This section presents the analysis of the performance of the system model in which closed-form expressions of the outage probability, throughput, ergodic rate and EE are determined in DTT and DLT modes.
Outage performance
Outage probability at \(D_{1}\)
User \(D_1\) is not in outage when it can decode both signals \(x_1\) and \(x_2\) received from the BS. The outage probability at \(D_1\) is thus obtained by
$$\begin{aligned} P_{D_1} = 1 - \Pr \left( {{\gamma _{2,{D_1}}}> \gamma _{t{h_2}},{} {\gamma _{1,{D_1}}} > \gamma _{t{h_1}}} \right) , \end{aligned}$$
(12)
where, \(\gamma _{t{h_1}} = {2^{2{R_1}}} - 1\) and \(\gamma _{t{h_2}} = {2^{2{R_2}}} - 1\) represent the threshold SNRs at \(D_{1}\) for detecting signals \(x_1\) and \(x_2\), respectively.
Theorem 1
The outage probability at \(D_{1}\) is given by
$$\begin{aligned} P_{D1} = 1 - {e^{ - \frac{{{\theta _1}}}{{{\Omega _1}}}}}, \end{aligned}$$
(13)
where, \({\theta _1} = \max ({\tau _1},{\nu _1}),{\tau _1} = \frac{{\gamma _{th2}}}{{\rho {\psi _I}({\alpha _2} - {\alpha _1}\gamma _{th2})}}\) and \({\nu _1} = \frac{{\gamma _{th1}}}{{{a_1}{\psi _I}\rho }}\) with \({\alpha _2} > {\alpha _1}\gamma _{th2}.\)
Proof
From (12), the outage probability at \(D_{1}\) can be determined by
$$\begin{aligned} {P_{{D_1}}}&= {} 1\!-\!Pr\left( {\frac{{{\psi _I}{{\left| {{h_1}} \right| }^2}{\alpha _2}\rho }}{{{\psi _I}{{\left| {{h_1}} \right| }^2}{\alpha _1}\rho \!+\!1}}\!>\!{\gamma _{t{h_2}}},{} {} {} {} \psi _I{{\left| {{h_1}} \right| }^2}{\alpha _1}\rho \!>\! {\gamma _{t{h_1}}}} \right) \nonumber \\&= {} 1 - Pr\left( {{{\left| {{h_1}} \right| }^2}> \,\frac{{{\gamma _{t{h_2}}}}}{{\rho {\psi _I}\left( {{\alpha _2} - {\alpha _1}{\gamma _{t{h_2}}}} \right) }},\,\,{{\left| {{h_1}} \right| }^2} > \,\frac{{{\gamma _{t{h_1}}}}}{{{\alpha _1}\rho {\psi _I}}}} \right) \nonumber \\&= {} 1 - Pr({\left| {{h_1}} \right| ^2} \ge {\theta _1})\nonumber \\&= {} 1 - \int _{{\theta _1}}^\infty {{f_{{{\left| {{h_1}} \right| }^2}}}(x)dx} \end{aligned}$$
(14)
Applying the following equation
$$\begin{aligned} {f_{{h_i}}}(x) = \frac{1}{{{\Omega _i}}}\exp ( - \frac{x}{{{\Omega _i}}}),{} {} {} {} i \in \{ SD_{1},D_{1}D_{2}\mathrm{{\} }} \end{aligned}$$
Eq. (14) can be obtained as follows
$$\begin{aligned} {P_{{D_1}}}&= {} 1 - \int _{{\theta _{1}}}^\infty {\frac{1}{{{\Omega _1}}}{e^{\frac{{ - x}}{{{\Omega _1}}}}}dx}\nonumber \\&= {} 1 - {e^{ - \,\,\frac{{{\theta _{1}}}}{{{\Omega _1}}}}} \end{aligned}$$
(15)
The proof is completed. \(\square\)
Corollary 1
From (15), the outage probability at \(D_{1}\) for high SNR \(\rho \rightarrow \infty\) is expressed by
$$\begin{aligned} P_{D_1}^\infty = \frac{{{\theta _1}}}{{{\Omega _1}}} \end{aligned}$$
(16)
Proof
From (12), when \(\rho \rightarrow \infty\), the outage probability at \(D_{1}\) with \(1 - {e^{-x}} \approx x\) is given by
$$\begin{aligned} P_{D_1}^{\infty }&= {} 1 - {P_r}\left( {{{\left| {{h_1}} \right| }^2} \ge \theta _1} \right) \nonumber \\&= {} 1 - {e^{ - \frac{{\theta _1}}{{\Omega {\,_1}}}}}\nonumber \\&= {} \frac{{{\theta _1}}}{{{\Omega _1}}} \end{aligned}$$
(17)
The proof is completed. \(\square\)
Based on (15) and \({\alpha _2} > {\alpha _1}\gamma _{th2}\), \(P_{D_1}\) depends on \(\tau _{1}\) and the random variable \(\Omega _{1}\) (\(|h_1|^2\)). The closer the d, the lower the \(P_{D_1}\). This means that a better transmission quality can be achieved, and vice versa.
