# Characterization of Energy Availability in RF Energy Harvesting Networks

Authors: Daniela Oliveira; Rodolfo Oliveira
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

The multiple nodes forming a Radio Frequency (RF) Energy Harvesting Network (RF-EHN) have the capability of converting received electromagnetic RF signals in energy that can be used to power a network device (the energy harvester). Traditionally the RF signals are provided by high power transmitters (e.g., base stations) operating in the neighborhood of the harvesters. Admitting that the transmitters are spatially distributed according to a spatial Poisson process, we start by characterizing the distribution of the RF power received by an energy harvester node. Considering Gamma shadowing and Rayleigh fading, we show that the received RF power can be approximated by the sum of multiple Gamma distributions with different scale and shape parameters. Using the distribution of the received RF power, we derive the probability of a node having enough energy to transmit a packet after a given amount of charging time. The RF power distribution and the probability of a harvester having enough energy to transmit a packet are validated through simulation. The numerical results obtained with the proposed analysis are close to the ones obtained through simulation, which confirms the accuracy of the proposed analysis.


---

## Body

## 1. Introduction

The nodes forming a Radio Frequency (RF) Energy Harvesting Network (RF-EHN) have the capability of converting received electromagnetic RF signals in energy. The RF energy is converted by an energy harvester device, which is composed of an RF antenna, a band pass filter parametrized to the RF signals, and a rectifying circuit able to convert RF to DC power [1]. In this way the converted RF signals are used to charge a battery (usually a supercapacitor) with finite capacity [2]. The harvested energy accumulated in the battery can then be used to transmit a packet. However, the transmission is only possible if the level of accumulated energy is higher than a given threshold representing the minimum level of energy required to complete a packet transmission.Recently, RF energy harvesting has attracted much attention and many efforts are being dedicated to develop innovative RF energy harvesting technologies as well as to investigate the performance of the networks formed by the harvesting devices. The RF energy harvesting literature dedicated to the efficient design of RF harvesting devices (see [3–7] for a few examples) is mainly focused on the minimization of the loss effects due to the RF-to-DC conversion and battery charging process. A different focus is also found in the literature, where the main goal is the study and characterization of RF-EHNs. Adopting a generic model for the RF energy harvesting devices, the goals are usually related with the scheduling of the harvesting devices in order to maximize the utilization of the RF energy and the frequency band constrained by specific throughput fairness policies [8]; the optimization of the harvester communication task to deal with the multiple tradeoffs associated with the physical and MAC layers [9, 10]; the characterization of the RF-EHN performance (throughput) and stability when RF energy harvesting is adopted [11]. Reference [12] investigates the performance (throughput) of a slotted Aloha random access wireless network consisting of two types of nodes: with unlimited energy supply and solely powered by an RF energy harvesting circuit. To illustrate the design considerations of RF-based harvesting networks, [13] points out the primary challenges of implementing and operating such networks, including nondeterministic energy arrival patterns, energy harvesting mode selection, and energy-aware cooperation among base stations. Reference [14] adopts a stochastic geometry framework based on the Ginibre model to analyze the performance of self-sustainable communications over cellular networks with general fading channels. The expectation of the RF energy harvesting rate, the energy outage probability, and the transmission outage probability are evaluated over Nakagami-m fading channels.RF-EHNs may also act as cognitive radio networks (CRNs), that is, using the spectrum in an opportunistic way without being licensed. Several works have explored these kinds of networks. Reference [15] provides an overview of the RF-EHNs including system architecture, RF energy harvesting techniques, and existing applications. The authors also explore various key design issues in the development of RF-EHNs, including cognitive radio networks. The work in [16] provides a comprehensive overview of recent development and challenges regarding the operation of cognitive radio networks powered by RF energy. Spectrum efficiency and energy efficiency are two critical issues in designing cognitive radio RF-EHNs. Reference [17] provides an overview of the RF-powered CRNs and discusses the challenges that arise for dynamic spectrum access in these networks. Focusing on the trade-off among spectrum sensing, data transmission, and RF energy harvesting, the authors discuss the dynamic channel selection problem in a multichannel RF-powered CRN. Reference [18] proposes a novel method for wireless networks coexisting where low-power mobiles in a secondary network, harvest ambient RF energy from transmissions by nearby active transmitters, while opportunistically accessing the spectrum licensed to the primary network. The authors analyze the transmission probability of harvesting terminals and the resulting spatial throughput. The optimal transmission power and terminals’ density are also derived for maximizing the throughput. The work in [19] considers an RF-powered green cognitive radio network, where a central node harvests energy from ambient sources and wirelessly delivers random harvested energy to cognitive users. The work evaluates the performance of such a network, showing the feasibility of the behavior if the energy transmission rate is below a certain threshold. Reference [20] considers a network where the unlicensed users can perform channel access to transmit a packet or to harvest RF energy when the selected channel is idle or occupied by the primary user, respectively. The work is mainly focused on finding the channel access policy that maximizes the throughput of the secondary user. Reference [21] analyzes an energy harvesting-based cognitive radio system to find the optimal spectrum sensing time, which maximizes the harvested energy. The work in [22] analyzes a cognitive and energy harvesting-based device-to-device (D2D) communication in cellular networks. The authors employ tools from stochastic geometry to evaluate the performance of the proposed communication system model with general path-loss exponent in terms of outage probability for D2D and cellular users. One of the work conclusions is that energy harvesting can be a reliable alternative to power cognitive D2D transmitters, while achieving acceptable performance.In this work we are particularly focused on the characterization of the RF power received by each harvester and its impact in terms of the probability of accumulating enough energy to transmit a packet. A generalized radio propagation environment is considered. Assuming that the sources of high power RF signals (e.g., base stations) are distributed according to a spatial Poisson process, we characterize the distribution of the received RF power from the multiple transmitters. Path loss, shadowing, and fading effects are considered. The distribution of the RF power is then used to derive the probability of a harvester node having enough energy to transmit a packet after a given period of time. A soft computational model (Gaussian approach) and a more complex model (non-Gaussian approach) are presented to compute the probability of a harvester node having enough energy to transmit. These are the main contributions of the paper. Considering multiple spatial and propagation scenarios, we validate the distribution of the RF power and the probability of a harvester have enough energy to transmit a packet. The numerical results obtained with the proposed analysis are close to the ones obtained through simulation, which confirms the accuracy of the proposed analysis. In this way, we provide a characterization of the battery charging time considering innovative assumptions, including the spatial distribution of the RF transmitters, the propagation effects, and the losses associated with the RF-to-DC conversion and battery charging process. The proposed model can thus be adopted to determine the probability of a harvester accumulating enough energy after a given period of time, which is a determinant condition to compute the throughput of RF-EHNs. As far as we known, this is the first work to derive such a probability when the multiple RF signals received by the harvesters are differently affected by multiple propagation effects.The rest of the paper is organized as follows. The system description is presented in Section2. The characterization of the received RF power is characterized in Section 3. Section 4 derives the probability of accumulating enough energy to transmit a packet through the Gaussian and non-Gaussian approaches. Finally, validation results are presented in Section 5 and conclusions are drawn in Section 6.


