ms thesis _bilal hasan qureshi
TRANSCRIPT
i
Spatio-Temporal Characteristics of Cellular Mobile
Channel using Directional Antennas
MS (Electronic Engineering)
Thesis Submitted by: Mr. Bilal Hasan Qureshi
MT 081008
Thesis submitted to the Department of Electronic Engineering in partial fulfillment of requirements for the Degree MS (Electronic Engineering)
December 2009
Department of Electronic Engineering
Mohammad Ali Jinnah University Islamabad
ii
Thesis Declaration December 2009
Certified that the work contained in this thesis entitled
Spatio-Temporal Characteristics of Cellular Mobile Channel using Directional Antennas
is totally my own work and no portion of the work referred in this thesis has been
submitted in support of an application for another degree or qualification of this or any
other institute of learning.
Bilal Hasan Qureshi
2009 by Mr. Bilal Hasan Qureshi. All rights reserved.
iii
Certificate of Approval December 2009
Thesis Submitted by: Mr. Bilal Hasan Qureshi
MT 081008
Certified that the work contained in this thesis entitled
Spatio-Temporal Characteristics of Cellular Mobile Channel using Directional Antennas
was carried out under my supervision and that in my opinion, it is fully Adequate, in scope and quality, for the degree of MS (Electronic Engineering)
Thesis Supervisor: _________________________________
Dr. Noor M Khan Associate Professor
Department of Electronic Engineering Mohammad Ali Jinnah University
Examiner 1: ___________________________________
Dr. Syed Isamil Shah
Professor and Associate Dean Department of Computing and Technology
Iqra University, Islamabad Campus Examiner 2: ___________________________________
Dr. Muhammad Mansoor Ahmed Professor and Executive Vice President (EVP)
Mohammad Ali Jinnah University
Dean of Faculty: __________________________________
Dr. Muhammad Abdul Qadir Professor and Dean Faculty of Engineering and Applied Sciences
Mohammad Ali Jinnah University
iv
Acknowledgments
All Thanks to ALMIGHTY ALLAH, the most gracious and the beneficent, who
always loves and cares us the most. May ALLAH Bestows Hazart Muhammad
(PBUH) with all His Blessings Who are ideal for all of us. First of all, I am thankful to
my supervisor Dr Noor M Khan Associate Professor (Department of Electronic
Engineering, Muhammad Ali Jinnah University Islamabad) who help and guide me in
completing this Thesis. He supervises and took keen interest in all the matters related
to my MS Thesis. I am grateful to another person Mr Syed Junaid Nawaz Assistant
Professor (Department of Electrical Engineering, Federal Urdu University of Arts
Science and Technology Islamabad) for immediate help and support in completing my
Thesis. I am also thankful to my all Family members including my Parents, my Brother
and my Sister for their timely help and moral support. Here I would like to mention the
name of the person to which i always remember in my life. He is my colleague
Mr Saeed Iqbal Wattoo Lecturer (Hamdard University, Islamabad Campus). I am
very thankful to him for providing me good company and encouragement during my
MS degree.
Last but not least, I would like to recall my all previous teachers who encourage me
during my academics. There is no way, no words, to express my love and gratitude for
them.
Bilal Hasan Qureshi
v
List of Publications
[1] Bilal Hasan Qureshi, Saeed Iqbal and Noor M. Khan “Effect of Directional Antennas at Both Ends of theLink on Spatial Characteristics of Cellular and Mobile Channel” 5th International IEEE Conference on Emerging Technologies, ICET, pp. 89-95, October 2009.
[2] Saeed Iqbal, Bilal Hasan Qureshi and Noor M. Khan “Effect of Directional
Antennas at Both Ends of the Link on Doppler Power Spectrum” 13th International IEEE Multitopic Conference, INMIC, pp. 388-391, December 2009.
[3] Syed Junaid Nawaz, Bilal Hasan Qureshi and Noor M. Khan “Angle of Arrival Statistics for 3-D Macrocell Environment using Directional Antenna at BS” 13th International IEEE Multitopic Conference, INMIC, pp. 210-214, December 2009.
[4] Syed Junaid Nawaz, Bilal Hasan Qureshi and Noor M. Khan “A Generalized 3-
D Scattering Model for Macrocell Environment with Directional Antenna at BS” Submitted to IEEE Trans. Vehicular Technol., Paper ID, VT-2009-01414.
[5] Syed Junaid Nawaz, Bilal Hasan Qureshi and Noor M. Khan “3D Spatial Characteristics of Macrocell Mobile Environment using Directional Antenna at BS” Submitted to 2nd International IEEE Conference on Future Computer and Communication (ICFCC-2010).
[6] Syed Junaid Nawaz, Bilal Hasan Qureshi and Noor M. Khan “Time of Arrival Statistics for 3D Macrocell Environment with Directional Antenna at Base Station” Submitted to 2nd International IEEE Conference on Future Computer and Communication (ICFCC- 2010).
vi
Abstract
This thesis presents the spatial and temporal characteristics of cellular mobile channel for
macrocell mobile environment using directional antennas. The directional antennas are used at both
ends of the link in 2D scattering model. Closed form expressions for the PDF of angle of arrival
(AoA) of multipath waves seen at BS and MS are found analytically with the assumption of uniform
and Gaussian scatterers around MS respectively. The behavior of the PDF of AoA seen at BS and MS
is observed and plotted by changing the separation between BS and MS and in the case of Gaussian
scatter density, the effect of varying the standard deviation is shown on the PDF of AoA at BS and
MS.
A 3D scattering model is also presented for the macrocell environment with MS located at the
center of a 3D scattering hemispheroid and a BS employing directional antenna located outside the
semispheroid. Closed form expressions for the joint and marginal PDF of AoA seen at MS and BS
both in azimuth and elevation planes are derived. Furthermore, closed form expressions for
propagation path delays and joint and marginal PDFs of Time of Arrival in correspondence with
azimuth and elevation angles are derived. The proposed 3D model is shown to deduce all previous
models that assume uniform distributions of scatterers around MS found in literature for macrocell
environment. It is shown that when the beamwidth of the directional antenna at BS is set to include
the whole scattering region of semispheroid, the spatial statistics are found to be the same as those
found in 3D model by Janaswamy. In a similar way, all 2D models that assume uniform distributions
of scatterers, whether directional or omnidirectional found in literature for macrocell environment can
be deduced from proposed 3D model by substituting elevation angle equal to zero. Finally, theoretical
results are compared with some notable 2D and 3D scattering model found in literature to validate the
generalization of the proposed 3D model. The derived spatial characteristics can be used to find the
second order statistics like level crossing rates (LCR), average fade durations (AFD) and spatial
correlations as well as the Doppler power spectrum of the mobile channel. These statistics help in the
design of high performance communication systems to achieve high data rates over fast fading time
varying channels.
vii
Table of Contents
Acknowledgments.................................................................................................................................. iv
List of Publications ................................................................................................................................. v
Abstract .................................................................................................................................................. vi
List of Figures ........................................................................................................................................ ix
List of Acronyms .................................................................................................................................... x
List of Notations ................................................................................................................................... xii
Chapter No. 1 .......................................................................................................................................... 1
Introduction to Spatial Characteristics of Cellular Mobile Channel ....................................................... 1
1.1 Overview ..................................................................................................................................................... 1
1.2 Capacity Demands in wireless Communications Systems .......................................................................... 1
1.3 Multipath Propagation ................................................................................................................................ 3
1.3.1 Two Dimensional Scattering Model ........................................................................................................ 3
1.3.2 Three Dimensional Scattering Model ...................................................................................................... 4
1.4 Problem Formulation .................................................................................................................................. 4
1.5 Methodology ............................................................................................................................................... 5
1.6 Organization of the Thesis .......................................................................................................................... 5
Chapter No. 2 .......................................................................................................................................... 6
Spatial Characteristics using Directional Antennas in 2D Scattering Model ......................................... 6
2.1 Introduction ................................................................................................................................................. 6
2.2 System Model for Directional Antennas at Both Ends of the Link ............................................................ 6
2.3 PDF of AoA using Uniform Scatter Density .............................................................................................. 7
2.3.1 PDF at MS ............................................................................................................................................ 7
2.3.2 PDF at BS ............................................................................................................................................ 8
2.4 PDF of AoA using Gaussian Scatter Density ............................................................................................. 9
2.4.1 PDF at MS ............................................................................................................................................ 9
2.4.1 PDF at BS .......................................................................................................................................... 11
2.5 Results and Descriptions ........................................................................................................................... 12
2.6 Conclusions ............................................................................................................................................... 15
Chapter No. 3 ........................................................................................................................................ 16
viii
Spatial Characteristics using Directional Antenna in 3D Scattering Model ......................................... 16
3.1 Introduction ............................................................................................................................................... 16
3.2 Directional Antenna in 3D Scattering Environment ................................................................................. 17
3.3 Angle of Arrival Statistics at MS .............................................................................................................. 23
3.3.1 Analytical Results of PDF at MS ....................................................................................................... 24
3.4 Angle of Arrival Statistics at BS ............................................................................................................... 27
3.4.1 Analytical Results of PDF at BS ........................................................................................................ 30
3.5 Conclusions ............................................................................................................................................... 32
Chapter No. 4 ........................................................................................................................................ 33
Time of Arrival for 3D Scattering Model ............................................................................................. 33
4.1 Introduction ............................................................................................................................................... 33
4.2 System Model for Time of Arrival Characteristics ................................................................................... 34
