Cooperative Nano Communication in the THz Gap Frequency Range using Wireless Power Transfer

Abstract

Advancements in nanotechnology and novel nano materials in the past decade have provided a set of tools that can be used to design and manufacture integrated nano devices, which are capable of performing sensing, computing, data storing and actuation. In this paper, we have proposed cooperative nano communication using Power Switching Relay (PSR) Wireless Power Transfer (WPT) protocol and Time Switching Relay (TSR) WPT protocol over independent identically distributed (i.i.d.) Rayleigh fading channels in the Terahertz (THz) Gap frequency band to increase the range of transmission. Outage Probability (OP) performances for the proposed cooperative nano communication networks have been evaluated for the following scenarios: A) A single decode-and-forward (DF) relay for PSR protocol and TSR protocol, B) DF multi-relay network with best relay selection (BRS) for PSR protocol and TSR protocol, and C) DF multi-relay network with multiple DF hops with BRS for PSR protocol and TSR protocol. The results have shown that the transmission distance can be improved significantly by employing DF relays with WPT. They have also shown that by increasing the number of hops in a relay the OP performance is only marginally degraded. The analytical results have been verified by Monte-Carlo simulations.

1. Introduction

 The advancements made in nanotechnology and novel nano materials in the past decade have made it possible for many innovative applications in different fields such as biomedical, environmental, military, etc. to be developed. Nanotechnology has enabled devices to be built in the range of 1 to a few hundred nanometers which are capable of performing basic tasks as sensing, relaying, etc. as discussed in [1] and [2].

 Richard Feynman first envisioned nanotechnology in 1959 in a speech titled "There's Plenty of Room at the Bottom". Nanotechnology have enabled engineers to design and manufacture integrated devices which are capable of performing many basic applications, but also more complex functions with communication, control and coordination among the nano machines [3]. A new type of wireless sensor network has been developed by combining nanotechnology and materials science theory, called bionanosensor networks in [4]. These bionanosensor networks are composed of a number of nanomachines with the ability to sense chemical signals. Nano sensors are actually nano machines that are capable of taking advantage of properties of novel nanomaterials, such as Graphene that can detect chemical compounds as low as one part per billion, presence of viruses and bacteria [5].

 Graphene, a single layer of carbon packed in a hexagonal lattice with a carbon-carbon distance of 0.142nm, is the first truly two-dimensional crystalline material, and is representative of a whole class of 2D materials. Graphene is extremely thin, mechanically very strong, flexible, transparent, and is a conductor. These properties make it interesting for applications in flexible electronics, gas sensing, and due to its low weight, for many applications in aircrafts, satellites, and others [6]. Graphene Nanoribbons and Carbon Nanotubes have enabled the development of  nano-batteries, nano-memories, nano-sensors, and nano-actuators, which take advantage of the unique properties observed in nanomaterials and its derivatives [7]. In [8], [9], and [10], THz frequency band nano antennas made of Graphene have been proposed and their performances have been investigated.

 In the electromagnetic spectrum located between the millimeter waves and infrared light waves are the THz waves, which have been rarely utilized except for astronomy and fields related to astronomy [11]. In [12] and [13], THz waves in the range of 0.1 THz - 10 THz for future wireless communication and technologies have been investigated. A review of the applications related to THz wave technologies and communications are discussed by Ho-Jin Song and Nagatsuma in [14]. The total path loss of a propagating wave in the terahertz band depends on the frequency, distance and the composition of the transmission medium at a molecular level [5]. Furthermore, in [5], a channel model for electromagnetic wireless nanonetworks in the terahertz band has been introduced, and the capacity of the nanonetwork has been analyzed.

 One of the hurdles for nano machines has been the amount of power that the nano machines are capable of producing, which has been around 800pJ so far [15], [16], [17], and [18]. However, by incorporating WPT techniques, it is possible to provide the necessary power that nano machines require for wireless data transmission. Nikola Tesla described the freedom to transfer energy between two points without the need for a physical connection to a power source as an “all-surpassing importance to man” [19] and [20]. Power transmission by radio waves dates back to the early work of Heinrich Hertz [21], where he demonstrated electromagnetic wave propagation in free space using a complete system with a spark gap to generate high-frequency power and to detect it at the receiving end. Power transmission is a three-step process where 1) dc electrical power is converted to RF power, 2) the RF power is then transmitted through free space to the receiver, and 3) the power is collected and converted back into dc power at the receiving point, [21]. In practice, WPT is usually implemented based on one of these three different technologies: 1) inductive coupling [22], 2) magnetic resonance coupling [23], or 3) microwave power transfer [21], respectively for short range (centimeter range), mid range (several meter range), and long range (up to tens of kilometer range), [24]. In [25], Varshney first proposed transmitting information and energy simultaneously. A time switching receiver (TSR) protocol and a power switching receiver (PSR) protocal are proposed in [26]. A practical point-to-point wireless simultaneous information and power transfer scheme is proposed, and the concept is further extended to multiple-input multiple-output (MIMO) systems in [27]. Energy harvesting is an effective and efficient means to prolong the life of wireless networks and increase the range of transmission with increase of power. In [28], a best cooperation mechanism for cooperative 5G network that can harvest energy and transmit data simulteneously in timeslot mode has been proposed.

 We have proposed in this paper using cooperative communication techniques with wireless power transfer to increase the transmission range of nano machines in the THz Gap frequency range for non-line-of-sight communication. By making advances in wireless nanosensor networks, it is possible to enable health monitoring, surveillance of weapons of mass destruction, etc [5].

The rest of this paper has been organized as follows: in Section II, THz wave propagation, wireless power transfer and relay models are explained; in Section III, outage probability has been calculated for: A) A single DF relay for PSR protocol and TSR protocol, B) multiple DF relays with BRS for PSR protocol and TSR protocol, and C)  multiple DF relays with multiple DF hops with BRS for PSR protocol, and TSR protocol; in Section IV, the numerical calculations are validated by simulations, and the results are investigated; finally, the paper is summarized and conclusions are presented in Section V.

