CMBX12 - repo.pw.edu.pl

Contents

1 Introduction 5

1.1 Challenges for future networks . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Mobile cellular networks . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 State of art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Aims of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Bases 12

3 Antenna arrays and waves propagation 15

3.1 Array concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Array pattern and gain . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Normalization to isotropic antenna . . . . . . . . . . . . . . . . . . . 17

3.5 Two ways of calculating received power . . . . . . . . . . . . . . . . . 18

3.6 Method of image as a set of useful rules . . . . . . . . . . . . . . . . . 19

3.7 Single element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.8 Analyse and implementing fading . . . . . . . . . . . . . . . . . . . . 25

3.9 Playing with parameters of phased arrays . . . . . . . . . . . . . . . . 27

3.9.1 Number of elements impacts on array pattern . . . . . . . . . 27

3.9.2 Influence of distance between elements on gain . . . . . . . . . 28

3.10 Array steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Optimization 30

4.1 Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Numerical algorithm: Nelder - Mead . . . . . . . . . . . . . . 43

4.3.2 Fuzzy logic algorithm . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Aiming at hot spot . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Rating predefined set of phases as method . . . . . . . . . . . 50

4.4.3 Nelder - Mead simulation . . . . . . . . . . . . . . . . . . . . 52

4.4.4 Modified FQLC simulation . . . . . . . . . . . . . . . . . . . . 53

5 Conclusions 55

4

1 Introduction

1.1 Challenges for future networks

Many subscribers all over the world connect to wireless networks. Wireless net-

works became a medium to conduct a conversation between mates and business

partners as well. Demand for connectivity is growing not only by the reason of

conversation but, first and foremost, popularity of mobile internet. The latter en-

hances significantly requirements for mobile wireless networks. Internet services

as social networking, emails, maps, navigation systems etc. are widely used by

customers. They expect to be connected everywhere to network facilities. More

precisely, users demand network coverage enlargement. On the other hand, some of

internet services and websites require carrying big amount of data through mobile

wireless network. Nowadays watching videos, listening music and viewing photos

on the internet become more popular. The amount of this type of subscribers is

increasing at a staggering pace. Enlarging popularity of tablets extremely stirs up

this process. Following graphs, presented by GSMA Company and A.T.Kearney,

show the growing number of subscribers and data traffic [1]:

5

Figure 1: Total number of mobile communications subscribers on the earth

(CAGR - Compound annual growth rate, CIS - Commonwealth of Independent

States - Russian Commonwealth ).

Source:”The Mobile Economy 2013” GSMA, A.T.Kearney by GSMA Wireless

Intelligence

6

Figure 2: Data traffic by application in mobile communication systems on the

earth (PB - Petabytes [1015 bytes],CAGR - Compound annual growth rate, VOIP -

Voice over Internet Protocol, M2M - Machine to machine).

Source:”The Mobile Economy 2013” GSMA, A.T.Kearney by Cisco VNI, 2013, A.

T. Kearney Analysis

Therefore the aim for future networks is to enlarge coverage and improve through-

put. However, these challenges are not effortless achievable. Perhaps, unsophis-

ticated solution occurs to us - improving number of base stations. Nevertheless,

there are some network providers expectations for future mobile networks. Their

purposes include decreasing (with respect to number of subscribers) the operational

expenditures (OPEX) and the capital expenditures (CAPEX) as well. The compa-

nies aim at their main goal - improving their profits. On the other hand, they still

need to attract customers. Hence, network providers have to care about user expe-

rience and this is obligatory for them to serve utilities as subscribers prefer. The

proverb says ”You cannot have your cake and eat it too”, but, in this case, I will

try! Self-organizing networks are expected to join all of this features. Challenges are

preparing enough configurable network and efficient optimization algorithm. Nowa-

days two dimensional antenna arrays are commercially available. Moreover they

can be steered (beam steering) by current phases. They enable more flexibility in

network configuration.

7

Figure 3: Argos two dimensional array for Multi User Multiple-Input

Multiple-Output project. Using similar arrays for beam steering enhances

flexibility of network.

Source: http://argos.rice.edu/

Moreover some optimization methods are available. Broadening idea of two di-

mensional arrays with some optimization methods I will attempt to join network

providers expectations and users wishes. Using this opportunity to enhance network

features, I took up the challenge to optimize throughput and coverage with already

mentioned two dimensional antenna arrays including modelling network and adapt-

ing algorithms. The question is ”if” and ”how much” I can work out the network

enhancement. Research is aimed at detail analysis of antenna arrays as well. It is

required for conscious network modelling.

1.2 Mobile cellular networks

In 1979 first cellular system was finished in Japan (Tokyo) [12]. Being 1G network,

speech transmission was delivered in analogue manner. In 1981 new communi-

cation technology reached Nordic countries in Europe [13]. And then, it spreads

through America, Israel and Australia. In USA omnidirectional antennas were used

in Advanced Mobile Phone System (AMPS) with 40 MHz bandwidth in 800-900

MHz range [12]. In the end of 1980’s 2 generation (2G) networks appeared. 2G

serves speech service as well as data service. Moreover two technological (schedul-

ing) achievements were implemented - TDMA (Time Division Multiple Access) and

CDMA (Code Division Multiple Access). Subsequently engineers prepared 3 gen-

eration (3G) system. One of the main features was standardization. Networks pa-

rameters was standardized globally by 3GPP (3rd Generation Partnership Project).

Furthermore, 3G network serves platform independent services. Network architec-

8

ture was changed from previous 2G solutions. The throughput and capacity en-

hanced. Then the network providers implemented newer solution - 4G network.

The main idea of new solution was all-IP networks. All-IP defines that all net-

work equipment can be reached through IP addresses. Moreover network staff as

RNC (Radio Network Controller) and BSC (Base Station Controller) is distributed

to BTS (Base Transceiver Station, EnodeB in LTE), servers and gateways [12].

Presently researchers work on 5th generation network. The vision present 5G as

”super-efficient”, ”super-fast”, more converged between fibre and wireless network

[14]. Furthermore there is a big impact on millimetre-waves bands for 4th and 5th

generation network [11].

Figure 4: Improvement of data throughput in network generations (1G could not

transfer data - 0 bit/s).

Source: http://www.laserfocusworld.com

1.3 State of art

Because of demand for technical solutions of these expectations, researcher all over

the world aim at the goal of improving network features. Nowadays there are some

approaches in antenna tilts configuration, antenna arrays and optimization as well.

As an example, Vlad-Ioan Bratu and Claes Beckman from Center for Wireless Sys-

tems KTH Royal Institute of Technology in Kista (Sweden) analyse impact of tilt on

load balancing [2]. They present that antenna tilts have an influence on the overall

cell throughput and users bit rates:

9

/articles/print/volume-49/issue-10/features/advances-in-communications-new-fso-provides-reliable-10-gbit-s-and-beyond-backhaul-connections.html

Figure 5: Cumulative Distribution Function that shows the relation of data rate

per Physical Resource Block in hexagonal network (distance between base stations

500 m) with different down-tilt angles.