## 2. System Description

This work considers a RF energy harvesting network, where each node accumulates energy from the base stations and other RF transmitters located in the neighborhood. A harvester node is able to initiate a packet transmission whenever the level of accumulated energy is above a transmission threshold.


### 2.1. Spatial Distribution of the RF Transmitters

We consider the scenario illustrated in Figure1, where the node N
H (the harvester node) accumulates energy from the transmitters that might be located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2. The area A can be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width d
χ and area 2
π
χ
d
χ and then integrating from χ
=
R
i
1 to χ
=
R
o
L; that is; A
=
∫
R
i
1
R
o
L
2
π
χ
d
χ. Using the Riemann sum, A can be approximated by the sum of the area of a finite number (L) of annuli of width ρ,(1)
A
≈
∑
l
=
1
L
A
l
,where A
l
=
π
(
R
o
l
)
2
-
(
R
i
l
)
2 denotes the area of the annulus l. R
o
l
=
(
R
i
1
+
l
ρ
) and R
i
l
=
(
R
i
1
+
(
l
-
1
)
ρ
) represent the radius of the larger and smaller circles of the annulus l, respectively.Figure 1
The harvester nodeN
H receives RF power from the transmitters located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.The number of transmitters located in a specific annulusl
∈
{
1
,
…
,
L
}, represented by the random variable (RV) X
l, is approximated by a Poisson process, being its Probability Mass Function (PMF) for a finite domain given by [23](2)
P
X
l
=
k
=
β
l
A
l
k
/
k
!
e
-
β
l
A
l
∑
i
=
0
n
β
l
A
l
i
/
i
!
e
-
β
l
A
l
,
k
=
0,1
,
…
,
n
,where β
l is the spatial density of the RF nodes transmitting in the annulus and n is the total number of mobile nodes.


### 2.2. Propagation Assumptions

We consider that the RF powerI
i received by the harvester N
H from the RF transmitter i is given by(3)
I
i
=
P
T
x
ψ
i
r
l
-
α
,where P
T
x is the transmitted power level of the ith RF transmitter (P
T
x
=
20
×
1
0
3 mW is assumed for each node) and   ψ
i is an instant value of the fading and shadowing gain observed in the channel between the receiver N
H and the transmitter node i. r
l represents the distance between the ith transmitter and the receiver. The values r
l and ψ
i represent instant values of the random variables R
l and Ψ
i, respectively. α represents the path-loss coefficient.The PDF ofR
l can be written as the ratio between the perimeter of the circle with radius x and the total area A
l, being represented as follows:(4)
f
R
l
x
=
2
π
x
A
l
R
i
l
<
x
<
R
o
l
,
0
,
otherwise
.To characterize the distribution ofΨ
i the small-scale fading (fast fading) and shadowing (slow fading) effects must be considered. The amplitude of the small-scale fading effect is assumed to be distributed according to a Rayleigh distribution, which is represented by(5)
f
ζ
x
=
x
σ
ζ
2
e
-
x
2
/
2
σ
ζ
2
,where x is the envelope amplitude of the received signal. 2
σ
ζ
2 is the mean power of the multipath received signal. 2
σ
ζ
2
=
1 is adopted in this work to consider the case of normalized power.Regarding the shadowing effect, we have assumed that it follows a log-normal distribution(6)
f
ξ
x
=
1
2
π
σ
ξ
x
e
-
ln
⁡
x
-
μ
2
/
2
σ
ξ
2
,where σ
ξ is the shadow standard deviation when μ
=
0. The standard deviation is usually expressed in decibels and is given by σ
ξ
d
B
=
10
σ
ξ
/
ln
⁡
(
10
). For σ
ξ
→
0, no shadowing results. Although (6) appears to be a simple expression, it is often inconvenient when further analyses are required. Consequently, [24] has shown that the log-normal distribution can be accurately approximated by a Gamma distribution, defined by(7)
f
ξ
x
≈
1
Γ
ϑ
ϑ
ω
s
ϑ
x
ϑ
-
1
e
-
x
ϑ
/
ω
s
,where ϑ is equal to 1
/
e
σ
ξ
2
-
1 and ω
s is equal to e
μ
ϑ
+
1
/
ϑ. Γ
(
·
) represents the Gamma function.The probability density function ofΨ
i is thus represented by(8)
f
Ψ
i
x
≈
f
ζ
2
x
·
f
ξ
x
≈
2
Γ
ϑ
ϑ
ω
s
ϑ
+
1
/
2
x
ϑ
-
1
/
2
K
ϑ
-
1
4
ϑ
x
ω
s
,which is the Generalized-K distribution, where K
ϑ
-
1
(
·
) is the modified Bessel function of the second kind.Due to the analytical difficulties of the Generalized-K distribution, an approximation of the PDF (8) by a more tractable PDF is needed. Reference [25] provides an approximation of the Generalized-K distribution by using the moment matching method to determine the parameters of the approximated Gamma distribution. With this method, [25] shows that the scale (θ
ψ) and shape (k
ψ) parameters of the Gamma distribution are given by(9)
θ
ψ
=
2
ϑ
+
1
ϑ
-
1
ω
s
,
(10)
k
ψ
=
1
2
ϑ
+
1
/
ϑ
-
1
,respectively.


