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Comprehensive Performance Analysis of Localizability in
Heterogeneous Cellular Networks
Tapan Bhandari
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
in
Electrical Engineering
Harpreet S. Dhillon, Chair
R. Michael Buehrer
Allen B. MacKenzie
June 1, 2017
Blacksburg, Virginia
Keywords: Stochastic Geometry, Localization, cellular network, proximate BS
measurements, Poisson point process
Copyright 2017, Tapan Bhandari
Comprehensive Performance Analysis of Localizability in Heterogeneous
Cellular Networks
Tapan Bhandari
(ABSTRACT)
The availability of location estimates of mobile devices (MDs) is vital for several important
applications such as law enforcement, disaster management, battlefield operations, vehicular
communication, traffic safety, emergency response, and preemption. While global positioning
system (GPS) is usually sufficient in outdoor clear sky conditions, its functionality is limited
in urban canyons and indoor locations due to the absence of clear line-of-sight between
the MD to be localized and a sufficient number of navigation satellites. In such scenarios,
the ubiquitous nature of cellular networks makes them a natural choice for localization
of MDs. Traditionally, localization in cellular networks has been studied using system level
simulations by fixing base station (BS) geometries. However, with the increasing irregularity
of the BS locations (especially due to capacity-driven small cell deployments), the system
insights obtained by considering simple BS geometries may not carry over to real-world
deployments. This necessitates the need to study localization performance under statistical
(random) spatial models, which is the main theme of this work.
In this thesis, we use powerful tools from stochastic geometry and point process theory to
develop a tractable analytical model to study the localizability (ability to get a location fix) of
an MD in single-tier and heterogeneous cellular networks (HetNets). More importantly, we
study how availability of information about the location of proximate BSs at the MD impacts
localizability. To this end, we derive tractable expressions, bounds, and approximations for
the localizability probability of an MD. These expressions depend on several key system
parameters, and can be used to infer valuable system insights. Using these expressions, we
quantify the gains achieved in localizability of an MD when information about the location
of proximate BSs is incorporated in the model. As expected, our results demonstrate that
localizability improves with the increase in density of BS deployments.
Comprehensive Performance Analysis of Localizability in Heterogeneous
Cellular Networks
Tapan Bhandari
(GENERAL AUDIENCE ABSTRACT)
Location based services form an integral part of vital day-to-day applications such as traffic
control, emergency response, and navigation. Traditionally, users have relied on the global
positioning system system (GPS) for localizing a device. GPS systems rely on the availability
of clear line-of-sight between the devices to be localized and a sufficient number of navigation
satellites. Since it is not possible to have these line-of-sight links, especially in urban canyons
and indoor locations, the ubiquity of cellular networks makes them a natural choice for
localization. Typically, localization using cellular networks is studied using simulations,
which are carried out by fixing the network configuration including the geometry of the
base stations (BSs) as well as the number of BSs that participate in localization. This
limits the scope of the results obtained since a change in the network configuration would
mean that one must do another set of time consuming simulations with the new network
parameters. This motivates the need to develop an analytical model to study the impact
of fundamental system-design factors such as BS geometries, number of participating BSs,
propagation effects, and channel conditions on localization in cellular networks. Such analysis
would make it convenient to infer how changing these system parameters affects localization.
In this thesis, we develop a general analytical model to study the localizability (ability of get a
location fix) of a device in a cellular network. In particular, we study how information about
the location of BSs in the proximity of the device to be localized affects localizability. We
derive expressions for metrics such as the localizability probability of a device. Our results help
quantify the gains achieved in localizability performance when information about the location
of BSs in the vicinity of the device to be localized is available at the device. Our results
concretely demonstrate that including this additional information significantly improves the
localizability performance, especially in regions with dense BS deployments.
Acknowledgments
First and foremost, I would like to express sincere gratitude to my advisor, Dr. Harpreet S.
Dhillon. It has been an honour to work for you during my Master’s. This work would not
have been possible without your ideas and guidance, which have helped at every step along
the way when I have been stuck. Working for you has helped me improve technically and
personally. Thanks a lot for everything.
I would like to thank my committee members Dr. Michael Buehrer and Dr. Alan MacKenzie
for their acceptance to be a part of my committee.
I would like to thank all my group members and friends in Durham 470. Thanks to Mehrnaz,
Priyabrata, Vishnu, Mustafa, Chiranjib, Kartheek and Shankar. It was great to have course
and research discussions with you all. Your presence definitely made the last two years a lot
easier and merrier. Thanks a lot.
I would like to thank all my friends outside the department. Special thanks to Viswanath,
Nikhil, Tarun, Ashwin, and Carol for making my stay in Blacksburg great fun.
Finally, I would like to thank my parents without whom I quite literally would not have
been where I am today. Thanks a lot Mom and Dad for supporting me through out the two
years.
iv
Contents
List of Figures ix
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Localizability in single-tier cellular networks, Chapters 2 and 3 . . . . 5
1.2.2 Localizability in HetNets, Chapter 4 . . . . . . . . . . . . . . . . . . 7
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The Impact of Proximate Base Station Measurements on Localizability in
Cellular Systems 10
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Localizability Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 14
v
2.4.2 Relevant Distance Distributions . . . . . . . . . . . . . . . . . . . . . 15
2.4.3 Localizability Probability . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Performance of Localizability in Cellular Networks 24
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Signal-to-interfernce ratio . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Localizability Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 Relevant Distance Distributions . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Localizability Probability . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.4 Localizability in the presence of shadowing . . . . . . . . . . . . . . . 32
3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Localizability in Heterogeneous Cellular Networks 37
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 New approach to study localizability in HetNets . . . . . . . . . . . . 38
vi
4.2.2 Displacement theorem-based analysis . . . . . . . . . . . . . . . . . . 38
4.2.3 Localizability probability . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Localizability Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.1 Specializing displacement theorem for PPPs with exclusion zone . . . 42
4.4.2 Applying Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.3 Distance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.4 Localizability Probability . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 59
4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Conclusion 64
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Appendix A 69
A.1 Proof of Lemma 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.2 Proof of Lemma 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.3 Proof of Lemma 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.4 Proof of Lemma 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.5 Proof of Lemma 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.6 Proof of Lemma 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
vii
A.7 Proof of Lemma 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.8 Proof of Lemma 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.9 Proof of Lemma 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.10 Useful results for non-homogeneous PPPs . . . . . . . . . . . . . . . . . . . . 75
Bibliography 78
viii
List of Figures
2.1 Illustration of the system model. The radiating BSs represent the active BSs.
The rest are inactive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 This figure compares Ploc for different values of dmax for L = 4, α = 4 and
p = q = 1. The markers correspond to simulation results and lines correspond
to analytical results using Theorem 1. . . . . . . . . . . . . . . . . . . . . . . 21
2.3 This figure compares Ploc for different values of λ for L = 4, α = 4, dmax = 20m
and p = q = 1. The markers correspond to simulation results and lines
correspond to analytical results using Theorem 1. . . . . . . . . . . . . . . . 22
3.1 Illustration of the system model. The MD is denoted by the triangle in the
center, dark circles denote active BSs and light circles denote inactive BSs.
The shaded region has activity factor p, and the region outside has activity
factor q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Effect of dmax on Ploc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Effect of λ on Ploc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Tightness of Lemma 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Tightness of Lemma 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix
4.1 Illustration of the system model. . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Illustration of the tightness of results in Theorem 6 . . . . . . . . . . . . . . 60
4.3 Illustration of the effect of increasing L . . . . . . . . . . . . . . . . . . . . . 61
4.4 Illustration of the effect of increasing the small cell density and the transmit
power of BSs in the small cell on Ploc. . . . . . . . . . . . . . . . . . . . . . . 62
x
Chapter 1
Introduction
1.1 Overview
The availability of accurate position estimates is essential in many fields such as law en-
forcement, vehicular communication, vehicular-pedestrian collision avoidance, emergency
response, and preemption [1, 2]. Not surprisingly, this area has received tremendous at-
tention from the research community in the past few decades [3–9]. Given the volume of
literature available in this field, it is natural for the uninitiated to think that the geolocation
puzzle has been solved, thereby diminishing the motivation to pursue research in this area.
This could not be farther from the truth. As an example, we are still somewhat short-handed
when it comes to providing positioning guarantees for E911 calls originating indoors. More
broadly, our understanding of the fundamental performance limits of localization systems is
still not as advanced as that of communication systems. As will be discussed shortly, this is
partly because of the restrictive system setups that are usually considered for the analysis
of geolocation systems.
Over the years, we have been extremely reliant on the global positioning system (GPS) to
determine the position of devices around us. A key factor in determining the position of a
1
2
device using GPS is the availability of a clear line-of-sight between a sufficient number of
satellites and the device. Increasing urbanization all over the globe has made it extremely
difficult to always maintain a line-of-sight between the devices and navigation satellites. Due
to this, determining the position of devices accurately using GPS has become challenging,
and this drawback is further amplified in indoor locations. However, the pressing need to
cater to all the location-dependent services has made cellular networks the natural choice
for localization of devices due to their ubiquity. Owing to all the above mentioned factors,
and applications, localization using cellular networks has attracted special interest from
the research community over the years [10–16]. Also, the availability of location data can
help network operators provide users with location-dependent services, and can also be
used to determine the location of devices during times of emergency. Until recently, Federal
Communications Commission (FCC) only mandated network operators to be able to localize
devices with a certain accuracy in outdoor locations [17, 18]. However, in early 2015, FCC
issued a mandate which requires network operators to locate an MD with a certain accuracy
even in indoor locations since most of the distress calls originate from indoor locations [19].
Since the functionality of GPS in indoor locations is limited, it has become necessary for
network operators to employ techniques that use measurements from the cellular network to
localize devices.
There is a rich body of literature which studies the performance of localization techniques
such as Time-of-Arrival (TOA), Observed-Time-Difference-of-Arrival (OTDOA), Received-
Signal-Strength (RSS), etc. in a cellular network [20,21]. Irrespective of the technique used
for localization, the performance of localization fundamentally depends on three main factors:
(i) number of BSs/anchors participating in the localization procedure, (ii) the geometry of
these anchors, and (iii) the accuracy of the positioning measurements from these anchors.
When the above mentioned factors are assumed to be deterministic, tools such as Cramer-
Rao Lower Bound (CRLB) can be used to determine the variance of the mean-squared
positioning error to study the performance of localization [22–25]. However, it would not be
reasonable to assume any of the above factors to be deterministic, and doing so limits the
3
scope of the results obtained to a specific configuration of the network. For example, fixing
the geometry of BSs, and the number of BSs participating in the localization procedure would
yield results that are specific to a given network configuration. Typically, when cellular-based
localization systems are studied, the hexagonal grid model is used. The major drawback of
doing such analysis is that results often lack generality. Often, complex simulators are also
used to study a localization system to gain useful insights using a set of standard system
parameters defined by 3GPP. However, such analysis makes it difficult to obtain insights into
how localizability of an MD is affected by system-design factors such as BS density, channel
conditions, and propagation effects. A cellular network is constantly evolving due to change
in the BS geometries, channel conditions, quality of measurements, and several other factors.
Ideally, it would be the best to study such a network by considering a general setup to account
for the inherent randomness associated with a cellular network. A realistic approach would
be to use a stochastic model for the locations of BSs in the network. Finally, averaging over
all the possibilities of BS geometries would yield general, and accurate results, which depend
on system parameters, and hence, provide a more holistic insight into system performance.
In this thesis, our main focus is on providing such a tractable analytical framework to study
the localizability of an MD in a HetNet.
Recently, stochastic geometry has emerged as a powerful mathematical tool to study the
performance of wireless networks. The underlying idea in the stochastic geometry-based
analyses is to model the locations of MDs, and BSs as point processes, which then lends
tractability to the analytical characterization of key performance metrics, such as coverage,
outage, and rate. In [26–29], tools from stochastic geometry are used to study the perfor-
mance of single-tier, and multi-tier HetNets by modeling the distribution of BS locations as
a Poisson point process (PPP). In this work, we use this approach to build a tractable ana-
lytical framework to study the localizability of an MD in single-tier, and multi-tier cellular
networks.
As discussed earlier, the performance of localization in cellular networks is dependent on the
number of participating BSs, the BS geometry, and the quality of measurements received
4
from the participating BSs. Naturally, the accuracy of the position estimate is directly
proportional to the number of BSs participating in the localization procedure. However,
the primary goal with which cellular networks are designed is for communication between
an MD, and its serving BS. The system is strategically designed to ensure that only one
BS is hearable at the MD to minimize the interference. This results in a conflict between
a localization system using the cellular network, and the communication between an MD
and BS in a cellular network, and this problem is usually referred to as the hearability
problem in the research community. Higher hearability of BSs at an MD would lead to higher
overall interference in the cellular network, which is not ideal for communication purposes.
Therefore, as a first step towards using cellular networks for localization, it is important
to characterize the number of BSs that are hearable at the MD to be localized. Since the
MD can be located anywhere in the network, it is important to derive the distribution of
the number of BSs that are hearable at all possible locations of the MD, which we define
as localizability. If we model the BS locations as a PPP, we can use the spatial ergodicity
of the model to position the MD to be localized at the origin, and derive the localizability
probability by averaging over all possible BS geometries. This spatial averaging is done using
tools from stochastic geometry. Stochastic geometry was first used to study the localizability
of an MD in a single-tier cellular network in [30–34]. The primary goal of this line of work
was to study the hearability problem in a single-tier cellular network, and gain useful insights
into how the number of participating BSs affects the localizability of an MD.
Apart from the number of BSs participating in the localization procedure, the accuracy of
location estimates is greatly influenced by other measurements such as information about
the position of BSs in the proximity of the MD. For example, in cellular network collaborated
localization techniques such as Cell-ID/Cell-of-Origin(COO), the MD reports to the network
important information such as the serving cell ID, the timing advance (difference between
its transmit, and receive time), estimated timing, and power of the detected neighbor cells.
Often, the MD connects to BSs in close proximity to it. Therefore, information about
the location of BSs in the proximity of the MD is available to the MD. In this thesis,
5
we characterize how measurements from proximate BSs impact localizability of an MD;
specifically, we study how the information about the location of the closest BS at the MD
impacts localizability.
1.2 Motivation and Contributions
In this section, we provide the motivation behind the work in each of the Chapters, and
briefly summarize the main contributions. The first part of the thesis, specifically, Chapters
2 and 3 focus on studying the impact of proximate BS measurements on the localizability of
an MD in a single-tier cellular network. In Chapter 4, we study the localizability of an MD
in a HetNet.
1.2.1 Localizability in single-tier cellular networks, Chapters 2
and 3
Motivation
As discussed above, the main challenge in using cellular networks for localization purposes
is the conflict in the purpose of the two networks; the purpose of the cellular network is
primarily for communication, and hence the interference at the MD must be minimized, while
the purpose of the localization system is ensuring that higher number of BSs participate in
the localization procedure. Therefore, canonical deterministic configurations, such as putting
anchors at the vertices of a polygon, won’t really be realistic for the analysis of cellular
networks. This factor coupled with the availability of information about the location of BSs
in the proximity of the MD, inspires us to first understand how many BSs are hearable at
a typical MD in realistic cellular deployments. Using this, we will study localizability of an
MD in a cellular network. There is significant body of literature which uses simulation, and
deterministic tools such as CRLB to study the performance of localization. Recently, an
6
analytical model was proposed to study the localizability of an MD in a single-tier cellular
network [30]. In [30], the primary goal is to study the hearability problem, and evaluate the
probability of localizing an MD in a cellular network. In these Chapters, we extend the work
in [30] to develop a general, and tractable analytical framework to study how localizability
is impacted when the MD is aware of the locations of BSs in its proximity.
Contributions
In Chapter 2, we develop an analytical model to study localization in cellular networks when
information about the location of the closest active BS in known to the MD. The locations
of the BSs are modeled according to a homogeneous PPP. An MD is said to be localizable
either when a certain fixed number of BSs participate in the localization procedure, or, when
the MD is within a certain predefined distance of the closest active BS in the network. Using
this definition for localizability of an MD, we use tools from stochastic geometry to derive
tractable expressions for the localizability probability of an MD. The expression depends on
key system parameters such as the density of BSs in the network, path loss, the number of
BSs participating in the localization procedure, the activity of BSs in the network used to
model the network load, and the predefined distance of the closest active BS from the MD.
In Chapter 2, the proposed model ignores the effects of large scale shadowing, and takes
into consideration only the distance of the closest active BS from the MD to be localized.
However, in Chapter 3, we consider the location information of the closest BS. That is, an
MD is said to be localizable either when a certain fixed number of BSs participate in the
localization procedure, or, when the location of the BS closest to the MD is known. We also
consider the effects of large scale shadowing in this model. It will be highlighted in more detail
as to why this becomes a technical challenge in Chapter 3. We develop tractable expressions
for localizability probability of the MD. We also derive useful bounds and approximations
for the localizability probability.
