Direction of Arrival Positioning Requirements for Location-Aware Beamforming in 5G mmWave UDN

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Direction of Arrival Positioning Requirements for
Location-Aware Beamforming in 5G mmWave UDN
Grigoriy Fokin
Laboratory of Software Defined Radio
The Bonch-Bruevich Saint Petersburg
State University of Telecommunications
Saint Petersburg, Russia
grihafokin@gmail.com
Ilya Grishin
Department of Communication Networks
The Bonch-Bruevich Saint Petersburg
State University of Telecommunications
Saint Petersburg, Russia
i.v.grischin@gmail.com
Abstract— The rapid growth in the number of simultaneously
operating transceivers in deployed radio access networks of the
fifth and subsequent generations leads to the problem of an
unacceptably high level of intra-system interference, provided
they are densified to one device per square meter. Adaptive
Location-Aware Beamforming (LAB) can potentially compensate
for high interference levels by maximizing the antenna pattern to
the source / receiver of the desired signal and minimizing the
antenna pattern to the source / receiver of the interfering signal.
The prerequisite for LAB is precise Direction of Arrival (DOA)
positioning of neighbor user equipment (UE). This study analyzes
the algorithm of multiple classification of UE signals from the
gNodeB (gNB) point of view in the F2 millimeter wave
(mmWave) frequency range. The content of this investigation can
be divided into two parts. In the first part mathematical model of
the DOA algorithm is given. In the second part simulation results
for DOA scenarios in 5G mmWave Ultra-Dense Networks (UDN)
are described. The contribution of this investigation is the 2D-
MUSIC resolution estimation in 5G UDN scenarios, which
reveals the dependence of the resolution threshold for 2D-
MUSIC algorithm on the number of elements in the antenna
array and Signal-to-Noise-and-Interference Ratio (SINR).
Simulation results reveal, that the resolution of the 2D-MUSIC
algorithm is achieved with two degrees angular separation of
devices, the number of antenna array elements more than 64 and
SINR more than 30 dB.
Keywords—5G, UDN, Multiple Signal Classication Algorithm,
2D-MUSIC, azimuth, elevation, spatial spectrum, DOA
I. INTRODUCTION
The first volumes of the IEEE Communications Magazine
at the beginning of 2022 year paid a lot of attention to the
issues of antenna systems for 5G and Beyond [1]–[4]. Authors
in [1] propose an efficient design methodology for V2X
antennas to provide the desired coverage. Investigation [2]
proposes mmWave-over-fiber based architecture with low-
complexity high-performance remote antenna units (RAUs) to
yield high throughput and reliable coverage. Article [3]
introduces the concept of intelligent omni-surface, which is
able to serve mobile users on both sides of the surface to
achieve full-dimensional communications by jointly
engineering its reflective and refractive properties with hybrid
beamforming scheme. Authors in [4] present modular massive
multiple-input multiple-output (mMIMO) as a candidate
technology for 6G to improve the spectral efficiency in low-
frequency bands. Common trend of observed investigations is
utilizing so called pencil beamforming [5] in 5G and Beyond
millimeter wave (mmWave) Ultra-Dense Networks (UDN).
One of the first mentions for Location based or Location-
Aware Beamforming (LAB) was introduced in [6]. Work [7]
show that by adjusting the beamforming and/or beam-training
parameters appropriately, the radio link capacity can be
optimized with respect to the prevailing positioning accuracy.
The problem for inter-beam interference for multi-beam-based
communication system is considered in [8]. A method to
evaluate the statistical properties of the reception angle,
considering the receiving Antenna Radiation Pattern (ARP)
and its beamwidth is investigated in [9]. 3D angle of arrival
(AOA) measurements processing model for positioning a
transmitter is proposed in [10]. Positioning accuracy limits
estimation of UE using AOA, Time Difference of Arrival
(TDOA) and TDOA/AOA measurement processing in three-
dimensional space with Cramer-Rao Lower Bound (CRLB) is
presented in [11], [12] and [13] respectively.
