UAV Relay Communications

Abstract

This section investigates the secrecy performance of a UAV-aided NOMA network in the context of underlay CR. Specifically, a multi-antenna secondary source (S) transmits a mixed signal incorporated with NOMA to two multi-antenna secondary destinations ($D_{i}$) via a UAV-enabled DF relay in the presence of an eavesdropper (E). In addition, both a transmit antenna selection scheme at S and a maximal ratio combining protocol at ${D_{i}}$ and E are considered. Considering all the links undergo Nakagami-m fading with path loss, we analyze the secrecy performance in terms of SOP for both users. Moreover, we also use Monte Carlo simulations to verify those expressions.

5.1 On Secrecy Analysis of Underlay Cognitive UAV-Aided NOMA Systems with Transmit Antenna Selection/Maximal-Ratio Combining (TAS/MRC)

This section investigates the secrecy performance of a UAV-aided NOMA network in the context of underlay CR. Specifically, a multi-antenna secondary source (S) transmits a mixed signal incorporated with NOMA to two multi-antenna secondary destinations ($D_{i}$) via a UAV-enabled DF relay in the presence of an eavesdropper (E). In addition, both a transmit antenna selection scheme at S and a maximal ratio combining protocol at ${D_{i}}$ and E are considered. Considering all the links undergo Nakagami-m fading with path loss, we analyze the secrecy performance in terms of SOP for both users. Moreover, we also use Monte Carlo simulations to verify those expressions.

The main contributions are as follows:

• (1) We study the secrecy performance for an underlay CR UAV-aided NOMA system. The scenario of a TAS scheme at a secondary source, the MRC protocol at the destinations, and the parallel PIC at the eavesdropper are considered.

• (2) The PDF and CDF of all links are first characterized when all the links experience both large-scale and small-scale fading. Then, utilizing those expressions, we derive both users’ closed-form expressions of the SOP.

• (3) We also investigate the impact of several parameters, i.e., the number of antennas, power distribution coefficient, the height of the UAV, and interference power threshold, on the secrecy performance.

• (4) Finally, we analyze the effect of the hovering position of the UAV and the flight trajectory on the secrecy performance.

5.2 System and Channel Models

As shown in Fig. 5.1, an underlay CR UAV-aided NOMA system is considered in this analysis, which includes a $L_{S}$-antenna secondary source (S), a single antenna UAV-relay (R), two multi-antenna secondary destinations, i.e., a $L_{D_{1}}$-antenna far user ($D_{1}$) and a $L_{D_{2}}$-antenna near user ($D_{2}$), and a $L_{E}$-antenna eavesdropper (E). Note that the UAV is a relay in this system to link the information transmission from S to destinations.

Fig. 5.1

System model

Unlike the traditional ground relaying system, due to the advantage of mobility, it is easier and faster to deploy the UAV-enabled relaying system, especially in natural disasters. Furthermore, the LoS link is more likely to be established with the UAV-enabled relay [1, 2]. It is assumed that there is no direct link between S and both two users due to deep fading and/or obstacle blocking.

We also assume that E can only overhear the second hop, and this assumption is reasonable. In a practical scenario, it may be difficult for E to find a suitable position to wiretap both hops as the UAV is allocated in a proper deployment position to build reliable links to serve both users well. Therefore, from the eavesdropping point of view and to guarantee that it can overhear the information of both users, E may adopt a strategy to locate itself on the destination side. The symbol, $d_{Rg}={\sqrt{{K_{Rg}^2}+{H^2}}}$, is used to indicate the distance between nodes R and g, where $g \in \{S,\,D_{i},\,E\}$, and H and $K_{Rg}$ denote the vertical distance of node R to the ground and the horizontal distance to node g, respectively. It is noted that the measured results from [3, 4] showed that in the suburban/open filed environment, the performance of the UAV communication system is significantly affected by path loss.

Moreover, according to the experimental results, the UAV channels in low-altitude applications can be more appropriately modeled as Nakagami-m distributions [5]. Therefore, all links are assumed to undergo Nakagami-m fading with a path loss, and $h_{ab}$ ($a, b\in \{P,\,S,\,R,\,D_{i},\,E\}$) is used to denote the channel coefficient between nodes a and b with a fading parameter $m_{ab}$ and an average channel power gain $\varOmega _{ab}=E\{|h_{ab}|^2\}$. Therefore, the channel power gains, $|h_{ab}|^2$, follows a gamma distribution with the PDF and CDF expressed as

$$\begin{aligned} {f_{|h_{ab}|^2}(x)}= \frac{\beta _{ab}^{m_{ab}} x^{m_{ab}-1}}{\varGamma (m_{ab})} \exp (-{\beta _{ab}}{x}) \end{aligned}$$

(5.1)

and

$$\begin{aligned} {F_{|h_{ab}|^2}(x)}=1-\sum _{i=0}^{m_{ab}-1} \frac{\beta _{ab}^i x^i}{i!} \exp (-{\beta _{ab}}{x}), \end{aligned}$$

(5.2)

where $\beta _{ab}= \frac{m_{ab}}{\varOmega _{ab}}$ and $\varGamma (\cdot )$ is defined as gamma function shown by [6, Eq. (8.339.1)].

In the first time slot, S adopts the selected antenna after performing the TAS scheme to transmit a mixed signal $x_{SR}=\sum \nolimits _{i=1}^2 {{\sqrt{{\alpha _{i}}{P_{s}}}}{x_{i}}}$ to R under the underlay cognitive radio model, where $\alpha _{i}$ means the power allocation factor satisfying ${{\alpha _{1}}+{\alpha _{2}}=1}$ and $\alpha _{1}>\alpha _{2}$, and $x_{i}$ means the signal of $D_{i}$ with E${|x_{i}|^2}$ $=$ 1. The transmit power of S is denoted as $P_{s}=\frac{I_{th}}{\left| h_{SP}\right| ^2}$, where $I_{th}$ is the interference threshold at the primary destination.

Similar to [7, 8], the interference signal from the primary transmitter is modeled as AWGN to simplify the theoretical analysis. Thus, the signal received by R is denoted as

$$\begin{aligned} {y_{SR}}={{\frac{h_{SR}^{\max }}{\sqrt{d_{SR}^\theta }}}{x_{SR}}+{n_{R}}}+{g_{P}}, \end{aligned}$$

(5.3)

where $n_{R}$ is the zero mean AWGN with a variance of $N_{R}$, and $g_{P}$ is the interference signal from PT, which is with $\mathcal{C}\mathcal{N}$(0, $\eta {N_{R}}$), and $\eta$ is the scaling coefficient of $g_P$, $|h_{SR}^{\max }|=\begin{array}{c} \max \\ {{1\le {l_S}\le {L_{S}}}} \end{array}{\left| {h_{SR}^{l_{S}}}\right| }$, with ${h_{SR}^{l_{S}}}$ means the channel coefficient between $l_{S}$-th antenna of S and R, and $\theta$ is the path loss exponent.