Outage probability at \(D_{2}\) for no direct link
Since \(D_1\) can not detect \(x_2\) as well as \(D_2\) can not recover the forwarded information from \(D_1\), the \(D_2\) is in outage. Hence, the outage probability at \(D_2\) is derived as (see (18)). By calculating \(J_2\) and \(J_3\), the outage probability for no direct link is determined by
$$\begin{aligned} \begin{array}{l} P_{{D_2},nodir} = \underbrace{\Pr \left( {{\gamma _{2,{D_1}}}< \gamma _{t{h_2}}^{HD}} \right) }_{{J_2}} + \underbrace{\Pr \left( {{\gamma _{2,{D_2}}} < \gamma _{t{h_2}}^{HD},{\gamma _{2,{D_1}}} > \gamma _{t{h_2}}^{HD}} \right) }_{{J_3}}, \end{array} \end{aligned}$$
(18)
Theorem 2
The outage probability at
\(D_{2}\)
can be obtained by
$$\begin{aligned} P_{{D_2},nodir} = 1 - {e^{ - \frac{{{\tau _1}}}{{{\Omega _1}}}}}+ \int \limits _{{\tau _1}}^\infty {\left( {1 - {{\mathop {e}\nolimits } ^{ - \frac{{\gamma _{t{h_2}}}}{{x{\psi _E}\rho {\Omega _2}}}}}} \right) \frac{1}{{{\Omega _1}}}\exp \left( {\frac{{- x}}{{{\Omega _1}}}} \right) } dx. \end{aligned}$$
(19)
Proof
Considering the Rayleigh fading channel, \(J_2\) can be given by
$$\begin{aligned} {J_2} = 1 - \exp \left( {\frac{{ - {\tau _1}}}{{{\Omega _1}}}} \right) . \end{aligned}$$
(20)
and \(J_3\) can be expressed as (see(21)).
$$\begin{aligned} \begin{array}{l} {J_3}=\Pr \left( {{{\left| {{h_2}} \right| }^2}{{\left| {{h_1}} \right| }^2}{\psi _E}\rho<\gamma _{t{h_2}},\frac{{{{\left| {{h_1}} \right| }^2}{\psi _I}{\alpha _2}\rho }}{{{\psi _I}{{\left| {{h_1}} \right| }^2}{\alpha _1}\rho +1}}>\gamma _{t{h_2}}} \right) \\ \\ \quad \,=\left\{ \begin{array}{l} \Pr \left( {{{\left| {{h_2}} \right| }^2}<\frac{{\gamma _{t{h_2}}}}{{{{\left| {{h_1}} \right| }^2}{\psi _E}\rho }}, {{\left| {{h_1}} \right| }^2}>\frac{{\gamma _{t{h_2}}}}{{{\psi _I}\rho \left( {{\alpha _2}-{\alpha _1}\gamma _{t{h_2}}} \right) }}} \right) ,{\alpha _2}> {\alpha _1}\gamma _{t{h_2}}\\ \\ 0,\,\,{\alpha _2} \le {a_1}\gamma _{t{h_2}} \end{array} \right. \\ \\ \quad \,= \int \limits _{\frac{{\gamma _{t{h_2}}}}{{{\psi _I}\rho \left( {{\alpha _2} - {\alpha _1}\gamma _{t{h_2}}} \right) }}}^\infty {\int \limits _0^{\frac{{\gamma _{t{h_2}}}}{{x{\psi _E}\rho }}} {{f_{{{\left| {{h_1}} \right| }^2}}}(x){f_{{{\left| {{h_2}} \right| }^2}}}(y)dxdy} }= \int \limits _{{\tau _1}}^\infty {\frac{1}{{{\Omega _{1}}}}\left[ {1 - \exp \left( {\frac{{ - \gamma _{t{h_2}}}}{{x{\psi _E}\rho {\Omega _{2}}}}} \right) } \right] \exp \left( {\frac{{ - x}}{{{\Omega _{1}}}}} \right) } dx. \end{array} \end{aligned}$$
(21)
The outage probability at \(D_2\) is given by
$$\begin{aligned} P_{{D_2},nodir} = \,{J_2}\, + \,{J_3}.\, \end{aligned}$$
(22)
\(\square\)
Corollary 2
The outage probability at \(D_{2}\) for high SNR can be determined as (see(23)), where \({K_1}(.)\) is the first order modified Bessel function of the second kind [55, Eq.(3.324.1)].