For more details, see source: ”Base Station Antenna Tilt for Load Balancing” by

Vlad-Ioan Bratu and Claes Beckman [2]

Moreover another research center - Ericsson Research in Goteborg (Sweden)-

aims at the impact of antenna tilt on coverage and capacity with two types of tilt

steering [3]. Fredrik Athley and Martin N. Johansson (Ericsson Research) in their

publication concluded that ”The results also confirm the previously known results

that total tilt has strong impact on both coverage and capacity” [3]. What is more

they present absorbing results of beam steering. Interestingly, there was not a big

difference between electrical and mechanical tilting concerning coverage. However

the choice of method has more significant impact on capacity - electrical tilt is more

appropriate for mean throughput and cell edges optimization, electrical-mechanical

tilt is better for peak rate improvement. Hence, we see that electrical tilting can

play a part in both optimization - that is promising for new research in view of

two dimensional arrays with electric steering. Furthermore some of researchers and

develop institutes as Bell Labs study the potential of using beam-forming in improv-

ing network features [4]. Appropriately steered lobes manage to improve throughput

and spectral efficiency. Beam steering aim at optimize lobes to achieve supreme gain

and trade off with respect to interferences. Nowadays antenna arrays give a huge

freedom of flexibility. There are some opportunities to read about their ability in

publications - as an example in ”Analysis of Beam-Steering and Directive Charac-

teristics of Adaptive Antenna Arrays for Mobile Communications ” by V. Kalnichev

from Samsung Electronics Co. Ltd. in Korea. The most popular type of antenna

array is linear one with uniform spacing. Using two dimensional planar arrays en-

hance flexibility of beam forming. Hence Advanced Smart Antenna Technologies

Research Group in Edinburgh analyses eventuality of using planar monopole an-

10

tenna array in LTE system [6]. The more flexible network is, the more ways of

optimization. ”A Roadmap from UMTS Optimization to LTE Self-Optimization”

[7] from Spain presents a framework for optimization with some KPIs (Key Per-

formance Indicators) description and process specification. Furthermore some more

detailed methods for determinate features optimization have been already investi-

gated. Some of them propose using numeric search algorithms and tilting as ”Joint

Throughput and Coverage Optimization Under Sparse System Knowledge in LTE-A

Networks” [8] presented on conference in Jeju in 2013. Another approach is to use

fuzzy logic - as an example Fuzzy Q-learning Controller in ”Cooperative Fuzzy Q-

Learning for Self-Organized Coverage and Capacity Optimization” [9] by Nasser ul

Islam M. and Andrea Mitschele-Thiel from Ilmenau University of Technology in Ger-

many. Aforesaid fuzzy method rely on antena tilts as well. However, several articles

go further and concentrate on Radio Resource Management [10]. On 18th Annual

IEEE International Symposium on Personal, Indoor and Mobile Radio Communi-

cations Technische Universitat Dresden and Actix GmbH introduce optimization

based on fuzzy logic controller that optimize handover parameter [10]. Nowadays

fuzzy logic that was firstly innovated in 60’s in XX century becomes widely used.

As we seen, it could enhance network features as well.

1.4 Aims of research

According to actual challenges in telecommunication, this research takes a step to-

wards network optimization. I would like to start with preparation of network model.

Providing an system model with two dimensional antenna arrays is a challenge which

makes possible to prepare the algorithms for network enhancement. Hence this re-

search is aim at analysing the planar antenna array as well. The experiment on

planar array model drifts towards the idea of employing real solutions in heteroge-

neous cellular network. Since they may improve the network performance. The goal

of research is to test some algorithms on prepared model. Implementation contains

fuzzy logic and numerical algorithms. They are expected to enhance network pa-

rameters.

To sum up, challenges are:

• Understanding and modelling planar antenna array,

• Preparing hexagonal network model with some planar antenna arrays,

• Implementing numerical and fussy logic algorithms and trying to optimize

modelled network.

11

2 Bases

In the following research two coordinate systems are required. The spherical co-

ordinate system is appropriate for calculating antenna patterns. The Cartesian

coordinate system better describes reality as subscribers distribution, base stations

locations etc. Let’s start with spherical coordinate system that specifies elements in

antenna array:

Figure 6: Description of identical and parallel elements locations in spherical

coordinates that will be used in this book.

Source: John N. Sahalos, ”Orthogonal Methods for Array Synthesis: Theory and

the ORAMA Computer Tool” [15]

A single point in spherical coordinate system is specified by [22]:

• θ ∈< 0, 180 >⇔ θ ∈< 0, π >

• φ ∈< 0, 360) ⇔ φ ∈< 0, 2π)

• r ≡ ρ, hence, r ∈< 0,∞)

Cartesian coordinate system simplifies network elements description. Hence latter

will be used to define network architecture and user locations. Therefore conversion

with appropriate quadrants locations is required (improvement of [23]):

θ =

arctan(

√x2+y2

z)| if z > 0

180− arctan(|√

x2+y2

z|) if z ≤ 0

φ =

arctan( yx) if x < 0 and y ≤ 0

180− arctan(| yx|) if x ≤ 0 and y ≥ 0

180 + arctan(| yx|) if x ≤ 0 and y > 0

360− arctan(| yx|) if x > 0 and y < 0

12

r =√

x2 + y2 + z2

Moreover let’s define antenna patterns as:

Vertical/elevation pattern: θ ∈< 0, 180 >, φ = 90 and θ ∈< 0, 180 >, φ = 270

- zy plane

Azimuth/horizontal pattern: θ = 90, φ ∈< 0, 360) - xy plane.

Furthermore half-wave dipole will be used. The following equation introduces half-

wave dipole pattern [27]

f = ~θcos(π

2cos(θ))

sin(θ)(1)

The figures present the antenna pattern of vertical dipole:

• Elevation antenna pattern (θ = 0 in the direction of current flow):

Figure 7: Source: My own research

13

• Horizontal antenna pattern:

Figure 8: Source: My own research

14

3 Antenna arrays and waves propagation

3.1 Array concepts

Antenna array were developed to enhance antenna features. Using a set of elements

gives an capability to steer beam. Furthermore array shapes influence on array

pattern. Most popular are linear array. They can be steered to broadside (main

beam perpendicular to array) or endfire (main beam in the some direction that array

elements spacing)[17]. Some other parameters have influence on array as well. In

the following research more complex two dimensional antenna array is used. For the

purpose of fully understanding antenna arrays, let’s introduce some basements.

Figure 9: Linear array steering as example.

