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Article

Radio Channel Capacity with Directivity Control of Antenna

Beams in Multipath Propagation Environment

Cezary Ziółkowski 1, Jan M. Kelner 1, * , Jarosław Krygier 1, Aniruddha Chandra 2and Aleš Prokeš 3

Citation: Ziółkowski, C.; Kelner,

J.M.; Krygier, J.; Chandra, A.; Prokeš,

A. Radio Channel Capacity with

Directivity Control of Antenna Beams

in Multipath Propagation

Environment. Sensors 2021,21, 8296.

https://doi.org/10.3390/s21248296

Academic Editor: Peter Han

Joo Chong

Received: 11 November 2021

Accepted: 8 December 2021

Published: 11 December 2021

Publisher’s Note: MDPI stays neutral

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Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

1Institute of Communications Systems, Faculty of Electronics, Military University of Technology,

00908 Warsaw, Poland; cezary.ziolkowski@wat.edu.pl (C.Z.); jaroslaw.krygier@wat.edu.pl (J.K.)

2Department of Electronics and Communication Engineering, National Institute of Technology,

Durgapur 713209, India; aniruddha.chandra@ieee.org

3Department of Radio Electronics, Brno University of Technology, 61600 Brno, Czech Republic;

prokes@vutbr.cz

*Correspondence: jan.kelner@wat.edu.pl; Tel.: +48-261-839-517

Abstract:

The basic technology that will determine the expansion of the technical capabilities of

ﬁfth generation cellular systems is a massive multiple-input-multiple-output. Therefore, assessing

the inﬂuence of the antenna beam orientations on the radio channel capacity is very signiﬁcant.

In this case, the effects of mismatching the antenna beam directions are crucial. In this paper, the

methodology for evaluating changes in the received signal power level due to beam misalignment

for the transmitting and receiving antenna systems is presented. The quantitative assessment of this

issue is presented based on simulation studies carried out for an exemplary propagation scenario.

For non-line-of-sight (NLOS) conditions, it is shown that the optimal selection of the transmitting

and receiving beam directions may ensure an increase in the level of the received signal by several

decibels in relation to the coaxial position of the beams. The developed methodology makes it

possible to analyze changes in the radio channel capacity versus the signal-to-noise ratio and distance

between the transmitter and receiver at optimal and coaxial orientations of antenna beams for various

propagation scenarios, considering NLOS conditions. In the paper, the inﬂuence of the directional

antenna use and their direction choices on the channel capacity versus SNR and the distance between

the transmitter and receiver is shown.

Keywords:

wireless communications; radio propagation; multi-elliptical propagation model; direc-

tional antennas; radio channel capacity; beam misalignment; non-line-of-sight (NLOS) conditions

1. Introduction

The effective increase in the capacity of wireless networks in relation to long term

evolution networks is one of the main goals in the development of ﬁfth generation (5G)

systems. This results from the dynamic increase in the number of users, both people and

devices in relation to the Internet of Things [

1

,

2

], and the growing demand for the number

of provided telecommunications services. On the other hand, network capacity is directly

related to the channel capacity (i.e., spectral efﬁciency) of the individual radio links, which

is directly proportional to the bandwidth of the transmitted signals. Thus, increasing the

capacity of radio channel conditions is necessary for achieving the above-mentioned aim.

Fifth generation systems will also operate in millimeter-wave bands in addition to

the lower frequency ranges of microwaves used so far, i.e., decimeter and centimeter

waves [

2

–

6

]. Generally, a path loss between a transmitter (TX) and receiver (RX) increases

with frequency. The fundamental way to compensate for this increase in attenuation is to

increase the energy gain of the antenna system, which is inversely proportional to the width

of the radiation pattern. For this reason, in 5G systems, especially for millimeter-wave

ranges, directional antennas or multi-antenna systems, including those based on a massive

multiple-input-multiple-output (massive-MIMO) technology, will be used [

6

–

8

]. These

Sensors 2021,21, 8296. https://doi.org/10.3390/s21248296 https://www.mdpi.com/journal/sensors

Sensors 2021,21, 8296 2 of 20

types of antenna systems enable beamforming [

6

,

9

], which makes a spatial multiplexing

technique more effective.

In the transmitting antenna systems, a signiﬁcant part of the energy is radiated in a

speciﬁc direction associated with an antenna power pattern direction, i.e., the direction of

the main lobe beam. To maximize a received signal power, i.e., minimizing the path loss for

the directional link, the direction of the receiving antenna beam should be appropriately

selected. Therefore, obtaining the maximum throughput in the directional wireless link

requires the implementation of an additional procedure that will ensure the optimal

orientation of the antenna beams.

Under line-of-sight (LOS) conditions, the beam directions of the transmitting and

receiving antennas are usually directed to each other to maximize the received signal

strength. However, in some cases, a beam alignment for LOS conditions is not possible.

For example, a base station beam (e.g., as the TX) is oriented in a speciﬁc sector direction,

while a mobile station (e.g., as the RX) moves along a street canyon, which does not match

the transmitting beam direction.

In non-LOS (NLOS) conditions characteristic for urban environments, the effect of a

beam misalignment is more visible and important for radio transmission achievement. In

this case, the orientation of the antenna beams on each other usually does not guarantee

minimizing the path loss. It may result from the occurrence of ﬁeld obstacles, e.g., buildings,

in the TX–RX direction. Therefore, ensuring proper matching of the beam directions of the

transmitting and receiving antennas is necessary to maximize the received signal power. It

will allow obtaining the optimal capacity in the given propagation conditions.

The main aim of this paper is to evaluate the impact of parameters and the optimal

selection of antenna orientation on the radio channel capacity under NLOS conditions.

Minimizing the power losses resulting from the mismatch of the antenna beams and optimal

choice of their direction is the basis for the radio link quality analysis deﬁned by the SNR.

The obtained results have a statistical nature and make it possible to assess the degradation

degree of channel capacity, considering the multipath environment propagation. This is

the basis for the effectiveness evaluation of the procedures for determining the optimal

orientation of both the transmitting and receiving antennas, ensuring maximum capacity

under speciﬁc propagation conditions. To assess changes in the received signal strength

for different antenna beam directions, a geometry-based multi-ellipsoidal propagation

model (MPM) [

10

,

11

] was used. This model is a three-dimensional (3D) version of the

multi-elliptical propagation model which takes into account only the azimuth plane [

11

].

The MPM considers the inﬂuence of the width and directions of beam patterns of the

transmitting and receiving antenna systems. It allows modifying the path loss model [

12

]

or received signal strength [

13

] of the directional link with beam misalignment. As a

result, the impact of the transmission parameters of the environment, antenna patterns,

and their spatial orientation on the channel capacity can be evaluated. This is the basis for

optimizing the antenna beam directions (e.g., in massive-MIMO systems), which ensures

the maximization of wireless link capacity under speciﬁc environmental conditions. Such

an approach to the analyzed problem, which takes into account the inﬂuence of a wide

range of environmental factors and factors related to the technical parameters of antenna

systems, determines the originality of the presented method for the assessment of the

channel capacity. The results presented in this paper refer to a strictly determined research

scenario, which is deﬁned by the mutual position of the objects (i.e., TX and RX), beam

parameters [

14

], and the transmission properties of the propagation environment described

by the power delay proﬁle (PDP) [

15

]. It is worth highlighting that this methodology used

for assessing the channel capacity, which considers the inﬂuence of parameters and patterns

of antenna beams, is universal and can be applied to various propagation environments

and scenarios.

Sensors 2021,21, 8296 3 of 20

2. Related Works

The channel capacity concept and subsequently formulated complete theory of infor-

mation and its transmission were developed by C.E. Shannon [

16

,

17

], based on the earlier

works of H. Nyquist and R. Hartley. Currently, this concept of communication channel ca-

pacity is called the Shannon–Hartley theorem (or Shannon capacity theorem) and is meant

as the theoretical upper bound on the information rate of data that can be transmitted at

an arbitrarily low error rate for the set signal-to-noise ratio (SNR). The analyzed channel

is classiﬁed as an additive white Gaussian noise (AWGN) and memoryless channel. In

the wireless link case, it should refer to the LOS and free space propagation conditions, an

isotropic antenna, so it does not consider the patterns and parameters of real antennas.