## 2.1. Spatial Distribution of the RF Transmitters

We consider the scenario illustrated in Figure1, where the node N
H (the harvester node) accumulates energy from the transmitters that might be located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2. The area A can be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width d
χ and area 2
π
χ
d
χ and then integrating from χ
=
R
i
1 to χ
=
R
o
L; that is; A
=
∫
R
i
1
R
o
L
2
π
χ
d
χ. Using the Riemann sum, A can be approximated by the sum of the area of a finite number (L) of annuli of width ρ,(1)
A
≈
∑
l
=
1
L
A
l
,where A
l
=
π
(
R
o
l
)
2
-
(
R
i
l
)
2 denotes the area of the annulus l. R
o
l
=
(
R
i
1
+
l
ρ
) and R
i
l
=
(
R
i
1
+
(
l
-
1
)
ρ
) represent the radius of the larger and smaller circles of the annulus l, respectively.Figure 1
The harvester nodeN
H receives RF power from the transmitters located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.The number of transmitters located in a specific annulusl
∈
{
1
,
…
,
L
}, represented by the random variable (RV) X
l, is approximated by a Poisson process, being its Probability Mass Function (PMF) for a finite domain given by [23](2)
P
X
l
=
k
=
β
l
A
l
k
/
k
!
e
-
β
l
A
l
∑
i
=
0
n
β
l
A
l
i
/
i
!
e
-
β
l
A
l
,
k
=
0,1
,
…
,
n
,where β
l is the spatial density of the RF nodes transmitting in the annulus and n is the total number of mobile nodes.


## 2.2. Propagation Assumptions

We consider that the RF powerI
i received by the harvester N
H from the RF transmitter i is given by(3)
I
i
=
P
T
x
ψ
i
r
l
-
α
,where P
T
x is the transmitted power level of the ith RF transmitter (P
T
x
=
20
×
1
0
3 mW is assumed for each node) and   ψ
i is an instant value of the fading and shadowing gain observed in the channel between the receiver N
H and the transmitter node i. r
l represents the distance between the ith transmitter and the receiver. The values r
l and ψ
i represent instant values of the random variables R
l and Ψ
i, respectively. α represents the path-loss coefficient.The PDF ofR
l can be written as the ratio between the perimeter of the circle with radius x and the total area A
l, being represented as follows:(4)
f
R
l
x
=
2
π
x
A
l
R
i
l
<
x
<
R
o
l
,
0
,
otherwise
.To characterize the distribution ofΨ
i the small-scale fading (fast fading) and shadowing (slow fading) effects must be considered. The amplitude of the small-scale fading effect is assumed to be distributed according to a Rayleigh distribution, which is represented by(5)
f
ζ
x
=
x
σ
ζ
2
e
-
x
2
/
2
σ
ζ
2
,where x is the envelope amplitude of the received signal. 2
σ
ζ
2 is the mean power of the multipath received signal. 2
σ
ζ
2
=
1 is adopted in this work to consider the case of normalized power.Regarding the shadowing effect, we have assumed that it follows a log-normal distribution(6)
f
ξ
x
=
1
2
π
σ
ξ
x
e
-
ln
⁡
x
-
μ
2
/
2
σ
ξ
2
,where σ
ξ is the shadow standard deviation when μ
=
0. The standard deviation is usually expressed in decibels and is given by σ
ξ
d
B
=
10
σ
ξ
/
ln
⁡
(
10
). For σ
ξ
→
0, no shadowing results. Although (6) appears to be a simple expression, it is often inconvenient when further analyses are required. Consequently, [24] has shown that the log-normal distribution can be accurately approximated by a Gamma distribution, defined by(7)
f
ξ
x
≈
1
Γ
ϑ
ϑ
ω
s
ϑ
x
ϑ
-
1
e
-
x
ϑ
/
ω
s
,where ϑ is equal to 1
/
e
σ
ξ
2
-
1 and ω
s is equal to e
μ
ϑ
+
1
/
ϑ. Γ
(
·
) represents the Gamma function.The probability density function ofΨ
i is thus represented by(8)
f
Ψ
i
x
≈
f
ζ
2
x
·
f
ξ
x
≈
2
Γ
ϑ
ϑ
ω
s
ϑ
+
1
/
2
x
ϑ
-
1
/
2
K
ϑ
-
1
4
ϑ
x
ω
s
,which is the Generalized-K distribution, where K
ϑ
-
1
(
·
) is the modified Bessel function of the second kind.Due to the analytical difficulties of the Generalized-K distribution, an approximation of the PDF (8) by a more tractable PDF is needed. Reference [25] provides an approximation of the Generalized-K distribution by using the moment matching method to determine the parameters of the approximated Gamma distribution. With this method, [25] shows that the scale (θ
ψ) and shape (k
ψ) parameters of the Gamma distribution are given by(9)
θ
ψ
=
2
ϑ
+
1
ϑ
-
1
ω
s
,
(10)
k
ψ
=
1
2
ϑ
+
1
/
ϑ
-
1
,respectively.


