In this section, we analyze the two main system metrics with the assumed channel models below.

### Channel model

Following the results in [33], the probability density function (PDF) of \(|\hat{h}_{\rm{R}}|^2\) is formulated by

$$\begin{aligned} f_{\left| {\hat{h}_{\rm{R}} } \right| ^2 } \left( x \right) = \alpha _{\rm{R}} e^{ - \beta _{\rm{R}} x} {_1 F_1} \left( {m_{\rm{R}} ,1,\delta _{\rm{R}} x} \right) , \end{aligned}$$

(9)

where \(\alpha _{\rm{R}} = \frac{{\left( {\frac{{2b_{\rm{R}} m_{\rm{R}} }}{{2b_{\rm{R}} m_{\rm{R}} + \varOmega _{\rm{R}} }}} \right) ^{m_{\rm{R}} } }}{{2b_{\rm{R}} }}\), \(\beta _{\rm{R}} = ({{2b_{\rm{R}} }})^{-1}\), \(\delta _{\rm{R}} = \frac{{\varOmega _{\rm{R}} }}{{2b_{\rm{R}} \left( {2b_{\rm{R}} m_{\rm{R}} + \varOmega _{\rm{R}} } \right) }}\), \(m_{\rm{R}}\) is the fading severity parameter, \(2b_{\rm{R}}\) and \(\varOmega _{\rm{R}}\) denote multipath components and the average power of light of sight (LOS), respectively, and \({_1 F_1}\left( {.,.,.} \right)\) is the confluent hypergeometric function of the first kind [46, Eq. 9.210.1]. Using [34], we can rewrite the PDF of \(|h_{\rm{R}}|^2\) as

$$\begin{aligned} f_{\left| {\hat{h}_R } \right| ^2 } \left( x \right) = \alpha _R \sum \limits _{k = 0}^{m_R - 1} {\xi \left( k \right) x^k e^{ - \varXi _R x} }, \end{aligned}$$

(10)

where \(\xi \left( k \right) = \frac{{\left( { - 1} \right) ^k \left( {1 - m_R } \right) _k \delta _R^k }}{{\left( {k!} \right) ^2 }}\), \(\varXi _R = \beta _R - \delta _R\) and \((.)_x\) denotes the Pochhammer symbol [46, p. xliii]. Based on [46, Eq.3.351.2], the the cumulative distribution function (CDF) of \(\left| {\hat{h}_R } \right| ^2\) can be obtained as

$$\begin{aligned} F_{\left| {\hat{h}_R } \right| ^2 } \left( x \right) = 1 - \alpha _R \sum \limits _{k = 0}^{m_R - 1} {\xi \left( k \right) } \sum \limits _{l = 0}^k {\frac{{k!x^l e^{ - \varXi _R x} }}{{l!\left( {\varXi _R } \right) ^{k - l + 1} }}}. \end{aligned}$$

(11)

The PDF and CDF of \(|h_i|^2\) are then, respectively, given as [35]

$$\begin{aligned} f_{\left| {\hat{h}_i } \right| ^2 } \left( x \right) = \left( {\frac{{m_i }}{{\varOmega _i }}} \right) ^{m_i } \frac{{x^{m_i - 1} e^{ - \left( {\frac{{m_i }}{{\varOmega _i }}} \right) x} }}{{\varGamma \left( {m_i } \right) }}, \end{aligned}$$

(12)

and

$$\begin{aligned} \begin{aligned} F_{{\left| {\hat{h}_{i} } \right| ^{2} }} \left( x \right)&= 1 - \frac{1}{{\varGamma \left( {m_{i} } \right) }}\varGamma \left( {m_{i} ,\frac{{m_{i} x}}{{\varOmega _{i} }}} \right) \\&= 1 - \sum \limits _{{b_{i} = 0}}^{{m_{i} - 1}} {\frac{1}{{b_{i} !}}} \left( {\frac{{m_{i} x}}{{\varOmega _{i} }}} \right) ^{{ + b_{i} }} e^{{ - \left( {\frac{{m_{i} }}{{\varOmega _{i} }}} \right) x}} \\ \end{aligned} \end{aligned}$$

(13)

where \(m_i\) and \(\varOmega _i\) are the fading severity parameter and the average power, respectively, and \(\varGamma (.,.)\) is the upper incomplete gamma function [46].