Later works deﬁned capacities for channels with non-dispersive fading and then

parallel channels, which also provided an introduction to MIMO channels and spatial

diversity systems for dispersive propagation environments. The ﬁrst works on the ergodic

capacity for the MIMO systems took into account Rayleigh fading and different types of

MIMO channels [

18

], e.g., uncorrelated, spatially correlated, double scattering, and keyhole.

On the other hand, in the literature, some papers focus on the capacity of channels with

Nakagami [19], Rician, Hoyt, or Weibull/log-normal fading [20].

For a few years, two (2D) or 3D geometry-based channel models have been used

to evaluate the channel capacity in multipath propagation environments characterized

by dispersion in time, frequency, and reception angle domains, e.g., [

21

]. This research

direction shows the inﬂuence of the patterns and parameters of antenna systems and the

angular dispersion occurring in the real multipath propagation environment on the channel

capacity determination. Recently, most of the research on capacity has been devoted to 5G

technologies (e.g., [

22

–

26

]), networks, and systems [

2

,

9

,

27

,

28

]. In particular, the analysis

of the signal propagation directions from the TX to the RX is crucial for systems based

on beamforming and massive-MIMO technologies [

25

,

26

]. The use of these technologies

in macro and micro-cells, as well as the creation of smaller, i.e., nano-, pico-, and femto-

cells (i.e., ultra-dense networks [

29

]) with the simultaneous use of spectral resources in

mm-wave bands [

5

,

30

] allows to signiﬁcantly increase (about 10-fold [

29

]) not only the

spectral efﬁciency of individual channels and links, but also the capacity of the entire

network [

2

,

9

,

27

,

28

]. This aspect was accurately summarized in [

2

]: “

. . .

capacity for wireless

communication depends on spectral efﬁciency and bandwidth. It is also related to cell size

. . .

Cell sizes are becoming small and physical layer technology is already at the boundary of Shannon

capacity . . . ”.

Antenna beam misalignment in emerging 5G systems, especially in NLOS conditions,

is a signiﬁcant issue from the viewpoint of effective beamforming and tracking procedures,

which ensure the achievement of the maximum capacity. Numerous papers, i.a., [

31

–

33

],

presenting both the effects and methods of reducing mismatch, testify to the importance

of this topic. The direction mismatch of the transmitting and receiving beams is the

reason for the increase in path loss. This fact is demonstrated in [

31

] by the results of

practically performed measurements. The effect of the increase in attenuation is the loss

of the received signal power, which results in a signiﬁcant decrease in the throughput of

the directional link. Power losses have a signiﬁcant impact on reducing the transmission

data rate. Examples of solutions that minimize the effects of mismatches for MIMO and

hybrid systems, non-orthogonal access systems based on the beamforming technique, are

presented in [32,33].

Hence, it may be seen that the analyzed area ﬁts well with the current research trends.

In the novel relationship of the channel capacity proposed in Section 3, two impact factors

of multipath propagation environment and antenna systems, respectively, were introduced.

These factors ensure the appropriate modiﬁcation of the capacity for the selected single-

channel deﬁned by Shannon [

16

,

17

]. In the case of MIMO channels, the obtained results

should be adequately diversiﬁed. This approach is innovative and original in relation to the

above-presented methods of channel capacity estimation. In this capacity evaluation, the

MPM [

11

] as a geometry-based channel model, which is based on any PDP, and consider

Sensors 2021,21, 8296 4 of 20

the parameters of antenna beams, was used. It allows for the analysis of the inﬂuence of

the angular spread of the received signals on the capacity in time-varying channels. On the

other hand, the MPM was veriﬁed based on empirical results, which provides the basis

for the correctness of the presented analyses. Often, the channel capacity is represented in

an analytical form and as a graph as a function of SNR, the number of antenna elements

in MIMO systems, or environmental parameters (e.g., for different fading distributions).

However, from a practical point of view, it is worth illustrating capacity as a function of the

TX–RX distance at a given SNR for the reference distance, which was done in this work too.

The remainder of this paper is organized as follows. Section 3includes the novel

approach to express the relationship between the channel capacity and the environmental

factors and antenna beam parameters. In Section 4, the MPM description and power

angular spectrum (PAS) estimation based on it is shown in short. Next, in Section 5, the

impact of the antenna beam directions on the received total power under LOS and NLOS

conditions is analyzed. Section 6depicts the inﬂuence of the antenna beam orientation

on the radio channel capacity. The results shown in Sections 5and 6were obtained based

on simulation studies for the selected spatial scenarios using the MPM and MATLAB

environment. Finally, a summary of the paper is contained in Section 7.

3. Capacity and Antenna Beam Parameters

The Shannon-Hartley theorem [

16

,

17

] introduces the fundamental relationship that

describes the relative capacity of the transmission channel,

Cf

, in particular, for the radio link

Cf(bit/s/Hz)=log2(1+SNR), (1)

where

SN R =Pf/Pn

is the ratio of the desired signal power

Pf

to the additive interference

power

Pn

in the AWGN form induced in the omnidirectional antenna. In this paper, the

above relationship is treated as a reference, which describes the capacity of the radio

channel with an omnidirectional antenna system under free-space propagation conditions.

In a multipath propagation environment, especially in NLOS conditions, the level

of the desired signal is signiﬁcantly reduced. It is the cause of the SNR reduction and

consequently of the channel capacity. Based on the Friis transmission equation [

34

], it can

be written

Pf(D)∝1/PL f(D)and Pm(D)∝1/PLm(D),(2)

where

Pf(D)

and

Pm(D)

are the desired signal powers received in free-space and multipath

propagation conditions versus distance

D

between the TX and RX, respectively, while

PL f(D)

and

PLm(D)

are environmental path loss under free-space and multipath propa-

gation conditions, respectively. For free-space propagation, the path loss in LOS conditions

has the form [34]

PL f(D)(dB)=20 log10 (4πD/λ), (3)

where

λ=c/fc

and

fc

are the wavelength and carrier frequency of the transmitted signal,

respectively, and cis the lightspeed.

For multipath propagation environments, the path loss may be represented by mul-

tiple propagation models. As examples of such models, MiWEBA (Millimetre-Wave

Evolution for Back-haul and Access) [

35

], METIS (Mobile and wireless communications

Enablers for the Twenty-twenty Information Society) [

36

], and 3GPP TR (3rd Generation

Partnership Project Technical Report) 38.901 [

15

] can be pointed out. The radio channel

capacity analysis presented in the remainder of the paper is based on the close-in (CI) free-

space reference distance path loss model [

14

], which does not affect the general character

of the proposed approach. The CI path loss model is shown in the following form:

PLm(D)(dB)=PLm(D0)+10PLE log10(D/D0), (4)

where

PLE

means a path loss exponent (PLE) and

D0

is the reference distance (for mm-

wave, usually

D0=

1

m

). In this case, the propagation conditions are deﬁned by the

Sensors 2021,21, 8296 5 of 20

appropriate selection of PLE values, which were determined based on empirical measure-

ments. For example, these coefﬁcient values for millimeter-waves and selected scenarios

are presented in [

14

]. The main drawback of this path loss determination approach for

radio links with narrow-beam antenna patterns is the fact that the measurement data used

for the

PLE

estimation are obtained for strictly determined parameters of the test-bed

antennas.

For the purposes of further analysis, the concept of the environmental factor

Ke(D)

is

introduced. It describes the relationship between the received signal powers in a multi-

path environment

Pm(D)

and in free-space conditions

Pf(D)

, considering omnidirectional

antenna systems. Based on Equation (2), this coefﬁcient can be expressed as

Ke(D)=Pm(D)

Pf(D)=PL f(D)

PLm(D). (5)

Therefore, assuming the same level of environmental interference (i.e., noise), the

channel capacity in the conditions of multipath propagation can be presented in the form

Cm=log2(1+KeSN R). (6)

To consider the inﬂuence of antenna system parameters on the channel capacity, the

antenna system factor is introduced

Ka(D)=Ps(D)

Pm(D), (7)

where

Ps(D)

represents the received signal power in the link with the narrow-beam antenna

system. This factor describes the relationship between the received signal powers in a mul-

tipath propagation environment using narrow-beam antenna systems and omnidirectional

antennas.