Investigation [14] provides simulation results of the
adaptive mMIMO beamforming (BF) according to the 5G
requirements and reveals that with the growth of the antenna
elements from 128 not only the accuracy of the BF increases
up to 4° resolution and accordingly meets 5G requirements up
to 5° precision. The well-known MUSIC algorithm has shown
their high efficiency in DOA/AOA and applicability for
systems with antenna arrays of any assembly [15]–[17].
Motivated by the described state-of-art, this work considers
multiple signal classification (MUSIC) DOA estimation in a
three-dimensional (3D) space with modifications of the above
algorithms in 5G mmWave UDN scenarios. The aim of this
work is estimation of MUSIC angular accuracy, which makes
it possible to increase UE spatial resolution in the mmWave
FR2 frequency range (24250–52600 GHz). The considered
FR2 suggests possibility of integrating mMIMO antennas into
UE and there are already technical solutions for ranges n257
(26.5-29.5 GHz), n259 (39.5-43.5 GHz), n260 (37-40 GHz)
and n261 (27.5-28.35 GHz) [18].

The paper is organized as follows. Section II formalizes
mathematical model for DOA estimation. Simulation results
are described in section III. Paper is concluded in Section IV.
II. MATHEMATICAL MODEL FOR DOA ESTIMATION
A. Direction of Arrival Estimation Problem Statement
The calculation of UE spatial coordinates is possible by
extracting necessary information from parameters of the radio
signal, received by BS, which propagates in Line-Of-Sight
(LOS) conditions. Assume, antenna arrays of the BS and UE
are Uniform Rectangular Arrays (URA) consisting of for
transmitting, and elements for receiving antenna. Also, for
DoA mathematical and simulation models suppose, that gNB
performs receive and UE performs transmit spatial signal
processing. gNB array with size ensures the formation
of a narrow antenna radiation pattern in both azimuth and
elevation planes, allowing UE direction of arrival estimation
with both azimuth φ and elevation θ angles. Denote complex
signal, emitted by UE array with elements as
which is described by:
, 

 
where denotes the vector
оf complex signals; NB denotes the number of beams;
is a directional beamforming matrix;
denotes beam steering vector of
beams; is the complex weight, applied
to the antenna element . The beam steering
vector of b beams is determined according to the equation:
, 2
where – the unit-norm array response vector for
azimuth and elevation angles. Best conditions for
transmission will be provided, when gNB angular coordinates,
chosen for vector of weight coefficients (2), correspond to the
true receiving gNB angular coordinates. Thus, the algorithm
for searching gNB angular coordinates should be analyzed. Let
there be a gNB, whose antenna array with elements
receive radio signal described by vector:
. The expression
connecting vector of transmitted-received signals is:
3
where M denotes the number of multipaths components (MPC)
by scattering of the signal; is the signal propagation time
along the m-th path; is the complex vector of Additive
White Gaussian Noise (AWGN); denotes
transmission matrix, which contains complex channel impulse
responses between the kth transmit antenna element and lth
receive antenna elements for mth path. Let be the main LOS
path transmission matrix, – path transmission
matrices for other M−1 paths which are associated with clusters
that can be scatterers or reflectors. The research results
presented in [19] showed that for the millimeter range the
maximum number of signal propagation paths is 4, which
allows using the Poisson distribution with the mathematical
expectation of 1.8 when modeling multipath signal propagation
in the 28 GHz frequency range [20]. At the same time, taking
into account the size of 5G UDN small cells, it can be assumed
that the signal levels for are negligible and may not be
taken into account in subsequent calculations. In this case, the
matrix (3) and is described by the expression:
 4
where is the complex gain of 1st LOS path; [ ]H is Hermitian
transpose; , – the unit-norm antenna
arrays response vectors of UE transmitter and gNB receiver.
These vectors, as well as vector from (2), can be calculated by:
 5
where index i is T, R or b; is matrix of
antenna array elements, Cartesian
coordinates, is the wavenumber vector:
 6
Example 3D scenario of signal propagation between two
uniform rectangular array (URA) is in Fig. 1.