$$\begin{aligned} P_{{D_2},nodir}^{\infty }&= {} \Pr \left( {\frac{{{\alpha _2}}}{{{\alpha _1}}}< \gamma _{t{h_2}}} \right) + \Pr \left( {{{\left| {{h_2}} \right| }^2}< \frac{{\gamma _{t{h_2}}}}{{{\psi _E}\rho {{\left| {{h_1}} \right| }^2}}},\frac{{{\alpha _2}}}{{{\alpha _1}}}> \gamma _{t{h_2}}} \right) \nonumber \\&= {} \Pr \left( {{{\left| {{h_2}} \right| }^2} < \frac{{\gamma _{t{h_2}}}}{{{\psi _E}\rho {{\left| {{h_1}} \right| }^2}}},\frac{{{\alpha _2}}}{{{\alpha _1}}} > \gamma _{t{h_2}}} \right) \nonumber \\&= {} \int \limits _0^\infty {\left[ {1 - \exp \left( {\frac{{ - \gamma _{t{h_2}}}}{{{\psi _E}\rho {\Omega _2}x}}} \right) } \right] } \frac{1}{{{\Omega _1}}}\exp \left( {\frac{{ - x}}{{{\Omega _1}}}} \right) dx \nonumber \\&= {} 1 - 2\sqrt{\frac{{\gamma _{t{h_2}}}}{{{\psi _E}\rho {\Omega _1}{\Omega _2}}}} {K_1}\left( {2\sqrt{\frac{{\gamma _{t{h_2}}}}{{{\psi _E}\rho {\Omega _1}{\Omega _2}}}} } \right) . \end{aligned}$$
(23)
Outage probability at \(D_{2}\) for User Relaying with Direct Link
When \(x_{2}\) can be detected at \(D_{1}\) but the SINR is smaller than the target SNR after MRC or both \(D_{1}\) and \(D_{2}\) can not detect \(x_{2}\), the outage probability will occur at \(D_{2}\) and is given by (see(24))
$$\begin{aligned} {P_{{D_2},dir}} = \underbrace{{P_r} (\gamma _{{D_2}}^{MRC}< {\gamma _{th{{}_2}}})}_{{J_4}}\underbrace{{P_r}\left( {{\gamma _{2,{D_1}}} > {\gamma _{th{{}_2}}}} \right) }_{{J_5}} + {} \underbrace{{P_r}({\gamma _{2,{D_1}}}< {\gamma _{th{{}_2}}},{\gamma _{1,{D_2}}} < {\gamma _{th{{}_2}}})}_{{J_6}}, \end{aligned}$$
(24)
Theorem 3
The outage probability at \(D_{2}\) can be given by (see(25))
$$\begin{aligned} P_{{D_2},dir}&= {} \int _0^\infty {\int _0^{\psi _{I}\,{\tau _1}} {\frac{1}{{{\Omega _0}{\Omega _1}}}\left( {1-{e^{-\frac{{\gamma _{t{h_2}}}}{{x\psi _{E}\rho {\Omega _2}}}+\frac{{y{\alpha _2}}}{{x\psi _{E}{\Omega _2}\left( {y{\alpha _1}\rho +1} \right) }}}}} \right) {e^{-\frac{x}{{\Omega {_1}}}\!-\!\frac{y}{{{\Omega _0}}}}}dxdy}}\times {e^{ - \frac{{{\tau _1}}}{{{\Omega _1}}}}}\,\nonumber \\&+ \left( {1 - {e^{ - \frac{{\tau _1}}{{{\Omega _1}}}}}} \right) \left( {1 - {e^{ - \,\,\frac{{{\tau _1}\,\psi _{I}}}{{{\Omega _0}}}}}} \right) \end{aligned}$$
(25)
Proof
From (24), the outage probability at \(D_{2}\) is determined by (see(26)), (see(27)), (see(28))