4.3 PDF of Time of Arrival using Directional Antenna .................................................................................. 36
4.4 Analytical Results ..................................................................................................................................... 38
4.5 Conclusions ............................................................................................................................................... 41
Chapter No. 5 ........................................................................................................................................ 42
Conclusions and Future Work .............................................................................................................. 42
5.1 Summary of the Thesis ............................................................................................................................. 42
5.2 Future Work .............................................................................................................................................. 43
5.2.1 Research Plan 1 .................................................................................................................................. 44
5.2.2 Research Plan 2 .................................................................................................................................. 44
5.2.3 Research Plan 3 .................................................................................................................................. 44
Bibliography ......................................................................................................................................... 45
Appendix A ........................................................................................................................................... 47
ix
List of Figures
Figure1.1 : A Typical Macrocell Mobile Environment ........................................................................................ 2 Figure1.2 : Circular Scattering Environment ....................................................................................................... 3 Figure1.3 : A typical 3D Scattering Model .......................................................................................................... 4 Figure 2.1 : System Model for uniform scatter density using directional antennas ............................................. 7 Figure 2.2 : Gaussian scatter density using directional antennas ....................................................................... 10 Figure 2.3 : PDF of AoA at MS assuming uniform scatter density ................................................................... 13 Figure 2.4 : PDF of AoA at BS assuming uniform scatters density .................................................................. 14 Figure 2.5 : PDF of AoA at MS assuming Gaussian scatters density ................................................................ 14 Figure 2.6 : PDF of AoA at BS assuming Gaussian scatters density ................................................................. 15 Figure 3.1 : A typical 3D Scattering Model ....................................................................................................... 17 Figure 3.2 : Geometry for volume of the illuminated region ............................................................................. 18 Figure 3.3 : Azimuth and elevation views of System Model ............................................................................ 19 Figure 3.4 : The threshold angle 1threshφ and 2threshφ as a function of elevation angles ...................................... 20
Figure 3.5 : Geometry for distance A of the scatterer form mobile station and the angle βthresh ........................ 21 Figure 3.6 : The threshold angle βthresh as a function of azimuth angles ............................................................. 22 Figure 3.7 : The Distance A of the scatter from MS .......................................................................................... 22 Figure 3.8 : Different elevation views of the Distance A of the scatter from MS .............................................. 23 Figure 3.9 : The joint PDF of AoA at MS .......................................................................................................... 25 Figure 3.10 : The PDF of AoA in elevation plane for different azimuth angles ................................................ 25 Figure 3.11 : The PDF of AoA in azimuth plane for different elevation angles ................................................ 26 Figure 3.12 : 3D PDF of AoA for zero elevation plane is compared with 2D [Petrus et al] ............................. 26 Figure 3.13 : 3D proposed model with & without directional antenna .............................................................. 26 Figure 3.14 : System Model for PDF of AoA at BS .......................................................................................... 28 Figure 3.15 : Elavation view of system model ................................................................................................... 29 Figure 3.16 : Marginal PDF of AoA in Azimuth plane seen at BS .................................................................... 31 Figure 3.17 : Marginal PDF of AoA in Elevation Plane .................................................................................... 31 Figure 3.18 : Marginal PDF of AoA in Elevation plane seen at BS (Ht = b) ..................................................... 32 Figure 4.1 : System Model for Time of Arrival ................................................................................................. 34 Figure 4.2 : The joint PDF of ToA in azimuth plane for α > αmax (Numerically integrated) ............................. 38 Figure 4.3 : The joint PDF of ToA in azimuth plane α = 2o (Numerically integrated) ...................................... 39 Figure 4.4 : The joint PDF of ToA in elevation plane for α = 2o ........................................................................ 39 Figure 4.5 : The marginal PDF of ToA for α ≥ αmax ............................................................................................ 39 Figure 4.6 : The joint propagation path delay in azimuth and elevation angle for α ≥ αmax ................................ 40 Figure 4.7 : The joint propagation path delay in azimuth and elevation angle for α = 4o ................................... 40 Figure 4.8 : The effect of directional antenna on propagation path delay in azimuth plane ............................... 40 Figure 4.9 : The effect of directional antenna on marginal function of path delay in elevation plane ................ 41
x
List of Acronyms
1 G First Generation of Land Mobile Systems 2 G Second Generation of Land Mobile Systems 3 G Third Generation of Land Mobile Systems 2D Two Dimensional (Scattering Model) 3D Three Dimensional (Scattering Model) AFD Average Fade Duration AoA Angle of Arrival AMPS Advanced Mobile Phone Services BLAST Bell Laboratories Layered Space Time BS Base Station CDF Commutative Density Function CDMA Code Division Multiple access DoA Direction of Arrival DoD Direction of Departure GBSBM Geometrical-based Single Bounce Macrocell Model GSM Global System for Mobile Communication ISI Inter Symbol Interference LCR Level Crossing Rate LOS Line of Sight MS Mobile Station
xi
OFDM Orthogonal Frequency Division Multiple Access PDF Probability Density Function TDMA Time Division Multiple Access ToA Time of Arrival UMTS Universal Mobile Telecommunication Systems V-BLAST Vertical Bell Laboratories Layered Space Time WCDMA Wideband Code Division Multiple Access
xii
List of Notations
Α Beamwidth of directional antenna used at base station in 2D & 3D scattering model
Β Beamwidth of directional antenna used at mobile station in 2D
scattering model
1threshφ & 2threshφ The angles in azimuth plane as a function of beamwidth of directional antenna used at BS
βb The elevation angle at BS in 3D scattering model βm The elevation angle at MS in 3D scattering model βthresh The threshold angle in elevation plane in 3D scattering model
bφ The azimuth angle at BS in 3D scattering model
mφ The azimuth angle at MS in 3D scattering model
βlim The angle that separate the region of spheroid having no effect of directional antenna
βmin The minimum angle in elevation plane in 3D scattering model βmax The maximum angle in elevation plane in 3D scattering model Ab_uniform Area of the illuminated scatterers at BS for uniform scatter
density D Distance between BS and MS Am_Gaussian Area of the illuminated scatterers at MS for Gaussian scatter
density Ab_Gaussian Area of the illuminated scatterers at BS for Gaussian scatter
density σ Standard Deviation of the Gaussian scatter density
xiii
( )θθF CDF of Angle of Arrival
( )θθf PDF of Angle of Arrival
r′ The effective strength of the radius of the scattering circle in
Gaussian scatters density VSpheroid Volume of the Spheroid
cφ The boundary angle which separate the clipped portion form the spheroid in 3D scattering model
maxφ The maximum angle in azimuth as a function of elevation in
3D scattering model
Lφ The limits of integration in azimuth plane in 3D scattering model
ht Height of the BS in 3D scattering model τ0 Time of Arrival along LOS τmax Maximum Time of Arrival in 3D scattering model
1
Chapter No. 1
Introduction to Spatial Characteristics of Cellular Mobile Channel
1.1 Overview
It is the need of hour to increase the capacity in cellular and mobile communication
systems. To achieve this objective, the resources of power and frequency have been utilized
efficiently with spectral signal processing techniques in the past years. However less attention
have been given to spatial aspects of the mobile channel [1]. The spatial aspects of the mobile
channel are Angle of Arrival (AoA) and Time of Arrival (ToA) characteristics. To exploit the
spatial domain parameters efficiently it is essential to have reliable understanding of radio
propagation characteristics of transmission path between BS and MS that leads to the design
of effective signal processing techniques [2]. Moreover, in the past years, it has been shown
in theory and practice that with the use of directional antennas the performance of the
wireless communication systems can be improved.
1.2 Capacity Demands in wireless Communications Systems
In wireless communication systems the increasing demand of capacity has always been
an important issue [18]. The concept of reuse of frequency was proposed by AT&T in 1968-
70 to achieve high capacity in analog cellular telephone system called the Advanced Mobile
Phone Services (AMPS). AMPS was the first U.S cellular telephone system relying on reuse
of FDMA to maximize the capacity. The analog cellular mobile systems of that age altogether
are known as the First Generation (1G) wireless technologies. Mobile systems have evolved
rapidly since then, incorporating digital communication technology and transformed to the
2
Figure1.1 : A Typical Macrocell Mobile Environment
new era of the Second Generation (2G) wireless technologies. The 2G wireless technologies
include Global System for Mobile Communication (GSM), IS – 136 and IS – 95. The GSM
evolved in 1990 using TDMA to accommodate a large number of users while IS – 136 and IS
– 95 uses Code Division Multiple Access (CDMA). The increasing demands of higher
spectral efficiency and data rates have led to the development of the Third Generation (3G)
wireless technologies. The 3G offers Universal Mobile telecommunication Systems (UMTS).
Wideband CDMA (WCDMA) and CDMA 2000 are primary standards of 3G wireless
technologies. The use of multiple antennas at the transmitter and receiver in wireless
systems, popularly known as MIMO (multiple-input multiple-output) technology, has rapidly
gained in popularity over the past decade due to its powerful performance-enhancing
capabilities. MIMO technology constitutes a breakthrough in wireless communication system
design [21]. In addition to the time and frequency dimensions that are exploited in
conventional single-antenna (single-input single-output) wireless systems, the leverages of
MIMO are realized by exploiting the spatial dimension (provided by the multiple antennas at
the transmitter and the receiver) [21].
3
Figure1.2 : Circular Scattering Environment
1.3 Multipath Propagation
In cellular mobile channel the signal is reflected and refracted from different obstacles
like trees, high rise buildings and mountains etc. which are called ‘scatterers’ as shown in
Figure1.1 and the phenomenon is called 'Multipath Propagation'. In order to observe the
spatial characteristics of the mobile channel in multipath propagation the understanding of
Physical channel is required essentially. To achieve this goal different 2D and 3D Geometric
models have been presented in literature for macrocell mobile environment. In macrocells the
multipath coming from distant scatterers are less important than form those scatterers which
are closer to MS. Furthermore, the single bounce scattering is assumed monotonously in
almost all 2D and 3D scattering model proposed in literature because multiple bounce
mitigate the signal power rapidly.