Using these results, we make some valuable inferences about the effects of BS density, prop-
7
agation, and the number of BSs participating in the localization procedure on system per-
formance. Note that our analyses are agnostic to the technique used for localization. The
expressions derived for localizability probability depend on system parameters such as the
density of BSs, number of participating BSs, and propagation effects. These expressions
help quantify the gains achieved in localizability performance when the MD is aware of the
location of BSs in its proximity. The key take away from these Chapters is that significant
gains are observed in the localizability performance when information about the location of
BSs in the proximity of the MD is available at the MD, particularly in dense BS deployments.
1.2.2 Localizability in HetNets, Chapter 4
Motivation
In Chapters 2 and 3, we study the localizability of an MD in a single-tier cellular network.
The ever-increasing densification of cellular networks is bringing BSs closer to the MDs.
The deployment of small cells, pico cells, and femto cells within each macro cell means that
cellular BSs are now within a few meters of the MDs. Despite the increasing popularity of
HetNets, to the best of our understanding there is no stochastic geometry-based analysis of
localization in these networks. There is however some work, which studies localization in a
HetNet in a WSN [35–37]. This work generalizes the BS geometries, but ignores the effect
of the self-interference in the network, and propagation effects. However, to account for the
fact that the cellular network must balance its requirements of catering to localization, and
communication with an MD, it is vital that interference is incorporated in the analysis of
localizability.
Contribution
In Chapter 4, we study the localizability of an MD in a HetNet scenario. A stochastic
geometry-based model to study HetNets was first proposed in [29] and [38]. Using this
8
model as a foundation, we analyze the localizability of an MD in a HetNet, where we model
tiers of BSs as homogeneous PPPs with different densities, and BSs belonging to different
tiers having different transmit powers. First, we study the hearability problem in which an
MD is localizable simply when a certain fixed number of BSs participate in the localization
procedure. We then study how availability of location information of the closest overall BS
among BSs of all the tiers affects localizability of an MD. We derive tractable expressions
for the localizability probability of an MD in such scenarios. From the model developed,
and the expressions derived, valuable system level insights can be gained by studying the
behavior of localizability probability of an MD on system parameters such as the number of
participating BSs, path loss, density of macro cell and small cells, transmit power of BSs in
different tiers, and the distance of the closest overall BS from the MD. Eventually, it can be
concluded that the accuracy with which an MD can be localized increases significantly when
the MD has information about the location of BSs in its proximity, particularly in dense BS
deployments.
1.3 Organization
Chapters 2, 3, and 4 contain all the technical contributions of this thesis. In Chapter 2, we
develop a tractable model to study the performance of localizability of an MD in a single-
tier cellular network when information about the location of the closest active BS in the
network is known at the MD. We derive the expression for localizability probability of an
MD. In Chapter 3, we study the performance of localizability of an MD in a single tier
cellular network when information about the location of the closest BS is known at the MD
in the presence of large scale shadowing. We derive useful bounds, and approximations for
localizability probability of an MD. In Chapter 4, we further enrich our analysis by studying
the localizability of an MD in a HetNet scenario. First, we study the hearability problem
followed by studying localizability of an MD when information about the location of the
closest overall BS among BSs of all tiers is known at the MD. Again, we derive some useful
9
approximations for the localizability probability of an MD in a HetNet scenario. Chapter 5
summarizes the results of the thesis, and discusses possible paths for future work.
Chapter 2
The Impact of Proximate Base
Station Measurements on
Localizability in Cellular Systems
2.1 Overview
As discussed in the previous Chapter, the ubiquity of cellular networks makes them a pre-
ferred choice for geolocation in places that lack GPS coverage. While the analysis of cellular-
based geolocation has traditionally been driven by simulation-based approaches, the increas-
ing irregularity in the BS locations facilitate the use of powerful mathematical tools from
stochastic geometry for their performance analysis. In this Chapter, we examine the im-
pact of proximate BS measurements on localizability (the ability to get a location fix) in
cellular-based systems. By proximate BS measurements we mean any measurements which
indicate that a specific BS is within a predefined distance to the mobile. In particular, we
derive a mathematically tractable expression for localizability probability, where localizabil-
ity is defined as the union of the two events: (i) at least L BSs are hearable at the device
10
Chapter 2. 11
to be localized, and (ii) the nearest active BS is within a certain predefined distance from
the device to be localized (i.e., there is a proximate BS). Using this result, we quantify the
gains achieved in localizability by incorporating measurements from the proximate BS in the
localization procedure.
2.2 Contributions
The ever increasing densification of cellular networks is bringing BSs closer to the MDs. Since
the true location of the BSs can be made available to the MDs (since they are anchors), the
presence of a BS close to the MD provides the location of the MD at least up to an accuracy
of the distance between the BS and the MD. Motivated by this, we define the localizability of
a device as a union of two events: (i) at least L BSs are hearable at the device to be localized,
and (ii) the nearest active BS is within a certain predefined distance dmax from the device
to be localized. The constant dmax can be chosen based on the target localization accuracy.
Modeling the BS locations as a PPP, we use tools from stochastic geometry to derive easy-to-
use expressions for the localizability probability. Using these results, we quantify the gains
in localizability that can be achieved by using the knowledge of the location of the proximate
BS.
2.3 System Model
We consider the same model as [1]. Key details are provided next. Interested readers can
refer to [1] for a more detailed discussion of the system model. Note that in this Chapter
the terms BS and anchor node are used interchangeably.
Consider a system in which the locations of the BSs are modeled as a homogenous PPP
Φ ∈ R2 with BS deployment density of λ BSs/m2. Due to the stationarity of the PPP,
the MD to be localized is assumed to be at the origin o. Let L be the number of BSs
Chapter 2. 12
participating in the localization procedure. These L BSs are selected based on the average
received power at the MD to be localized. As shown in Fig. 2.1, we model the activity of the
BSs in the network using two activity factors p and q. Let xi and xi denote the location of
the closest and closest active BSs, respectively, and let Ri and Ri denote the distances of the
closest BS and closest active BS from the origin o, respectively. As illustrated in Fig. 2.1, the
activity factor p represents the probability that a BS located in b (o,RL) is active (models
the effect of BS coordination), and q represents the probability that a BS located in bc (o,RL)
is active (models the effect of network load). Here, b (o, r) is a ball of radius r centered at o
and bc (o, r) is its complement. Therefore, the number of active BSs in b (o,RL) is given by
Ω =∑L−1
i=0 ai, where ai is an indicator function taking value of 1 with probability p. Thus,
the number of active BSs in b (o,RL) is a binomial random variable
fΩ(ω) =
(L− 1
ω
)pω(1− p)L−ω−1, ω ∈ 0, ..., L− 1 . (2.1)
The SINR observed at the MD when it connects to a BS xk ∈ Φ for k ∈ 1, ..., L is:
SINRk(L) =P‖xk‖−α
L∑i=1i 6=k
aiP‖xi‖−α +∞∑
j=L+1
bjP‖xj‖−α + σ2
, (2.2)
where xk is the location of the kth closest BS from the MD, bj is an indicator function taking
value of 1 with probability q, P is the transmit power, σ2 is the noise variance, α > 2 is the
path loss exponent, and ‖(·)‖ denotes the `2-norm. The expression in (2.2) gives the SINR
that is observed prior to any processing gain. As discussed in [33], it is worth highlighting
that the expression for SINR in (2.2) does not contain a small-scale fading term because it
is assumed that the processing at the receiver averages out the small-scale fading effects.
The effect of shadowing on SINR can be handled by using the displacement theorem as
demonstrated in [39]. We use an approach similar to [33], where the terms in the SINR
expression for the interference due to the BSs other than the closest active BS in (2.2) are
approximated by their means. In interference limited networks, SINR at the Lth BS can be
Chapter 2. 13
RL
R1
Activity factor p
Activity factor q
dmax
Figure 2.1: Illustration of the system model. The radiating BSs represent the active BSs.
The rest are inactive.
replaced by signal-to-interference-ratio for the link from the Lth farthest BS given by
SIRL(L) =PR−αL
PR−α1 + E[I1|R1, RL,Ω] + E[I2|RL], (2.3)
where I1 =∑Ω
i=2 P‖xi‖−α is the aggregate interference due to all the active BSs in b (o,RL) \
b(o, R1
)and I2 =
∑∞j=L+1 bjP‖xj‖−α is the aggregate interference due to all the active BSs
in bc (o,RL). It must be noted here that since the Lth farthest BS from the origin is the
serving BS, the power received from this BS will not be a part of the aggregate interference.
The means of I1 and I2 are given by Lemmas 4 and 5 of [33], respectively. Using these
results, the SIR in (2.3) can be expressed as
SIRL(L) =R−αL
R−α1 + 2(Ω−1)2−α .
R2−αL −R2−α
1
R2L−R
21
+ 2πqλα−2
R2−αL
.
Chapter 2. 14
2.4 Localizability Performance
2.4.1 Definitions and Preliminaries
The performance of a localization system fundamentally depends on the number of BSs that
participate in the localization procedure, the accuracy of measurements from these BSs and
the locations of these BSs relative to the MD being localized. It is well-known that the
positioning accuracy usually improves when more anchor nodes are able to participate in
the localization procedure [33]. In our setup, it is therefore important to determine the
probability of having at least L hearable BSs at the MD. This was termed as L-localizability
in [33], which was formally defined as
PL = E[1
(SIRL(L) ≥ β
γ
)], (2.4)
where β is the post-processing SINR threshold to qualify whether a BS is hearable at the
MD and can hence successfully participate in the localization process, 1(·) is the indicator
function, and γ is the processing gain. Please refer to [33] for more details on the formal
treatment of this metric. In this Chapter, we enrich this metric by incorporating additional
information about the locations of the BSs in the vicinity of the MD. There are several ways
in which this information can be incorporated in the localization procedure. For instance, the
Cell-ID based localization approach used in cellular networks associates a mobile’s location
with its serving BS. We simply enhance our definition of localizability by using the Cell-ID as
the location of the device only if the closest active BS is within a certain predefined distance
from the MD. Since we deal with L BSs participating in the localization procedure, we only
consider the location of the closest active BS in b (o,RL). Another approach could be to use
information about the location of the BS closest to the MD (i.e., not necessarily the closest
active BS). When information about the location of a BS is accurately known, a decision can
be made about the location of the device at least up to an accuracy of the distance between
the BS and the MD. Thus, we define an MD to be localizable when either of the following
events are true: (i) at least L BSs participate in the localization process, and (ii) when the
Chapter 2. 15
closest active BS amongst all BSs located in b (o,RL) is located in b (o, dmax). When (ii)
holds, the MD can be localized with an accuracy of at least dmax. As discussed in detail in
Section 4.5, the fact that the location of the closest active BS in b (o,RL) is in b (o, dmax)
becomes significantly more impactful in dense BS deployments. In [33], information about
the location of the closest active BS relative to the MD has not been considered while
evaluating the localizability probability of the MD. The probability of localizability in this
case can therefore be defined formally as:
Ploc = E[1
(SIRL(L) ≥ β
γ
⋃R1 ≤ dmax
)]= P
[SIRL(L) ≥ β
γ
]︸ ︷︷ ︸
Term-1: Probability of L-localizabilityPL(p,q,α,β,γ,λ)
+P[R1 ≤ dmax
]︸ ︷︷ ︸
Term-2
−
P[
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax
]︸ ︷︷ ︸
Term-3
P[R1 ≤ dmax
]. (2.5)
It can be observed by comparing (3.4) and (2.5) that when information about the location
of the closest active BS in b (o,RL) is incorporated in the definition of localizability of an
MD, the localizability at a given post processing SINR threshold improves. In other words,
Ploc ≥ PL. Let us now proceed to the evaluation of Ploc.
2.4.2 Relevant Distance Distributions
From our defnition of localizability coupled with the fact that the decision about BS partic-
ipation is made based on the average received power, which depends mainly on the distance
of the BS from the MD, it is important to model the distributions of distances of some
important locations from o. Note that SIRL given by (2.3) is a function of three random
variables: R1, RL, and Ω. It is therefore natural to determine the distributions of these three
random variables first. We begin with RL whose distribution is given by [1]
fRL(r) = e−λπr2 2 (λπr2)
L
r(L− 1)!. (2.6)
Chapter 2. 16
As is evident from (2.5) and the expression for SIR, evaluation of Term-1 and Term-3 in
Ploc makes it necessary to determine the distribution of R1 conditioned on RL and Ω, and
the distribution of R1 conditioned on R1 ≤ dmax, RL and Ω respectively. The cumulative
distribution function of the distance of the closest active BS from o, R1, conditioned on RL
and Ω is [33]
FR1|RL,Ω(r|RL,Ω) = 1−(R2L − r2
R2L
)Ω
. (2.7)
We now study the distribution of R1 conditioned on the fact that the closest active BS in
the region b (o,RL) is located in b (o, dmax) given RL and Ω. This will be an important
intermediate result in the localizability analysis.
Lemma 1. Conditioned on the fact that the closest active BS in the region b (o,RL) is located
in b (o, dmax), the cumulative distribution function of the distance of the closest active BS to
the origin, when, p ≤ 1, q ≤ 1 and Ω ≥ 1 is
FR1|R1≤dmax,RL,Ω(r|R1 ≤ dmax, RL,Ω) =
R2ΩL − (R2
L − r2)Ω
R2ΩL − (R2
L − d2max)Ω
, 0 ≤ r ≤ min (dmax, RL) . (2.8)
Proof. The CDF to be determined is P[R1 ≤ r|R1 ≤ dmax, RL,Ω
]=
P[R1 ≤ r|R1 ≤ dmax, RL,Ω
]=
FR1|RL,Ω(r|RL,Ω)
FR1|RL,Ω(dmax|RL,Ω), 0 ≤ r ≤ min (dmax, RL) .
The result follows by substituting the expression from (2.7).
In order to evaluate Term-2 in (2.5), the distribution of the distance of the closest active BS
from the origin, R1 is necessary. The distribution of R1 is given by
fR1(r) =
L−1∑ω=0
∫ ∞0
A (r, rL, ω) drL, (2.9)
where A (r, rL, ω) = fR1|RL,Ω(r|rL, ω)fRL(rL)fΩ(ω).
Chapter 2. 17
2.4.3 Localizability Probability
We now proceed to the evaluation of Ploc. For notational simplicity, we use the following
definitions:
Ψ(r1, rL, q, α, β, γ, λ) = 1
r−αL
r−α1 + 2(ω−1)2−α .
r2−αL −r2−α
1
r2L−r
21
+ 2πqλα−2
r2−αL
≥ β
γ
and
Θ(α, β, γ, q, L) =
1−L−1∑l=0
e−α−2
2qβ/γ
(α−2
2qβ/γ
)ll!
fΩ(0).
The expression for localizability probability in (2.5) has three terms in it. Term-1 is known
directly from [33] as the probability of L−localizability as
PL(p, q, α, β, γ, λ) = P[SIRL(L) ≥ β
γ
]= Θ(α, β, γ, q, L)+
L−1∑ω=1
∫ ∞0
∫ rL
0
Ψ(r, rL, q, α, β, γ, λ)A(r, rL, ω)drLdr. (2.10)
For the special case of α = 4, this expression simplifies to [33]:
PL(p, q, 4, β, γ, λ) =
X∑ω=0
fΩ(ω)
∫ C(ω)
0B (r, ω) fRL(r)dr, (2.11)
where B (r, ω) =
(1− 1√
γ/β−πqλr2+(ω−1)2
4−ω−1
2
)ω, C (ω) =
√γ/β−ωπqλ
and
X = min (L− 1, bγ/βc). Term-2 follows directly from (2.9). Term-3 corresponding to the
cases when p ≤ 1, q ≤ 1 and p = q = 1 is evaluated in the following Lemma and its Corollary.
Lemma 2. Conditioned on the fact that the closest active BS in b (o,RL) is located in
b (o, dmax), the probability that an MD can be localized when p ≤ 1 and q ≤ 1 is
P[
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax
]= Θ(α, β, γ, q, L)
+L−1∑ω=1
fΩ(ω)
∫ ∞0
∫ min(rL,dmax)
0
Ψ(r, rL, q, α, β, γ, λ)Y(r, rL, ω)drLdr, where (2.12)
Y(r, rL, ω) = fR1|R1≤dmaxRL,Ω(r|rL, ω)fRL(rL)fΩ(ω).
Chapter 2. 18
Proof. Case (i) : Ω = 0. In this case R1 in (2.3) has no meaning. This is equivalent to the
case when p = 0 and Term-3 in (2.5) simply reduces to P[
SIRL(L) ≥ βγ
]∣∣∣∣p=0
which has
been evaluated in Proposition 1 of [33]. Therefore,
P[
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax
]∣∣∣∣p=0
= Θ(α, β, γ, q, L). (2.13)
Case (ii) : Ω ≥ 1. In this case,
P[
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax
]= EΩ
[ERL
[ER1
[1
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax, RL,Ω]]]
,
where the SIRL term is given by (2.3), the expression for fR1|R1≤dmax,RL,Ω(r|r ≤ dmax, rL, ω)
is given by (2.8) and the expression for fRL(rL) is given by (2.6).