 

2. Propagation, Wireless Power Transfer and Relaying Models

2.1 Terahertz Propagation Model

 A propagation model for Graphene-based nano-tranceivers in the Terahertz band (0.1 - 10.0 THz) is given in [5], where path loss and noise power have been calculated as follows  

 

2.1.1 Path loss

 The total path loss for an electromagnetic wave in the Terahertz band can be written as follows

\(A(f, d)=A_{ {spread}}(f, d)+A_{ {absorption}}(f, d)\)       #(1)

where A_spread (f,d) denotes the spreading loss of the signal and A_absorption (f,d) denotes the signal attenuation due to molecular absorption. The spreading loss can be written as

\(A_{ {spread}}(f, d)=20 log \left(\frac{4 \pi f d}{c}\right)\)       #(2)

and the molecular absorption attenuation can be written as

\(A_{ {absorption}}=\frac{1}{\tau(f, d)}=e^{k(f) d}\)       #(3)

Transmittance of medium, τ , has been obtained using the Beer-Lambert Law as

\(\tau(f, d)=\frac{P_{0}}{P_{i}}=e^{-k(f) d}\)       #(4)

where  f  is the frequency of the electromagnetic wave considered, d  is the total path length, P₀ and P_i are the radiated and incident powers, and k  is the medium absorption coefficient. The absorption coefficient for the composition of molecules found along the path can be defined as

\(k(f)=\sum_{i, g} k^{i, g}(f)\)       #(5)

where absorption coefficient of the isotopologue i of gas g is k^i,g in m^-1 and is calculated as

\(k^{i, g}(f)=\frac{p}{p_{0}} \frac{T_{s t p}}{T} Q^{i, g} \sigma^{i, g}(f)\)       #(6)

 Q^i,g, p, T, p₀, T_stp and σ^i,g are respectively the volumetric density in molecules⁄m³ ,  pressure, temperature, standard pressure, standard temperature, and the absorption cross section for the isotopologue i of gas g in m^2⁄molecule. The total number of molecules per unit volume, Q^i,g, of the isotopologue i of gas g for the given gas mixture can be calculated using the Ideal Gas Law as

\(Q^{i, g}=\frac{p}{R T} q^{i, g} N_{A}\)       #(7)

where R is the gas constant, q^(i,g) is the mixing ratio for the isotopoloque i of gas g, and N_A stand for the Avogadro constant.  σ^(i,g) can be defined as

\(\sigma^{i, g}=S^{i, g} G^{i, g}(f)\)       #(8)

where S^i.g is the line intensity which can be directly obtained from the High Resolution Transmission (HITRAN) database. S^i.g defines the strength of the absorption by a specific type of molecules. gi.g  is the line shape; the continuum absorption at the far ends of the line shape can be accounted as proposed in [29] and subsequently used in [5] as

\(G^{i, g}(f)=\frac{f}{f_{c}^{i, g}} \frac{tanh \left(\frac{h c f}{2 k_{B} T}\right)}{tanh \left(\frac{h c f_{c}^{i, g}}{2 k_{B} T}\right)} F^{i, g}(f)\)       #(9)

where h is the Planck constant, c is the speed of light in a vacuum, k_b is the Boltzmann constant, and T is the system temperature. To obtain the line shape, G^(i,g), the resonant frequency f^i.g_c) for the isotopologue i of gas g has to be determined as follows

\(f_{c}^{i, g}=f_{c 0}^{i, g}+\delta^{i, g} \frac{p}{p_{0}}\)       #(10)

where f^i.g_c0is the zero-pressure position of the resonance, and δ^i.g is the linear pressure shift. These parameters can be directly obtained from the HITRAN database. Van Vleck-Wisskopf asymmetric line shape [30] has been considered for the representation of molecular absorption

\(F^{i, g}(f)=100 c \frac{\alpha_{L}^{i, g}}{\pi} \frac{f}{f_{c}^{i, g}}\left[\frac{1}{\left(f-f_{c}^{i, g}\right)^{2}+\left(\alpha_{L}^{i, g}\right)^{2}}+\frac{1}{\left(f+f_{c}^{i, g}\right)^{2}+\left(\alpha_{L}^{i, g}\right)^{2}}\right]\)       #(11)

where f, c, α_L^i.g, and f_c^(i,g) are respectively the frequency considered, speed of light in a vacuum, Lorentz half-width coefficient for isotopologue i of gas g, and resonant frequency for the isotopologue i of gas g. Lorentz half-width can be obtained as a function of air-broadened half-width α^air₀ and self-broadened half-width α^i.g₀ as follows

\(\alpha_{L}^{i, g}=\left[\left(1-q^{i, g}\right) \alpha_{0}^{i, g}+q^{i, g} \alpha_{0}^{i, g}\right]\left(\frac{p}{p_{0}}\right)\left(\frac{T_{0}}{T}\right)^{\gamma}\)       #(12)

where q^i.g, p, p₀, T₀, T, and γ are respectively the mixing ratio for the isotopologue i of gas g,  system pressure, reference pressure, reference temperature, system temperature, and temperature broadening coefficient. The values for γ, α₀^air, and α^i.g₀ can be directly obtained from the HITRAN database.

 

2.1.2 Noise Power

 The total noise of the system contains molecular absorption noise, antenna noise temperature, system noise temperature, etc.

\(T_{ {noise}}=T_{ {system}}+T_{ {molecular}}+T_{ {other}}\)       #(13)

 The noise power at the receiver can be calculated as follows for a given bandwidth

\(P_{n}(f, d)=\int_{B} N(f, d) d f=k_{B} \int_{B} T_{n o i s e}(f, d) d f\)       #(14)

where f , d , N, k_B, and T_noise are respectively the frequency considered, transmission distance, noise power spectral density, Boltzmann constant, and equivalent noise temperature. The molecular absorption noise introduced during the propagation of the electromagnetic waves in the terahertz band can be calculated as follows. The parameter that measures this phenomenon is known as the emissivity of the channel, ε

\(\varepsilon(f, d)=1-\tau(f, d)\)       #(15)

where τ is the transmissivity of the medium given from equation (4) [5]. The noise temperature due to molecular absorption T_molecular that an omnidirectional antenna detects from the medium can be obtained as follows in Kelvin

\(T_{ {molecular}}(f, d)=T_{0} \varepsilon(f, d)\)       #(16)

T_0 and ε are respectively the reference temperature and emissivity of the channel [5]. It is assumed for this paper that graphene based electronics devices are very low noise, and therefore, only molecular noise absorption is considered.