1.3.1 Two Dimensional Scattering Model
In 2D scattering model MS is assumed to be located at the center of the scattering
region which may be a circle or an ellipse in azimuth plane, while the BS is usually located
above the ground. The BS antenna is assumed to be surrounded by scattering free region
while MS is assumed to be surrounded by scattering objects. The circular scattering model
[3] and the elliptical scattering model [5,6] are the most popular 2D scattering models
proposed in literature which are shown in Figure1.2.
4
Figure1.3 : A typical 3D Scattering Model
1.3.2 Three Dimensional Scattering Model
A typical 3D scattering environment allows the angular statistics to be distributed in
both azimuth and elevation plane. In almost in all the 3D propagation models found in
literature the macrocell environment has been visualized rigorously with low MS antenna
which is assumed to be located at the center of the semispheroid above the ground and BS is
located in scattering free region at some height Ht above the ground. It can be observed that
3D scattering model has a close resemblance with realistic sub urban mobile environment.
1.4 Problem Formulation
In order to meet the increasing demand of capacity, the resources of frequency and
power has been used extensively. The spectral signal processing techniques alone cannot
meet the increasing demand of capacity [1]. The spatial characteristics of the mobile channel
are proven to be helpful in order to cater to such needs. Therefore, an understanding of the
physical channel is required to exploit these spatial characteristics [2]. It has been observed
that PDF of AoA at BS and MS is found rigorously in 2D and 3D scattering model. Some
authors use the directional antennas at BS in 2D scattering model. However to the best of our
5
knowledge the use of directional antennas at both ends of the link is never seen in the
literature in 2D scattering model to investigate the spatial characteristics of cellular mobile
channel. Moreover in 3D scattering model the PDF of AoA in closed form simultaneously in
azimuth and elevation plane is never observed using directional antenna.
The problem can be formulated in three parts:
1) To investigate Effect of Directional Antennas used at Both Ends of the Link on the
spatial characteristics of cellular mobile channel in 2D scattering model.
2) To investigate the spatial characteristics in closed form simultaneously in azimuth and
elevation plane using directional antenna at BS in 3D scattering model.
3) To investigate the temporal characteristics of cellular mobile channel using Directional
Antenna at BS in 3D scattering model.
1.5 Methodology
To derive the expression for PDF of AoA for macrocell mobile environment while
directional antennas are used at both ends of the Radio Link, the mathematical derivations
found in [2] are used. The 2D geometrical models [3,4] are also used for modeling and
characterization of mobile radio channels. However in case of 3D scattering model the
direction antenna is employed at BS in the semi-spheroid model proposed by Janaswamy [11]
for the derivation of PDF of AoA in closed form in azimuth and elevation plane.
1.6 Organization of the Thesis
The rest of the thesis is organized as follows: The derivations of PDFs of AoA at MS
and BS are presented in Chapter No. 2 using directional antennas in 2D scattering model. In
Chapter No. 3 the PDF of AoA at MS and BS is presented using directional antenna at BS for
3D scattering model. The PDF of ToA for macrocell mobile environment using directional
antenna in 3D scattering model is illustrated in Chapter No. 4. Finally, conclusions are given
in Chapter No. 5 which accompanied with three research plans for the extension of this thesis.
6
Chapter No. 2
Spatial Characteristics using Directional Antennas in 2D Scattering Model
2.1 Introduction
The angle of arrival statistics have been observed extensively in azimuth plane for
macrocell mobile environment. To reduce the effect of interference between multipaths
adaptive antennas with phase shift mechanism are proposed in literature. However to achieve
this goal fixed beam directional antennas are also equally capable. The PDF of AoA at BS
and MS have been found in [3] using directional antenna at the BS with the assumption that
uniform distributed scatterers are confined in a circle around MS in azimuth plane. Similarly
PDF for AoA & ToA is found in [4] using 2D elliptical model, where marginal PDF in angle
and time is found from joint distribution of angle and time. A similar kind of work is done in
[8,9] by using Gaussian scatter density around MS where PDF of AoA is found at BS and MS
respectively while directional antenna is used at BS.
In this Chapter, the directional antennas are proposed at both ends of the radio link in
2D scattering model. The rest of the Chapter is arranged as follows: System model for
directional antennas at both ends of the link is described in section 2.1. The derivation of PDF
of AoA of multipath at MS and BS using uniform and Gaussian scatter densities are given in
section 2.3 and section 2.4 respectively. Results and descriptions are shown in section 2.5 and
conclusions are made in section 2.6.
2.2 System Model for Directional Antennas at Both Ends of the Link
In this section, a macrocell environment is modeled using directional antennas at both
ends of the link in 2D scattering model. The distance between BS and MS is d as shown in
Figure 2.1. The radius of the circle in which scatterers are confined in azimuth plane is R.
7
Figure 2.1 : System Model for uniform scatter density using directional antennas
When a directional antenna of beamwidth α is used only at BS the scatterers present in the
region JKEFGO would be illuminated. The length LM is r and the angles θ₁ and θ₂ are the
same angles in azimuth plane as obtained in [3].
⎪⎪⎩
⎪⎪⎨
⎧
≤<
≤<+
≤<
=
πθθ
θθθαθθ
αθθ
2
21
1
;
; tancossin
tan0 ;
R
dR
r (2.1)
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−+= − αααθ 2
221
1 sin1cossincosRd
Rd (2.2)
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−−= − αααθ 2
221
2 sin1cossincosRd
Rd (2.3)
In addition to directional antenna used at BS if another directional antenna of beamwidth β is
used at MS the scatterers in the region JKLMNO would illuminated as shown in Figure 2.1.
2.3 PDF of AoA using Uniform Scatter Density
The work in this section is presented in two parts. The PDF of AoA at MS is found in
section 2.3.1 while section 2.3.2 presents the PDF of AoA at BS.
2.3.1 PDF at MS
In this section we derive the PDF of AoA at MS with assumption of uniform scatter
density around MS using directional antennas are used at both ends of the link. The CDF of
AoA at MS using uniform scatter density is given in (2.4).
8
βθβθθβ
βθ <<−= ∫+
−;
21)(
2
m_unifrom
drA
F (2.4)
Where r is the radius of the circle in which scatterers are confined which can be computed by
(2.1) under the limit - β < θ < β. The area of the region JKLMNO is Am_unifrom in uniform
distribution of scatterers. This area is actually twice the areas of the sector JKM and the
triangle KLM as shown in the Figure 2.1.
( )⎭⎬⎫
⎩⎨⎧ −+= 11
2 sin 21
212m_unifrom θβθ rRRA (2.5)
In the above equation r and θ₁ are computed using (2.1) & (2.2) with θ = β because the
beamwidth β is such that θ₁ < β < θ₂. By substituting the values Am_unifrom can be simplified as
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛−+−×
⎭⎬⎫
⎩⎨⎧
++
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−+=
−
−
αααβ
αββαααα
22
21
22
212
sin1cossincossin
tancossin tan sin1cossincosm_unifrom
Rd
Rd
dRRd
RdRA
(2.6)
The PDF of AoA at MS is obtained by differentiating (2.4) over θ. The parameter Ω used in
(2.7) below is a normalizing factor such that the area under the curve is unity.
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≤<⎭⎬⎫
⎩⎨⎧
+Ω
≤<Ω
=
|| ||; 2
tancossin tan
- ; 2
)(
1m_uniform
2
11m_uniform
2
βθθαθθα
θθθ
θθ
A
d
AR
f (2.7)
2.3.2 PDF at BS
The PDF of AoA at BS assuming uniform scatter density around MS is found by
computing the area of the strip of length KL and width Δθ, whose scatters are illuminated by
the beamwidth of directional antenna used at BS with truncation according to the directional
antenna used at MS. The width Δθ is infinitely small such that the length LL' and KK' are
more like a straight lines. The length of strip KL is the difference of x₁ and x₂ as shown in the
Figure 2.1. Where x₁ is taken form [3] and x2 can be solved form triangle BML.
9
22221 coscos Rdddx +−−= αα (2.8)
βcos2222 drrdx −+= (2.9)
[ ] θααβθθ
θdRddddrrdA 2222
22b_uniform coscos cos2 +−+−−+= ∫
Δ+ (2.10)
The Area of the strip KK'LL' with length ( x₂ - x₁ ) and width Δθ can be found in (2.10) where
the parameter r is computed using (2.1). The CDF of the AoA at BS using uniform scatter
density is found as under.
αθαθπ
θα
αθ <<−= ∫+
−;
2)(
2b_uniform d
RA
F (2.11)
Where πR² is the area of the circle in which scatterers are confined uniformly around the MS.
Finally PDF of AoA at BS if directional antennas are used at both ends of the link is found by
differentiating (2.11) over θ.
αθαπ
θθ <<−= ;2
)( 2b_uniform
RA
f (2.12)
Combining (2.10) and (2.12) the PDF of AoA at BS is simplified in (2.13). The parameter Ω
used in (2.13) below is a normalizing factor such that the area under the curve is unity.
⎥⎥⎦
⎤+−+−
⎭⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎢⎢⎣
⎡
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+Ω
=
2222
2/122
2
coscos
cos tancossin
tan 2 tancossin
tan2
)(
Rddd
dddDR
f
θθ
βθββ
θθββ
θπ
θθ
(2.13)
2.4 PDF of AoA using Gaussian Scatter Density
The work in this section is presented in two parts. The closed form expression for PDF
of AoA at MS is found in section 2.4.1 while section 2.4.2 presents the PDF of AoA at BS.