Corollary 1. Conditioned on the fact that the closest active BS in b (o,RL) is located in
b (o, dmax), the probability that an MD can be localized when p ≤ 1 and q ≤ 1 and α = 4 is
P[
SIRL(L) ≥ β
γ
∣∣∣∣R1 ≤ dmax
]∣∣∣∣α=4
= 1−Θ(4, β, γ, q, L)−
X∑ω=1
fΩ(ω)
∫ C(ω)
0
[1− B(r, ω)
D(r, ω)
]fRL(r)dr,
where X = min (L− 1, bγ/βcc) and D(r, ω) = 1−(r2−d2
max
r2
)ω.
Proof. The SIRL can be expressed as:
SIRL(a)=
1
Y 2 + (Ω− 1)Y + πqλR2L
where (a) follows by defining X =(RLR1
), Y = X2 and α = 4. Now, SIRL ≥ β/γ =⇒
Y 2 + (Ω− 1)Y ≤ κ−1
=⇒(Y +
(Ω− 1)
2
)2
≤ κ−1 +(Ω− 1)2
4
=⇒ RL√√κ−1 + (Ω−1)2
4− Ω−1
2
≤ R1 ≤ RL,
Chapter 2. 19
where κ−1 = γ/β − πλR2L. Therefore for Ω ≥ 1,
P[
SIRL(L) ≥ β
γ
∣∣∣∣ R1 ≤ dmax
]= 1− EΩ
[ERL
[FR1|R1≤dmax,RL,Ω
(r|R1 ≤ dmax, RL,Ω
)]]evaluated at r = RL√√
κ−1+(Ω−1)2
4−Ω−1
2
. The expression for Ω = 0, is given by (2.13).
Remark 1. Conditioned on the fact that the closest active BS in b (o,RL) is located in
b (o, dmax), the probability that an MD can be localized for the case when all BSs are active,
that is, p = q = 1 can be evaluated directly from (2.12). Since, p = 1, the first term in (2.12)
disappears and the summand need only be evaluated at ω = L− 1. When α = 4, this result
can be evaluated using Corollary 1 by evaluating the summand at ω = L− 1 and substituting
q = 1.
Substituting the results from Theorem 2 of [33], Lemma 1 and Lemma 2 into (2.5), we derive
the following result.
Theorem 1. The probability that an MD can be localized either by L hearable BSs or by the
fact that the location of the closest active BS in b (o,RL) is in b (o, dmax) is:
Ploc(p, q, α, β, γ, λ, dmax) = Θ(α, β, γ, q, L) +L−1∑ω=1
∫ ∞0
∫ rL
0Ψ(r, rL, q, α, β, γ, λ)A(r, rL, ω)drLdr
+
L−1∑ω=1
∫ dmax
0
∫ ∞0A (r, rL, ω) drLdr
[1−
[Θ(α, β, γ, q, L)+
L−1∑ω=1
∫ ∞0
∫ min(rL,dmax)
0Ψ(r, rL, q, α, β, γ, λ)Y(r, rL, ω)drLdr
]].
Substituting the results from Corollary 2.2 of [33], Lemma 1 and Corollary 1 into (2.5), we
derive the following result.
Corollary 2. The probability that an MD can be localized either by L hearable BSs or by the
Chapter 2. 20
fact that the location of the closest active BS in b (o,RL) is in b (o, dmax) when α = 4 is:
Ploc(p, q, 4, β, γ, λ, dmax) =X∑ω=0
fΩ(ω)
∫ C(ω)
0B(r, ω)fRL(r)dr +
[L−1∑ω=1
∫ dmax
0
∫ ∞0A (r, rL, ω) drLdr
][
Θ(4, β, γ, q, L) +X∑ω=1
fΩ(ω)
∫ C(ω)
0
[1− B(r, ω)
D(r, ω)
]fRL(r)dr
]
where X = min (L− 1, bγ/βcc).
Remark 2. Using Theorem 1, the result for the case when all BSs are active, that is,
p = q = 1 and Ω = L−1 can be evaluated directly. It must be noted that in the case when all
BSs are active, R1 can be simply replaced by R1. The terms for the special case when p = 0
disappear and the summands need to be evaluated only at ω = L− 1.
Remark 3. In this Chapter, the definition of localizability of an MD uses information about
the location of the closest “active” BS within b (o,RL) along with the fact that L BSs success-
fully partcipate in the localization procedure. In this approach it was sufficient to model the
distribution of R1 conditioned on the fact that R1 ≤ dmax given RL and Ω. As mentioned in
section 2.4, another approach would be to simply incorporate information about the location
of the closest BS in b (o,RL). In this approach it becomes necessary to model the distribu-
tion of R1 conditioned on R1 ≤ dmax, RL and Ω making it more mathematically intensive
compared to the approach used in this Chapter. It must be noted that for the case when all
the BSs are active, that is, p = q = 1, both approaches would converge to the same result.
2.5 Results and Discussion
In this section, we validate the analytical results derived in Section 2.4 and discuss the
gains in the localization performance when information about the location of the closest
active BS is used in the localization procedure. First, we briefly describe the simulation
setup. All the simulations for the plot in Fig. 2.2 are obtained using a BS deployment
density of λ = 1/(100)2 BSs/m2. The analytical expressions for Ploc are derived with the
Chapter 2. 21
−30 −25 −20 −15 −10 −5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pre−processing SINR, β/γ (dB)
Plo
c
Ploc
with dmax
= 0m
Ploc
with dmax
= 5m
Ploc
with dmax
= 10m
Ploc
with dmax
= 15m
Ploc
with dmax
= 20m
Figure 2.2: This figure compares Ploc for different values of dmax for L = 4, α = 4 and
p = q = 1. The markers correspond to simulation results and lines correspond to analytical
results using Theorem 1.
approximation that the interference due to BSs located in b (o,RL) \b(o, R1
)and the BSs
located in bc (o,RL) are approximated by their means. However, no such approximations
have been used while compiling the simulation results for Ploc. All the simulation results
have been compiled for L = 4, α = 4 and p = q = 1 (all BSs active). First, let us consider
the results shown in Fig. 2.2 in which the results are compiled for different values of dmax,
ranging from 0 m to 20 m. It can be observed that the simulation results exactly match the
analytical results. When dmax = 0 m, information about the location of the closest active
BS is not used in the localization procedure. It can be observed from the trends that the
probability of localizability of the MD starts to improve as the value of dmax increases which
is as expected. For the case of dmax = 20 m significant gains are observed. Let us now
consider the results illustrated in Fig. 2.3 which show the comparison of Ploc at dmax = 20 m
at post processing SINRs of −10 dB and −14 dB over a range of BS deployment densities
from 10−6/m2 to 10−2/m2. It can be observed from the trends that we begin to see significant
Chapter 2. 22
10−6
10−5
10−4
10−3
10−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BS Deployment Density, λ (BSs/m2)
Plo
c
Ploc
at dmax
= 20m,β/γ = −10dB
Ploc
at dmax
= 20m,β/γ = −14dB
Ploc
at dmax
= 0m,β/γ = −10dB
Ploc
at dmax
= 0m,β/γ = −14dB
Figure 2.3: This figure compares Ploc for different values of λ for L = 4, α = 4, dmax = 20m
and p = q = 1. The markers correspond to simulation results and lines correspond to
analytical results using Theorem 1.
gains in terms of the localization performance for density values of λ = 10−4/m2 and greater.
These results show that localization performance would get better in more dense networks
which is consistent with intuition.
2.6 Summary
The ever-increasing densification of cellular networks is bringing MDs closer to their serving
BSs. Since it is reasonable to assume that the BSs know their true locations (they are
anchors), the presence of a BS in the vicinity of an MD can help in finding its position fix.
Motivated by this, we defined localizability of an MD as a union of two events: (i) at least
L BSs are hearable at the device to be localized, and (ii) the nearest active BS is within
distance dmax from the MD to be localized. Using tools from stochastic geometry, we derived
tractable expressions for the localizability probability by modeling the BS locations as a
Chapter 2. 23
PPP. This work generalizes the analysis of [33] where the information about the location of
the proximate BS was not explicitly incorporated in localization procedure. This work has
several extensions. One natural extension is to characterize the localizability performance
when the MD is aware of the location of the closest overall BS rather than the location of
the closest active BS. In the next Chapter, we extend the analysis presented in this Chapter
to determine the localizability probability of an MD when information about the location
of the closest BS is available to the MD. As discussed in detail in Remark 3, the analysis
to characterize the localizability probability when the MD is aware of the location of closest
BS is mathematically more complicated compared to the analysis when the MD is aware of
the location of the closest active BS.
Chapter 3
Performance of Localizability in
Cellular Networks
3.1 Overview
As already discussed in the previous Chapter, in localization techniques which use measure-
ments from a cellular network to localize an MD, the network is aware of the locations of
BSs to which the MD has connected, since this information is communicated to the network
by the MD. Typically, an MD connects to BSs in close proximity to it. Therefore, we can
safely infer that the network and MD are aware of the location of the BS closest to the MD.
Since information about the location of the closest BS is available to the MD, irrespective
of whether that BS is active or not, it is more natural to study how availability of location
of the closest BS impacts localizability. In the previous Chapter, to preserve tractability,
we discussed the impact of proximate BS measurements on the localizability of an MD in a
cellular network, by assuming that an MD is aware of the location of the closest active BS
(rather than the closest BS). In this Chapter, we enrich our approach further by studying
how localizability is affected when the MD is aware of the location of the closest BS. As will
be evident in the sequel, this makes the analysis a lot more challenging compared to the
24
Chapter 3. 25
analysis in Chapter 2. Again, we evaluate the localizability probability of an MD, where the
localizability is defined as a union of the following events: (i) at least L BSs are hearable at
the MD, and (ii) the geographically closest BS is within a certain predefined distance from
the MD. First, we derive an expression for the localizability probability by taking into ac-
count the coordination between BSs participating in localization procedure in the absence of
shadowing. Finally, we derive some useful bounds and approximations for the localizability
probability in the presence of shadowing.
3.2 Contributions
In localization techniques such as Cell-ID (CID)/Cell-of-Origin (COO), the MD is aware
of locations of BSs around it. When an MD associates to a particular BS (which is most
probably a BS in close proximity), information about its location is communicated to the
MD. In this Chapter, we incorporate information about location of the BS closest to the
MD in the definition of localizability. We define an MD to be localizable as a union of
following events: (i) at least L BSs are hearable at the MD, and (ii) the closest BS is
within a certain predefined distance dmax from the MD. Using tools from stochastic geometry,
we determine the localizability probability of an MD by accounting for the network load
and coordination between BSs. We also derive some easy-to-use approximations for the
localizability probability in the presence of shadowing. These results help provide some key
insights into system design aspects of localizability in a cellular network.
3.3 System Model
Consider a system in which locations of BSs are modeled as a homogenous PPP Φ ∈ R2 with
intensity λ and BSs transmitting at power P . The MD to be localized is assumed to be at
the origin o due to the stationarity of PPPs. Let L be the number of BSs that participate
Chapter 3. 26
R1R1
RL RL
dmax dmax
Case(i): Occurs with probability p
Case(ii): Occurs with probability 1-p
Figure 3.1: Illustration of the system model. The MD is denoted by the triangle in the
center, dark circles denote active BSs and light circles denote inactive BSs. The shaded
region has activity factor p, and the region outside has activity factor q.
in the localization procedure. The BSs that participate in the localization procedure are
selected based on average received power observed at the MD. As shown in Figure 4.1, the
coordination of BSs and network load is modeled using activity factors p and q. Let xi denote
the position of the ith closest BS from the origin in Φ and Ri denote its distance from o. Let
x1 and R1 denote the position and distance of the closest active point from o, respectively.
The activity factor p is the probability that a BS is active in b (o,RL) and q is the probability
that a BS is active outside b (o,RL). The number of BSs active in b (o,RL) is denoted by
Ω =∑L−1
i=1 ai, where ai is a Bernoulli random variable with parameter p. Therefore, Ω follows
a binomial distribution. That is, fΩ (ω) =(L−1ω
)pω (1− p)ω.
Chapter 3. 27
3.3.1 Signal-to-interfernce ratio
In this Chapter, we assume that the network is interference limited, that is, interference
at MD is much higher than noise power. Therefore, a suitable metric that can be used to
select BSs that participate in the localization procedure is SIR observed at the MD. If the
SIR observed is greater than a certain threshold θ, then a BS participates in the localization
procedure. In the absence of shadowing, the SIR depends upon the distance of BS from the
MD and path loss α. Therefore, L BSs participate in the localization procedure if the SIR
from each of the L closest BSs is greater than θ. However, we use the fact that if SIR for
the Lth closest BS is greater than θ, then SIR due to all active BSs closer than the Lth BS
will also be greater than θ. Therefore, if the SIR due to the Lth closest BS is greater than θ,
L BSs participate in the localization procedure. The SIR observed at the MD due to a BS
xk ∈ Φ, where k ∈ [1, L] is
SIRk(L) =PR−αk
L∑i=1i 6=k
aiPRi−α +
∞∑j=L+1
bjPRj−α, (3.1)
where bj indicates whether a BS outside b (o,RL) is active. As was the case in the previous
Chapter, small scale fading is assumed to be averaged out at the receiver and hence ignored
in the analysis. In the first part of this chapter, we study the localizability of an MD in
the absence of shadowing. Hence, shadowing is also ignored in (3.3). In the sequel, we
derive some approximations to study localizability in the presence of shadowing. As done
in [33,40–44], and the previous Chapter, we approximate SIR using the dominant interferer
approach, where the aggregate interference due to all BSs except for the dominant one is
approximated by its mean. Therefore, SIR observed at the MD for the link with the Lth BS
is [33]
SIRL(L) =R−αL
R−αd + 2(Ω−1)2−α .
R2−αL −R2−α
d
R2L−R
2d
+ 2πqλα−2
R2−αL
, (3.2)
Chapter 3. 28
where Rd is the distance of dominant interferer from o. Now, the dominant interferer can
either be at x1 with probability p or at x1 with probability 1− p. Therefore,
SIRL(L) =
S1 =
R−αL
R−α1 +2(Ω−1)
2−α .R2−αL
−R2−α1
R2L−R2
1+ 2πqλα−2
R2−αL
; prob. p
S2 =R−αL
R−α1 +2(Ω−1)
2−α .R2−αL
−R2−α1
R2L−R2
1+ 2πqλα−2
R2−αL
; prob. 1− p(3.3)
3.4 Localizability Performance
3.4.1 Definitions and Preliminaries
As discussed in Chapter 2, the performance of localization depends on the number of BSs
that participate in the localization procedure, which directly drives the number of different
positioning measurements, the accuracy of measurements from the participating BSs and
the geometry of these BSs relative to the MD being localized. It is obvious that significant
gains in localization performance are obtained as the number of participating BSs increases.
As discussed in [33], we define the L−localizability probability of the MD as the probability
that at least L BSs are hearable at the MD.
PL = P [SIRL (L) ≥ θ] = E [1 (SIRL(L) ≥ θ)] (3.4)
In contrast to Chapter 2, here we quantify the gains in localizability probability of an MD
when information about the location of the BS closest is known to the MD. An MD is said
to be localizable if, (i) at least L BSs partcipate in the localization procedure, or (ii) if
the closest BS is within a certain predefined distance dmax from the MD to be localized.
Mathematically, the localizability probability of an MD can be given as
Ploc = P[SIRL(L) ≥ θ
⋃R1 ≤ dmax
]= PL + P [R1 ≤ dmax]− P [SIRL(L) ≥ θ, R1 ≤ dmax] . (3.5)
Chapter 3. 29
3.4.2 Relevant Distance Distributions
In order to evaulate Ploc we would first need to evaluate some distance distributions. First,
we give some well known results. The distribution of RL is [33]
fRL (rL) = exp(−λπr2
L
) 2 (λπr2L)L
(L− 1)!; 0 ≤ rL ≤ ∞. (3.6)
The distribution of R1 conditioned on RL and Ω is given by [33]
fR1|RL,Ω(r1
∣∣rL, ω) =2ωr1
(r2L − r2
1
)ω−1
r2ωL
; 0 ≤ r1 ≤ rL. (3.7)
Lemma 3. Conditioned on R1, RL and Ω, the distribution of R1 when Ω ≥ 1 and the BS
at x1 is not active is
fR1|R1,RL,Ω
(r1
∣∣r1, rL, ω)
=2ωr1 (r2
L − r21)ω−1
(r2L − r2
1)ω , (3.8)
for 0 ≤ r1 ≤ r1 ≤ rL.
Proof. When the dominant interferer is x1, we have that 0 ≤ R1 ≤ R1 ≤ RL. The CDF of
R1 conditioned on R1, RL and Ω can be given by FR1|R1,RL,Ω
(rd∣∣r1, rL, ω
)= 1− P
[R1 > rd
∣∣r1, rL, ω]
= 1− P[min x : r1 ≤ ‖x‖ ≤ rL > rd
∣∣r1, rL]
(a)= 1−
∏x:≤r1‖x‖≤rL
P[‖x‖ > rd
∣∣rL](b)= 1−
∏x:≤r1‖x‖≤rL
r2L − r2
d
r2L − r2
1
= 1−(r2L − r2
d
r2L − r2
1
)ω, (3.9)
where (a) follows from the fact that points are iid in b (o, rL) \b (o, r1) and (b) follows from
the fact that points are uniformly distributed in b (o, rL) \b (o, r1).