 

2.2 Wireless Power Transfer

 Energy harvesting is an effective means to prolong the life and range of a wireless network. A simultaneous wireless information and power transfer scheme is proposed in [31] and [25]. The harvested energy stored in the battery, denoted by Q  in joule, is given by

\(Q=\zeta \mathbb{E}\left[i_{D C}(t)\right]=\zeta h P\)       #(17)

where ζ is the conversion efficiency 0<ζ<1 and P is the received signal power. Two practically realizable receiver designs have been proposed in [32], namely time-switching receivers and power-switching receivers.

 

2.2.1 Time Switching-Based Relaying Protocol  

Fig. 1. TSR protocol

For the TSR Protocol in Fig. 1, T is the block of time during which information and power are transmitted;  α is the fraction of time that is used for harvesting energy, and the remaining time (1- α)T is used for information transmission, where α is  0 <α < 1. All the energy harvested during the harvesting phase is consumed during the transmission of the information. The harvested energy can be written as follows from [32]

\(E_{h}=\frac{\eta P_{b}|h|^{2}}{A_{i}} \alpha T\)       #(18)

where η is the energy conversion efficiency, 0 < η <1, which depends on rectification process and energy harvesting circuitry, h denotes the channel coefficient of the wireless power transmission link, P_b is the transmitted power from the power beacon [33], and 〖 A〗_i  represents the total pathloss of the electromagnetic wave transmission.

 

2.2.2 Power Switching-Based Relaying Protocol

Fig. 2. PSR protocol

 From Fig. 2, it is possible to see that T is the total block time at the node. During the first half of the block, ρP_b is used for energy harvesting, and (1- ρ) P_b is used for information transmission.  ρ is the choice of power fraction used at the node for energy harvesting where ρ is 0 < ρ <1. It is assumed that all the harvested energy is consumed during the information transmission. The harvested energy at the node can be given as follows from [32]

\(E_{h}=\frac{\eta \rho P_{b}|h|^{2}}{A_{i}}(T / 2)\)       #(19)

where η is the energy conversion efficiency, 0 < η <1, which depends on rectification process and energy harvesting circuitry, h denotes the channel coefficient of the wireless power transmission link, P_b is the transmitted power from the power beacon [33], and  A_i  represents the total pathloss of the electromagnetic wave transmission.

 

2.3 Relay Models

 In this paper, decode-and-forward relaying scheme has been employed at all relays and hops. Where there are multiple relays, BRS scheme has been employed, and PSR protocol and TSR protocol have been used for WPT to increase the range of transmission over Rayleigh fading channels. P_PB   represents the transmitted signal power from the power beacon, A_i represents the total pathloss of the electromagnetic wave in the Terahertz band due to molecular absorption attenuation and spreading loss,  and N_i is the total noise.

2.3.1 Single DF Relay

Fig. 3. Single DF Relay using WPT

 A cooperative nano relaying network as in Fig. 3, where data is transmitted from the source nano machine to the destination nano machine through a relaying nano machine, has been considered. hsr denotes the channel coefficient of the Source to Relay link, hrd denotes the channel coefficient of the Relay to Destination link, g_(PB_s ) denotes the channel coefficient of the Power Beacon to Source link, and g_(PB_R ) denotes the channel coefficient of the Power Beacon to Relay link. The instantaneous end-to-end signal-to-noise ratio (SNR) can be written as

\(\gamma_{d}=\min \left(\alpha_{s r}\left|h_{s r}\right|^{2}\left|g_{P B_{S}}\right|^{2}, \alpha_{r d}\left|h_{r d}\right|^{2}\left|g_{P B_{R}}\right|^{2}\right)\)       #(20)

where for TSR Protocol \(\alpha_{i}=\frac{2 \eta \alpha P_{P B}}{A_{i} A_{P B_{i}}(1-\alpha) N_{i}}\), and for PSR Protocol \(\alpha_{i}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{i} A_{P B_{i}} N_{i}}\). \(\gamma_{s r}=\alpha_{s r}\left|h_{s r}\right|^{2}\left|g_{P B_{S}}\right|^{2}\)  represents Source to Relay SNR, and \(\gamma_{r d}=\alpha_{r d}\left|h_{r d}\right|^{2}\left|g_{P B_{R}}\right|^{2}\)  represents Relay to Destination SNR. Furthermore, γ_d  represents the end-to-end SNR for S→R →D.

 

2.3.2 Multiple DF Relays with BRS

Fig. 4. Multiple DF Relays with BRS using WPT

 A cooperative nano relaying network as in Fig. 4 has been considered. The data is transmitted from the source nano machine to the destination nano machine through multiple relaying nano machines, where the best relay is selected for information transmission. h_srk denotes the channel coefficient of the linkS → R_k, h_rkd denotes the channel coefficient of the link R_k→D, g_PBs) denotes the channel coefficient of the link PB → S, and g_PBRk denotes the channel coefficient of the link PB_RK)→R_k. The end-to-end SNR can be written as

\(\gamma_{d}={k}=\overset{max}{1,2}, \ldots, K(\gamma_k)\)       #(21)

and γ_k  can be determined by

\(\gamma_{k}={k}=\overset{max}{1,2}, \ldots, K\left(\alpha_{s r_{k}}\left|h_{s r_{k}}\right|^{2}\left|g_{P B_{S}}\right|^{2}, \alpha_{r_{k} d}\left|h_{r_{k} d}\right|^{2}\left|g_{P B_{R_{ k}}}\right|^{2}\right)\)       #(22)

where for TSR Protocol α_i=  \(\alpha_{i}=\frac{2 \eta \alpha P_{P B}}{A_{i} A_{P B_{i}}(1-\alpha) N_{i}}\), and for PSR Protocol  \(\alpha_{i}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{i} A_{P B_{i}} N_{i}}\) represents \(S \rightarrow R_{k}\) SNR, and \(\gamma_{r_{k} d}= \alpha_{r_{k} d}\left|h_{r_{k} d}\right|^{2}\left|g_{P B_{R_{k}}}\right|^{2}\) represents   \(R_{k} \rightarrow D \text { SNR }\). γ_k represents the SNR for each particular relay. Furthermore, γ_d  represents the end-to-end SNR for \(S \rightarrow R_{k} \rightarrow D\).