Similar to [9, 10], perfect SIC is employed in this analysis. Therefore, the far user’s signal, $x_{1}$, is first decoded at R, and then $x_{1}$ is removed from the mixed signal before decoding $x_{2}$. Hence, it has the SNR and SINR for $D_{2}$ and $D_{1}$ at R as

$$\begin{aligned} {\gamma _{SR}^{D_{2}}}={\frac{{\alpha _{2}}{\rho _{SR}}{|h_{SR}^{\max }|^2}}{(\eta +1){d_{SR}^{\theta }}}} \end{aligned}$$

(5.4)

and

$$\begin{aligned} {\gamma _{SR}^{D_{1}}}={\frac{{\alpha _{1}}{\rho _{SR}}{|h_{SR}^{\max }|^2}}{{{{\alpha _{2}}{\rho _{SR}}{|h_{SR}^{max}|^2}}}+{(\eta +1){d_{SR}^{\theta }}}}}, \end{aligned}$$

(5.5)

respectively, where ${\rho _{SR}={\frac{P_{s}}{N_{R}}}}$.

Next, the CDF of ${|h_{SR}^{max}|^2}$ can be obtained as

$$\begin{aligned} {F_{|h_{SR}^{max}|^2}(x)}=\left[ 1-\sum _{i=0}^{m_{SR}-1} \frac{\beta _{SR}^i x^i}{i!}\exp (-{\beta _{SR}}{x})\right] ^{L_{S}}. \end{aligned}$$

(5.6)

In the second time slot, the DF protocol, which eliminates the influence of the first hop on the second hop in the dual-hop communication, is adopted at R to process the received signal. Therefore, R first decodes the received signal and then forwards the re-encoded one to $D_{i}$ in the presence of E. Hence, the received signals at $l_{D_{i}}$-th antenna of $D_{i}$ and at $l_{E}$-th antenna of E can be written as

$$\begin{aligned} {y_{R{D_{i}}}^{l_{D_{i}}}}={{\frac{h_{RD_{i}}^{l_{D_{i}}}}{\sqrt{d_{RD_{i}}^\theta }}}{x_{RD}}+{n_{D_{i}}}} \end{aligned}$$

(5.7)

and

$$\begin{aligned} {y_{RE}^{l_{E}}}={{\frac{h_{RE}^{l_{E}}}{\sqrt{d_{RE}^\theta }}}{x_{RD}}+{n_{E}}}, \end{aligned}$$

(5.8)

respectively, where $n_{D_{i}}$ and $n_{E}$ are the zero mean AWGN with variances of $N_{D_{i}}$ and $N_{E}$ at $D_{i}$ and E, and $x_{RD}=\sum \nolimits _{i=1}^2{{\sqrt{{\alpha _{i}}{P_{R}}}}{x_{i}}}$, and $P_{R}$ is the transmit power at R. Moreover, $h_{RD_{i}}^{l_{D_{i}}}$ and $h_{RE}^{l_{E}}$ denote the channel coefficients for the $l_{D_{i}}$-th antenna of the $D_{i}$ and $l_{E}$-th antenna of E, respectively.

Next, $D_{i}$ adopts the MRC scheme to process the received signals. Meanwhile, SIC is also performed. Specifically, the far user $D_{1}$ decodes its signal while treating $D_{2}$’s signal as interference. While the near user, $D_{2}$, first decodes and subtracts $D_{1}$’s signal and then processes its signal.

Therefore, it has the instantaneous SINR at $D_{1}$ and the SNR at $D_{2}$ as

$$\begin{aligned} {\gamma _{RD_{1}}}={\frac{{\rho _{RD_{1}}}{\alpha _{1}}{|h_{RD_{1}}|^2}}{{{{\alpha _{2}}{\rho _{RD_{1}}}{|h_{R{D_{1}}}|^2}}}+{{d_{R{D_{1}}}^{\theta }}}}} \end{aligned}$$

(5.9)

and

$$\begin{aligned} {\gamma _{RD_{2}}}={\frac{\rho _{RD_{2}}{\alpha _{2}}{|h_{RD_{2}}|^2}}{d_{RD_{2}}^{\theta }}}, \end{aligned}$$

(5.10)

respectively, where ${\rho _{RD_{i}}=\frac{P_{R}}{N_{D_{i}}}}$ and $|h_{RD_{i}}|^2=\sum \nolimits _{l_{D_{i}}=1}^{L_{D_{i}}}{|h_{RD_{i}}^{l_{D_{i}}}|^2}$.

Making use of Eqs. (5.2), (5.9), and (5.10), the CDFs of $\gamma _{RD_{1}}$ and $\gamma _{RD_{2}}$ can be obtained as

$$\begin{aligned} {F_{\gamma _{R\!D_{1}}}(x)}&\!=\! {\left\{ \begin{array}{ll} 1\!-\!{{\sum \limits _{i=0}^{{m_{R\!D_{1}}{L_{D_{1}}}}-1}}{\frac{{({d_{R\!D_{1}}^\theta }{\beta _{R\!D_{1}}})^i}{x^i}}{{\left[ {\rho _{R\!D_{1}}}({{\alpha _{1}}-{{\alpha _{2}}x}})\right] ^i}i!}}}{\exp \left( -{\frac{\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta }x}{{\rho _{R\!D_{1}}}({\alpha _{1}}-{\alpha _{2}x})}}\right) },&{}{\text{ if } \; x\!<\!{\frac{\alpha _{1}}{\alpha _{2}}}};\\ {1},&{}\mathrm{{else}} \end{array}\right. } \end{aligned}$$

(5.11)

and

$$\begin{aligned} &{F_{\gamma _{RD_{2}}}(x)}=1-\!{\sum _{j=0}^{{m_{RD_{2}}{L_{D_{2}}}}-1}{\frac{{({\beta _{RD_{2}}}{d_{RD_{2}}^\theta })^j}{x^j}}{{({\alpha _{2}\rho _{RD_{2}}})^j}j!}}}{{\exp \left( -{\frac{{\beta _{RD_{2}}}{d_{RD_{2}}^\theta }x}{\alpha _{2}\rho _{RD_{2}}}}\right) }}, \end{aligned}$$

(5.12)

respectively.