Moreover, the PDF of \(\gamma _C\) is calculated with corresponding severity parameters \(\{m_{Cn}\}^N_n\) and average powers \(\{\varOmega _{Cn}\}^N_n\). Therefore, we can express the PDF of \(\gamma _C\) as [36, 37] and [24]

$$\begin{aligned} f_{\gamma _C } \left( x \right) = \left( {\frac{{m_I }}{{\varOmega _I }}} \right) ^{m_I } \frac{{x^{m_I - 1} }}{{\varGamma \left( {m_I } \right) }}e^{ - \left( {\frac{{m_I }}{{\varOmega _I }}} \right) x}, \end{aligned}$$

(14)

where the parameters \(m_I\) and \(\varOmega _I\) are obtained from moment based estimators. For this, we define \(\varTheta = \sum \nolimits _{n = 1}^I {\left| {h_{nR} } \right| ^2 }\), and without loss of generality, we assume no power control is used, i.e., \(P_{Cn} = P_C\) or \(\rho _{Cn} = \rho _C\). Then, we have \(\varOmega _I = \rho _C \varOmega _C\), where \(\varOmega _C = E\left[ \varTheta \right] = \sum \nolimits _{n = 1}^N {\varOmega _{Cn} }\) and \(m_I = \frac{{\varOmega _C^2 }}{{E\left[ {\varTheta ^2 } \right] - \varOmega _C^2 }}\). From this, the exact moments of \(\varTheta\) can be obtained as

$$\begin{aligned} \begin{aligned} E\left[ {\varTheta ^{n} } \right]&\approx \sum \limits _{{n_{1} = 0}}^{n} {\sum \limits _{{n_{2} = 0}}^{{n_{1} }} \cdots } \sum \limits _{{n_{{N - 1}} = 0}}^{{n_{{N - 2}} }} {\left( \begin{gathered} n \\ n_{1} \\ \end{gathered} \right) \left( \begin{gathered} n_{1} \\ n_{2} \\ \end{gathered} \right) \left( \begin{gathered} n_{{N - 2}} \\ n_{{N - 1}} \\ \end{gathered} \right) } \\&\quad \times E\left[ {\left| {h_{{1R}} } \right| ^{{2\left( {n - n_{1} } \right) }} } \right] E\left[ {\left| {h_{{1R}} } \right| ^{{2\left( {n_{1} - n_{2} } \right) }} } \right] \cdots E\left[ {\left| {h_{{1R}} } \right| ^{{2\left( {n_{{N - 1}} } \right) }} } \right] \\ \end{aligned} \end{aligned}$$

(15)

where \(E\left[ {\left| {h_{iR} } \right| ^n } \right] = \frac{{\varGamma \left( {m_{Cn} + \frac{n}{2}} \right) }}{{\varGamma \left( {m_{Cn} } \right) }}\left( {\frac{{m_{Cn} }}{{\varOmega _{Cn} }}} \right) ^{ - \frac{n}{2}}\).

### Outage probability of \(D_1\)

An outage event of \(D_1\) is given when *R* and \(D_1\) cannot detect \(x_1\) correctly. Then, the outage probability of \(D_1\) is given as

$$\begin{aligned} \begin{aligned} P_{D_1 }&= \Pr \left( {\min \left( {\varGamma _{R \rightarrow x_1 } ,\varGamma _{D_1 \rightarrow x_1 } } \right) < \gamma _1 } \right) \\&= 1 - \underbrace{\Pr \left( {\varGamma _{R \rightarrow x_1 }> \gamma _1 } \right) }_{B_1 }\underbrace{\Pr \left( {\varGamma _{D_1 \rightarrow x_1 } > \gamma _1 } \right) }_{B_2 } \end{aligned}, \end{aligned}$$

(16)

where \(\gamma _{i } = 2^{2R_i } - 1\), and \(R_i\) is the target rate.