Introducing the coefﬁcients

Ke(D)

and

Ka(D)

makes it possible to determine the func-

tional relationship between the signal powers

Ps(D)

and

Pf(D)

received in the links with

narrow-beam and omnidirectional antennas in the multipath and free-space propagation

environments, respectively,

Ps(D)=Ka(D)Ke(D)Pf(D). (8)

Hence, the channel capacity

Cs

considering the multipath propagation environment

and the narrow-beam antenna system can be expressed by the following formula:

Cs=log2(1+KeKaSN R). (9)

Equation (5) shows that the propagation models for free-space and for multipath

conditions are the basis for determining the environmental factor

Ke(D)

. On the other

hand, the evaluation of the average received power is necessary for determining the

antenna system factor,

Ka(D)

. The proposed assessment method of channel capacity uses

the MPM to determine

Ps(D)

in the radio link with the narrow-beam antenna system,

considering the multipath propagation conditions. Thanks to this, the inﬂuence of both the

antenna parameters and the transmission properties of the propagation environment on

the channel capacity can be mapped.

In further analysis, as references, Shannon measures of the channel capacities,

Cf

and

Cd

, deﬁned for free-space conditions, omnidirectional and directional antennas, respec-

tively, are used.

Cf

is deﬁned by Equation (1), whereas based on the Friis transmission [

34

],

Cd

considers the change in gains (in linear measure) of the transmitting

GT

and receiving

GRantennas relative to the omnidirectional antennas

Cd=log2(1+GTGRSN R), (10)

Sensors 2021,21, 8296 6 of 20

where

SN R

is determined for the omnidirectional antennas according to the above deﬁni-

tion. In the case of Cd, the antenna beams are oriented to each other.

4. Multi-Elliptical Propagation Model and Power Angular Spectrum Estimation

The MPM provides the estimation of a PAS,

p(θR,ϕR)

, or probability density function

(PDF),

f(θR,ϕR)

, of angle of arrival (AOA),

(θR,ϕR)

where

θR

and

ϕR

are the angles in the

elevation and azimuth planes, respectively. This model is a 3D geometry-based statistical

approach to modeling the spatial scattering of the received signal [

10

,

11

]. Figure 1shows

the MPM geometry, i.e., potential scattering areas represented by confocal semi-ellipsoids

or ellipses in 3D or 2D model versions, respectively [

11

]. This geometry results from a

PDP deﬁning transmission properties of the analyzed channel. On the other hand, powers

deﬁned in the PDP are the basis for determining the power of each propagation path. This

solution was ﬁrstly used by J.D. Parsons and A.S. Bajwa [

37

], and next by C. Oestges, V.

Erceg, and A.J. Paulraj [

38

]. In the MPM, it is assumed that the received signal is a sum of

delayed components related to the scatterers occurring on the appropriate semi-ellipsoids.

Sensors 2021, 21, 8296 7 of 22

Figure 1. Scattering geometry of MPM.

The geometric structure parameters of the MPM are closely related to the transmis-

sion properties of the propagation environment described by the PDP. In the case of en-

vironments with multipath propagation, the presence of several or a dozen local extremes

of the PDP function can be observed. This means that as a result of scattering on obstacles,

the electromagnetic wave reaches the RX through various propagation paths. This is the

reason why many components of the received signal arrive at the RX with different delays.

In practice, the components derived from the single scatterings determine the received

signal level. Thus, the semi-ellipsoids can be used to map the most likely positions of the

scattering elements. Obviously, the number of ellipsoids is equal to the number of PDP

extremes that come from the components of the received signal that form time-clusters

with similar delay. If the TX–RX distance is equal to ,D then the major xn

a and minor

,

yn

b zn

c half-axes of the nth semi-ellipsoid have the form [39]:

() ()

=+ == +

11

and 2 ,

22

xn n yn zn n n

acτDbccτcτD (11)

where n

τ is the delay of the nth time-cluster. These delays are determined as arguments

of the PDP local extrema. Each of such extremum represents the time-cluster of the reach-

ing propagation paths.

The geometric structure of the MPM has been described in detail in [10,11,39]. The

3D MPM model can be reduced to a 2D multi-elliptical model, in which the propagation

phenomena dominate in the azimuth plane [11]. This modeling approach in relation to

other geometry-based channel models ensures the minimization of the PAS estimation

Figure 1. Scattering geometry of MPM.

The geometric structure parameters of the MPM are closely related to the transmission

properties of the propagation environment described by the PDP. In the case of environ-

ments with multipath propagation, the presence of several or a dozen local extremes of

the PDP function can be observed. This means that as a result of scattering on obstacles,

the electromagnetic wave reaches the RX through various propagation paths. This is the

reason why many components of the received signal arrive at the RX with different delays.

In practice, the components derived from the single scatterings determine the received

signal level. Thus, the semi-ellipsoids can be used to map the most likely positions of the

scattering elements. Obviously, the number of ellipsoids is equal to the number of PDP

extremes that come from the components of the received signal that form time-clusters

Sensors 2021,21, 8296 7 of 20

with similar delay. If the TX–RX distance is equal to

D

, then the major

axn

and minor

byn

,

czn half-axes of the nth semi-ellipsoid have the form [39]:

axn =1

2(cτn+D)and byn =czn =1

2pcτn(cτn+2D),(11)

where

τn

is the delay of the nth time-cluster. These delays are determined as arguments of

the PDP local extrema. Each of such extremum represents the time-cluster of the reaching

propagation paths.

The geometric structure of the MPM has been described in detail in [

10

,

11

,

39

]. The

3D MPM model can be reduced to a 2D multi-elliptical model, in which the propagation

phenomena dominate in the azimuth plane [

11

]. This modeling approach in relation to

other geometry-based channel models ensures the minimization of the PAS estimation

error as shown in [

40

]. The efﬁciency of channel modeling using multi-ellipsoidal geometry

is also shown in [

41

] for the real vehicular-to-infrastructure scenario in the 60 GHz band

described in [42].

Estimation of

p(θR,ϕR)

consists in determining the trajectories of the propagation

paths coming from the TX and reaching the RX. These paths consider the multi-ellipsoidal

geometry of the scatterer positions. As mentioned, the geometric structure of the MPM

maps the potential locations of the scattering elements. Thus, the intersection of the

radiated propagation path with the individual semi-ellipses indicates the positions of the

scattering elements. Based on the angle of departure (AOD),

(θT,ϕT)

, where

θT

and

ϕT

are

the angles in the elevation and azimuth planes, respectively, the radial coordinate of the

scatterer in the spherical system with the origin in the TX can be determined [39]

rT=−1

2ab2

yDsin θTcos ϕT+1

2asb2

yDsin θTcos ϕT2+4ab2

ya2

x−D2

4, (12)

where

a=bysin θTcos ϕT2+a2

xcos2θT+(sin θTsin ϕT)2

,

ax=axn

, and

by=byn

.

Equation (12) is the result of solving the equation system describing the selected semi-

ellipsoid and the propagation path line from the TX for the analyzed AOD, (θT,ϕT)[39].