In addition, we conduct this analysis under the assumption that E has the largest decoding ability. Specifically, the MRC technique is applied at E to process the received signal. The PIC is also employed to decode the superimposed signals, i.e., the interference from $x_{1}$ ($x_{2}$) can be eliminated when decoding $x_{2}$ ($x_{1}$). Hence, the SNRs for the signals of $D_1$ and $D_2$ at E can be expressed as

$$\begin{aligned} {\gamma _{RE}^{D_{1}}}={\frac{\alpha _{1}{\rho _{RE}}{|h_{RE}|^2}}{d_{RE}^{\theta }}}=\alpha _{1}{\gamma _{RE}} \end{aligned}$$

(5.13)

and

$$\begin{aligned} {\gamma _{RE}^{D_{2}}}={\frac{\alpha _{2}{\rho _{RE}}{|h_{RE}|^2}}{d_{RE}^{\theta }}}=\alpha _{2}{\gamma _{RE}}, \end{aligned}$$

(5.14)

where $\rho _{RE}=\frac{P_{R}}{N_{E}}$, $|h_{RE}|^2=\sum \nolimits _{l_{E}=1}^{L_{E}}{|h_{RE}^{l_{E}}|^2}$, and ${\gamma _{RE}}={\frac{{\rho _{RE}}{|h_{RE}|^2}}{d_{RE}^{\theta }}}$ with the PDF of $\gamma _{RE}$ denoted as

$$\begin{aligned} f_{{\gamma }_{RE}}(x)={\frac{{({{\beta _{RE}}}{{d_{RE}^\theta }})^{m_{RE}L_{E}}}{x^{{m_{RE}L_{E}}-1}}}{{{\rho _{RE}}^{m_{RE}L_{E}}}{\varGamma ({m_{RE}L_{E}})}}}\exp {\left( -{\frac{{\beta _{RE}}{{d_{RE}^\theta }}x}{\rho _{RE}}}\right) }. \end{aligned}$$

(5.15)

5.3 Secrecy Performance Analysis

In this section, the derivation of SOPs for two users is presented.

5.3.1 SOP Analysis

5.3.1.1 SOP of $D_{1}$

The instantaneous SC of $D_{1}$ for the two links can be expressed as

$$\begin{aligned} {C_{SR}^{D_{1}}}\,=\,{\frac{1}{2}}{{\log _{2}\left( {1+{\gamma _{SR}^{D_{1}}}}\right) }} \end{aligned}$$

(5.16)

and

$$\begin{aligned} {C_{RD_{1}}}={\left\{ {{\frac{1}{2}}{{\log _{2}\frac{{1+{\gamma _{RD_{1}}}}}{{1+{\gamma _{RE}^{D_{1}}}}}}}}\right\} ^+}, \end{aligned}$$

(5.17)

where${\left\{ x\right\} ^+}={\max \left\{ x,0\right\} }$.

The worst hop in the DF relaying system dominates the system capacity. Hence, the SC of $D_{1}$ as

$$\begin{aligned} C_{D_{1}}={\min \left\{ {C_{SR}^{D_1}},C_{RD_{1}}\right\} }. \end{aligned}$$

(5.18)

A secrecy outage occurs when the instantaneous SC exceeds the user’s threshold requirement, $C_{th_{1}}$. Therefore, it has the SOP of $D_{1}$ as

$$\begin{aligned} {P_{SOP}^{D_{1}}}&=\text {Pr}\left\{ C_{D_{1}}<{C_{th_{1}}}\right\} \nonumber \\ {\qquad }&={\text {Pr}\left\{ \min \left\{ {C_{SR}^{D_{1}}},{C_{RD_{1}}}\right\} <{C_{th_{1}}}\right\} } \nonumber \\ {\qquad }&={1-{\text {Pr}\left\{ {C_{SR}^{D_{1}}}\ge {C_{th_{1}}},{C_{RD_{1}}\ge {C_{th_{1}}}}\right\} }} \nonumber \\ {\qquad }&={1-\text {Pr}\left\{ {C_{SR}^{D_{1}}}\ge {C_{th_{1}}}\right\} \text {Pr}\left\{ {C_{RD_{1}}}\ge {C_{th_{1}}}\right\} } \nonumber \\ {\qquad }&=1-{\text {Pr}\left\{ {\gamma _{SR}^{D_{1}}}\ge {{C_{T_{1}}}-1}\right\} }{\text {Pr}\left\{ {\gamma _{RD_{1}}}\ge {{C_{T_{1}}}(1+{\gamma _{RE}^{D_{1}}})-1}\right\} }, \end{aligned}$$

(5.19)

where $C_{T_{1}}={2^{2{C_{th_{1}}}}}$.

Based on Eqs. (5.1), (5.5), and (5.6), we have

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{SR}^{D_{1}}}\ge {{C_{T_{1}}}-1}\right\} }&={\text {Pr}\left\{ {|h_{SR}^{\max }|^2}\ge {\gamma _{th_{1}}|h_{SP}|^2}\right\} }\nonumber \\ &{\quad }={\int _{0}^{\infty }\!\left[ 1-{F_{|h_{SR}^{\max }|^2}}{(\gamma _{th_{1}}x)}\right] {f_{|h_{SP}|^2}(x)}dx} \nonumber \\ &{\quad }=1-{\int _{0}^{\infty }{\frac{{\beta _{SP}^{m_{SP}}}{x^{m_{SP}-1}}}{\varGamma (m_{SP})}\exp (-\beta _{SP}x)}}\nonumber \\ &{\qquad }\times {{\underbrace{\left[ {1-{\exp (-\beta _{SR}\gamma _{th_{1}}x)}{\sum _{i=0}^{m_{SR}-1}}}{\frac{(\beta _{SR}\gamma _{th_{1}}x)^i}{i!}}\right] ^{L_{S}}}_{J}}dx}, \end{aligned}$$

(5.20)

where $\gamma _{th_{1}}={\frac{(C_{T_{1}}-1)(\eta +1)d_{SR}^{\theta }}{\rho _{I}(\alpha _{1}-{\alpha _{2}(C_{T_{1}}-1)})}}$ and $\rho _{I}={\frac{I_{th}}{N_{R}}}$.

Applying the binomial theorem to the term, J, in Eq. (5.20), it has

$$\begin{aligned} &\!J\!={\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\exp (-k\beta _{SR}{\gamma _{th_{1}}}x)}{{\left[ {{\sum _{i=0}^{m_{SR}-1}}{\frac{(\beta _{SR}\gamma _{th_{1}}x)^i}{i!}}}\right] ^k}}\nonumber \\ &={\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\exp (-k\beta _{SR}{\gamma _{th_{1}}}x)}{\prod _{i=0}^{m_{SR}-2}{\big (\frac{(\beta _{SR}\gamma _{th_{1}})^i}{i!}\big )^{k_{i+1}}}}\nonumber \\ &{\quad }\times {\sum _{k_{1}=0}^{k}}{\sum _{k_{2}=0}^{k-k_{1}}}{\sum _{k_{m_{SR}-1}=0}^{k-{k_{1}}-\cdots -{k_{m_{SR}-2}}}}{{\genfrac(){0.0pt}0{k}{k_{1}}}{\genfrac(){0.0pt}0{k-k_{1}}{k_{2}}}\ldots {\genfrac(){0.0pt}0{k-k_{1}-\cdots -{k_{m_{SR}-2}}}{k_{m_{SR}}-1}}}\nonumber \\ &{\quad }\times {{\left( \frac{(\beta _{SR}\gamma _{th_{1}})^{m_{SR}-1}}{(m_{SR}-1)!}\right) }^{k-k_{1}-\cdots -k_{m_{SR}-1}}}{x^\varTheta }, \end{aligned}$$

(5.21)

where $\varTheta ={(m_{SR}-1)(k-k_{1})}$ ${-(m_{SR}-2)}{k_{2}}{-\cdots -{2k_{m_{SR}-2}}-{k_{m_{SR}-1}}}$.