$$\begin{aligned} {J_4}&= {} \Pr \left( {{{\left| {{h_2}} \right| }^2}< \frac{{\gamma _{t{h_2}}}}{{{{\left| {{h_1}} \right| }^2}\psi _{E}\rho }} - \frac{{{{\left| {{h_0}} \right| }^2}{\alpha _2}}}{{{{\left| {{h_1}} \right| }^2}\psi _{E}\left( {{{\left| {{h_0}} \right| }^2}{\alpha _1}\rho + 1} \right) }},{{\left| {{h_0}} \right| }^2} < {\tau _1}\,\psi _{I}} \right) \nonumber \\&= {} \int _0^\infty {\int _0^{\psi _{I}\,{\tau _1}} {\int \limits _0^{\frac{{\gamma _{t{h_2}}}}{{x\,\psi _{E}\rho }}\,\, - \,\,\frac{{y{\alpha _2}}}{{x\,\psi _{E}\left( {y\,{\alpha _1}\rho + 1} \right) }}} {{f_{{{\left| {h{\,_1}} \right| }^2}}}\left( x \right) {f_{{{\left| {{h_0}} \right| }^2}}}\left( y \right) {f_{{{\left| {h{\,_2}} \right| }^2}}}\left( z \right) dxdydz} } }\nonumber \\&= {} \int _0^\infty {\int _0^{\psi _{I}\,{\tau _1}} {\frac{1}{{{\Omega _0}{\Omega _1}}}\left( {1 - {e^{ - \,\,\frac{{\gamma _{t{h_2}}}}{{x\,\psi _{E}\rho \,\Omega {\,_2}}}\,\, + \,\,\frac{{y{\alpha _2}}}{{x\,\psi _{E}\Omega {\,_2}\left( {y\,{\alpha _1}\rho + 1} \right) }}}}} \right) {e^{ - \,\,\frac{x}{{\Omega {\,_1}}}\,\, - \,\,\frac{y}{{{\Omega _0}}}}}dxdy} } \end{aligned}$$
(26)
$$\begin{aligned} {J_5}&= {} {\left| {{h_1}} \right| ^2} > {\tau _1}=\int _{\tau _1}^\infty {{f_{{{\left| {h{\,_1}} \right| }^2}}}\left( x \right) dx}= {e^{ - \,\,\frac{{\tau _1}}{{\Omega {\,_1}}}}} \end{aligned}$$
(27)
$$\begin{aligned} {J_6}&= {} \Pr \left( {{{\left| {{h_0}} \right| }^2}< \psi _{I}{} {} {\tau _1},{{\left| {{h_1}} \right| }^2} < {\tau _1}} \right) = \int _0^{{\tau _1}} {\int _0^{\psi _{I}{} {\tau _1}} {{f_{{{\left| {h{{}_1}} \right| }^2}}}\left( x \right) } } {f_{{{\left| {{h_{{} 0}}} \right| }^2}}}\left( y \right) dxdy\;\;\;{} {} {} {} {} {}\nonumber \\&= {} \left( {1 - {e^{ - \frac{{\psi _{I}{} {\tau _1}}}{{{\Omega _0}}}}}} \right) \left( {1 - {e^{ - \frac{{{\tau _1}}}{{{\Omega _1}}}}}} \right) \end{aligned}$$
(28)
\(\square\)
Throughput for DLT mode
User relaying without direct link
With a given constant R, the transmitted information of the source node depends on the outage probability performance due to wireless fading channels. Therefore, the throughput of the system is determined by
$$\begin{aligned} \tau _{t,nodir} = \left( {1 - P_{{D_1}}} \right) {R_1} + \left( {1 - P_{{D_2},nodir}} \right) {R_2}, \end{aligned}$$
(29)
where \(P_{D_1}\) and \(P_{{D_2},nodir}\) can be achieved from (15) and (19), respectively.
