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Energy harvesting cognitive radio networks: security analysis for Nakagami-m fading

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Abstract

Energy harvesting has lately been of particular attention to researchers. In addition, cognitive radio networks (CRNs) are recognized as an attainable measure for the problem of radio spectrum shortage in next generation radio access. A combination of these two technologies, which forms energy harvesting CRNs (EHCRNs), allows wireless communication terminals to prolong their operation time in limited spectrum scenarios. Nonetheless, that CRNs create opportunities for secondary users to access primary users’ spectrum induces vulnerability of message security. So far, security capability analysis of EHCRNs has been limited to Rayleigh fading whilst Nakagami-m fading is more common than Rayleigh fading and better reflects distinct fading severity degrees in practical scenarios. Accordingly, this paper firstly offers the precise security capability analysis of EHCRNs under interference power constraint, Nakagami-m fading, maximum transmit power constraint, and primary interference. Then, the offered analysis is ratified by computer simulations. Ultimately, multiple results reveal that the security capability is considerably improved with smaller primary interference and lower required security threshold. Moreover, the security capability is significantly impacted by channel severity and is optimized with appropriate selection of time percentage.

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Acknowledgements

This paper was funded by the scientific research fund of Thu Dau Mot University through a scientific topic called Physical Performance Information Security Analysis in Cognitive Radio Network.

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Authors and Affiliations

  1. Thu Dau Mot University, Thu Dau Mot City, Vietnam

    Thiem Do-Dac

  2. Ho Chi Minh City University of Technology, VNU-HCM, Ho Chi Minh City, Vietnam

    Khuong Ho-Van

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  1. Thiem Do-Dac
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  2. Khuong Ho-Van
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Correspondence to Khuong Ho-Van.

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Appendices

Appendix 1: Proof of Theorem 1

Rewrite \(\varUpsilon \left( {{P_s}} \right) \) in (18) as

$$\begin{aligned} \varUpsilon \left( {{P_s}} \right) = \int \limits _0^\infty {\int \limits _0^{{2^{{C_0}}}\left( {1 + y} \right) - 1} {{f_{{\varPsi _d},{\varPsi _e}}}\left( {\left. {x,y} \right| {P_s}} \right) dxdy} }. \end{aligned}$$
(45)

Conditioned on \(P_s\), \(\varPsi _d\) and \(\varPsi _e\) are statistically independent. Accordingly, the jointly conditional PDF of \(\varPsi _d\) and \(\varPsi _e\), \({{f_{{\varPsi _d},{\varPsi _e}}}\left( {\left. {x,y} \right| {P_s}} \right) }\), can be rewritten as a product of marginal PDFs: \({f_{{\varPsi _d},{\varPsi _e}}}\left( {\left. {x,y} \right| {P_s}} \right) = {f_{{\varPsi _d}}}\left( {\left. x \right| {P_s}} \right) {f_{{\varPsi _e}}}\left( {\left. y \right| {P_s}} \right). \) Inserting this result into (45), one has

$$\begin{aligned} \begin{aligned} \varUpsilon \left( {{P_s}} \right)&= \int \limits _0^\infty {\left[ {\int \limits _0^{{2^{{C_0}}}\left( {1 + y} \right) - 1} {{f_{{\varPsi _d}}}\left( {\left. x \right| {P_s}} \right) dx} } \right] } {f_{{\varPsi _e}}}\left( {\left. y \right| {P_s}} \right) dy \\&= \int _0^\infty {{F_{{\varPsi _d}}}\left( {\left. {{2^{{C_0}}}\left[ {1 + y} \right] - 1} \right| {P_s}} \right) } {f_{{\varPsi _e}}}\left( {\left. y \right| {P_s}} \right) dy. \end{aligned} \end{aligned}$$
(46)

To numerically evaluate (46), one needs to obtain two expressions of \({F_{{\varPsi _d}}}\left( {\left. \rho \right| {P_s}} \right) \) and \({f_{{\varPsi _e}}}\left( {\left. y \right| {P_s}} \right) \). In the sequel, they are derived.