Defining

$$\begin{aligned} \varPhi =&{{\sum \limits _{k_{1}=0}^{k}}{\sum \limits _{k_{2}=0}^{k-k_{1}}}\ldots {\sum \limits _{k_{m_{SR}-1}=0}^{k-{k_{1}}-\cdots -{k_{m_{SR}-2}}}}}{{\genfrac(){0.0pt}0{k}{k_{1}}}}{\genfrac(){0.0pt}0{k-k_{1}}{k_{2}}}\ldots {\genfrac(){0.0pt}0{k-k_{1}-\cdots -{k_{m_{SR}-2}}}{k_{m_{SR}}-1}}, \end{aligned}$$

(5.22)

$$\begin{aligned} \varPsi _{1} =&{{\left( \frac{(\beta _{SR}\gamma _{th_{1}})^{m_{SR}-1}}{(m_{SR}-1)!}\right) }^{k-k_{1}-\cdots -k_{m_{SR}-1}}}{\prod \limits _{i=0}^{m_{SR}-2}{\left( {\frac{(\beta _{SR}\gamma _{th_{1}})^i}{i!}}\right) ^{k_{i+1}}}}, \end{aligned}$$

(5.23)

and substituting Eq. (5.21) into Eq. (5.20), we have

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{SR}^{D_{1}}}\ge {{C_{T_{1}}}-1}\right\} }&=1-{\int _{0}^{\infty }}{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\exp (-k\beta _{SR}{\gamma _{th_{1}}}x)}{\varPhi \varPsi _{1}}{x^{\varTheta }} \nonumber \\ &~~~\times {\frac{{\beta _{S\!P}^{m_{S\!P}}}{x^{m_{S\!P}-1}}}{\varGamma (m_{S\!P})}\exp (-\beta _{S\!P}x)}dx \nonumber \\ &=1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{1}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}} \nonumber \\ &~~~\times {\int _{0}^{\infty }}{x^{\varTheta +m_{S\!P}-1}}{\exp \left[ {-({k\beta _{SR}{\gamma _{th_{1}}}+\beta _{S\!P}})x}\right] dx}\nonumber \\ &=1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{1}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}}\ (\varTheta +m_{S\!P}-1)!\nonumber \\ &~~~\times {({k\beta _{SR}{\gamma _{th_{1}}}+\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}.\nonumber \\ \end{aligned}$$

(5.24)

Based on Eqs. (5.11), and (5.15), it has

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{R\!D_{1}}}\ge {{C_{T_{1}}}(1+{\gamma _{R\!E}^{D_{1}}})-1}\right\} }=&1\!-\!\bigg [{\int _{0}^{t}\!{F_{\gamma _{R\!D_{1}}}}{(C_{T_{1}}{(1+{\alpha _{1}x})}\!-\!1)}{f_{{\gamma _{RE}}}(x)}{dx}}\nonumber \\ &+{{{\int _{t}^{\infty }}}\!{f_{{\gamma _{R\!E}}}(x)}dx}\bigg ]\nonumber \\ &={\int _{0}^{t}{\left[ {1\!-\!{F_{\gamma _{R\!D_{1}}}{({C_{T_{1}}({1+\alpha _{1}x})\!-\!1})}}}\right] {f_{\gamma _{R\!E}}(x)}}\,dx}\nonumber \\ &=\int \limits _{0}^{t}{\exp \left[ -{\frac{{\beta _{R\!D_{1}}}{d_{R\!D_{1}}^\theta }{({C_{T_{1}}}-1+{C_{T_{1}}{\alpha _{1}}x})}}{\rho _{R\!D_{1}}({\alpha _{1}}-{{\alpha _{2}}({C_{T_{1}}}-1+{C_{T_{1}}{\alpha _{1}}x})})}}\right] }\nonumber \\ &\cdot \!\sum _{j=0}^{m_{R\!D_{1}}{{L\!_{D_{1}}}}\!-\!1}\!{\frac{{\beta _{R\!D_{1}}^j}{({d_{R\!D_{1}}^\theta })^j}{({C_{T_{1}}-1+{C_{T_{1}}\alpha _{1}}x})^j}}{{j!}{{({\rho _{R\!D_{1}} ({{\alpha _{1}}-{{\alpha _{2}}{(\!{C_{T_{1}}\!-\!1\!+\!{{C_{T_{1}}}{\alpha _{1}}x}})}}})})}^j}}}\nonumber \\ &\cdot {\frac{{({\beta _{R\!E}}{d_{R\!E}^\theta })^{m_{R\!E}L_{E}}}{x^{m_{R\!E}L_{E}-1}}}{{\rho _{R\!E}^{m_{R\!E}L_{E}}}{\Gamma ({m_{R\!E}}L_{E})}}}{\exp \left( -{\frac{{\beta _{R\!E}}{d_{R\!E}^\theta }x}{\rho _{R\!E}}}\right) }\,dx, \end{aligned}$$

(5.25)

where $t\!=\!{\frac{{\frac{\alpha _{1}}{\alpha _{2}}}+1-{C_{T_{1}}}}{\alpha _{1}{C_{T_{1}}}}}$.

Since the closed-form expression for Eq. (5.25) is difficult to obtain, the Gaussian Chebyshev quadrature [6] is applied to obtain an approximated expression as

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{R\!D_{1}}}\ge {{C_{T_{1}}}(1+{\gamma _{R\!E}^{D_{1}}})-1}\right\} }&={\frac{{({\beta _{R\!E}}{d_{R\!E}^\theta })^{\mu _{R\!E}}}}{{\rho _{R\!E}^{\mu _{R\!E}}}{\varGamma ({\mu _{R\!E}})}}}{{\sum _{j=0}^{\mu _{R\!D_{1}}-1}} {\frac{{(\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta })}^j}{j!}}}{\frac{t}{2}}{\frac{\pi }{N}}\nonumber \\ &{\quad }\cdot {\sum _{n=1}^{N}}{\frac{({{C_{T_{1}}-1}+{C_{T_{1}}{\alpha _{1}}{\tau }}})^j}{\left[ {\rho _{R\!D_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}\right] ^j}}\nonumber \\ &{\quad }\cdot {\exp \left[ {\frac{(1-C_{T_{1}}-C_{T_{1}}\alpha _{1}\tau ){\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta }}}{{\rho _{RD_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}}}\right] }\nonumber \\ &{\quad }\cdot \exp {\left( {-{\frac{\beta _{R\!E}{d_{R\!E}^\theta }}{\rho _{R\!E}}}}\tau \right) }{\tau ^{\mu _{RE}-1}}{\sqrt{1-{{\phi }^2}}}, \end{aligned}$$

(5.26)

where N is a factor of the Gaussian-Chebyshev term, and ${\phi }\!=\!{\cos ({{\frac{2{n}-1}{2{N}}}{\pi }})}$, ${\tau }\!=\!{{\frac{t}{2}}{(\phi +1)}}$, $\mu _{RD_{i}}=m_{RD_{i}}{L_{D_{i}}}$, $\mu _{RE}=m_{RE}L_{E}$.

