Secure communication with energy harvesting multiple half-duplex DF relays assisted with jamming

Abstract

Physical layer security in relay based cooperative networks is a promising approach to maintain confidentiality of information. In this paper, secrecy performance of a dual hop network, with multiple energy harvesting decode and forward (DF) relays, has been analyzed where an eavesdropper also receives the information from the transmission of a selected relay. A selected DF relay harvests energy based on power splitting (PS) scheme in the first time slot. In the second time slot, the selected DF relay transmits the information as well as jamming signals, only when it has harvested sufficient power as decided by a threshold, based on the outage constraint of the network. The secrecy outage probability (SOP) under some assumed conditions, termed as a conditional SOP (CSOP), has been evaluated in closed form. Performance regarding the CSOP increases with increase in transmits power of the source, threshold outage rate, number of relay and energy conversion efficiency, whereas it decreases with an increase in threshold secrecy rate. We indicate the optimal value of PS factor for harvesting energy and the optimal value of a fraction of harvested energy devoted to information signal transmission at which the CSOP is minimum. The SOP without any condition has also been evaluated with and without jamming. The results shows that the CSOP yields better results than the SOP. It is observed that the CSOP with jamming is better than the SOP with jamming by 49.12% for 1 bits/s/Hz threshold secrecy rate and by 81.94% for a threshold secrecy rate of 0.5 bits/s/Hz, respectively at 10 dBW of the source transmits power. A MATLAB based simulation is used to verify our analytical works of the CSOP and the SOP with jamming.

Appendix

The $I^{'}$ from Eq. (24) is given as:

$$\begin{aligned} I^{'}=\frac{1}{{{\varOmega _{\mathcal {RD}}}}}\int \limits _0^{\frac{{\left( {1 - \beta } \right) }}{{\alpha \eta \beta }}} {\exp \left\{ { - \left( {\frac{{A_1{N_0}}}{{\alpha \eta \beta {P_{\mathcal {S}}}{\varOmega _\mathcal {SR}}y}} + \frac{y}{{{\varOmega _{\mathcal {RD}}}}}} \right) } \right\} dy}. \end{aligned}$$

(31)

For simplicity in expressing the expression, we assume the some constant as: $A_1=2^{2R_{TH}^{SEC}} \left( 1+\gamma _{\mathcal {E}_i}\right) -1$, $A_2=\varOmega _{\mathcal {RD}}$, $A_3=\frac{{\left( {1 - \beta } \right) }}{{\alpha \eta \beta }}$ and $A_4=\frac{{A_1{N_0}}}{{\alpha \eta \beta {P_{\mathcal{S}}}{\varOmega _{{\mathcal{S}}{\mathcal{R}}}}}}$. The Eq. (31) can be re-expressed under assumed constant as:

$$\begin{aligned} I^{'}=\frac{1}{A_2}\int \limits _0^{A_3} {\exp \left\{ { - \left( {\frac{A_4}{y} + \frac{y}{A_2}} \right) } \right\} dy}. \end{aligned}$$

(32)

Split the Eq. (32) in two parts to make suitable for the solution and expressed as:

$$\begin{aligned} {I^{'}} = \underbrace{\frac{1}{{{A_2}}}\int \limits _0^\infty {\exp \left\{ { - \left( {\frac{{{A_4}}}{y} + \frac{y}{{{A_2}}}} \right) } \right\} dy} }_{I_1^{'}} - \underbrace{\frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\exp \left\{ { - \left( {\frac{{{A_4}}}{y} + \frac{y}{{{A_2}}}} \right) } \right\} dy} }_{I_2^{'}}. \end{aligned}$$

(33)

With the help of [31], eq. 3.324.1], the $I_1^{'}$ can be expressed as:

$$\begin{aligned} \begin{array}{c} I_1^{'} = \frac{1}{{{A_2}}}\int \limits _0^\infty {\exp \left\{ { - \left( {\frac{{{A_4}}}{y} + \frac{y}{{{A_2}}}} \right) } \right\} dy} = \sqrt{\frac{{4{A_4}}}{{{A_2}}}} {K_1}\left( {\sqrt{\frac{{4{A_4}}}{{{A_2}}}} } \right) \end{array} \end{aligned}$$

(34)

where $K_1 (.)$ is the modified Bessel function of first order second kind.