Fig. 1. Scenario of of signal propagation between two URAs in 3D
For frequency range of 28 GHz the wavelength is λ=0,011
m, so the wave front in the vicinity of the receiving antenna can
be considered flat. Modern 5G networks provide for a high
density of UE connection, implying that URA will be
simultaneously exposed to several U radio waves:
 7
where denotes transmission matrix (4) for uth UE,
denotes the signal transmitted by the uth UE. The task, which is
posed in this case on the receiver side, is to determine the
number of sources and then find their angular coordinates in
order to subsequently form suitable ARP. In this case, it is
necessary to take into account the possibility that the angular
distance between UEs in some cases may be less than the
beamwidth formed by URA. The solution of this problem
makes it possible to obtain projection algorithms, in particular,
the multiple signal classification (MUSIC) algorithm and the
2D-MUSIC modification, the principle of which is based on
the orthogonality of signal and noise subspaces.
B. MUSIC DOA Algorithm Formalization
According to MUSIC algorithm, the covariance matrix of
the received signal (7) should be found. This matrix
can be represented as:
 8
where is the covariance matrix of the signal ;
denotes the identity matrix; denotes the
variance of the Gaussian white noise; denotes the
spectral decomposition of the covariance matrix;
denotes the matrix of eigenvectors,
; , denotes the
diagonal matrix of positive eigenvalues, .
The value allows you to set a threshold value ,
according to which the number of eigenvalues in that
exceeds determines the possible number of signal sources
from (1): , where ,
. The remaining eigenvalues are taken
equal to . The spectral decomposition of the matrix can be
represented by the sum of the signal submatrix and the noise
submatrix:
, 9
where is the matrix of the first
eigenvectors of ; denotes
the matrix of first eigenvalues of ;
denotes matrix of last
eigenvectors of .;
denotes matrix of last eigenvalues of .
The system of vectors forms an
orthonormal basis in the space of dimension NR, which,
according to (9, 10), can be represented by a signal -
dimensional subspace and its -dimensional
orthogonal complement, which is a subspace of noise.
Projection operators on the subspace of signals and the
subspace of noise:
, 0
The URA response vectors are orthogonal to the noise
subspace. This fact makes it possible to determine the elevation
angle θ and azimuth φ of UTs by finding the maxima of the
spatial spectrum function:
, 
where denote estimates of position and azimuth angles;
is estimation of the response vector BS URA
elements. In practical applications, the calculation of the
covariance matrix is carried out for L signal samples:
, 2

Calculations (8)–(11) are made based on the matrix.
The complexity of the 2D-MUSIC algorithm is quite high and
is estimated as [21]:
 3
It should also be noted that the stability of the algorithm is
reduced if there are correlations between the received signals.
III. SIMULATION MODEL FOR DOA ESTIMATION
In order to determine the accuracy characteristics and
resolution of the 2D-MUSIC algorithm, a number of
computational experiments were carried out using the
developed simulation program for various values of the noise
level, number of URA elements, and angular spacing of UE.
The distance between the elements of the URA is λ/2. Obtained
simulation results allows us to draw a conclusion about
algorithm resolution with Tables I-IV and 3D images of spatial
spectra with its isolines in Fig. 2-4, illustrating the resolution of
two UE sources. In Fig. 2-4 initial values of position and
azimuth angles for signal sources are marked in red. The
measurement results for one UE in Table 1 showed that the
2D-MUSIC algorithm provides zero error in calculating the
angular coordinates of the UE in the case of using an antenna
system with a grid consisting of more than 16 elements for
values of SINR ≤ 5dB. The antenna system with a grid of 16
elements for the given values of SINR determines the angular
coordinates with an error, which for urban micro (UMi) can be
considered as an acceptable indicator [22]. The subsequent
reduction in the number of elements in the antenna array of the
gNB leads to a decrease in accuracy, which makes impossible
to apply a number of positioning scenarios in 5G UDN
according to 3GPP TS 38.305 [23].