Applying Eqs. (5.21), and (5.25), we express the analytical expression of SOP for $D_{1}$ as

$$\begin{aligned} P_{SOP}^{D_{1}}& ={1-}\left[ 1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{1}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}}(\varTheta \!+\!m_{S\!P}\!-\!1)!\right. \nonumber \\ &{\quad } \times {({k\beta _{SR}{\gamma _{th_{1}}}\!+\!\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}\Bigg ]{\frac{{({\beta _{R\!E}}{d_{R\!E}^\theta })^{\mu _{R\!E}}}}{{\rho _{R\!E}^{\mu _{R\!E}}}{\varGamma ({\mu _{R\!E}})}}}{{\sum _{j=0}^{\mu _{R\!D_{1}}-1}} {\frac{{(\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta })}^j}{j!}}}\nonumber \\ &{\quad }\times {\frac{t}{2}}{\frac{\pi }{N}}{\sum _{n=1}^{N}}{\frac{({{C_{T_{1}}-1}+{C_{T_{1}}{\alpha _{1}}{\tau }}})^j}{\left[ {\rho _{R\!D_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}\right] ^j}}\nonumber \\ &{\quad }\times {\exp \left[ {\frac{(1-C_{T_{1}}-C_{T_{1}}\alpha _{1}\tau ){\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta }}}{{\rho _{RD_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}}}\right] }\nonumber \\ &{\quad }\times \exp {\left( {-{\frac{\beta _{R\!E}{d_{R\!E}^\theta }}{\rho _{R\!E}}}}\tau \right) }{\tau ^{\mu _{RE}-1}}{\sqrt{1-{{\phi }^2}}}. \end{aligned}$$

(5.27)

5.3.1.2 SOP of $D_{2}$

The instantaneous SC for $D_{2}$ is

$$\begin{aligned} C_{D_{2}}={\min \left\{ {C_{SR}^{D_2}},C_{RD_{2}}\right\} }, \end{aligned}$$

(5.28)

where

$$\begin{aligned} {C_{SR}^{D_{2}}}\,=\,{\frac{1}{2}}{{\log _{2}\left( {1+{\gamma _{SR}^{D_{2}}}}\right) }} \end{aligned}$$

(5.29)

and

$$\begin{aligned} {C_{RD_{2}}}={\left\{ {{\frac{1}{2}}{{\log _{2}\frac{\left( {1+{\gamma _{RD_{2}}}}\right) }{\left( {1+{\gamma _{RE}^{D_{2}}}}\right) }}}}\right\} ^+}. \end{aligned}$$

(5.30)

Next, the SOP of $D_{2}$ is expressed as

$$\begin{aligned} {P_{SOP}^{D_{2}}}&=\text {Pr}\left\{ C_{D_{2}}<{C_{th_{2}}}\right\} \nonumber \\ &={\text {Pr}\left\{ \min ({C_{SR}^{D_{2}}},{C_{RD_{2}}})<{C_{th_{2}}}\right\} } \nonumber \\ &={1-{\text {Pr}\left\{ {C_{SR}^{D_{2}}}\ge {C_{th_{2}}},{C_{RD_{2}}\ge {C_{th_{2}}}}\right\} }} \nonumber \\ &={1-\text {Pr}\left\{ {C_{SR}^{D_{2}}}\ge {C_{th_{2}}}\right\} \text {Pr}\left\{ {C_{RD_{2}}}\ge {C_{th_{2}}}\right\} } \nonumber \\ &=1-{\text {Pr}\left\{ {\gamma _{SR}^{D_{2}}}\ge {{C_{T_{2}}}-1}\right\} }{\text {Pr}\left\{ {\gamma _{RD_{2}}}\ge {{C_{T_{2}}}(1+{\gamma _{RE}^{D_{2}}})-1}\right\} }, \nonumber \\ \end{aligned}$$

(5.31)

where $C_{T_{2}}={2^{2{C_{th_{2}}}}}$ and $C_{th_2}$ is the capacity threshold of $D_{2}$.

From Eqs. (5.1), (5.4), and (5.6), we have

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{SR}^{D_{2}}}\ge {{C_{T_{2}}}-1}\right\} } &={\text {Pr}\left\{ {|h_{SR}^{max}|^2}\ge {\gamma _{th_{2}}|h_{SP}|^2}\right\} }\nonumber \\ &={\int _{0}^{\infty }\!\left[ 1-{F_{|h_{SR}^{max}|^2}}{(\gamma _{th_{2}}x)}\right] {f_{|h_{SP}|^2}(x)}dx}\nonumber \\ &=1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{2}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}}(\varTheta +m_{S\!P}-1)!\nonumber \\ &~~~\times {({k\beta _{SR}{\gamma _{th_{2}}}+\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}, \end{aligned}$$

(5.32)

where $\varPsi _{2}={\prod \limits _{i=0}^{m_{SR}-2}{\big (\frac{(\beta _{SR}\gamma _{th_{2}})^i}{i!}\big )^{k_{i+1}}}}{{\big (\frac{(\beta _{SR}\gamma _{th_{2}})^{m_{SR}-1}}{(m_{SR}-1)!}\big )}^{k-k_{1}-\cdots -k_{m_{SR}-1}}}$ and $\gamma _{th_{2}}={\frac{(C_{T_{2}}-1)(\eta +1)d_{SR}^{\theta }}{\rho _{I}\alpha _{2}}}$.

After that, from Eqs. (5.12) and (5.15), and (3.326.2) in [6], it has

$$\begin{aligned} {\text {Pr}\left\{ {\gamma _{RD_{2}}}\ge {{C_{T_{2}}}(1+{\gamma _{RE}^{D_{2}}})-1}\right\} }&={{\int _{0}^{\infty }}{\left[ {1\!-\!{F_{\gamma _{R\!D_{2}}}{({C_{T_{2}}({1+\alpha _{2}x})-1})}}}\right] {f_{\gamma _{R\!E}}(x)}}\,dx} \nonumber \\ &={\int _{0}^{\infty }}{\sum _{j=0}^{\mu _{RD_{2}}-1}}{\frac{{{(\beta _{RD_{2}}{d_{RD_{2}}^\theta })^j}{(C_{T_{2}}-1+{C_{T_{2}}\alpha _{2}x})^j}}}{{(\alpha _{2}\rho _{RD_{2}})^j}{j!}}} \nonumber \\ &~~\cdot {\exp \left( -{\frac{{\beta _{RD_{2}}{d_{RD_{2}}^\theta }}{(C_{T_{2}}-1+{C_{T_{2}}\alpha _{2}x})}}{\alpha _{2}\rho _{RD_{2}}}}\right) } \nonumber \\ &~~\cdot {\frac{{\beta _{RE}^{\mu _{RE}}}{({d_{RE}^\theta })^{\mu _{RE}}}{x^{{\mu _{RE}-1}}}}{{\rho _{RE}^{\mu _{RE}}}{\varGamma ({\mu _{RE}})}}} {\exp \left( -{\frac{\beta _{RE}{d_{RE}^\theta }x}{\rho _{RE}}}\right) }\,dx \nonumber \\ &={\frac{{({\beta _{RE}}{d_{RE}^\theta })^{\mu _{RE}}}}{{\rho _{RE}^{\mu _{RE}}}{\varGamma ({\mu _{RE}})}}} {\exp \left( -\frac{{\beta _{RD_{2}}}{d_{RD_{2}}^\theta }{(C_{T_{2}}-1)}}{{\alpha _{2}\rho _{RD_{2}}}}\right) } \nonumber \\ &~~~\cdot {\sum _{j=0}^{\mu _{RD_{2}}-1}}{\frac{(\beta _{RD_{2}}{d_{RD_{2}}^\theta })^j}{(\alpha _{2}\rho _{RD_{2}})^j{j!}}} {\sum _{q=0}^{j}}{{\genfrac(){0.0pt}0{j}{q}}{(C_{T_{2}}-1)^{j-q}}} \nonumber \\ &~~~\times {{(C_{T_{2}}\alpha _{2})^{q}}}{\frac{\varGamma (\mu _{RE}+q)}{\left( \beta _{RD_{2}}d_{RD_{2}}^\theta \frac{C_{T_{2}}}{\rho _{RD_{2}}}\!+\!\frac{\beta _{RE}d_{RE}^\theta }{\rho _{RE}}\right) ^{\mu _{RE}+q}}}. \end{aligned}$$

