BS antennas are arranged
arbitrarily. In the figure, the wavelength numbers denote the distances between the
antennas. The four BS antennas are isotropic antennas. When the angular spread of the
incident wave is sufficiently narrow, the spatial correlation (absolute value of the complex
correlation) between the antennas can be calculated as follows [31]:
ρ=
exp j2πd
λcos φs−4π2d2sin2φsσ2
2λ2!
, (5)
where dis the distance between the BS antennas,
φs
is the angle of the incident wave
relative to the line that connects the individual BS array, and
σ
is the angular spread of
the incident wave. In this paper, the 4
×
4 correlation matrix was calculated assuming
σ= 1.5◦[29].
Sensors 2021,21, 6184 8 of 20
Sensors 2021, 21, x FOR PEER REVIEW 8 of 21
Figure 5. Arrangement of the BS antennas.
In the case where the incident wave angle for the x-axis
φ
is 0°, the spatial coefficient
between BS #1 and BS #2,
ρ
12, is 0.987 (using Equation (5)), and the distance between the
virtual point sources of BS #1 and BS #2 to realize
ρ
12 = 0.987 is 0.037 λ (using the bisection
method mentioned in Section 3). Similarly,
ρ
13,
ρ
14,
ρ
23,
ρ
24, and
ρ
34 are 0.614, 0.805, 0.713,
0.885, and 0.947, then d13, d14, d23, d24, and d34 are 0.209 λ, 0.144 λ, 0.177 λ, 0.109 λ, and 0.074
λ, respectively.
4.2. Arrangement of Virtual Point Sources in the Implementation Model
To realize a 4 × 4 correlation matrix, the arrangement of virtual point sources that
generate the initial phase in the implementation model in order to achieve the desired BS
correlation is examined. It should be noted that the implementation model used to realize
the BS correlation is a physical model that obtains the 4 × 4 correlation matrix using a
zeroth-order Bessel function of the first kind that expresses the relationship between the
spatial correlation and the antenna separation in Jakes’ model. In this paper, the two-point
source model depicted in Figure 3 [21] is extended to a four-point source model.
Figure 6 shows the relationship between the channel capacity of a 2 × 2 MIMO system
calculated using Shannon’s theorem and the spatial correlation [32]. The signal-to-noise
ratio (SNR) is set to 30 dB. The analytical results in Figure 6 show that the impact of the
change in spatial correlation on the MIMO channel capacity is very large in high-correla-
tion situations.
Figure 6. Channel capacity vs. spatial correlation.
yBS#1
BS#2
BS#3
BS#4
x
5.1λ
6.7λ
5.4λ
8.5λ
6.3λ
5λ
Incident Wave
23deg
σ
=
s
φ
φ
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
12
14
16
18
20
22
Spatial Correlation
ρ
Channel Capacity [bits/s/Hz]
Small
Large
SNR = 30dB
Figure 5. Arrangement of the BS antennas.
In the case where the incident wave angle for the x-axis
φ
is 0
◦
, the spatial coefficient
between BS #1 and BS #2,
ρ12
, is 0.987 (using Equation (5)), and the distance between the
virtual point sources of BS #1 and BS #2 to realize
ρ12
= 0.987 is 0.037
λ
(using the bisection
method mentioned in Section 3). Similarly,
ρ13
,
ρ14
,
ρ23
,
ρ24
, and
ρ34
are 0.614, 0.805, 0.713,
0.885, and 0.947, then d
13
,d
14
,d
23
,d
24
, and d
34
are 0.209
λ
, 0.144
λ
, 0.177
λ
, 0.109
λ
, and
0.074 λ, respectively.
4.2. Arrangement of Virtual Point Sources in the Implementation Model
To realize a 4
×
4 correlation matrix, the arrangement of virtual point sources that
generate the initial phase in the implementation model in order to achieve the desired BS
correlation is examined. It should be noted that the implementation model used to realize
the BS correlation is a physical model that obtains the 4
×
4 correlation matrix using a
zeroth-order Bessel function of the first kind that expresses the relationship between the
spatial correlation and the antenna separation in Jakes’ model. In this paper, the two-point
source model depicted in Figure 3[21] is extended to a four-point source model.
Figure 6shows the relationship between the channel capacity of a 2
×
2 MIMO system
calculated using Shannon’s theorem and the spatial correlation [
32
]. The signal-to-noise
ratio (SNR) is set to 30 dB. The analytical results in Figure 6show that the impact
of the change in spatial correlation on the MIMO channel capacity is very large in
high-correlation situations.
Sensors 2021, 21, x FOR PEER REVIEW 8 of 21
Figure 5. Arrangement of the BS antennas.
In the case where the incident wave angle for the x-axis
φ
is 0°, the spatial coefficient
between BS #1 and BS #2,
ρ
12, is 0.987 (using Equation (5)), and the distance between the
virtual point sources of BS #1 and BS #2 to realize
ρ
12 = 0.987 is 0.037 λ (using the bisection
method mentioned in Section 3). Similarly,
ρ
13,
ρ
14,
ρ
23,
ρ
24, and
ρ
34 are 0.614, 0.805, 0.713,
0.885, and 0.947, then d13, d14, d23, d24, and d34 are 0.209 λ, 0.144 λ, 0.177 λ, 0.109 λ, and 0.074
λ, respectively.
4.2. Arrangement of Virtual Point Sources in the Implementation Model
To realize a 4 × 4 correlation matrix, the arrangement of virtual point sources that
generate the initial phase in the implementation model in order to achieve the desired BS
correlation is examined. It should be noted that the implementation model used to realize
the BS correlation is a physical model that obtains the 4 × 4 correlation matrix using a
zeroth-order Bessel function of the first kind that expresses the relationship between the
spatial correlation and the antenna separation in Jakes’ model. In this paper, the two-point
source model depicted in Figure 3 [21] is extended to a four-point source model.
Figure 6 shows the relationship between the channel capacity of a 2 × 2 MIMO system
calculated using Shannon’s theorem and the spatial correlation [32]. The signal-to-noise
ratio (SNR) is set to 30 dB. The analytical results in Figure 6 show that the impact of the
change in spatial correlation on the MIMO channel capacity is very large in high-correla-
tion situations.