Source: ” Adaptive Antennas and Phased Arrays for Radar and Communications”

Fenn, Alan J. [18]

3.2 Electric field

Antenna array could be treated like a set of vibrators. Hence the total electric field

can be calculated as the sum of electric fields of single elements [15]:

E(r) =N∑

n=1

En(r) (2)

Furthermore the electric far field of typical element can be expressed as [15]:

E(r) = −jωµe−jβr

4πrfn(θ, φ) (3)

where:

µ - the magnetic permeability of the space,

ω - the angular frequency,

β = 2πλ

- the free space wave number,

15

r - distance from antenna,

fn(θ, φ) - directional characteristics of n-th element electric field. Assuming that

antennas are the same type we have [15]:

fn(θ, φ) = Inf(θ, φ) (4)

where

In - complex excitation for element, f(θ, φ) - single element pattern. Using these

equations we obtain [15]:


4πrf(θ, φ)

N∑

n=1

Inejβrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)] (5)

Moreover we know that current for an element is [18]:

In = I0ejΨn (6)

where

I0 - amplitude [A]

Ψ - phase [rad]

Finally, we achieve two parameters that we can change. However, in our research we

will change only phase offset because of difficulties in changing current amplitude

in real system. What is more a part of obtained total electric field equation is an

Array Factor [15]:

AF (θ, φ) =N∑

n=1

Inejβrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)] =

=N∑

n=1

I0ej(Ψn+βrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)])

(7)

Array factor is an equation that characterize the antenna array. If all elements are

an isotropic antennas, the array factor is an array pattern.

3.3 Array pattern and gain

In any antenna research it is obligatory to find antenna pattern. The antenna pattern

draws a picture of radiation of the set of elements. Consequently, there is an ability

to specify some more parameters as an example received power in user equipment

(it will be discussed in next chapter)). Based on some references ([18] and [15]) we

know that antenna pattern is an multiplication of array factor and single element

pattern. Hence it is defined by [18]:

F (θ, φ) = f(θ, φ)AF (θ, φ) (8)

16

Moreover having antenna pattern, there is an opportunity to calculate radiation

intensity [18] :

U(θ, φ) = |F (θ, φ)|2 (9)

In our research we will need antenna gain. Following equation joins radiation inten-

sity with gain [18]:

G(θ, φ) =4πU(θ, φ)

Pin

(10)

In addition, we can recalculate it into decibels [18]: G[dB] = 10log10(G) .

3.4 Normalization to isotropic antenna

Due to the fact that gain shall be described in relation to isotropic antenna - dBi,

antenna modelling entails gain normalization. As we already know from (9) and

(10) [18]:

G(θ, φ)[linear] =4π|F (θ, φ)|2

Ptx

(11)

Hence, for one isotropic element:

G(θ, φ)[linear] =4π|AF (θ, φ)|2

Ptx

=4π|Iej(Ψ+βrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)])|2

Ptx

(12)

Antenna arrays reinforce main beam, respectively, isotropic element does not inten-

sify signal:

G(θ, φ)[dBi] = 0dBi (13)

G(θ, φ)[linear] = 1 (14)

Thus we achieve:

4π|Iej(Ψ+βrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)]))|2 = Ptx (15)

Let’s assume isotropic element in point (0,0,0), so:

4π|I|2 = Ptx (16)

Hence (remember that I is an amplitude I = |I|):

I =

√

Ptx

4π(17)

As an example if we assume that PTOTALtx = 29dBm = 0.7943282 Watts for all

elements. Hence power of one element has severally :

17

Type of array Power per element Current

2x2 0.1986 [W] 0.1257[A]

3x3 0.0883 [W] 0.0838 [A]

4x4 0.0496 [W] 0.0628[A]

8x8 0.0124 [W] 0.0314 [A]

During these configuration test, antenna model achieved |G[dBi]| < 0.01dBi for all

of items.

3.5 Two ways of calculating received power

Received power may be calculated in two ways. One of approaches is more funda-

mental and employs electric-magnetic waves propagation. Nevertheless, this prop-

agation theory make derivation of formula more complex. R. K. Shevgaonkar in

”Electromagnetic Waves” [19] presents this approach:

Pavgdensity =1

2Re( ~E × ~H) (18)

Additionally, power flow require fulfilling some conditions [19]:

• directions vectors of ~E and ~H ought to be perpendicular,

• they can not be in time quadrature as well

Based on equation (5) and time-variant full version of current formula In =

I0ej(ωt+Ψ) we obtain:


4πrf(θ, φ)

N∑

n=1

ejβrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)]I0ej(ωt+Ψ) (19)

and then:


4πrf(θ, φ)(

N∑

n=1

ejβrn[sin(θ) sin(θn) cos(φ−φn)+cos(θ) cos(θn)]I0ejΨ)ejωt (20)

Furthermore we know that (directions are assumed) [19]:

~E = E0ejωt~x (21)

~H = H0ejωt~y =

E0

ηejωt~y (22)

18

Thus, for uniform plane of wave average power equation is useful [19]:

Pavgdensity =|E0|22

Re(1

η)[W/m2] (23)

Hence, for a specific material:

• In loss-less dielectric medium η is real (η =√

µ

ǫ) - we assumed that air is

loss-less dielectric medium!

Pavgdensity =|E0|22

1

η[W/m2] (24)

• lossy medium: η complex, E and H not in phase.

Pavgdensity =|E0|22

Re(1

η)[W/m2] (25)

• good conductor: σ >> ωǫ, η ≈√

ωµ

2σ+ i

√

ωµ

2σ, the phase angle between E and

H is approximately 45.

Pavgdensity =|E0|22

√

σ

2ωµ[W/m2] (26)

where:

η - intrinsic impedance

σ - conductivity [ Sm

= 1Ωm

]

ǫ - permittivity [ Fm]

µ - permeability [Hm]

Finally, there is an opportunity to calculate received power Prx using area of the

effective antenna aperture [16] (Prx = APavgdensity).

Another approach concentrate on system model features. The latter uses antenna

gains (Gtx - transmiter antenna gain [dBi], Grx - receiver antenna gain [dBi]) and

attenuation model PL [dB] [20]. A link budget formula is availed (based on [20]):

Prx = Ptx +Gtx +Grx − PL (27)

where Prx and Ptx are respectively received and transmitted power. Link budget ap-

proach simplifies modelling and it is widely used in mobile wireless network systems

modelling (as in 3GPP standards [21]) . Hence this model will be used.

3.6 Method of image as a set of useful rules

To address requirements for cellular systems antenna, reflector antenna for base sta-

tion is obligatory. Hence, let’s introduce method that describes influence of reflected

19

surface behind antenna on pattern. In our analysis we assume infinite flat sheet as

an reflected surface (as perfect ground).Method of image is described by several

rules [25]

1. the ground can be replaced by an image of the antenna,

2. image is at the same distance from ground but behind it,

3. current in image has same magnitude but phase is shifted by 180 ,

4. antennas avail same amount of power.

