Power Distribution of D2D Communications in Case of Energy Harvesting Capability over κ-μ Shadowed Fading Conditions

The Internet of Things (IoT) has revolutionized the world through its centric concepts, such as augmented reality, high-resolution video streaming, self-driven cars, smart environment, e-health care, etc. This new paradigm is based on various technologies, such as the combining of sensing devices and embedded systems with cyber-physical systems, device-to-device (D2D) communications and 5G wireless systems [1]. One of the most important features of 5G networks is that it allows seamless connectivity for any type of devices, applications and heterogeneous networks. On the other hand, D2D communication is a promising technology that enables two or more neighbor devices to communicate directly, in either a stand-alone or a network-coordinated fashion. The potential of D2D communication in 6G cellular networks for applications of smart factories (such as industry 5.0) and vehicular communications, according to [2], has recently been underlined (e.g., autonomous driving and vehicular platooning). Infotainment is also one of the services provided among D2D. D2D communication provides some advantages, such as cellular coverage extension, data offloading, information sharing and energy efficiency [3]. This kind of communication can be also used for safety applications (rescue and search missions, road safety, etc.) [4]. While in [5], the authors tackled the problem of connectivity and security in terms of public safety based on D2D systems.

Regarding the spectrum allocated for D2D communications, two strategies can be envisioned [6]: (i) in-band, where cellular communication and D2D communication share the same spectrum licensed to the cellular network. This spectrum can be divided into non-overlapping or shared bands (respectively, overlaying and underlaying); (ii) out-band, where D2D communication exploits an unlicensed spectrum (ISM band). The overlay mode is easy to implement; even so, the underlay one is more efficient for spectrum usage. The coexistence of D2D and cellular communications in the same frequency band is challenging due to the hardness of interference handling as illustrated in Figure 1. This problem can be solved through power control and resource allocation [7]. This power control mechanism also contributes in device battery usage optimization. The issue of energy consumption in the context of D2D communication is very active in the research community [8,9,10]. In addition, energy harvesting techniques can substantially improve the operation duration of D2D equipments through the additional energy gathered [11].

In this paper, we focus on the power control mechanism using stochastic geometry tools while considering general fading channel conditions. Further, we analyze the energy-harvesting capability in the D2D case. The contributions of this paper are summarized as follows:

The remainder of this paper is structured as follows: In Section 2 we discuss some related works. Section 3 presents the system model for D2D communications under $\kappa$-$\mu$ shadowed fading channels. Section 4 analytically provides the distribution of transmit power and derives some relevant special cases. In Section 5, we present an RF energy-harvesting model based on the cellular ambient RF transmissions. Section 6 provides numerical results for both simulations and analytical derivations. Finally, the Section 7 concludes the paper.

In order to deal with the interference issue, generally, transmit power control uses channel state information (CSI) between the sender and receiver to meet the required signal-to-noise ratio [12]. In order to model the channel, many recent works use some basic distributions, but the fitting of the shadowed $\kappa$-$\mu$ distribution to experimental data is better than that achieved by the classical distributions previously mentioned. It provides a general multi-path model for a line-of-sight (LOS) propagation scenario controlled by two shape parameters $\kappa$ and $\mu$, in which, the LOS component is subject to shadowing. In addition, some classical fading distributions are included in the shadowed $\kappa$-$\mu$ distribution as particular cases, e.g., one-sided Gaussian, Rayleigh, Nakagami-m, Rician and $\kappa$-$\mu$ [13]. More recently, other generalized fading models are proposed, such as $\alpha$-$\kappa$-$\mu$, $\alpha$-$\eta$-$\mu$ and $\alpha$-$\kappa$-$\mu$ shadowed [14,15].

D2D equipments are battery-powered leading to consider the energy as a main concern for the communication networks. Some literature works have energy management and optimization as the main concern ([16,17,18,19]). Recently, in [20], the authors proposed a technique to handle the battery non-linearity in order to extend the network usage duration.