A. The expression of \({F_{{\varPsi _d}}}\left( {\left. \rho \right| {P_s}} \right) \)

Conditioned on \({P_s}\), the CDF of \(\varPsi _d\) is derived by using its explicit form in (11) as

$$\begin{aligned} \begin{aligned} {F_{{\varPsi _d}}}\left\{ {\left. \rho \right| {P_s}} \right\}&= \Pr \left\{ {\left. {{\varPsi _d}< \rho } \right| {P_s}} \right\} \\&= \Pr \left\{ {\left. {\frac{{{P_s}{g_3}}}{{{P_p}{g_6} + {N_0}}} < \rho } \right| {P_s}} \right\} \\&= \int \limits _0^\infty {{F_{{g_3}}}\left( {\frac{{\left[ {{P_p}x + {N_0}} \right] \rho }}{{{P_s}}}} \right) } {f_{{g_6}}}\left( x \right) dx. \end{aligned} \end{aligned}$$
(47)

Using (1) for \({f_{{g_6}}}\left( \cdot \right) \) and (3) for \({F_{{g_3}}}\left( \cdot \right) \), one rewrites (47) as

$$\begin{aligned} \begin{aligned}&{F_{{\varPsi _d}}}\left( {\left. \rho \right| {P_s}} \right) = \int \limits _0^\infty \left( {1 - \sum \limits _{t = 0}^{{m_3} - 1} {\frac{{{{\left[ {\left( {{P_p}x + {N_0}} \right) {\beta _3}\rho /{P_s}} \right] }^t}}}{{t!{e^{\left( {{P_p}x + {N_0}} \right) {\beta _3}\rho /{P_s}}}}}} } \right) \\&\quad \frac{{\beta _6^{{m_6}}{x^{{m_6} - 1}}}}{{{e^{{\beta _6}x}}\varGamma \left( {{m_6}} \right) }}dx \\&\qquad = 1 - \frac{{{e^{ - {N_0}{\beta _3}\rho /{P_s}}}}}{{\beta _6^{ - {m_6}}\varGamma \left( {{m_6}} \right) }}\sum \limits _{t = 0}^{{m_3} - 1} {\frac{1}{{t!}}{{\left( {\frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) }^t}} \int \limits _0^\infty {{\left( {x + \frac{{{N_0}}}{{{P_p}}}} \right) }^t}\\&\quad \frac{{{x^{{m_6} - 1}}}}{{{e^{\left( {{\beta _6} + \frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) x}}}} dx. \end{aligned} \end{aligned}$$
(48)

Utilizing the binomial expansion in [24, Eq. (1.111)], (48) is simplified as

$$\begin{aligned} \begin{aligned} {F_{{\varPsi _d}}}\left( {\left. \rho \right| {P_s}} \right)&= 1 - \frac{{{e^{ - {N_0}{\beta _3}\rho /{P_s}}}}}{{\beta _6^{ - {m_6}}\varGamma \left( {{m_6}} \right) }}\sum \limits _{t = 0}^{{m_3} - 1} {\frac{1}{{t!}}{{\left( {\frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) }^t}} \\&\quad \times \int \limits _0^\infty {\left[ {\sum \limits _{k = 0}^t {\left( {\begin{array}{c} t \\ k \\ \end{array}} \right) {x^k}{{\left( {\frac{{{N_0}}}{{{P_p}}}} \right) }^{t - k}}} } \right] } {x^{{m_6} - 1}}{e^{ - \left( {{\beta _6} + \frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) x}}dx \\&= 1 - \frac{{{e^{ - {N_0}{\beta _3}\rho /{P_s}}}}}{{\beta _6^{ - {m_6}}\varGamma \left( {{m_6}} \right) }}\sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\frac{1}{{t!}}{{\left( {\frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) }^t}\left( {\begin{array}{c} t \\ k \\ \end{array}} \right) {{\left( {\frac{{{N_0}}}{{{P_p}}}} \right) }^{t - k}} } } \\&\quad \times \int \limits _0^\infty {{x^{k + {m_6} - 1}}{e^{ - \left( {{\beta _6} + \frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) x}}} dx. \end{aligned} \end{aligned}$$
(49)