2.4.1 PDF at MS
The PDF of AoA at MS assuming Gaussian distribution of scatterers can be found
using similar kind of derivation as section 2.3.1 with a difference that the length r depends on
Gaussian scatter density. Hence PDF of AoA using Gaussian scatter density can be found by
replacing r with r′ which is the effective strength of the length LM using Gaussian scatters
10
Figure 2.2 : Gaussian scatter density using directional antennas
dxxrr
∫ ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−=
0
2
21exp
214'
σπσσ (2.14)
In Figure 2.2 a circle with virtual boundary of radius R is shown around the Gaussian
scatterers for simplifications in derivation. The term 4σ is the radius of the virtual circular
region in which scatters are present. The significance of 4σ is that 99.9% of the scatterers are
present within the circle of radius 4σ. In rest of the equations of this section we use 4σ as
radius of the circle. Substituting the value of r in (2.14), r′ can be simplified as
( )( )
⎪⎩
⎪⎨
⎧
≤<⎟⎠⎞
⎜⎝⎛ +
≤<−=
|| || ; 2
sin csc erf 2
; 22 erf 2'
1
11
βθθσ
αθασ
θθθσ
dr (2.15)
( )⎭⎬⎫
⎩⎨⎧ −+= )(sin')4(
214
212 1
21m_Gaussian θβσσθ rA (2.16)
The effective area of the region JKLMNO in Gaussian distributed scatterers can be found
using r′. Substituting the values of r′ and θ₁, Am_Gaussian can be simplified as shown below.
( ) ( ) ( )
( ) ( )⎪⎭
⎪⎬⎫
⎟⎟
⎠
⎞−−
⎪⎩
⎪⎨⎧
⎜⎜⎝
⎛−×
⎭⎬⎫
⎩⎨⎧ +
+⎪⎭
⎪⎬⎫
−−⎩⎨⎧
=
−
−
2
2221
22
22212
m_Gaussian
sin1cossin cossin
2
sincsc erf8sin1cossin cos16
Rd
Rd
dR
dR
dA
αααβ
σαβασααασ
(2.17)
The PDF of AoA of at MS assuming Gaussian scatter density around MS is found in (2.18)
11
βθβθθ <<−Ω
= ;)'(2
)( 2
m_Gaussain
rA
f (2.18)
Finally, the PDF of AoA can be found in closed form is shown below. The parameter Ω used
in (2.18) and (2.19) is a normalizing factor such that the area under the curve is unity.
( )( )
( ) ( ) ( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) ( )( )
( )
|| || ;
sin 1cos
sin
cossin2
sin csc erf4sin 1cossin cos8
2sin csc erf
;
sin 1cos
sin
cossin2
sin csc 4sin 1cossin cos8
22erf
)(
1
2
22
2
122
22212
22
11
2
22
2
122
22212
22
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
≤<⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
−−
−⎥⎦
⎤⎢⎣
⎡ ++
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛ +Ω
≤<−⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
−−
−⎥⎦
⎤⎢⎣
⎡ ++
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−
Ω
=
−−
−−
βθθ
αα
α
βσ
αβασααασ
σαθασ
θθθ
αα
α
βσ
αβασααασ
σ
θθ
Rd
Rd
dR
dR
d
d
Rd
Rd
derfR
dR
d
f
(2.19)
2.4.1 PDF at BS
The PDF of AoA at BS with assumption of Gaussian scatter density around MS is
found by follow the similar kind of derivation as section 2.3.2 with a difference that the area
of the strip KK'LL' dependents on Gaussian scatter density. We define r as under
)(4 12 xxr −−= σ (2.20)
In above equation substituting the values of x₁ and x₂ from (2.8) and (2.9), r can be simplified
in (2.21). Similarly the effective strength of the strip KL which is r′′ shown in (2.22).
αθαβθββ
θθββ
θ
θθσ
<<−⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
⎩⎨⎧
+−−+=
;cos tancossin
tan2 tancossin
tan
cos cos 4
22
2222
dddd
ddRdr
(2.21)
σσπσ
421exp
211''
0
2
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−−= ∫ dxxr
r (2.22)
12
Where the value of r is taken from (2.21) r′′ is simplified in closed form as shown below.
αθασ
σ <<−⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛−= ;
2erf
2114'' rr (2.23)
The effective area of the strip kk'LL' is Ab_Gaussian which is actually area of the rectangular
region of length and width Δα can be found as under.
∫Δ+
=θθ
θθ
b_Gaussian '' drA (2.24) Finally the PDF of AoA at BS assuming Gaussian scatter density around MS using directional
antennas at both ends of the link can be found as
2b_Gaussian
)4(2)(
σπθθ
Af = ; -α < θ < α (2.25)
Where π (4σ)² is area of the circle with virtual boundary 4σ. Combining (2.24) and (2.25) the
PDF of AoA at BS is simplified in (2.26) below. The parameter Ω used in (2.26) is a
normalizing factor such that the area under the curve is unity.
( )
⎥⎥⎦
⎤
⎭⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
+−−⎢⎣
⎡ +Ω
−Ω
=
2/122
222
cos tancossin
tan2 tancossin
tan21
2cos
2cos 22 erf
168)(
βθββ
θθββ
θσ
σθ
σθ
πσπσθθ
dDdd
ddRdf
(2.26)
2.5 Results and Descriptions
This section presents the results of the derivations in section 2.3 and section 2.4. The
PDF of AoA at MS assuming uniform and Gaussian scatter densities are given in (2.7) and
(2.19) respectively. Their results are shown in Figure 2.3 and Figure 2.5 respectively. In
Figure 2.3 α = 5o, β = 80o, R = 400m, d = 2000m, d = 2500m, d = 3000m, d = 3500m, while
in Figure 2.5 α = 5o, β = 80o, R = 400m, d = 2000m, σ = 100m, σ = 200m, σ = 300m,
σ = 400m are used. The results show that in case of uniform scatter density the PDF of AoA
at MS becomes flat as distance between MS and BS increases as shown in Figure 2.3 which
means that the fact of directional antennas is reducing with an increase in the distance
between BS and MS thus tending towards Clark's model [10] with the truncation according to
13
Figure 2.3 : PDF of AoA at MS assuming uniform scatter density
the beamwidth of directional antenna at MS. The same behavior can also be seen in case of
Gaussian scatter density where PDF of AoA at MS becomes more and more flat as σ of
Gaussian scatter distribution decreases. It is due to the fact that with decrease in σ of the
distribution the scatterers tends towards compactness and hence the effect of directional
antenna is negligible.
Figure 2.4 and Figure 2.6 show the plots of PDF of AoA at BS assuming uniform and
Gaussian scatter densities as derived in (2.13) and (2.26) respectively. In Figure 2.4 α =10o,
β = 90o, R = 400m, d = 1000m, d = 1500m, d = 2000m while in Figure 2.6, α = 10 o, β = 90o,
R = 400m, d = 2000m, σ = 100m, σ = 120m, σ = 140m are used. The result show that the
behavior of the PDF of AoA at BS can also be explained in the same manner as explained in
the case at MS. The PDF of AoA at BS using uniform scatter density becomes flat as the
distance between the MS and BS decreases with a truncation according to α as shown in
Figure 2.4. A reverse behavior is seen with an increase in the distance between MS and BS
where the hump of the PDF curve rises. In case of Gaussian scatter density the PDF of AoA
at BS becomes more and more flat as σ of the scatterers distribution increases with the
truncation α as shown in Figure 2.6.
-80 -60 -40 -20 0 20 40 60 802
3
4
5
6
7
8
9
10
11
12x 10-3
Angle (Degrees)
PD
F of
AoA
at M
S u
sing
Uni
form
Sca
tter D
ensi
ty
d = 2000m d = 2500m d = 3000m d = 3500m
14
Figure 2.4 : PDF of AoA at BS assuming uniform scatters density
Figure 2.5 : PDF of AoA at MS assuming Gaussian scatters density
-10 -8 -6 -4 -2 0 2 4 6 8 104
4.5
5
5.5
6
6.5
7x 10-3
Angle (Degrees)
PD
F of
AoA
at B
S u
sing
Uni
form
Sca
tter D
ensi
ty
d =1000md =1500md =2000m
-80 -60 -40 -20 0 20 40 60 800
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Angle (Degrees)
PD
F of
AoA
at M
S u
sing
Gau
ssia
n S
catte
r Den
sity
σ = 100mσ = 200mσ = 300mσ = 400m
15
Figure 2.6 : PDF of AoA at BS assuming Gaussian scatters density
2.6 Conclusions
In this Chapter we have derived the closed form expression for PDF of AoA of
multipath at BS and MS while directional antennas are used at both ends of the link. We
modeled the environment by assuming uniform and Gaussian distribution of scatterers around
MS. Four scenarios of the PDF of AoA seen at BS and MS using uniform and Gaussian
scatter densities have been explained. The results have been shown by changing the distance
between BS and MS in the case of uniform scatter density, while in case of Gaussian scatter
density the effect of changing the σ has shown in the plot.
-10 -8 -6 -4 -2 0 2 4 6 8 100.03
0.035
0.04
0.045
0.05
0.055
0.06
Angle (Degrees)
PD
F O
f AoA
at B
S u
sing
Gau
ssia
n S
catte
r Den
sity
σ = 100mσ = 120mσ = 140m
16
Chapter No. 3
Spatial Characteristics using Directional Antenna in 3D Scattering Model
3.1 Introduction
It has been observed that the cellular mobile channel of the suburban macrocell mobile
environment can be completely visualized, rigorously using 3D scattering model, which
offers more precise spatial and temporal statistics. A 3D Geometric model is proposed in [11]
to derive the PDF of AoA of multipath components as seen from BS and MS simultaneously
in azimuth and elevation planes. A similar kind of model for 3D scattering environment is
presented in [15] using ellipsoidal model for the derivation of direction of arrival (DoA) and
direction of departure (DoD) in azimuth and elevation planes. Another 3D Geometric channel
model is illustrated in [12], which is derived from a 2D Geometrical based single bounce
macrocell (GBSBM) model, where the comparisons of 2D and 3D models published in
literature have been shown in comparison with the experimental data. In [13] uplink/downlink
PDF of DoA and Time of Arrival (ToA) statistics are derived analytically with the
assumption that scatterers are uniformly distributed in a 3D semispheroid with a flat circular
base centered at MS. The power spectral density and PDF of AoA with non zero elevation
plane is derived theoretically in [14] using 3D scattering model, where theoretical results are
compared with the field measurement.