Figure 6. Channel capacity vs. spatial correlation.
yBS#1
BS#2
BS#3
BS#4
x
5.1λ
6.7λ
5.4λ
8.5λ
6.3λ
5λ
Incident Wave
23deg
σ
=
s
φ
φ
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
12
14
16
18
20
22
Spatial Correlation
ρ
Channel Capacity [bits/s/Hz]
Small
Large
SNR = 30dB
Figure 6. Channel capacity vs. spatial correlation.
Sensors 2021,21, 6184 9 of 20
In the implementation model, the distances between the six virtual point sources vary
depending on the desired BS correlation. Then, there may not be a geometric arrangement
of virtual point sources that represents the distances between all the virtual point sources.
In this paper, to evaluate the performance of the uplink channel for MIMO terminals, the
coordinates of the four virtual point sources were determined with priority given to high
correlation; thus, the MIMO channel capacity using that arrangement could be close to the
actual MIMO channel capacity.
Figure 7shows the arrangement of the virtual point sources in a 4
×
4 MIMO system.
By placing this model in the center of Figure 3, the geometrical phase difference can be
calculated using Equation (3) to generate the initial phase matrix described in
Equation (4)
.
The arrangement of the virtual point sources in Figure 7varies in an arbitrary fashion by
changing the distance relationships between the virtual point sources according to the
desired 4 ×4 correlation matrix.
Sensors 2021, 21, x FOR PEER REVIEW 9 of 21
In the implementation model, the distances between the six virtual point sources vary
depending on the desired BS correlation. Then, there may not be a geometric arrangement
of virtual point sources that represents the distances between all the virtual point sources.
In this paper, to evaluate the performance of the uplink channel for MIMO terminals, the
coordinates of the four virtual point sources were determined with priority given to high
correlation; thus, the MIMO channel capacity using that arrangement could be close to the
actual MIMO channel capacity.
Figure 7 shows the arrangement of the virtual point sources in a 4 × 4 MIMO system.
By placing this model in the center of Figure 3, the geometrical phase difference can be
calculated using Equation (3) to generate the initial phase matrix described in Equation
(4). The arrangement of the virtual point sources in Figure 7 varies in an arbitrary fashion
by changing the distance relationships between the virtual point sources according to the
desired 4 × 4 correlation matrix.
Figure 7. Arrangement of virtual point sources.
The relationship between the four virtual point sources (A, B, C, and D) in Figure 7
and the BS antennas (BS #1, #2, #3, and #4) is determined according to the desired 4 × 4
correlation matrix. First, the maximum distance is identified from the six distances between
the four virtual point sources, and these are labeled, for example, as dCD in Figure 7. In the
case of Figure 5, d13 is dCD; thus, points C and D are either BS #1 or #3. As a result, points
A and B become either BS #2 or #4, and dAB is set to d24. Second, the maximum distance is
identified from the remaining four distances, and this is set to dAD, which is the diagonal
of the implementation model. In Figure 5, d23 is dAD. Therefore, points A, B, C, and D are
uniquely determined to be BS #2, BS #4, BS #1, and BS #3, respectively. Finally, the coor-
dinates of the four virtual point sources are established. The coordinates of point A (xA,
yA) are used as the origin. The coordinates of point B (xB, yB) may be any point on the
circumference of radius dAB centered on point A; however, those are set to (dAB, 0) on the
x-axis. The coordinates of point C (xC, yC) are the intersection of the circumference of radius
dAC centered on point A and the circumference of radius dBC centered on point B, as illus-
trated by the red arcs in Figure 7. However, when dBC > dAC + dBA, the coordinates of point
C are set to (−dAC, 0) so that dAC and dAB are satisfied, and dBC is set to the longest possible
value. Similarly, the coordinates of point D (xD, yD) are at the intersection of the circum-
ference of radius dAD centered on point A and the circumference of radius dBD centered on
point B, as illustrated by the blue arcs in Figure 7. As before, when dAD > dAB + dBD, the
coordinates of point D are set to (dAB + dBD, 0) so that dAB and dBD are satisfied, and dAD is
set to the longest possible value. The virtual point source arrangement of the implemen-
tation model in Figure 5 is shown in Figure 8a.
A (x
A
, y
A
)
C (x
C
, y
C
)
B (x
B
, y
B
)
D(x
D
, y
D
)
x
y
AC
dBD
d
CD
d
AB
d
BC
dAD
d
Figure 7. Arrangement of virtual point sources.
The relationship between the four virtual point sources (A, B, C, and D) in Figure 7
and the BS antennas (BS #1, #2, #3, and #4) is determined according to the desired 4
×
4
correlation matrix. First, the maximum distance is identified from the six distances between
the four virtual point sources, and these are labeled, for example, as d
CD
in Figure 7. In
the case of Figure 5,d
13
is d
CD
; thus, points C and D are either BS #1 or #3. As a result,
points A and B become either BS #2 or #4, and d
AB
is set to d
24
. Second, the maximum
distance is identified from the remaining four distances, and this is set to d
AD
, which is the
diagonal of the implementation model. In Figure 5,d
23
is d
AD
. Therefore, points A, B, C,
and D are uniquely determined to be BS #2, BS #4, BS #1, and BS #3, respectively. Finally,
the coordinates of the four virtual point sources are established. The coordinates of point
A (x
A
,y
A
) are used as the origin. The coordinates of point B (x
B
,y
B
) may be any point on
the circumference of radius d
AB
centered on point A; however, those are set to (d
AB
, 0) on
the x-axis. The coordinates of point C (x
C
,
yC
) are the intersection of the circumference of
radius d
AC
centered on point A and the circumference of radius d
BC
centered on point B,
as illustrated by the red arcs in Figure 7. However, when
dBC >dAC +dBA
, the coordinates
of point C are set to (
−
d
AC
, 0) so that d
AC
and d
AB
are satisfied, and d
BC
is set to the
longest possible value. Similarly, the coordinates of point D (x
D
,y
D
) are at the intersection
of the circumference of radius d
AD
centered on point A and the circumference of radius
d
BD
centered on point B, as illustrated by the blue arcs in
Figure 7
. As before, when
dAD >dAB +dBD
, the coordinates of point D are set to (d
AB
+d
BD
, 0) so that d
AB
and d
BD
are
satisfied, and d
AD
is set to the longest possible value. The virtual point source arrangement
of the implementation model in Figure 5is shown in Figure 8a.