The coordinate transformation involving the translation of the coordinate system

origin to the RX allows for the AOA determination of individual propagation paths [39]

θR=arctan q(rTsin θTcos ϕT+D)2+(rTsin θTsin ϕT)2

rTcos θT

, (13)

ϕR=arctan rTsin θTsin ϕT

rTsin θTcos ϕT+D. (14)

In the simulation procedure for estimating

p(θR,ϕR)

, the normalized radiation pattern

of the transmitting antennas,

|gT(θT,ϕT)|2

, is used to generate the AODs, (

θT

,

ϕT

). Since

these patterns meet the probability density axioms [

43

], the PDF of AOD can be written

as [39]

fT(θT,ϕT)=1

4π|gT(θT,ϕT)|2sin θT

for θT∈h0, π/2)and ϕT∈h−π,π).(15)

In the MPM, the local scattering phenomenon that occurs in the vicinity of the trans-

mitting and receiving antennas is also taken into account. In this case, the two-dimensional

von Mises distribution is used to describe the AOA statistical properties [11,39]

f0(θR,ϕR)=C0exp(γθcos(π/2−θR))

2πI0(γθ)·exp(γϕcos ϕR)

2πI0(γϕ)

for θR∈h0, π/2)and ϕR∈h−π,π),

(16)

where

γθ

and

γϕ

deﬁne the angular dispersion of the local scattering components in

the elevation and azimuth planes, respectively,

I0(·)

is the zero-order modiﬁed Bessel

Sensors 2021,21, 8296 8 of 20

function of an imaginary argument, and

C0

represents the normalizing constant such that

(C0/2πI0(γθ))

π/2

R0

exp(γθcos(π/2 −θR))dθR=1.

In the simulation procedure, the powers of the received signal components that are

associated with the individual propagation paths are determined from the PDP. To generate

these powers, an exponential distribution whose parameters (i.e., mean values

pn

) are the

local extremes of the PDP is adopted

fp(e

p)=(1/pn)exp(e

p/pn)for e

p≥0,

0 for e

p<0, (17)

where

pn

is the nth local extreme of the PDP that corresponds to the propagation paths

from the nth semi-ellipsoid.

As a result of the simulation, an ordered set of AOAs,

(θR,ϕR)

, and the corresponding

powers

~

p

are obtained. This set is the basis for the estimation the PAS,

pR(θR,ϕR)

in the

vicinity of the receiving antenna [

39

]. To obtain the PAS at the output of the receiving

antenna,

p(θR,ϕR)

, spatial ﬁltering of

pR(θR,ϕR)

using the normalized pattern of the

receiving antenna,

|gR(θR,ϕR)|2

, should be realized [

10

,

11

]. A similar procedure of spatial

ﬁltering is described in [

15

]. A detailed description of the practical implementation of

the estimation procedure can be found in [

11

]. The PAS at the output of the receiving

antenna,

p(θR,ϕR)

, are the basis for determining the received power

Ps(D)

according to

the relationship [11]

Ps=

π

Z

−π

π/2

Z

0

p(θR,ϕR)dθRdϕR=

π

Z

−π

π/2

Z

0

pR(θR,ϕR)|gR(θR,ϕR)|2dθRdϕR. (18)

Equations (7) and (18) show that the calculation of the antenna system factor,

Ka(D)

,

comes down to the determination of p(θR,ϕR).

The above description shows that many factors related to electromagnetic wave prop-

agation, which signiﬁcantly affect the received signal level, are included in the proposed

method of the PAS estimation. The transmission properties of the propagation environment

characterizing the PDP determine the geometrical structure of the MPM and its spatial

parameters. The mapping of the spatial ﬁltration phenomenon by the antenna systems is

realized by the utilization of their normalized radiation/reception patterns in the genera-

tion procedure of AODs, AOAs, and powers of the propagation paths. This approach to the

analyzed problem allows to consider the inﬂuence of antenna parameters (i.e., directions

of maximum radiation/reception, half-power beamwidths (HPBWs), pattern shape) on the

received signal level and, as a result, on the radio channel capacity. This is important in

NLOS conditions especially.

5. Antenna Orientation and Received Power for LOS/NLOS Conditions

The MPM does not directly provide path loss prediction and only gives us the pos-

sibility of assessing the PAS as a normalized function. In practice, many models derived

from statistically averaged measurement data can be used to evaluate the path loss. How-

ever, these models are deﬁned for the beams directed on each other and selected HPBWs,

e.g., [

14

,

44

]. Presented in [

12

,

13

], the MPM-based methodology provides the modiﬁcation

of the path loss and power balance for different HPBWs and orientations of the antenna

beams. A relative power factor,

K

, is its basis. It represents a relative power for the analyzed

beam mismatch and alignment conditions, as follows

K(α,β,D)(dB)=10 log10

Ps(α,β,D)

Ps(α=180◦,β=0◦,D), (19)

Sensors 2021,21, 8296 9 of 20

where

Ps(D)→Ps(α,β,D)

is the received power for the

α

and

β

directions of the trans-

mitting and receiving antenna beams (determined with respect to the OX axe in Figure 1),

respectively, and the selected distance

D

. This power is calculated based on Equation (18)

and the PAS obtained in the MPM.

5.1. Assumptions for Simulation Studies

The evaluation of the power losses resulting from the mismatch of the antenna beams

in the directional link and optimal selection of their orientation especially in NLOS condi-

tions, is based on the simulation tests. Additionally, simulation results for LOS conditions

to verify the simulation procedure correctness and to show the more complex nature of

the propagation phenomenon under NLOS conditions are presented. In the paper, all

presented simulation studies were performed based on the MPM implementation prepared

in the MATLAB environment.

In simulation studies, a spatial scenario as shown in Figure 1was analyzed. The

adopted scenario may suit communications in microcell between the 5G New Radio

gNodeB base station and user equipment operating in the millimeter-wave band. The

following assumptions were considered:

•carrier frequency is equal to fc=28 GHz;

•

PDPs are based on tapped-delay line (TDL) models from the 3GPP TR 38.901 stan-

dard [

15

], i.e., the TDL-B and TDL-D for NLOS and LOS conditions, respectively;

these TDLs are adopted for analyzed

fc

and rms delay spread,

στ

, for so-called the

normal-delay proﬁle and urban macro (UMa) scenario, i.e., στ=266 ns;

•

Rician factor deﬁning the direct path component in the scenario for LOS conditions is

appropriate for TDL-D [15], i.e., κ=13.3 dB;

•

intensity coefﬁcients of the local scattering components, i.e., the 2D von Mises distri-

bution parameters, are equal to γθ=γϕ=60;

•distance between the TX and RX is equal to D=50 m;

•

beam power patterns consider only the main lobe of the antenna systems. These

patterns are modeled by a Gaussian model [

43

] for the appropriate beam parameters,

i.e., HPBWs and gain.

•

HPBWs of the transmitting and receiving antennas are the same in the azimuth and

elevation planes, i.e.,

HPBWTθ,Rθ=HPBWTϕ,Rϕ=

10

◦

based on the real antenna

parameters used in [14,44];

•

gains of the transmitting and receiving antennas are calculated based on the following

formula [45,46]:

GT,R=41253η

HPBWTθ,RθHPBWTϕ,Tϕ

, (20)

where

η=

0.7 is a typical average antenna efﬁciency. By extension, the gains are equal

to GT=GR=24.6 dBi for the transmitting and receiving antennas, respectively.

•

Low heights of the transmitting (7 m) and receiving (1.5 m) antennas are based on

measurement scenarios [14];

•beam alignment is deﬁned for α=180◦and β=0◦(see Figure 1);

•

analyzed ranges of beam directions are as follows: 90

◦≤α≤

270

◦

and

−

90

◦≤β≤

90

◦

;

•steps of changing the antenna directions in simulation studies are ∆α=∆β=1◦;

•

to obtain average statistical results in the MPM,

L=

10 paths are generated at the

TX for each time-cluster (semi-ellipsoid). On the other hand,

M=

360 Monte-Carlo

simulations were run for each analyzed scenario; in this case, the average resolution

of generating the AODs is about 0.1◦.

In accordance with the purpose of simulation tests, the mismatch effects of direc-

tions between the transmitting and receiving beams are presented. As a measure of the

power loss of the received signal, which results from the beam misalignment, the factor

K(α,β,D)→K(α,β)deﬁned by Equation (19) was used.