Chapter 3. 30
3.4.3 Localizability Probability
Next, we determine, the localizability probability Ploc. The expression for PL in (4.2) is [33]
PL =
1−L−1∑l=0
e−α−2
2qβ/γ
(α−2
2qβ/γ
)ll!
fΩ(0) +L−1∑ω=1
∫ ∞0
∫ rL
0
1 (SIRL (L) ≥ θ)A (r1, rL, ω) drLdr1,
(3.10)
where A (r1, rL, ω) = fR1|RL,Ω(r1|rL, ω)fRL(rL)fΩ(ω). Now we move on to determine
P [SIRL(L) ≥ θ, R1 ≤ dmax]. We will handle the cases for p = 0 and p 6= 0 separately.
Lemma 4. Conditioned on the fact that the closest BS is located in b (o, dmax), the probability
that an MD is localizable when p = 0 is
P [SIRL(L) ≥ θ, R1 ≤ dmax]p=0 =
∫ ∞0
∫ min(rL,dmax)
0
1 (C ≥ θ) fR1,RL (r1, rL) dr1drL,
where C =[α−22πqλ
]r−2L for α > 2 and fR1,RL (r1, rL) follows from multiplying (3.6) and (3.7)
evaluated with ω = L− 1.
Proof. When p = 0 =⇒ Ω = 0, and only the BSs outside b (o,RL) contribute to the
interference and the SIR is C = SIRL (L)∣∣Ω=0
=[α−22πqλ
]R−2L for α > 2. Note that C only
depends on RL. Therefore,
P [SIRL(L) ≥ θ, R1 ≤ dmax]p=0 = ER1,RL [1 (C ≥ θ) , R1 ≤ dmax]
and the proof follows.
Lemma 5. Conditioned on the fact that the closest BS is located in b (o, dmax), the probability
that an MD is localizable when p 6= 0 is
P [SIRL(L) ≥ θ, R1 ≤ dmax]p6=0 =p
[L−1∑ω=1
∫ ∞0
∫ M
0
1 (S1 ≥ θ)P (r1, rL) dr1drL
]+ (1− p)[
L−1∑ω=1
∫ ∞0
∫ rL
0
∫ N
0
1 (S2 ≥ θ)Q (r1, r1, rL, ω) dr1dr1drL
],
Chapter 3. 31
where M = min (rL, dmax), N = min (r1, dmax), P (r1, rL) = fR1|RL(r1
∣∣rL) fRL (rL) fΩ (ω),
and Q (r1, r1, rL, ω) = fR1|R1,RL,Ω
(r1
∣∣r1, rL, ω)fR1|RL,Ω
(r1
∣∣rL, ω) fRL (rL) fΩ (ω).
Proof. When p 6= 0, the joint probability is given by
P [SIRL(L) ≥ θ, R1 ≤ dmax]p6=0 = E [1 (SIRL (L) ≥ θ, R1 ≤ dmax)]
(a)= pE [1 (S1 ≥ θ, R1 ≤ dmax)] + (1− p)E [1 (S2 ≥ θ, R1 ≤ dmax)]
(b)= pEΩ [ER1,RL [1 (S1 ≥ θ, R1 ≤ dmax)]] + (1− p)EΩ
[ER1,R1,RL
[1 (S2 ≥ θ, R1 ≤ dmax)]],
where (a) follows by substitution for SIRL (L) from (3.3), and (b) follows from the fact that
S1 is conditioned on R1, RL and Ω, and S2 is conditioned on R1, R1, RL and Ω. The proof
follows by using distributions from (3.6), (3.7) and (3.8).
Theorem 2. The joint probability of SIRL(L) ≥ θ and R1 ≤ dmax is given by
P [SIRL (L) ≥ θ, R1 ≤ dmax] =
[∫ ∞0
∫ M
0
1 (C ≥ θ) fR1,RL (r1, rL) dr1drL
]fΩ (0) +
p
[L−1∑ω=1
∫ ∞0
∫ M
0
1 (S1 ≥ θ)P (r1, rL) dr1drL
]+ (1− p)[
L−1∑ω=1
∫ ∞0
∫ rL
0
∫ N
0
1 (S2 ≥ θ)Q (r1, r1, rL, ω) dr1dr1drL
]
Proof. The proof follows by using the results from Lemma 4 and Lemma 5, and applying
total probability.
Theorem 3. The localizability probability of an MD is
Ploc = PL +(1− exp
(−λπd2
max
))+
[∫ ∞0
∫ M
0
1 (C ≥ θ) fR1,RL (r1, rL) dr1drL
]fΩ (0) +
p
[L−1∑ω=1
∫ ∞0
∫ M
0
1 (S1 ≥ θ)P (r1, rL) dr1drL
]+ (1− p)[
L−1∑ω=1
∫ ∞0
∫ rL
0
∫ N
0
1 (S2 ≥ θ)Q (r1, r1, rL, ω) dr1dr1drL
]
Proof. The proof follows by substituting results from Theorem 2, (3.10), and that
P [R1 ≤ dmax] = 1− exp (−λπd2max) in (4.2).
Chapter 3. 32
Remark 4. The result derived in Theorem 3 can be used to evaluate the localizability prob-
ability for the case when all BSs are active, that is, p = q = 1 and Ω = L − 1. Note that
when p = 1, the closest active BS is simply the closest BS and the case for p = 0 need not
be evaluated.
3.4.4 Localizability in the presence of shadowing
Next, we study localization performance in the presence of shadowing. The effect of shad-
owing can be incorporated by using displacement theorem [39], in which case we get an
equivalent homogenous PPP Ψ with density λe = λE[S2/α
], where S represents the large
scale shadowing modeled as a lognormal random variable, that is, S = 10S10 and S ∼ N (µ, σ),
where µ and σ are the mean and standard deviation in dB of S. Using the moment generat-
ing function, it can be shown that, E[S2/α
]= exp
(ln(10)
5µα
+ 12
(ln(10)
5σα
)2)
for the lognormal
shadowing. Let the location of ith closest BS in Ψ be represented by xi and its distance from
o be represented by Ri. Let x1 represent the position of the closest active BS in Ψ and
R1 represents its distance from o. Use of displacement theorem ”perturbs” the location of
the geographically closest point. Therefore, in this Chapter, we derive few approximations
and bounds to study the effect of shadowing. The exact analysis is deferred to future work.
First, let us consider the case when shadowing is very high.
Lemma 6. When shadowing variance is very high, the localizability probability can be simply
given by
Ploc u PL + P [R1 ≤ dmax] (1− PL) ,
where PL is evaluated using the equivalent PPP Ψ.
Proof. When shadowing variance is very high, the events SIRL(L) ≥ θ and R1 ≤ dmax are
almost independent. Therefore, (4.2) simplifies to
Ploc u PL + P [R1 ≤ dmax]− P [SIRL(L) ≥ θ]P [R1 ≤ dmax] .
Chapter 3. 33
Lemma 7. A lower bound for Ploc is
Ploc ≥ PL + P [R1 ≤ dmax]− ES[P[SIR ≥ θ, R1 ≤ S−
1αdmax
]],
where P[SIRL (L) ≥ θ, R1 ≤ S−
1αdmax
]can be evaluated using Theorem 2 for Ψ and replacing
dmax with S−1αdmax.
Proof. When the displacement theorem is applied to Φ, the position of a BS located at x ∈ Φ
changes as x = S− 1α
x x where x ∈ Ψ. We now transform the closest point in Ψ back to its
original location in Φ, the closest point in Φ will definitely be closer than this point, that is,
R1 ≤ S1α R1 This would help give a lower bound on Ploc. Therefore,
Ploc ≥PL + P [R1 ≤ dmax]− ES[P[SIRL (L) ≥ θ, S
1α R1 ≤ dmax
]]= PL + P [R1 ≤ dmax]− ES
[I (SIRL (L) ≥ θ) , R1 ≤ S−
1αdmax
].
3.5 Results and Discussion
In this section, we ratify the analytical results and study the tightness of results derived
in Lemmas 6 and 7. Note that all analytical expressions are derived using the dominant
interferer approach. However, the simulations are compiled without using any approxima-
tions. The results in Figure 3.2 are developed for λ = 1/ (1002) BSs/m2, L = 4, α = 4,
p = 1/2, q = 2/3 and dmax ranging from 0m to 25m. It is clear that the analytical results
closely match the simulation results. As it can be observed, there is significant gain in the
localizability performance as dmax increases. This is as expected, since the value of dmax is
inversely proportional to the accuracy of location estimate. Therefore, higher value of dmax
results in higher localizability probability, but lower accuracy. Figure 3.3 shows the effect of
λ on Ploc. It can be observed that higher BS density results in higher value of Ploc, which is
Chapter 3. 34
as expected since higher value of λ results in a denser BS deployment, which increases the
probability that the closest BS is within the distance dmax from the MD. Figure 3.4 shows
that when the shadowing standard deviation is high, the approximation derived in Lemma 6
is tight and the tightness increases as σ increases which is as expected. Figure 3.5 shows
that lower bound derived in Lemma 7 is fairly tight when compared to the exact simulation
results. These simple approximations and bounds preserve tractability and can be used to
obtain key system design insights.
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
SimulationTheorem-3
Increasing dmax
dmax
= 0m to 25m
Figure 3.2: Effect of dmax on Ploc.
Chapter 3. 35
10−6
10−5
10−4
10−3
10−2
0.4
0.5
0.6
0.7
0.8
0.9
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
Density of BSs, λ (in BS/m2)
SimulationTheorem-3
Increasing dmax
dmax
= 10m, 15m
Figure 3.3: Effect of λ on Ploc.
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
ExactLemma-6
Increasing σσ = 20dB, 25dB, 30dB
Figure 3.4: Tightness of Lemma 6.
Chapter 3. 36
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
ExactLemma-7
Figure 3.5: Tightness of Lemma 7.
3.6 Summary
In this Chapter, we use tools from stochastic geometry to derive an expression for the
localizability probability of an MD in a single-tier cellular network by taking into account
the coordination between BSs participating in the localization procedure when information
about the location of the BS closest to the MD is known to the MD. We also derive some
approximations for the localizability probability in the presence of shadowing. A meaningful
extension to this work is to use this model to study the localizability of an MD in a multi-tier
heterogeneous network, which is discussed in more detail in the following Chapter.
Chapter 4
Localizability in Heterogeneous
Cellular Networks
4.1 Overview
In Chapters 2 and 3, we study the localizability of an MD when information about the
location of the closest active BS and closest BS is available to the MD, respectively. In
this Chapter, we use mathematical tools from point process theory and stochastic geometry
to develop a tractable framework to study the impact of proximate BS measurements on
the localizability of an MD in a heterogeneous cellular network (HetNet). An MD is said
to be localizable when either of the following events occur: (i) atleast L BSs are hearable
at the MD to be localized, and (ii) the geographically closest BS to the MD (among all
tiers) is within a certain predefined distance from the MD to be localized. The fact that the
analysis involves hearability of L BSs along with measurements which indicate the location
of the geographically closest BS, leads to the development of a holistic model that can be
used to characterize localizability performance in HetNets. Using these tools, we derive
expressions for the localizability probability of an MD. Our results help quantify the gains in
localizability performance, when the MD is aware about the location of the closest BS. Our
37
Chapter 4. 38
results concretely demonstrate that localizability of an MD in a HetNet improves significantly
when the MD is aware of the location of the closest BS, particularly in dense BS deployments.
4.2 Contributions
4.2.1 New approach to study localizability in HetNets
Using the tools developed in [45], [33], and [42], we develop a tractable approach to study
the performance of localizability in a HetNet. The locations of BSs in the macro cell and
small cell tiers are modeled as homogeneous PPPs with the BSs belonging to different tiers
transmitting with different transmit powers, and each tier having different density. We define
the localizability of an MD as a union of the following events: (i) atleast L BSs participate
in the localization procedure, and (ii) the closest BS is within a certain predefined distance
dmax from the MD to be localized. Note that the distance dmax is simply a representation of
accuracy of the location estimate.
4.2.2 Displacement theorem-based analysis
The two events that constitute the localizability probability have a stark difference; one of
the events deals with the hearability of L BSs involves analysis of the Lth strongest BS over-
all, and the other deals with the proximity of the closest BS to the MD involves analysis of
the geographically closest BS. This leads to a scenario in our analysis which requires the need
to perturb BS locations in a PPP with an exclusion zone at center. Although the displace-
ment theorem is defined for general PPPs [46], its application to the relevant literarure on
localization has been limited to homogeneous PPP [39]. Contrary to this, our work requires
application of displacement theorem to a non-homogeneous PPP (as an intermediate step),
which leads to a non homogeneous PPP, and requires a slightly more careful treatment. In
this Chapter, we characterize the resulting non-homoegenoues PPP (that results from the
Chapter 4. 39
application of the displacement theorem) and use it for the localizability analysis.
4.2.3 Localizability probability
We define a metric to measure localizability performance by taking into account, the hear-
ability of atleast L BSs and information about the location of the closest BS. Using this
metric, we derive tractable expressions for the localizability probability of an MD. Using
these results, we provide key insights into the performance of localizability in a cellular net-
work and quantify the gains in localizability performance as a function of density of BSs
around the MD.
4.3 System Model
RGC
dmax
Mobile DeviceLth strongest BSClosest BS
Tier-1
Tier-2
Figure 4.1: Illustration of the system model.
Consider a 2−tier HetNet as shown in Fig. 4.1, in which BS locations are modeled using
homogeneous PPPs Φ1 and Φ2 with intensities λ1 and λ2, respectively. All BSs, which belong
to Φ1 have transmit power P1, and the BSs that belong to Φ2 have transmit power P2. The
Chapter 4. 40
MD to be localized is assumed to be at the origin due to the stationarity of PPPs. Let xij
denote the location of the ith closest BS in the jth tier, and Rij denote the distance of xij
from the origin o. Due to dense urban deployment of BSs, HetNets are typically assumed
to be interference limited and hence, the noise power can be ignored. Therefore, the signal-
to-interference ratio (SIR) observed by an MD at the origin due to a link with the BS xij
is
SIRxij =Pj‖xij‖−α∑
n Pn∑
xm∈Φn\xij ‖xmn‖−α, (4.1)
where α > 2 is the pathloss exponent. It is assumed that processing at the receiver averages
out the effects of small scale fading and hence does not appear in the above expression. In this
work, since the primary goal is to analyze the performance of localization in a HetNet, the
treatment of large scale shadowing is deferred to future work. Using displacement theorem, it
is possible to define an ”equivalent” homogeneous PPP Φe with intensity λe =∑K
i=1 λiP2/αi
in which each BS transmits at the same power (assumed henceforth as unity) and which
leads to the same SIR distribution as the original PPP. Interested readers are advised to
refer to [47] for a more detailed exposition of this general idea. Let the position of the BSs
in Φe be represented by zi, which corresponds to the ith closest BS, and let Ri represent the
distance of zi from the origin. Let b (o, r) represent a ball of radius r centered at o.
4.4 Localizability Performance
As discussed in Chapters 2 and 3, information about the location of BSs in the proximity
of the MD is available to the MD. In this Chapter, we incorporate information about the
number of BSs that participate in the localization procedure and the location of the closest
BS relative to the MD in our analysis in a HetNet scenario. We define that an MD is
localizable when either of the following events are true: (i) atleast L BSs participate in the
localization procedure, or (ii) when the geographically closest BS among all tiers is located
in b (o, dmax). A BS is said to participate in the localization procedure if the SIR due to its
Chapter 4. 41
link with the MD is greater than a certain threshold θ. Therefore, when all BSs transmit
with the equal transmit power, L BSs participate in the localization procedure if the SIR
due to the link with the Lth closest point is greater than θ.
Definition 1. Let SIRL denote the SIR due to the Lth strongest point. Then, the localizability
probability of an MD is given by
Ploc = P [SIRL ≥ θ ∪ RGC ≤ dmax]
= P [SIRL ≥ θ] + P [RGC ≤ dmax] + P [SIRL ≥ θ, RGC ≤ dmax] , (4.2)
where RGC is the distance of the geographically closest BS from the MD. We define PL =
P [SIRL ≥ θ] and P(RGC≤dmax)L = P [SIRL ≥ θ, RGC ≤ dmax].
Remark 5. As established in Defnition 1, PL is simply the probability that L BSs participate
in the localization procedure, that is, it is the probability that SIRL ≥ θ. This can be handled
by the superposing the two PPPs, Φ1 and Φ2 such that all BSs have unity transmission
power. In the superposed PPP, Φe, the Lth strongest BS is simply the Lth closest. However,
evaluation of P(RGC≤dmax)L is significantly more complicated and requires careful treatment.