Another interesting technology that can significantly help in increasing the D2D device lifetime is radio frequency energy harvesting (RF-EH). It is defined as the capability to convert the received RF signals into electricity in order to help the device in its information processing and transmission [21]. This technique is commonly utilized in energy-constrained wireless networks, as they have a limited lifetime that largely confines the network performance. It can be applied in multiple fields, such as wireless sensor networks, wireless body networks and wireless charging systems. In RF-EH, radio signals with frequencies from 3 kHz to 300 GHz are used as a medium to carry energy in an electromagnetic radiation form. The received signal strength decays inversely proportionally to the traveled distance, specifically at 20 dB per decade of the distance [22]. Hence, the RF-EH depends on three main factors: the transmit power, the distance between the energy source and the harvester and the wavelength of the RF signals [23]. It has been shown that it is possible to harvest 3.5 mW of power at a distance of 0.6 m and 1 $\mathsf{\mu }$W at a distance of 11 m using a Powercast RF energy harvester module operating at 915 MHz [24]. RF-EH was discussed in [25] for cognitive radio networks. An RF power conversion circuit extracts DC power from the received RF signals. The circuit is activated only when the received RF power is greater than a predefined threshold, which depends on circuit sensitivity. Similarly, in [26], an energy-harvesting region was considered for relay users, where they harvest ambient RF energy from access points. By considering energy harvesting, the distribution of relays is derived in order to increase the D2D transmission opportunities. Authors in [27] have proposed an IoT energy-harvesting model that does not require battery storage, nor a voltage converter. The implementation tests of the proposed system showed an 8% increase in the amount of harvested power and a 60% increase in the device lifetime. More recently, in [28], the authors conducted experiments with zero energy IoT devices, where the needed power supply was scavenged from the RF signal. In the context of energy harvesting, RF sources can be devised into two categories: dedicated RF sources and ambient ones. Dedicated RF sources use ISM frequency bands. They are used when a more predictable energy supply is needed, but they have some inconveniences, such as generating a high deployment cost, and their power can be limited by federal regulations due to health and safety concerns about RF radiation. On the other side, as the ambient RF sources are not intended specifically for RF energy transfer, they provide free energy [22]. More recently, in [29], authors considered the impact of the RF power density variation on the instantaneous charging capacity of the energy buffer in the RF energy-harvesting system. Although RF-EH has the lowest energy intensity compared to other sources of energy, it has some advantages. In particular, it does not depend on the weather or geographical conditions (unlike solar and wind energy); it can be used in any location where there is ambient RF waves; and it can serve multiple device at the same time [24]. It is important to note that a wireless power transfer can be used for equipments that required an important power supply, as experimented in [30].

In Table 1, we summarize the contribution of the present work as a comparison to the literature review.

The list of all of the abbreviations cited in this paper is summarized in the following Table 2:

In this work, we considered an underlaid scenario where multiple macro-cellular UEs and D2D users share the same frequency band as depicted in Figure 2. Independent homogeneous Poisson point processes (PPP) ${\mathsf{\Phi }}_C$ and ${\mathsf{\Phi }}_D$ with constant intensity ${\lambda }_C$ and ${\lambda }_D$ were used to model the location of cellular and D2D equipment, respectively. In order to transmit, each UE uses independent and identically distributed (i.i.d.) transmit powers.

We assumed that all links are affected by i.i.d. $\kappa$-$\mu$ shadowed fading, and we denoted by H the fading power. For the sake of brevity, we say that: $H\sim S(\beta ,\kappa ,\mu ,m)$, where $\beta =\mathbb{E}\left[H\right]$ is the mean of H, and $\kappa ,\mu ,m$ are the parameters of the $\kappa ,\mu$ shadowed distribution, with the conditions that: $\mu ,m\in \mathbb{N}$, and $\mu \le m$.

With this setting, and from [35], eqn. (4b)], the PDF of H is expressed as: with parameters defined as $d={\displaystyle \frac{m\mu {(1+\kappa )}^2}{m+\mu {\kappa }^2+2m\kappa }}$, and $\omega ={\displaystyle \frac{d}{\beta }}$.

In addition, the CCDF of H (from [36], Equation (13)]) is given as:with $c={\displaystyle \frac{m\mu (1+\kappa )}{\beta (\mu \kappa +m)}}$, and $b_i={\displaystyle \frac{(m-\mu )!}{i!(m-\mu -i)!}}.{\left({\displaystyle \frac{m}{\mu \kappa +m}}\right)}^i{\left({\displaystyle \frac{\mu \kappa }{\mu \kappa +m}}\right)}^{m-\mu -i}$.