Sensors 2021,21, 6184 10 of 20
Sensors 2021, 21, x FOR PEER REVIEW 10 of 21
Figure 8. Arrangement of the virtual point sources to achieve the BS correlation at
φ
= 0° in Fig-
ure 5: (a) before translating; (b) after translating.
The maximum distance dCD is not utilized when determining the coordinates of the
four virtual point sources. This is because there may not always be a geometrical arrange-
ment of the four virtual point sources that realizes all the distances between the virtual
point sources; therefore, the shorter distances between virtual point sources, that is, a
higher BS correlation, are given priority to establish the coordinates of the virtual point
sources. As a result, the arrangement of the virtual point sources may satisfy all the dis-
tances required to enable the desired 4 × 4 correlation matrix. Accordingly, the arrange-
ment of the virtual point sources, shown in Figure 8a, satisfies all the distances between
the virtual point sources, resulting in the desired 4 × 4 correlation matrix.
The initial phase matrix for each BS antenna is created by calculating the geometric
phase difference for the designed implementation model, as shown in Figure 8a, with ref-
erence to the initial phase matrix of BS #1. Therefore, the coordinates of the virtual point
source for BS #1 is translated to the origin, as presented in Figure 8b. Using the above
method, the arrangement of the virtual point sources to complete the 4 × 4 correlation
matrix, which is dependent on the arrangement of the BS antennas, the direction of the
incident wave, and the angular spread of the incident wave, can be determined.
4.3. Design of Implementation Model for the Conventional BS Antenna Arrangement
Figure 9 shows the relationships between conventional BS antenna arrangements and
the incoming wave angle
φ
. In this paper, two types of four-element BS antennas are
considered: a circular and a linear array installed at equal intervals, as illustrated in Figure
9a,b, respectively [33]. The distance between adjacent antennas was set to 5 λ.
Figure 9. Relationship between the BS antenna arrangement and the incident wave: (a) circular
array; (b) linear array.
x
y
BS#1
BS#2
BS#3
BS#4
φ
5λ
23deg
S
σ
=
Base Station
Incident Wave
BS#1
BS#2
BS#3 BS#4
φ
5λ
Base Station
23deg
S
σ
=
Incident Wave
(b)
(a)
Figure 8.
Arrangement of the virtual point sources to achieve the BS correlation at
φ
= 0
◦
in Figure 5:
(a) before translating; (b) after translating.
The maximum distance d
CD
is not utilized when determining the coordinates of
the four virtual point sources. This is because there may not always be a geometrical
arrangement of the four virtual point sources that realizes all the distances between the
virtual point sources; therefore, the shorter distances between virtual point sources, that
is, a higher BS correlation, are given priority to establish the coordinates of the virtual
point sources. As a result, the arrangement of the virtual point sources may satisfy all
the distances required to enable the desired 4
×
4 correlation matrix. Accordingly, the
arrangement of the virtual point sources, shown in Figure 8a, satisfies all the distances
between the virtual point sources, resulting in the desired 4 ×4 correlation matrix.
The initial phase matrix for each BS antenna is created by calculating the geometric
phase difference for the designed implementation model, as shown in Figure 8a, with
reference to the initial phase matrix of BS #1. Therefore, the coordinates of the virtual point
source for BS #1 is translated to the origin, as presented in Figure 8b. Using the above
method, the arrangement of the virtual point sources to complete the 4
×
4 correlation
matrix, which is dependent on the arrangement of the BS antennas, the direction of the
incident wave, and the angular spread of the incident wave, can be determined.
4.3. Design of Implementation Model for the Conventional BS Antenna Arrangement
Figure 9shows the relationships between conventional BS antenna arrangements
and the incoming wave angle
φ
. In this paper, two types of four-element BS antennas
are considered: a circular and a linear array installed at equal intervals, as illustrated in
Figure 9a,b, respectively [33]. The distance between adjacent antennas was set to 5 λ.
Sensors 2021, 21, x FOR PEER REVIEW 10 of 21
Figure 8. Arrangement of the virtual point sources to achieve the BS correlation at
φ
= 0° in Fig-
ure 5: (a) before translating; (b) after translating.
The maximum distance dCD is not utilized when determining the coordinates of the
four virtual point sources. This is because there may not always be a geometrical arrange-
ment of the four virtual point sources that realizes all the distances between the virtual
point sources; therefore, the shorter distances between virtual point sources, that is, a
higher BS correlation, are given priority to establish the coordinates of the virtual point
sources. As a result, the arrangement of the virtual point sources may satisfy all the dis-
tances required to enable the desired 4 × 4 correlation matrix. Accordingly, the arrange-
ment of the virtual point sources, shown in Figure 8a, satisfies all the distances between
the virtual point sources, resulting in the desired 4 × 4 correlation matrix.
The initial phase matrix for each BS antenna is created by calculating the geometric
phase difference for the designed implementation model, as shown in Figure 8a, with ref-
erence to the initial phase matrix of BS #1. Therefore, the coordinates of the virtual point
source for BS #1 is translated to the origin, as presented in Figure 8b. Using the above
method, the arrangement of the virtual point sources to complete the 4 × 4 correlation
matrix, which is dependent on the arrangement of the BS antennas, the direction of the
incident wave, and the angular spread of the incident wave, can be determined.
4.3. Design of Implementation Model for the Conventional BS Antenna Arrangement
Figure 9 shows the relationships between conventional BS antenna arrangements and
the incoming wave angle
φ
. In this paper, two types of four-element BS antennas are
considered: a circular and a linear array installed at equal intervals, as illustrated in Figure
9a,b, respectively [33]. The distance between adjacent antennas was set to 5 λ.
Figure 9. Relationship between the BS antenna arrangement and the incident wave: (a) circular
array; (b) linear array.
x
y
BS#1
BS#2
BS#3
BS#4
φ
5λ
23deg
S
σ
=
Base Station
Incident Wave
BS#1
BS#2
BS#3 BS#4
φ
5λ
Base Station
23deg
S
σ
=
Incident Wave
(b)
(a)
Figure 9.
