Outage probability of dual-hop cooperative communication networks over the Nakagami-m fading channel with RF energy harvesting

Abstract

Cooperative communication systems with wireless power transfer are being investigated for future advanced wireless networks. In this research, we propose applying a wireless power transfer to dual-hop cooperative communication systems, in which a relay node harvests the energy from the radiofrequency in order to forward the received signal, and a source node can communicate with a destination node directly or through the selected relay nodes. The system performance is evaluated by an outage probability that is calculated over independent and identically distributed (i.i.d) Nakagami-m distributions in two scenarios, i.e., integer m and arbitrary m. Furthermore, the closed forms of the outage probability expressions are derived in the case of both the amplify-and-forward (AF) and decode-and-forward (DF) protocols. The Monte Carlo method is utilized to simulate the system with the aims of evaluating the system performance and verifying the theoretical analysis. The numerical results highlight that the proposed calculation method and the closed form of the outage probability are accurate. We also compare the system performance of both the AF and DF protocols and show the effect of parameter m on the performance of the system.

Appendix: The proof of term \({\mathbb J}(a,b) \)

Combining (8), (9), and (11), function ${\mathbb J}(a,b)$ can be rewritten as:

$$ \begin{array}{@{}rcl@{}} {\mathbb J}(a,b) &=& 1 - \Pr \left( {{\gamma_{{\text{SR}}}} > {\gamma_{{\text{th}}}}, {\gamma_{{\text{RD}}}} > {\gamma_{{\text{th}}}}} \right)\\ & =& 1 - \Pr \left( {X > a, XY > b } \right), \end{array} $$

(41)

where ${\gamma _{{\text {th}}}} = {2^{\frac {{2{\mathcal R}}}{{1 - \alpha }}}} - 1$; $a = \frac {{{\gamma _{\text {{th}}}}}}{{{P_{\text {S}}}}}, {and} b = \frac {{\left ({1 - \alpha } \right ){\gamma _{\text {{th}}}}}}{{2\alpha \eta {P_{\text {S}}}}}$.

The expansion of the incomplete gamma function by a series development is applied for $\gamma \left ({m,z} \right )$ [25, Eq. 5].

$$ \begin{array}{@{}rcl@{}} \gamma \left( {m,z} \right) = {e^{- z}}\sum\limits_{k = 0}^{\infty} {\frac{{\Gamma \left( m \right){z^{m + k}}}}{{\Gamma \left( {m + k + 1} \right)}}}. \end{array} $$

(42)

From Eq. 42, it is clear that when the Nakagami-m fading parameter is not an integer, the CDF can be expressed as a single infinity series of incomplete gamma functions.

Substituting (42) into (18) and after some algebraic manipulations, we obtain the PDF of the instantaneous SNR over the first hop.

$$ \begin{array}{@{}rcl@{}} {f_{X}}\left( x \right) &=& {\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)^{N{m_{1}}}}\frac{{N{x^{N{m_{1}}\! - \!1}}}}{{\Gamma \left( {{m_{1}}} \right)}}\exp \left( {\! -\! \frac{{N{m_{1}}x}}{{{\lambda_{1}}}}} \right)\\ &&\times{\left[ {\sum\limits_{k = 0}^{\infty} {{{\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)}^{k}}\frac{{{x^{k}}}}{{\Gamma \left( {{m_{1}} \!+ \!k + 1} \right)}}} } \right]^{N\! -\! 1}}. \end{array} $$

(43)

Based on [17, 0.314], Eq. 43 is rewritten as:

$$ \begin{array}{@{}rcl@{}} {f_{X}}\left( x \right) = \sum\limits_{k = 0}^{\infty} {{c_{k}}} {\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)^{N{m_{1}}}}\frac{{N{x^{N{m_{1}} + k - 1}}}}{{\Gamma \left( {{m_{1}}} \right)}}\exp \left( { - \frac{{N{m_{1}}x}}{{{\lambda_{1}}}}} \right), \end{array} $$

(44)

where c_k is defined as

$$ \begin{array}{@{}rcl@{}} &&{c_{0}} = {\left[ {\frac{1}{{\Gamma \left( {{m_{1}} + 1} \right)}}} \right]^{N - 1}}, {\text{for}} k = 0, \end{array} $$

(45a)

$$ \begin{array}{@{}rcl@{}} &&{c_{k}} = \sum\limits_{\ell = 1}^{k} {\frac{{\Gamma \left( {{m_{1}} + 1} \right)}}{k}{{\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)}^{\ell} }} \frac{{\left( {\ell N - k} \right){c_{k - \ell }}}}{{\Gamma \left( {{m_{1}} + \ell + 1} \right)}},{\text{for}} k \ge 1. \end{array} $$

(45b)