(5.33)

Finally, from Eqs. (5.31) and (5.32), the SOP of $D_{2}$ can be derived as

$$\begin{aligned} {P_{SOP}^{D_{2}}}&=1-\left( {1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{2}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}}}(\varTheta +m_{S\!P}-1)!\right. \nonumber \\ &~~~\times {{({k\beta _{SR}{\gamma _{th_{2}}}+\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}}\Bigg ){\frac{{({\beta _{RE}}{d_{RE}^\theta })^{\mu _{RE}}}}{{\rho _{RE}^{\mu _{RE}}}{\varGamma ({\mu _{RE}})}}}\nonumber \\ &~~~\times {\exp \left( -\frac{{\beta _{RD_{2}}}{d_{RD_{2}}^\theta }{(C_{T_{2}}-1)}}{{\alpha _{2}\rho _{RD_{2}}}}\right) }{\sum _{j=0}^{\mu _{RD_{2}}-1}}{\frac{(\beta _{RD_{2}}{d_{RD_{2}}^\theta })^j}{(\alpha _{2}\rho _{RD_{2}})^j{j!}}} {\sum _{q=0}^{j}}{\genfrac(){0.0pt}0{j}{q}}\nonumber \\ &~~~\times {(C_{T_{2}}-1)^{j-q}}{{(C_{T_{2}}\alpha _{2})^{q}}}{\frac{\varGamma (\mu _{RE}+q)}{\left( \beta _{RD_{2}}d_{RD_{2}}^\theta \frac{C_{T_{2}}}{\rho _{RD_{2}}}\!+\!\frac{\beta _{RE}d_{RE}^\theta }{\rho _{RE}}\right) ^{\mu _{RE}+q}}}. \end{aligned}$$

(5.34)

5.3.2 Asymptotic Analysis

To enhance the study, a theoretical insight in terms of asymptotic analysis for both users is provided in this section to show the significance of the analytical results in the high SNR regimes, revealing the system’s performance limits.

For $D_1$, let ${\rho _{I}}\rightarrow {\infty }$ (i.e.,${\gamma _{th_{1}}}\rightarrow {0}$), while $\rho _{RD_{1}}$ keeps fixed and finite, it has ${\lim \limits _{{\gamma _{th_1}}\rightarrow 0}}{\exp (-k{\beta _{SR}}{\gamma _{th_{1}}}x)}=1$. Therefore, the asymptotic expression can be obtained as

$$\begin{aligned} P_{SOP_1}^{{\rho _{I}\rightarrow {\infty }}}&={1-}\big (1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{{\beta _{S\!P}^{m_{S\!P}}}}{({k\beta _{SR}{\gamma _{th_{1}}}\!+\!\beta _{S\!P}})^{-m_{S\!P}}}\big )\nonumber \\ &~~~ \times {\frac{{({\beta _{R\!E}}{d_{R\!E}^\theta })^{\mu _{R\!E}}}}{{\rho _{R\!E}^{\mu _{R\!E}}}{\varGamma ({\mu _{R\!E}})}}}{{\sum _{j=0}^{\mu _{R\!D_{1}}-1}} {\frac{{(\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta })}^j}{j!}}}{\frac{t}{2}}{\frac{\pi }{N}}\nonumber \\ &~~~\times {\sum _{n=1}^{N}}{\frac{({{C_{T_{1}}-1}+{C_{T_{1}}{\alpha _{1}}{\tau }}})^j}{\left[ {\rho _{R\!D_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}\right] ^j}}\nonumber \\ &~~~\times {\exp \left[ {\frac{(1-C_{T_{1}}-C_{T_{1}}\alpha _{1}\tau ){\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta }}}{{\rho _{RD_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}}}\right] }\nonumber \\ &~~~\times \exp {({-{\frac{\beta _{R\!E}{d_{R\!E}^\theta }}{\rho _{R\!E}}}}\tau )}{\tau ^{\mu _{RE}-1}}{\sqrt{1-{{\phi }^2}}}. \end{aligned}$$

(5.35)

Next, let ${\rho _{RD_{1}}}\rightarrow \infty$ while $\rho _{I}$ keeps fixed and finite. Making use of the first two terms of the Taylor series expansion, the asymptotic expression of $D_1$ can be obtained as

$$\begin{aligned} P_{SOP_1}^{{\rho _{RD_{1}}\rightarrow \infty }}\,=\,&{1-}\big (1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{1}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}} \nonumber \\ & \times (\varTheta \!+\!m_{S\!P}\!-\!1)!{({k\beta _{SR}{\gamma _{th_{1}}}\!+\!\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}\big )\nonumber \\ & \times {\frac{{({\beta _{R\!E}}{d_{R\!E}^\theta })^{\mu _{R\!E}}}}{{\rho _{R\!E}^{\mu _{R\!E}}}{\varGamma ({\mu _{R\!E}})}}}{{\sum _{j=0}^{\mu _{R\!D_{1}}-1}} {\frac{{(\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta })}^j}{j!}}}{\frac{t}{2}}{\frac{\pi }{N}}\nonumber \\ &\times {\sum _{n=1}^{N}}{\frac{({{C_{T_{1}}-1}+{C_{T_{1}}{\alpha _{1}}{\tau }}})^j}{\left[ {\rho _{R\!D_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}\right] ^j}}\nonumber \\ &\times {\left[ 1+{\frac{(1-C_{T_{1}}-C_{T_{1}}\alpha _{1}\tau ){\beta _{R\!D_{1}}{d_{R\!D_{1}}^\theta }}}{{\rho _{RD_{1}}(\alpha _{1}-{\alpha _{2}} ({C_{T_{1}}}+{C_{T_{1}}\alpha _{1}\tau }-1))}}}\right] }\nonumber \\ &\times \exp {({-{\frac{\beta _{R\!E}{d_{R\!E}^\theta }}{\rho _{R\!E}}}}\tau )}{\tau ^{\mu _{RE}-1}}{\sqrt{1-{{\phi }^2}}}. \end{aligned}$$