Relationship between the BS antenna arrangement and the incident wave: (
a
) circular array;
(b) linear array.
Sensors 2021,21, 6184 11 of 20
First, the 4
×
4 correlation matrix is derived using Equation (5), considering the angle
of the incident wave and the distances between the individual BS antennas. Figure 10
shows the BS correlation as a function of the incoming wave angle. As shown in Figure 10,
the 4
×
4 correlation matrix varies significantly depending on the BS antenna arrangement
and the incoming wave angle φ.
Sensors 2021, 21, x FOR PEER REVIEW 11 of 21
First, the 4 × 4 correlation matrix is derived using Equation (5), considering the angle
of the incident wave and the distances between the individual BS antennas. Figure 10
shows the BS correlation as a function of the incoming wave angle. As shown in Figure
10, the 4 × 4 correlation matrix varies significantly depending on the BS antenna arrange-
ment and the incoming wave angle
φ
.
(a)
(b)
Figure 10. BS Correlation characteristics vs. incoming wave angle: (a) circular array; (b) linear ar-
ray.
In the case of the circular array, illustrated in Figure 9a, the angle of the incident wave
for the BS array constructed by BS #1 and BS #2 at the incoming wave angle
φ
is the same
as that constructed by BS #3 and BS #4, whereas the angle of the incident wave for the BS
array constructed by BS #1 and BS #4 at
φ
is the same as that constructed by BS #2 and
BS #3. Consequently,
ρ
12 and
ρ
14 coincide with
ρ
34 and
ρ
23, respectively, regardless of the
incoming wave angle
φ
, and there are four types of BS correlation characteristics, as
shown in Figure 10a.
However, in the case of a linear array, illustrated in Figure 9b, the angle of the inci-
dent wave for any BS array is the same; however, the distance between the BS antennas is
different. The distance between BS #1 and #2 is the same as that between BS #2 and #3 or
between BS #3 and #4, whereas the distance between BS #1 and #3 is the same as that
between BS #2 and #4. Consequently, there are three types of BS correlation characteristics,
as exhibited in Figure 10b.
045 90 135 180 225 270 315 360
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming Wa ve Angle
φ
[deg]
Correlation Coefficient |
ρ
|
BS#1-BS#3BS#2-BS#4
BS#1-BS#2
BS#3-BS#4
BS#1-BS#4
BS#2-BS#3
BS#1
BS#2
BS#3 BS#4
φ
045 90 135 180 225 270 315 360
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming Wav e Angle
φ
[deg]
Correlation Coefficient |
ρ
|
BS#1-BS#4
BS#1-BS#2
BS#2-BS#3
BS#3-BS#4
BS#1-BS#3
BS#2-BS#4
BS#1
BS#2
BS#3
BS#4
φ
Figure 10.
BS Correlation characteristics vs. incoming wave angle: (
a
) circular array; (
b
) linear array.
In the case of the circular array, illustrated in Figure 9a, the angle of the incident wave
for the BS array constructed by BS #1 and BS #2 at the incoming wave angle
φ
is the same
as that constructed by BS #3 and BS #4, whereas the angle of the incident wave for the BS
array constructed by BS #1 and BS #4 at
φ
is the same as that constructed by BS #2 and
BS #3. Consequently,
ρ12
and
ρ14
coincide with
ρ34
and
ρ23
, respectively, regardless of the
incoming wave angle
φ
, and there are four types of BS correlation characteristics, as shown
in Figure 10a.
However, in the case of a linear array, illustrated in Figure 9b, the angle of the incident
wave for any BS array is the same; however, the distance between the BS antennas is
different. The distance between BS #1 and #2 is the same as that between BS #2 and #3
or between BS #3 and #4, whereas the distance between BS #1 and #3 is the same as that
between BS #2 and #4. Consequently, there are three types of BS correlation characteristics,
as exhibited in Figure 10b.
Second, based on the BS correlation characteristics shown in Figure 10, the distances
between the virtual point sources are estimated via the bisection method using Jakes’
Sensors 2021,21, 6184 12 of 20
model. Figure 11 shows the virtual point source separation as a function of the incoming
wave angle, which ranges from 0
◦
to 90
◦
considering the symmetry of the BS arrangement.
Sensors 2021, 21, x FOR PEER REVIEW 12 of 21
Second, based on the BS correlation characteristics shown in Figure 10, the distances
between the virtual point sources are estimated via the bisection method using Jakes’
model. Figure 11 shows the virtual point source separation as a function of the incoming
wave angle, which ranges from 0° to 90° considering the symmetry of the BS arrangement.
(a)
(b)
Figure 11. Virtual point source separation vs. incoming wave angle: (a) circular array; (b) linear
array.
As described in Figure 11, the virtual point source separations according to the de-
signed implementation model, indicated by dots in the figure, are in good agreement with
the ideal distance that is required to achieve the desired BS correlation shown in Figure
10. Therefore, the 4 × 4 correlation matrix to be mounted on the MIMO-OTA apparatus to
embody the desired BS correlation can be generated by the proposed method, which de-
termines the implementation model depending on the arrangement of the BS antenna and
the incoming wave angle.
As shown in the circular array, illustrated in Figure 9a, when the incoming wave
angle is 45° the spatial correlation
ρ
13 is 1 because the array formed by BS #1 and #3 is
parallel to the incident wave. Therefore, the virtual point sources for BS #1 and BS #3 have
the same coordinates; therefore, there are only three virtual point sources in the imple-
mentation model. Furthermore, when the incoming wave angle is 0°, the spatial correla-
tions
ρ
12 and
ρ
34 are 1 because two arrays (BS #1 and #2, BS #3 and #4) are parallel to the
incident wave. Hence, there are only two virtual point sources in the implementation
015 30 45 60 75 90
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Incoming Wav e Angle
φ
[deg]
Virtual Point Source Separation in Wavelength
BS#1-BS#3
BS#2-BS#4
BS#1-BS#2
BS#3-BS#4
BS#1-BS#4
BS#2-BS#3
BS#1
BS#2
BS#3 BS#4
φ
015 30 45 60 75 90
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Incoming Wave Angle
φ
[deg]
Virtual Point S ource Separation in Wavelength
BS#1-BS#4
BS#1-BS#2
BS#2-BS#3
BS#3-BS#4
BS#1-BS#3
BS#2-BS#4
BS#1
BS#2
BS#3
BS#4
φ
Figure 11.