Fig. 2. 2D-MUSIC spectrum and it isolines in the area of angular coordinates
coordinates NR=16 (44); SNR=10 dB; θR=12o; φR=51o; L=300 snapshots
Fig. 3. 2D-MUSIC spectrum and it isolines in the area of angular coordinates
NR=256; SINR=30 dB; θ1=52°; φ1=54°; θ2=50°; φ2=56°; L=300 snapshots
TABLE I. DOA ESTIMATION: ΘR=12 °; ΦR=51°;
NR№
SNR
20, dB 10, dB 5, dB 0, dB
4
112o51o12o50 12o48o12o53o
212o51o12o51 12o49o13o49o
312o51o12o52 11o52o13o52o
412o51o12o50 11o49o12o48o
512o51o12o50 12o52o12o51o
total θR±0oφR±0oθR±0oφR±1oθR±1oφR±3oθR±1oφR±4o
1
6
112o51o12o51o12o52o12o51o
212o51o12o51o12o51o12o51o
312o51o12o51o12o51o12o50o
412o51o12o51o12o50o12o52o
512o51o12o51o12o51o12o52o
total θR±0oφR±0oθR±0oφR±0oθR±0oφR±1oθR±0oφR±1o
6
4
112o51o12o51o12o51o12o51o
212o51o12o51o12o51o12o51o
312o51o12o51o12o51o12o51o
412o51o12o51o12o51o12o51o
512o51o12o51o12o51o12o51o
total θR±0oφR±0oθR±0oφR±0oθR±0oφR±0oθR±0oφR±0o
TABLE II. DOA ESTIMATION: θ1=52 °; φ1=54°; θ2=49°; φ2=55°
SINR,
dB
NR =16; equal powers UTs: Р1= Р2
30 0,575 2,78710–3 6,85810–5 54° 54° 47° 55°
20 0,575 2,80910–3 1,67010–3 54° 54° 46° 55°
10 0,575 – 0,019 50° 54° – –
NR =64; equal powers UTs: Р1= Р2
30 2,414 0,044 7,37010–5 53° 54° 47° 55°
20 2,415 0,044 4,21310–4 53° 54° 47° 55°
10 2,415 0,045 4,70810–3 53° 54° 47° 55°
NR =256; equal powers UTs: Р1= Р2
30 11,575 0,585 1,21010–5 53° 54° 48° 55°
20 11,580 0,584 4,24810–4 53° 54° 48° 55°
10 11,570 0,589 2,32210–3 53° 54° 48° 55°
NR =16; unequal powers UTs: Р1= 0,25 Р2
30 1,144 3,53810–4 2,02110–4 54° 54° 47° 55°
20 1,143 – 4,60810–3 – – 47° 55°
10 0,711 – 0,036 – – 47° 55°
NR =64; unequal powers UTs: Р1= 0,25 Р2

30 4,644 5,75710–3 4,75810–5 53° 54° 47° 55°
20 4,644 5,90410–3 1,22410–3 53° 54° 47° 55°
10 4,646 – 0,011 – – 47° 55°
NR =256; unequal powers UTs: Р1= 0,25 Р2
30 19,675 0,086 1,21110–5 53° 54° 48° 55°
20 19,678 0,086 2,96810–4 53° 54° 48° 55°
10 19,664 0,088 2,68810–3 52° 54° 48° 55°
To determine the potential capabilities of the 2D-MUSIC
algorithm, cases of separation by 1°, 2°, 3° or more degrees in
azimuth and elevation angle for P1=P2 and P1≠P2 were
considered (P1, P2 denote the powers of UE1 and UE2).
Resolution refers to the minimum possible offset in the angular
coordinates of the UE sources relative to each other, at which
their separate observation is possible. The Rayleigh criterion is
considered as a measure of resolution, in which there is a dip
between the peaks of the spatial spectrum equal to or greater
than 2 dB [24]. The number of radio sources is automatically
determined by the number of eigenvalues of the matrix
exceeding the power value of the receiver 's own noise. The
results of individual experiments are presented in the form of
drawings of spatial spectra and isolines of their fragments, and
also summarized in Tables II–IV.
As the results of the experiments were revealed, for the step
of enumerating the estimation of angular coordinates of 1°, the
resolution of the 2D-MUSIC algorithm in azimuth and
elevation is 3°, presented in Table II. As can be seen from
Table II, regardless of the SINR indicators and the number of
URA elements, the errors in estimating the elevation angle in
the average level are Δθ = ±2°. For an array consisting of 16
elements, stable operation of the algorithm for a given angular
separation of the UTs is observed for SINR=30 dB.
