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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

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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.

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