The rest of this thesis is organized as follows. In chapter 2, we review the point process, some mathematical tools, and connection setup procedure for some existing wireless communication standards. The following chapter 3 introduces system model and the procedures to set up a D2D communication link. SDLS analysis and analytical results under Binomial point process and Poisson point process models are described respectively in chapter 4 and chapter 5. Finally, we draw our conclusion and future work in chapter 6. In the table 1.1, we give some symbol descriptions of our thesis.
Table 1.1: Summary of Notations
Symbol Description
Rd d-dimensional space
R+ positive real numbers
A∩ B the intersection of set A and set B
P(A) probability of event A
E(X) expectation of random variable X
α path loss exponent
d number of dimensions of the network
|A| Lebesgue measure of set A. For d = 1, 2, or 3, it consists with the standard measure of length, area, or volume.
· Euclidean norm
cd volume of the d-dim. unit ball
LX(s) = E(e−sX) Laplace transform of random variable X
Φ ={Xi} ⊂ Rd point process
FX(x) =P(X ≤ x) cumulative distribution function of random variable X
PD common transmit power of active devices
N0 noise power
Chapter 2
Preliminaries
The signal-to-interference-plus-noise-ratio (SINR) is the primary signal quality met-ric that must be satisfied by a communication link. The interference is an important term in wireless network. However, it is a function of network topology, path loss, and channel fading effect. There are two main tools to help us analyze the interference:
stochastic geometry [21] and random geometric graphs [23]. Because the nodes appear-ing in the wireless network may follow some spatial distribution, stochastic geometry is useful in modeling the network topology, and analyzing the average behavior over many spatial realizations of a network. Then, we will introduce some concepts about stochastic geometry. They called it point process (PP). Later, there will be some math-ematical tools and functions to help us compute some properties that we are interested in of wireless communication systems. Finally, the connection procedure of some wire-less communication standards such as FlashLinQ, Bluetooth, WLAN, and ZigBee are introduced.
2.1 Point Process
In the stochastic geometry, the basic objects are point processes. A PP can be described as a stochastic collection of points in the space. For a detailed concept and analysis of PP, readers can refer to [21-22, 24]. Why do we consider the PP? When we describe the locations of nodes, a PP can be used to model these nodes on the space
that we are interested in. It can help us do some mathematical analysis of the wireless network.
φ(B) denotes the number of points of φ in B ⊂ Rd. N denotes the smallest sigma algebra such that the mappings φ→ φ(B) are measurable for all Borel sets B ⊂ Rd. Definition 1 (Point process, [24]) It is a measurable mapping function Φ onRdfrom the probability space (Ω,A, P) to (N, N ).
Φ : Ω→ N (2.1)
A point process is a stochastic variable which takes the values in the set of the sequences N. There are some terminologies about PP Φ in the following.
1. If a PP is said to be simple ⇒ no two points with the same location (at most one point at a location).
|A|→0lim
P(Φ(A) ≥ 1)
P(Φ(A) = 1) = 1 (2.2)
where Φ(A) is the number of points in set A.
2. If a PP is said to be stationary ⇒ the regulation of the PP is invariant by trans-lation.
Φ + s ={u + s|u ∈ Φ} ∼ Φ, s ∈ Rd (2.3) 3. If a PP is said to be isotropic ⇒ the regulation of the PP is invariant by rotation.
RΦ ={Ru|u ∈ Φ} ∼ Φ ∀R ∈ Md×d ={d × d Orthogonal Matrix} (2.4) where R denotes the rotation around the origin.
If the PP is both stationary and isotropic, it is also called a motion-invariant PP.
Definition 2 (Intensity Measure, [24]) The intensity measure of the PP Φ is equal to the average number of points in the set B ⊂ Rd.
Λ(B) =E [Φ(B)] (2.5)
0 0.2 0.4 0.6 0.8 1
Definition 3 (Probability Generating Functional, [24]) Let ψ(x) : Rd → [0, ∞) be measurable. The probability generating functional (PGFL) of the PP can be defined.
G(ψ) =E
The subsequent subsections will introduce two types of PP, namely Binomial point process (BPP) and Poisson point process (PPP).
2.1.1 Binomial Point Process
In a bounded domain, the BPP is generally used to construct the location of wireless nodes with a fixed number n. It is a quite simple point process. The n points are inde-pendently and uniformly distributed in a bounded and closed set B ⊂ Rd(d-dimensional space). The BPP can be illustrated in Fig. 2.1. The probability that there are k points in A⊂ B is given by where Φ(A) is defined as the number of points in set A and |A| stands for the Lebesgue measure of the set A.
0 0.2 0.4 0.6 0.8 1
Figure 2.2: PPP with set S = [0, 1]× [0, 1]; left side: homogeneous, λ=100; right side:
inhomogeneous, λ(x, y) = 200y.
Note that the Φ(A) and Φ(B) are not independent even if A∩B = ∅. This property makes the analysis of the received interference in the wireless network more difficult.
The PGFL of BPP Φ is equal to
The PPP is a most well-studied point process. Its importance comes from its ease to analyze. Therefore, the PPP provides a handy computational framework for different performance metric of interest.
1. If a PP is said to be a homogeneous PPP ⇒ the value of intensity keeps the same across space. It is isotropic and stationary process. It may be the simplest point process. It can be illustrated on the left side in Fig. 2.2.
2. If a PP is said to be an inhomogeneous PPP ⇒ the value of intensity doesn’t keep the same across space, and it can be used to model the distribution with non-uniform type across space. Like in a city, the intensity is not the constant everywhere. It can be illustrated on the right side in Fig. 2.2.
The homogeneous PPP Φ with intensity (or density) λ is characterized by the below properties:
1. For all the disjoint sets A1, . . . , Ak, the random variables Φ(Ai) are independent.
2. All the random variables Φ(Ai) are Poisson random variables with mean λ|Ai|.
P(Φ(A) = k) = e−λ|A|(λ|A|)k
k! (2.9)
Conditioned on the fact that there are n points in set A, the n points are independently and uniformly distributed in set A, namely BPP. There are some important properties of the PPP as the following.
1. The superposition of two or more independent PPPs with intensities λ1, . . . , λk is also a PPP with intensity Σki=1λi
2. The independent thinning of the PPP with intensity λ results in independent PPPs with intensity κiλ such that K
i=1κi = 1.
3. If Φ is a homogeneous PPP with unit intensity (λ=1), then λ−1/dΦ is a homoge-neous PPP with intensity λ.
4. The PGFL of PPP Φ is equal to
G(ψ) = exp
−
Rd
(1− ψ(x))Λ(dx)
(2.10)