 

2.3.3 Multiple DF Relays with Multiple DF Hops using BRS

A cooperative nano relaying network as in Fig. 5, where the data is transmitted from the source nano machine to the destination nano machine through relaying nano machines, has been considered. \(h_{s r_{1 k}}\)  denotes the channel coefficient of the link  \(S \rightarrow R_{1 k}, h_{r_{N k} d}\)  denotes the channel coefficient of the \(R_{N k} \rightarrow D, g_{P B_{s}}\)  denotes the channel coefficient of the link  \(P B \rightarrow S, \text { and } g_{P B_{N k}}\)denotes the channel coefficient of the link \(P B \rightarrow R_{n, k}\)

Fig. 5. Multiple DF Relays with Multiple DF Hops with BRS using WPT

 The end-to-end SNR can be written as

\(\gamma_{d}={k}=\overset{max}{1,2}, \ldots, K(\gamma_k)\)       #(23)

and γ_k can be determined by

\(\gamma_{k}={k}=\overset{max}{1,2}, \ldots, K\left(\alpha_{s r_{1 k}}\left|h_{s r_{1 k}}\right|^{2}\left|g_{P B_{S}}\right|^{2}, \ldots, \alpha_{r_{N k} d}\left|h_{r_{N k} d}\right|^{2}\left|g_{P B_{R_{N k}}}\right|^{2}\right)\)       #(24)

where for TSR Protocol\(\alpha_{i}=\frac{2 \eta \alpha P_{P B}}{A_{i} A_{P B_{i}}(1-\alpha) N_{i}}\), and for PSR Protocol  \(\alpha_{i}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{i} A_{P B_{i}} N_{i}}\) represents \(S \rightarrow R_{1 k} \text { SNR }\), and \(\gamma_{N k}=\alpha_{s r_{N k}}\left|h_{s r_{N k}}\right|^{2}\left|g_{P B_{R_{N k}}}\right|^{2}\) represents\(R_{N k} \rightarrow D \text { SNR }\). Furthermore, γ_d  represents the end-to-end SNR for S → R_Nk  →D.

 

3. Outage Probability

 Outage probability is the probability that the instantaneous SNR at the receiver falls below a threshold SNR γ_th. P_PB is the transmitted power from the power beacon, A_i (f,d) is the total pathloss of the electromagnetic wave in the Terahertz band due to molecular absorption attenuation and spreading loss, N_i (f,d) is the noise power, η is the energy conversion efficiency, α is the fraction of time that is used for harvesting energy, and ρ is the choice of power fraction used at the node for energy harvesting.\(\sigma_{y_{1}}, \sigma_{y_{2}}, \sigma_{x_{1}}, \text { and } \sigma_{x_{2}}\)  are the Rayleigh channel fading parameters. Function K_υ (z) is as explained in [34].

 

3.1 Single DF Relay

 The OP for a single DF Relay using WPT, as in Fig. 3, can be calculated as follows. Since γ_sr and γ_rd are independent Rayleigh distributed random variables, the cumulative distributive functions (CDFs) for γ_sr and γ_rd can be written as

\(F_{\gamma_{s r}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{1}}\left(\frac{\gamma_{t h}}{\alpha_{s r} x_{1}}\right) f_{x_{1}}\left(x_{1}\right) d x_{1}\)       #(25)

\(F_{\gamma_{r d}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{2}}\left(\frac{\gamma_{t h}}{\alpha_{r d} x_{2}}\right) f_{x_{2}}\left(x_{2}\right) d x_{2}\)       #(26)

where y_1=|h_sr |^2,  y_2=|h_rd |^2, x_1=|g_(〖PB〗_s ) |^2, and x_2=|g_(〖PB〗_R ) |^2. After some mathematical manipulations (25) and (26) can be written as

\(F_{\gamma_{s r}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}} d x_{1}-\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{s r} x_{1}}\right)^{2} /_{2 \sigma_{y_{1}}^{2}}}dx_1\)       #(27)

\(F_{\gamma_{r d}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}} d x_{2}-\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{r d} x_{2}}\right)^{2} /_{2 \sigma_{y_{2}}^{2}}}dx_2\)       #(28)

(27) and (28) can be calculated using [ [34], page 337, Eq. 3.326, 2^10] and [ [34], page 370, Eq. 3.478, 4], and can be written as

\(F_{\gamma_{s r}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{s r} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\)       #(29)

\(F_{\gamma_{r d}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{r d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(30)

The end-to-end CDF of  γ_d of the relay can be calculated as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\left[1-\left(1-F_{\gamma_{s r}}\left(\gamma_{t h}\right)\right)\left(1-F_{\gamma_{r d}}\left(\gamma_{t h}\right)\right)\right]\)       #(31)

After substituting (29), (30) in (31), F_(γ_d ) (γ_th ) can be written as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=1-\frac{\gamma_{t h}^{2}}{\alpha_{s r} \alpha_{r d} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(32)

 

3.1.1 OP for TSR protocol

 The OP for single DF relay using WPT protocol TSR can be written as follows by substituting the values for \(\alpha_{s r}=\frac{2 \eta \alpha P_{P B}}{A_{s r} A_{P B_{S}}(1-\alpha) N_{s r}} \text { and } \alpha_{r d}=\frac{2 \eta \alpha P_{P B}}{A_{r d} A_{P B_{R}}(1-\alpha) N_{r d}}\) in (32).