Moreover we know that total electric field is [15] (as in equation (2)):

Etotal(θ, φ, r) =n

∑

i=1

En(θ, φ, r) (28)

where En(θ, φ, r) is a field from n-th element. Hence, we can add the electric field of

element and the electric field of image. Consequently, we obtain a good basement

for understanding and preparing reflector antenna.

3.7 Single element

Demand for system model of heterogeneous cellular system entails necessity of reflec-

tor antenna. Therefore antenna with flat sheet meet the requirements. According

to ”Antennas” by John D. Krause [25] we know that this scenario is identical with

horizontal antenna above ground. Moreover we want to use λ2dipole antenna. Thus

we can use a pattern of horizontal λ2antenna above ground ([25]: p. 466, equation

10). Assumption comprehends that infinite flat sheet is used.

f(θ, α) =cos(π

2cos(φ) cos(α))

√

1− cos2(φ) cos2(α)sin(hr sin(α)) (29)

where

• hr =2πhλ

• α is an angle recalculated from θ - following figure defines it:

20

Figure 10: Scheme of horizontal dipole antenna above ground (at height h) in

coordinates used in previous equation.

Source: John D. Krause, ”Antennas”, TATA McGRAW-HILL Edition, Second

edition, New Delphi 1997 [25]

Because of specification of our system, we expect to have the maximum radi-

ation in the direction of the plane sheet normal. Therefore we suppose (based on

”Antennas” by John D. Krause [25]) distance from antenna to the ground as λ4.

However, we need also to rotate our calculation point by −90 by x-axis. Hence, we

recalculate coordinates to Cartesian coordinates and rotate it by equation based on

[26] :

Y = y cos(Θ) + z sin(Θ) (30)

Z = −y sin(Θ) + z cos(Θ) (31)

where Θ is a rotation angle (counted from y axis), z,y are old coordinates and Z,Y

are new coordinates. Thus we achieve transformation as X = x, Y = −z, Z = y.

Hence, we achieve:

21

Figure 11: Approximated (by points) pattern of half-wave dipole in front of infinite

plane sheet

Source: My own research

Firstly, we rotated it by 90 degrees by y axis (X = −z, Y = −x, Z = y) as well

(because of the fact that we prefer wider lobe in xy plane). However, I will present

in further research that finally we decide to use another rotation by y axis.

We used horizontal dipole and rotations instead of vertical dipole, because vertical

one is not enough particularly described by its equation. According to detailed

description (by equation) we can set our dipole element in any direction.

Going into details, there is a necessity to mention that antenna pattern for finite

sheet is not so perfect. In order to being meticulous, we will discuss effects of finite

sheet. Some physical phenomena affect on pattern. Therefore 3 regions exist [25]:

1. Region 1 : Directed waves and reflected waves arrive.

2. Region 2 : Only direct field (in our model - pattern of half-wave dipole).

3. Region 3 : Only diffracted field.

22

Following schemes present these regions:

Figure 12: 3 regions of finite flat sheet. Last scheme present a modification of flat

sheet edges for reduce this effect.

Source: John D. Krause, ”Antennas”, TATA McGRAW-HILL Edition, Second

edition, New Delphi 1997 [25]

23

Thus we conduct an experiment to fully understand this effects:

• Region 1’st was modelled as equation 29 defines.

• Region 2’nd was implemented by modified horizontal dipole pattern:

John D. Krause [25] presents derivation formula from (1) to coordinates the

same as in (29). However, we ought to take into consideration that there is a

flat sheet in this case. Thus method of image kicks in. The image strengthens

the electric field because of constructive interference. The main lobe reinforces

two times (considering waves superposition). Hence second region pattern

equation is horizontal dipole pattern formula times 12:

f(θ, α) =1

2

cos(π2cos(φ) cos(α))

√

1− cos2(φ) cos2(α)(32)

• Region 3’rd require Geometrical Theory of Diffraction [25]. Electric dipole

could be modeled as Hertzian dipole as well [28]. Thus equations from ”Scat-

tering of dipole field by perfectly conducting disk” apply [29]. In region 3

we assume that rear lobe is 0.1 times main lobe, because of figure 15th from

already mentioned publication [29] for R = 4λ.

The results of the experiment for four different distances between ground and dipole

are shown below:

Figure 13: Elevation and horizontal patterns for half-wave dipole above finite

circular flat sheet (h - height above ground, flat sheet radius is 4λ) with regions

approximation


24

The figure shows that antenna with h equal λ/4 has most directional character-

istic (as it was in [25]). Using method of image, there is not a problem to explain

it - a constructive interference appears. These waves are a superposition of antenna

waves and image waves.

Nevertheless, antenna pattern in third region is not extremely high in relation to re-

gion 1 (especially, when we do not count on antenna plane - θ is not 90). Moreover

region second is only in narrow θ range. In addition, array will be put on one flat

sheet. Hence, flat sheet will be really big considering single element size. Addition-

ally, calculating more than one region takes a lot of time. Hence in our research we

will use only first one. At the same breath, antenna arrays regions analysis could

be very interesting topic for further research.

3.8 Analyse and implementing fading

Getting into basements of antennas, fading should be discussed. Shadow fading

describes the spatial fluctuations of average power density (hence, it introduces

fluctuations of received power as well) of electro-magnetics waves [24]. Reflection,

diffraction and transition through obstacle cause that waves spread and their parts

go through many different paths (multipath propagation). Therefore they have

different delays. This effect could randomly boosts or abates received signal. As

an example, OFDMA (Orthogonal Frequency-Division Multiplexing) scheduling in

LTE was introduced for reducing consequences of multipath propagation. Thus this

technique requires amplifiers with wide range of linearity (more complex amplifiers)

because of probability of signal superposition. Owing to simultaneous changes in

propagation environment (obstacle locations are changing), received power fluctu-

ates in space as well as in time. In the purpose of getting into fading effect, we

analyse it from scratch. Most simplified wave propagation environment include two

antennas and earth:

Figure 14: Transmit and receive antennas above earth.

Source: Sophocles J. Orfanidis, ”Electromagnetic Waves and Antennas” - chapter

19, ECE Department, Rutgers University 2008 [17]

25

Earth can be modelled as imperfect ground. Moreover we can assume parameters

for earth from RECOMMENDATION ITU-R P.527-3 (fig. 1., p. 2) [30]. Method

of image can be used as similar approach to [17]. The only difference is that we

already assumed in image field that it’s phase is shifted by 180.