These studies relied on the Gaussian model [

43

] for the main lobe of the antenna

pattern. However, it should be highlighted that the MPM may consider any pattern

Sensors 2021,21, 8296 10 of 20

shape. For example, in [

47

,

48

], the actual pattern of 5G New Radio gNodeB base station

antenna system based on the massive-MIMO technology was implemented in the MPM

for downlink and uplink inter-beam interference analysis.

5.2. LOS Conditions

First, the effects of beam direction mismatch for LOS conditions (i.e., for TDL-D [15])

are presented. In Figures 2and 3,

K(α,β)

as a function of

α

and

β

directions of the

transmitting and receiving antenna beams is illustrated.

Sensors 2021, 21, 8296 11 of 22

• analyzed ranges of beam directions are as follows: 90 270α°≤ ≤ ° and

90 90 ;

β

−°≤≤ °

• steps of changing the antenna directions in simulation studies are ΔΔ1;αβ==°

• to obtain average statistical results in the MPM, =10L paths are generated at the

TX for each time-cluster (semi-ellipsoid). On the other hand, 360

M

= Monte-

Carlo simulations were run for each analyzed scenario; in this case, the average res-

olution of generating the AODs is about 0.1 .°

In accordance with the purpose of simulation tests, the mismatch effects of directions

between the transmitting and receiving beams are presented. As a measure of the power

loss of the received signal, which results from the beam misalignment, the factor

()()

,

,,KαβDKαβ→ defined by Equation (19) was used.

These studies relied on the Gaussian model [43] for the main lobe of the antenna

pattern. However, it should be highlighted that the MPM may consider any pattern shape.

For example, in [47,48], the actual pattern of 5G New Radio gNodeB base station antenna

system based on the massive-MIMO technology was implemented in the MPM for down-

link and uplink inter-beam interference analysis.

5.2. LOS Conditions

First, the effects of beam direction mismatch for LOS conditions (i.e., for TDL-D [15])

are presented. In Figures 2 and 3,

()

,

Kαβ as a function of α and

β

directions of the

transmitting and receiving antenna beams is illustrated.

Figure 2. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under LOS conditions (3D graph).

Figure 2.

Relative power factor K(

α

,

β

) versus

α

and

β

directions of transmitting and receiving beams

under LOS conditions (3D graph).

Sensors 2021, 21, 8296 12 of 22

Figure 3. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under LOS conditions (2D graph).

These graphs clearly show that when the beams are directed at each other, a domi-

nant received power is obtained. In addition, it may be seen that if the transmitting beam

is not directed at the RX (i.e., 180α≠± °), the occurring power losses can be partially com-

pensated by the optimal selection of the receiving beam direction .

β

The compensation

efficiency of the receiving beam direction is depicted in Figure 3. Additionally, in Figure

4, the power losses that occur while maintaining a constant receiving angle 0

β

=°

(red

dashed line) are shown.

Figure 4. Received signal power losses due to mismatch of transmitting beam direction under LOS

conditions.

In the analyzed LOS conditions, the Rician factor is equal to 13.3 dB. This means that

the first time-cluster in the PDP is dominant. Thus, the direct path (i.e., 00τ=) signifi-

cantly determines the received power for beam misalignment as well, which results from

the Friis equation [34]. Figures 2 and 3 show that the maximum power is obtained for

beam alignment, which is obvious. The direction of the receiving antenna has a decisive

influence on the power level. Despite the direction changes of the transmitting antenna,

the extremum power is ensured when the RX antenna is pointed at the TX.

For individual ,α the graph of the optimal reception angle (i.e., the optimal direc-

tion of the receiving beam), max ,β which ensures the maximization of the received signal

power, i.e.,

Figure 3.

Relative power factor K(

α

,

β

) versus

α

and

β

directions of transmitting and receiving beams

under LOS conditions (2D graph).

These graphs clearly show that when the beams are directed at each other, a dominant

received power is obtained. In addition, it may be seen that if the transmitting beam

is not directed at the RX (i.e.,

α6=±

180

◦

), the occurring power losses can be partially

compensated by the optimal selection of the receiving beam direction

β

. The compensation

efﬁciency of the receiving beam direction is depicted in

Figure 3.

Additionally, in

Figure 4,

the power losses that occur while maintaining a constant receiving angle

β=

0

◦

(red

dashed line) are shown.

Sensors 2021,21, 8296 11 of 20

Sensors 2021, 21, 8296 12 of 22

Figure 3. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under LOS conditions (2D graph).

These graphs clearly show that when the beams are directed at each other, a domi-

nant received power is obtained. In addition, it may be seen that if the transmitting beam

is not directed at the RX (i.e., 180α≠± °), the occurring power losses can be partially com-

pensated by the optimal selection of the receiving beam direction .

β

The compensation

efficiency of the receiving beam direction is depicted in Figure 3. Additionally, in Figure

4, the power losses that occur while maintaining a constant receiving angle 0

β

=°

(red

dashed line) are shown.

Figure 4. Received signal power losses due to mismatch of transmitting beam direction under LOS

conditions.

In the analyzed LOS conditions, the Rician factor is equal to 13.3 dB. This means that

the first time-cluster in the PDP is dominant. Thus, the direct path (i.e., 00τ=) signifi-

cantly determines the received power for beam misalignment as well, which results from

the Friis equation [34]. Figures 2 and 3 show that the maximum power is obtained for

beam alignment, which is obvious. The direction of the receiving antenna has a decisive

influence on the power level. Despite the direction changes of the transmitting antenna,

the extremum power is ensured when the RX antenna is pointed at the TX.

For individual ,α the graph of the optimal reception angle (i.e., the optimal direc-

tion of the receiving beam), max ,β which ensures the maximization of the received signal

power, i.e.,

Figure 4.

Received signal power losses due to mismatch of transmitting beam direction under LOS

conditions.

In the analyzed LOS conditions, the Rician factor is equal to 13.3 dB. This means

that the ﬁrst time-cluster in the PDP is dominant. Thus, the direct path (i.e.,

τ0=

0)

signiﬁcantly determines the received power for beam misalignment as well, which results

from the Friis equation [

34

]. Figures 2and 3show that the maximum power is obtained for

beam alignment, which is obvious. The direction of the receiving antenna has a decisive

inﬂuence on the power level. Despite the direction changes of the transmitting antenna,

the extremum power is ensured when the RX antenna is pointed at the TX.

For individual

α

, the graph of the optimal reception angle (i.e., the optimal direction

of the receiving beam),

βmax

, which ensures the maximization of the received signal power,

i.e.,

Kmax =K(α,βmax)=max

−90◦≤β≤90◦K(α,β), (21)

is presented in Figure 5.

Sensors 2021, 21, 8296 13 of 22

() ()

max max 90 90

,

max , ,

β

KKαβ Kαβ

−°≤≤°

== (21)

is presented in Figure 5.

Figure 5. Optimal reception angle for different transmitting beam direction under LOS conditions.

The charts in Figures 4 and 5 prove the obvious conclusion graphically. For LOS con-

ditions, the optimal direction of the receiving beam is generally constant and equal

max 0.β≅°

Practically, it means that in this case it has no way of compensating the effects

of mismatching the direction of the transmitting beam by proper selection of the receiving

beam direction. This is due to the presence of the delayed components (i.e., time-clustes

for 0

n

τ>) in addition to the dominant direct path occurring under LOS conditions. In

free space conditions, the direct path appears only. In this case, the graph in Figure 5

would have a constant value for max 0.β=°

5.3. NLOS Conditions

Under NLOS conditions (i.e., for TDL-B [15]), the multipath propagation phenome-

non makes it necessary to search for optimal α and

β

directions of the transmitting and

receiving antenna beams, which will ensure the maximization of the received signal level.

Figures 6 and 7 illustrate the mismatch effects between the TX and RX beam directions.

Figure 5. Optimal reception angle for different transmitting beam direction under LOS conditions.