User relaying with direct link
The throughput of system is given by
$$\begin{aligned} \tau _{t,dir} = \left( {1 - P_{{D_1}}} \right) {R_1} + \left( {1 - P_{{D_2},dir}} \right) {R_2}, \end{aligned}$$
(30)
where \(P_{{D_1}}\) and \(P_{{D_2},dir}\) can be achieved from (15) and (25), respectively.
Ergodic rate for DTT mode
Ergodic rate at \(D_{1}\)
The achievable rate at \(D_{1}\) where \(D_{1}\) can detect \(x_{2}\) is given by
$$\begin{aligned} {R_{{D_1}}} = \frac{1}{2}{\log _2}\left( {1 + {\gamma _{{D_1}}}} \right) . \end{aligned}$$
(31)
Theorem 4
The ergodic rate at \(D_{1}\) is determined by
$$\begin{aligned} R_{{D_1}} = \frac{{ - \exp \left( {\frac{1}{{{\psi _I}{\alpha _1}\rho {\Omega _1}}}} \right) }}{{2\ln 2}}Ei\left( {\frac{{ - 1}}{{{\psi _I}{\alpha _1}\rho {\Omega _1}}}} \right) , \end{aligned}$$
(32)
where Ei(.) indicates the exponential integral function [55, Eq.(3.354.4)].
Proof
See Appendix 1. \(\square\)
Ergodic rate at \(D_{2}\) for User Relaying Without Direct Link
Since \(x_2\) needs to be detected at both \(D_1\) and \(D_2\), the achievable rate at \(D_{2}\) is given by
$$\begin{aligned} {R_{{D_2},nodir}}{} = {} \frac{1}{2}{\log _2}\left( {1 + \min \left( {{\gamma _{2,{D_1}}},{\gamma _{2,{D_2}}}} \right) } \right) . \end{aligned}$$
(33)
Theorem 5
The ergodic rate at
\(D_{2}\)
is given by
$$\begin{aligned} R_{{D_2},nodir} = \frac{1}{{2\ln 2}}\int \limits _0^{\frac{{{\alpha _2}}}{{{\alpha _1}}}} {\left[ {\frac{{{e^{ - \frac{x}{{{\psi _I}\rho \left( {{\alpha _2} - {\alpha _1}x} \right) {\Omega _1}}}}}}}{{1 + x}}} \right. } \left. {+ \frac{{\int _{\frac{x}{{{\psi _I}\rho \left( {{\alpha _2} - {\alpha _1}x} \right) }}}^\infty {\frac{1}{{{\Omega _1}}}\left( {1 - {e^{ - \frac{x}{{y\rho {\psi _E}{\Omega _2}}}}}} \right) {e^{ - \frac{y}{{{\Omega _1}}}}}dy} }}{{1 + x}}} \right] dx. \end{aligned}$$
(34)
Proof
See Appendix 2 . \(\square\)
Remark 1
The ergodic rate in the asymptotic expression at \(D_{2}\) for high SNR region \(\rho \rightarrow \infty\) is obtained by
$$\begin{aligned} R_{{D_2},nodir}^{\infty }=\frac{1}{{2\ln 2}}\int \limits _0^\infty {\frac{{1 - {F_X}(x)}}{{1 + x}}dx}. \end{aligned}$$
(35)
From the analytical result in (35), this expression can be deployed by
$$\begin{aligned} R_{{D_2},nodir}^{\infty }\!=\!\frac{1}{{2\ln 2}}\!\int \limits _0^{\frac{{{\alpha _2}}}{{{\alpha _1}}}} {\frac{{\!2\sqrt{\!\frac{x}{{{\psi _E}\rho \,{\Omega _1}{\Omega _2}}}} \,{K_1}\left( {\!2\sqrt{\!\frac{x}{{{\psi _E}\rho \,{\Omega _1}{\Omega _2}}}} } \right) }}{{1 + x}}dx}. \end{aligned}$$
(36)
Proof
See Appendix 3. \(\square\)
Ergodic rate at \(D_{2}\) for user relaying with direct link
The ergodic rate at \(D_{2}\) is given by