Virtual point source separation vs. incoming wave angle: (
a
) circular array; (
b
) linear array.
As described in Figure 11, the virtual point source separations according to the de-
signed implementation model, indicated by dots in the figure, are in good agreement with
the ideal distance that is required to achieve the desired BS correlation shown in
Figure 10
.
Therefore, the 4
×
4 correlation matrix to be mounted on the MIMO-OTA apparatus to
embody the desired BS correlation can be generated by the proposed method, which deter-
mines the implementation model depending on the arrangement of the BS antenna and the
incoming wave angle.
As shown in the circular array, illustrated in Figure 9a, when the incoming wave angle
is 45
◦
the spatial correlation
ρ13
is 1 because the array formed by BS #1 and #3 is parallel to
the incident wave. Therefore, the virtual point sources for BS #1 and BS #3 have the same
coordinates; therefore, there are only three virtual point sources in the implementation
model. Furthermore, when the incoming wave angle is 0
◦
, the spatial correlations
ρ12
and
ρ34
are 1 because two arrays (BS #1 and #2, BS #3 and #4) are parallel to the incident wave.
Hence, there are only two virtual point sources in the implementation model. Even in the
case of few virtual point sources, their arrangement can be set using the proposed method,
as shown in Figure 12a,b.
Sensors 2021,21, 6184 13 of 20
Sensors 2021, 21, x FOR PEER REVIEW 13 of 21
model. Even in the case of few virtual point sources, their arrangement can be set using
the proposed method, as shown in Figure 12a,b.
(a) (b) (c)
Figure 12. Arrangement of virtual point sources: (a)
φ
= 45° in a circular array; (b)
φ
= 0° in a
circular array; (c)
φ
= 0° in a linear array.
In the linear array, illustrated in Figure 9b, the arrangement of the virtual point
sources required to complete the BS correlation is determined when the incoming wave
angle is 0°, as shown in Figure 12c. Then, d14 in the implementation model is set to 0.361
λ, which is 0.007 λ less than the desired distance, as shown in Figure 11b. The calculated
spatial correlation
ρ
14 is 0.07, which is 0.02 larger than the desired BS correlation 0.05;
however, the effect of this difference on the MIMO channel capacity is considered to be
very small (see Figure 6). It is concluded from these analytical results that the initial phase
matrix implemented on the MIMO-OTA apparatus representing the desired BS correla-
tion can be generated using the proposed method, providing an OTA method for the eval-
uating the performance of the uplink channel from a MIMO mobile terminal with respect
to the BS correlation characteristics.
5. Experimental Verification
5.1. Bidirectional Fading Emulator
Figure 13 shows the configuration of the developed two-dimensional bidirectional
fading emulator. To achieve a good multipath propagation environment, 14 scatterers are
arranged at equal angular intervals on a circle of radius 1.2 m [34]. The probes comprise
vertically polarized half-wavelength sleeve dipole antennas. The MIMO terminal is lo-
cated at the center of the emulator. This fading emulator functions as an operating algo-
rithm based on the same formulation described in [24]. The instruments, such as the net-
work analyzer (Keysight, HP8753E), power combiner/divider (mini circuit, ZAPD-2-S+
and ZB8PD-252-S+), and phase shifter (mini circuit, JSPHS-23+), have bidirectional char-
acteristics at the measurement frequency. This apparatus generates a fading environment
by radio-frequency (RF) processing, and it can be implemented even if transmission and
reception are reversed; thus, the performance of the uplink channel from the MIMO ter-
minal can be evaluated.
Figure 12.
Arrangement of virtual point sources: (
a
)
φ
= 45
◦
in a circular array; (
b
)
φ
= 0
◦
in a circular
array; (c)φ= 0◦in a linear array.
In the linear array, illustrated in Figure 9b, the arrangement of the virtual point sources
required to complete the BS correlation is determined when the incoming wave angle is
0
◦
, as shown in Figure 12c. Then, d
14
in the implementation model is set to 0.361
λ
, which
is 0.007
λ
less than the desired distance, as shown in Figure 11b. The calculated spatial
correlation
ρ14
is 0.07, which is 0.02 larger than the desired BS correlation 0.05; however,
the effect of this difference on the MIMO channel capacity is considered to be very small
(see Figure 6). It is concluded from these analytical results that the initial phase matrix
implemented on the MIMO-OTA apparatus representing the desired BS correlation can be
generated using the proposed method, providing an OTA method for the evaluating the
performance of the uplink channel from a MIMO mobile terminal with respect to the BS
correlation characteristics.
5. Experimental Verification
5.1. Bidirectional Fading Emulator
Figure 13 shows the configuration of the developed two-dimensional bidirectional
fading emulator. To achieve a good multipath propagation environment, 14 scatterers are
arranged at equal angular intervals on a circle of radius 1.2 m [
34
]. The probes comprise
vertically polarized half-wavelength sleeve dipole antennas. The MIMO terminal is located
at the center of the emulator. This fading emulator functions as an operating algorithm
based on the same formulation described in [
24
]. The instruments, such as the network
analyzer (Keysight, HP8753E), power combiner/divider (mini circuit, ZAPD-2-S+ and
ZB8PD-252-S+), and phase shifter (mini circuit, JSPHS-23+), have bidirectional character-
istics at the measurement frequency. This apparatus generates a fading environment by
radio-frequency (RF) processing, and it can be implemented even if transmission and re-
ception are reversed; thus, the performance of the uplink channel from the MIMO terminal
can be evaluated.
To measure the uplink channel properties, a wave transmitted from a network an-
alyzer is emitted from the MIMO terminal antenna and is then received at each probe.
The received signals are controlled using phase shifters operated by a digital-to-analog
converter to emulate the Rayleigh fading channel. These are compounded by a power
combiner and then measured using a network analyzer. This apparatus has a high time
correlation characteristic of approximately 0.995; thus, each channel response can be
measured individually.