## 3. RF Received Power

### 3.1. RF Received Power due to the Transmitters Located within the Annulusl

The amount of RF power received by the harvester nodeN
H located in the centre of an annulus l is given by(11)
Σ
=
∑
i
=
1
n
l
I
i
,where I
i is the RF power received from the ith transmitter and n
l is the total number of transmitters in the annulus l.LetM
I
i
(
s
) represent the MGF of the ith transmitter located within the annulus (i
=
1
,
…
,
n
l) given by(12)
M
I
i
s
=
E
I
i
e
s
I
i
=
E
Ψ
i
E
R
l
e
s
I
i
.Using the PDF of the distance given in (4) and the PDF of the small-scale fading and shadowing effects in (8), the MGF of the power received by the node N
H from the ith transmitter in (12) can be written as follows:(13)
M
I
i
s
=
∫
0
+
∞
∫
R
i
l
R
o
l
e
s
I
i
f
R
l
r
l
f
Ψ
i
ψ
i
d
r
l
d
ψ
i
,which using (3), (9), (10), and (4) can be simplified to(14)
M
I
i
s
=
2
π
A
l
2
+
k
ψ
α
P
T
x
θ
ψ
s
k
ψ
·
R
o
l
2
+
k
ψ
α
ϱ
R
o
l
-
R
i
l
2
+
k
ψ
α
ϱ
R
i
l
,where ϱ
(
x
)
=
F
2
1
k
ψ
,
k
ψ
+
2
/
α
,
1
+
k
ψ
+
2
/
α
,
-
x
α
/
P
T
x
θ
ψ
s and 2
F
1 represents the Gauss Hypergeometric function [26].Departing from the fact that the individual powerI
i is independent and identically distributed when compared to the other transmitters, the PDF of the aggregate RF power I given a total of k active transmitters is the convolution of the PDFs of each I
i. Following this rationale, the MGF of I is given by(15)
M
I
/
k
s
=
M
I
1
s
×
M
I
2
s
×
⋯
×
M
I
k
s
=
M
I
i
s
k
.Using the law of total probability, the PDF of the RF powerI can be written as(16)
f
I
j
=
∑
k
=
0
n
f
I
j
∣
X
l
=
k
P
X
l
=
k
,leading to the MGF of the aggregate power, I, which can be written as(17)
E
e
s
I
=
∑
k
=
0
n
P
X
l
=
k
∫
-
∞
+
∞
e
s
j
f
I
j
∣
X
l
=
k
d
j
=
∑
k
=
0
n
P
X
l
=
k
M
I
/
k
s
.Using (15), the MGF of I is given as follows:(18)
E
e
s
I
=
∑
k
=
0
n
P
X
l
=
k
e
k
l
n
M
I
i
s
.Using the MGF of the Poisson distribution in (18), the MGF of I is finally given by(19)
E
e
s
I
=
e
β
l
A
l
M
I
i
s
-
1
.The first- and second-order statistics of the aggregate RF power received by N
H from the transmitters located within the annulus l are an important tool. E
[
I
], the expected value of the aggregate RF power, can be determined by using the Law of Total Expectation. It can be shown that(20)
E
I
=
E
E
I
∣
X
l
=
2
π
β
l
P
T
x
e
σ
ξ
2
R
o
l
2
-
α
-
R
i
l
2
-
α
2
-
α
.Making similar use of the Law of Total Variance, the variance of the aggregate RF power can be described as(21)
V
a
r
I
=
V
a
r
I
i
E
X
l
+
E
I
i
2
V
a
r
X
l
.Since X
l is given by a Poisson distribution (with mean β
l
A
l), the variance of the RF power is given as follows:(22)
V
a
r
I
=
β
l
A
l
∂
2
M
I
i
0
∂
s
2
=
π
β
l
P
T
x
2
k
ψ
θ
ψ
2
1
+
k
ψ
R
o
l
2
-
2
α
-
R
i
l
2
-
2
α
1
-
α
.The first and second moments can be matched with the respective moments of a given distribution to obtain a closed-form approximation for the aggregate received RF power. As shown in [27], the aggregate RF power due to path loss, fast fading, and shadowing effect can be approximated by a Gamma distribution. Consequently, the shape and the scale parameters of the Gamma distribution, denoted by k
l and θ
l, are, respectively, given by(23)
k
l
≈
E
I
2
V
a
r
I
,
(24)
θ
l
≈
V
a
r
I
E
I
.


### 3.2. RF Received Power due to the Transmitters Located withinL Annuli

As shown in the previous subsection, the RF powerI received from the transmitters located within the lth annulus is approximated by a Gamma distribution, with MGF M
I
l
(
s
)
=
(
1
-
θ
l
s
)
-
k
l. Since the annulus of width R
o
L
-
R
i
1 where the transmitters are located can be expressed as a summation of L annuli of width ρ, the MGF of the aggregate power received from the transmitters located within the L annuli is given by(25)
M
I
a
g
g
s
=
∏
l
=
1
L
1
-
θ
l
s
-
k
l
.Finally the expectation of the aggregate RF power can be computed as follows:(26)
E
I
a
g
g
=
∂
M
I
a
g
g
0
∂
s
.


### 3.3. Distribution of the Aggregate RF Power

The aggregate RF power may be stated as being the summation of theL individual aggregated RF powers received from the transmitters located within each annulus. Expressions for the PDF and the CDF of the summation of L independent Gamma random variables were initially derived by Mathai in [28]. Those were simplified in [29] in order to be computed more efficiently.Let{
Z
l
}
l
=
1
L be independent but not necessarily identically distributed Gamma variables with parameters k
l (shape) and θ
l (scale). The PDF of the aggregate RF power is written as I
a
g
g
=
∑
l
=
1
L
‍
Z
l, which can be approximated by [29](27)
f
I
a
g
g
x
≈
∏
l
=
1
L
θ
1
θ
l
k
l
∑
w
=
0
+
∞
δ
w
x
∑
l
=
1
L
k
l
+
w
-
1
exp
⁡
-
x
/
θ
1
θ
1
∑
l
=
1
L
k
l
+
w
Γ
∑
l
=
1
L
k
l
+
w
,where θ
1
=
m
i
n
l
{
θ
l
}, δ
w coefficients are computed recursively, δ
w
+
1
=
1
/
w
+
1
∑
i
=
1
w
+
1
∑
l
=
1
L
k
l
1
-
θ
1
/
θ
l
i
δ
w
+
1
-
i, and δ
0
=
1. Γ
(
·
) is the Gamma function. Finally, the CDF of I
a
g
g, F
I
a
g
g
(
x
)
=
∫
-
∞
x
‍
f
I
a
g
g
(
z
)
d
z, is computed as follows [29]:(28)
F
I
a
g
g
x
≈
∏
l
=
1
L
θ
1
θ
l
k
l
∑
w
=
0
∞
δ
w
θ
1
∑
l
=
1
L
k
l
+
w
Γ
∑
l
=
1
L
k
l
+
w
×
∫
0
x
z
∑
l
=
1
L
k
l
+
w
-
1
exp
⁡
-
z
θ
1
d
z
.