\(P_{ {Out}}(f, d)=1-\frac{\gamma_{ {th}}^{2} A_{ {sr}} A_{P B_{S}} A_{ {rd}} A_{P B_{R}}(1-\alpha)^{2} N_{ {sr}} N_{r d}}{4 \eta^{2} \alpha^{2}\left(P_{P B}\right)^{2} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}}\times K_{1}\left(\frac{\gamma_{t h} A_{s r} A_{P B_{S}}(1-\alpha) N_{s r}}{2 \eta \alpha P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h} A_{r d} A_{P B_{R}}(1-\alpha) N_{r d}}{2 \eta \alpha P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(33)

 

3.1.2 OP for PSR Protocol

 The OP for single DF relay using WPT protocol PSR can be written as follows by substituting the values for \(\alpha_{s r}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{s r} A_{P B_{S}} N_{s r}} \text { and } \alpha_{r d}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{r d} A_{P B_{R}} N_{r d}}\) in (32).

\(P_{ {Out}}(f, d)=1-\frac{\gamma_{ {th}}^{2} A_{ {sr}} A_{P B_{S}} A_{r d} A_{P B_{R}} N_{s r} N_{r d}}{\eta^{2} \rho^{2}(1-\rho)^{2}\left(P_{P B}\right)^{2} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}}\times K_{1}\left(\frac{\gamma_{t h} A_{s r} A_{P B_{S}} N_{s r}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h} A_{r d} A_{P B_{R}} N_{r d}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(34)

 

3.2 Multiple DF Relays with BRS

 The OP for multiple DF relays with BRS using WPT, as in Fig. 4, can be calculated as in [35], [36]

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\prod_{k=1}^{K} F_{\gamma_{k}}\left(\gamma_{t h}\right)\)       #(35)

where \(F_{\gamma_{k}}\left(\gamma_{t h}\right)\)can be express as

\(F_{\gamma_{k}}\left(\gamma_{t h}\right)=\left[1-\left(1-F_{\gamma_{s r_{k}}}\left(\gamma_{t h}\right)\right)\left(1-F_{r_{r_{k} d}}\left(\gamma_{t h}\right)\right)\right]\)       #(36)

Since \(\gamma_{S r_{k}}\) and \(V r_{k} d\) are independent Rayleigh distributed random variables, the CDFs of\(F_{\gamma_{s r_{k}}}\left(\gamma_{t h}\right)\)  and \(F_{\gamma_{r_{k} d}}\left(\gamma_{t h}\right)\)  can be written as

\(F_{\gamma_{s r_{k}}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{1}}\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} x_{1}}\right) f_{x_{1}}\left(x_{1}\right) d x_{1}\)       #(37)

\(F_{\gamma _{r_k} d}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{2}}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} x_{2}}\right) f_{x_{2}}\left(x_{2}\right) d x_{2}\)       #(38)

where \(y_{1}=\left|h_{s r_{k}}\right|^{2}, \quad y_{2}=\left|h_{r_{k} d}\right|^{2}, x_{1}=\left|g_{P B_{s}}\right|^{2}, \text { and } x_{2}=\left|g_{P B_{R_{k}}}\right|^{2}\). After some mathematical manipulations (37) and (38) can be written as

\(F_{\gamma_{s r_{k}}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}} d x_{1}-\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} x_{1}}\right)^{2} /_{2 \sigma_{y_{1}}^{2}}}d x_{1}\)       #(39)

\(F_{\gamma_{r_{k} d}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}} d x_{2}-\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{r_{k}} d x_{2}}\right)^{2} /_{2 \sigma_{y_{2}}^{2}}}dx_2\)       #(40)

(39) and (40) can be calculated using [ [34], page 337, Eq. 3.326, 2^10] and [ [34], page 370, Eq. 3.478, 4], and can be written as

\(F_{\gamma_{s r_{k}}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\)       #(41)

\(F_{\gamma_{r_{k} d}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(42) 

By substituting (41), (42) in (36) it is possible to write F_(γ_k ) (γ_th ) as

\(F_{\gamma_{k}}\left(\gamma_{t h}\right)=1-\frac{\gamma_{t h}^{2}}{\alpha_{s r_{k}} \alpha_{r_{k} d} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(43)

by substituting (43) in (35), it can be stated as follows

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\prod_{k=1}^{K} 1-\frac{\gamma_{t h}^{2}}{\alpha_{s r_{k}} \alpha_{r_{k} d} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(44)

For i.i.d. Rayleigh fading channels, it is possible to further simplify (44) to

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\left[1-\frac{\gamma_{t h}^{2}}{\alpha_{s r_{k}} \alpha_{r_{k} d} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{K}\)       #(45)

Applying the following mathematical identity

\((1+x)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) x^{k}\)       #(46)

to (45), it can be written as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\sum_{j=0}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h}^{2}}{\alpha_{s r_{k}} \alpha_{r_{k} d} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\right.\left.\left(\frac{\gamma_{t h}}{\alpha_{s r_{k}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{k} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{j}\)       #(47)

 

3.2.1 OP for TSR protocol

The OP for multiple DF relays with BRS using WPT protocol TSR can be written as follows by substituting the values for  \(\alpha_{s r_{k}}=\frac{2 \eta \alpha P_{P B}}{A_{s r_{k} A P B_{S}}(1-\alpha) N_{s r_{k}}} \text { and } \alpha_{r_{k} d}=\frac{2 \eta \alpha P_{P B}}{A_{r_{k} d A_{P B} R_{k}}(1-\alpha) N_{r_{k} d}}\)  in (47).

\(P_{O u t}(f, d)=1+\sum_{j=1}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h}^{2} A_{s r_{k}} A_{P B_{S}} A_{r_{k} d} A_{P B_{R_{k}}}(1-\alpha)^{2} N_{s r_{k}} N_{r_{k} d}}{4 \eta^{2} \alpha^{2}\left(P_{P B}\right)^{2} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}}\right.\left.\times K_{1}\left(\frac{\gamma_{t h} A_{s r_{k}} A_{P B_{S}}(1-\alpha) N_{s r_{k}}}{2 \eta \alpha P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right) K_{1}\left(\frac{\gamma_{t h} A_{r_{k} d} A_{P B_{R_{k}}}(1-\alpha) N_{r_{k} d}}{2 \eta \alpha P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^j\)       #(48)

 

3.2.2 OP for PSR Protocol

The OP for multiple DF relays with BRS using WPT protocol PSR can be written as follows by substituting the values for  \(\alpha_{s r_{k}}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{s r_{k}} A_{P B_{S}} N_{s r_{k}}} \text { and } \alpha_{r_{k} d}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{r_{k} d} A_{P B_{B_{H}}} N_{r_{k} d}}\)  in (47).