Etotal = Earray + ρTMEimage (33)

Moreover we can use equation from [17]:

ρTM =

√

n2 − sin2(α)− n2 cos(α)√

n2 − sin2(α) + n2 cos(α)(34)

where

• n2 = ǫr − j η02Πσλ

• ǫr =ǫǫ0

The antenna was half-wave dipole in front of infinite sheet on 15 meters (coordinates

x=0,y=0,z=15) with current in direction of z-axis. After simulation we achieved

following results on 1.5 meter height (user equipment height):

Figure 15: Power density on user height (1.5 meter) in [Wattsm2 ] for single element

antenna (half-wave dipole antenna, λ4in front of the infinite flat sheet).


26

Figure 16: Power density on user height (1.5 meter) in [Wattsm2 ] for single element

antenna above the earth (half-wave dipole antenna, λ4in front of the infinite flat

sheet, earth as imperfect ground with params er = 4, σ = 15 from ITU-R [30] -

”medium dry ground”).


The interferences cause difference. The signal is strengthened in some locations

and weakened in another locations. It seems like random process in space. Addi-

tionally, environment in real world is still changing: obstacles are changing location,

weather is changing etc. Therefore fading is not only random in space but also ran-

dom in time. Hence, many propagation models treat them as random variable. As

an example, ”Millimeter Wave Channel Modeling and Cellular Capacity Evaluation”

[20] propose equation (to use in link budget):

PL(d)[dB] = α + β10log10(d) + ξ, ξ ∼ N(0, σ2) (35)

where α, β, σ are parameters defined by author for predefined frequence.

This model is appropriate for urban environment. We want to put our model in the

city. Hence, we will use it.

3.9 Playing with parameters of phased arrays

3.9.1 Number of elements impacts on array pattern

The antenna array is a set of elements. The number of them is expected to influence

on array pattern. Hence, we test some configuration:

27

Figure 17: Elevation pattern (on the left) and horizontal pattern (on the right) of

antenna array with different amount of elements. Distance between elements is

still the same and equals λ2


The more elements we have, the more directional is antenna. Moreover more

lobes appear on array gain plot. Consequently, there are weakened areas. Looking

for analogy in books, ”Array and Phased Array Antenna Basics” [31] presents that

directivity increase for array in comparison to one element. Furthermore ”Electro-

magnetic Waves and Antennas” [17] figures shows that three elements array has

more lobes than two elements one.

3.9.2 Influence of distance between elements on gain

Elevation and horizontal patterns with different distances between antenna elements

are presented on the following figures:

Figure 18: Elevation patterns of antenna array with different distances between

elements (p). The number of elements is still the same and equals 16.


28

Figure 19: Horizontal patterns of antenna array with different distances between

elements (p). The number of elements is still the same and equals 16.


Distances between elements affect on number of lobes. Distance is defined as

distance between centres of elements. Hence, minimum distance in the direction

of current flow is λ2. The results are comparable to ”Electromagnetic Waves and

Antennas” conclusions for linear array.

3.10 Array steering

Using antenna arrays we expected to steer main lobe and, respectively, power gain.

The expectation is that there will be a ability to focus on hot spot. The way to steer

beam is based on waves interferences. The physical phenomena of waves constructive

or destructive interferences create the lobes. Hence, the rules as in ”Double-split

experiment” kick in [32]

we have equation:

∆s =ax

d(36)

so in our research in elevation plane:

∆λθ =p(henodeb − hue)

y(37)

in horizontal plane:

∆λφ =p|x|y

(38)

where p is a distance between elements. The phase shift should be identical every

two elements in line. Having ∆λθ and ∆λφ = p|x|y

we manage to calculate values of

phase shifts.

Based on identical rules and waves superposition ”Adaptive Antennas and Phased

29

Figure 20: Double-slit experiment

Source: ”Doppelspaltexperiment”, Wikipedia

Arrays for Radar and Communications” [18] introduce (p. 195, equation 8.3):

Ψns = −β sin θs(xn cosφs + yn sinφs)− βzncosθs (39)

The latter equation will be used in our research.

4 Optimization

4.1 Scenario

Scenario assumes 57 cells. The distribution of cell considers hexagonal grid. Distance

between base stations was set to 500 meters. Every base station is equipped by three

antennas. The antennas divides area into three sectors. Every sector is another cell.

The antennas’ locations are 34 meter above ground (earth). The following picture

present the locations of base stations and the direction of antennas. Please notice

”calculation area” and ”target area” that will be used in the following chapters. The

”calculation area” is the area that is precisely calculated (including user equipment

connections to cells). The ”target area” is the area where hot spots will be put.

Additionally, throughput in ”target area” will be calculated.I prepared the following

graph to present the structure of network.

30

Figure 21: Distribution of base stations and antennas in modelled network. Red rectangle - ”target area”, green rectangle - ”calculation

area”. Grid and locations are based on ”A mathematical perspective of self-optimizing wireless networks” by Ingo Viering (Nomor

Research GmbH), Martin Dottling, Andreas Lobinger (Nokia Siemens Networks) [33]. Graph: My own research

31

Moreover, base stations parameters are assumed as:

Cell

ID

Antenna

type

Location

(Y) [m]

Location

(X) [m]

Antenna

height

(Z) [m]

Antenna

direction*

[]

Transmit

power

[dBm]

Down-

tilt

[]

Maximum

antenna

gain

[dBi]

