Statistical Evaluation of the Azimuth and Elevation Angles Seen at the Output of the Receiving Antenna

Abstract

A method to evaluate the statistical properties of the reception angle seen at the input receiver that considers the receiving antenna pattern is presented. In particular, the impact of the direction and beamwidth of the antenna pattern on distribution of the reception angle is shown on the basis of 3D simulation studies. The obtained results show significant differences between distributions of angle of arrival and angle of reception. This means that the presented new method allows assessing the impact of the receiving antenna pattern on the correlation and spectral characteristics at the receiver input in simulation studies of wireless channel. The use of this method also provides an opportunity for analysis of a co-existence between small cells and wireless backhaul, what is currently a significant problem in designing 5G networks.

Full text

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 1
Statistical Evaluation of the Azimuth and Elevation Angles
Seen at the Output of the Receiving Antenna
Cezary Ziółkowski and Jan M. Kelner
Abstract—A method to evaluate the statistical properties of the reception
angle seen at the input receiver that considers the receiving antenna
pattern is presented. In particular, the impact of the direction and
beamwidth of the antenna pattern on distribution of the reception angle is
shown on the basis of 3D simulation studies. The obtained results show
significant differences between distributions of angle of arrival and angle of
reception. This means that the presented new method allows assessing the
impact of the receiving antenna pattern on the correlation and spectral
characteristics at the receiver input in simulation studies of wireless
channel. The use of this method also provides an opportunity for analysis
of a co-existence between small cells and wireless backhaul, what is
currently a significant problem in designing 5G networks.
Index Terms—Angle of arrival, angle of reception, angle spread,
antenna radiation pattern, azimuthal and elevational planes, channel
models, channel modeling, directional receiving antenna, geometric
channel models, half power beamwidth.
I. INTRODUCTION
The direction and spatial shape of the pattern of the receiving
antenna significantly affect the statistical properties of the signal
reception angle. These properties that are described by probability
density function (PDF) deform the correlation and spectral
characteristics of the signals transmitted in wireless channels [1-3].
Therefore, mapping the spatial distribution of the reception angle
“seen at the output of the receiving antenna”, is required to obtain
the convergence of the simulation studies and actual measurements.
The use of this method also provides an opportunity for analysis of a
co-existence between small cells and wireless backhaul (WB).
Especially, it will allow to evaluate the interference caused by 5G
access point towards WB receiver needed to determine a minimum
distance between 5G and WB deployments. This current and
significant problem in designing 5G networks stems from
narrowbeam antenna patterns, which ensure minimizing power
consumption and increase the range of radio links.
The directivity of the receiving antenna results in a spatial
selection of propagation paths. Therefore, the signal arriving at the
input of the receiver is a superposition of signals from all
propagation paths and their levels are formed by the receiving
antenna pattern. This is the cause of differentiation of the statistical
properties of the angle of arrival (AOA) in the surroundings of the
receiving antenna and the angle of reception (AOR) that is “seen at
the output of the receiving antenna”. In the case of an
omnidirectional antenna in the azimuth plane and its large
beamwidth in the elevation plane, PDF of AOA and PDF of AOR
are convergent. In literature, we can find many models and mapping
methods of the statistical properties of angle that consider the
scattering phenomenon both on the azimuth and elevation planes
[4-8]. However, these models and methods focus only on mapping
AOA and are mainly based on omnidirectional antennas. Only a few
Manuscript received April 24, 2017.
C. Ziółkowski and J. M. Kelner are with the Institute of
Telecommunications, Faculty of Electronics, Military University of
Technology, 00-908 Warsaw, Poland (e-mail: cezary.ziolkowski@wat.edu.pl;
jan.kelner@wat.edu.pl).
Digital Object Identifier 10.1109/TAP.2018.2796719
of them consider the sector antennas but merely on the transmission
side and in a simplified manner [9,10]. Therefore, these models and
methods can be used in simulation only in scenarios where PDFs of
AOA and AOR are convergent. For the sectoral and narrowbeam
antennas, the effect consideration of spatial filtering antenna is
required to evaluate the correlation and spectral signal properties at
the input of the receiver. 3GPP channel model gives such
possibilities but only to a limited scope [11]. These limitations are
the result of using only a few strictly defined scenarios that specify
the parameters of propagation phenomena. The solution that is
presented in this communication, addresses this problem and is
applicable in channel simulation studies and in empirical data
analysis. In this case, the developed method enables to use any
propagation scenario that is defined by the power delay profile
(PDP) or power delay spectrum (PDS). In addition, the consideration
of the transmitter and receiver antenna patterns in the angular power
distribution is an innovative contribution of this paper.
Here, we present a method to assess the statistical properties of
AOR that consider the impact of the receiving antenna pattern on the
direction of signal reception in 3D, and the beamwidth of the antenna
pattern on PDF of AOR is shown for the simulation scenarios, whose
parameters are defined on the basis of the 3GPP channel model [11].
The remainder of this communication is organized as follows. The
system geometry and 3D channel model for AOA generation is
presented in Section II. Determination of AOR and estimation of its
PDF is described in Section III. The next section includes the results
of the simulation studies that show the effects of the direction and
beamwidth of the antenna pattern on spread of AOR. Section IV
provides some concluding remarks.
II. 3D CHANNEL MODEL
The geometrical model of the channel is used to generate sets of
AOAs that describe the space scattering of signals as a set of half-
ellipsoids. The number and spatial parameters of these half-ellipsoids
are defined on the basis of the power delay profile (PDP) or power
delay spectrum (PDS). The local extremes of these characteristics
represent signal components that arrive at the receiver with the same
delay and form so-called time clusters. The amount and the position
in time domain of these extremes define the number and size of the
individual half-ellipsoids. The geometry of the channel model is
shown in Fig. 1.
Adopted geometry is a basis to generate the angular parameters of
the propagation paths arriving at the receiver with a delay relative to
the direct path. These parameters are sets of angles in the azimuth
and elevation planes, and the levels of power that characterize the
intensity of each path. In addition to the parameters of the paths that
represent delayed components of the signal, the parameters of the
local scattering paths are generated. In this case, the von Mises PDF
is used. As a result, we obtain the sets of the angles in the elevation
Θ and azimuth Φ planes, and power levels P that describe each
0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 5
propagation path arriving at the receiver.
     