## 3.1. RF Received Power due to the Transmitters Located within the Annulusl

The amount of RF power received by the harvester nodeN
H located in the centre of an annulus l is given by(11)
Σ
=
∑
i
=
1
n
l
I
i
,where I
i is the RF power received from the ith transmitter and n
l is the total number of transmitters in the annulus l.LetM
I
i
(
s
) represent the MGF of the ith transmitter located within the annulus (i
=
1
,
…
,
n
l) given by(12)
M
I
i
s
=
E
I
i
e
s
I
i
=
E
Ψ
i
E
R
l
e
s
I
i
.Using the PDF of the distance given in (4) and the PDF of the small-scale fading and shadowing effects in (8), the MGF of the power received by the node N
H from the ith transmitter in (12) can be written as follows:(13)
M
I
i
s
=
∫
0
+
∞
∫
R
i
l
R
o
l
e
s
I
i
f
R
l
r
l
f
Ψ
i
ψ
i
d
r
l
d
ψ
i
,which using (3), (9), (10), and (4) can be simplified to(14)
M
I
i
s
=
2
π
A
l
2
+
k
ψ
α
P
T
x
θ
ψ
s
k
ψ
·
R
o
l
2
+
k
ψ
α
ϱ
R
o
l
-
R
i
l
2
+
k
ψ
α
ϱ
R
i
l
,where ϱ
(
x
)
=
F
2
1
k
ψ
,
k
ψ
+
2
/
α
,
1
+
k
ψ
+
2
/
α
,
-
x
α
/
P
T
x
θ
ψ
s and 2
F
1 represents the Gauss Hypergeometric function [26].Departing from the fact that the individual powerI
i is independent and identically distributed when compared to the other transmitters, the PDF of the aggregate RF power I given a total of k active transmitters is the convolution of the PDFs of each I
i. Following this rationale, the MGF of I is given by(15)
M
I
/
k
s
=
M
I
1
s
×
M
I
2
s
×
⋯
×
M
I
k
s
=
M
I
i
s
k
.Using the law of total probability, the PDF of the RF powerI can be written as(16)
f
I
j
=
∑
k
=
0
n
f
I
j
∣
X
l
=
k
P
X
l
=
k
,leading to the MGF of the aggregate power, I, which can be written as(17)
E
e
s
I
=
∑
k
=
0
n
P
X
l
=
k
∫
-
∞
+
∞
e
s
j
f
I
j
∣
X
l
=
k
d
j
=
∑
k
=
0
n
P
X
l
=
k
M
I
/
k
s
.Using (15), the MGF of I is given as follows:(18)
E
e
s
I
=
∑
k
=
0
n
P
X
l
=
k
e
k
l
n
M
I
i
s
.Using the MGF of the Poisson distribution in (18), the MGF of I is finally given by(19)
E
e
s
I
=
e
β
l
A
l
M
I
i
s
-
1
.The first- and second-order statistics of the aggregate RF power received by N
H from the transmitters located within the annulus l are an important tool. E
[
I
], the expected value of the aggregate RF power, can be determined by using the Law of Total Expectation. It can be shown that(20)
E
I
=
E
E
I
∣
X
l
=
2
π
β
l
P
T
x
e
σ
ξ
2
R
o
l
2
-
α
-
R
i
l
2
-
α
2
-
α
.Making similar use of the Law of Total Variance, the variance of the aggregate RF power can be described as(21)
V
a
r
I
=
V
a
r
I
i
E
X
l
+
E
I
i
2
V
a
r
X
l
.Since X
l is given by a Poisson distribution (with mean β
l
A
l), the variance of the RF power is given as follows:(22)
V
a
r
I
=
β
l
A
l
∂
2
M
I
i
0
∂
s
2
=
π
β
l
P
T
x
2
k
ψ
θ
ψ
2
1
+
k
ψ
R
o
l
2
-
2
α
-
R
i
l
2
-
2
α
1
-
α
.The first and second moments can be matched with the respective moments of a given distribution to obtain a closed-form approximation for the aggregate received RF power. As shown in [27], the aggregate RF power due to path loss, fast fading, and shadowing effect can be approximated by a Gamma distribution. Consequently, the shape and the scale parameters of the Gamma distribution, denoted by k
l and θ
l, are, respectively, given by(23)
k
l
≈
E
I
2
V
a
r
I
,
(24)
θ
l
≈
V
a
r
I
E
I
.


## 3.2. RF Received Power due to the Transmitters Located withinL Annuli

As shown in the previous subsection, the RF powerI received from the transmitters located within the lth annulus is approximated by a Gamma distribution, with MGF M
I
l
(
s
)
=
(
1
-
θ
l
s
)
-
k
l. Since the annulus of width R
o
L
-
R
i
1 where the transmitters are located can be expressed as a summation of L annuli of width ρ, the MGF of the aggregate power received from the transmitters located within the L annuli is given by(25)
M
I
a
g
g
s
=
∏
l
=
1
L
1
-
θ
l
s
-
k
l
.Finally the expectation of the aggregate RF power can be computed as follows:(26)
E
I
a
g
g
=
∂
M
I
a
g
g
0
∂
s
.