\(P_{O u t}(f, d)=1+\sum_{j=1}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h}^{2} A_{s r_{k}} A_{P B_{S}} A_{r_{k} d} A_{P B_{R_{k}}} N_{sr_{k}} N_{r_{k} d}}{\eta^{2} \rho^{2}(1-\rho)^{2}\left(P_{P B}\right)^{2} \sigma_{x_{1}} \sigma_{y_{1}} \sigma_{x_{2}} \sigma_{y_{2}}}\right.\times K_{1}\left(\frac{\gamma_{t h} A_{s r_{k}} A_{P B_{S}} N_{s r_{k}}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\left.K_{1}\left(\frac{\gamma_{t h} A_{r_{k}d} A_{P B_{R_{k}}} N_{r_{k} d}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{j}\)       #(49)

3.3 Multiple DF Relays with Multiple DF Hops using BRS

 The OP for multiple DF relays with multiple DF hops using WPT as in Fig. 5, can be calculated as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\prod_{k=1}^{K} F_{\gamma_{k}}\left(\gamma_{t h}\right)\)       #(50)

where \(F_{\gamma_{k}}\left(\gamma_{t h}\right)\) is the end-to-end CDF for the particular relay, which can be stated as

\(F_{\gamma_{k}}\left(\gamma_{t h}\right)=\left[1-\left(1-F_{\gamma_{s r_{1}, k}}\left(\gamma_{t h}\right)\right)\right.\left.\prod_{n=1}^{N}\left(1-F_{\gamma r_{n, k} d}\left(\gamma_{t h}\right)\right)\right]\)       #(51)

Since \(Y_{S \boldsymbol{r}_{1, k}}\) and \(\gamma_{r_{n, k} d}\) are independent Rayleigh distributed random variables, the CDFs of \(F_{S r_{1, k}}\left(\gamma_{t h}\right)\) and \(F_{r_{n, k} d}\left(\gamma_{t h}\right)\) can be written as follows

\(F_{\gamma_{s r_{1, k}}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{1}}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} x_{1}}\right) f_{x_{1}}\left(x_{1}\right) d x_{1}\)       #(52)

\(F_{\gamma_{r_{n, k} d}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} F_{y_{2}}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} x_{2}}\right) f_{x_{2}}\left(x_{2}\right) d x_{2}\)       #(53)

where \(y_{1}=\left|h_{s r_{1, k}}\right|^{2}, \quad y_{2}=\left|h_{r_{n, k} d}\right|^{2}, x_{1}=\left|g_{P B_{s}}\right|^{2}, \text { and } x_{2}=\left|g_{P B_{R_{n, k}}}\right|^{2}\), After some mathematical manipulations (52) and (53) can be written as

\(F_{\gamma_{s r_{1, k}}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}} d x_{1}-\int_{0}^{\infty} \frac{x_{1}}{\sigma_{x_{1}}^{2}} e^{-x_{1}^{2} / 2 \sigma_{x_{1}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} x_{1}}\right)^{2} /_{2 \sigma_{y_{1}}^{2}}}dx_1\)       #(54)

\(F_{\gamma_{r_{n, k} d}}\left(\gamma_{t h}\right)=\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}} d x_{2}-\int_{0}^{\infty} \frac{x_{2}}{\sigma_{x_{2}}^{2}} e^{-x_{2}^{2} / 2 \sigma_{x_{2}}^{2}}e^{-\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} x_{2}}\right)^{2} /_{2 \sigma_{y_{2}}^{2}}}d x_{2}\)       #(55)

(54) and (55) can be calculated using [ [34], page 337, Eq. 3.326, 2^10] and [ [34], page 370, Eq. 3.478, 4], and can be written as

\(F_{\gamma_{s r_{1, k}}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\)       #(56)

\(F_{\gamma_{r_{n, k} d}}\left(\gamma_{t h}\right)=\Gamma(1)-\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\)       #(57)

By substituting (56) and (57) in (51),  F_(γ_k ) (γ_th ) can be written as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\left[1-\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) \prod_{n=1}^{N} \frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]\)       #(58)

and by substituting (58) in (50),  \(F_{\gamma_{k}}\left(\gamma_{t h}\right)\) can be stated as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\prod_{k=1}^{K}\left[1-\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) \prod_{n=1}^{N} \frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]\)       #(59)

For i.i.d. Rayleigh fading channels, the above equation can be further simplified to

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\left[1-\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) \prod_{n=1}^{N} \frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{k}\)       #(60)

Applying the mathematical identity in (46) to (60),  F_(γ_d ) (γ_th )  can be written as

\(F_{\gamma_{d}}\left(\gamma_{t h}\right)=\sum_{j=0}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{s r_{1}} \sigma_{x_{1}} \sigma_{y_{1}}}\right) \prod_{n=1}^{N} \frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}} K_{1}\left(\frac{\gamma_{t h}}{\alpha_{r_{n} d} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{j}\)       #(61)

 

3.3.1 OP for TSR protocol

 The OP for multiple DF relays with multiple DF hops with BRS using WPT protocol TSR  can be written as follows by substituting the values for \(\alpha_{r_{n, k} d}=\frac{2 \eta \alpha P_{P B}}{A_{r_{n, k} d} A_{P B_{R_{n, k}}}(1-\alpha) N_{r_{n, k} d}}\) and \(\alpha_{r_{n, k} d}=\frac{2 \eta \alpha P_{P B}}{A_{r_{n, k} d A_{P B} R_{n, k}}(1-\alpha) N_{r_{n, k} d}}\)  in (61).