Now, $I_2^{'}$ from the Eq. (33) can be expressed as:

$$\begin{aligned} I_2^{'} = \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\exp \left\{ { - \left( {\frac{{{A_4}}}{y} + \frac{y}{{{A_2}}}} \right) } \right\} dy} \end{aligned}$$

(35)

The $I_2^{'}$ can be re-expressed with the help of [32], eq. 46] as:

$$\begin{aligned} I_2^{'}&= \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\exp \left( { - \frac{{{A_4}}}{y}} \right) \exp \left( { - \frac{y}{{{A_2}}}} \right) dy} = \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\sum \limits _{\nu = 0}^\infty {\frac{1}{{\nu !}}{{\left( { - \frac{{{A_4}}}{y}} \right) }^\nu }} \exp \left( { - \frac{y}{{{A_2}}}} \right) dy}\nonumber \\&= \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\left\{ {1 - \frac{{{A_4}}}{y} + \sum \limits _{\nu = 2}^\infty {\frac{1}{{\nu !}}{{\left( { - \frac{{{A_4}}}{y}} \right) }^\nu }} } \right\} \exp \left( { - \frac{y}{{{A_2}}}} \right) dy} \nonumber \\&= \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\exp \left( { - \frac{y}{{{A_2}}}} \right) dy} - \frac{{{A_4}}}{{{A_2}}}\int \limits _{{A_3}}^\infty {\frac{{\exp \left( { - \frac{y}{{{A_2}}}} \right) }}{y}} dy + \frac{1}{{{A_2}}}\int \limits _{{A_3}}^\infty {\sum \limits _{\nu = 2}^\infty {\frac{1}{{\nu !}}{{\left( { - \frac{{{A_4}}}{y}} \right) }^\nu }} \exp \left( { - \frac{y}{{{A_2}}}} \right) dy} \nonumber \\&= \exp \left( { - \frac{{{A_3}}}{{{A_2}}}} \right) +\frac{{{A_4}}}{{{A_2}}} Ei\left[ { - \frac{{{A_3}}}{{{A_2}}}} \right] + \frac{1}{{{A_2}}}\sum \limits _{\nu = 2}^\infty {\frac{{{{\left( { - {A_4}} \right) }^\nu }}}{{\nu !}}\underbrace{\int \limits _{{A_3}}^\infty {\frac{{\exp \left( { - \frac{y}{{{A_2}}}} \right) }}{{{y^\nu }}}dy} }_{I_2^{''}}} \end{aligned}$$

(36)

where Ei(.) is the exponential integral function, $\sum \limits _{Initial\,Value}^{{Final\,Value}} (.)$ indicates the sum of the series. With the help of [31], eq. 3.351.4], we can express the $I_2^{''}$ in closed form as:

$$\begin{aligned} \begin{array}{ll} I_2^{''} &{}= \int \limits _{{A_3}}^\infty {\frac{{\exp \left( { - \frac{y}{{{A_2}}}} \right) }}{{{y^\nu }}}dy} \\ &{}= {\left( { - 1} \right) ^\nu }\frac{{{{\left( {\frac{1}{{{A_2}}}} \right) }^{\nu - 1}}Ei\left[ { - \frac{{{A_3}}}{{{A_2}}}} \right] }}{{\left( {\nu - 1} \right) !}} + \frac{{\exp \left( { - \frac{{{A_3}}}{{{A_2}}}} \right) }}{{{A_3}^{\left( {\nu - 1} \right) }}}\sum \limits _{\lambda = 0}^{\left( {\nu - 2} \right) } {\frac{{{{\left( { - \frac{{{A_3}}}{{{A_2}}}} \right) }^\lambda }}}{{\left( {\nu - 1} \right) \left( {\nu - 2} \right) \ldots \left( {\nu - \lambda } \right) }}} \end{array} \end{aligned}$$

(37)

As per Eq. (33), $I^{'}$ can be expressed in closed form with the help of equation from (34) to (37) as:

$$\begin{aligned} {I^{'}}&= \sqrt{\frac{{4{A_4}}}{{{A_2}}}} {K_1}\left( {\sqrt{\frac{{4{A_4}}}{{{A_2}}}} } \right) \nonumber \\&- \left[ {\begin{array}{l} {\exp \left( { - \frac{{{A_3}}}{{{A_2}}}} \right) + \frac{{{A_4}}}{{{A_2}}}Ei\left[ { - \frac{{{A_3}}}{{{A_2}}}} \right] }\\ { + \frac{1}{{{A_2}}}\sum \limits _{\nu = 2}^\infty {\frac{{{{\left( { - {A_4}} \right) }^\nu }}}{{\nu !}}\left[ {\begin{array}{l} {{{\left( { - 1} \right) }^\nu }\frac{{{{\left( {\frac{1}{{{A_2}}}} \right) }^{\nu - 1}}Ei\left[ { - \frac{{{A_3}}}{{{A_2}}}} \right] }}{{\left( {\nu - 1} \right) !}} + \frac{{\exp \left( { - \frac{{{A_3}}}{{{A_2}}}} \right) }}{{{A_3}^{\left( {\nu - 1} \right) }}}}\\ {\sum \limits _{\lambda = 0}^{\left( {\nu - 2} \right) } {\frac{{{{\left( { - \frac{{{A_3}}}{{{A_2}}}} \right) }^\lambda }}}{{\left( {\nu - 1} \right) \left( {\nu - 2} \right) \ldots \left( {\nu - \lambda } \right) }}} } \end{array}} \right] } } \end{array}} \right] , \end{aligned}$$

(38)

This is the closed form expression of $I^{'}$ which is used in Eq. (25).