iii MN
ji
ij
MN
ji
ij
MN
ji
ij p,
1,0
,
1,0
,
1,0 ,,   PΦΘ

(1)
where: i is the number of the time cluster (half-ellipsoid), j is the
number of component in the ith time cluster, N represents the
number all time clusters (half-ellipsoids), and Mi means the number
of the components (propagation paths) in the ith time cluster. The
elements with i = 0 represent the local scattering components.
Fig. 1. Geometry of 3D channel model.
This channel model is one of a few that considers the impact of
radiation transmitter antenna on the statistical properties of AOA. A
detailed description of the generation procedure of Θ, Φ, and P is
presented in [12]. In this publication, the evaluation of the statistical
properties of the generated sets of the angles shows the compliance
with the measurement data. This justifies the use of this model to
determine the angle and power parameters of the received signal
components in simulation studies of channel. The used geometric
channel model is an extension of our previous works, i.a., [13,14].
III. ESTIMATION OF AOR DISTRIBUTION
Evaluation of the influence of the receiving antenna at AOR is
based on the input data that are Θ, Φ, P, and the power pattern of
this antenna,
 
ijijR
g

,
2
. The sets of the angles and powers that
represent the parameters of the propagation paths, can be written in
integrated form as
  
i
MN
ji
ijijij
p,
1,0
,

. This form shows that each
element of this set represents the power that arrives at the receiver
from
 
ijij

,
direction. Thus, for the jth path of the ith half-
ellipsoid, the signal power,
 
ijijRij
p

,
, at the output of the
receiving antenna expresses the following relationship:
     