## 3.3. Distribution of the Aggregate RF Power

The aggregate RF power may be stated as being the summation of theL individual aggregated RF powers received from the transmitters located within each annulus. Expressions for the PDF and the CDF of the summation of L independent Gamma random variables were initially derived by Mathai in [28]. Those were simplified in [29] in order to be computed more efficiently.Let{
Z
l
}
l
=
1
L be independent but not necessarily identically distributed Gamma variables with parameters k
l (shape) and θ
l (scale). The PDF of the aggregate RF power is written as I
a
g
g
=
∑
l
=
1
L
‍
Z
l, which can be approximated by [29](27)
f
I
a
g
g
x
≈
∏
l
=
1
L
θ
1
θ
l
k
l
∑
w
=
0
+
∞
δ
w
x
∑
l
=
1
L
k
l
+
w
-
1
exp
⁡
-
x
/
θ
1
θ
1
∑
l
=
1
L
k
l
+
w
Γ
∑
l
=
1
L
k
l
+
w
,where θ
1
=
m
i
n
l
{
θ
l
}, δ
w coefficients are computed recursively, δ
w
+
1
=
1
/
w
+
1
∑
i
=
1
w
+
1
∑
l
=
1
L
k
l
1
-
θ
1
/
θ
l
i
δ
w
+
1
-
i, and δ
0
=
1. Γ
(
·
) is the Gamma function. Finally, the CDF of I
a
g
g, F
I
a
g
g
(
x
)
=
∫
-
∞
x
‍
f
I
a
g
g
(
z
)
d
z, is computed as follows [29]:(28)
F
I
a
g
g
x
≈
∏
l
=
1
L
θ
1
θ
l
k
l
∑
w
=
0
∞
δ
w
θ
1
∑
l
=
1
L
k
l
+
w
Γ
∑
l
=
1
L
k
l
+
w
×
∫
0
x
z
∑
l
=
1
L
k
l
+
w
-
1
exp
⁡
-
z
θ
1
d
z
.


## 4. Probability of Transmission

### 4.1. Gaussian Approach

Departing from the fact that the RF power received from the transmitters located in the annulusl can be approximated by a Gamma distribution,(29)
G
a
m
m
a
k
l
,
θ
l
,with k
l and θ
l given by (23) and (24), respectively, the envelope signal (amplitude) received from the transmitters is given by the square root of a Gamma distributed random variable, which is given by a Generalized Gamma distribution with the following parameters,(30)
A
I
l
=
G
G
θ
l
,
2
k
l
,
2
.Since a Gamma distribution, with shape k
l and scale θ
l, is the sum of k Exponential (
1
/
θ
l
) distributions, using the Central Limit Theorem (CLT), when k
l is large, the Generalized Gamma distribution can be approximated by a normal distribution [30]. In these conditions the amplitude of the aggregate signals received by the harvester N
H from the transmitters located in the annulus l can be also approximated by a normal distribution represented by(31)
A
I
l
≈
N
μ
A
I
l
,
σ
A
I
l
2
≈
N
η
θ
l
Γ
k
l
+
1
/
2
Γ
k
l
,
η
2
θ
l
Γ
k
l
+
1
Γ
k
l
-
Γ
k
l
+
1
/
2
2
Γ
k
l
2
,where the loss factor 0
<
η
<
1 represents the losses associated with the RF-to-DC conversion and battery charging efficiency.During the battery charging period, the received power(
A
I
l
)
2 is accumulated in a discrete period of time Δ
t. The amount of energy stored in the battery of the harvester node during n
t time intervals is given by(32)
ϵ
l
=
Δ
t
∑
n
=
1
n
t
A
I
l
2
.Considering the unit variance random variable ϵ
l
′
=
∑
n
=
1
n
t
A
I
l
/
σ
A
I
l
2, with(33)
ϵ
l
=
ϵ
l
′
×
σ
A
I
l
2
,and considering Δ
t
=
1 for the sake of simplicity, ϵ
l
′ follows a noncentral Chi-squared distribution with noncentrality parameter λ
l
=
∑
k
=
1
n
t
(
μ
A
I
l
/
σ
A
I
l
)
2. When n
t is large enough, it is possible to use the Central Limit Theorem to approximate the Chi-square distribution to a Gaussian distribution [31], and the following approximation holds(34)
ϵ
l
′
≈
N
n
t
+
λ
l
,
2
n
t
+
2
λ
l
.Using (33), the energy accumulated in the battery of the harvester node N
H due to the transmitters located in the annulus l follows the following Gaussian distribution:(35)
ϵ
l
≈
N
μ
l
,
σ
l
2
≈
N
σ
A
I
l
2
n
t
+
λ
l
,
σ
A
I
l
4
2
n
t
+
2
λ
l
.Because L annuli are considered, the energy accumulated in the battery of the harvester node N
H due to the transmitters located in the L annuli is given by(36)
ϵ
=
∑
l
=
1
L
ϵ
l
,and ϵ follows the following distribution(37)
ϵ
≈
N
∑
l
=
1
L
μ
l
,
∑
l
=
1
L
σ
l
2
≈
N
μ
Σ
L
,
σ
Σ
L
2
.Therefore, denotingγ as the level of battery charge (accumulated energy) required to transmit a packet, the probability of reaching a γ level of accumulated energy after n
t units of time is given by(38)
P
C
=
Q
γ
-
μ
Σ
L
σ
Σ
L
2
,where Q
(
x
)
=
1
/
2
π
∫
x
∞
‍
e
-
u
2
/
2
d
u is the complementary distribution function of the standard normal.


### 4.2. Non-Gaussian Approach

In the last subsection the CLT was used to approximate the amplitude of the aggregate signals received by the harvesterN
H from the transmitters located in the annulus l, A
I
l, as represented in (31). However, when k
l is small, the CLT does not hold and, consequently, the Gaussian approach is not valid. In what follows we present a formulation when CLT does not hold. While the formulation is more computationally complex, it exhibits higher accuracy for low k
l values.Departing from the MGF in (25), the Characteristic Function (CF) of the RF power received from the annulus l is written as(39)
φ
l
t
=
∏
l
=
1
L
1
-
η
θ
l
i
t
k
l
,where 0
<
η
<
1 is the loss factor. When the aggregate power from the L annuli is considered the CF is written as follows:(40)
φ
I
a
g
g
t
=
∏
l
=
1
L
1
-
η
θ
l
i
t
k
l
.Because the amount of energy received in n
t samples is expressed as(41)
ϵ
=
∑
n
=
1
n
t
I
a
g
g
t
,the CF of ϵ is as follows:(42)
φ
ϵ
t
=
∏
b
=
1
n
t
φ
I
a
g
g
t
=
∏
b
=
1
n
t
∏
l
=
1
L
1
-
η
θ
l
i
t
κ
l
=
∏
l
=
1
L
1
-
η
θ
l
i
t
κ
l
n
t
.Using the Fourier Transform, the PDF of ϵ is written as(43)
f
ϵ
x
=
1
2
π
∫
-
∞
+
∞
e
-
i
t
x
φ
ϵ
t
d
t
.Again, the probability of reaching a γ level of accumulated energy after n
t units of time is given by(44)
P
C
ϵ
>
γ
=
1
-
P
ϵ
≤
γ
=
1
-
∫
-
∞
γ
f
ϵ
x
d
x
.