1 3GPP** 0 0 32 0 + 90 46 15 14

2 3GPP** 0 0 32 120+ 90 46 15 14

3 Planar

array

0 0 32 240+ 90 29*** -**** 18

4 3GPP** 500 0 32 0+ 90 46 15 14

5 3GPP** 500 0 32 120+ 90 46 15 14

6 3GPP** 500 0 32 240+ 90 46 15 14

7 3GPP** 250 -433.0 32 0+ 90 46 15 14

8 3GPP** 250 -433.0 32 120+ 90 46 15 14

9 3GPP** 250 -433.0 32 240+ 90 46 15 14

10 3GPP** -250 -433.0 32 0+ 90 46 15 14

11 3GPP** -250 -433.0 32 120+ 90 46 15 14

12 3GPP** -250 -433.0 32 240+ 90 46 15 14

13 Planar

array

-500 0 32 0+ 90 29*** -**** 18

14 3GPP** -500 0 32 120+ 90 46 15 14

15 3GPP** -500 0 32 240+ 90 46 15 14

16 3GPP** -250 433.01 32 0+ 90 46 15 14

17 Planar

array

-250 433.01 32 120+ 90 29*** -**** 18

18 3GPP** -250 433.01 32 240+ 90 46 15 14

19 3GPP** 250 433.01 32 0+ 90 46 15 14

20 3GPP** 250 433.01 32 120+ 90 46 15 14

21 3GPP** 250 433.01 32 240+ 90 46 15 14

22 3GPP** 1000 0 32 0+ 90 46 15 14

23 3GPP** 1000 0 32 120+ 90 46 15 14

24 3GPP** 1000 0 32 240+ 90 46 15 14

25 3GPP** 750 -433.01 32 0+ 90 46 15 14

26 3GPP** 750 -433.01 32 120+ 90 46 15 14

27 3GPP** 750 -433.01 32 240+ 90 46 15 14

28 3GPP** 500 -

866.025

32 0+ 90 46 15 14

29 3GPP** 500 -

866.025

32 120+ 90 46 15 14

30 3GPP** 500 -

866.025

32 240+ 90 46 15 14

31 3GPP** 0 -

866.025

32 0+ 90 46 15 14

32 3GPP** 0 -

866.025

32 120+ 90 46 15 14

33 3GPP** 0 -

866.025

32 240+ 90 46 15 14

34 3GPP** -500 -

866.025

32 0+ 90 46 15 14

35 3GPP** -500 -

866.025

32 120+ 90 46 15 14

36 3GPP** -500 -

866.025

32 240+ 90 46 15 14

37 3GPP** -750 -433.01 32 0+ 90 46 15 14

38 3GPP** -750 -433.01 32 120+ 90 46 15 14

32

Cell

ID

Antenna

type

Location

(Y) [m]

Location

(X) [m]

Antenna

height

(Z) [m]

Antenna

direction*

[]

Transmit

power

[dBm]

Down-

tilt

[]

Maximum

antenna

gain

[dBi]

39 3GPP** -750 -433.01 32 240+ 90 46 15 14

40 3GPP** -1000 0 32 0+ 90 46 15 14

41 3GPP** -1000 0 32 120+ 90 46 15 14

42 3GPP** -1000 0 32 240+ 90 46 15 14

43 3GPP** -750 433.01 32 0+ 90 46 15 14

44 3GPP** -750 433.01 32 120+ 90 46 15 14

45 3GPP** -750 433.01 32 240+ 90 46 15 14

46 3GPP** -500 866.025 32 0+ 90 46 15 14

47 3GPP** -500 866.025 32 120+ 90 46 15 14

48 3GPP** -500 866.025 32 240+ 90 46 15 14

49 3GPP** 0 866.025 32 0+ 90 46 15 14

50 3GPP** 0 866.025 32 120+ 90 46 15 14

51 3GPP** 0 866.025 32 240+ 90 46 15 14

52 3GPP** 500 866.025 32 0+ 90 46 15 14

53 3GPP** 500 866.025 32 120+ 90 46 15 14

54 3GPP** 500 866.025 32 240+ 90 46 15 14

55 3GPP** 750 433.01 32 0+ 90 46 15 14

56 3GPP** 750 433.01 32 120+ 90 46 15 14

57 3GPP** 750 433.01 32 240+ 90 46 15 14

* considering used coordinate system 2 and prepared antenna pattern [details below table]

** antenna pattern for macro-cell (3GPP TR 36.814 V9.0.0 (2010-03), page 59 [34]) [details below table]

*** with normalization showed in 3.4.

**** started with 0 and will be changed.

Going into details, 3GPP antennas is specified by [34]:

• horizontal pattern:

AH(φ) = −min[12(φ

φ3dB

)2, Am] (40)

where φ3dB = 70, Am = 25dB,

• vertical pattern:

AV (θ) = −min[12(θ − θetiltθ3dB

)2, SLAV ] (41)

where θ3dB = 10, SLAV = 20dB

3D pattern is created by equation:

A(φ, θ) = −min[−[AH(φ) + AV (θ), Am] (42)

After implementing it for current research (main beam at 90), following patterns

was achieved:

33

Figure 22: 3GPP pattern used in current research.

Based on: 3rd Generation Partnership Project, Technical Specification Group

Radio Access Network, Evolved Universal Terrestrial Radio Access (E-UTRA),

Further advancements for E-UTRA physical layer aspects. (Release 9) [34]

Figure 23: 3GPP pattern used in current research.

Based on: 3rd Generation Partnership Project, Technical Specification Group

Radio Access Network, Evolved Universal Terrestrial Radio Access (E-UTRA),

Further advancements for E-UTRA physical layer aspects. (Release 9) [34]

Moreover planar array antenna was finally prepared. Maximum gain equals 18

dBi. Having real problem with distances (distance between elements cannot be

smaller than λ2because of antenna size λ

2), enhancement was implemented. Turning

dipoles by 45solves mentioned problem. In addition, there is now a chance to even

add second perpendicular dipole. However, in this research additional antennas will

not be added. Furthermore the channels now are less coherent because of larger

34

distance in z-axis direction [35]. Thus following schemes present prepared planar

antenna array:

Figure 24: Sheme of prepared (modelled) antenna array.


Figure 25: Antenna arrays on base station and user equipments.


35

Getting into details of planar array pattern final version:

Figure 26: Modelled horizontal planar array pattern.


Figure 27: Modelled vertical planar array pattern.


36

Both figures are identical because of single element pattern (half-wave dipole

antenna above infinite sheet, height above ground is λ4, turned by 45):

Figure 28: Pattern of single element rotated by 45


Furthermore, the frequency specification and attenuation model is based on milime-

ter waves (mmW) parameters presented by Sundeep Rangan, Theodore S. Rap-

paport, Elza Erkip in ”Milimeter Wave Cellular Wireless Networks: Potentials and

Challenges” [11]. Since these frequencies are expected in the future networks. Hence

following parameters are used [11]:

Frequency 73 GHz

Bandwidth 1 GHz

Attenuation PL(d)[dB] = α + β10log10(d) + ξ, ξ ∼ N(0, σ2)*

Parameters: α = 86.6, β = 2.45, σ = 8 from [11]

*shadowing will not be used in optimization tests to make results less random.

37

The motivation for 73 GHz in recently mentioned paper is presented on following

figure from [36]:

Figure 29: The attenuation on the sea level considering gigahertz frequencies.

Source: ”State of the Art in 60-GHz Integrated Circuits and Systems for Wireless

Communications”, Theodore S. Rappaport, James N. Murdock, Felix Gutierrez

Moreover assumption of infinite sheet behind antenna array is more realistic because

of frequencies. Since the wave length (λ) is only 0.0041 meter (41 milimeters)

and single antenna has 0.00205 meter length. Flat sheet on base stations can be

significantly bigger, as an example 2x2 meter (0.1292 meter x 0.1292 meter is a

minimum for 64 elements planar array as modelled).

As in real system, user equipments choose cells by received power (calculated by

(27)). What is more, there is necessity to find throughput values for evaluate network

(by ”Shannon’s Capacity Theorem” as in [41]):

C = Wlog2(1 + SNR) (43)

The current research requires SINR rather than SNR. Hence we will use SINR -

C = W ∗ log2(1 + SINR). Since interferences (signals from antennas that user

equipment is not connected to) are assumed as noise. Following SINR is calculated

by:

SINR =Prxc

∑

i 6=c Prxi+N

(44)

where Prxc, Prxi

are received power from base stations, and thermal noise, N, is

recalculated form N = −174 + 10log10(δf) [32] to linear value.