In evaluating the joint probability of events (i) and (ii) in a multi-tier setup it becomes
necessary to perturb the position of BSs such that all BSs have the same transmit power
since we would need to determine the Lth strongest BS overall while retaining the location of
the geographically closest BS. The closest BS can belong to either tier. We condition on the
fact that the closest BS belongs to the jth tier, which means that the closest BS is transmitting
at a power Pj. This results in an exclusion zone in R2 in the region b (o,R1j). With this
exclusion zone, we displace the locations of all the BSs such that all BSs transmit at power
Pj. This ensures that the position of x1j is retained and all BSs transmit at the same power
(and the distribution of SIR is preserved). To this end, we develop Theorem 4 to displace
the positions of points in a PPP with an exclusion zone such that the distribution of the SIR
is preserved even after perturbing the locations of points.
Chapter 4. 42
4.4.1 Specializing displacement theorem for PPPs with exclusion
zone
Theorem 4. Let Φ1, ...,ΦK ∈ R2 denote K homogeneous PPPs with locally finite and diffuse
intensity measures Λi, i ∈ [1, K] and intensities λi, i ∈ [1, K]. Let each PPP have its own
constant mark denoted by Pi, i ∈ [1, K]. Conditioned on the fact that the closest point to
the origin belongs to Φj, if each tier is transformed such that all tiers have a constant mark
Pj (and the resultant SIR distribution is preserved), the resultant PPPs Φi’s will be non-
homogeneous with intensities
λi (r) =
0 , for r <
R1j√Cij,
λiCij , for r ≥ R1j√Cij,
where R1j is the distance of the closest point in Φj from the origin, and Cij =(PiPj
)2/α
.
Proof. The closest overall point belongs to the jth-tier and the distance of this point from the
origin is R1j. This means that there are no points inside b(o,R1j). Conditioned on this fact,
we now displace the points from each of the K-tiers such that each point in the displaced
PPP has a mark equal to Pj. This would ensure that the position of the points belonging to
the jth-tier does not change. The points, which belong to Φi are transformed to y ∈ R2 such
that y =(PiPj
)− 1αx = K−
1α
ij x, where y is the location of the points in the displaced PPP. The
underlying process is non-homogeneous in R2 with intensity at a distance of r from o given
by
λ(r)(a)=
0 , for r < R1j,
λi , for r ≥ R1j.
(4.3)
where (a) follows from Slivnyak’s theorem [46]. Using displacement theorem, the transfor-
mation of a PPP with density given by (4.3) with probability kernel h (x,A) is a PPP with
Chapter 4. 43
intensity measure [46]
Λi (A) =
∫R2
h(x,A)λ(x)dx(b)= E
∫R2
I(K−
1α
ij x ∈ A)λ(x)dx = E
∫R2
I(x ∈ AK
1αij
)λ(x)dx
= E∫R2⋂b(o,R1j)
I(x ∈ AK
1αij
)λ(x)dx+ E
∫R2⋂bc(o,R1j)
I(x ∈ AK
1αij
)λ(x)dx
(c)= E
∫R2⋂bc(o,R1j)
I(x ∈ AK
1αij
)λidx = λi
∣∣K 2αijA\b(o,R1j)
∣∣,where (b) follows from the transformation of x to y, (c) follows from the kernel specific to this
lemma, and |.| is the Lebesgue measure. Let Cij = K2αij =⇒ Λi (A) = λi|CijA\b(o,R1j)|.
Now, let the set A over, which we calculate intensity measure be an infinitesimal ring of
thickness dr and radius r with its center at o. The Lebesgue measure of the set A is
|A| = 2πrdr. Then, |CijA\b(o,R1j)| can be calculated as the Lebesgue measure of the set
CijA without the area of intersection between CijA and the ball b(o,R1j). There could be
three possible cases: (i) R1j ≤√Cijr, (ii)
√Cijr ≤ R1j ≤
√Cijr +
√Cijdr, and (iii)
R1j ≥√Cijr +
√Cijdr. The intensity measure can then be given as
Λi (A) = λi|CijA\b(o,R1j)| =
2πλiCijrdr , for r ≥ R1j√Cij,
λi
[(2πCijrdr)−
(π(R2
1j − Cijr2))]
, for r ≤ R1j√Cij≤ r + dr,
0 , for r + dr ≤ R1j√Cij.
Now, the intensity λi(r) can be derived as
λi(r) = limdr→0
Λi (A)
2πrdr=
0 , for r ≤ R1j√
Cij,
λiCij , for r ≥ R1j√Cij.
(4.4)
This is equivalent to a homogeneous PPP in the region bc(o,
R1j√Cij
).
Note that the PPP Φi is conditioned on R1j and the fact that the closest point belongs to
the jth-tier.
Remark 6. The probability that the closest overall point belongs to the kth tier is Ck = λk∑Ki=1 λi
.
Chapter 4. 44
4.4.2 Applying Theorem 4
We now use Theorem 4 to displace the locations of all BSs in both PPPs conditioned on
the position of the closest BS such that all BSs have the same transmit power as the closest
BS (and the distribution of SIR is preserved). The closest BS could belong either to Φ1 or
Φ2. Let us condition on the fact that the closest BS belongs to the jth tier. Let n represent
the tier to which the closest BS belongs. Therefore, we now have an exclusion zone in R2
since there are no points in b (o,R1j). Using Theorem 4, we displace all the points in Φj
and Φi, i, j ∈ 1, 2 , i 6= j except x1j such that all points have the same mark Pj and the
intensities of the displaced PPPs Φi and Φj given by λi and λj respectively, are determined.
Therefore, we now have two non-homogeneous PPPs Φi and Φj in which all BSs have the
same transmit power given by Pj. Note that Cij =(PiPj
)2/α
. We now superpose these PPPs
to obtain a single equivalent PPP and determine its intensity. As it can be observed, if
P1 6= P2, 0 < Cij < 1 or Cij > 1 depending on the ratio of transmit powers of BSs in the two
tiers. The superposition of Φi and Φj would depend on the range in which Cij is present.
When 0 < Cij < 1, the intensity of the superposed PPP, ΦS is
λS (r) =
0 , for 0 ≤ r < R1j ,
λj , for R1j ≤ r < R1j√Cij,
λj + λiCij , forR1j√Cij≤ r ≤ ∞,
(4.5)
and when Cij > 1, the intensity of ΦS is
λS (r) =
0 , for 0 ≤ r < R1j√Cij,
λiCij , forR1j√Cij≤ r < R1j ,
λj + λiCij , for R1j ≤ r ≤ ∞.
(4.6)
The PPP ΦS is conditioned on R1j and the fact that n = j. Note that we now have a single
non-homogeneous PPP ΦS, and an atom at x1j. Applying Theorem 4 causes the locations
of points to change, but the position of x1j is retained. In ΦS, x1j can lie anywhere in R2
since application of Theorem 4 may perturb some points such that they move closer to the
Chapter 4. 45
MD than x1j. It is in fact important to note that x1j can now also be the Lth strongest BS
overall. Hence, we will have to handle the atom at x1j separately in our analysis. Now, let
us consider the case when the transmit powers of BSs belonging to both tiers is the same,
that is, P1 = P2 = P . In this case, the two homogeneous PPPs Φ1 and Φ2 can simply
be superposed to obtain a homogeneous PPP ΦS with intensity λS =∑2
q=1 λq with BSs
transmitting at power P . In our analysis we will handle the 3 cases for 0 < Cij < 1, Cij > 1
and P1 = P2 separately. When P1 6= P2, we first establish the equivalent superposed PPP
appropriately, evaluate the joint probability that the SIR due to the Lth strongest BS is
greater than θ and that R1j ≤ dmax, and decondition on n at the end. When P1 = P2 = P ,
ΦS is a homogeneous PPP and P(RGC≤dmax)L is simple to evaluate. Next, we evaluate the SIR
due to the Lth strongest BS in each of the cases.
Remark 7. Similar to the analyses in Chapters 3 and 4, the SIR is accurately approximated
using the dominant interferer approach, in which the interference due to the closest BS is
considered exactly, and the aggregate interference due to all other BSs is approximated by its
mean.
In general, given a PPP Φ, the SIR observed at the MD due to a link with the Lth strongest
BS is
SIRL =P‖yS‖−α
P‖yD‖−α +∑
yi∈b(o,RS)\b(o,RD) P‖yi‖−α +∑
yj∈bc(o,RS) P‖yj‖−α, (4.7)
where yS is the position of the serving BS, yD is the position of the dominant interferer,
I1 =∑
yi∈b(o,‖yS‖)\b(o,‖yD‖) P‖yi‖−α is the interference due to all BSs in the annular region
b (o, ‖yS‖) \b (o, ‖yD‖) and I2 =∑
yj∈bc(o,‖yS‖) P‖yj‖−α is the interference due to all BSs
outside the region b (o, ‖yS‖). We use the dominant interferer approximation, in which I1
and I2 are approximated by their means. That is, (4.7) can be reduced to
SIRL u‖yS‖−α
‖yD‖−α + E [I1] + E [I2]. (4.8)
Note that in (4.8), we denote the position of the Lth strongest BS above by yS and the
dominant interferer by yD because, multiple cases are encountered in the analysis which will
Chapter 4. 46
be handled separately. This will be evident in the sequel. We now determine SIRL, which
we will use to evaluate PL in section 4.4.4. In Φe, all BSs have unity transmission power.
The serving BS is the Lth strongest BS, which is the Lth closest BS overall, located at zL.
The dominant interferer is the closest overall BS located at z1. We now determine the SIR
of the link between the Lth strongest BS in the PPP Φe and the MD at the origin in the
following lemma.
Lemma 8. In Φe, the SIR due to the link between the Lth strongest BS and the MD condi-
tioned on R1 and RL is
SIRL =R−αL
R−α1 + 2(L−2)2−α
R2−αL −R2−α
1
R2L−R
21
+ 2πλeα−2R
2−αL
, (4.9)
for α > 2.
Proof. In this scenario, zL is the serving BS, and z1 is the dominant interferer. There are L−2
points i.i.d and uniformly distributed in the annular region b (o,RL) \b (o,R1). The mean
of I1 can be evaluated as E [I1] = E[∑L−1
i=2 PjR−αi
]=∑L−1
i=2 E[R−αi
]= (L− 2)E
[R−αi
]=
(L− 2)∫ RLR1
r−α 2rR2L−R
21dr = 2(L−2)
R2L−R
21
∫ RLR1
r1−αdr = 2(L−2)2−α
R2−αL −R2−α
1
R2L−R
21
. Next, let us evaluate the
mean of I2 as E [I2] = E[∑
z∈Φe\b(o,RL) ‖z‖−α]
=∫ 2π
0
∫∞RLr−αλ (r) rdrdθ =
2π∫∞RLr1−αλ (r) dr = 2πλe
∫∞RLr1−αdr = 2πλe
2−αR2−αL for α > 2. Substitution in (4.8) completes
the proof.
Next, we evaluate the SIR, which we will use to evaluate P(RGC≤dmax)L in section 4.4.4. Recall
that in this case, the location of the closest point must be retained. Hence, the anlaysis
progresses differently compared to the analysis for PL. We consider the cases for 0 < Cij < 1,
Cij > 1 and P1 = P2 separately. When 0 < Cij < 1, the atom at x1j is always the
dominant interferer since no points can lie in the region b (o,R1j) as can be inferred from
(4.5). Therefore, xL−1 is the Lth strongest BS. However, multiple cases are encountered
depending on the range in whichR1j√Cij
is present. The interference evaluated depends on
the range in whichR1j√Cij
is present. We determine the SIR in each of these cases along with
probabilities of occurence of these cases.
Chapter 4. 47
Lemma 9. The SIR due to the link between the MD and the Lth strongest BS conditioned
on RL−1, R1j and n = j when 0 < Cij < 1 is
SIR(Cij<1)L =
L1 , with prob. P
(Cij<1)1 ,
L2 , with prob. P(Cij<1)2 ,
L3 , with prob. P(Cij<1)3 ,
(4.10)
where L1 =R−αL−1
R−αij + 2(L−2)2−α
R2−αL−1−
(R1j√Cij
)2−α
R2L−1−
R21j
Cij
+2π(λj+λiCij)
α−2R2−αL−1
,
L2 =R−αL−1
R−αij + 2(L−2)2−α
R2−αL−1−R
2−α1j
R2L−1−R
21j
+ 2πα−2
[λjR
2−αL−1 + λiCij
(R1j√Cij
)2−α] ,L3 =
R−αL−1
R−αij + R−α1 + 2(L−3)2−α
(λj+λiCij)R2−αL−1−λjR
2−α1 −λiCij
(R1j√Cij
)2−α
(λj+λiCij)R2L−1−λjR
21−λiCij
(R1j√Cij
)2
+2π(λj+λiCij)
α−2R2−αL−1
,
P(Cij<1)1 = ER1j |n=j
[1− FR1|R1j ,n=j
(R1j√C1j
)],
P(Cij<1)2 = ER1j |n=j
[FRL−1|R1j ,n=j
(R1j√Cij
)− FRL−1|R1j ,n=j (R1j)
], and
P(Cij<1)3 = 1− P (Cij<1)
1 − P (Cij<1)2 .
Proof. When 0 < Cij < 1, it can be inferred from the expression of λS in (4.5) that x1j is
closest to the origin. That is, x1j is the dominant interferer and the Lth strongest point is
xL−1. In this scenario, there are three possible cases depending on the range, in whichR1j√Cij
is present:
1. R1j ≤ R1j√Cij≤ R1, which occurs with probability
P(Cij<1)1 = ER1j |n=j
[1− FR1|R1j ,n=j
(R1j√C1j
)]. There are L − 2 points i.i.d and uni-
formly distributed in the annular region b(o, RL−1
)\b(o,
R1j√Cij
), and we have E [I1]
(a)=
Chapter 4. 48
2Pj(L−2)
2−α
R2−αL−1−
(R1j√Cij
)2−α
R2L−1−
R21j
Cij
, where (a) follows from Lemma 22, and
E [I2](b)=
2πPj(λj+λiCij)
α−2R2−αL−1 for α > 2, where (b) follows from Lemma 25.
2. R1j ≤ RL−1 ≤ R1j√Cij
, which occurs with probability
P(Cij<1)2 = ER1j |n=j
[FRL−1|R1j ,n=j
(R1j√Cij
)− FRL−1|R1j ,n=j (R1j)
]. Here L−2 points are
i.i.d and uniformly distributed in the annular region b(o, RL−1
)\b (o,R1j), and we
have E [I1](e)=
2Pj(L−2)
2−αR2−αL−1−R
2−α1j
R2L−1−R
21j
, where (e) follows from Lemma 22, and E [I2](f)=
2πPjα−2
[λjR
2−αL−1 + λiCij
(R1j√Cij
)2−α]for α > 2, where (f) follows from Lemma 24.
3. R1j ≤ R1 ≤ R1j√Cij≤ RL−1, which occurs with probability P
(Cij<1)3 = 1 − P
(Cij<1)1 −
P(Cij<1)2 . Here there are L − 3 points, which are i.i.d, but not uniformly distributed
between R1 and the serving BS, RL−1. Therefore we have,
E [I1](c)=
2Pj(L−3)
2−α
λj((
R1j√Cij
)2−α−R2−α
1
)+(λj+λiCij)
(R2−αL−1−
(R1j√Cij
)2−α)
λj
((R1j√Cij
)2
−R21
)+(λj+λiCij)
(R2L−1−
(R1j√Cij
)2)
, where (c) follows
from Lemma 23, and E [I2](d)=
2πPj(λj+λiCij)
α−2R2−αL−1 for α > 2, where (d) follows from
Lemma 25.
Substitution in (4.8) completes the proof.
When Cij > 1, multiple cases are encountered depending on the location of x1j in R2. In this
case, there are no points in the region b(o,
R1j√Cij
)as it can be inferred from the expression
for density of ΦS when Cij > 1 in (4.6). The location of R1j determines the serving BS
and the dominant interferer. As we will see in the following lemma, four different cases are
encountered when Cij > 1 and we determine the SIR in each of the cases depending on the
dominant interferer and the Lth strongest BS overall. As done in Lemma 9, we determine
the probability of occurence of each of these cases.
Lemma 10. The SIR due to the link between the MD and the Lth strongest BS conditioned
Chapter 4. 49
on R1, RL−1, RL, R1j and n = j when Cij > 1 is
SIR(Cij>1)L =
G1 , with prob. P(Cij>1)1 ,
G2 , with prob. P(Cij>1)2 ,
G3 , with prob. P(Cij>1)3 ,
G4 , with prob. P(Cij>1)4 ,
(4.11)
where G1 =R−αL−1
R−α1j + 2(L−2)2−α
R2−αL−1−R
2−α1j
R2L−1−R
21j
+2π(λj+λiCij)
α−2R2−αL−1
,
G2 =R−α1j
R−α1 + 2(L−2)2−α
R2−α1j −R
2−α1
R21j−R2
1+
2π(λj+λiCij)
α−2R2−α
1j
,
G3 =R−αL
R−α1 + R−α1j + 2(L−2)2−α
R2−αL −R2−α
1
R2L−R
21
+ 2πα−2
[λiCijR
2−αL + λj
(R1j√Cij
)2−α] ,G4 =
R−αL−1
R−α1 + R−α1j + 2(L−3)2−α
[(λj+λiCij)R
2−αL−1−λiCijR
2−α1 −λjR2−α
1j
(λj+λiCij)R2L−1−λiCijR
21−λjR2
1j
]+
2π(λj+λiCij)
α−2R2−αL−1
,
P(Cij>1)1 = ER1j |n=j
[1− FR1|R1j ,n=j (R1j)
],
P(Cij>1)2 = ER1j |n=j
πλiCij(R2
1j −R2
1j
Cij
)L−1
(L− 1)!exp
(πλiCij
(R2
1j −R2
1j
Cij
)) ,P
(Cij>1)3 = ER1j |n=j
[FRL|R1j ,n=j
(R1j
)− FRL|R1j ,n=j
(R1j√Cij
)], and
P(Cij>1)4 = 1− P (Cij>1)
1 − P (Cij>1)2 − P (Cij>1)
3 .
