### DF scheme outage analysis

In this second phase, if *R* decides to forward the detected symbols, *X*_{
r
}, to *D* after regenerating, the received signal at *D* is

$$\begin{array}{@{}rcl@{}} y_{D2df}= \sqrt{P_{r}} h_{rd}X_{r} + n_{D} \end{array} $$

(9)

where *P*_{
r
}=(1−*ρ*)*P*_{
R
F
}+*E*_{max}, where *P*_{
R
F
} denotes the energy harvested by relay node through RF and *E*_{max} denotes the natural energy harvested by relay node from ambient environment (e.g., solar energy, wind energy).

Accordingly, the SNR of the received signal at *D* can be expressed as

$$\begin{array}{@{}rcl@{}} \gamma_{rd}= \frac{P_{r} |h_{rd}|^{2}}{N_{0}(1-\alpha d)^{2}} \end{array} $$

(10)

The probability density function (PDF) of |*h*_{
r
d
}|^{2} can be given as

$$ p(|h_{rd}|^{2})=\frac{{m_{rd}}^{m_{rd}}{\gamma_{rd}}^{m_{rd}-1}}{\bar{\gamma_{rd}}^{m_{rd}}\Gamma(m_{rd})}\exp \left(-\frac{m_{rd}}{\bar{\gamma_{rd}}}\gamma_{rd}\right) $$

(11)

where the average SNR of the received signal at destination can be written as

$$\begin{array}{@{}rcl@{}} \bar{\gamma_{rd}}=P_{r} \frac{E\left(|h_{rd}|^{2} \right)}{N_{0}\left(1-\alpha d\right)^{2}} =P_{r} \frac{E_{h3}}{N_{0}\left(1-\alpha d\right)^{2}} \end{array} $$

(12)

The outage probability of the consider system for the DF scheme can be expressed as

$$\begin{array}{@{}rcl@{}} P_{\text{out}}(\gamma_{th}) &= P{\left[\gamma_{sr}>\gamma_{th}\right]}P{\left[\gamma_{d}\leq \gamma_{th}|\gamma_{sr}>\gamma_{th}\right]} \\ & + P{\left[\gamma_{sr} \leq \gamma_{th}\right]}P{\left[\gamma_{D1}\leq \gamma_{th}\right]} \end{array} $$

(13)

The probability with the SNR of *S*-*R* link *γ*_{
s
r
} can be expressed as

$$ {{}\begin{aligned} P\left[\gamma_{sr}\leq \gamma_{th}\right] &=\int_{0}^{\gamma_{th}}\frac{{m_{sr}}^{m_{sr}}{\gamma_{sr}}^{m_{sr}-1}}{\bar{\gamma_{sr}}^{m_{sr}}\Gamma(m_{sr})}\exp \left(-\frac{m_{sr}}{\bar{\gamma_{sr}}}{\gamma_{sr}}\right)d\gamma_{sr}\\ &=1-\exp{\left(-\frac{m_{sr}}{\bar{\gamma_{sr}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sr}-1} \left(\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)^{n}\frac{1}{n!} \end{aligned}} $$

(14)

Similarly, the probability with the SNR of *S*-*D* link *γ*_{
s
d
} can be expressed as

$$ {{}\begin{aligned} P\left[\gamma_{sd}\leq \gamma_{th}\right] &=\int_{0}^{\gamma_{th}}\frac{{m_{sd}}^{m_{sd}}{\gamma_{sd}}^{m_{sd}-1}}{\bar{\gamma_{sd}}^{m_{sd}} \Gamma(m_{sd})}\exp \!\left(\! -\frac{m_{sd}}{\bar{\gamma_{sd}}}{\gamma_{sd}}\! \right)d\gamma_{sd} \\ &=1-\exp{\left(-\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sd}-1} \left(\frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}}\right)^{n}\frac{1}{n!} \end{aligned}} $$

(15)

The probability with the SNR of the combined signal at D, *γ*_{
D
}, can be expressed as

$$ \begin{aligned} P{\left[\gamma_{d}\leq \gamma_{th}|\gamma_{sr}>\gamma_{th}\right]}&=P{\left[\gamma_{D}\leq \gamma_{th}\right]}= P{\left[{\max}{\{\gamma_{D1},\gamma_{D2}\}}\leq \gamma_{th}\right]} \\ &=\left[1-\exp{\left(-\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sd}-1} \left(\frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}}\right)^{n}\frac{1}{n!} \right] \\ &\quad \times \left[\! 1-\exp{\left(\! -\frac{m_{rd}}{\bar{\gamma_{rd}}}\gamma_{th}\! \right)}\sum_{n=0}^{m_{rd}-1} \left(\frac{m_{rd}\gamma_{th}}{\bar{\gamma_{rd}}}\right)^{n}\frac{1}{n!}\right] \end{aligned} $$