\(P_{ {Out }}(f, d)=1+\sum_{j=1}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h} A_{s r_{1, k}} A_{P B_{S}}(1-\alpha) N_{s r_{1, k}}}{2 \eta \alpha P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right.\times K_{1}\left(\frac{\gamma_{t h} A_{s r_{1, k}} A_{P B_{S}}(1-\alpha) N_{S r_{1, k}}}{2 \eta \alpha P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\prod_{n=1}^{N} \frac{\gamma_{t h} A_{r_{n, k} d} A_{P B_{R_{n, k}}}(1-\alpha) N_{r_{n, k} d}}{2 \eta \alpha P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\times K_{1}\left(\frac{\gamma_{t h} A_{r_{n, k} d} A_{P B_{R_{n, k}}}(1-\alpha) N_{r_{n, k} d}}{2 \eta \alpha P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)^{j}.\)       #(62)

 

3.3.2 OP for PSR Protocol

 The OP for multiple DF relays with multiple DF hops with BRS using WPT protocol PSR can be written as follows by substituting the values for \(\alpha_{S r_{1, k}}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{S r_{1, k}} A_{P B_{S}} N_{S r_{1, k}}}\) and \(\alpha_{r_{n, k} d}=\frac{\eta \rho(1-\rho) P_{P B}}{A_{r_{n, k} d} A_{P B_{R_{n, k}}} N_{r_{n, k} d}}\)  in (61).

\(P_{{Out}}(f, d)=1+\sum_{j=1}^{K}\left(\begin{array}{l} K \\ j \end{array}\right)(-1)^{j}\left[\frac{\gamma_{t h} A_{s r_{1}, k} A_{P B_{S}} N_{s r_{1}, k}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right.\times K_{1}\left(\frac{\gamma_{t h} A_{s r_{1}, k} A_{P B_{S}} N_{s r_{1}, k}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{1}} \sigma_{y_{1}}}\right)\prod_{n=1}^{N} \frac{\gamma_{t h} A_{r_{n, k} d} A_{P B_{R_{n, k}}} N_{r_{n, k} d}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\left.\times K_{1}\left(\frac{\gamma_{t h} A_{r_{n, k} d} A_{P B_{R_{n, k}}} N_{r_{n, k} d}}{\eta \rho(1-\rho) P_{P B} \sigma_{x_{2}} \sigma_{y_{2}}}\right)\right]^{j}.\)       #(63) 

 

4. Results and Discussion

 A standard medium of 1 percent water vapor molecules has been considered for all scenarios. Furthermore, for the scenarios considered, the value of Rayleigh fading parameter is σ_i=5 unless stated otherwise. The transmission windows have been chosen after taking into consideration the following papers [12], [37], [38], and [39]. According to [12], transmission windows with minimum attenuation are located at 300GHz, 350GHz, 410GHz, 670GHz, and 850GHz bands. Monte-Carlo simulations have verified all analytical results. 

Table 1. Parameters and Constants of Molecular Absorptions

 

4.1 Single DF Relay

 Fig. 6 and Fig. 7 are respectively OP vs ρ and α. From Fig. 6, it is evident that the best OP performances have been obtained at ρ=0.5, and we can observe in Fig. 7 that the OP performances have improved as the value of α has increased.

Fig. 6. OP vs ρ, 350GHz, 410GHz, 670GHz with PSR protocol, PPB = 1, η = 1.

Fig. 7. OP vs α, 350GHz, 410GHz, 670GHz with TSR protocol, PPB = 1, η = 1.

 Fig. 8 and Fig. 9 are respectively OP vs η for PSR protocol and TSR protocol, and it is possbile to infer that OP performances improve as the value of η is increased from 0 to 1.

Fig. 8. OP vs η, 350GHz, 410GHz, 670GHz with PSR protocol, PPB = 1, ρ = 0.5.

Fig. 9. OP vs η, 350GHz, 410GHz, 670GHz with TSR protocol, PPB = 1, α = 0.9.

 

Fig. 10. OP vs Distance, 350GHz, 410GHz, 670GHz with PSR protocol, PPB = 10 * 10-3,  η = 0.8, ρ = 0.5

Fig. 11. OP vs Distance, 350GHz, 410GHz, 670GHz with TSR protocol, PPB = 10 * 10-3,  η = 0.8, ρ = 0.6

 

Fig. 12. OP vs Distance, 350GHz with PSR protocol, PPB = 10 * 10-3, η = 0.8, ρ = 0.5

Fig. 13. OP vs Distance, 350GHz with TSR protocol, PPB = 10 * 10-3, η = 0.8, ρ = 0.6

 From Fig. 10 and Fig. 11, OP vs distance performances for single DF relay with WPT over 350GHz, 410GHz, and 670GHz bands have been compared respectively for PSR protocol and TSR protocol. It is possible to see that the best OP performances have been obtained in the 350GHz band for both PSR protocol and TSR protocol. Furthermore, from Fig. 12 and Fig. 13, it is evident that the OP performances for PSR and TSR protocols for a single DF relay decrease as expected when the power beacon is further away from the nano machines.

 

4.2 Multiple DF Relays with BRS

 OP for multiple DF relays using WPT over i.i.d. Rayleigh fading channels with BRS scheme have been analyzed for the scenarios of 2, 3, and 5 relays using WPT based on TSR protocol and PSR protocol, and are presented from Fig. 14 to Fig. 21. It can be inferred from the above mentioned figures that by increasing the number of relays in a network, it is possible to obtain better OP performances. From Fig. 20 and Fig. 21, it is evident that the best OP performances have been achieved for both PSR protocol and TSR protocol in the 350GHz transmission window. The best OP performances for PSR protocol have been obtained when ρ=0.5. For TSR protocol, as the number of relays is increased, the required faction of time of α is decreased. Furthermore, it is evident from the following figures that as η is increased, OP performances for PSR protocol and TSR protocol are improved.

 

Fig. 14. OP vs ρ, over 350GHz with PSR protocol for Multiple DF Relays,

PPB = 1, η = 1, d = 1.