The charts in Figures 4and 5prove the obvious conclusion graphically. For LOS

conditions, the optimal direction of the receiving beam is generally constant and equal

βmax ∼

=

0

◦

. Practically, it means that in this case it has no way of compensating the effects

of mismatching the direction of the transmitting beam by proper selection of the receiving

beam direction. This is due to the presence of the delayed components (i.e., time-clustes

for

τn>

0) in addition to the dominant direct path occurring under LOS conditions. In free

space conditions, the direct path appears only. In this case, the graph in Figure 5would

have a constant value for βmax =0◦.

Sensors 2021,21, 8296 12 of 20

5.3. NLOS Conditions

Under NLOS conditions (i.e., for TDL-B [

15

]), the multipath propagation phenomenon

makes it necessary to search for optimal

α

and

β

directions of the transmitting and receiving

antenna beams, which will ensure the maximization of the received signal level. Figures 6

and 7illustrate the mismatch effects between the TX and RX beam directions.

Sensors 2021, 21, 8296 13 of 22

() ()

max max 90 90

,

max , ,

β

KKαβ Kαβ

−°≤≤°

== (21)

is presented in Figure 5.

Figure 5. Optimal reception angle for different transmitting beam direction under LOS conditions.

The charts in Figures 4 and 5 prove the obvious conclusion graphically. For LOS con-

ditions, the optimal direction of the receiving beam is generally constant and equal

max 0.β≅°

Practically, it means that in this case it has no way of compensating the effects

of mismatching the direction of the transmitting beam by proper selection of the receiving

beam direction. This is due to the presence of the delayed components (i.e., time-clustes

for 0

n

τ>) in addition to the dominant direct path occurring under LOS conditions. In

free space conditions, the direct path appears only. In this case, the graph in Figure 5

would have a constant value for max 0.β=°

5.3. NLOS Conditions

Under NLOS conditions (i.e., for TDL-B [15]), the multipath propagation phenome-

non makes it necessary to search for optimal α and

β

directions of the transmitting and

receiving antenna beams, which will ensure the maximization of the received signal level.

Figures 6 and 7 illustrate the mismatch effects between the TX and RX beam directions.

Figure 6.

Relative power factor K(

α

,

β

) versus

α

and

β

directions of transmitting and receiving beams

under NLOS conditions (3D graph).

Sensors 2021, 21, 8296 14 of 22

Figure 6. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under NLOS conditions (3D graph).

Figure 7. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under NLOS conditions (2D graph).

The graphs presented in Figure 8 show that in NLOS conditions for a limited range

of changes 90α<°

and for assumed parameters of antennas and propagation environ-

ments, a 6 dB increase in the received power can be achieved by optimal selection of the

TX and RX beam angles. This means that under these propagation conditions the mutual

coaxiality of the beams does not provide the highest received signal level.

Figure 8. Compensation effectiveness for beam misalignment under NLOS conditions.

Furthermore, Figure 8 shows that for the analyzed scenario, in the absence of coaxi-

ality of the transmitting beam direction, which exceeds ±15°, the optimal selection of the

receiving beam direction provides the possibility of increasing the received power in re-

lation to the beam alignment. Therefore, in systems using the massive-MIMO and beam-

forming technologies, the optimal selection of the antenna beam directions should be pro-

vided.

The statistical evaluation of an optimal direction of the receiving beam corresponding

to the transmitting beam misalignment is presented in Figure 9.

Figure 7.

Relative power factor K(

α

,

β

) versus

α

and

β

directions of transmitting and receiving beams

under NLOS conditions (2D graph).

The graphs presented in Figure 8show that in NLOS conditions for a limited range of

changes

|α|<

90

◦

and for assumed parameters of antennas and propagation environments,

a 6 dB increase in the received power can be achieved by optimal selection of the TX and

RX beam angles. This means that under these propagation conditions the mutual coaxiality

of the beams does not provide the highest received signal level.

Furthermore, Figure 8shows that for the analyzed scenario, in the absence of coaxiality

of the transmitting beam direction, which exceeds

±

15

◦

, the optimal selection of the

receiving beam direction provides the possibility of increasing the received power in

relation to the beam alignment. Therefore, in systems using the massive-MIMO and

beamforming technologies, the optimal selection of the antenna beam directions should be

provided.

Sensors 2021,21, 8296 13 of 20

Sensors 2021, 21, 8296 14 of 22

Figure 6. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under NLOS conditions (3D graph).

Figure 7. Relative power factor K(α,β) versus α and β directions of transmitting and receiving beams

under NLOS conditions (2D graph).

The graphs presented in Figure 8 show that in NLOS conditions for a limited range

of changes 90α<°

and for assumed parameters of antennas and propagation environ-

ments, a 6 dB increase in the received power can be achieved by optimal selection of the

TX and RX beam angles. This means that under these propagation conditions the mutual

coaxiality of the beams does not provide the highest received signal level.

Figure 8. Compensation effectiveness for beam misalignment under NLOS conditions.

Furthermore, Figure 8 shows that for the analyzed scenario, in the absence of coaxi-

ality of the transmitting beam direction, which exceeds ±15°, the optimal selection of the

receiving beam direction provides the possibility of increasing the received power in re-

lation to the beam alignment. Therefore, in systems using the massive-MIMO and beam-

forming technologies, the optimal selection of the antenna beam directions should be pro-

vided.

The statistical evaluation of an optimal direction of the receiving beam corresponding

to the transmitting beam misalignment is presented in Figure 9.

Figure 8. Compensation effectiveness for beam misalignment under NLOS conditions.

The statistical evaluation of an optimal direction of the receiving beam corresponding

to the transmitting beam misalignment is presented in Figure 9.

Sensors 2021, 21, 8296 15 of 22

Figure 9. Optimal reception angle for different transmitting beam direction under NLOS conditions.

Figures 7 and 8 show that under NLOS conditions, the received signal obtains the

statistically highest power level for the TX beam direction equal to 90 .α=± ° However,

in this case, the beam direction of the receiving antenna to achieve this power level should

be equal max 23 .β≅°

The lack of the direct path (i.e., the Rician factor equal to 0) under NLOS conditions

is the principal cause of the difference in results in relation to those obtained for LOS con-

ditions. For NLOS conditions, the majority of the received power comes from the delayed

components scattered on the semi-ellipsoids. Therefore, the global maximum of the re-

ceived power does not appear for 180α=°

and 0

β

=°

(see Figures 6–8).

6. Antenna Orientation and Radio Channel Capacity

The results presented in Section 5 show that under NLOS propagation conditions, to

obtain the maximum received signal level on the radio link with directional antennas, it

is necessary to determine the optimal orientation of the antenna beams. Of course, in LOS

conditions, this problem does not arise since the direction determined by the TX and RX

positions is the direction of the maximum received signal level. In this section, an assess-

ment of the impact of both environmental transmission properties and antenna orienta-

tions on the channel capacity under NLOS conditions is presented. As reference data, the

Shannon channel capacities f

C and d

C with the omnidirectional and directional an-

tenna systems under free-space propagation conditions, respectively, were assumed.

The assessment of the received signal level, and consequently the SNR, which di-

rectly determines the channel capacity, is based on the MPM use in the simulation test

procedure, considering antennas with narrow radiation/reception beams. These studies

are carried out for the assumptions described in Section 5, taking into account the changes

in the distance .D The transmission properties of the propagation environment are re-

flected in the adopted CI path loss model [14] described by Equation (4).

This influence maps the channel capacity variation with the omnidirectional antenna

systems as a function of the reference SNR, which corresponds to free-space propagation

conditions. In this case, the SNR is directly proportional to the emitted signal level. For

free-space LOS (blue line), multipath LOS (red line), and multipath NLOS (black line)

propagation conditions, the graphs of capacity as an SNR function are shown in Figure

10.

Figure 9.

Optimal reception angle for different transmitting beam direction under NLOS conditions.

Figures 7and 8show that under NLOS conditions, the received signal obtains the

statistically highest power level for the TX beam direction equal to

α=±

90

◦

. However, in

this case, the beam direction of the receiving antenna to achieve this power level should be

equal βmax ∼

=23◦.