### Proposition 1

Here, the closed-form of \(B_1\) is given as

$$\begin{aligned} \begin{aligned} B_1&= \alpha _R \sum \limits _{k = 0}^{m_R - 1} {\sum \limits _{l = 0}^k {\sum \limits _{p = 0}^l {\left( {\begin{array}{*{20}c} l \\ p \\ \end{array} } \right) } \frac{{\xi \left( k \right) k!\varGamma \left( {m_I + p} \right) }}{{l!\varGamma \left( {m_I } \right) \left( {\varXi _R } \right) ^{k - l + 1} }}} } e^{ - \varXi _R \phi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) } \\&\quad \times \left( {\phi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) } \right) ^l \left( {1 + \frac{{\varOmega _I \varXi _R \phi _1 }}{{m_I }}} \right) ^{ - m_I - p} \left( {\frac{{\varOmega _I }}{{m_I \left( {\rho _S \sigma _{e_R }^2 + 1} \right) }}} \right) ^p \\ \end{aligned} \end{aligned}$$

(17)

###
*Proof*

See Appendix A.

Next, using (6), \(B_2\) is rewritten as

$$\begin{aligned} \begin{aligned} B_{2}&= \Pr \left( {\frac{{\rho _{R} A_{1} \left| {\hat{h}_{1} } \right| ^{2} }}{{\rho _{R} A_{2} \left| {\hat{h}_{1} } \right| ^{2} + \rho _{R} \sigma _{{e_{1} }}^{2} + 1}}> \gamma _{1} } \right) \\&= \Pr \left( {\left| {\hat{h}_{1} } \right| ^{2} > \phi _{2} } \right) \\&= 1 - F_{{\left| {\hat{h}_{1} } \right| ^{2} }} \left( {\phi _{2} } \right) \\ \end{aligned} \end{aligned}$$

(18)

where \(\phi _2 = \frac{{\left( {\rho _R \sigma _{e_1 }^2 + 1} \right) \gamma _1 }}{{\left( {A_1 - A_2 \gamma _1 } \right) \rho _R }}\). Based on the CDF of \(\hat{h} _i\) in (13), \(B_2\) can be expressed as

$$\begin{aligned} B_2 = \sum \limits _{b_1 = 0}^{m_1 - 1} {\frac{{e^{ - \frac{{m_1 \phi _2 }}{{\varOmega _1 }}} }}{{b_1 !}}} \left( {\frac{{m_1 \phi _2 }}{{\varOmega _1 }}} \right) ^{b_1 }. \end{aligned}$$

(19)

Finally, substituting (17) and (19) into (16), \(P_{D_1}\) can be obtained by

$$\begin{aligned} \begin{gathered} P_{D_1 } = 1 - \alpha _R \sum \limits _{k = 0}^{m_R - 1} {\sum \limits _{l = 0}^k {\sum \limits _{p = 0}^l {\sum \limits _{b_1 = 0}^{m_1 - 1} {\left( {\begin{array}{*{20}c} l \\ p \\ \end{array} } \right) \frac{{\xi \left( k \right) k!}{\varGamma \left( {m_I + p} \right) e^{ - \varXi _R \phi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) - \frac{{m_1 \phi _2 }}{{\varOmega _1 }}} }}{{b_1 !l!\varGamma \left( {m_I } \right) }{\left( {\varXi _R } \right) ^{k - l + 1} \left( {\phi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) } \right) ^{ - l} }}} } } } \\ \times \left( {\frac{{m_1 \phi _2 }}{{\varOmega _1 }}} \right) ^{b_1 } \left( {1 + \frac{{\varOmega _I \varXi _R \phi _1 }}{{m_I }}} \right) ^{ - m_I - p} \left( {\frac{{\varOmega _I }}{{m_I \left( {\rho _S \sigma _{e_R }^2 + 1} \right) }}} \right) ^p \\ \end{gathered} \end{aligned}$$