$$\begin{aligned} {R_{{D_2},dir}} = E\left[ {\frac{1}{2}{{\log }_2}\left( {1 + \min \left( {{\gamma _{2,{D_1}}},\gamma _{{D_2}}^{MRC}} \right) } \right) } \right] . \end{aligned}$$
(37)
Theorem 6
From (37), the ergodic rate at \(D_{2}\) can be computed by (see(38))
$$\begin{aligned} R_{{D_2},dir}\!=\!\frac{1}{{2\ln 2}}\int \limits _0^{\frac{{{\alpha _2}}}{{{\alpha _1}}}} {\left[ {\frac{{{e^{\!-\!\frac{x}{{\psi _{I}\rho \left( {{\alpha _2} - {\alpha _1}x} \right) {\Omega _1}}}}}}}{{1\!+\!x}}} \right. } \left. {\!-\!\frac{{\int _0^\infty {\int _{\frac{x}{{\psi _{I}\rho \left( {{\alpha _2}\!-\!{\alpha _1}x} \right) }}}^\infty {\frac{1}{{{\Omega _1}{\Omega _0}}}\left( {1\!-\!{e^{\!-\!\frac{{x\left( {y{\alpha _1}\rho \!+\!1} \right) \!+\!y{\alpha _2}\rho }}{{z\rho \psi _{E}{\Omega _2}\left( {y{\alpha _1}\rho \!+\!1} \right) }}}}} \right) {e^{\!-\!\frac{y}{{{\Omega _0}}}\!-\!\frac{z}{{{\Omega _1}}}}}dydz} } }}{{1\!+\!x}}} \right] dx. \end{aligned}$$
(38)
Proof
See Appendix 4. \(\square\)
Remark 2
The ergodic rate in the asymptotic expression at \(D_{2}\) for high SNR region \(\rho \rightarrow \infty\) is given by
$$\begin{aligned} R_{{D_2},dir}^{\infty } = \frac{1}{{2\ln 2}}\int \limits _0^\infty {\frac{{1 - {F_X}\left( x \right) }}{{1 + x}}dx} \end{aligned}$$
(39)
From (39), this expression can be deployed by
$$\begin{aligned} R_{{D_2},dir}^{\infty } = \frac{1}{{2\ln 2}}\int \limits _0^{\frac{{{\alpha _2}}}{{{\alpha _1}}}} {\frac{{2\sqrt{\frac{x}{{{\psi _E}\rho \,{\Omega _1}{\Omega _2}}}} \,\!{K_1}\!\left( {2\sqrt{\frac{x}{{{\psi _E}\rho \,{\Omega _1}{\Omega _2}}}} } \right) }}{{1 + x}}dx} \end{aligned}$$
(40)
Proof
See Appendix 5. \(\square\)
Ergodic rate of the system for user relaying without direct link
The ergodic rate of system is determined by
$$\begin{aligned} \tau _{r,nodir} = R_{{D_1}} + R_{{D_2},nodir}, \end{aligned}$$
(41)
where \(R_{{D_1}}\) and \(R_{{D_2},nodir}\) can be obtained from (32) and (34), respectively.
Ergodic rate of the system for user relaying with direct link
The ergodic rate of system is thus expressed by
$$\begin{aligned} \tau _{r,dir} = R_{{D_1}} + R_{{D_2},dir}, \end{aligned}$$
(42)
where \(R_{{D_1}}\) and \(R_{{D_2},dir}\) can be obtained from (32) and (38), respectively.
Energy efficiency
The EE can be determined as the ratio of the total data rate over the total consumed power in entire network, which is given by \(\mathrm{{EE}} \buildrel \Delta \over = \frac{R}{{{P_S} + {P_r}}}\). The energy efficiency of user relaying systems can be given as
$$\begin{aligned} E{E_\phi } = \frac{{2\tau _\phi ^{HD}}}{{\rho \left( {1 + {\psi _E}{\Omega _1}} \right) }}, \end{aligned}$$
(43)
where \(\phi \in \left( {t,r} \right)\), denotes the system energy efficiency in DLT mode and DTT mode, respectively.