2
, , ,
Rij ij ij ij ij ij R ij ij
p p g
     

(2)
In simulation studies, the Gaussian beam is commonly
parametrized power pattern of the antennas [15,16]
   
22
222
π2
, exp exp
RR gg
gG


 




  





(3)
where:
R
G
means the boresight gain of the antenna,


g
and


g
represent beamwidths of the antenna pattern in the elevation and
azimuth planes, respectively. These parameters are closely related to
the half power beamwidths (HPBWs) in the respective planes (see
[15]).
These relationships are the basis for the transformation of the
signal power from the surroundings of the antenna to the input of the
receiver.
Let
     
 
 
 
εε:,,  ijij
jiO
, where

ε
and

ε
are the neighborhoods of θ and φ, respectively. Thus,
 
 



,,
OijijRij
p
represents the total power of the signal that arrives
at the input of the receiver from
 
 
ε,ε
sector. Let Q means
the number of all propagation paths reaching the receiver and let in
the neighborhood of ever θ and φ is at least one angle that describes
the path. Then, for Q → ∞, the size of the neighborhood approaches
zero for each angle (εθ → 0 and εφ → 0). Consequently, each sum
related to εθ and εφ represents the power that falls on the elementary
interval of the angle. Therefore, in the limit, we obtain the power
angle spectrum,
 

,
R
P
 
 
 
 
,
ε 0 ε 0
,
,
4εε
Rij ij ij
R
Q
p
P






  


O
(4)
It means that these finite sums can be treated as an estimator of
 

,
R
P
. Note also that the signal average power, P0, at the input of
the receiver is estimated by
 
 
01 0
ε 0 ε 0
,
4εε
i
M
N
Rij ij ij
ij
Q
p
P





  


(5)
In practice, we can present
 

,
R
P
in the form
   

,, 0RR fPP 
(6)
where
 

,
R
f
means PDF of AOR. On the basis of (6), the
estimator of PDF of AOR,
 

,
~R
f
, is
 
 
 
 


 
N
i
M
jijijRij
ijijRij
Rip
p
f
0 1
,
0
,
,
C,
~


 
O
(7)
where C0 is a normalizing constant that is associated with εθ, εφ, and
provides a condition
 
1d d ,
~
lim °90
0
°180
°180
0ε0ε
 






R
f
.
In the real environment, scattering phenomena that occur in the
elevation and azimuth planes are independent. Hence, marginal
PDFs of AOR in the elevation and azimuth planes can be represented
in the following forms:
 
 
 
 
 
 
 
 
0 1 0 1
C , C
ii
Rij ij Rij ij
RR
MM
NN
Rij ij Rij ij
i j i j
pp
ff
pp





   


 
KL
(8)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 5
(a)
(b)
(c)
Fig. 2. (a) Power pattern of widebeam antenna for α = –30º, 0º, 60º in azimuth
plane (linear scale), (b) PR( φ ), and (c) PR( θ ) for scenario 2.
(a)
(b)
(c)
Fig. 3. (a) Power pattern of narrowbeam antenna for α = –30º, 0º, 60º in
azimuth plane (linear scale), (b) PR( φ ), and (c) PR( θ ) for scenario 2.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 5
where
 

R
f
~
and
 

R
f
~
are the estimators of PDFs of AOR in the
elevation and azimuth planes, respectively,
     
 


ε:,  ij
jiK
,
   
 
 


ε:,  ij
jiL
, whereas
Cθ and Cφ fulfill conditions
 
1d
~
lim °90
0
0ε




R
f
and
 
1d
~
lim °180
°180
0ε





R
f
.
In simulation and empirical studies of channel, (2) and (8) are the
basis for the practical assessment of the impact of the receiving
antenna pattern on the statistical properties of AOR.
IV. ANTENNA PATTERN PARAMETERS AND PDF OF AOR
In this section, we show the impact of the receiving antenna on the
statistical properties of AOR that is viewed from the receiver input.
Here, we focus on the changes of PDF of AOR that result from
changes in both the direction,