Simulation results revealed, that for 1° step of enumerating
the estimation of angular coordinates, the resolution of the 2D-
MUSIC algorithm in azimuth and elevation angle is 3°. As can
be seen from Table II, regardless of the SINR indicators and
the number of URA elements, the errors in estimating the
elevation angle in the average level are Δθ=±2°. For an array,
consisting of 16 elements, stable operation of algorithm for a
given angular separation of UEs is observed for SINR=30 dB.
The 2D-MUSIC algorithm loses resolution characteristics at
lower values of angular spacing between devices in the case of
a search step equal to 1° even for URA with NR=256 elements
and SINR=30 dB (Table III). In this case errors may occur
during the calculation of the number of signal sources.
TABLE III. SPATIAL SPECTRUM, IN THE AREA OF ANGULAR
COORDINATES OF UTS WITH EQUAL POWERS
53° 54° 55° 56° 57° 58°
47°
–
24,72
–
23,56
–
22,12
–
20,49
–
19,03 –18,56
48°
–
23,50
–
21,92
–
19,88
–
17,32
–
14,90 –15,04
49°
–
21,92
–
19,76
–
16,74
–
12,22 –8,20 –12,9
50°
–
19,83
–
16,74
–
11,74 0 –8,38 –15,23
51° –
16,98
–
11,94
–0,59 –7,74 –
14,64
–18,41
52°
–
12,95 –1,62 –7,42
–
14,65
–
18,39 –20,87
53° –9,07 –7,26
–
14,28
–
18,29
–
20,86 –22,68
54°
–
11,98
–
14,27
–
17,99 –20,7
–
22,62 –24,03
λ1=36,319; λ2=0,041; λ3=1,49510–5
Subsequent calculations in the region of the highest peak of
spatial spectrum (Table III) with a search step of 0.25° shows,
that the 2D-MUSIC algorithm provides a resolution of 2° in
azimuth φ and elevation θ angle. The simulation results for
sources of equal power and URA with NR =256 elements are
shown in Table IV. Simulation results showed that resolution
of the 2D-MUSIC algorithm is preserved at SINR > 10 dB for
URA with NR=256.
Simulation results revealed that the resolution of equally
powerful and unequally powerful UTs for NR=64 and the
separation of sources by angular coordinates by 2° is possible
at values of SINR ≥ 30 dB. For antenna array with 16 elements
2D-MUSIC algorithm loses resolution characteristics.
The resolution of the algorithm was also evaluated when
the UE was spaced by angular coordinates by a value of 1°.
Measurements have shown that when the UTs are separated by
angular coordinates by 1°, the resolution of the sources is
impossible (Fig. 4).
TABLE IV. VALUES OF THE SPATIAL SPECTRUM,DB IN THE AREA OF
ANGULAR COORDINATES OF UTS θ1=52°; φ1=54°; θ2=50°; φ2=56°; NR=256
54,00° 54,25° 54,50° 54,75° 55,00° 55,25° 55,50° 55,75° 56,00°
SINR=30 dB
50,00° –31,33 –30,19 –28,86 –27,27 –25,32 –22,79 –19,18 –12,92 –0,53
50,25° –30,21 –28,89 –27,32 –25,40 –22,92 –19,42 –13,53 –1,10 –13,83
50,50° –28,93 –27,37 –25,47 –23,05 –19,72 –14,59 –8,45 –14,55 –19,70
50,75° –27,41 –25,52 –23,14 –19,93 –15,36 –11,21 –15,47 –20,00 –23,16
51,00° –25,57 –23,18 –20,01 –15,62 –12,20 –16,02 –20,26 –23,31 –25,61
51,25° –23,22 –19,99 –15,47 –11,97 –16,05 –20,34 –23,39 –25,68 –27,49
51,50° –19,98 –15,04 –10,41 –15,51 –20,19 –23,36 –25,69 –27,52 –29,02
51,75° –14,75 –6,76 –14,33 –19,80 –23,19 –25,61 –27,49 –29,02 –30,31
52,00° 0 –12,53 –19,19 –22,89 –25,45 –27,4 –28,97 –30,28 –31,41
SINR=20 dB