## 4.1. Gaussian Approach

Departing from the fact that the RF power received from the transmitters located in the annulusl can be approximated by a Gamma distribution,(29)
G
a
m
m
a
k
l
,
θ
l
,with k
l and θ
l given by (23) and (24), respectively, the envelope signal (amplitude) received from the transmitters is given by the square root of a Gamma distributed random variable, which is given by a Generalized Gamma distribution with the following parameters,(30)
A
I
l
=
G
G
θ
l
,
2
k
l
,
2
.Since a Gamma distribution, with shape k
l and scale θ
l, is the sum of k Exponential (
1
/
θ
l
) distributions, using the Central Limit Theorem (CLT), when k
l is large, the Generalized Gamma distribution can be approximated by a normal distribution [30]. In these conditions the amplitude of the aggregate signals received by the harvester N
H from the transmitters located in the annulus l can be also approximated by a normal distribution represented by(31)
A
I
l
≈
N
μ
A
I
l
,
σ
A
I
l
2
≈
N
η
θ
l
Γ
k
l
+
1
/
2
Γ
k
l
,
η
2
θ
l
Γ
k
l
+
1
Γ
k
l
-
Γ
k
l
+
1
/
2
2
Γ
k
l
2
,where the loss factor 0
<
η
<
1 represents the losses associated with the RF-to-DC conversion and battery charging efficiency.During the battery charging period, the received power(
A
I
l
)
2 is accumulated in a discrete period of time Δ
t. The amount of energy stored in the battery of the harvester node during n
t time intervals is given by(32)
ϵ
l
=
Δ
t
∑
n
=
1
n
t
A
I
l
2
.Considering the unit variance random variable ϵ
l
′
=
∑
n
=
1
n
t
A
I
l
/
σ
A
I
l
2, with(33)
ϵ
l
=
ϵ
l
′
×
σ
A
I
l
2
,and considering Δ
t
=
1 for the sake of simplicity, ϵ
l
′ follows a noncentral Chi-squared distribution with noncentrality parameter λ
l
=
∑
k
=
1
n
t
(
μ
A
I
l
/
σ
A
I
l
)
2. When n
t is large enough, it is possible to use the Central Limit Theorem to approximate the Chi-square distribution to a Gaussian distribution [31], and the following approximation holds(34)
ϵ
l
′
≈
N
n
t
+
λ
l
,
2
n
t
+
2
λ
l
.Using (33), the energy accumulated in the battery of the harvester node N
H due to the transmitters located in the annulus l follows the following Gaussian distribution:(35)
ϵ
l
≈
N
μ
l
,
σ
l
2
≈
N
σ
A
I
l
2
n
t
+
λ
l
,
σ
A
I
l
4
2
n
t
+
2
λ
l
.Because L annuli are considered, the energy accumulated in the battery of the harvester node N
H due to the transmitters located in the L annuli is given by(36)
ϵ
=
∑
l
=
1
L
ϵ
l
,and ϵ follows the following distribution(37)
ϵ
≈
N
∑
l
=
1
L
μ
l
,
∑
l
=
1
L
σ
l
2
≈
N
μ
Σ
L
,
σ
Σ
L
2
.Therefore, denotingγ as the level of battery charge (accumulated energy) required to transmit a packet, the probability of reaching a γ level of accumulated energy after n
t units of time is given by(38)
P
C
=
Q
γ
-
μ
Σ
L
σ
Σ
L
2
,where Q
(
x
)
=
1
/
2
π
∫
x
∞
‍
e
-
u
2
/
2
d
u is the complementary distribution function of the standard normal.


## 4.2. Non-Gaussian Approach

In the last subsection the CLT was used to approximate the amplitude of the aggregate signals received by the harvesterN
H from the transmitters located in the annulus l, A
I
l, as represented in (31). However, when k
l is small, the CLT does not hold and, consequently, the Gaussian approach is not valid. In what follows we present a formulation when CLT does not hold. While the formulation is more computationally complex, it exhibits higher accuracy for low k
l values.Departing from the MGF in (25), the Characteristic Function (CF) of the RF power received from the annulus l is written as(39)
φ
l
t
=
∏
l
=
1
L
1
-
η
θ
l
i
t
k
l
,where 0
<
η
<
1 is the loss factor. When the aggregate power from the L annuli is considered the CF is written as follows:(40)
φ
I
a
g
g
t
=
∏
l
=
1
L
1
-
η
θ
l
i
t
k
l
.Because the amount of energy received in n
t samples is expressed as(41)
ϵ
=
∑
n
=
1
n
t
I
a
g
g
t
,the CF of ϵ is as follows:(42)
φ
ϵ
t
=
∏
b
=
1
n
t
φ
I
a
g
g
t
=
∏
b
=
1
n
t
∏
l
=
1
L
1
-
η
θ
l
i
t
κ
l
=
∏
l
=
1
L
1
-
η
θ
l
i
t
κ
l
n
t
.Using the Fourier Transform, the PDF of ϵ is written as(43)
f
ϵ
x
=
1
2
π
∫
-
∞
+
∞
e
-
i
t
x
φ
ϵ
t
d
t
.Again, the probability of reaching a γ level of accumulated energy after n
t units of time is given by(44)
P
C
ϵ
>
γ
=
1
-
P
ϵ
≤
γ
=
1
-
∫
-
∞
γ
f
ϵ
x
d
x
.