(5.36)

Applying the approaches, we can respectively obtain the asymptotic expressions when $\rho _{I}\rightarrow \infty$ and $\rho _{RD_2}\rightarrow \infty$ for $D_2$ as

$$\begin{aligned} P_{SOP_2}^{{\rho _{I}\rightarrow {\infty }}}=&\,{1-}\big (1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{{\beta _{S\!P}^{m_{S\!P}}}}\nonumber \\ & \times {({k\beta _{SR}{\gamma _{th_{2}}}\!+\!\beta _{S\!P}})^{-m_{S\!P}}}\big )\nonumber \\ &\times {\frac{{({\beta _{RE}}{d_{RE}^\theta })^{\mu _{RE}}}}{{\rho _{RE}^{\mu _{RE}}}{\varGamma ({\mu _{RE}})}}} {\exp \left( -\frac{{\beta _{RD_{2}}}{d_{RD_{2}}^\theta }{(C_{T_{2}}-1)}}{{\alpha _{2}\rho _{RD_{2}}}}\right) }\nonumber \\ &\times {\sum _{j=0}^{\mu _{RD_{2}}-1}}{\frac{(\beta _{RD_{2}}{d_{RD_{2}}^\theta })^j}{(\alpha _{2}\rho _{RD_{2}})^j{j!}}} {\sum _{q=0}^{j}}{{\genfrac(){0.0pt}0{j}{q}}{(C_{T_{2}}-1)^{j-q}}}\nonumber \\ &\times {{(C_{T_{2}}\alpha _{2})^{q}}}{\frac{\varGamma (\mu _{RE}+q)}{\left( \beta _{RD_{2}}d_{RD_{2}}^\theta \frac{C_{T_{2}}}{\rho _{RD_{2}}}\!+\!\frac{\beta _{RE}d_{RE}^\theta }{\rho _{RE}}\right) ^{\mu _{RE}+q}}} \end{aligned}$$

(5.37)

and

$$\begin{aligned} {P_{SOP_2}^{\rho _{RD_2}\rightarrow \infty }}= &\, 1\!-\big ({1-{\sum _{k=0}^{L_{S}}}{\genfrac(){0.0pt}0{L_{S}}{k}}{(-1)^k}{\varPhi \varPsi _{2}}{\frac{{\beta _{S\!P}^{m_{S\!P}}}}{{\varGamma (m_{S\!P})}}}}\nonumber \\ &\times {(\varTheta +m_{S\!P}-1)!{({k\beta _{SR}{\gamma _{th_{2}}}+\beta _{S\!P}})^{-\varTheta -m_{S\!P}}}}\big )\nonumber \\ &\times {\frac{{({\beta _{RE}}{d_{RE}^\theta })^{\mu _{RE}}}}{{\rho _{RE}^{\mu _{RE}}}{\varGamma ({\mu _{RE}})}}} {\left( 1-\frac{{\beta _{RD_{2}}}{d_{RD_{2}}^\theta }{(C_{T_{2}}-1)}}{{\alpha _{2}\rho _{RD_{2}}}}\right) }\nonumber \\ &\times {\sum _{j=0}^{\mu _{RD_{2}}-1}}{\frac{(\beta _{RD_{2}}{d_{RD_{2}}^\theta })^j}{(\alpha _{2}\rho _{RD_{2}})^j{j!}}} {\sum _{q=0}^{j}}{{\genfrac(){0.0pt}0{j}{q}}{(C_{T_{2}}-1)^{j-q}}}\nonumber \\ &\times {{(C_{T_{2}}\alpha _{2})^{q}}}{\frac{\varGamma (\mu _{RE}+q)}{\left( \frac{\beta _{RE}d_{RE}^\theta }{\rho _{RE}}\right) ^{\mu _{RE}+q}}}. \end{aligned}$$

(5.38)

5.4 Performance Evaluation

5.4.1 Numerical Results and Discussions

In this section, we discuss the numerical and simulation results. The parameters are set as $\rho _{I}=15$ dB, $\rho _{R\!E}=10$ dB, $\alpha _{1}=0.85$, $\alpha _{2}=0.15$, $\theta =2$, $K_{S\!R}= 80$ m, $K_{R\!D_{1}}=80$ m, $K_{R\!D_{2}}=60$ m, ${K_{R\!E}=280}$ m, $H=60$ m, $m_{S\!P}=m_{S\!R}=m_{R\!D_{1}}=m_{R\!D_{2}}=m_{R\!E}=2$, $\varOmega _{S\!P}=\varOmega _{S\!R}=\varOmega _{R\!D_{1}}=\varOmega _{R\!D_{2}}=\varOmega _{R\!E}=1$, $C_{th_{1}}=0.0005$ bits/s/Hz, $C_{th_{2}}=0.001$ bits/s/Hz, $N=50$, and $L_{S}=L_{D_{1}}=L_{D_{2}}=L_{E}=2$.

Fig. 5.2

SOP of $D_{1}$ versus $\rho _{RD_{1}}$ for various H

Fig. 5.3

SOP of $D_{2}$ versus $\rho _{RD_{2}}$ for various H

Figures 5.2 and 5.3 plot the impact of $\rho _{RD_{1}}$ and $\rho _{RD_{2}}$ on the SOP performance for $D_{1}$ and $D_{2}$ under different H, respectively. We can observe from those two figures that the secrecy performance is enhanced as $\rho _{RD_{1}}$ ($\rho _{RD_{2}}$) increases due to the channel advantage of the legitimate link over the eavesdropping one. Furthermore, the asymptotic results are approximate with the exact ones at the high SNR regions. However, it is also observed that when $\rho _{RD_{1}}$ and $\rho _{RD_{2}}$ are in the comparably high SNR region, i.e., $\rho _{RD_{1}}>28$ dB in Fig. 5.2 and $\rho _{RD_{2}}>24$ dB in Fig. 5.3, the SOP performance almost remains unchanged with $\rho _{RD_{1}}$ ($\rho _{RD_{2}}$) increases. This is because the worst hop dominates the performance in the dual-hop link. In addition, one can see that a larger value of H corresponds to a worse SOP performance as a higher path loss resulted.

Fig. 5.4

SOP of $D_{1}$ versus $\rho _{RD_{1}}$ for various $\alpha _1$

Fig. 5.5

SOP of $D_{2}$ versus $\rho _{RD_{2}}$ for various $\alpha _2$

The SOP performance versus $\rho _{RD_{1}}$ and $\rho _{RD_{2}}$ for varying $\alpha _{1}$ and $\alpha _{2}$ are presented in Figs. 5.4 and 5.5. One can observe that the SOP performance enhances as $\alpha _{1} (\alpha _2)$ increases. A higher $\alpha _{1} (\alpha _2)$ means more power is allocated to the corresponding user.