Sensors 2021,21, 6184 14 of 20
Sensors 2021, 21, x FOR PEER REVIEW 14 of 21
Figure 13. Two-dimensional bidirectional MIMO-OTA apparatus.
To measure the uplink channel properties, a wave transmitted from a network ana-
lyzer is emitted from the MIMO terminal antenna and is then received at each probe. The
received signals are controlled using phase shifters operated by a digital-to-analog con-
verter to emulate the Rayleigh fading channel. These are compounded by a power com-
biner and then measured using a network analyzer. This apparatus has a high time corre-
lation characteristic of approximately 0.995; thus, each channel response can be measured
individually.
5.2. Measurement of the BS Correlation
In the first step of the investigation, the proposed method was evaluated via OTA
testing with a uniform azimuthal angular power spectrum using a 4 × 4 correlation matrix
according to the desired BS correlation using a two-dimensional bidirectional fading em-
ulator. To evaluate the antenna performance on the terminal side, the MIMO terminal was
placed at the center of the bidirectional fading emulator.
The arrangement of the MIMO terminal using a half-wavelength dipole antenna is a
quasi-linear array with a half-wavelength spacing aligned along the x-axis, as presented
in Figure 14. Table 2 lists the measurement and analytical conditions used to perform the
OTA testing and Monte Carlo simulations. The frequency used in the measurement is 1.95
GHz, which is the center frequency of the uplink channel in the 2 GHz band in Japan. The
polarization is assumed to be vertical because only vertically polarized dipole antennas
are used in the bidirectional fading emulator.
Figure 13. Two-dimensional bidirectional MIMO-OTA apparatus.
5.2. Measurement of the BS Correlation
In the first step of the investigation, the proposed method was evaluated via OTA
testing with a uniform azimuthal angular power spectrum using a 4
×
4 correlation
matrix according to the desired BS correlation using a two-dimensional bidirectional fading
emulator. To evaluate the antenna performance on the terminal side, the MIMO terminal
was placed at the center of the bidirectional fading emulator.
The arrangement of the MIMO terminal using a half-wavelength dipole antenna is a
quasi-linear array with a half-wavelength spacing aligned along the x-axis, as presented
in Figure 14. Table 2lists the measurement and analytical conditions used to perform the
OTA testing and Monte Carlo simulations. The frequency used in the measurement is
1.95 GHz
, which is the center frequency of the uplink channel in the 2 GHz band in Japan.
The polarization is assumed to be vertical because only vertically polarized dipole antennas
are used in the bidirectional fading emulator.
Sensors 2021, 21, x FOR PEER REVIEW 15 of 21
Figure 14. Configuration of the four-element MIMO terminal.
Table 2. Measurement and analytical conditions.
Frequency 1.95 GHz
Number of MIMO antennas 4
Arrangement of MIMO antenna Quasi-liner array
MIMO antenna element Half-wavelength dipole
Interval of MIMO antenna 0.5 λ
Number of BS antennas 4
Angular spread of the incident wave 1.5°
Number of probes 14
Traveling distance 5000
Number of samplings 5000
XPR Vertical polarization only
Method of EM analysis Method of moments
Figure 15 shows the BS correlation calculated using Equation (6) as a function of the
incoming wave angle. The symbols are for OTA test results using the proposed method
to realize the BS correlation, whereas the solid and broken curves indicate the analytical
outcomes from the Monte Carlo simulations and the calculated correlation using Equation
(5), respectively.
*** *
11 22 33 44
** ** ** **
11 11 22 22 33 33 44 44
1
4
pq pq pq pq
pq
pp qq pp qq pp qq pp qq
hh hh hh hh
hh hh hh hh hh hh h h hh
ρ
=+++
(6)
where hpr indicates the channel response between the BS antenna #p and the MIMO termi-
nal antenna #r, and the asterisk (*) denotes the complex conjugate.
It can be observed in Figure 15 that the experimental results obtained using the bidi-
rectional fading emulator are in good agreement with both the Monte Carlo analysis re-
sults and the theoretical BS correlations obtained from Equation (5). As described in Sec-
tion 4.3, the error in the BS correlation at ϕ = 0° in the linear array is owing to the error
that occurs when creating the arrangement of the virtual point sources, as shown in Figure
12c. This confirms that the desired BS correlation in a 4 × 4 MIMO system can be achieved
using the initial phase matrix, depending on the virtual point source arrangement.
x
y
0.5
λ
#1#2#3#4
Figure 14. Configuration of the four-element MIMO terminal.
Sensors 2021,21, 6184 15 of 20
Table 2. Measurement and analytical conditions.
Frequency 1.95 GHz
Number of MIMO antennas 4
Arrangement of MIMO antenna Quasi-liner array
MIMO antenna element Half-wavelength dipole
Interval of MIMO antenna 0.5 λ
Number of BS antennas 4
Angular spread of the incident wave 1.5◦
Number of probes 14
Traveling distance 5000
Number of samplings 5000
XPR Vertical polarization only
Method of EM analysis Method of moments
Figure 15 shows the BS correlation calculated using Equation (6) as a function of the
incoming wave angle. The symbols are for OTA test results using the proposed method
to realize the BS correlation, whereas the solid and broken curves indicate the analyt-
ical outcomes from the Monte Carlo simulations and the calculated correlation using
Equation (5), respectively.
ρpq =1
4
hp1hq1∗
qhp1hp1∗qhq1hq1∗+hp2hq2∗
qhp2hp2∗qhq2hq2∗+hp3hq3∗
qhp3hp3∗qhq3hq3∗+hp4hq4∗
qhp4hp4∗qhq4hq4∗
(6)
where h
pr
indicates the channel response between the BS antenna #pand the MIMO terminal
antenna #r, and the asterisk (*) denotes the complex conjugate.
It can be observed in Figure 15 that the experimental results obtained using the
bidirectional fading emulator are in good agreement with both the Monte Carlo analysis
results and the theoretical BS correlations obtained from Equation (5). As described in
Section 4.3, the error in the BS correlation at
φ
= 0
◦
in the linear array is owing to the
error that occurs when creating the arrangement of the virtual point sources, as shown in
Figure 12c
. This confirms that the desired BS correlation in a 4
×
4 MIMO system can be
achieved using the initial phase matrix, depending on the virtual point source arrangement.
