Joint effects of non-linear energy harvesting, primary interference, and full-duplex destination-assisted jamming on spectrum sharing networks

Abstract

Spectrum sharing (SS) and energy harvesting (EH) basically improve spectrum-and-energy efficiencies required by modern communications systems. Nonetheless, most publications have focused on the linear (L) EH model which did not characterize practically non-linearities of EH circuits. Moreover, SS induces secondary receivers to suffer primary interference (PI) and facilitates eavesdroppers to overhear legitimate data. To secure communications in energy harvesting spectrum sharing networks (EHSSNs), this paper proposes a full-duplex (FD) secondary destination which not only receives legitimate signal but also jams signal reception of the eavesdropper. Joint effects of non-linear (NL) EH, PI, and FD destination-assisted jamming on security-and-reliability performances of EHSSNs are first analyzed in terms of outage and intercept probabilities that are then endorsed by Monte-Carlo simulations. Next, the proposed analysis is exploited to achieve the asymptotic analysis and the security-and-reliability analysis for two baseline EHSSNs (half-duplex (HD) destination and L EH). Finally, numerous results demonstrate joint effects of NL EH, PI, and FD destination on data security and communication reliability by comparing three EHSSNs: (i) L EH and FD destination; (ii) NL EH and FD destination; (iii) NL EH and HD destination. Notably, the proposed communications scheme with the FD destination is more secure than the baseline scheme with the HD destination. In addition, the reliability of the communications scheme with the NL EH is inferior to that with the L EH but the data security of the former is better than the latter. Additionally, the security-reliability trade-off exists for all the considered communications schemes. Moreover, the PI enhances the security but degrades the reliability. Furthermore, the time fraction for EH can be optimized to attain the best security-reliability trade-off.

Appendix A: Proof for the PDF of \(P_s\) in (28)

To obtain the PDF of $P_s$, one firstly needs the CDF of $P_s$. Plugging (10) into the definition of the CDF results in

$$\begin{aligned} {F_{{P_s}}}\left( x \right)= & {} \mathbb {P} \left\{ {\min \left( {{{\hat{P}}_s},\frac{{{Q}}}{{{g_{sp}}}}} \right) \le x} \right\} \nonumber \\= & {} 1 - \mathbb {P} \left\{ {\min \left( {{{\hat{P}}_s},\frac{{{Q}}}{{{g_{sp}}}}} \right)> x} \right\} \nonumber \\= & {} 1 - \underbrace{\mathbb {P} \left\{ {{{\hat{P}}_s}> x} \right\} }_{{{\mathcal{P}}_{1}}}\underbrace{\mathbb {P} \left\{ {\frac{{{Q}}}{{{g_{sp}}}} > x} \right\} }_{{{\mathcal{P}}_2}}. \end{aligned}$$

(51)

The $\mathcal{P}_2$ term can be rewritten in terms of the CDF of $g_{ps}$ as

$$\begin{aligned} {{\mathcal{P}}_2} = \mathbb {P} \left\{ {{g_{sp}} < \frac{{{Q}}}{x}} \right\} = 1 - {e^{ - \frac{{{Q}}}{{{\lambda _{sp}}x}}}}. \end{aligned}$$

(52)

Given $\hat{P}_s$ in (1), the $\mathcal{P}_1$ term is further decomposed as

$$\begin{aligned} {{\mathcal{P}}_1}= & {} \mathbb {P} \left\{ {0> x,{P_p}{g_{ps}} \le {P_l}} \right\} \nonumber \\&+\, \mathbb {P} \left\{ {{A_s}{P_p}{g_{ps}}> x,{P_l}< {P_p}{g_{ps}} \le {P_u}} \right\} \nonumber \\&+\, \mathbb {P} \left\{ {{A_s}{P_u}> x,{P_p}{g_{ps}}> {P_u}} \right\} \nonumber \\= & {} \mathbb {P} \left\{ {{g_{ps}}> \frac{x}{{{A_s}{P_p}}},\frac{{{P_l}}}{{{P_p}}} < {g_{ps}} \le \frac{{{P_u}}}{{{P_p}}}} \right\} \nonumber \\&+\, \mathbb {P} \left\{ {{A_s}{P_u}> x} \right\} \mathbb {P} \left\{ {{g_{ps}} > \frac{{{P_u}}}{{{P_p}}}} \right\} . \end{aligned}$$

(53)

To complete the derivation of (53), three cases should be considered as follows:

Case 1: $x \le {A_s}{P_l}$

This case simplifies (53) as

$$\begin{aligned} {{\mathcal{P}}_1}= & {} \mathbb {P} \left\{ {\frac{{{P_l}}}{{{P_p}}}< {g_{ps}} \le \frac{{{P_u}}}{{{P_p}}}} \right\} + \mathbb {P} \left\{ {{g_{ps}} > \frac{{{P_u}}}{{{P_p}}}} \right\} \nonumber \\= & {} \mathbb {P} \left\{ {\frac{{{P_l}}}{{{P_p}}} < {g_{ps}}} \right\} \nonumber \\= & {} {e^{ - \frac{{{P_l}}}{{{P_p}{\lambda _{ps}}}}}}. \end{aligned}$$

(54)

Case 2: ${A_s}{P_l} < x \le {A_s}{P_u}$

This case simplifies (53) as

$$\begin{aligned} {{\mathcal{P}}_1}= & {} \mathbb {P} \left\{ {\frac{x}{{{A_s}{P_p}}}< {g_{ps}} \le \frac{{{P_u}}}{{{P_p}}}} \right\} + \mathbb {P} \left\{ {{g_{ps}} > \frac{{{P_u}}}{{{P_p}}}} \right\} \nonumber \\= & {} \mathbb {P} \left\{ {\frac{x}{{{A_s}{P_p}}} < {g_{ps}}} \right\} \nonumber \\= & {} {e^{ - \frac{x}{{{A_s}{P_p}{\lambda _{ps}}}}}}. \end{aligned}$$

(55)

Case 3: $x > {A_s}{P_u}$

This case simplifies (53) as

$$\begin{aligned} {{\mathcal{P}}_1} = 0. \end{aligned}$$

(56)

Plugging ${{\mathcal{P}}_1}$ and ${{\mathcal{P}}_2}$ into (51), one obtains the CDF of $P_s$ as

$$\begin{aligned} {F_{{P_s}}}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1 - {e^{ - \frac{{{P_l}}}{{{P_p}{\lambda _{ps}}}}}}\left( {1 - {e^{ - \frac{{{Q}}}{{{\lambda _{sp}}x}}}}} \right) }&{}{,x \le {A_s}{P_l}}\\ {1 - {e^{ - \frac{x}{{{A_s}{P_p}{\lambda _{ps}}}}}}\left( {1 - {e^{ - \frac{{{Q}}}{{{\lambda _{sp}}x}}}}} \right) }&{}{,{A_s}{P_l} < x \le {A_s}{P_u}}\\ 1&{}{,x > {A_s}{P_u}} \end{array}} \right. \end{aligned}$$

(57)

The derivative of ${F_{P_s}}\left( x \right)$, which is simplified as (28), is the PDF of $P_s$, completing the proof.