To achieve the objective of higher performance in terms of capacity in wireless
systems we propose the use of directional antenna at BS in 3D scattering model for spatial
characteristics of mobile channel. The rest of the Chapter is organized as follows: Proposed
3D scattering model with directional antenna used at BS is described in section 3.2.
17
Figure 3.1 : A typical 3D Scattering Model
The section 3.3 shows the joint and marginal PDFs of AoA at MS in azimuth and elevation
planes. Similarly, the joint and marginal PDFs of AoA at BS in azimuth and elevation planes
are shown in section 3.4. Finally conclusions are shown at the end of the Chapter on the basis
of analytical results in section 3.5.
3.2 Directional Antenna in 3D Scattering Environment
In this section, we describe the proposed 3D scattering model for macrocell
environment which assumes uniform distributions of scattering objects around MS that are
confined in a semispheroid and the BS is equipped with a directional antenna. The proposed
3D scattering model is shown in Figure 3.1, where major and minor dimensions of the
semispheriod are a and b respectively and the BS is employed with a directional antenna of
beamwidth α at height ht above the ground. The angles made by the direction of signal arrival
in azimuth and elevation planes at MS are symbolized by mφ and βm and at BS are symbolized
by bφ and βb respectively. The scatterers present in whole spheroid would not be illuminated
when BS equipped with directional antenna, which means that the semispheroid is partial
18
Figure 3.2 : Geometry for volume of the illuminated region
illuminated. The volume of the region, whose scatterers are illuminated, is represented as V
and the volume of the region, whose scatterers are not illuminated by the beamwidth of the
directional antenna, is V₁. The geometry of the illuminated and clipped region is shown in
Figure 3.2. The following derivations are used for the volume of the illuminated region.
341
34ellipsoidofvolume
111ellip1
111ellip
cbaVV
cbaV
π
π
==
==
(3.1)
ααα sin;sin;sin 1222
1222
1 dacdaabbdaa −=−=−= (3.2)
( ) ( ) sin sin sin
3222222
1 αααπ dadaabdaV −⎟
⎠⎞
⎜⎝⎛ −−=
(3.3)
ba
34 , 2
22
spheriod1spheriod π=−= VV
VV
(3.4)
In the above equation, VSpheroid is the volume of the whole spheroid. The volume V can be
rewritten in the closed form expression as
( )( )a
dadadbV3
sin sin sin2 2 αααπ −+=
(3.5)
When the beam width of the directional antenna is set equal or greater than αmax all the
scatterers inside semispheroid get illuminated. If we substitute α = αmax in (3.5) the volume
deduces to V = 2/3 π a2 b which is the volume of the semispheroid.
⎟⎠⎞
⎜⎝⎛= −
da1
max sinα (3.6)
19
Figure 3.3 : Azimuth and elevation views of System Model
The portions R1, R2 and R3 of the illuminated region can further be grouped into two partitions,
i.e. R2 alone and the union of R1 and R3 as shown in Figure 3.1. In azimuth plane, the
threeshold angles 1threshφ and 2threshφ , seperates these two different portions of illuminated
region. These angles can be found as a function of elevation angle and beamwidth.
( ) ( )
⎪⎪⎩
⎪⎪⎨
⎧
≤≤−
<≤⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=−
−−
2sincos;
2
sincos0; sincoscoscossin
coscos
1
12221
1threshπβααπ
αβαββαα
αφ
m
mmm
ad
adda
aad
(3.7)
( ) ( )
⎪⎪⎩
⎪⎪⎨
⎧
≤≤−
<≤⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=−
−−
2sincos;
2
sincos0; sincoscoscossin
coscos
m1
12221
2threshπβααπ
αβαββαα
αφ
m
mmm
ad
adda
aad
(3.8)
The angles 1φ and 2φ , which are shown in Figure 3.3 are the above thereshold angles
computed for βm = 0o i.e for zero elevation angle. The threeshold angles 1threshφ and 2threshφ are
ploted in Figure 3.4 with parameters ht = 100m, d = 800m, a = 100m, b = 50m and α = 2o as a
function of elevation angles. Similarly, in elevation plane, βthresh is the threshold angle which
separates two illuminated region in elevation plane are shown in Figure 3.3. The threshold
elevation angle βthresh can be found with the help of the geometry illustrated in Figure 3.5.
20
Figure 3.4 : The threshold angle 1threshφ and 2threshφ as a function of elevation angles
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=αφφ
αtancossin
tan
mm
dPet
(3.9)
( ) ( )( ) ( )2222
2
222
sinsincsc
sin
ααφα
α
ddx
dPetx
m −+=
−=
(3.10)
( ) αφα 222222
212 sincsc mda
abxa
aby +−=−=
(3.11)
⎟⎠⎞
⎜⎝⎛= −
Pety21
thresh tanβ
(3.12)
The threshold elevation angle βthresh can be found in closed form after doing tedious
simplifications, as a function of azimuth angle and beamwidth α of the directional antenna.
( )( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧≤≤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−
+
=
−
otherwise ; 0
|||;sincsc
sincsccot 212222
1
thresh
φφφαφα
αφα
β
m
m
m
dabda
(3.13)
The threeshold angle βthresh is ploted in Figure 3.6 as a function of azimuth angles with
parameters ht = 100m, d = 800m, a = 100m, b = 50m and α = 2o.
The rest of the equations of
this section are simplified under the following assumptions. These assumptions are valid for a
realistic 3D scattering environment using directional antenna at base station, 0 ≤ βthresh ≤ π/2,
| 1φ | ≤ | 1φ |, 0 ≤ | 1φ | ≤ ⎟⎠⎞
⎜⎝⎛−
da1cos and ⎟
⎠⎞
⎜⎝⎛−
da1cos ≤ | 2φ | ≤ π.
21
Figure 3.5 : Geometry for solving βthresh and rm2
The limits for illuminated region (R1, R2 and R3) are defined in correspondence with azimuth
and elevation angles. As discussed earlier, the region R1 and R3 are grouped as P1
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
≤≤
≤≤→→
2thresh m1thresh
thresh
311
or
0 &
φφφ
ββm
RRp
Similarly the limits for the region R2 can be written as P2
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
≤≤≤≤−
≤≤
→→
2thresh 2thresh
1thresh 1thresh
thresh
22
-
or 2
φφφφφφ
πββ
m
m
m
Rp
The distance from the scattering boundary to the MS is rm, which is further symbolized as rm1
and rm2 for the regions P1 and P2 respectively. The distance rm1 has been found in [11] and
the distance rm2 can be found by solving the geometry as shown in Figure 3.5.
mmm ab
barββ 2222
22
1 sincos +=
(3.14)
mm
Petrβcos2 =
(3.15)
Finally the distance rm, form MS can be found in closed form in (3.16). The distance rm is
ploted in Figure 3.7 in azimuth and elecation angles with parameters ht = 100m, d = 800m,
a = 100m, b = 50m and α = 2o. It can be observed in Figure 3.7 that for any particular
azimuth angle there is a threshold angle in elevation plane plane βthresh , as derived earlier in
22
Figure 3.6 : The threshold angle βthresh as a function of azimuth angles
Figure 3.7 : The Distance rm of the scatter from MS
( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+
+=
p ; sinseccsc
p ; sincos
2
12222
22
αβφα
ββ
mm
mm
m
d
abba
r
(3.16)
(3.13). The distance rm, is seen to be uniform in Figure 3.7 for whole azimth plane, for
βm > βthresh , which follows the result as derived in [11]. In other words for the elevation
angle βm > βthresh all the scatterers would illuminated in azimuth plane and the distance form
23
Figure 3.8 : Different elevation views of the Distance rm of the scatter from MS
the MS to the boundry of semispheriod is same for whole azimuth plane. In order to elaborate
this effect, the same distance rm is reploted in Figure 3.8, in azimuth plane for some
particular elavation angles for example, βm = 0o, 30o, 45 o and 55 o.
3.3 Angle of Arrival Statistics at MS
The Joint density function in angles seen at MS and radius rm can be written in (3.17).
The Jacobean transformation J (x,y,z) is given in Appendix A.
( )mm
mmmmmm
rzryrx
mmm zyxJzyxfrp
βφβφβ
βφ
sinsincoscoscos),,(
),,(,,
===
=
(3.17)
( )mm
mmm
mmmmmmmm
mmmmmmmm
rr
rrrr
zyxJβ
ββφβφβφβφβφβφβ
cos1
cos0sinsinsincoscossincoscossinsincoscoscos
,, 2
1
=−−−
=
−
(3.18)
When scatterers are uniformly distributed, the scatter density function can be written as
( )⎪⎩
⎪⎨⎧ ∈
=otherwise;0
&,;1),,( RegionIzyx
Vzyxf
(3.19)
Combing the above equations the joint density function can be written as
( )V
rrp mmmmm
ββφ cos,,2
=
(3.20)
24
The joint PDF of AoA in azimuth and elevation planes is found by integrating above equation
over rm, which is presented in (3.21) in closed form. Similarly, marginal PDF of AoA seen at
MS can be found by integrating (3.21) with appropriate limits as shown in (3.22) and (3.23).