For a transmitter–receiver distance R, the path loss has a standard singular model of $R^{-\alpha }$ with a path-loss exponent $\alpha$ with $2<\alpha <6$.

We supposed that each D2D receiver only connects to the nearest D2D transmitter and that each D2D transmitter communicates with exactly one D2D receiver. This is ensured if the D2D receiver outnumbers the D2D transmitter, if not, some D2D transmitters will have no receiver, and these D2D transmitters will be inactive [32].

Without a loss of generality, we are interested in a typical D2D link where the transmitter is located at the origin $\mathit{O}$, and denote it as subscript 0. Let $P_0$ be the typical D2D transmitter transmit power, $H_0$ be the fading power in the typical D2D communication channel and the distance between the typical D2D transmitter and receiver be $R_0$. Hence, we can express the received power in the typical D2D link as

At the typical D2D receiver, the total interference caused by the other D2D transmitters is given as:and the interference caused by all cellular transmitters is given by the following expression:where $P_{i,.}$, $H_{i,.}$ and $R_{i,.}$ represent the transmit power, the channel fading power and the locations of the cellular user (subscript C) and D2D users (subscript D), respectively. In the expression of D2D interference $I_D$, we exclude the signal power generated by our D2D typical transmitter. Note that $I_C$ and $I_D$ are independents, as we consider that the transmit power, the fading power and the location of cellular and D2D users are all independent of each other.

In this paper, we focus our study on one macro-cell area, and we consider that the interference from other neighbor cells is negligible. Thus, the $SINR$ at the typical D2D receiver is given by where $N_0$ is the noise power and T is the minimum $SINR$ threshold required for a successful communication between a D2D pair. Table 3 recaps all variables used in our system model:

Starting with the first expectation over I and putting $s=cT{R}_0^{\alpha }/p$, we obtain

Since the total interference is: $I=I_D+I_C$ and $I_D$ and $I_C$ are independent, we have

Thus, in the first step, we try to calculate ${\mathcal{L}}_{I_D}\left(s\right)$:

Based on the probability-generating functional $\left(PGFL\right)$ theorem for $PPP$, and neglecting the small integration from 0 to $R_0$, we obtain:

By the same way, we obtain for the Laplace transform ${\mathcal{L}}_{I_C}(s)$ with the following result: hence, the final expression of ${\mathbb{E}}_{I_1}$ is: now, passing to the second expectation over I in (8), denoted by ${\mathbb{E}}_{I_2}$:

We note that $C_k^t={\displaystyle \frac{k!}{t!(k-t)!}}$. The last step comes from the fact that the expected value of the product of independent variables is the product of expected values.

Thus, we start by calculating the first term in the last equation, denoted by ${\mathbb{E}}_{I_{2,D}}$. Using Theorem 2 in [37] for $t\in {\mathbb{N}}^{*}$, we have where $C_M={\displaystyle \frac{t!}{{\prod }_{r=1}^l{m}_r!}}$, ${\mathbb{N}}^l$ is the set of vectors M with l natural elements, ${\Vert M\Vert }_1={\sum }_{i=1}^l{m}_i=t$ and D is the disk of origin $O$ and radius $R_0$.

Let us calculate the expressions of ${\mathbb{E}}_{H,P}[1]$ and ${\mathbb{E}}_{H,P}[2]$, related to the expectations given in (11), respectively:

The second expectation can be defined as:

Based on (11) and (12), the expression of ${\mathbb{E}}_{I_{2,D}}$ is given as:

We express both integrals with respect to x:

The second integral is:

Thus, the final form of ${\mathbb{E}}_{I_{2,D}}$ is given as:

Using the same method as done with ${\mathbb{E}}_{I_{2,D}}$, we have:

Now, we can calculate the total expression of ${\mathbb{E}}_{I_2}$ based on (13), (14) and (10): where $E_D={\mathbb{E}}_P[P_D^{\frac{2}{\alpha }}]$ and $E_C={\mathbb{E}}_P[P_C^{\frac{2}{\alpha }}]$.

We can then have the expression of our $CDF$ using the $PDF$ of $R_0$ expressed in [38]: $f_{R_0}(r)=2\pi {\lambda }_{D}r.e^{-\pi {\lambda }_D.r^2}$.