Current research uses calculations that can be called ”pixel” based model as in[37].

It means that every pixel (25 meters x 25 meters area) has density (0.0015 for

38

normal pixel and 131 times 0.0015 for hot spot pixel). Furthermore these ”pixels”

are connected to cell. Implemented calculations of 5 percentile throughput and mean

throughput assume ”pixel” model.

To sum up this chapter, let’s look on the received power distribution and the user

equipment connectivity distribution in ”calculation area” (defined by 21) in modelled

system without planar arrays beam steering:

Figure 30: Received power [dBm] distribution in ”calculation area” with planar

arrays antennas in cell 3,13,17 (defined by 21).


39

Figure 31: User equipment attachment to cells in ”calculation area” with planar

arrays antennas in cell 3,13,17 (defined by 21).


In order to compare, the following pictures present system without planar arrays

(all cells are 3GPP standard antenna):

Figure 32: Received power [dBm] distribution in ”calculation area” (defined by 21)

with 3GPP antennas only.


40

Figure 33: User equipment attachment to cells in ”calculation area” (defined by 25

Source: My own research) with 3GPP antennas only.

4.2 Cost function

In a view of optimization, creating cost function is necessity. This function should

describe the quality of network services. In this research the KPIs (Key Performance

Indicators) are assigned to hot spot throughput and ”target area” coverage. Having

defined priority, the cost function should be aimed at describing this two values.

The aforesaid function should be addicted to the way of optimization. The numerical

minimization methods, as Nelder-Mead method presented in next chapter, aim at

looking for minimum function arguments. Hence, they require cost function that

decrease denote the enhancement of system parameters. Therefore sum of following

cost functions is used in research that uses numerical method (4.3.1). The functions

are inspired by [8].

41

Figure 34: Cost function describes coverage.


Figure 35: Cost function describes 5th percentile of throughput.


As expected, the cost functions decrease with 5th percentile throughput and

coverage increase.

Although, in fuzzy logic optimization algorithms there are states rather than func-

tions. At the same breath, states are connected with network quality. Hence, we

expect to put in them some parameters that have impact on throughput and cover-

age. Thus, 5th percentile of hot spot throughput and mean value of throughput are

used. The details are described in 4.3.2.

42

4.3 Algorithms

4.3.1 Numerical algorithm: Nelder - Mead

The Nelder - Mead algorithm was presented in 1965 in ”The Computer Journal” [38].

These algorithm is a simplex search method. It looks for minimum of function. In

first iteration there is a necessity to create a simplex. N+1 points describe simplex in

N dimensional space. Every simplex is created by starting point and some additional

points. New points are find randomly in this iteration. In every iteration algorithm

counts cost function values and decides which direction should be chosen. The other

decision can be decreasing searching area. Following figure present the actions in

Nelder-Mead method:

Figure 36: Defined actions in Nelder - Mead

Source: http://www.sciencedirect.com [40]

43

Algorithm choose actions by comparing values of function in specific points. The

points and decision are defined in figure 37.

Figure 37: Actions and decisions in Nelder Mead algorithm.

Source: ”A simplex method for function minimization” by J. A. Nelder, R. Mead,

The Computer Journal 7(4)308-313, 1965 [38]

On the latter figure P is a centroid of simplex. ”h” and ”l” indexes mean respectively

high and low values. More details in [38].

4.3.2 Fuzzy logic algorithm

Fuzzy Q-learning Component contains a few parts (as in [9] and [40]):

• states

• actions and strategies

• membership/label function

• rating system

• rules

The states are defines by parameters of network. In my algorithm it is parametrized

by phases, mean value of hot spot throughput and 5th percentile value of hot spot

throughput. States are associated with KPIs (Key Performance Indicators). There-

fore we want to change to appropriate state that enhance network parameters. Ac-

tions and strategies are expected to increase the network rates. During optimization,

algorithm learning which action should be done to achieve a goal. The other neces-

sary part is the membership (label) that gives us the probability that current state

44

is the same as one of predefined (fuzzification). Therefore, we know which set of

actions can be enable. The rating system introduce some quality of changing states

in evaluation. Hence, it describes how good the action was. Thus, we can use the

knowledge in next evaluation. What is more some rules are defined too. They define

the mapping from states to actions (defuzzification). This rules present what are

the options during making the decisions.

My fuzzy logic algorithm is based on Fuzzy Q-Learning Component. This compo-

nent is based on some rules, probability and set of possible actions. My version

of FQLC has two states and 20 operations. However in every state algorithm can

chooses from 12 operations.

Firstly, algorithm identify the state (”label” function). State are defined as (si ∈ S)

:

• State 1: s1=[phases, mean value of throughput, 5th percentile of throughput]

⇔ 5th percentile of throughput is less than 100 times mean value of through-

put,

• State 2: s2=[phases, mean value of throughput, 5th percentile of throughput]

⇔ 5th percentile of throughput is more than or equal 100 times mean value

of throughput,

The set of operations contains (o ∈ O) :

1. Make main lobe wider horizontally and steer to up (state 1, subset A),

2. Make main lobe wider horizontally and steer to the left (state 1, subset A),

3. Make main lobe wider horizontally and steer to the right (state 1, subset A),

4. Make main lobe wider horizontally and steer to the down (state 1, subset A),

5. Make main lobe wider horizontally and do not steer (state 1, subset A),

6. Make main lobe narrower horizontally and steer to up (state 2, subset B),

7. Make main lobe narrower horizontally and steer to the left (state 2, subset B),

8. Make main lobe narrower horizontally and steer to the right (state 2, subset

B),

9. Make main lobe narrower horizontally and steer to down (state 2, subset B),

10. Make main lobe narrower horizontally and do not steer (state 2, subset B),

45

11. Make main lobe wider vertically and steer to up (state 1, subset A),

12. Make main lobe wider vertically and steer to the left (state 1, subset A),

13. Make main lobe wider vertically and steer to the right (state 1, subset A),

14. Make main lobe wider vertically and steer to the down (state 1, subset A),

15. Make main lobe wider vertically and do not steer(state 1, subset A),

16. Make main lobe narrower vertically and steer to up (state 2, subset B),

17. Make main lobe narrower vertically and steer to the left (state 2, subset B),

18. Make main lobe narrower vertically and steer to the right (state 2, subset B),

19. Make main lobe narrower vertically and steer to down (state 2, subset B),

20. Make main lobe narrower vertically and do not steer (state 2, subset B),

The action is defined as chosen operation in according to current state. During the

iteration, algorithm is learning by rating the actions (as in [9]).