( )( )( )
( )( )( )
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
−++
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
1
2
22322
2
23
2222
22
; sinsin2sinseccsc
; sinsin2sincos
csccos
,
pdadab
da
pdadadb
abbaa
p
mm
mmm
mm
ααπαβφα
ααπββ
αβ
βφ
(3.21)
πφββφββφφπ
β
β20 ; ),(),()( 2
0 thresh
thresh ≤≤+= ∫∫ mmmmmmmm dpdpp (3.22)
20 ; ),( ),(2),()( 2thresh
2thresh
2
1
1thresh
1thresh
πβφβφφβφφβφβφ
φ
φ
φ
φ
φ≤≤++= ∫∫∫
+
−
+
+
+
− mmmmmmmmmmm dpdpdpp thresh
thresh
(3.23)
The above equations for marginal PDF of AoA at MS in azimuth and elevation planes can be
simplified in closed form expressions as shown below.
πφααπ
ββαβαφα
φ
20 ; ))sin(sin(2
)2cos()(sin21csctansin)(csc
)( 2thresh
2222thresh4
thresh233
≤≤−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−++−++
=
m
m
m dadabdabbaa
baad
p (3.24)
( )2
0 ; sincos
csccos
2tan
2cotln4
2sec
2csc
)cos(12
1)cos(2
8sinsec
))sin(sin(1)(
2thresh1thresh
23
2222
22
2thresh1thresh2thresh21thresh2
1thresh2thresh
223
2
πβφφββ
αβ
φαφαφαφα
φαφααβ
ααπβ
≤≤⎪⎭
⎪⎬⎫
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
+
⎟⎟⎠
⎞⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛ ++
++−
⎪⎩
⎪⎨⎧
⎜⎜⎝
⎛
−+−+=
mmm
m
mm
abbaa
addadabd
p
(3.25)
3.3.1 Analytical Results of PDF at MS
In this section, the results of the marginal and the joint statistics of angle of arrival
simultaneously in elevation and azimuth plane are shown. The parameters used in all the plots
shown in this section are d = 800m, a = 100m, b = 50m and α = 2o. The joint PDF of AoA at
MS is shown in Figure 3.9. The marginal PDF of AoA in elevation and azimuth planes are
25
Figure 3.9 : The joint PDF of AoA at MS
Figure 3.10 : The PDF of AoA in elevation plane for different azimuth angles
shown in Figure 3.10 and Figure 3.11 respectively. In Figure 3.10 the marginal PDF of AoA
at MS is shown in elevation plane for different azimuth angle like mφ = 0o, 16o, 22o, 40o
similarly in Figure 3.11 the marginal PDF of AoA at MS in azimuth plane is shown for
βm = 0o, 25o, 40o and βm ≥ βthresh. The results obtained for βm = 0o are shown with the PDF
results of 2D scattering model [3] as shown Figure 3.12. Similarly, when the beam width of
the directional antenna is taken equal or greater than αmax there is no clipping and result
follows [11] as shown in Figure 3.13.
26
Figure 3.11 : The PDF of AoA in azimuth plane for different elevation angles
Figure 3.12 : 3D PDF of AoA for zero elevation plane is compared with 2D [Petrus et. al]
Figure 3.13 : 3D proposed model with & without directional antenna
27
3.4 Angle of Arrival Statistics at BS
In this section, we present the PDF of AoA at seen at BS using directional antenna in
3D scattering model. The system model is shown in Figure 3.14. The distance ρb is the
projection of rb on azimuth plane which can be defined as bbb r βρ cos= . The distance from
the BS to intersection points of simispheriod are rb1 and rb2 for a particluar direction defined
by bφ and βb. The plane developed by the longitudial crossection of conical beam for different
values of βb is varied. If the varing geomatry containing the illuminated plane of scatterers is
analyzed, we see that before it touches the ground, the geomatry forms spherical segments of
varying dimensions as shown in Figure 3.14. If the process of varying βb continued after the
illuminated scattereing plane touches the ground, the shape of the plane becomes some area
bounded by the arcs 'NQN , NO , 'OO and '' NO , as shown in Figure 3.14. Arc 'OO increases form its initial value of zero, to its maximum at the far edge of the scattering
semispheriod and then reduces gradually to zero at point Q. The angle cφ is the angle
subtended by the arc 'OO , which can be written in simplified form [11] as
⎭⎬⎫
⎩⎨⎧ −+
= −
b
b2222
1c tan2
tan)(cosβ
βφdh
adh
t
t (3.26)
The angle cφ takes the value from zero to maxφ , where maxφ is the maximum azimuth angle for
some particular angle βb.
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−+
= −
dbah
baadh
ba
btbt ββφ
22
22
2
222
2
2
1max
tan1tancos
(3.27)
The distances ρb1and ρb2 are the projection of rb1and rb2 on the azimuth plane respectively.
The distance ρb1 follow the arc 'NQN . Similarly, the distance ρb2 follow three different arcs
NO , 'OO and '' NO depending upon the limits of the bφ .
PPRQQ −−
=2
b1ρ ,
⎪⎪⎩
⎪⎪⎨
⎧
−><<
≤≤−+
=
22t
bct
maxc
2
b2
tan|| 0 ; tan
;
adhh
PPRQQ
bb
b
βφφβ
φφφρ
(3.28)
28
Figure 3.14 : System Model for PDF of AoA at BS
Parameters P, Q and R are the same as by Janaswamy in [11]. These distances can formulate
in closed form expression after doing substitution and simplification.
2
222
2
22
2
2
;tancos;tan1b
haadRb
hadQbaP t
bt
bb +−=+=+= βφβ
(3.29)
(
⎟⎠⎞+−++−−
++
=
)cos)tancos2()tan()(a
costantan
1
2222222222
22222b1
btbbtb
bbtb
dbhhdaabdb
dbhaab
φβφβ
φββ
ρ
(3.30)
(
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
−><<
≤≤
⎟⎠⎞+−++−+
++
=
22ct
maxc
2222222222
22222
b2
tan|| 0 ; tan
;
)cos)tancos2()tan()(a
costantan
1
adhh
dbhhdaabdb
dbhaab
tbb
b
b
btbbtb
bbtb
βφφβ
φφφ
φβφβ
φββ
ρ
(3.31)
If the azimuth angle bφ is taken equal to maxφ , the distances ρb1and ρb2 becomes equal. The
angles βmin and βmax defines the limits for the arrival of multipath in elevation plane, which
can be expressed as
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−+−
= 22
22222t
min adadbahdhtβ , ⎟
⎠⎞
⎜⎝⎛
−= −
adht1
max tanβ (3.32)
The angle βlim is the function of azimuth beamwidth α, which operates as a threshold angle to
exert the effect of directional antenna on the geometry explained in Figure 3.15.
29
Figure 3.15 : Elavation view of system model
( )22
22222222
lim 2cos2))2cos(2())((2
daahdaddabhadba tt
−
−+−−+=
ααβ
(3.33)
The elevation angle for LOS path is denoted by βLOS and the angle β1 corresponds to the
elevation angle for the longest propagation from BS to MS as shown below.
⎟⎠⎞
⎜⎝⎛= −
dht1
LOS tanβ , ⎟⎠⎞
⎜⎝⎛
+= −
adht1
1 tanβ (3.34)
The angles β2, β3 and β4 are seen when bφ is set as α as shown in Figure 3.15. These angles
represents the rotated beam in azimuth plane to touches the boundary of the illuminated
scattering region clipped by α, and can be expressed as
⎟⎠⎞
⎜⎝⎛
+= −
adht1
2 tanβ,
⎟⎠⎞
⎜⎝⎛= −
αβ
costan 1
3 dht (3.35)
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−= −
ααβ
222
14
sincostan
dadht (3.36)
The maximum angle seen in azimuth plane for a fixed elevation angle, βb, when α clips the
boundary of illuminated scattering region, is other then maxφ , shown by Lφ as
⎩⎨⎧
≤≤<≤
=maxlim
limminmaxL ;
;βββαβββφ
φb
b
(3.37)
The joint density in correspondence with the angles seen at BS and as a function of distance
rb is given below.
Vrrp bb
bbbββφ cos),,(
2
= , b
bbbb V
rpβ
ρβφcos
),,(2
= (3.38)
The above equation integrated over rb under the limit rb1 & rb2 has the following solution.
30
2
13cos),(
3 b
b
r
r
bbbb V
rp ββφ =
( )b
bb Vp
βρρβφ 2
3b1
3b2
cos3),( −=
(3.39)
The volume of illuminated scattering region, V is derived earlier, when substituted in above
equation the solution of joint PDF of AoA can be simplified as follows.
( )( )( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧≤≤≤≤−
−+−
=otherwise;0
&; sinsin2
seccsc
),(
maxmin2
3b1
3b2
2
βββαφαααπ
ρρβα
βφ
bbb
bb
dadadba
p
(3.40)
Marginal PDF of AoA in azimuth plane seen at BS can be obtained by integrating above
equation over βb for appropriates limits. However, the closed-form solution can also be
obtained in a similar way as in [11].