The CDF and PDF of the individual links are already obtained by Eqs. 2 and 44; consequently, the CDF of link S-R-D can be derived as:

$$ \begin{array}{@{}rcl@{}} \mathbb{J}\left( {a,b} \right)& = &1 - \int\limits_{a}^{\infty} {\left[ {1 - {F_{Y}}\left( {\frac{b}{x}} \right)} \right]} {f_{X}}\left( x \right)dx\\ &= &1\! - \!\left[ {\int\limits_{a}^{\infty} {{f_{X}}\left( x \right)dx - \int\limits_{a}^{\infty} {{F_{Y}}\left( {\frac{b}{x}} \right){f_{X}}\left( x \right)dx} } } \right], \end{array} $$

(46)

where $b = \frac {{\left ({1 - \alpha } \right ){\gamma _{{\text {th}}}}}}{{2\alpha \eta {P_{\text {S}}}}} {\text {and}} a = \frac {{{\gamma _{{\text {th}}}}}}{{{P_{\text {S}}}}}$.

With high transmission power, parameter a can be approximated as $a = \frac {{{\gamma _{{\text {th}}}}}}{{{P_{\text {S}}} \to \infty }} \approx 0$. Hence, $\mathbb {J}\left ({a,b} \right )$ is changed.

$$ \begin{array}{@{}rcl@{}} \mathbb{J}\left( {a,b} \right) &\le& \sum\limits_{k = 0}^{{{\mathcal N}_{t}}}{\sum\limits_{j = 0}^{{{\mathcal N}_{t}}}{{{\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)}^{N{m_{1}}}}{{\left( {\frac{{{m_{2}}b}}{{{\lambda_{2}}}}} \right)}^{{m_{2}} + j}}\frac{{{c_{k}}}}{{\Gamma \left( {{m_{2}} + j + 1} \right)}}} }\\ &\times& \frac{N}{{\Gamma \left( {{m_{1}}} \right)}}\int\limits_{0}^{\infty} {{x^{v}}\exp \left( { - \frac{{{m_{2}}b}}{{{\lambda_{2}}x}} - \frac{{N{m_{1}}x}}{{{\lambda_{1}}}}} \right)dx}. \end{array} $$

(47)

with v = Nm₁ + k − m₂ − j − 1.

Applying [17, Eq:3.471.9], we have

$$ \begin{array}{@{}rcl@{}} &&\int\limits_{0}^{\infty} {{x^{v}}\exp \left( { - \frac{{{m_{2}}b}}{{{\lambda_{2}}x}} - \frac{{N{m_{1}}x}}{{{\lambda_{1}}}}} \right)dx}=\\ && 2{\left( {\frac{{{m_{2}}b{\lambda_{1}}}}{{N{m_{1}}{\lambda_{2}}}}} \right)^{\frac{{N{m_{1}} + k-{m_{2}}-j}}{2}}}{\mathcal{K}_{N{m_{1}} + k-{m_{2}}-j}}\left( {2\sqrt {\frac{{N{m_{1}}{m_{2}}b}}{{{\lambda_{1}}{\lambda_{2}}}}} } \right), \end{array} $$

(48)

and then the approximation of $\mathbb {J}\left ({a,b} \right )$ in Eq. 47 is changed, as in Eq. 27.

To analyze the exact outage probability expression, Eqs. 2 and 44 are substituted into Eq. 46, and we obtain function $\mathbb {J}\left ({a,b} \right )$.

$$ \begin{array}{@{}rcl@{}} \mathbb{J}(a,b) &=& \underbrace {1\! -\! \sum\limits_{k = 0}\!^{\infty}\left( \frac{m_{1}}{\lambda_{1}} \right)^{Nm_{1}} \frac{c_{k}N}{\Gamma (m_{1})}\int\limits_{a}^{\infty} x^{Nm_{1} + k\! -\! 1}\exp\left( - \frac{Nm_{1}x}{\lambda_{1}} \right)dx}_{\mathbb{J}_{1}} \\ &&+\underbrace {\sum\limits_{k = 0}^{\infty} {{{\left( {\frac{{{m_{1}}}}{{{\lambda_{1}}}}} \right)}^{N{m_{1}}}}} \frac{{N{c_{k}}}}{{\Gamma \left( {{m_{1}}} \right)}}\int\limits_{a}^{\infty} {{F_{Y}}\left( {\frac{b}{x}} \right){x^{N{m_{1}} + k \!- \!1}}\exp \left( { - \frac{{N{m_{1}}x}}{{{\lambda_{1}}}}} \right)dx} }_{{\mathbb{J}_{2}}}.\\ \end{array} $$

(49)

To calculate $\mathbb {J}_{1}$, [17, 3.351.2] can be used, while to calculate $\mathbb {J}_{2}$, the variable x is replaced with x = au, and then we obtain (24). Finally, we consider the system model in the two cases of index power v ≥ 0andv < 0, in which [17, 3.351.2] is used for the case v ≥ 0 and the exponential integral function is used for the case v < 0. We completely prove the proposition 1 here.