, and beamwidths, HPBWθ,φ, of the
transmitting and receiving antenna patterns. To include these
relations we introduce following designations in the analytical
description of PDFs:
   


RR ff ~~
and
   


RR ff ~~
, where
 


,
,HPBW
. In the simulation studies, PDPs from 3GPP non-
line-of-sight propagation scenarios [11, Table 7.7.2-2] are assumed
to design the geometric channel model. As an example, we used
three urban macro (UMa) scenarios for 28 GHz. The scenarios 1, 2,
and 3 are defined as short-, normal-, and long-delay profiles [11,
Table 7.7.3-2] that represent the average results of measurement
campaigns in diverse urban environments. All tests are carried out at
narrowbeam and widebeam antennas for 28 GHz whose parameters
are as follows: the widebeam antenna –
dBi0.15
R
G
,
 0.30

HPBW
, and
28.8HPBW


, the narrowbeam antenna –
dBi5.24
R
G
,
 6.8

HPBW
, and
10.9HPBW


, respectively.
These parameters are adopted on the basis of measurement
campaigns [16-18], which aim was to assess the propagation
phenomena in wireless links of 5G networks. To evaluate 3D
modeling procedure, 200 Monte Carlo runs were carried out in the
Matlab.
In the first step, we consider the impact of

that is the angle
between the signal source and receiving antenna pattern direction in
the azimuth plane. Based on (6) and (8), the graphs of
 

,
R
P
in
the azimuth,
 

R
P
, and elevation,
 

R
P
, planes for the widebeam
and narrowbeam antennas, and scenario 2 are presented in Figs. 2
and 3, respectively. In each of the figures, a case of AOA is
considered to assess the impact of
 
ijijR
g

,
2
on
 

R
P
and
 

R
P
.
In addition, the angular positions of the receiving antenna patterns
are shown in Figs. 2 a) and 3 a), respectively. For the widebeam
antenna, the increase of

to 120º causes a decrease of 30 dB and
27 dB in the maximum of
 

R
P
and
 

R
P
. Whereas for the
narrowbeam antenna, this reduction is 46 dB and 40 dB,
respectively.
As we can see, the change in

is reflected not only in level
changes, but also in the changes of the angular dispersion of the
receiving signal power. For scenario 2, the influence of

on the
statistical properties of the power dispersion that represent PDFs, are
illustrated in Fig. 4.
As can be seen in Fig. 4 b), PDFs are asymmetrical for
0

,
which is clearly visible with increasing

. To quantify the
influence of

on the intensity of the AOR scattering phenomenon,
the standard deviations are used. For the elevation and azimuth
planes, these parameters are determined, respectively, as
     
2
11
2~
1
~
1







  



J
nnRn
J
nnRn f
J
f
J
 
(9)
     
2
11
2~
1
~
1







  



J
nnRn
J
nnRn f
J
f
J
 
(10)
where J is the number of data from the simulation results.
(a)
(b)
Fig. 4. PDFs of AOR for α = –30º, 0º, 60º in (a) elevation and (b) azimuth
planes.
For the widebeam and narrowbeam antennas, the graphs of
 



and
 



versus

are shown in Fig. 5. For the widebeam and
narrowbeam transmitting antennas, the standard deviations of AOA
are (
14


,
 34


), and (
6



,
19


), respectively.
Fig. 5 shows that
 



and
 



reach minimum for
0

. It
means that the beamwidth of receiving antenna limits the intensity of
the power dispersion by approximately 11°, 27° and 4°, 15°
compared to AOA for the widebeam and narrowbeam antenna,
respectively. For
°30

, the scattering intensity of AOR
significantly increases particularly in the azimuth plane.
The relationships (2) and (8) also provide an opportunity to
analyze the changes in PDF of AOR as a function of the antenna
pattern beamwidth. In this case, HPBW is analyzed as a parameter of
PDFs for angles in the elevation and azimuth planes. The example of
mapping the impact of the antenna pattern beamwidth on PDF of