( ) ( )b
be
bbbbbb A
Vdpdzd
Vp φφ
φφρρφ ,Ellipse
cos ; 1== ∫∫
(3.41)
Where beA φ, is the area of scattering ellipse seen for a fixed angle bφ . Finally, the closed-form
expression for the marginal PDF of azimuth angle of arrival seen at BS can be expressd as
( )
( )( )( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧≤≤−
−+−
=otherwise;0
;sinsin4sincsccos3
2
22
αφαααφαφ
φ
bbb
b
dadada
p
(3.42)
Similarly, the marginal PDF of AoA in elevation seen at base station [11] can be written as
( )∫− −= L
Lb
bb d
Vp
φ
φφρρ
ββ 3
b13b22cos3
1)(
(3.43)
Substituting the value of V, the above equation can be written as
( )( ) ( )∫− −−+
= L
Lb
bb d
dadadbap
φ
φφρρ
ααπβαβ 3
b13b22
2
sinsin2seccsc)(
(3.44)
3.4.1 Analytical Results of PDF at BS
In this section, we describe the results of marginal PDF of AoA at BS both in azimuth
and elevation planes. The parameters used in all the plots shown in this section are d = 800m,
a = 100m, b = 50m and ht = 100m. Figure 3.16 shows the PDF of AoA in azimuth plane seen
31
Figure 3.16 : Marginal PDF of AoA in Azimuth plane seen at BS
Figure 3.17 : Marginal PDF of AoA in Elevation Plane
at BS for different values of beamwidth i.e. α = 3o, 4o, 5o and αmax. In Figure 3.17 the PDF of
AoA at BS in elevation plane is shown for α = 2oand αmax. Similarly in Figure 3.18 the result
for PDF of AoA in elevation plane are shown when the antenna height ht of BS antenna is set
equal to the elevation axis, b of scattering region, i.e. ht = b = 100m. It has been observed that
when the beamwidth α is set equal or greater than maximum beamwidth i.e. α ≥ αmax there is
no clipping of scattering region and the PDF is found to be same as in [11], which proves the
generalization and validity of proposed model.
32
Figure 3.18 : Marginal PDF of AoA in Elevation plane seen at BS (ht = b)
3.5 Conclusions
The closed form expression of angle of arrival statistics at MS and BS has been
presented for 3D macrocell mobile environment with directional antenna mounted at elevated
base station. The result has been shown for the joint and marginal PDF of AoA at MS and BS
respectively with different azimuth and elevation angles. Finally in order to prove the validity
and generality of proposed model, comparison has been made with some notable 2D and 3D
scattering models which are proposed in literature. The results have been compared by taking
beamwidth of the direction antenna α ≥ αmax which illuminates the whole scattering region,
the PDF of AoA both in azimuth and elevation planes have been found to be same as by
Janaswamy [11].
33
Chapter No. 4
Time of Arrival for 3D Scattering Model
4.1 Introduction
To meet the challenges of present and future in wireless communication systems the
spatial and temporal characteristics are proven to be useful in literature. The geometrically
based models for macrocell mobile environment illustrated in Chapter No. 2 and Chapter No.
3 are some typical and adequate solutions in this regard. In [19] the temporal statistics of
cellular mobile channel are observed for picocell, microcell, and macrocell environments
using 2D scattering model. The results shown in [19] help in the design of efficient equalizers
to combat inter symbol interference (ISI) for wideband systems. In [4] the joint and marginal
PDF of AoA and ToA are derived for the 2D elliptical and circular models. A Geometrical
model is considered in [13] with hollow-disc centered at the MS, uplink/downlink PDF of
ToA/AoA are shown by varying the thickness of hollow disc’s which degenerates to the well
known uniform-ring or uniform-disc densities. In [7] Gaussian scatter density around MS is
assumed for AoA and ToA using 2D circular and elliptical scattering model where the results
are compared with experimental measurements.
A 3D Geometric model is considered in [11] for angular arrival of multipath waves in
the azimuth and elevation planes, where the closed form expressions are derived for the PDF
of AoA with first and second order statistics. In [13] the uplink/downlink trivariate
distributions of ToA and AoA has been found using a 3D model similar to [11] but could not
be taken for macrocell mobile environment because of the fact that the BS and MS could not
assumed to be at same height, moreover the scatterers around the MS are assumed to be
confined in a spherical region with same radius R along azimuth and elevation plane. The
actual scenario is described completely by taking the scattering spheroid with different
lengths of major and minor axis ( minor axis along elevation plane ) and elevated BS to better
34
Figure 4.1 : System Model for Time of Arrival
model the macrocell environment. The geometrically based single bounce macrocell
(GBSBM) channel model using directional antenna at BS is presented in [3] which illustrate
the power of the multipath components in addition to PDF of AoA and ToA of multipath
components. It has been shown in [3] that the level crossing rate of the fading envelope
reduces and the envelope correlation increases significantly if a directional antenna is
employed at BS.
In this Chapter, we illustrate the temporal for proposed 3D scattering model with
directional antenna to be employed at elevated BS. The rest of the chapter is organized as
follows: System model for temporal characteristics using directional antenna at BS is
described in section 4.2. The derivation of PDF of ToA is presented in section 4.3. The
analytical results with descriptions are given in section 4.4. Finally conclusions are made in
section 4.5 on the basis of analytical results.
4.2 System Model for Time of Arrival Characteristics
This section illustrates the system model for ToA characteristics using directional
antenna at BS in 3D semispheroid model. The system model is shown in Figure 4.1 which is
similar to the model used in Chapter No. 3 for AoA statistics at BS and MS. The azimuth and
elevation angles seen at BS and MS are the same as described in Chapter No. 3. Furthermore
35
the relation for αmax, βthresh and the volume V of the illuminated area are the same as found
earlier, these relations are rewritten in the following equations which would be used in the
derivation of PDF of ToA.
⎟⎠⎞
⎜⎝⎛= −
da1
max sinα (4.1)
( )( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧≤≤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−
+
=
−
otherwise ; 0
||||;sincsc
sincsccot 212222
1
thresh
φφφαφα
αφα
β
m
m
m
dabda
(4.2)
( )( )a
dadadbV3
sin sin sin2 2 αααπ −+=
(4.3)
The LOS distance d los from MS to BS can be found as under. 22
los thdd += (4.4)
The distance of the scatterer form MS and BS respectively can be expressed in closed form
using the angles at MS side.
( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
<≤+
≤≤
+
=
thresh
thresh
2222
22
0; sinseccsc
2;
sincos
),(
ββαβφα
πββ
ββ
m
mm
m
mm
mmm
d
abba
βφr
(4.5)
( )mmmmmmmmb hdrdr , β,rr βφβφ sincoscos2)( t2
los2 +−+=
(4.6)
Substituting the value of rm form (4.5) in (4.6) the distance rb can be simplified as
( )
( )( ) ( ) ( )( )⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
<≤
++−++
≤≤
++−
++
=
thresh
t22
los
thresh
2222
22
t2222
222
los
0 ; sinseccscsincoscos2 sinseccsc
2 ;
cossin
sincoscos2 cossin
)(
ββαβφαβφβαβφα
πββ
βββφβ
ββ
φ
m
mmmmmmm
m
mmmmm
mm
mmb
dhddd
babahd
babad
, βr (4.7)
The propagation path delay for d los is τo, and the maximum propagation path delay for longest
distance is τmax, can be found using the velocity of propagation as
36
caDH
cD t
22
maxlos
0
)(,
++== ττ
(4.28)
Moreover, the propagation path delay of waves reflected form the scatterers located at the
boundary of scattering region for a particular azimuth and elevation angle is symbolized by
τlim.
c , βr , βr , β mmbmmm
mm)()()( lim
φφφτ +=
(4.8)
4.3 PDF of Time of Arrival using Directional Antenna
This section presents, the PDF of ToA of macrocell mobile environment using
directional antenna in 3D scattering model. Substituting (4.6) in (4.8) and solving for the
distance rm as function of τ and angles on MS side, gives the following solution.
( )mtmmmmm hdc
dc , βrβφβτ
τφτsincoscos22
),(2
los22
+−−
=
(4.9)
Similarly, the distance rb can be expressed as a function of τ and angles of BS.
( )bbbbbb hdc
dcrβφβτ
τβφτsincoscos22
) , ,(t
2los
22
+−−
= (4.10)
The joint ToA/AoA PDF can be written using [4] and [7].
( ) ( )( )mmm
mmmmm
rJrp
pβφβφ
βφτ,,
,,,, =
(4.11)
( )1
,,−
∂∂
=τ
βφ mmmm
rrJ
(4.12)
( )))sincoscos(2(
)sincoscos(2,,
2222
2
mtmmt
mtmmmmm hdcchdc
hcdrJ
βφβττβτφβ
βφ+−++
+−= (4.13)
( )
mmmmmmmm
rzryrx
mmm zyxJzyxfrp
ββφβφ
βφ
sincossincoscos),,(
),,(,,
===
=
(4.14)
( )
1
cos0sinsinsincoscossincoscossinsincoscoscos
,,
−
−−−
=
mmm
mmmmmmmm
mmmmmmmm
rrrrr
zyxJββ
φβφβφβφβφβφβ
(4.15)
( )mmr
zyxJβcos
1,, 2=
(4.16)
When scatterers are uniformly distributed in illuminated region (IRegion) of volume V, then the
scatter density function can be written as
37
( )⎪⎩
⎪⎨⎧ ∈
=otherwise ;0
&,1),,( RegionIzyx
Vzyxf
(4.17)
The joint PDF of AoA can be written as
( )V
rrp mmmmm
ββφ cos,,2
=
(4.18)
After simplification, the joint function of ToA in correspondence with AoA can be written as
( ) ( )4
t
t222
los2222
los
)sincoscos(8cos)sincoscos(2)(,,
mmm
mmmmmm hcdV
hdccdcdcpβτφβ
ββφβτττβφτ
+−+−+−
=
(4.19)
Similarly, joint function found in correspondence with angles seen at BS can be expressed as
( ) ( )4
222los
2222los
)sincoscos(8cos)sincoscos(2)(
,,btbb
bbtbbbb hcdV
hdccdcdcp
βτφβββφβτττ
βφτ+−
+−+−= (4.20)
The joint PDF of ToA in azimuth and elevation plane can be found by integrating above
equation over elevation and azimuth angles respectively.