(16)

Therefore, the outage probability can be obtained by substituting (14), (15), and (16) into (13) as

$$ {{}\begin{aligned} P_{\text{out}}^{(df)}(\gamma_{th})&=\exp{\left(-\frac{m_{sr}}{\bar{\gamma_{sr}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sr}-1} \left(\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)^{n}\frac{1}{n!}\\ &\quad \times\left[\! 1-\exp{\left(\! -\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\! \right)}\sum_{n=0}^{m_{sd}-1} \left(\!\frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}}\! \right)^{n}\frac{1}{n!}\! \right]\\ &\quad \times\left[\! 1-\exp{\left(\! -\frac{m_{rd}}{\bar{\gamma_{rd}}}\gamma_{th}\! \right)}\sum_{n=0}^{m_{rd}-1} \left(\! \frac{m_{rd}\gamma_{th}}{\bar{\gamma_{rd}}}\! \right)^{n}\frac{1}{n!}\! \right]\\ &\quad +\left[\! 1-\exp{\left(\! -\frac{m_{sr}}{\bar{\gamma_{sr}}}\gamma_{th} \!\right)}\sum_{n=0}^{m_{sr}-1} \left(\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)^{n}\frac{1}{n!}\right]\\ &\quad \times\left[\! 1-\exp{\left(\! -\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\! \right)}\sum_{n=0}^{m_{sd}-1} \left(\! \frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}} \!\right)^{n}\frac{1}{n!}\! \right] \end{aligned}} $$

(17)

### AF scheme outage analysis

In this second phase, if the node *R* decides to forward the detected symbols, *X*_{
r
}, to the node *D* after amplifying, the received signal at node *D* can be expressed as

$$\begin{array}{@{}rcl@{}} y_{D2af}&=&Gh_{rd}y_{R}+n_{D}\\ &=&Gh_{rd}\left[\sqrt\rho \left(\sqrt{P_{s}} h_{sr}X + n_{R} \right) + z_{R}\right]+n_{D} \end{array} $$

(18)

where \(G=\frac {\sqrt {P_{r}}}{\sqrt {P_{s}|h_{sr}|^{2}+N_{0}}}\) is the amplifying factor at *R*. Therefore, the SNR of the received signal at *D* can be expressed as

$$\begin{array}{@{}rcl@{}} \gamma_{rd2}&=&\frac{\frac{{P_{s}P_{r}}\rho|h_{rd}|^{2}|h_{sr}|^{2}}{{P_{s}|h_{sr}|^{2}+N_{0}}}} {\frac{P_{s}|h_{rd}|^{2}}{P_{s}|h_{sr}|^{2}+N_{0}}\left(\rho N_{0}+\sigma^{2}\right)+N_{0}}\\ &=&\frac{\frac{\rho P_{s}{|h_{sr}|^{2}}}{\rho N_{0}+\sigma^{2}}\frac{P_{r}|h_{rd}|^{2}}{N_{0}}}{\frac{1}{\rho}\frac{\rho P_{s}|h_{sr}|^{2}}{\rho N_{0}+\sigma^{2}}+\frac{P_{r}|h_{rd}|^{2}}{N_{0}}+\frac{N_{0}}{\rho N_{0}+\sigma^{2}}}\\ &=&\frac{\gamma_{sr}\gamma_{rd}}{a\gamma_{sr}+\gamma_{rd}+b}=\frac{\gamma_{1}\gamma_{2}}{a\gamma_{1}+\gamma_{2}+b} \end{array} $$