The lack of the direct path (i.e., the Rician factor equal to 0) under NLOS conditions

is the principal cause of the difference in results in relation to those obtained for LOS

conditions. For NLOS conditions, the majority of the received power comes from the

delayed components scattered on the semi-ellipsoids. Therefore, the global maximum of

the received power does not appear for α=180◦and β=0◦(see Figures 6–8).

6. Antenna Orientation and Radio Channel Capacity

The results presented in Section 5show that under NLOS propagation conditions,

to obtain the maximum received signal level on the radio link with directional antennas,

it is necessary to determine the optimal orientation of the antenna beams. Of course, in

LOS conditions, this problem does not arise since the direction determined by the TX

and RX positions is the direction of the maximum received signal level. In this section,

an assessment of the impact of both environmental transmission properties and antenna

Sensors 2021,21, 8296 14 of 20

orientations on the channel capacity under NLOS conditions is presented. As reference

data, the Shannon channel capacities

Cf

and

Cd

with the omnidirectional and directional

antenna systems under free-space propagation conditions, respectively, were assumed.

The assessment of the received signal level, and consequently the SNR, which directly

determines the channel capacity, is based on the MPM use in the simulation test procedure,

considering antennas with narrow radiation/reception beams. These studies are carried

out for the assumptions described in Section 5, taking into account the changes in the

distance

D

. The transmission properties of the propagation environment are reﬂected in

the adopted CI path loss model [14] described by Equation (4).

This inﬂuence maps the channel capacity variation with the omnidirectional antenna

systems as a function of the reference SNR, which corresponds to free-space propagation

conditions. In this case, the SNR is directly proportional to the emitted signal level. For

free-space LOS (blue line), multipath LOS (red line), and multipath NLOS (black line)

propagation conditions, the graphs of capacity as an SNR function are shown in Figure 10.

Sensors 2021, 21, 8296 16 of 22

Figure 10. Capacity versus SNR for omnidirectional antenna pattern and different propagation con-

ditions.

The obtained results are a graphical representation of Equations (1) and (6). For mul-

tipath propagation, the e

K coefficient is determined based on Equations (4) and (5),

where LOS and NLOS propagation conditions are determined by the PLE values equal to

2.1PLE = and 3.4,PLE = respectively [14]. These graphs show that the radio channel

capacity under LOS conditions in relation to free-space is only 0.5 bit/s/Hz less. On the

other hand, the NLOS conditions significantly reduce the channel capacity even several

times.

The use of antenna systems with narrow-beam radiation/reception patterns is one

way to minimize the negative effects of multipath propagation under NLOS conditions.

The effects of using spatially selective antenna systems are shown in Figure 11.

Figure 11. Capacity versus SNR for directional antenna pattern and under NLOS conditions.

Figure 10.

Capacity versus SNR for omnidirectional antenna pattern and different propagation

conditions.

The obtained results are a graphical representation of Equations (1) and (6). For

multipath propagation, the

Ke

coefﬁcient is determined based on Equations (4) and (5),

where LOS and NLOS propagation conditions are determined by the PLE values equal to

PLE =

2.1 and

PLE =

3.4, respectively [

14

]. These graphs show that the radio channel

capacity under LOS conditions in relation to free-space is only 0.5 bit/s/Hz less. On the

other hand, the NLOS conditions signiﬁcantly reduce the channel capacity even several

times.

The use of antenna systems with narrow-beam radiation/reception patterns is one

way to minimize the negative effects of multipath propagation under NLOS conditions.

The effects of using spatially selective antenna systems are shown in Figure 11.

For LOS conditions, the alignment of the transmitting and receiving antenna beams

(i.e., beams oriented to each other) provides statistically multiple increases in the radio chan-

nel capacity. Of course, this increase depends on the gains of the antennas. For the analyzed

radio link with the directional antennas whose gains are equal to

GT=GR=24.6 dBi,

the

capacity is higher by 1.5

÷

3.5 bit/s/Hz in relation to the radio link with the omnidi-

rectional antenna systems and free-space propagation conditions. The analysis results

presented in Figure 11 show that the radio channel capacity also depends on the distance

between the TX and RX. The double distance reduction increases the radio channel capacity

by about 1.5 bit/s/Hz in the whole analyzed range of SNR variability.

Sensors 2021,21, 8296 15 of 20

Sensors 2021, 21, 8296 16 of 22

Figure 10. Capacity versus SNR for omnidirectional antenna pattern and different propagation con-

ditions.

The obtained results are a graphical representation of Equations (1) and (6). For mul-

tipath propagation, the e

K coefficient is determined based on Equations (4) and (5),

where LOS and NLOS propagation conditions are determined by the PLE values equal to

2.1PLE = and 3.4,PLE = respectively [14]. These graphs show that the radio channel

capacity under LOS conditions in relation to free-space is only 0.5 bit/s/Hz less. On the

other hand, the NLOS conditions significantly reduce the channel capacity even several

times.

The use of antenna systems with narrow-beam radiation/reception patterns is one

way to minimize the negative effects of multipath propagation under NLOS conditions.

The effects of using spatially selective antenna systems are shown in Figure 11.

Figure 11. Capacity versus SNR for directional antenna pattern and under NLOS conditions.

Figure 11. Capacity versus SNR for directional antenna pattern and under NLOS conditions.

Under NLOS conditions, the alignment of the transmitting and receiving antenna

beams does not provide to achieve the maximum received power. Therefore, under

these propagation conditions, the massive-MIMO system should supply a beam steering

mechanism to the direction of the maximum level of the received signal. The justiﬁcation

for the application of such a solution is illustrated in Figure 12.

Sensors 2021, 21, 8296 17 of 22

For LOS conditions, the alignment of the transmitting and receiving antenna beams

(i.e., beams oriented to each other) provides statistically multiple increases in the radio

channel capacity. Of course, this increase depends on the gains of the antennas. For the

analyzed radio link with the directional antennas whose gains are equal to

24.6 dBi,

TR

GG== the capacity is higher by 1.5 ÷ 3.5 bit/s/Hz in relation to the radio link

with the omnidirectional antenna systems and free-space propagation conditions. The

analysis results presented in Figure 11 show that the radio channel capacity also depends

on the distance between the TX and RX. The double distance reduction increases the radio

channel capacity by about 1.5 bit/s/Hz in the whole analyzed range of SNR variability.

Under NLOS conditions, the alignment of the transmitting and receiving antenna

beams does not provide to achieve the maximum received power. Therefore, under these

propagation conditions, the massive-MIMO system should supply a beam steering mech-

anism to the direction of the maximum level of the received signal. The justification for

the application of such a solution is illustrated in Figure 12.

Figure 12. Capacity versus SNR for βmax under NLOS conditions.

It can see that the use of the direction of the maximum signal level ensures an addi-

tional increase in the capacity by 2 ÷ 3 bit/s/Hz. The increment increases as the TX–RX

distance is greater. For 50 m,D= the selection of the direction of the maximum level en-

sures a statistical increase in the channel capacity by about 2 bit/s/Hz, while for

200 m,D=this increase is reduced to 3 bit/s/Hz.

The channel capacity change versus the distance between the TX and RX, considering

the stability of the emitted power (i.e., its constant value), makes it possible to practically

assess the reception effectiveness for max.β However, changing the TX/RX position makes

it necessary to search for the reception direction of the maximum signal level. Changes in

max

β as a function of D for 90α=°

are shown in Figure 13.

Figure 12. Capacity versus SNR for βmax under NLOS conditions.

Sensors 2021,21, 8296 16 of 20

It can see that the use of the direction of the maximum signal level ensures an addi-

tional increase in the capacity by 2

÷

3 bit/s/Hz. The increment increases as the TX–RX

distance is greater. For

D=

50

m,

the selection of the direction of the maximum level en-

sures a statistical increase in the channel capacity by about 2 bit/s/Hz, while for

D=

200

m,

this increase is reduced to 3 bit/s/Hz.

The channel capacity change versus the distance between the TX and RX, considering

the stability of the emitted power (i.e., its constant value), makes it possible to practically

assess the reception effectiveness for

βmax

. However, changing the TX/RX position makes

it necessary to search for the reception direction of the maximum signal level. Changes in

βmax as a function of Dfor α=90◦are shown in Figure 13.