5.3. Measurement of the 4 ×4 MIMO Uplink Channel Capacity
The MIMO channel capacity was measured using a two-dimensional bidirectional
fading emulator to evaluate the performance of the four-element MIMO terminal uplink
channel. Figure 16 shows the 4
×
4 MIMO (four elements in the mobile terminal and
four elements in the BS) channel capacity as a function of the incoming wave angle with
the SNR as a parameter. The MIMO terminal illustrated in Figure 14 was placed at the
center of the bidirectional fading emulator. The round and square symbols denote the OTA
measurement results for the circular and linear arrays at the BS, respectively. The solid and
broken curves indicate the Monte Carlo simulation results for the circular and linear arrays
at the BS, respectively. The SNR was changed from 10 to 40 dB in 10 dB intervals [
35
]. The
frequency was 1.95 GHz, and the XPR was set to 50 dB, which is equivalent to a vertically
polarized propagation environment.
Sensors 2021,21, 6184 16 of 20
Sensors 2021, 21, x FOR PEER REVIEW 16 of 21
(a)
(b)
Figure 15. Characteristics of the BS correlation as a function of the incoming wave angle: (a) circu-
lar array; (b) linear array.
5.3. Measurement of the 4 × 4 MIMO Uplink Channel Capacity
The MIMO channel capacity was measured using a two-dimensional bidirectional
fading emulator to evaluate the performance of the four-element MIMO terminal uplink
channel. Figure 16 shows the 4 × 4 MIMO (four elements in the mobile terminal and four
elements in the BS) channel capacity as a function of the incoming wave angle with the
SNR as a parameter. The MIMO terminal illustrated in Figure 14 was placed at the center
of the bidirectional fading emulator. The round and square symbols denote the OTA
measurement results for the circular and linear arrays at the BS, respectively. The solid
and broken curves indicate the Monte Carlo simulation results for the circular and linear
arrays at the BS, respectively. The SNR was changed from 10 to 40 dB in 10 dB intervals
[35]. The frequency was 1.95 GHz, and the XPR was set to 50 dB, which is equivalent to a
vertically polarized propagation environment.
015 30 45 60 75 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming Wave Angle
φ
[deg]
Correlation Coefficient |
ρ
|
OTA measurement
Monte Carlo
Analysis (Eq. (5))
symbol
ρ
12
ρ
34
ρ
23
ρ
14
ρ
13
ρ
24
015 30 45 60 75 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming Wave Angle
φ
[deg]
Correlation Coefficient |
ρ
|
ρ
12
ρ
23
ρ
34
ρ
13
ρ
24
ρ
14
OTA measurement
Monte Carlo
Analysis (Eq. (5))
symbol
Figure 15.
Characteristics of the BS correlation as a function of the incoming wave angle: (
a
) circular
array; (b) linear array.
It can be observed from Figure 16 that the results measured via OTA testing are in
good agreement with the analytical results obtained from the Monte Carlo simulations
regardless of the BS arrangement and incoming wave angle. In the case of
φ
= 90
◦
in the
linear array, the measured channel capacity is slightly larger than the analytical value.
A possible cause of this phenomenon is the BS correlation being insufficiently close to
1 because this apparatus cannot achieve a time correlation of 1. As shown in Figure 6,
when the correlation coefficient approaches 1, the MIMO channel capacity deteriorates
significantly. However, it is possible to evaluate the performance of the MIMO terminal
because a 0.99 BS correlation can be achieved using this bidirectional fading emulator.
Similarly, with a circular array at the BS, the same phenomenon was observed when the BS
correlation was 1 depending on the incoming wave angle.
Sensors 2021,21, 6184 17 of 20
Sensors 2021, 21, x FOR PEER REVIEW 17 of 21
Figure 16. 4 × 4 MIMO channel capacity as a function of the incoming wave angle.
It can be observed from Figure 16 that the results measured via OTA testing are in
good agreement with the analytical results obtained from the Monte Carlo simulations
regardless of the BS arrangement and incoming wave angle. In the case of
φ
= 90° in the
linear array, the measured channel capacity is slightly larger than the analytical value. A
possible cause of this phenomenon is the BS correlation being insufficiently close to 1 be-
cause this apparatus cannot achieve a time correlation of 1. As shown in Figure 6, when
the correlation coefficient approaches 1, the MIMO channel capacity deteriorates signifi-
cantly. However, it is possible to evaluate the performance of the MIMO terminal because
a 0.99 BS correlation can be achieved using this bidirectional fading emulator. Similarly,
with a circular array at the BS, the same phenomenon was observed when the BS correla-
tion was 1 depending on the incoming wave angle.
In the case of a circular array at the BS, shown in Figure 15, the six BS correlations do
not have the same characteristic because the correlation increases or decreases depending
on the incoming wave angle. Therefore, the MIMO channel capacity does not change sig-
nificantly with respect to the incoming wave angle.
In the case of a liner array at the BS, the six BS correlations are inclined in the same
direction depending on the incoming wave angle. Consequently, the MIMO channel ca-
pacity varies significantly with respect to the incoming wave angle compared with the
case of a circular array. In particular, when the SNR is 40 dB, there is a difference of ap-
proximately 35 bits/s/Hz between the incoming wave angles of 0° and 90°.
Moreover, as shown in Figure 16, it can be seen that the channel capacity of the linear
array is higher in the range of 0°–65° regardless of the SNR, whereas that of the circular
array is higher in the range of 65°–90°. Comparing Figure 15a,b, the BS correlation for the
linear array is lower than that of the circular array in the range of 0°–65°; thus, the channel
capacity is higher. On the other hand, in the range of 65°–90°, the channel capacity is high
because the BS correlation for the circular array is low. Considering the BS correlation at
ϕ = 65°, there are three correlation values of approximately 0.93, two of approximately
0.76, and one of approximately 0.53 for both the circular and linear arrays. Therefore, the
channel capacity is the same regardless of the arrangement of the BS antenna.