(20)

### Outage probability of \(D_2\)

The outage events of \(D_2\) occurs when *R* and \(D_2\) cannot detect \(x_2\) correctly. Therefore, the outage probability of \(D_2\) is given as

$$\begin{aligned} \begin{aligned} P_{D_2 }&= \Pr \left( {\min \left( {\varGamma _{R \rightarrow x_2 } ,\varGamma _{D_2 \rightarrow x_2 } } \right) < \gamma _{2 } } \right) \\&= 1 - \Pr \left( {\varGamma _{R \rightarrow x_2 }> \gamma _{2 } } \right) \Pr \left( {\varGamma _{D_2 \rightarrow x_2 } > \gamma _{2} } \right) \\ \end{aligned} \end{aligned}$$

(21)

### Proposition 2

The closed-form outage probability of \(P_{D_2}\) is obtained as

$$\begin{aligned} \begin{gathered} P_{D_2 } = 1 - \alpha _R \sum \limits _{k = 0}^{m_R - 1} {\sum \limits _{l = 0}^k {\sum \limits _{p = 0}^l {\sum \limits _{b_2 = 0}^{m_2 - 1} {\left( {\begin{array}{*{20}c} l \\ p \\ \end{array} } \right) } \frac{{k!\xi \left( k \right) }{\varGamma \left( {m_I + p} \right) e^{ - \varXi _R \psi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) - \frac{{m_2 \psi _2 }}{{\varOmega _2 }}} }}{{b_2 !l!\varGamma \left( {m_I } \right) }{\left( {\varXi _R } \right) ^{k - l + 1} \left( {\psi _1 \left( {\rho _S \sigma _{e_R }^2 + 1} \right) } \right) ^{ - l} }}} } } \\ \times \left( {\frac{{m_2 \psi _2 }}{{\varOmega _2 }}} \right) ^{b_2 } \left( {1 + \frac{{\varOmega _I \varXi _R \psi _1 }}{{m_I }}} \right) ^{ - m_I - p} \left( {\frac{{\varOmega _I }}{{m_I \left( {\rho _S \sigma _{e_R }^2 + 1} \right) }}} \right) ^p \\ \end{gathered} \end{aligned}$$

(22)

###
*Proof*

See Appendix B.

### Diversity order

To gain some insight, we derive under the asymptotic outage probability of \(D_i\) under a high SNR \((\rho = \rho _S=\rho _R \rightarrow \infty )\). The diversity order is defined as [38]

$$\begin{aligned} d = - \mathop {\lim }\limits _{\rho \rightarrow \infty } \frac{{\log \left( {P_{D_i }^\infty } \right) }}{{\log \left( \rho \right) }}, \end{aligned}$$

(23)

where \({P_{D_i }^\infty }\) is the asymptotic outage probability of \(D_i\).

### Proposition 3

The asymptotic outage probability of \(D_1\) is given as

$$\begin{aligned} \begin{aligned} P_{D_1 }^\infty&= 1 - \left( {1 - \frac{1}{{\varGamma \left( {m_1 + 1} \right) }}\left( {\frac{{m_1 \phi _2 }}{{\varOmega _1 }}} \right) ^{m_1 } } \right) \\&\quad \times \left( {1 - \alpha _R \phi _1 \left( {\frac{{\left( {m_I } \right) !}}{{\varGamma \left( {m_I } \right) }}\left( {\frac{{\varOmega _I }}{{m_I }}} \right) + \left( {\rho _S \sigma _{e_R }^2 + 1} \right) } \right) } \right) \\ \end{aligned} \end{aligned}$$

(24)

###
*Proof*

See Appendix C.