(19)

where \(a=\frac {1}{\rho }\), \(b=\frac {N_{0}}{\rho N_{0}+\sigma ^{2}}\), *γ*_{
s
r
}=*γ*_{1}, and *γ*_{
r
d
}=*γ*_{2}. Let \(P_{srd}^{(\text {out})}\) denotes the outage probability of *γ*_{rd2}, so it can be expressed as

$$ \begin{aligned} P_{srd}^{(\text{out})}&=P\left[\gamma_{rd2}\leq\gamma_{th}\right]=\int_{0}^{\gamma_{th}} P\left[\gamma_{2}\geq\frac{(ax+b)\gamma_{th}}{x-\gamma_{th}}|\gamma_{1}\right]\cdot {f}_{\gamma_{1}}(x)dx \\ & \quad+\int_{\gamma_{th}}^{\infty} P\left[\gamma_{2}\leq\frac{(ax+b)\gamma_{th}}{x-\gamma_{th}}|\gamma_{1}\right]\cdot {f}_{\gamma_{1}}(x)dx \\ &=1-\frac{2(m_{rd}-1)!m_{sr}^{m_{sr}}}{\bar{\gamma_{sr}}^{m_{sr}}\Gamma(m_{sr})\Gamma(m_{rd})} \exp{\left(-\frac{am_{rd}\gamma_{th}}{\bar{\gamma_{rd}}}-\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)} \\ &\quad \times\sum_{n=0}^{m_{sr}-1}\sum_{k=0}^{m_{rd}-1}\sum_{l=0}^{k}\left(\begin{array}{c} k\\ l \end{array}\right)\left(\begin{array}{l} {m_{sr}-1}\\ n \end{array}\right)a^{u}\left(a\gamma_{th}+b\right)^{v} \left(\frac{m_{sr}}{\bar{\gamma_{sr}}}\right)^{\frac{w}{2}}\\ &\quad \times\left(\frac{m_{rd}}{\bar{\gamma_{rd}}}\right)^{p}K_{-w}\left(2\sqrt{\frac{m_{sr}m_{rd}\gamma_{th}\left(a\gamma_{th}+b\right)}{\bar{\gamma_{sr}} \bar{\gamma_{rd}}}}\right)\gamma_{th}^{q}\frac{1}{k!} \end{aligned} $$

(20)

where *u*=*k*−*l*, \(v=\frac {n+l+1}{2}\), *w*=*l*−*n*−1, \(p=\frac {2k+n-l+1}{2}\), \(q=\frac {2k+2m_{sr}-n-l-1}{2}\).

Therefore, the outage probability of system can be obtained by substituting (14), (15), and (20) into (13) as

$$ \begin{aligned} P_{\text{out}}^{(af)}\left(\gamma_{th}\right)&=\exp{\left(-\frac{m_{sr}}{\bar{\gamma_{sr}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sr}-1} \left(\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)^{n}\frac{1}{n!}\\ &\times\left[1-\exp{\left(-\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sd}-1} \left(\frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}}\right)^{n}\frac{1}{n!}\right]\\ &\times\left[1-\frac{2\left(m_{rd}-1\right)!m_{sr}^{m_{sr}}}{\bar{\gamma_{sr}}^{m_{sr}}\Gamma(m_{sr}) \Gamma(m_{rd})}\exp{\left(-\frac{am_{rd}\gamma_{th}}{\bar{\gamma_{rd}}}-\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)}\right.\\ &\times\sum_{n=0}^{m_{sr}-1}\sum_{k=0}^{m_{rd}-1}\sum_{l=0}^{k}\left(\begin{array}{l} k\\ l \end{array}\right)\left(\begin{array}{l} {m_{sr}-1}\\ {n} \end{array}\right)a^{u}\left(a\gamma_{th}+b\right)^{v} \left(\frac{m_{sr}}{\bar{\gamma_{sr}}}\right)^{\frac{w}{2}}\\ &\left.\times\left(\frac{m_{rd}}{\bar{\gamma_{rd}}}\right)^{p}K_{-w}\left(2\sqrt{\frac{m_{sr}m_{rd}\gamma_{th}\left(a\gamma_{th}+b\right)} {\bar{\gamma_{sr}}\bar{\gamma_{rd}}}}\right)\gamma_{th}^{q}\frac{1}{k!}\right]\\ &+\left[1-\exp{\left(-\frac{m_{sr}}{\bar{\gamma_{sr}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sr}-1} \left(\frac{m_{sr}\gamma_{th}}{\bar{\gamma_{sr}}}\right)^{n}\frac{1}{n!}\right]\\ &\times\left[1-\exp{\left(-\frac{m_{sd}}{\bar{\gamma_{sd}}}\gamma_{th}\right)}\sum_{n=0}^{m_{sd}-1} \left(\frac{m_{sd}\gamma_{th}}{\bar{\gamma_{sd}}}\right)^{n}\frac{1}{n!}\right] \end{aligned} $$

(21)