Sensors 2021, 21, 8296 18 of 22

Figure 13. Statistical value of βmax versus TX–RX distance under NLOS conditions.

The presented result of max

β versus D has a statistical nature because the simula-

tion studies using the MPM are based on the statistical transmission characteristics of the

channel (i.e., TDLs from the 3GPP standard [15]). On the other hand, this statistical nature

arises from averaging the results over several simulation cycles. As can be seen, as the

distance increases, the reception directions for the maximum signal level converge to

max 0.β→°

Determining the radio channel capacity for optimal receiving beam direction makes

it possible to evaluate the system effectiveness for selecting the signal reception direction.

The comparison of the capacity change for the straight (i.e., 180α=°

and 0

β

=°

) and

optimal (i.e., 90α=°

and max

ββ=) directions of the antenna beams is shown in Figure

14.

Figure 14. Channel capacity versus TX–RX distance for straight and optimal directions of antenna

beams under NLOS conditions.

Figure 13. Statistical value of βmax versus TX–RX distance under NLOS conditions.

The presented result of

βmax

versus

D

has a statistical nature because the simulation

studies using the MPM are based on the statistical transmission characteristics of the

channel (i.e., TDLs from the 3GPP standard [

15

]). On the other hand, this statistical nature

arises from averaging the results over several simulation cycles. As can be seen, as the

distance increases, the reception directions for the maximum signal level converge to

βmax →0◦.

Determining the radio channel capacity for optimal receiving beam direction makes it

possible to evaluate the system effectiveness for selecting the signal reception direction.

The comparison of the capacity change for the straight (i.e.,

α=

180

◦

and

β=

0

◦

) and

optimal (i.e.,

α=

90

◦

and

β=βmax

) directions of the antenna beams is shown in Figure 14.

The exemplary graphs are obtained assuming that, at

D=

50

m,

the level of the

received signal provides the

SN R =

20

dB

. It is obvious that as the TX–RX distance

increases, the level of the desired signal decreases. Thus, the radio channel capacity

decreases. In the case of the optimal direction of the antenna beams, a six-fold increase

in the distance causes only about a 2.5-fold reduction in the capacity. On the other hand,

with the increase in the distance, maintaining the straight direction results in a 4.4-fold

decrease in the capacity. This shows that the use of steering and selection of the optimal

beam direction in the antenna system (e.g., massive-MIMO) mitigate the degrading effect

of distance on the radio channel capacity.

Sensors 2021,21, 8296 17 of 20

Sensors 2021, 21, 8296 18 of 22

Figure 13. Statistical value of βmax versus TX–RX distance under NLOS conditions.

The presented result of max

β versus D has a statistical nature because the simula-

tion studies using the MPM are based on the statistical transmission characteristics of the

channel (i.e., TDLs from the 3GPP standard [15]). On the other hand, this statistical nature

arises from averaging the results over several simulation cycles. As can be seen, as the

distance increases, the reception directions for the maximum signal level converge to

max 0.β→°

Determining the radio channel capacity for optimal receiving beam direction makes

it possible to evaluate the system effectiveness for selecting the signal reception direction.

The comparison of the capacity change for the straight (i.e., 180α=°

and 0

β

=°

) and

optimal (i.e., 90α=°

and max

ββ=) directions of the antenna beams is shown in Figure

14.

Figure 14. Channel capacity versus TX–RX distance for straight and optimal directions of antenna

beams under NLOS conditions.

Figure 14.

Channel capacity versus TX–RX distance for straight and optimal directions of antenna

beams under NLOS conditions.

7. Conclusions

This paper focuses on the inﬂuence assessment of the antenna system parameters,

with particular emphasis on their orientation, on the radio channel capacity under NLOS

propagation conditions. The need to take up such topics is related to implementing new

technologies in antenna systems with beamforming and tracking technologies, e.g., massive-

MIMO. The performed evaluation has a statistical nature and is based on simulation studies.

In this case, the MPM was used to map the effects of propagation phenomena. The obtained

results show that under NLOS conditions, it is desirable to use directional antennas as this

provides signiﬁcant compensation for signal attenuation. The effect of this is as follows:

•

a dozen or so times increase in the radio channel capacity compared to the omnidirec-

tional antenna;

•

the direction selection of the maximum received signal level increase by about 2 bit/s/Hz

the channel capacity regardless of the TX–RX distance;

•

the control system for selecting the reception direction of the maximum signal level

increases the capacity of the link, and its efﬁciency increases with increasing distance.

The issues presented in this article are of signiﬁcant practical importance. The pro-

posed procedure provides a quantitative assessment of the efﬁciency of using beam-steering

antenna systems under NLOS conditions. The use of the geometry-based MPM in the

simulation tests presents the possibility to consider not only the parameters and patterns

of the antenna system, but also the type of propagation environment. Thanks to this, the

method of analyzing the capacity of the directional radio links enables the evaluation of

the spatial range of the implementation of complex telecommunication services. This is

important in the process of planning the area covered by base stations. This determines

the originality of the radio channel capacity analysis method described in this paper in

comparison to the methods presented so far in the literature.

In the future, the authors plan to conduct empirical research for selected scenarios

that will allow to verify the approach presented in this paper. Additionally, the authors

also want to compare the impact of utilizing a simpliﬁed antenna pattern (i.e., Gaussian

model for the main lobe) with a real pattern (i.e., considering the side-lobes) on various

Sensors 2021,21, 8296 18 of 20

parameters of the directional radio link, including throughput, interference, energy balance,

and angular spread.

Author Contributions:

Conceptualization, C.Z. and J.M.K.; methodology, C.Z., J.M.K., J.K., A.C.

and A.P.; software, J.M.K. and J.K.; validation, C.Z., J.M.K., J.K, A.C. and A.P.; formal analysis, C.Z.

and J.M.K.; investigation, C.Z., J.M.K., J.K, A.C. and A.P.; resources, J.M.K.; data curation, J.M.K.;

writing—original draft preparation, C.Z., J.M.K., J.K, A.C. and A.P.; writing—review and editing,

C.Z., J.M.K., J.K., A.C. and A.P.; visualization, J.M.K.; supervision, C.Z., J.M.K., J.K., A.C. and A.P.;

project administration, J.M.K., A.C. and A.P.; funding acquisition, J.M.K., A.C. and A.P. All authors

have read and agreed to the published version of the manuscript.

Funding:

This work was developed within a framework of the research grants: grant no. GBMON/

13996/2018/WAT sponsored by the Polish Ministry of Defense, grant no. UGB/22-854/2021/WAT

sponsored by the Military University of Technology, Poland, grant no. CRG/2018/000175 spon-

sored by SERB, DST, Government of India, project no. 17-27068S sponsored by the Czech Science

Foundation, grant no. LO1401 sponsored by the National Sustainability Program, the Czech Republic.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement:

The data presented in this study are available on request from the

corresponding author. The data are not publicly available due to project restrictions.

Acknowledgments:

The authors would like to express their great appreciation to the Sensors jour-

nal Editors and anonymous Reviewers for their valuable suggestions, which have improved the

manuscript quality.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

2D two dimensional

3D three dimensional

3GPP 3rd Generation Partnership Project

5G ﬁfth-generation

AOA angle of arrival

AOD angle of departure

AWGN additive white Gaussian noise

CI close-in free-space reference distance (path loss model)

HPBW half-power beamwidth

LOS line-of-sight

METIS Mobile and wireless communications Enablers for the Twenty-twenty Information Society

MIMO multiple-input-multiple-output

MiWEBA Millimetre-Wave Evolution for Back-haul and Access

MPM multi-elliptical propagation model

NLOS non-line-of-sight

PAS power angular spectrum

PDF probability density function

PDP power delay proﬁle

PL path loss

PLE path loss exponent

RX receiver

SNR signal-to-noise ratio

TDL tapped-delay line

TR technical report

TX transmitter

UMa urban macro

Sensors 2021,21, 8296 19 of 20

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