Proof. When Cij > 1, the serving BS could be x1j, xL−1 or xL depending on the location of
x1j. There are four possible scenarios in this case:
1.R1j√Cij≤ R1j ≤ R1, which occurs with probability
P(Cij>1)1 = ER1j |n=j
[1− FR1|R1j ,n=j (R1j)
]. This follows from the fact that there are no
Chapter 4. 50
points in the annular region b (o,R1j) \b(o,
R1j√Cij
). Here, the BS at x1j is the dominant
interferer and xL−1 is the serving BS. There are L − 2 points, which are uniformly
distributed and i.i.d in the annular region b(o, RL−1
)\b (o,R1j). Therefore, E [I1]
(a)=
2Pj(L−2)
2−αR2−αL−1−R
2−α1j
R2L−1−R
21j
, where (a) follows from Lemma 22, and E [I2](b)=
2πPj(λj+λiCij)
α−2R2−αL−1,
where (b) follows from Lemma 25.
2.R1j√Cij≤ RL−1 ≤ R1j ≤ RL, wich occurs with probability
P(Cij>1)2 = ER1j |n=j
πλiCij(R2
1j−R2
1jCij
)L−1
(L−1)!exp
(πλiCij
(R2
1j −R2
1j
Cij
)). This follows from
the fact that there are exactly L−1 points in the annular region b (o,R1j) \b(o,
R1j√Cij
).
Here, x1 is the dominant interferer, and x1j is the serving BS. There are L − 2
points uniformly distributed and i.i.d in the annular region b (o,R1j) \b(o, R1
). There-
fore, E [I1](e)=
2Pj(L−2)
2−αR2−α
1j −R2−α1
R21j−R2
1, where (e) follows from Lemma 22, and E [I2]
(f)=
2πPj(λj+λiCij)
α−2R2−α
1j , where (f) follows from Lemma 25.
3.R1j√Cij≤ RL ≤ R1j, which occurs with probability
P(Cij>1)3 = ER1j |n=j
[FRL|R1j ,n=j
(R1j
)− FRL|R1j ,n=j
(R1j√Cij
)]. Here, x1 is the dominant
interferer, and xL is the serving BS. There are L− 2 points uniformly distributed and
i.i.d in the annular region b(o, RL
)\b(o, R1
). Therefore, E [I1]
(g)=
2Pj(L−2)
2−αR2−αL −R2−α
1
R2L−R
21
,
where (g) follows from Lemma 22, and E [I2](h)=
2πPjα−2
[λiCijR
2−αL + λjR
2−α1j
], where (h)
follows from Lemma 24. Note that there is a point at x1j, which needs to be handled
separately while calculating the overall interference.
4.R1j√Cij≤ R1 ≤ R1j ≤ RL−1, which occurs with probability
P(Cij>1)4 = 1−P (Cij>1)
1 −P (Cij>1)2 −P (Cij>1)
3 . Here, x1 is the dominant interferer, and xL−1
is the serving BS. There are L − 3 points in the annular region b(o, RL−1
)\b(o, R1
),
which are i.i.d but not uniformly distributed. Therefore,
E [I1](c)=
2Pj(L−3)
2−α
[(λj+λiCij)R
2−αL−1−λiCijR
2−α1 −λjR2−α
1j
(λj+λiCij)R2L−1−λiCijR
21−λjR2
1j
], where (c) follows from Lemma 23,
Chapter 4. 51
and E [I2](d)=
2πPj(λj+λiCij)
α−2R2−αL−1, where (d) follows from Lemma 25. Note that there
is a point at x1j, which needs to be handled separately while calculating the overall
interference.
Substitution in (4.8) completes the proof.
Remark 8. Note that when P1 = P2, BSs belonging to both tiers have the same transmit
power P . Therefore, ΦS is simply a homogeneous PPP with intensity λ0 =∑2
k=1 λk and
all BSs have a transmit power P . The serving BS is xL and the dominant interferer is x1.
Therefore, SIRL conditioned on RL and R1 is given by
SIR(P1=P2)L =
R−αL
R−α1 + 2(L−2)2−α
R2−αL −R2−α
1
R2L−R
21
+ 2πλ0α−2 R
2−αL
. (4.12)
4.4.3 Distance Distributions
In this section, we derive the relevant distance distributions we will use to evaluate the
localizability probability in section 4.4.4. Note that we need to derive all the distance
distributions needed to decondition the SIR expressions evaluated in (4.9), (4.11), (4.10),
and (4.12). These distance distributions are derived for the superposed PPP ΦS. First we
provide some well known results. Given the homogeneous PPP Φe with intensity λe, the pdf
of RL is [48]
fRL (rL) =2(λeπr
2L
)Lexp
(−λeπr2
L
)rL (L− 1)!
for 0 ≤ rL ≤ ∞. (4.13)
The pdf of R1 conditioned on RL is given by [33]
fR1|RL(r1
∣∣rL) =2 (L− 1) r1
r2(L−1)L
(r2L − r2
1
)L−2for 0 ≤ r1 ≤ rL. (4.14)
The cdf of distance of the geographically closest point from the origin, RGC is
FRGC (r) = 1− exp(−λ0πr
2), λ0 =
2∑k=1
λk. (4.15)
Lemma 11. The probability density function of R1j conditioned on the fact that n = j is
fR1j |n=j
(r∣∣n = j
)= 2λ0πre
−λ0πr2
. (4.16)
Chapter 4. 52
Proof. See Appendix A.1.
Now, we derive the distance distribution of the kth closest point conditioned on R1j and
n = j when 0 < Cij < 1 and Cij > 1. These distributions will be useful in evaluating several
other distance distributions in the sequel. It is also used to evaluate the probabilities of
occurences of each of the cases encountered in (4.10) and (4.11).
Lemma 12. The pdf of Rk conditioned on R1j and n = j when 0 < Cij < 1 is
fRk|R1j ,n=j
(r∣∣R1j , n = j
)=
0 , for 0 ≤ r < R1j ,
2πλjr(πλj(r2−R21j))
k−1
(k−1)! e(−(πλj(r2−R21j))) , for R1j ≤ r < R1j√
Cij,
2π(λj+λiCij)r((λiCij+λj)πr2−(λi+λj)πR
21j)
k−1
(k−1)! e(−((λiCij+λj)πr2−(λi+λj)πR
21j)) , for
R1j√Cij≤ r ≤ ∞.
(4.17)
Proof. See Appendix A.2
Lemma 13. The pdf of Rk conditioned on R1j and n = j when Cij > 1 is
fRk|R1j ,n=j
(r∣∣R1j , n = j
)=
0 , for 0 ≤ r < R1j√Cij,
2πλiCijr(πλi(Cijr2−R21j))
k−1
(k−1)! e(−(πλi(Cijr2−R21j))) , for
R1j√Cij≤ r < R1j ,
2π(λj+λiCij)r((λiCij+λj)πr2−(λi+λj)πR
21j)
k−1
(k−1)! e(−((λiCij+λj)πr2−(λi+λj)πR
21j)) , for R1j ≤ r ≤ ∞.
.
(4.18)
Proof. The proof follows along similar lines as the proof for Lemma 12.
Remark 9. The distribution of Rk conditioned on n = j when 0 < Cij < 1 and Cij > 1 can
be obtained as follows
fRk|n=j
(r∣∣n = j
)=
∫ ∞0
fRk|R1j ,n=j
(r∣∣r1j, n = j
)fR1j |n=j
(r1j
∣∣n = j)
dr1j, (4.19)
where fRk|R1j ,n=j
(r∣∣R1j, n = j
)follows from (4.17) if 0 < Cij < 1 and from (4.18) if Cij > 1,
and fR1j |n=j
(r1j
∣∣n = j)
follows from (4.16).
Chapter 4. 53
Next, we derive the relevant distance distributions for each of the cases encountered when
0 < Cij < 1 and Cij > 1. Lemmas 14, 16 and 15 deal with the cases encountered when
0 < Cij < 1. The distance distributions evaluated in these lemmas will be used to decondition
each of the three cases encountered in (4.10) separately and then use total probability to
determine the result for the case when 0 < Cij < 1.
Lemma 14. The joint pdf of RL−1 and R1j conditioned on n = j when R1j ≤ R1j√Cij≤ R1 ≤
RL−1 and 0 < Cij < 1 is fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
=
4πLλ0 (λj + λiCij)L−1
(L− 2)!rL−1r1j
(r2L−1 −
r21j
Cij
)L−2
exp
(−π[(λj + λiCij) r
2L−1 +
λj (Cij − 1)
Cijr2
1j
]).
(4.20)
Proof. See Appendix A.3.
Lemma 15. The joint pdf of RL−1 and R1j conditioned on n = j when R1j ≤ RL−1 ≤ R1j√Cij
and 0 < Cij < 1 is fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
=
rL−1r1j
(πλj
(r2L−1 − r2
1j
))k−1exp
(−πλj r2
L−1 − πλir21j
)∫∞
0
∫ r1j√C1j
r1j rL−1r1j
(πλj
(r2L−1 − r2
1j
))k−1exp
(−πλj r2
L−1 − πλir21j
)drL−1dr1j
. (4.21)
Proof. See Appendix A.4.
Lemma 16. The joint pdf of R1, RL−1 and R1j conditioned on n = j when R1j ≤ R1 ≤R1j√Cij≤ RL−1 and 0 < Cij < 1 is
fR1,RL−1,R1j |n=j
(r1, rL−1, r1j
∣∣n = j)
=L−3∑τ=0
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)fR1,R1j |n=j
(r1, r1j
∣∣n = j), (4.22)
where fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
follows from (A.6),
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)follows from (A.5) and fR1,R1j |n=j
(r1, r1j
∣∣n = j)
follows from
(A.7).
Chapter 4. 54
Proof. See Appendix A.5.
Next, we derive the distance distributions for every case encountered when Cij > 1. As it
can be seen from (4.11), four cases are encountered depending on the location of x1j ∈ R2
and we derive the distance distributions to decondition each of these expression separately
in lemmas 17, 20, 18 and 19, and use total probability to arrive at the result for Cij > 1.
Lemma 17. The joint pdf of RL−1 and R1j conditioned on n = j whenR1j√Cij≤ R1j ≤ R1 ≤
RL−1 and Cij > 1 is fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
=
4πLλ0 (λj + λiCij)L−1 rL−1r1j
(r2L−1 − r2
1j
)L−2exp
(−π (λj + λiCij) r
2L−1
)exp
(πλi (Cij − 1) r2
1j
)(L− 2)!
.
(4.23)
Proof. See Appendix A.6.
Lemma 18. The joint pdf of R1 and R1j conditioned on n = j whenR1j√Cij≤ RL−1 ≤ R1j ≤
RL and Cij > 1 is
fR1,R1j |n=j
(r1, r1j
∣∣n = j)
= fR1|R1j ,n=j
(r1
∣∣r1j, n = j)fR1j |n=j
(r1j
∣∣n = j), (4.24)
where fR1|R1j ,n=j
(r1
∣∣r1j, n = j)
follows from (A.10) and fR1j |n=j
(r1j
∣∣n = j)
follows from
(4.16).
Proof. See Appendix A.7.
Lemma 19. The joint pdf of R1, RL and R1j conditioned on n = j whenR1j√Cij≤ RL ≤ R1j
and Cij > 1 is
fR1,RL,R1j |n=j
(r1, rL, r1j
∣∣n = j)
= fR1|RL,R1j ,n=j
(r1
∣∣rL, r1j , n = j)fRL,R1j |n=j
(rL, r1j
∣∣n = j),
(4.25)
where fR1|RL,R1j ,n=j
(r1
∣∣rL, r1j, n = j)
follows from (A.12), fRL,R1j |n=j
(rL, r1j
∣∣n = j)
follows
from (A.13).
Chapter 4. 55
Proof. See Appendix A.8.
Lemma 20. The joint pdf of R1, RL−1 and R1j conditioned on n = j whenR1j√Cij≤ R1 ≤
R1j ≤ RL−1 and Cij > 1 is
fR1,RL−1,R1j |n=j
(r1, rL−1, r1j
∣∣n = j)
=L−3∑τ=0
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)fR1,R1j |n=j
(r1, r1j
∣∣n = j), (4.26)
where fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
follows from (A.16),
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)follows from (A.15), fR1,R1j |n=j
(r1, r1j
∣∣n = j)
follows from (A.17).
Proof. See Appendix A.9.
4.4.4 Localizability Probability
Now that we have the expressions for SIR and the distance distributions, we first evaluate
the expression for PL and P(RGC≤dmax)L . Using these results, we evaluate the localizability
probability Ploc and then provide an easy-to-use approximation for Ploc.
Lemma 21. The L−localizability probability of an MD is given by
PL = ER1,RL [SIRL ≥ θ] , (4.27)
where SIRL follows from (4.9) and the joint distribution of R1 and RL follows from (4.13)
and (4.14).
Proof. The L−localizability probability of an MD is given by
PL = P [SIRL ≥ θ](a)= ER1,RL [SIRL ≥ θ] ,
where (a) follows from the fact that SIRL in (4.9) is conditioned on R1 and RL and hence
we need to jointly decondition on these two random variables using the pdfs from (4.13) and
(4.14).
Chapter 4. 56
Corollary 3. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when 0 < Cij < 1 is
PL =2∑j=1
Cj
[3∑
a=1
PLaP(Cij<1)a
]1 (0 < Cij < 1) , (4.28)
where
PL1 = ERL−1,R1j |n=j [1 (L1 ≥ θ)] , is evaluated using the joint distribution (4.20),
PL2 = ERL−1,R1j |n=j [1 (L2 ≥ θ)] , is evaluated using the joint distribution (4.21),
PL3 = ER1,RL−1,R1j |n=j [1 (L3 ≥ θ)] , is evaluated using the joint distribution (4.22),
with R1j ∈ [0, dmax] and P (Cij<1)a , a ∈ 1, 2, 3 follow from Lemma 9.
Proof. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when 0 < Cij < 1 is given
by PL = P[SIR
(Cij<1)L ≥ θ, RGC ≤ dmax
]. To determine this we simply need to decondition
over the SIR(Cij<1)L given by (4.10). Since there are three possible cases L1, L2, and L3, which
occur with probabilities P(Cij<1)1 , P
(Cij<1)2 , and P
(Cij<1)3 , respectively, we first decondition
each case and use total probability to arrive at the result.
Corollary 4. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when Cij > 1 is
PG =2∑j=1
Cj
[4∑b=1
PGbP
(Cij>1)b
]1 (Cij > 1) , (4.29)
where
PG1 = ERL−1,R1j |n=j [1 (G1 ≥ θ)] , is evaluated using the joint distribution (4.23),
PG2 = ER1,R1j |n=j [1 (G2 ≥ θ)] , is evaluated using the joint distribution (4.24),
PG3 = ER1,RL,R1j |n=j [1 (G3 ≥ θ)] , is evaluated using the joint distribution (4.25),
PG4 = ER1,RL−1,R1j |n=j [1 (G4 ≥ θ)] , is evaluated using the joint distribution (4.26),
with R1j ∈ [0, dmax] , and P(Cij>1)b , b ∈ 1, 2, 3, 4 follow from Lemma 10.
Proof. This proof follows along similar lines as Corollary 3.
Chapter 4. 57
Corollary 5. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when P1 = P2 is
PE = ER1,RL
[SIR
(P1=P2)L ≥ θ
]1 (P1 = P2) , (4.30)
where SIR(P1=P2)L follows from (4.12), the joint distribution of R1 and RL follows from (4.13)
and (4.14), and R1 is deconditioned over the range 0− dmax.
Proof. The proof follows along similar lines as Lemma 21 except for the fact that R1 is
deconditioned over the range 0− dmax.
Next, we combine the results of Corollaries 3, 4, and 5 to derive the expression for P(RGC≤dmax)L .
Theorem 5. The joint probability that SIRL ≥ θ and that RGC ≤ dmax is
P(RGC≤dmax)L = PL + PE + PG (4.31)
where PL follows from (4.28), PE follows from (4.30), and PG follows from (4.29).
Proof. This proof simply follows by collating the results of Corollaries 3, 4, and 5.
We now derive the localizability probability by substituting the results of Lemma 21 and
Theorem 5 in (4.2).
Theorem 6. The probability that an MD is localizable either by L hearable BSs or by the
fact that the geographically closest BS is in b (o, dmax) is
Ploc = PL +[1− exp
(−λ0πd
2max
)]+ P
(RGC≤dmax)L (4.32)
where PL follows from (4.27) and P(RGC≤dmax)L follows from (4.31).