Similarly, the asymptotic of \(D_2\) can be expressed by

$$\begin{aligned} \begin{aligned} P_{{D_{2} }}^{\infty }&= 1 - \left( {1 - \frac{1}{{\varGamma \left( {m_{2} - 1} \right) }}\left( {\frac{{m_{2} \psi _{2} }}{{\varOmega _{2} }}} \right) ^{{m_{2} }} } \right) \left( 1 \right. \\&\quad - \left. {\alpha _{R} \psi _{1} \left( {\frac{{\left( {m_{I} } \right) !}}{{\varGamma \left( {m_{I} } \right) }}\left( {\frac{{\varOmega _{I} }}{{m_{I} }}} \right) + \left( {\rho _{S} \sigma _{{e_{R} }}^{2} + 1} \right) } \right) } \right) \\ \end{aligned} \end{aligned}$$

(25)

The results in (24) and (25) refer to limits of outage performance in the region of high SNR. It can be predicted that the outage performance of two ground users encounters the lower bound even though we improve other system parameters. As discussed, the diversity is then zero.

### Ergodic capacity of \(D_1\)

The ergodic capacity of \(x_i\) is expressed as [39]

$$\begin{aligned} R_{x_1 } = \frac{1}{{2\log \left( 2 \right) }}\int \limits _0^{\frac{{A_1 }}{{A_2 }}} {\frac{{1 - F_{Q_1 } \left( x \right) }}{{1 + x}}} dx, \end{aligned}$$

(26)

where \(Q_1 = \min \left( {\varGamma _{R \rightarrow x_1 },\varGamma _{D_1 \rightarrow x_1 } } \right)\).

### Proposition 4

The closed-form ergodic capacity of \(x_1\) is given as (27), where \(\varPsi _1 = \frac{{\left( {\rho _S \sigma _{e_R }^2 + 1} \right) \varXi _R \left( {1 + \theta _p } \right) }}{{A_2 \rho _S \left( {1 - \theta _p } \right) }} + \frac{{m_1 \left( {\rho _R \sigma _{e_1 }^2 + 1} \right) \left( {1 + \theta _p } \right) }}{{A_2 \varOmega _1 \rho _R \left( {1 - \theta _p } \right) }}\).

$$\begin{aligned} \begin{aligned} R_{{x_{1} }}&\approx \frac{{\alpha _{R} }}{{2\ln \left( 2 \right) }}\sum \limits _{{k = 0}}^{{m_{R} - 1}} {\sum \limits _{{l = 0}}^{k} {\sum \limits _{{p = 0}}^{l} {\sum \limits _{{b_{1} = 0}}^{{m_{1} - 1}} {\left( {\begin{array}{*{20}c} l \\ p \\ \end{array} } \right) \frac{{\xi \left( k \right) k!}}{{b_{1} !l!\varGamma \left( {m_{I} } \right) }}} } } } \frac{{\varGamma \left( {m_{I} + p} \right) }}{{\left( {\varXi _{R} } \right) ^{{k - l + 1}} }} \\&\quad \times \left( {\frac{{\left( {\rho _{R} \sigma _{{e_{1} }}^{2} + 1} \right) m_{1} }}{{\varOmega _{1} \rho _{R} }}} \right) ^{{b_{1} }} \left( {\frac{{\varOmega _{I} }}{{m_{I} \left( {\rho _{S} \sigma _{{e_{R} }}^{2} + 1} \right) }}} \right) ^{p} \left( {\frac{{\left( {\rho _{S} \sigma _{{e_{R} }}^{2} + 1} \right) }}{{\rho _{S} }}} \right) ^{l} \\&\quad \times \frac{\pi }{P}\sum \limits _{{p = 0}}^{P} {\frac{{A_{1} \sqrt{1 - \theta _{p}^{2} } e^{{ - \varPsi _{1} }} }}{{2A_{2} + A_{1} \left( {1 + \theta _{p} } \right) }}} \left( {1 + \frac{{\varOmega _{I} \varXi _{R} \left( {1 + \theta _{p} } \right) }}{{A_{2} m_{I} \rho _{S} \left( {1 - \theta _{p} } \right) }}} \right) ^{{ - m_{I} - p}} \left( {\frac{{\left( {1 + \theta _{p} } \right) }}{{A_{2} \left( {1 - \theta _{p} } \right) }}} \right) ^{{b_{1} + l}} \\ \end{aligned} \end{aligned}$$

