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 = Eleft[ varTheta right] = sum nolimits _{n = 1}^N {varOmega _{Cn} }) and (m_I = frac{{varOmega _C^2 }}{{Eleft[ {varTheta ^2 } right] – varOmega _C^2 }}). From this, the exact moments of (varTheta) can be obtained as
$$begin{aligned} begin{aligned} Eleft[ {varTheta ^{n} } right]&approx sum limits _{{n_{1} = 0}}^{n} {sum limits _{{n_{2} = 0}}^{{n_{1} }} cdots } sum limits _{{n_{{N – 1}} = 0}}^{{n_{{N – 2}} }} {left( begin{gathered} n \ n_{1} \ end{gathered} right) left( begin{gathered} n_{1} \ n_{2} \ end{gathered} right) left( begin{gathered} n_{{N – 2}} \ n_{{N – 1}} \ end{gathered} right) } \&quad times Eleft[ {left| {h_{{1R}} } right| ^{{2left( {n – n_{1} } right) }} } right] Eleft[ {left| {h_{{1R}} } right| ^{{2left( {n_{1} – n_{2} } right) }} } right] cdots Eleft[ {left| {h_{{1R}} } right| ^{{2left( {n_{{N – 1}} } right) }} } right] \ end{aligned} end{aligned}$$
(15)
where (Eleft[ {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]
$$begin{aligned} R_{x_1 } = frac{1}{{2log left( 2 right) }}int limits _0^{frac{{A_1 }}{{A_2 }}} {frac{{1 – F_{Q_1 } left( x right) }}{{1 + x}}} dx, end{aligned}$$
(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) }}).
$$begin{aligned} begin{aligned} R_{{x_{1} }}&approx frac{{alpha _{R} }}{{2ln left( 2 right) }}sum limits _{{k = 0}}^{{m_{R} – 1}} {sum limits _{{l = 0}}^{k} {sum limits _{{p = 0}}^{l} {sum limits _{{b_{1} = 0}}^{{m_{1} – 1}} {left( {begin{array}{*{20}c} l \ p \ end{array} } right) frac{{xi left( k right) k!}}{{b_{1} !l!varGamma left( {m_{I} } right) }}} } } } frac{{varGamma left( {m_{I} + p} right) }}{{left( {varXi _{R} } right) ^{{k – l + 1}} }} \&quad times left( {frac{{left( {rho _{R} sigma _{{e_{1} }}^{2} + 1} right) m_{1} }}{{varOmega _{1} rho _{R} }}} right) ^{{b_{1} }} left( {frac{{varOmega _{I} }}{{m_{I} left( {rho _{S} sigma _{{e_{R} }}^{2} + 1} right) }}} right) ^{p} left( {frac{{left( {rho _{S} sigma _{{e_{R} }}^{2} + 1} right) }}{{rho _{S} }}} right) ^{l} \&quad times frac{pi }{P}sum limits _{{p = 0}}^{P} {frac{{A_{1} sqrt{1 – theta _{p}^{2} } e^{{ – varPsi _{1} }} }}{{2A_{2} + A_{1} left( {1 + theta _{p} } right) }}} left( {1 + frac{{varOmega _{I} varXi _{R} left( {1 + theta _{p} } right) }}{{A_{2} m_{I} rho _{S} left( {1 – theta _{p} } right) }}} right) ^{{ – m_{I} – p}} left( {frac{{left( {1 + theta _{p} } right) }}{{A_{2} left( {1 – theta _{p} } right) }}} right) ^{{b_{1} + l}} \ end{aligned} end{aligned}$$
(27)
Proof
See Appendix D.
Ergodic capacity (D_2)
Similarly, the ergodic capacity of (x_2) is written as
$$begin{aligned} R_{x_2 } = frac{1}{{2ln left( 2 right) }}int limits _0^infty {frac{{1 – F_{Q_2 } left( y right) }}{{1 + y}}} dy, end{aligned}$$
(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].
$$begin{aligned} begin{aligned} R_{{x_{2} }}&= frac{{alpha _{R} }}{{2ln left( 2 right) }}sum limits _{{k = 0}}^{{m_{R} – 1}} {sum limits _{{l = 0}}^{k} {sum limits _{{p = 0}}^{l} {sum limits _{{b_{2} = 0}}^{{m_{2} – 1}} {left( {begin{array}{*{20}c} l \ p \ end{array} } right) } frac{{k!xi left( k right) left( {varXi _{R} } right) ^{{ – k + l – 1}} }}{{b_{2} !l!varGamma left( {m_{I} } right) }}} } } \&quad times left( {frac{{m_{2} left( {rho _{R} sigma _{{e_{2} }}^{2} + 1} right) }}{{varOmega _{2} rho _{R} A_{2} }}} right) ^{{b_{2} }} left( {frac{{left( {rho _{S} sigma _{{e_{R} }}^{2} + 1} right) }}{{rho _{R} A_{2} }}} right) ^{l} left( {frac{{varOmega _{I} }}{{m_{I} left( {rho _{S} sigma _{{e_{R} }}^{2} + 1} right) }}} right) ^{p} \&quad times G_{{1,left[ {1:1} right] ,0,left[ {1:1} right] }}^{{1,1,1,1,1}} left[ {begin{array}{*{20}c} {frac{{varOmega _{I} varXi _{R} }}{{m_{I} rho _{R} A_{2} varPsi _{2} }}} \ {frac{1}{{varPsi _{2} }}} \ end{array} left| {begin{array}{*{20}c} {1 + l + b_{2} } \ {1 – m_{I} – p} \ – \ {0,0} \ end{array} } right. } right] \ end{aligned} end{aligned}$$
(29)
Proof
See Appendix E.
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