Proof. The proof simply follows by substituting (4.27) and (4.31) in (4.2) and the fact that
P [RGC ≤ dmax](a)= 1− exp (−λ0πd
2max), where (a) follows from (4.15).
Chapter 4. 58
It can be observed from the expression of Ploc in (4.32) and the expressions of (4.28) and
(4.29), that the expression for Ploc is computationally intensive. To this end, we now derive
a simple, yet tight approximation for the localizability probability Ploc.
Remark 10. We work with a 2-tier setup, in which the location of BSs are homogeneous
PPPs Φ1 and Φ2 with intensities λ1 and λ2 respectively, The BSs in Φ1 transmit with power
P1 and BSs in Φ2 transmit with power P2. Now, one of these tiers is a macro cell and the
other is a small-cell. If Φj is a macro cell then Φi is the small cell, where i, j ∈ 1, 2 , i 6= j.
The density of the small cell is always higher than the macro cell and the transmit power of
BSs in the macro cell is higher than the transmit power of BSs in the small cell. That is,
λj > λi and Pi > Pj. In such a scenario, the expressions for SIR(Cij<1)L and SIR
(Cij>1)L from
(4.10) and (4.11) respectively, can be approximated as follows:
SIR(Cij<1)L u L1 =
R−αL−1
R−αij + 2(L−2)2−α
R2−αL−1−
(R1j√Cij
)2−α
R2L−1−
R21j
Cij
+2π(λj+λiCij)
α−2R2−αL−1
(4.33)
SIR(Cij>1)L u G1 =
R−αL−1
R−α1j + 2(L−2)2−α
R2−αL−1−R
2−α1j
R2L−1−R
21j
+2π(λj+λiCij)
α−2R2−αL−1
(4.34)
The approximations in (4.33) and (4.34) can be used since effectively since these are the cases
that occur that with high probability and hence other cases can be ignored. It is important
to note that SIR(Cij<1)L and SIR
(Cij>1)L depend only on three random variables RL−1 and R1j,
and n.
We now derive the approximate expression for P(RGC≤dmax)L in the following theorem.
Theorem 7. An approximation for the joint probability of SIRL ≥ θ and RGC ≤ dmax, that
is, P(RGC≤dmax)L is
P(RGC≤dmax)L u PE1 (P1 = P2)
+2∑j=1
Cj
ERL−1,R1j |n=j [1 (L1 ≥ θ)]︸ ︷︷ ︸Expression-1
1 (0 < Cij < 1) + ERL−1,R1j |n=j [1 (G1 ≥ θ)]︸ ︷︷ ︸Expression-2
1 (Cij > 1)
,(4.35)
Chapter 4. 59
where Expression-1 is deconditioned using the pdf in (4.20), and Expression-2 is decondi-
tioned using the pdf in (4.23) with R1j in the range 0− dmax.
Proof. The proof simply follows from the fact that the case L1 occurs with a high probability
when 0 < Cij < 1 and G1 occurs with a high probability when Cij > 1. Once this is
established, the proof follows along the same lines as corollaries 3 and 4.
We now use this result to give the approximate expression for Ploc.
Corollary 6. The probability that an MD is localizable either by L hearable BSs or by the
fact that the geographically closest BS is in b (o, dmax) is
Ploc u PL +[1− exp
(−λ0πd
2max
)]+ P
(RGC≤dmax)L (4.36)
where where PL follows from (4.27) and P(RGC≤dmax)L follows from (4.35).
Proof. The proof follows along similar lines as Theorem 6, except that P(RGC≤dmax)L follows
from (4.35).
4.5 Numerical Results and Discussions
In this section, we verify our theoretical results and study the tightness of the approximation
provided in Corollary 6. We also provide some key insights and trends that are observed by
varying the system parameters. First, we provide a brief outline of the simulation framework.
Let us consider tier-1 to be the macro cell and tier-2 to be the small-cell. The simualtion
results of Fig. 4.2a are evaluated using λ2 = 103λ1, P1 = 102P2, dmax = 4m, L = 4, and α = 4.
The analytical results are calculated using the expressions from Theorem 6 and Corollary 6.
Note that the analytical results are derived with an approximation that only the interference
due to the dominant interferer is considered exactly and the aggregate interference due to
all other BSs is approximated by its mean. However, this approximation is not used while
Chapter 4. 60
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
SIR Threshold, θ (in dB)
Loca
lizab
ility
Pro
babi
lity
Plo
c
SimulationTheorem−6Corollary−6
(a) Tightness of approximation
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
SimulationCorollary-6
Increasing dmax
dmax
= 0m to 4m
(b) Effect of increasing dmax.
Figure 4.2: Illustration of the tightness of results in Theorem 6
and Corollary 6 with true simuation and the effect of increasing dmax on Ploc.
evaluating the simulation results. It can be observed from the plot in Fig. 4.2a that the results
from Theorem 6 exactly match the simulation and the approximation of Corollary 6 remains
tight over the entire range of the SIR threshold, θ. Next, we study the impact of increasing
dmax on the localizability probability as illustrated in Fig. 4.2b. The results in this figure are
obtained using λ2 = 103λ1, P1 = 102P2, L = 4, α = 4, and increasing dmax from 0m to 4m.
The case for dmax = 0m corresponds to the scenario when information about the location of
the closest BS is not used in the localization procedure. It can be said that dmax is essentially
the accuracy upto which the position estimate of the MD can be obtained. It is intuitive that
as this constraint is relaxed, that is, dmax is increased, the localizability probability of the
MD increases as it can be observed from the results in Fig. 4.2b. Now, we study the impact
of increasing L on the localizability probability as illustrated in Fig. 4.3a. The results in this
figure are compiled using λ2 = 103λ1, P1 = 102P2, dmax = 4m, α = 4 and increasing L from 4
to 10. It can be observed from the results that, the localizability probability decreases with
increase in the value of L. This is because, increasing the value of L leads to an increase
in the interference due to the BSs in the near field, that is, the number of BSs contributing
to the interference increases. Due to this, some form of interference mitigation techniques
need to be employed in order to achieve improvement in performance at higher values of
Chapter 4. 61
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
SimulationCorollary-6
Increasing LL = 4,6,8,10
(a) Effect of increasing L.
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
SIR Threshold, θ (in dB)
SimulationCorollary-6
Increasing αα = 3,4,5,6
(b) Effect of increasing α.
Figure 4.3: Illustration of the effect of increasing L
and α on Ploc.
L. Note that even as L increases, Ploc saturates to a constant at higher SIR thresholds.
This happens because of incorporating information about the location of the closest BS in
the localization procedure. Therefore, the performance degradation due to increasing L, is
significantly dampened by using proximate BS measurements. Next, we study the impact of
increasing α on Ploc as illustrated in Fig. 4.3b. It can be observed that increasing the value
of α causes a degradation in the geolocation performance. This occurs because of the fact
that the effective SIR reduces as α increases. Next, we discuss the impact of increasing the
density of BSs in the small cell on localizability probability as illustrated in Fig. 4.4a. These
results are compiled for P1 = 102P2, L = 4, α = 4 and an SIR threshold of θ = −20dB. The
distance dmax is increased from 0m to 8m. The trends in the figure clearly show that the
geolocation performance improves significantly as the density of BSs is increased. This is as
expected since the probability of the closest BS lying inside the region b (o, dmax) increases
for a given value of dmax as the density increases. This result confirms that, the performance
of Cell-ID/COO based localization approaches improves significantly in areas with dense BS
deployments. Finally, we also, observed the effect of increasing the transmission power of
BSs in the small cell on geolocation performance as illustrated in Fig. 4.4b. These results
show that there is a negligible change in the performance of localization as the transmit
Chapter 4. 62
10−5
10−4
10−3
10−2
10−1
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
Density of BSs in Small−Cell (in BS/m2)
SimulationCorollary-6
Increasing dmax
dmax
= 0m,4m,6m,8m
(a) Effect of increasing small cell density.
−30 −25 −20 −15 −10 −50
0.2
0.4
0.6
0.8
1
Loca
lizab
ility
Pro
babi
lity,
Plo
c
P2/P
1 (in dB)
SimulationCorollary-6
Increasing dmax
dmax
= 0m,8m,10m,12m
(b) Effect of increasing small cell transmit power.
Figure 4.4: Illustration of the effect of increasing the small cell density and the transmit
power of BSs in the small cell on Ploc.
power of BSs is increased. This is also as expected since the performance is primarily driven
by the ratio of the densities of the macro cell and the small cell rather than the ratio of the
powers.
4.6 Concluding Remarks
Accurate positioning of an MD in urban canyons and indoor locations using GPS has become
increasingly challenging due to the absence of a direct line-of-sight between the MD and a
minimum number of navigation satellites. This has sparked tremendous interest in studying
geolocation performance in a cellular network. Motivated by this, we study the performance
of localization in a heterogeneous cellular network using tools from stochastic geometry and
point process theory. We develop a new tractable model to study the localizability of an MD
when information about the location of a proximate BS is known. We defined an MD to be
localizable as a union of the following events: (i) atleast L BSs participate in the localization
procedure, which is essentially the event that the Lth strongest BS is hearable, and (ii) the
geographically closest BS is within a distance dmax from the MD. The contrasting nature of
Chapter 4. 63
the above events; one, in which we need the Lth strongest BS and the other, in which we
need the closest BS, led us to apply the displacement theorem to PPPs with an exclusion
zone. Using this result, we derive expressions for localizability probability of an MD in a
heterogeneous network scenario. We also develop a tight approximation for the localizability
probability, which can be effectively used to study the performance of localization in this
setup.
With the increasing developments in 5G standard involving unification of networks such as
LTE Advanced and WiFi, to pave way for the internet-of-things, localization is an integral
part of this system. In the presence of WiFi access points, indoor localization need not rely
only on cellular networks. The WiFi standard also has procedures such as Round-Trip-Time
to calculate the location estimate of the MD relative to the location of the access points. A
logical extension of this model would be to incorporate the presence of WiFi access points
as a finite network along with the presence of a heterogeneous cellular network to analyze
the performance of localization.
Chapter 5
Conclusion
5.1 Summary
Availability of high resolution position estimates of devices around us, especially in urban
canyons, and indoor locations using GPS is challenging due to an absence of a clear line-of-
sight between the MD, and the navigation satellites. Due to this, cellular networks become
a natural choice for localization due to their ubiquity. This has sparked tremendous in-
terest in the research community to study localization in cellular networks. Traditionally,
localization in cellular networks is studied using complex system level simulations, and by
using deterministic tools such as CRLB by fixing the geometry of BSs. A major deficiency
of such analysis is that the scope of the results is limited as the models are not general
enough, and do not account for the randomness associated with a cellular network. Besides
all these factors, there is unfortunately no literature on studying the effect of proximate BS
measurements on the localizability of an MD in a cellular network for a general BS geometry,
and number of participating BSs. In this thesis, we use tools from stochastic geometry,
and point process theory to develop a general, yet tractable analytical model to study the
performance of localizability of an MD in a single-tier cellular network when information
about the location of proximate BSs is known to the MD. We then develop a model to study
64
Chapter 5. 65
the localizability of an MD in a HetNet scenario. The value of the developed model lies in
the fact that it results in tractable expressions for metrics used to study localizability of an
MD. These expressions finally depend on system parameters such as BS density, number of
participating BSs, propagation effects such as path loss, activity of BSs, and quality of the
measurements received by the MD. This makes it very simple to make inferences about how
system parameters will alter the system performance.
In Chapter 2, we develop an analytical model to study the performance of localizability of an
MD in a single-tier cellular network when information about the location of the closest active
BS is known to the MD. The activity of BSs is used to model the network load. Information
about the location of the closest active BS at the MD can help localize the MD at least up
to an accuracy of the distance between the BS, and the MD. This would definitely help in
improving the accuracy of the position estimate of the MD, especially when the quality of
measurements received at the MD is below par. The main goal of this work is to quantify the
gains in localizability performance due to the availability of information about the location of
a BS in the vicinity of the MD. We define an MD to be localizable either when a fixed number
of BSs participate in the localization procedure, or, when the MD is within a certain fixed
predefined distance from the closest active BS. Using this definition, we derive analytical
expressions for the localizability probability of an MD, that is, the probability that an MD
can be localized. In Chapter 3, we study the localizability of an MD in single-tier cellular
network when information about the location of the closest BS is known, irrespective of
whether that BS is active or not. In this Chapter, we also study the effects of large scale
shadowing on the performance of localizability. We again derive some analytical expressions
for the localizability probability. We also derive useful bounds and approximations for the
localizability probability when large scale shadowing is incorporated in the system model.
It is shown that significant gains are achieved in localizability probability when information
about the location of the closest active BS/closest BS is incorporated in the model compared
to the case when this information is not known at the MD. Also, significant gains are observed
when information about the location of the closest BS is used rather than the location
Chapter 5. 66
information of the closest active BS, which is as expected. It is shown that, the localizability
probability improves as the density of BSs increases. This is due to the fact that higher
BS density means BSs are closer together, and the probability that an MD is within a fixed
distance from a BS in its proximity increases.
Finally, in Chapter 4, we develop a tractable model to study the performance of localizability
of an MD in a HetNet scenario. Multiple tiers of BSs are modeled using homogeneous
PPPs of different densities, and BSs of different tiers have different transmit powers. An
MD is said to be localizable either when a certain fixed number of BSs participate in the
localization procedure, or, when the MD is within a certain predefined distance from the
closest overall BS among all tiers. We derive an expression for the localizability probability of
an MD in a HetNet. We also develop a simple, yet tight, approximation for the localizability
probability. It is shown that, localizability probability increases with increase in the BS
densities, particularly with increase in the density of small cells. It is also shown that, as
the number of BSs that participate in the localization procedure increases, the localizability
probability decreases. This is as expected because, the probability that a higher number
of BSs participate in the localization procedure would require the SIR measurements from
a higher number of BSs to be above the set threshold. We also study the performance of
localizability under other system parameters such as the predefined distance between the
MD, and the closest overall BS, BS transmit powers, and propagation effects such as path
loss.
5.2 Future Work
In this section, we summarize some of the possible extensions, and future directions for this
line of work.
Stochastic geometry-based approach to study localization performance
The analyses presented in this thesis are primarily meant to study the performance of localiz-
Chapter 5. 67
ability of an MD in a cellular network. That is, we study the probability of being able to get
a location fix of an MD in a cellular network. Typically, the performance of localization in a
network is dependent on the measurements that are used to localize the MD, and the location
estimators used to determine the position from these measurements. Some of the measure-
ments that are used often are time of arrival (TOA), time difference of arrival (TDOA),
direction of arrival (DOA), and received signal strength (RSS). These measurements are
then used by estimators such as least squares estimator, maximum-likelihood (ML) estima-
tor, etc. to determine the position of the MD. Localization performance for such systems
is typically characterized by determining bounds on the variance of the mean-squared po-
sitioning error by using tools such as CRLB. However, the performance of these techniques
is normally studied by fixing the BS geometries, and the number of participating BSs. A
valuable extension would be to use tools from stochastic geometry to study the performance
of above techniques for random BS geometries, and evaluate bounds and expressions for the
variance of mean-squared positioning error.
Develop a more precise analytical framework
In our analyses, all the expressions are derived using an approximation, which is popularly
known as the dominant interferer approach in the stochastic geometry community [30,40–44].
In this approximation, the total interference observed at the MD is approximated as the sum
of the interference due to the dominant interferer, and the mean of the sum of interference
due to all other BSs. Although, the current set of results obtained under this approximation
are quite tight, it would be a worthwhile extension, and mathematical challenge to charac-
terize the interference more precisely than the approach followed in this work.
Develop simple bounds and approximations
In all the technical Chapters, we have derived analytical expressions for localizability prob-
ability, which are fairly tight when compared to the simulations. The only approximation
used to maintain tractability in the analysis is the dominant interferer approach. Inspite
of this, it can be observed that, the expressions are fairly complex, involve multiple large
integrals, and are not in closed form. It would be a valuable extension to develop simple,
Chapter 5. 68
yet tight bounds, and approximations for the localizability probability to get better system
level insights.
Appendix A
A.1 Proof of Lemma 11
The cdf is given by P[R1j ≤ r
∣∣n = j]
= 1−P[R1j>r,n=j]
P[n=j]= 1− 1
CjP[R1j > r,R1j < min i
i 6=jR1i
](a)= 1− 1
Cj
∫ ∞r
K∏i=1i 6=j
P [R1i > x] fR1j(x)dr = 1− 2πλj
Cj
∫ ∞r
x
[K∏i=1
e−λiπx2
]dx
= 1− 2πλj
Cj
∫ ∞r
xe−λ0πx2
dx,
where (a) follows from the fact that R1is are independent, because Φis are independent.