(27)

###
*Proof*

See Appendix D.

### Ergodic capacity \(D_2\)

Similarly, the ergodic capacity of \(x_2\) is written as

$$\begin{aligned} R_{x_2 } = \frac{1}{{2\ln \left( 2 \right) }}\int \limits _0^\infty {\frac{{1 - F_{Q_2 } \left( y \right) }}{{1 + y}}} dy, \end{aligned}$$

(28)

where \(Q_2 = \min \left( {\varGamma _{R \rightarrow x_2 },\varGamma _{D_1 \rightarrow x_2 } } \right)\).

### Proposition 5

The closed-form ergodic capacity of \(x_1\) is given as (29), where \(\varPsi _2 = \frac{{\varXi _R \left( {\rho _S \sigma _{e_R }^2 + 1} \right) }}{{\rho _R A_2 }} + \frac{{m_2 \left( {\rho _R \sigma _{e_2 }^2 + 1} \right) }}{{\varOmega _2 \rho _R A_2 }}\) and \(G_{1,\left[ {1:1} \right] ,0,\left[ {1:1} \right] }^{1,1,1,1,1}[.,.]\) denotes the Meijer-G function with two variables [42].

$$\begin{aligned} \begin{aligned} R_{{x_{2} }}&= \frac{{\alpha _{R} }}{{2\ln \left( 2 \right) }}\sum \limits _{{k = 0}}^{{m_{R} - 1}} {\sum \limits _{{l = 0}}^{k} {\sum \limits _{{p = 0}}^{l} {\sum \limits _{{b_{2} = 0}}^{{m_{2} - 1}} {\left( {\begin{array}{*{20}c} l \\ p \\ \end{array} } \right) } \frac{{k!\xi \left( k \right) \left( {\varXi _{R} } \right) ^{{ - k + l - 1}} }}{{b_{2} !l!\varGamma \left( {m_{I} } \right) }}} } } \\&\quad \times \left( {\frac{{m_{2} \left( {\rho _{R} \sigma _{{e_{2} }}^{2} + 1} \right) }}{{\varOmega _{2} \rho _{R} A_{2} }}} \right) ^{{b_{2} }} \left( {\frac{{\left( {\rho _{S} \sigma _{{e_{R} }}^{2} + 1} \right) }}{{\rho _{R} A_{2} }}} \right) ^{l} \left( {\frac{{\varOmega _{I} }}{{m_{I} \left( {\rho _{S} \sigma _{{e_{R} }}^{2} + 1} \right) }}} \right) ^{p} \\&\quad \times G_{{1,\left[ {1:1} \right] ,0,\left[ {1:1} \right] }}^{{1,1,1,1,1}} \left[ {\begin{array}{*{20}c} {\frac{{\varOmega _{I} \varXi _{R} }}{{m_{I} \rho _{R} A_{2} \varPsi _{2} }}} \\ {\frac{1}{{\varPsi _{2} }}} \\ \end{array} \left| {\begin{array}{*{20}c} {1 + l + b_{2} } \\ {1 - m_{I} - p} \\ - \\ {0,0} \\ \end{array} } \right. } \right] \\ \end{aligned} \end{aligned}$$

(29)

###
*Proof*

See Appendix E.