Fig. 15. OP vs α, over 350GHz with TSR protocol for Multiple DF Relays,

 PPB = 1, η = 1, d = 1.

 

Fig. 16. OP vs η, over 350GHz with PSR protocol for Multiple DF Relays,

PPB = 1, ρ = 0.5, d = 1.

Fig. 17. OP vs η, over 350GHz with TSR protocol for Multiple DF Relays,

PPB = 1,  α = 0.6, d = 1.

 

Fig. 18. OP vs Distance, 350GHz with PSR protocol for Multiple DF Relays,

PPB = 10 * 10-3, η = 0.8, ρ = 0.5.

Fig. 19. OP vs Distance, 350GHz with TSR protocol for Multiple DF Relays,

PPB = 10 * 10-3, η = 0.8, α = 0.6.

 

Fig. 20. OP vs Distance, 350GHz, 410GHz, 670GHz,  with PSR protocol for Multiple DF Relays, Relays = 5, PPB = 10 * 10-3, η = 0.8, ρ = 0.5.

Fig. 21. OP vs Distance, 350GHz, 410GHz, 670GHz, with TSR protocol for Multiple DF Relays, Relays = 5, PPB = 10 * 10-3, η = 0.8, α = 0.6.

 

4.3 Multiple DF Relays with Multiple DF Hops using BRS

 OP for multiple DF relays with multiple DF hops per relay over i.i.d. Rayleigh fading channels with BRS scheme using WPT are investigated from Fig. 22 to Fig. 27. In Fig. 22 and Fig. 23, a single DF relay with multiple DF hops with WPT using PSR protocol and TSR protocol have been respectively investigated. It is possible to see from the graphs that by increasing the number of hops from 2 to 5, the OP performances have only been degraded minutely. In Fig. 26, multiple DF relays with multiple DF hops using PSR protocol, and in Fig. 27, multiple DF relays with multiple DF hops using TSR protocol have been investigated. For the scenarios considered in the graphs mentioned, there are 5 hops per relay. Futhermore, it can be observed from Fig. 26 and Fig. 27 that by increasing the number of relays from 1 to 5, the OP performances are increased significantly respectively for PSR protocol and TSR protocol. Moreover from Fig. 24 and Fig. 25, it is possible to see that the best OP performances have been obtained in the 350GHz transmission window.

 

Fig. 22. OP vs Distance, 350GHz using PSR protocol for Single Relay With Multiple DF Hops, PPB = 10 * 10-3, ρ = 0.5, η = 0.8.

Fig. 23. OP vs Distance, 350GHz using TSR protocol for Single Relay With Multiple DF Hops, PPB = 10 * 10-3, α = 0.6, η = 0.8.

 

Fig. 24. OP vs Distance, 350GHz, 410GHz, 670GHz using PSR protocol for Single Relay With Multiple DF Hops, Hops = 5, PPB = 10 * 10-3, ρ = 0.5, η = 0.8.

Fig. 25. OP vs Distance, 350GHz, 410GHz, 670GHz using TSR protocol for Single Relay With Multiple DF Hops, Hops = 5, PPB = 10 * 10-3, α = 0.6, η = 0.8.

 

Fig. 26. OP vs Distance, 350GHz window using PSR protocol for Multiple Relay With Multiple  Hops, PPB = 10 * 10-3, ρ = 0.5, η = 0.8.

Fig. 27. OP vs Distance, 350GHz window using TSR protocol for Multiple Relays With Multiple  Hops, PPB = 10 * 10-3, α = 0.6, η = 0.8.

 

5. Conclusion

 One of the major hurdles associated with nano communications is the limited amount of transmission power available for nano devices. In this paper, we have proposed using cooperative communication techniques with wireless power transfer protocols TSR and PSR to extend the transmission distances through relaying. Terahertz band is capable of very high capacity and very high transmission bit rates. Furthermore, the Terahertz band is rarely utilized currently. Therefore, cooperative communication among nano devices in the Terahertz band with wireless power transfer can provide significant improvements to the capabilities of nano-devices / machines.

 The results have shown that by increasing the number of relays, it is possible to get better OP performance. In addition, it has shown that by increasing the number of hops per relay from 2 to 5, the OP performances were not affected significantly. Therefore, the range between the source nano machine and the destination nano machine can be significantly boosted through cooperative communication with wireless power transfer, which will enable nano technology in diverse fields ranging from national security application to health care monitoring systems. Furthermore, it is evident from the results that the best OP performances are observed in the 350GHz transmission window.

 From the results obtained, it is possible to see that as the energy conversion efficiency increases from 0 to 1, the OP performances improve, and that the energy conversion efficiency required for same outage probability decreases as the number of relays increases. It is also possible to observe that as the transmission frequency increases, the required energy conversion efficiency increases to obtain the same outage probability.

 For wireless power transfer using power switching relaying protocol, the best outage probability performances have been obtained when ρ=0.5, and for wireless power transfer using time switching relay protocol, when the fraction of time used for energy harvesting (α) increases. Furthermore, the outage probability performances improve as the number of relays increases and the fraction of time needed for energy harvesting decreases. For time switching relaying protocol, as the fraction of time used for energy harvesting, α increases, the time available for information transmission decreases, and consequently the throughput of the network decreases. Therefore, when deciding the value of α, throughput required for the cooperative nano communication network needs to be taken into consideration.

 When the distance between the power beacon and the nano machines increases, the OP performances degrade. 

 Molecular absorption is dependent on frequency and distance. From the results obtained, it is possible to see that when the transmission frequency increases, the attenuation due to molecular absorption increases and the OP performances as well as the transmission distances decrease. Therefore, the relaying distances are dependent on the attenuation due to molecular absorption which is dependent on the frequency of operation and transmission energy. The amount of energy piezoelectric nano-generators are able to produce constantly is around 800pJ at the moment [15], [16], [17], and [18], therefore through wireless power transfer, it is possible to increase the power available for data transmission, etc. In the future, with further improvements in nanotechnology and THz band devices, the transmission distances could be further increased and better performances could be obtained. The analytical results have been verified by Monte-Carlo simulations for all the scenarios in this paper.