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 5
(a)
(b)
Fig. 5. Graphs of σθ(Ω) and σφ(Ω) versus | α | for widebeam and narrowbeam
antennas in (a) elevation and (b) azimuth planes.
(a)
(b)
Fig. 6. PDFs of AOR and AOA for widebeam and narrowbeam antennas in
(a) elevation and (b) azimuth planes.
AOR is considered for
0

. For the widebeam and narrowbeam
antennas, the graphs of PDFs of AOR in the elevation and azimuth
planes are shown in Figs. 6 a) and 6 b), respectively. Additionally,
PDFs of AOA at the surrounding the receiving antenna with
regarding the transmitting antenna pattern are presented to compare
the statistical properties of the reception angle.
It is obvious, that the AOR dispersion is reduced along with a
decrease in the pattern beamwidth of the receiving antenna.
However, quantitative assessment of the effect is possible because of
the analysis of the simulation data that is presented in this
communication. Here, we limit this analysis only to assess the
impact of changes in the pattern beamwidth in the azimuth plane. In
Fig. 7,
 



is shown as a function of
HPBW

for three UMa
scenarios. Simulation test are performed on the assumptions
0

and
30HPBW


.
Fig. 7. Graphs of σφ(Ω) versus HPBWφ for UMa scenarios (α = 0,
HPBWθ = 30°).
Based on the graphs of σφ(Ω) for AOA, we can see that the type of
environment significantly differentiated the intensity of the scattering
phenomenon. This environmental impact is already present when the
HPBWφ of the transmitting antenna exceeds 5°. Applying the
narrowbeam pattern of the receiving antenna substantially reduces the
impact of the environment on the AOR distribution. In this case, the
environmental impact appears only for
40HPBW


and it is
insignificant compared to AOA.
V. CONCLUSION
This communication presents methods to map and assess the
impact of the receiving antenna on the statistical properties of AOR.
In the case of using the receiving sector antennas, the obtained
results show significant differences between PDFs of AOA and
AOR. This means that in both simulation and empirical studies of
channel, the assessment of the correlation and spectral characteristics
of the signal at the receiver input requires consideration of the
receiving antenna pattern. The modeling procedure presented in this
communication, is a new method of mapping the statistical
characteristics of AOR that considers the receiving antenna
parameters and type of propagation environment. The use of this
processing data in simulation studies of wireless system channels
significantly reduce the approximation error of the modeling results
with respect to actual measurements of signal at the output of the
receiving antenna. The developed method gives us the opportunity to
analyze the co-existence between small cells and WB, what is
currently a significant problem in designing 5G networks. The