46

The following graph presents my algorithm structure:

Figure 38: My algorithm which is based on Fuzzy Q-learning Component [9]

47

Learning process is defined by a few equations [9]:

1. The degree of truth of all activated rules (degree of truth of state equals 1):

V (st+1) =∑

p∈Pst+1

maxk∈Kq(Lp, okp) (45)

Pst+1 is a set of active rules for state st+1. K describe a set of possible actions,

Lp represents state (as label).

2. State Quality:

SQt = Mtphs +Q5tphs(46)

3. New State Quality:

SQt+1 = Mtphs t+1 +Q5tphs t+1(47)

4. Reward:

rt+1 = SQt+1 − SQt; (48)

5. Difference between state qualities:

∆Q = rt+1 + γV −∑

p∈Pst

q(Lp, op) (49)

γ describes the impact of the long period reward. q(Lp, op) express quality of

operation op in state represented by Lp.

6. Finally, update is as follow:

qt+1(Lp, op) = qt(Lp, op) + β∆Q (50)

β express how big influence has new information on knowledge.

Therefore algorithm is still learning. Assuming this research, the learning process

has impact on the robustness and quickness of network enhancement.

4.4 Simulations

4.4.1 Aiming at hot spot

Let’s assume the case where we know were hot spot is. Hot spot is a set of 13 pixels.

The following figure shows a shape of it and its location.

48

Figure 39: Shape and location of hot spot.


Calculating parameters of network for default case (planar arrays are not steered),

we achieved:

Parameter Value

5th percentile of hot spot throughput ≈ 13.458 Mbps

Mean value of hot spot throughput ≈ 20.619 Mbps

5th percentile of ”target area” throughput ≈ 6.161 kbps

Changing array phases has an impact of main lobe direction. Hence, when more

power is aimed at hot spot, hot spot has better throughput:

Parameter Value

5th percentile of hot spot throughput ≈ 21.258 Mbps

Mean value of hot spot throughput ≈ 234.195 Mbps !

5th percentile of ”target area” throughput ≈ 5.236 kbps

49

Figure 40: Power density with lobe aimed at single hot spot.


Figure 41: Power density with lobe not aimed at single hot spot.


4.4.2 Rating predefined set of phases as method

The approach is based on looking in a few directions and decide which one is the best.

Moreover the main lobe can be a superposition of two half-power lobes. The phases

are calculated by (39) for both. Next, it is assumed that (considering half-power

assumption):

ejΨ =1

2ejΨ1 +

1

2ejΨ2 (51)

50

Hence, there is a possibility to find Ψ values. Ψ1 and Ψ2 are calculated from the set

of direction:

φ1, θ1, φ2, θ2:

• φ1 ∈ 15180

π, 165180

π

• and φ2 ∈ 15180

π, 165180

π

• and θ1 ∈ 110180

π, 165180

π

• and θ2 ∈ 110180

π, 165180

π.

Every configuration is tested for every planar array. However we change only set of

phases in one planar array at the same time. Therefore there are 30 configurations

for 3 planar arrays. The cost function as in 4.2 for numerical algorithm is used to

rate network configuration. Thus, the results are (one hot spot as in 4.4.1):

Figure 42: Improvement of 5th percentile throughput through number of iterations.


51

Figure 43: Improvement of mean value of throughput through number of iterations.


What is more we see that in this approach in every situation all user were covered.

These results will be discussed in chapter ”Conclusions”.

4.4.3 Nelder - Mead simulation

The Nelder - Mead method looks for optimum by searching through all dimension.

The dimensions contain the set of angles. In this field it is similar approach as in

recent chapter. There are sets of 4 angles for every planar array and the beam is

created by considering two beams. Hence twelve dimensions kick in. Furthermore

some modifications are obligatory. The antenna pattern consider some constraints:

0 < φ < π and 0 < θ < π (it is made by absolute value of modulo function in system

model). In addition, the users are on the ground. Therefore antenna is aimed at the

ground by dividing it by 2 and adding π2. Finally, it produce following results (one

hot spot in ”target area” as in 4.4.1, parameters: α = 2/3, β = 1/4, γ = 4):

52



Results are not perfect. However, hole ”target area” is still covered. The Nelder-

Mead algorithm need to have specified some parameters which are response for step

length: α, β, γ. Perhaps, there is a possibility to enhance. Therefore I tried:

• α = 2/3, β = 1/2, γ = 4

• α = 1/2, β = 1/2, γ = 4

• α = 1, β = 1/2, γ = 4

• α = 1, β = 1, γ = 20

Unfortunately, changing parameters did not improve results. It will be further

summed up in ”Conclusions”.

4.4.4 Modified FQLC simulation

Fussy control in many fields gives good result. Prepared algorithm was checked on

my system model. The following results was achived:

53





This approach enhances network parameters. What is more ”target area” was

covered. Furthermore is only one approach in this research that use not only 5th

percentile of throughput but also the mean value. The conclusion will be presented

in the following section.

54

5 Conclusions

The antenna arrays give us a big flexibility. Many shapes and patterns can be

used. Additionally, we can achieve expected pattern considering some parameters.

However the modelling of antenna array is really difficult. The physical phenomena

as diffraction, reflection and transition influence on wave propagation. Therefore

the sheet and ground has an impact on power propagation. The effects are even

stronger in network modelling. Since larger distances between user equipment and

base station are assumed. More direction characteristic of prepared planar antenna

could be a solution especially with appropriate steering. My research propose three

strategies of steering. The first approach is based on predefined set. Hence, there

is a possibility to improve received gain in predefined cell area. When there is a

hot spot, the results make good impression. We delight the enhancement of mean

throughput in this approach. The improvement of 5th percentile of throughput

satisfy as well. The computational complexity is not as complicated as in other

approaches. The Nelder-Mead method disappoints. The outcomes does not satisfy

completely. I expected this approach to be really efficient. However, algorithm did

not enhance network (the improvement is approximately zero). The main reason

seems to appear in antenna pattern. The pattern has many lobes. Therefore many

local minimums come out. Furthermore there is an impression that algorithm makes

”shrink” operation and lose better solutions. It even surprises more because in an-

other research [8] similarly prepared algorithm was one of the best. Thus I assume

that this algorithm is not appropriate for antenna planar array systems becaues of

antenna pattern. The last approach was the one that I expected to be suitable for

the future cellular networks. The fuzzy logic approach can learn. Hence, the im-

provement shall appear and it is. The enhancement is not spectacular but the mean

value increase about 5%. The 5th percentile of throughput grows as well. However,

we need to consider that the learning is based on the sum of these two values and

one of them can decrease (as we saw in the pictures).

Looking at research challenges 1.4 I analyse what was achieved. The network was

modelled. Some additional experiments show us the planar array capabilities. Fi-

nally,two from the set of three algorithms achieved improvement of network. Thus,

considering this research, there is a big possibility to join subscribers and network

operations wishes. The growing number of user equipments can be covered. What

is more some algorithms and network configuration could provide high throughput

as well.

55

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59

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