50,00° –27,09 –25,94 –24,61 –23,03 –21,09 –18,57–15,01 –9,05 –0,51
50,25° –25,97 –24,65 –23,08 –21,16 –18,69 –15,22 –9,52 –0,06 –9,82
50,50° –24,69 –23,13 –21,23 –18,81 –15,50 –10,47 –4,78 –10,46 –15,5
50,75° –23,17 –21,28 –18,90 –15,70 –11,16 –7,19 –11,31 –15,78–18,93
51,00° –21,33 –18,94 –15,78 –11,43 –8,09 –11,83–16,03 –19,07–21,36
51,25° –18,99 –15,77 –11,29 –7,87 –11,89 –16,11–19,15 –21,43–23,24
51,50° –15,78 –10,91 –6,44 –11,35 –15,97 –19,11 –21,44 –23,27–24,77
51,75° –10,69 –3,41 –10,23 –15,59 –18,95 –21,37–23,25 –24,77–26,05
52,00° 0 –8,61 –15,01 –18,67 –21,21 –23,15–24,72 –26,03–27,15
SINR=10 dB
50,00° –18,88 –17,57 –16,02 –14,13 –11,73 –8,49 –3,85 0 –18,87
50,25° –17,61 –16,06 –14,19 –11,82 –8,65 –4,15 –0,24 –4,31 –17,61
50,50° –16,12 –14,25 –11,91 –8,82 –4,58 –1,18 –4,65 –8,85 –16,12
50,75° –14,32 –11,98 –8,92 –4,84 –1,79 –4,98 –8,99 –11,98 –14,32
51,00° –12,07 –8,99 –4,89 –1,83 –5,04 –9,05 –12,05 –14,31 –12,08
51,25° –9,13 –4,87 –1,36 –4,76 –8,95 –12,01 –14,31 –16,13 –9,13
51,50° –5,07 –0,72 –4,16 –8,70 –11,89 –14,25 –16,11 –17,63 –5,07
51,75° –0,93 –3,52 –8,31 –11,67 –14,12 –16,03 –17,59 –18,89 –0,93
52,00° –3,50 –7,96 –11,40 –13,94 –15,91 –17,51 –18,84 –19,98 –3,50
λ1=36,319; λ2=0,041

Fig. 4. 2D-MUSIC spectrum and it isolines in the area of angular coordinates
NR=256; SINR=30 dB; θ1=52°; φ1=54°; θ2=51°; φ2=55°; L=300 snapshots
IV. CONCLUSION
In this paper, we considered the 2D-MUSIC algorithm and
explored its capabilities in terms of the dependence of the
resolution threshold and algorithm accuracy on the number of
elements in the antenna array and the SINR value. The results
of simulation, carried out using the developed program,
showed, that the resolution of the 2D-MUSIC algorithm is
achieved with an angular separation of devices of 2o and the
number of antenna array elements NR≥64. These conditions
make it possible to provide the accuracy required for the vast
majority of positioning scenarios in fifth generation networks.
However, in the case of antenna arrays with NR=64, the
requirements for SINR≥30 dB must be met. It is also necessary
to take into account that amplitude-phase errors can occur due
to a number of factors like: errors in the geometric arrangement
of the array elements, the mutual influence of the array
elements, which has a complex dependence on the direction of
signal arrival, and a number of others. These errors adversely
affect the accuracy and resolution of the 2D-MUSIC algorithm.
It should also be noted that estimation accuracy of the UEs
angular coordinates and algorithm resolution decrease as the
number of accompanied UEs approaches the number of URA
elements. At the same time, the planned density of devices in
5G communication networks can reach 106/km2. This makes it
possible that the number of sources, detected by the gNB can
reach several tens or hundreds. In connection with the
foregoing, we can conclude that in 5G mmWave UDN should
use antenna arrays with more than 256 antenna elements.
ACKNOWLEDGMENT
This research is supported by the Russian Science
Foundation Grant No. 22-29-00528, https://rscf.ru/project/22-
29-00528/
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