## 5. Validation Results

This section describes a set of simulations and numerical results to validate the analytical methodology proposed in the paper. The simulated scenario considered a spatial circular areaA as described in Section 2 with R
i
1
=
10 m and R
o
L
= 410 m. The multiple nodes were spread over the area A according to the spatial Poisson process and 4 different spatial densities were simulated, β
l
=
{
1,2
,
3,4
}
×
1
0
-
4 nodes/m2. In each simulation the RF propagation scenario described in Section 2 was parametrized with α
=
2, and σ
ξ
d
B
= 4.5 dB. Finally, we have considered the battery operation voltage equal to 1 V, the loss factor η
=
0.5, and the required energy threshold to transmit a packet (γ) equal to 10 mAh. The parameters used in the validation are summarized in Table 1.Table 1
Parameters adopted in the validation and simulations.
ρ
400,200,100,4 m
L
2,4
,
100
β
l
1,2
,
3,4
×
10
-
7 nodes/m2
α
2
η
0.5
Δ
t
1 minute
R
i
1
10 m
R
o
L
410 m
P
T
x
20 W
σ
ξ
d
B
4.5 dB
γ
10 mAh
n
t
20The first results, presented in Figure2, compare the CDF of the RF received power computed with (28). The simulated results were obtained for the spatial density value β
l
=
1
×
1
0
-
4. In the model, a different number (L
=
{
2,4
,
100
}) of annuli were adopted to compute the model and compare the accuracy of the model for different number of annuli. As can be seen, the accuracy of the model increases with the number of annuli considered in the model. This is because as more annuli are considered for the same circular area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2, the width of each annulus ρ decreases, leading to a more accurate value of the mean and variance (E
[
I
] and Var[
I
], resp.) of the RF power received from the transmitters located in a single annuli. This fact increases the accuracy of the conditions in (23) and (24), leading to a more accurate characterization of the distribution of the received RF power. From the results plotted in Figure 2, we observe that for L
=
100 the numerical results are close to the results obtained through simulation, confirming the accuracy of the proposed model.Figure 2
CDF of the RF power received by the nodeN
H from the transmitters located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.The numerical results presented in Figure3 compare the CDF of the RF received power (computed with (28)) for different spatial density values (β
l
=
{
1
×
1
0
-
4
,
2
×
1
0
-
4
,
3
×
1
0
-
4
,
4
×
1
0
-
4
}). The numerical results were obtained considering L
=
100 (consequently ρ
=
4). As expected, the results confirm that the average of the RF power received by the harvester increases with the spatial density of the transmitters.Figure 3
CDF of the RF power received by the nodeN
H considering different densities of transmitters (β
l in nodes per square meter) located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.In Figure4 we compare the probability of having enough energy accumulated in the harvester battery to transmit a packet (P
C). The probability was computed using (38); that is, the Gaussian approach was adopted. The simulated values were obtained for the spatial density value β
l
=
4
×
1
0
-
4 nodes/m2. The charging threshold γ was defined to 10 mAh and we have considered a battery voltage of 1 V. The model was computed for L
=
1 (ρ
=
400), L
=
2 (ρ
=
200), and L
=
40 (ρ
=
10). As can be observed, the results computed with the model do not match with the ones obtained by simulation. This fact was intentionally exploited to show that while for different parameterizations the model and the simulation results match, for the specific parameterization adopted in the validation scenario the CLT does not hold because the k
l values are too small. Consequently, (30) is not an accurate approximation and a large deviation of the P
C’s model is observed. In this case, it would be better to adopt the non-Gaussian approach, because it leads to more accurate model results.Figure 4
Gaussian approach model: probability of having enough energy accumulated in the harvester battery to transmit a packet (P
C) for different density β
=
4
×
1
0
-
4 of transmitters (β
l in nodes per square meter) located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.In Figure5 we compare the probability of having enough energy accumulated in the harvester battery to transmit a packet (P
C). The probability was computed using (44), that is the non-Gaussian approach, for L
=
2 (ρ
=
200), L
=
4 (ρ
=
100), and L
=
40 (ρ
=
10). The simulated values are the same as depicted in Figure 4; that is, the considered scenario is the same. As can be observed, the results computed with the model do not match with the ones obtained by simulation for L
=
2 and L
=
4. However, if more annuli are used, the model accurately characterizes P
C, as is the case for L
=
40 in the figure. This fact is due to the approximation of k
l and θ
l in (23) and (24), respectively. As the number of annuli (L) increases, E
[
I
] and Var[
I
] in (23) and (24) become more accurate.Figure 5
Non-Gaussian approach model: probability of having enough energy accumulated in the harvester battery to transmit a packet (P
C) for different density β
=
4
×
1
0
-
4 of transmitters (β
l in nodes per square meter) located in the area A
=
π
(
R
o
L
)
2
-
(
R
i
1
)
2.As can be observed, the results computed with the model forL
=
40 are close to the ones obtained by simulation. Moreover, the probability of reaching a battery charging level equal to the γ threshold increases over time, as expected. The results confirm the accuracy of the proposed characterization, which may be easily adopted to evaluate the probability of charging over time.Finally, we highlight that the mean aggregate RF power (I
a
g
g) considered in the validation scenario is low to show the error of the Gaussian approach. For higher I
a
g
g values, the error of the Gaussian approach becomes smaller and the model becomes more accurate. The non-Gaussian approach is generally a better solution (because it does not depend on the CLT); however it exhibits a higher computational complexity.


## 6. Final Remarks

In this paper we have characterized the battery charging time of a harvester node that accumulates the received RF energy in a battery. Admitting that the transmitters are spatially distributed according to a spatial Poisson process, we use the distribution of the received RF power from multiple transmitters to derive the probability of a harvester having enough energy to transmit a packet after a given amount of charging time. The distribution of the RF power and the probability of a harvester node having enough energy to transmit a packet are validated through simulation. The numerical results obtained with the proposed analysis are close to the ones obtained through simulation, which confirms the accuracy of the proposed analysis.



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*Source: 7849175-2016-10-17.xml*