( ) ( ) πφπτττββφτφτπ
<<<<= ∫ mmmmm dpp - & ; ,,, max 0
2
0
(4.21)
( ) ( )2
0 & ; ,,, max 0πβτττφβφτβτ
π
π
<<<<= ∫−
mmmmm dpp (4.22)
( ) ( ) maxmaxmax 0 - & ; ,,,max
min
φφφτττββφτφτβ
β
<<<<= ∫ bbbbb dpp (4.23)
( ) ( ) maxminmax 0 & ; ,,,max
max
βββτττφβφτβτφ
φ
<<<<= ∫−
bbbbb dpp (4.24)
Where the limits βmin, βmax and mφ are the same as found in last Chapter. If we substitute ht = 0
and a = b, the proposed model deduces to the model in [13], Olenko et.al., and the temporal
statistics are found similar. Moreover, if we substitute zero for βb the proposed model deduces
to the 2D model given in [4] and the joint function of ToA/AoA is found same as
( ) ( )4
222los
2222los
)cos(8cos2)(,
τφφτττ
φτcdA
dccdcdcpmc
mm −
−+−= (4.25)
If we integrate (4.25) over azimuth angle for appropriate limits, the expression can be
obtained in closed-form for the marginal PDF of ToA for the case of 2D scattering model [4].
Where Ac, is the area of illuminated scattering plane centered at MS in the base of the
scattering semispheroid, which can be expressed as function of azimuth beamwidth α, as
ααφφπ 22221
2 sinsin2)( dadaAc −+−+= (4.26)
38
Figure 4.2 : The joint PDF of ToA in azimuth plane for α > αmax (Numerically integrated)
If we substitute α > αmax, the angles 1φ and 2φ becomes equal and the equation reduces to 2aAc π= (i.e. the area of circle with radius a).
4.4 Analytical Results
The analytical results of time of arrival statistics are shown in detail in this section.
The joint statistics of ToA with azimuth plane is show in Figure 4.2 and Figure 4.3 for
α > αmax and α = 2o. It can be observed in Figure 4.2 that for time τo (for LOS) the statistics are
symmetrically distributed around zero azimuth angle, moreover as the time runs form τo to
τmax the hump of the PDF decreases. The sharp transition in PDF of ToA around τo is more
visible as shown in Figure 4.3 for α = 2o. The PDF of ToA with elevation angles is shown in
Figure 4.4 and the marginal PDF of ToA is shown in Figure 4.5, which describes that the
probability of the multipath signals decreases as the time τ increases, which means that
multipath signals are more probable to arrive earlier near time τ0 of LOS and signals are less
probable to arrive with longer delays. The propagation path delays are shown in Figure 4.6 to
Figure 4.9. The propagation path delays in azimuth and elevation angles seen at MS for
α > αmax and α > 2o are shown in Figure 4.6 and Figure 4.7 respectively, which describes as
elevation angle increase, the hump of the curve decreases. Similarly, the marginal
propagation path delay in azimuth and elevation angles are shown in Figure 4.8 and Figure
4.9 respectively, which clearly demonstrate the effect of directional antenna on propagation
path delay which further leads to affect the PDF of ToA.
39
Figure 4.3 : The joint PDF of ToA in azimuth plane α = 2o (Numerically integrated)
Figure 4.4 : The joint PDF of ToA in elevation plane for α = 2o
Figure 4.5 : The marginal PDF of ToA for α ≥ αmax
40
Figure 4.6 : The joint propagation path delay in azimuth and elevation angle for α ≥ αmax
Figure 4.7 : The joint propagation path delay in azimuth and elevation angle for α = 4o
Figure 4.8 : The effect of directional antenna on propagation path delay in azimuth plane
41
Figure 4.9 : The effect of directional antenna on marginal function of path delay in elevation plane
4.5 Conclusions
In this chapter, closed form expressions have been derived for joint PDF of ToA
in correspondence with azimuth and elevation angles seen at MS and BS. Macrocell
environment has been modeled using directional antenna at elevated BS in 3D semispheroid
model with MS located at its center. The closed form expression for propagation path delay as
function of azimuth and elevation angles seen at MS has been derived. Finally, theoretical
results have been shown to illustrate the effect of directional antenna on temporal
characteristics of proposed model.
42
Chapter No. 5
Conclusions and Future Work
This Chapter presents a summary of the thesis with some directives for the extension
of the proposed results given at the end of this Chapter.
5.1 Summary of the Thesis
The spatial and temporal characteristics for cellular mobile channel have been
presented in this thesis using directional antenna. The 2D scattering model is used to
investigate the effect of directional antennas employed at both ends of the radio link on the
PDF of AoA seen at BS and MS respectively, with assumption of uniform and Gaussian
distributions of scatterers around MS. However in case of 3D scattering model directional
antenna is employed only at BS to observe the spatial and temporal characteristics of mobile
channel.
In Chapter No. 1, we have addressed the issue of physical channel modeling for the
cellular mobile communication systems. We have described the 2D and 3D Geometric
models proposed in literature for multipath propagations in macrocell environments. We have
extensively studied the previous approaches used for modeling cellular mobile channel in
macrocell environments.
In Chapter No. 2, we have proposed the directional antennas used at both ends of the
radio link to observe spatial characteristics for mobile channel. A 2D scattering model is used
to investigate the effect of directional antennas on the spatial characteristics of cellular mobile
channel. The closed form expressions for the PDF of AoA of multipath at BS and MS have
been derived using directional antennas at both ends of the link. We have thoroughly
discussed the macrocell environments with the assumption that uniform and Gaussian
scatterers have been distributed around mobile station. Four scenarios of the PDF of AoA at
43
BS and MS have been illustrated using uniform and Gaussian scatter density. The results have
been shown by changing the distance between BS and MS for uniform scatter density, while
in case of Gaussian scatter density the effect of changing the σ has been shown.
In Chapter No. 3, a macrocell environment is modeled using 3D hemispheroid model
with MS is located at the center and BS is equipped with a directional antenna. The angle of
arrival statistics of multipath waves, seen at MS and BS have been presented in closed form
for macrocell mobile environments. The theoretical results of the joint and marginal PDF of
AoA, seen at MS and BS have been plotted for different azimuth and elevation angles. We
have compared the proposed theoretical results of the PDF of AoA, with some previous
models found in literature to elaborate the effect of directional antenna. The proposed
theoretical results, with azimuth beamwidth of the direction antenna, α ≥ αmax are seen similar
to Janaswamy [11], which illuminates the whole scattering region. Moreover, the theoretical
results obtained with zero elevation angle, is seen similar to Petrus [3], where a 2D scattering
model is used to derive PDF of AoA of multipath signals.
In Chapter No. 4, the closed form density function of joint ToA for proposed 3D
scattering model in correspondence with azimuth and elevation angles seen at MS and BS
respectively have been derived. Finally, the proposed theoretical results for PDF of ToA have
been shown, with azimuth beamwidth of the direction antenna, α ≥ αmax to illustrate the effect
of directional antenna on temporal characteristics.
5.2 Future Work
Future directives, for further research, to extend the work presented in this thesis, may
involve the investigation of the effect of directional antennas on Doppler power spectrum,
Angular spread, spatial correlations and second order statistics like LCR (level crossing rate)
and AFD (average fade duration). To pursue this concept three research plans are proposed as
under:
44
5.2.1 Research Plan 1
Doppler Spectrum:
The angle of arrival statistics found in Chapter No. 2 and Chapter No. 3 using
directional antennas may be used for characterization and tracking of time varying fading
channels. The effect of directional antenna may be seen on the Doppler power spectrum [3],
using 2D and 3D scattering model for high speed communication channels.
5.2.2 Research Plan 2
Angular Spread:
The AoA statistics obtained using the proposed model, employing directional antenna
may be used to find the angular energy distribution in azimuth and elevation plane. The
Angular Spread parameters [16], like Shape factor, Angular Constriction and Orientation
Parameter may be found from the theoretical angular energy distribution which can be
compared with those using the real time data, acquired by the measurement campaigns.
Moreover the effect of directional antenna can be observed on second order statistics like
LCR and ADF.
5.2.3 Research Plan 3
Spatial Correlations:
The effect of multipath interference can be reduced using directional antennas. The closed
form expressions for the AoA statistics, described in Chapter No. 2 and Chapter No. 3 may be
used to investigate the spatial correlations between multipath components of the received
signal for the design of high performance MIMO communication links, in order to enhance
data rates.
45
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47
Appendix A Proof of the Jacobin transformations
( )
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
mmm
zzrz
yyry
xxrx
rJ
βφ
βφ
βφ
φβ
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=,,
mmm
mmmm
mmmm
rzryrx
βφβφβ
sinsincoscoscos
===
( )mmm
mmmmmmmm
mmmmmmmm
mmm
rrrrr
rJββ
φβφβφβφβφβφβ
φβcos0sin
sinsincoscossincoscossinsincoscoscos
,, −−−
=
( ) ( ) ( )( ) sincoscos - 0 cossin -
sinsinsincossincoscoscoscoscos,, 2222
mmmmmmm
mmmmmmmmmmmmmmmmm
rrrrrrrJ
βφβφβφβφβφβφβφβφβ ++=
( ) sincoscos
sinsincossincoscoscos,,222
222232232
mmmm
mmmmmmmmmmmmm
r
rrrrJ
βφβ
φββφβφβφβ
+
++=
( ) ( ) ( )mmmmmmmmmmmm rrrJ φφββφφβφβ 22222232 cos sinsincossincoscos,, +++=
( )( ) ( )( ) mmmmm
mmmmmmm
mmmmmmmm
rrJ
rrJ
rrrJ
βφβ
βββφβ
βββφβ
cos,,
sincoscos,,
sincoscos,,
2
222
2232
=
+=
+=