Thus, (8) becomes:

Finally, the transmit power CDF is expressed as follows: where $k_1=\frac{cT{N}_0}{p}$, $k_2=\pi {\lambda }_D+\pi {(\frac{cT}{\omega p})}^{2/\alpha }.\frac{\mathsf{\Gamma }(1-\frac{2}{\alpha }).\mathsf{\Gamma }(d+\frac{2}{\alpha })}{\mathsf{\Gamma }(d)}({\lambda }_D{E}_D+{\lambda }_C{E}_C).$

Therefore, we have the first integral multiplier $\sum _{i,j}f(i,j)={\displaystyle \sum _{i=0}^{m-\mu }}{b}_i{\displaystyle \sum _{j=0}^{m-i-1}}\frac{2\pi {\lambda }_D}{j!}{(\frac{Tc}{p})}^j.N_0^j$, the second $\begin{array}{l}{\sum }_{i,j,k,l,M}g(i,j,k,l,M)={\displaystyle \sum _{i=0}^{m-\mu }}{b}_i{\displaystyle \sum _{j=1}^{m-i-1}}\frac{2\pi {\lambda }_D}{j!}{(\frac{Tc}{p})}^j{\displaystyle \sum _{k=1}^j}{C}_j^k{N}_0^{j-k}\\ {\displaystyle \sum _{l=1}^k}{\displaystyle \sum _{M\in {\mathbb{N}}^l}}{\displaystyle \frac{C_M}{l!}}{\prod }_{i=1}^l\frac{2\pi }{\alpha \mathsf{\Gamma }(d)}.{(\frac{1}{\omega })}^{2/\alpha }{(\frac{cT}{p})}^{2/\alpha -m_i}\mathsf{\Gamma }(m_i-\frac{2}{\alpha })\mathsf{\Gamma }(d+\frac{2}{\alpha })({\lambda }_D{E}_D+{\lambda }_C{E}_C),\end{array}$ and the last integral multiplier$\begin{array}{l}{\sum }_{i,j,k,t,l,M}h(i,j,k,t,l,M)={\displaystyle \sum _{i=0}^{m-\mu }}{b}_i{\displaystyle \sum _{j=1}^{m-i-1}}\frac{2\pi {\lambda }_D}{j!}{(\frac{Tc}{p})}^j{\displaystyle \sum _{k=1}^j}{C}_j^k{N}_0^{j-k}{\displaystyle \sum _{t=1}^{k-1}}{C}_k^t{\displaystyle \sum _{l=1}^t}{\displaystyle \sum _{s=1}^{k-t}}{({\lambda }_D{E}_D)}^l{({\lambda }_C{E}_C)}^s\\ ({\displaystyle \sum _{M\in {\mathbb{N}}^l}}{\displaystyle \frac{C_M}{l!}}{\prod }_{i=1}^l\frac{2\pi }{\alpha \mathsf{\Gamma }(d)}.{(\frac{1}{\omega })}^{2/\alpha }{(\frac{cT}{p})}^{2/\alpha -m_i}\mathsf{\Gamma }(m_i-\frac{2}{\alpha })\mathsf{\Gamma }(d+\frac{2}{\alpha }))\\ ({\displaystyle \sum _{M\in {\mathbb{N}}^s}}{\displaystyle \frac{C_M}{s!}}{\prod }_{i=1}^s\frac{2\pi }{\alpha \mathsf{\Gamma }(d)}.{(\frac{1}{\omega })}^{2/\alpha }{(\frac{cT}{p})}^{2/\alpha -m_i}\mathsf{\Gamma }(m_i-\frac{2}{\alpha })\mathsf{\Gamma }(d+\frac{2}{\alpha }))\end{array}$

with: $A_1=cT{N}_0;A_2=\pi {\lambda }_D;$ $A_3=\pi {(\frac{cT}{\omega })}^{\frac{2}{\alpha }}.\frac{\mathsf{\Gamma }(1-\frac{2}{\alpha }).\mathsf{\Gamma }(d+\frac{2}{\alpha })}{\mathsf{\Gamma }(d)}{\lambda }_C{E}_C;$ $A_4=\pi {(\frac{cT}{\omega })}^{\frac{2}{\alpha }}$$.\frac{\mathsf{\Gamma }(1-\frac{2}{\alpha }).\mathsf{\Gamma }(d+\frac{2}{\alpha })}{\mathsf{\Gamma }(d)}{\lambda }_D;$