69
Chapter 5. 70
A.2 Proof of Lemma 12
When 0 < Cij < 1, the cdf of Rk conditioned on R1j and n = j is FRk|R1j ,n=j
(r∣∣R1j, n = j
)
(a)=
0 , for 0 ≤ r < R1j ,
1−∑k−1
q=0 e(−πλj(r2−R2
1j)) (πλj(r2−R21j))
q
q! , for R1j ≤ r < R1j√Cij,
1−∑k−1
q=0 e(−((λiCij+λj)πr
2−(λi+λj)πR21j)) ((λiCij+λj)πr
2−(λi+λj)πR21j)
q
q! , forR1j√Cij≤ r ≤ ∞,
(b)=
0 , for 0 ≤ r < R1j ,
1− Γ(k,(πλj(r2−R21j)))
(k−1)! , for R1j ≤ r < R1j√Cij,
1− Γ(k,((λiCij+λj)πr2−(λi+λj)πR
21j))
(k−1)! , forR1j√Cij≤ r ≤ ∞,
(A.1)
where (a) follows from the fact that there are k − 1 points inside b(o, Rk
), and (b) follows
by substituting for the incomplete gamma function. Differentiation using Leibniz’s rule
completes the proof.
A.3 Proof of Lemma 14
The joint pdf of RL−1 and R1j conditioned on n = j is
fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
= fRL−1|R1j ,n=j
(rL−1
∣∣r1j, n = j)fR1j |n=j
(r1j
∣∣n = j). (A.2)
In this scenario, there are L− 2 points i.i.d and uniformly distributed in the annular region
b(o, RL−1
)\b(o,
R1j√Cij
). Therefore, the cdf of RL−1 conditioned on R1j and n = j follows
from (A.18) by substititing k = L − 1 and db =r1j√Cij
and fR1j |n=j
(r1j
∣∣n = j)
follows from
(11). Substitution in (A.2) completes the proof.
Chapter 5. 71
A.4 Proof of Lemma 15
The joint pdf of RL−1 and R1j conditioned on n = j can be obtained by multiplying (4.17)
and (4.16) with 0 ≤ RL−1 ≤ ∞ and 0 ≤ R1j ≤ ∞. But in this case, R1j ≤ RL−1 ≤ R1j√Cij
.
Therefore the joint pdf of RL−1 and R1j conditioned on n = j with 0 ≤ R1j ≤ RL−1 ≤R1j√Cij≤ ∞ is simply a truncated pdf given by fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
=
4π2λj(λi+λj)rL−1r1j(πλj(r2L−1−r
21j))
k−1exp(−πλj r2
L−1−πλir21j)
(L−2)!∫∞0
∫ r1j√C1j
r1j
4π2λj(λi+λj)rL−1r1j(πλj(r2L−1−r
21j))
k−1exp(−πλj r2
L−1−πλir21j)
(L−2)!drL−1dr1j
(A.3)
for r1j ≤ rL−1 ≤ r1j√Cij
and 0 ≤ r1j ≤ ∞.
A.5 Proof of Lemma 16
The joint pdf of R1, RL−1 and R1j conditioned on n = j is given by
fR1,RL−1,R1j |n=j
(r1, rL−1, r1j
∣∣n = j)
=L−3∑τ=0
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)fR1,R1j |n=j
(r1, r1j
∣∣n = j)
(A.4)
Next, there are L − 3 points i.i.d in the annular region b(o, RL−1
)\b(o, R1
). As it can be
observed from the expression of intensity in (4.5), the intensities are different in the regions
b
(o,
R1j√Cij
)\b(o, R1
)and b
(o, RL−1
)\b(o,
R1j√Cij
)and the number of points in these two re-
gions must sum to L−3. Note that there is a PPP in the annular region b
(o,
R1j√Cij
)\b(o, R1
)and the number of points in this region cannot exceed L − 3. Let T represent the number
of points in b
(o,
R1j√Cij
)\b(o, R1
), then T ∈ 0, ..., L− 3. Therefore, the probability mass
function of T follows a truncated Poisson distribution given by
fT |R1,R1j ,n=j
(τ∣∣r1, r1j , n = j
)=
[πλiCij
(r2
1 −r21j
Cij
)]ττ !
/L−3∑q=0
[πλiCij
(r2
1 −r21j
Cij
)]qq!
. (A.5)
Chapter 5. 72
Now, let us condition on the fact that there are τ points in the annular region b
(o,
R1j√Cij
)\
b(o, R1
). This would mean that there are L − 3 − τ points i.i.d and uniformly distributed
in the annular region b(o, RL−1
)\b(o,
R1j√Cij
). The pdf of RL−1 conditioned on T = τ , R1j,
R1 and n = j follows from (A.18) by substituting k = L− 3− τ and db = R1j, and we have
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j , r1, T = τ, n = j)
=
2π (λj + λiCij) rL−1
(π (λj + λiCij)
(r2L−1 −
r21j
Cij
))L−3−τe
(−(π(λj+λiCij)
(r2L−1−
r21jCij
)))
(L− 3− τ)!. (A.6)
Deconditioning on T gives the pdf fRL−1|R1j ,R1,n=j
(rL−1
∣∣r1j, r1, n = j). Note that
R1j√Cij≤
R1 ≤ R1j. The proof for joint pdf of R1 and R1j conditioned on n = j is along the same
lines as Lemma 15 with k = 1. That is,
fR1,R1j |n=j
(r1, r1j
∣∣n = j)
=4π2λj (λi + λj) r1r1j exp
(−πλj r2
1 − πλir21j
)∫∞
0
∫ r1j√Cij
r1j 4π2λj (λi + λj) r1r1j exp(−πλj r2
1 − πλir21j
)dr1dr1j
.
(A.7)
forr1j√Cij≤ r1 ≤ r1j and 0 ≤ r1j ≤ ∞.
A.6 Proof of Lemma 17
The joint pdf of RL−1 and R1j conditioned on n = j is
fRL−1,R1j |n=j
(rL−1, r1j
∣∣n = j)
= fRL−1|R1j ,n=j
(rL−1
∣∣r1j, n = j)fR1j |n=j
(r1j
∣∣n = j). (A.8)
In this scenario, there are L− 2 points i.i.d and uniformly distributed in the annular region
b(o, RL−1
)\b (o,R1j). Therefore, the cdf of RL−1 conditioned on R1j and n = j follows from
(A.18) by substititing k = L − 1 and db = r1j, and fR1j |n=j
(r1j
∣∣n = j)
follows from (11).
Substitution in (A.8) completes the proof.
Chapter 5. 73
A.7 Proof of Lemma 18
The joint pdf of R1 and R1j conditioned on n = j is
fR1,R1j |n=j
(r1, r1j
∣∣n = j)
= fR1|R1j ,n=j (r1|r1j, n = j) fR1j |n=j (r1j|n = j) . (A.9)
In this scenario, there are L− 1 points i.i.d and uniformly distributed in the annular region
b (o,R1j) \b(o,
R1j√Cij
). That is, R1j is the Lth closest point from the origin. Therefore, the
pdf of R1 conditioned on R1j and n = j is
fR1|R1j ,n=j
(r1
∣∣r1j, n = j)
=2 (L− 1) r1(r2
1j −r21j
Cij
)L−1
(r2
1j − r21
)L−2. (A.10)
The pdf fR1j |n=j
(r1j
∣∣n = j)
follows from (4.16). Substitution in (A.9) completes the proof.
A.8 Proof of Lemma 19
Here, the joint pdf of R1, RL and R1j conditioned on n = j is given by
fR1,RL,R1j |n=j
(r1, rL, r1j
∣∣n = j)
=fR1|RL,R1j ,n=j
(r1
∣∣rL, r1j , n = j)fRL.R1j |n=j
(rL, r1j
∣∣n = j).
(A.11)
In this scenario, there are L− 1 points i.i.d and uniformly distributed in the annular region
b (o,RL) \b(o,
R1j√Cij
). Therefore, the pdf of R1 conditioned on RL, R1j and n = j is
fR1|RL,R1j ,n=j
(r1
∣∣rL, r1j, n = j)
=2 (L− 1) r1(r2L −
r21j
Cij
)L−1
(r2L − r2
1
)L−2. (A.12)
The proof for joint pdf of RL and R1j conditioned on n = j is along the same lines as
Lemma 15. Therefore fRL,R1j |n=j
(rL, r1j
∣∣n = j)
=
4π2λiCij(λi+λj)rLr1j(πλi(Cij r2L−r
21j))
L−1exp(−πλiCij r2
L−πλjr21j)
(L−1)!∫∞0
∫ r1jr1j√C1j
4π2λiCij(λi+λj)rLr1j(πλi(Cij r2L−r
21j))
L−1exp(−πλiCij r2
L−πλjr21j)
(L−1)!drLdr1j
. (A.13)
forr1j√Cij≤ rL ≤ r1j and 0 ≤ r1j ≤ ∞. Substitution in (A.11) completes the proof.
Chapter 5. 74
A.9 Proof of Lemma 20
The joint pdf of R1, RL−1 and R1j conditioned on n = j is given by
fR1,RL−1,R1j |n=j
(r1, rL−1, r1j
∣∣n = j)
=L−3∑τ=0
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j, r1, T = τ, n = j)
fT |R1,R1j ,n=j
(τ∣∣r1, r1j, n = j
)fR1,R1j |n=j
(r1, r1j
∣∣n = j)
(A.14)
Next, there are L − 3 points i.i.d in the annular region b(o, RL−1
)\b(o, R1
). As it can be
observed from the expression of intensity in (4.6), the intensities are different in the regions
b (o,R1j) \b(o, R1
)and b
(o, RL−1
)\b (o,R1j) and the number of points in these two regions
must sum to L − 3. Note that there is a PPP in the annular region b (o,R1j) \b(o, R1
)and the number of points in this region cannot exceed L − 3. Let T represent the number
of points in b (o,R1j) \b(o, R1
), then T ∈ 0, ..., L− 3. Therefore, the probability mass
function of T follows a truncated Poisson distribution given by
fT |R1,R1j ,n=j
(τ∣∣r1, r1j , n = j
)=
[πλiCij
(r2
1 − r21j
)]ττ !
/L−3∑q=0
[πλiCij
(r2
1 − r21j
)]qq!
. (A.15)
Now, let us condition on the fact that there are τ points in the annular region b (o,R1j) \b(o, R1
). This would mean that there are L − 3 − τ points i.i.d and uniformly distributed
in the annular region b(o, RL−1
)\b (o,R1j). The pdf of RL−1 conditioned on T = τ , R1j, R1
and n = j follows from (A.18) by substituting k = L − 3 − τ and db = R1j, and we have
fRL−1|R1j ,R1,T ,n=j
(rL−1
∣∣r1j , r1, T = τ, n = j)
=
2π (λj + λiCij) rL−1
(π (λj + λiCij)
(r2L−1 − r2
1j
))L−3−τexp
(−(π (λj + λiCij)
(r2L−1 − r2
1j
)))(L− 3− τ)!
.
(A.16)
Deconditioning on T gives the pdf fRL−1|R1j ,R1,n=j
(rL−1
∣∣r1j, r1, n = j). Note that
R1j√Cij≤
R1 ≤ R1j. The proof for joint pdf of R1 and R1j conditioned on n = j is along the same
Chapter 5. 75
lines as Lemma 15 with k = 1. That is,
fR1,R1j |n=j
(r1, r1j
∣∣n = j)
=4π2λiCij (λi + λj) r1r1j exp
(−πλiCij r2
1 − πλjr21j
)∫∞
0
∫ r1jr1j√Cij
4π2λiCij (λi + λj) r1r1j exp(−πλiCij r2
1 − πλjr21j
)dr1dr1j
.
(A.17)
for r1j ≤ r1 ≤ r1j√Cij
and 0 ≤ r1j ≤ ∞.
A.10 Useful results for non-homogeneous PPPs
Consider a system, in which the the locations of BSs are modeled using a non-homogeneous
PPP Φ with intensity
λ (r) =
0 , for 0 ≤ r < da,
λa , for da ≤ r < db,
λb , for db ≤ r ≤ ∞.
The transmit power of all BSs is P . The location of the ith closest point from the origin is
xi and the distance of this point from the origin is di. The kth strongest point is the serving
BS. The location of the dominant interferer is xD and the location of the serving BS is xS.
Lemma 22. The mean interference due to all BSs in the annular region b (o, dS) \b (o, dD)
when 0 ≤ da ≤ dD ≤ dS ≤ db or 0 ≤ da ≤ db ≤ dD ≤ dS is E [I1] = 2PN2−α
d2−αS −d2−α
D
d2S−d
2D
where N
is the number of points in the annular region b (o, dS) \b (o, dD).
Proof. Note that in this case, the locations of xD and xS are such that the points in the
annular region b (o, dS) \b (o, dD) will be uniform and i.i.d. Consider that there are N points
in the annular region b (o, dS) \b (o, dD). The distance distribution of each of these points
is fd (r) = 2rd2S−d
2D
for dD ≤ r ≤ dS. Therefore, E [I1] = E[∑
x∈b(o,dS)\b(o,dD) P‖x‖−α]
=
P∑
x∈b(o,dS)\b(o,dD) E [‖x‖−α]
(a)= PNE
[‖x‖−α
]= PN
∫ dS
dD
r−α2r
d2S − d2
D
dr =2PN
d2S − d2
D
∫ dS
dD
r1−αdr =2PN
2− αd2−αS − d2−α
D
d2S − d2
D
.
Chapter 5. 76
where (a) follows from the fact that all N points are i.i.d.
Lemma 23. The mean interference due to all BSs in the annular region b (o, dS) \b (o, dD)
when 0 ≤ da ≤ dD ≤ db ≤ dS is E [I1] = 2PN2−α
[λa(d2−α
b −d2−αD )+λb(d2−α
S −d2−αb )
λa(d2b−d
2D)+λb(d2
S−d2b)
]where N is the
number of points in the annular region b (o, dS) \b (o, dD).
Proof. Note that in this case, the points are i.i.d but not uniformly distributed in the annular
region b (o, dS) \b (o, dD). The distance distribution of a point in this annular region can be
given by
fd (r) =
2r
d2b−d
2DP [x ∈ b (o, db) \b (o, dD)] for dD ≤ r ≤ db
2rd2S−d
2bP [x ∈ b (o, dS) \b (o, db)] for db ≤ r ≤ dS
=
2r
d2b−d
2D
λa(d2b−d
2D)
[λa(d2b−d
2D)+λb(d2
S−d2b)]
for dD ≤ r ≤ db
2rd2S−d
2b
λb(d2S−d
2b)
[λa(d2b−d
2D)+λb(d2
S−d2b)]
for db ≤ r ≤ dS
Therefore, E [I1] = E[∑
x∈b(o,dS)\b(o,dD) P‖x‖−α]
= P∑
x∈b(o,dS)\b(o,dD) E [‖x‖−α]
(a)= PNE
[‖x‖−α
]= PN
∫ dS
dD
r−αfd (r) dr =2PN
2− α
[λa(d2−αb − d2−α
D
)+ λb
(d2−αS − d2−α
b
)λa (d2
b − d2D) + λb (d2
S − d2b)
].
where (a) follows from the fact that all N points are i.i.d.
Lemma 24. The mean interference due to all BSs in the region bc (o, dS) when 0 ≤ da ≤
dS ≤ db is E [I2] = 2πPα−2
[λad
2−αS + (λb − λa) d2−α
b
]for α > 2.
Proof. E [I2] = E[∑
x∈bc(o,dS) P‖x‖−α]
(a)=∫ 2π
0
∫∞dSPr−αλ (r) rdrdθ = 2πP
∫∞dSr1−αλ (r) dr
= 2πP
[∫ db
dS
r1−αλadr +
∫ ∞db
r1−αλbdr
]=
2πP
α− 2
[λad
2−αS + (λb − λa) d2−α
b
]for α > 2.
where (a) follows from Campbell’s theorem [46].
Lemma 25. The mean interference due to all BSs in the region bc (o, dS) when 0 ≤ da ≤
db ≤ dS is E [I2] = 2πPλbα−2
d2−αS for α > 2.
Chapter 5. 77
Proof. E [I2] = E[∑
x∈bc(o,dS) P‖x‖−α]
(a)=∫ 2π
0
∫∞dSPr−αλ (r) rdrdθ = 2πP
∫∞dSr1−αλ (r) dr
= 2πP
[∫ ∞dS
r1−αλbdr
]=
2πPλbα− 2
d2−αS for α > 2.
where (a) follows from Campbell’s theorem [46].
Lemma 26. The distance distribution of dk when da ≤ db ≤ d1 ≤ dk is
fdk|db(r∣∣db) =
2π (λb) r(π (λb)
(r2 − d2
b
))k−1exp
(−(π (λb)
(r2 − d2
b
)))(k − 1)!
for db ≤ dk ≤ ∞ (A.18)
Proof. Since d1 ≥ db =⇒ there are k − 1 points i.i.d and uniformly distributed in the
annular region b (o, dk) \b (o, db). Therefore,
Fdk|db(r∣∣db) = P
[dk ≤ r
∣∣db] = 1−k−1∑i=0
exp
(2π
∫ r
db
λ (t) tdt
) (2π∫ rdbλ (t) tdt
)ii!
= 1−k−1∑i=0
exp(−π (λb)
(r2 − d2
b
)) (π (λb) (r2 − d2b))
i
i!= 1− Γ (k, (π (λb) (r2 − d2
b)))
(k − 1)!
Differentiation using Leibniz’s rule completes the proof.
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