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
+Emax, where P
R
F
denotes the energy harvested by relay node through RF and Emax 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
=γ1, and γ
r
d
=γ2. Let \(P_{srd}^{(\text {out})}\) denotes the outage probability of γrd2, 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)