Fig. 5.6

SOP of $D_{1}$ versus $\rho _I$ for various $\rho _{RD_{1}}$

Fig. 5.7

SOP of $D_{2}$ versus $\rho _I$ for various $\rho _{RD_{2}}$

The SOP versus $\rho _{I}$ for $D_{1}$ and $D_{2}$ under various $\rho _{RD_{1}}$ and $\rho _{RD_{2}}$ are depicted in Figs. 5.6 and 5.7. We can see that the SOP performance enhances with the increase of $\rho _I$, as more power can be used for information transmission at S. Furthermore, as expected, the asymptotic results are approximate with the exact ones at the high SNR regimes.

Fig. 5.8

SOP of $D_{1}$ versus $\rho _{RD_{1}}$ for various $\rho _{RE}$

Fig. 5.9

SOP of $D_{2}$ versus $\rho _{RD_{2}}$ for various $\rho _{RE}$

Figures 5.8 and 5.9 plot the SOP versus $\rho _{RE}$ for varying $\rho _{RD_{1}}$ and $\rho _{RD_{2}}$. One can reveal that the SOP performance improves as the $\rho _{RE}$ value decreases. This is because a smaller value of $\rho _{RE}$ means a poorer natural communication environment for the eavesdropper.

Fig. 5.10

SOP of $D_{1}$ versus $\rho _{RD_{1}}$ for various $L_{D_{1}}$

Fig. 5.11

SOP of $D_{2}$ versus $\rho _{RD_{2}}$ for various $L_{D_{2}}$

The impact of the number of antennas on the SOP performance is illustrated in Figs. 5.10 and 5.11. One can see that the value of SOP decreases as $L_{D_{i}}$ increases when $\rho _{RD_{1}}<24$ dB in Fig. 5.10 and $\rho _{RD_{2}}< 36$ dB in Fig. 5.11. This is because a higher diversity gain can be obtained when more antennas are available on the device. When $\rho _{RD_{i}}$ exceeds a certain value, i.e., $\rho _{RD_{1}}>24$ dB and $\rho _{RD_{2}}>36$ dB, the secrecy performance is almost similar for the different number of antennas, which is because the first hop dominates the system performance.

Fig. 5.12

SOP of $D_1$ versus $\rho _{RD_1}$ for various $\eta$

Fig. 5.13

SOP of $D_2$ versus $\rho _{RD_2}$ for various $\eta$

The impact of scaling coefficient $\eta$ on the SOP performance is presented in Figs. 5.12 and 5.13. As illustrated in those two figures, it is observed that the SOP performance improves as the value of $\eta$ decreases. This is because a smaller value of $\eta$ means less interference from the primary transmitter, which can result in better SOP performance.

5.4.2 SOP for the UAV with Different Spatial Positions and a Constant Speed

This section presents the SOP performance when the UAV has different spatial positions and flies at a constant speed. For convenience, as shown in Fig. 5.14, the positions of S, R, ${\text{ D }}_{i}$, and E are presented in a three-dimensional coordinate, which are ($x_S,y_S,z_S$), ($x_R,y_R,z_R$), ($x_{D_i},y_{D_i},z_{D_i}$), and ($x_E,y_E,z_E$). Unless otherwise specified, the initial coordinates of those nodes are set as S  (0, 0, 0), R  (30, 30, 60), ${\text{ D }}_{1}$  (90, 90, 0), ${\text{ D }}_{2}$  (70, 70, 0), and E  (230, 230, 0), and the unit is meter (m).

Fig. 5.14

Three-dimensional coordinate model and the flight trajectory of the UAV with a constant speed

Scenario 1: UAV is with different spatial positions.

In this scenario, the distance between the UAV and the node g ($g \in \{S,\,D_{1},\,D_{2},\,E\}$) can be re-expressed using the three-dimensional coordinates as

$$\begin{aligned} d_{Rg}={\sqrt{{(x_R-x_g)^2}+{(y_R-y_g)^2}+{(z_R-z_g)^2}}}. \end{aligned}$$

(5.39)

The results are presented in Figs. 5.15 and 5.16. Those two figures show that an optimal UAV deployment position exists for each curve in which the secrecy outage performance is the best.

Fig. 5.15

SOP of $D_1$ versus the UAV deployment position under various $z_R$ with $\alpha _1=0.8$

Fig. 5.16

SOP of $D_2$ versus the UAV deployment position under various $z_R$ with $\alpha _1=0.8$

Scenario 2: UAV flies with a constant speed.

As shown in Fig. 5.14, a simple scenario in which the UAV flies horizontally along with the y-axis to the right with a constant speed v is considered. Therefore, the distance between the UAV and the node g is

$$\begin{aligned} d_{Rg}\!=\!{\sqrt{{(x_R\!-\!x_g)^2}+{(y_R\!+\!vt\!-\!y_g)^2}+{(z_R\!-\!z_g)^2}}}, \end{aligned}$$

(5.40)

where t is the flight time. As illustrated in Fig. 5.17, one can observe that the SOP performance for both users first improves and then decreases as the UAV flies. This is because the UAV first approaches and then flies away from the users, which decreases the distance between the UAV and the users and increases.

From the above two scenarios, we can conclude that it is better to allocate the UAV in a proper position to result in the best secrecy performance of the system.

Fig. 5.17

SOP versus the trajectory of the UAV with $D_1$ (90, 70, 0), and $v=5$ m/s

Last but not least, compared with previous works on multi-antenna scenarios for UAV-NOMA systems, it is noted that the purpose of adopting multi-antenna is to cover different sectors [11], or to realize the beamforming [12,13,14], or to employ directional modulation [15]. Unlike those previous works, our focus of employing multi-antenna in this study is on diversity-combining techniques regarding SOP performance, i.e., both TAS at the source and MRC at the destinations. The primary technical challenge of the analysis includes but is not limited to obtaining the PDF and CDF expressions for the TAS output in the presence of underlay cognitive radio and the MRC output, as well as doing the asymptotic analysis. Numerical results from the analytical and asymptotic expressions show that the asymptotic results approximate the exact ones at the high SNR regions. Furthermore, it is also found that those figures have an error floor because another hop with fixed SNRs dominates the system performance.

5.5 Conclusion

This section analyzes the secrecy performance for an underlay CR-based UAV-aided NOMA system over Nakagami-m fading channels. Both users’ analytical and asymptotic expressions of SOP have been obtained when the TAS scheme is adopted at a secondary source and the MRC protocol is deployed at the destinations. Furthermore, we have analyzed the effects of the number of antennas, power distribution coefficient, height of the UAV, and interference power threshold on the secrecy performance. Numerical results illustrate that increasing the average SNR can enhance the secrecy performance, and the worse hop dominates the secrecy performance of the whole link. Moreover, increasing the number of receiving antennas, interference power threshold, and power distribution coefficient can also enhance the SOP performance of the system. In contrast, the system performance deteriorates as the height of the UAV increases. In addition, an optimal UAV deployment position makes the secrecy outage performance the best.