As described above, the fluctuations in the channel capacity characteristics with re-
gard to the incoming wave angle, shown in Figure 16, can be considered via association
with the BS correlation characteristics, shown in Figure 15. Thus, it is possible to use a
015 30 45 60 75 90
0
10
20
30
40
50
60
Incoming Wave Angle
φ
[deg]
Channel Capacity [bits/s/Hz]
Ver. pol. only
f = 1.95GHz
SNR = 40dB
30dB
20dB
10dB
Circular array Line array
OTA measuremet
Monte Carlo
35 bits/s/Hz
Figure 16. 4×4 MIMO channel capacity as a function of the incoming wave angle.
In the case of a circular array at the BS, shown in Figure 15, the six BS correlations do
not have the same characteristic because the correlation increases or decreases depending
on the incoming wave angle. Therefore, the MIMO channel capacity does not change
significantly with respect to the incoming wave angle.
In the case of a liner array at the BS, the six BS correlations are inclined in the same
direction depending on the incoming wave angle. Consequently, the MIMO channel
capacity varies significantly with respect to the incoming wave angle compared with the
case of a circular array. In particular, when the SNR is 40 dB, there is a difference of
approximately 35 bits/s/Hz between the incoming wave angles of 0◦and 90◦.
Moreover, as shown in Figure 16, it can be seen that the channel capacity of the linear
array is higher in the range of 0
◦
–65
◦
regardless of the SNR, whereas that of the circular
array is higher in the range of 65
◦
–90
◦
. Comparing Figure 15a,b, the BS correlation for the
linear array is lower than that of the circular array in the range of 0
◦
–65
◦
; thus, the channel
capacity is higher. On the other hand, in the range of 65
◦
–90
◦
, the channel capacity is high
because the BS correlation for the circular array is low. Considering the BS correlation at
φ
= 65
◦
, there are three correlation values of approximately 0.93, two of approximately
0.76, and one of approximately 0.53 for both the circular and linear arrays. Therefore, the
channel capacity is the same regardless of the arrangement of the BS antenna.
As described above, the fluctuations in the channel capacity characteristics with regard
to the incoming wave angle, shown in Figure 16, can be considered via association with the
BS correlation characteristics, shown in Figure 15. Thus, it is possible to use a bidirectional
fading emulator to conduct an OTA evaluation of the uplink from a MIMO terminal
antenna that considers the BS correlation characteristics.
6. Conclusions
This paper presents a method of implementing a 4
×
4 correlation matrix to realize
the BS correlation characteristics in order to evaluate an uplink channel using bidirectional
MIMO-OTA apparatus. The initial phases of the secondary sources were generated by
arranging virtual point sources based on Jakes’ model according to the BS correlation. An
examination of the effectiveness of the proposed method using bidirectional MIMO-OTA
apparatus made it clear that the desired BS correlation characteristics can be achieved only
by controlling the initial phase of the probe. Hence, the performance of the uplink from a
MIMO terminal considering the BS correlation can be evaluated with high accuracy.
Sensors 2021,21, 6184 18 of 20
In this paper, only the vertical polarization component was examined. However, the
MIMO terminal antenna is expected to have both vertical and horizontal polarization
components. It is considered that the BS correlation can be controlled by setting the
initial phases of the secondary sources for both the vertical and horizontal polarization
components, which will be investigated in future work.
Furthermore, the practical propagation environment model for the uplink channel is
assumed to be a three-dimensional channel model where radio waves propagate over the
full-solid angle. In one of my previous studies [
36
], a methodology for controlling the BS
correlation for a 2
×
2 MIMO system realized by a three-dimensional bidirectional fading
emulator was reported. The results showed that the measured BS correlation of the uplink
channel agreed well with the desired values, confirming the effectiveness of the proposed
method for a three-dimensional channel model. The accuracy of the 4
×
4 correlation
matrix will be improved because the flexibility of the virtual point source arrangement
increases by placing the virtual point source in three-dimensional space. As a result, the
difficulty of realizing all the BS correlations, as shown in Figure 12c, may be resolved.
The proposed OTA testing method will be applied to the evaluation of communication
performance considering the correlation coefficient between the received signals for other
wireless sensor networks, such as the Internet of Things (IoT) and emerging Vehicle-to-
Everything (V2X) technologies [
37
–
39
]. To ensure the success of upcoming connected car
systems, the author is currently developing a 256
×
256 MIMO antenna system that utilizes
circular array beam steering technology [
40
]. However, in large-scale MIMO systems, there
are some difficulties in terms of realizing OTA measurements. One of the difficulties is
that the number of BS correlations is very large compared with the number of BS antennas.
Another is that a large number of scatterers are necessary to create the full-rank channel
matrix. Therefore, not only the scope of applications of the proposed method for uplink
channels, but also OTA testing methods for downlink channels [
41
] need to be examined,
which will be investigated in future work.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The author declares no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
MIMO Multiple-input multiple-output
OTA Over-the-air
BS Base station
5G Fifth-generation
3GPP 3rd Generation Partnership Project
CTIA Cellular Telecommunication and Internet Association
XPR Cross-polarization power ratio
i.i.d. Independent and identically distributed
DUT Device under test
NLOS Non-line-of-sight
SNR Signal-to-noise ratio
RF Radio-frequency
IoT Internet of Things
V2X Vehicle-to-everything
Sensors 2021,21, 6184 19 of 20
Notations
KNumber of secondary wave sources
Pmi(t)Phase-shift of the carrier wave from the i-th scatterer according to the m-th BS antenna
φiAzimuth angle of the i-th scatterer
φvDirection of movement of the DUT antenna
fDMaximum Doppler frequency
vSpeed of the DUT antenna
λWavelength of the carrier wave
ϕmi Initial phase of the i-th scatterer with respect to the m-th BS antenna
ρSpatial correlation between the antennas
dDistance between the isotropic antennas
ΦmInitial phase matrix corresponding to BS #m
αmi Geometrical phase difference between the virtual point source of BS #mand probe #i
|∆ρ|Difference between the calculated spatial correlation and the desired BS correlation
σAngular spread of the incident wave
φRelationships between conventional BS antenna arrangements and the incoming
wave angle
hpr Channel response between the BS antenna #pand the MIMO terminal antenna #r
(*) Complex conjugate
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