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 5
significance of this problem is presented in [19]. In this case, the
3GPP [11] model was used, which defines determined spatial
scenarios and considers antenna patterns in a simple filtering
procedure. The use of our method gives the opportunity to consider
the actual environmental conditions that are defined by measurement
PDPs.
REFERENCES
[1] G. L. Stüber, Principles of mobile communication, 3rd ed. New York,
USA: Springer, 2011.
[2] A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model for the
distribution of the angle of arrival and the associated correlation
function and power spectrum at the mobile station,” IEEE Trans. Veh.
Technol., vol. 51, no. 3, pp. 425–434, May 2002.
[3] B. T. Sieskul, C. Kupferschmidt, and T. Kaiser, “Spatial fading
correlation for local scattering: A condition of angular distribution,”
IEEE Trans. Veh. Technol., vol. 60, no. 3, pp. 1271–1278, Mar. 2011.
[4] J. Zhang, C. Pan, F. Pei, G. Liu, and X. Cheng, “Three-dimensional
fading channel models: A survey of elevation angle research,” IEEE
Commun. Mag., vol. 52, no. 6, pp. 218–226, Jun. 2014.
[5] A. Ahmed, S. J. Nawaz, and S. M. Gulfam, “A 3-D propagation model
for emerging land mobile radio cellular environments,” PLoS ONE, vol.
10, no. 8, p. e0132555, Aug. 2015.
[6] S. J. Nawaz, M. Riaz, N. M. Khan, and S. Wyne, “Temporal analysis of
a 3D ellipsoid channel model for the vehicle-to-vehicle communication
environments,” Wirel. Pers. Commun., vol. 82, no. 3, pp. 1337–1350,
Jan. 2015.
[7] A. Y. Olenko, K. T. Wong, S. A. Qasmi, and J. Ahmadi-Shokouh,
“Analytically derived uplink/downlink TOA and 2-D-DOA distributions
with scatterers in a 3-D hemispheroid surrounding the mobile,” IEEE
Trans. Antennas Propag., vol. 54, no. 9, pp. 2446–2454, Sep. 2006.
[8] A. Y. Olenko, K. T. Wong, and S. A. Qasmi, “Distribution of the uplink
multipaths’ arrival delay and azimuth-elevation arrival angle because of
‘bad urban’ scatterers distributed cylindrically above the mobile,”
Trans. Emerg. Telecommun. Technol., vol. 24, no. 2, pp. 113–132, Mar.
2013.
[9] N. M. Khan, M. T. Simsim, and P. B. Rapajic, “A generalized model for
the spatial characteristics of the cellular mobile channel,” IEEE Trans.
Veh. Technol., vol. 57, no. 1, pp. 22–37, Jan. 2008.
[10] L. Jiang and S. Y. Tan, “Simple geometrical-based AOA model for
mobile communication systems,” Electron. Lett., vol. 40, no. 19, pp.
1203–1205, Sep. 2004.
[11] “3GPP TR 38.901 V14.2.0 (2017-09). Study on channel model for
frequencies from 0.5 to 100 GHz (Release 14).” 3rd Generation
Partnership Project (3GPP), Technical Specification Group Radio
Access Network, Valbonne, France, Sep. 2017.
[12] C. Ziółkowski and J. Kelner, “Antenna pattern in three-dimensional
modelling of the arrival angle in simulation studies of wireless
channels,” IET Microw. Antennas Propag., vol. 11, 2017, to be
published.
[13] C. Ziółkowski and J. M. Kelner, “Estimation of the reception angle
distribution based on the power delay spectrum or profile,” Int. J.
Antennas Propag., vol. 2015, p. e936406, Dec. 2015.
[14] C. Ziółkowski, J. M. Kelner, L. Nowosielski, and M. Wnuk, “Modeling
the distribution of the arrival angle based on transmitter antenna
pattern,” in 2017 11th European Conference on Antennas and
Propagation (EuCAP), Paris, France, 2017, pp. 1582–1586.
[15] R. Vaughan and J. Bach Andersen, Channels, propagation and antennas
for mobile communications. London: Institution of Electrical Engineers,
2003.
[16] M. K. Samimi and T. S. Rappaport, “3-D millimeter-wave statistical
channel model for 5G wireless system design,” IEEE Trans. Microw.
Theory Tech., vol. 64, no. 7, pp. 2207–2225, Jul. 2016.
[17] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun,
“Wideband millimeter-wave propagation measurements and channel
models for future wireless communication system design,” IEEE Trans.
Commun., vol. 63, no. 9, pp. 3029–3056, Sep. 2015.
[18] S. Sun, G. R. MacCartney, M. K. Samimi, and T. S. Rappaport,
“Synthesizing omnidirectional antenna patterns, received power and
path loss from directional antennas for 5G millimeter-wave
communications,” in 2015 IEEE Global Communications Conference
(GLOBECOM), San Diego, CA, USA, 2015, pp. 1–7.
[19] S. Kim, E. Visotsky, P. Moorut, K. Bechta, A. Ghosh, and C. Dietrich,
“Coexistence of 5G with the incumbents in the 28 and 70 GHz bands,”
IEEE J. Sel. Areas Commun., vol. 35, no. 6, pp. 1254–1268, Jun. 2017.