The last integral in (49) is computed with the aid of [24, Eq. (3.381.4)] as

$$\begin{aligned} \begin{aligned} {F_{{\varPsi _d}}}\left( {\left. \rho \right| {P_s}} \right)&= 1 - \frac{{\beta _6^{{m_6}}}}{{\varGamma \left( {{m_6}} \right) }}{e^{ - \frac{{{N_0}{\beta _3}\rho }}{{{P_s}}}}}\sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\frac{1}{{t!}}{{\left( {\frac{{{P_p}{\beta _3}\rho }}{{{P_s}}}} \right) }^t} } } \\&\quad \times \left( {\begin{array}{c} t \\ k \\ \end{array}} \right) {\left( {\frac{{{N_0}}}{{{P_p}}}} \right) ^{t - k}}\frac{{\varGamma \left( {k + {m_6}} \right) }}{{{{\left( {{\beta _6} + {P_p}{\beta _3}\rho /{P_s}} \right) }^{k + {m_6}}}}}. \end{aligned} \end{aligned}$$
(50)

B. The expression of \({f_{{\varPsi _e}}}\left( {\left. x \right| {P_s}} \right) \)

Following the derivation of (50), one can obtain the CDF of \(\varPsi _e\) as

$$\begin{aligned} \begin{aligned} {F_{{\varPsi _e}}}\left( {\left. x \right| {P_s}} \right)&= 1 - \frac{{\beta _7^{{m_7}}}}{{\varGamma \left( {{m_7}} \right) }}\sum \limits _{l = 0}^{{m_4} - 1} {\sum \limits _{n = 0}^l {\frac{{\varGamma \left( {n + {m_7}} \right) }}{{l!}}{{\left( {\frac{{{P_s}}}{{{P_p}{\beta _4}}}} \right) }^{n + {m_7} - l}} } } \\&\quad \times \left( {\begin{array}{c} l \\ n \\ \end{array}} \right) {{\left( {\frac{{{N_0}}}{{{P_p}}}} \right) }^{l - n}}\frac{{{x^l}{e^{ - {N_0}{\beta _4}x/{P_s}}}}}{{{{\left( {x + {\beta _7}{P_s}/{{P_p}{\beta _4}}} \right) }^{n + {m_7}}}}}. \end{aligned} \end{aligned}$$
(51)

By taking the derivative of \({F_{{\varPsi _e}}}\left( {\left. x \right| {P_s}} \right) \) with respect to x, one represents the conditional PDF of \(\varPsi _e\) as

$$\begin{aligned} {f_{{\varPsi _e}}}\left( {\left. x \right| {P_s}} \right)&= \sum \limits _{l = 0}^{{m_4} - 1} {\sum \limits _{n = 0}^l {{H_1}} } \left( {{{{{\bar{H}}} }_5}\frac{x}{{x + {{{{\bar{H}}} }_4}}} + {{{{\bar{H}}} }_6}x - {{{{\bar{H}}} }_2}} \right) \\&\frac{{{x^{l - 1}}{e^{ - {{{{\bar{H}}} }_3}x}}}}{{{{\left( {x + {{{{\bar{H}}} }_4}} \right) }^{n + {m_7}}}}}, \end{aligned}$$
(52)

where \(H_1\) is given by (38) and

$$\begin{aligned} {{{\bar{H}}} _2}&= P_s^{n + {m_7} - l}l, \end{aligned}$$
(53)
$$\begin{aligned} {{{\bar{H}}} _3}&= {{{N_0}{\beta _4}}}/{{{P_s}}}, \end{aligned}$$
(54)
$$\begin{aligned} {{{\bar{H}}} _4}&= {{{\beta _7}{P_s}}}/\left( {{{P_p}{\beta _4}}}\right) , \end{aligned}$$
(55)
$$\begin{aligned} {{{\bar{H}}} _5}&= P_s^{n + {m_7} - l}\left( {n + {m_7}} \right) , \end{aligned}$$
(56)
$$\begin{aligned} {{{\bar{H}}} _6} &= P_s^{n + {m_7} - l - 1}{N_0}{\beta _4}. \end{aligned}$$
(57)

C. The expression of \(\varUpsilon \left( {{P_s}} \right) \)

Substituting \(\rho ={{2^{{C_0}}}\left( {1 + x} \right) - 1}\) into (50) results in

$$\begin{aligned} \begin{aligned}&{F_{{\varPsi _d}}}\left( {\left. {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right| {P_s}} \right) = 1 - \frac{{\beta _6^{{m_6}}}}{{\varGamma \left( {{m_6}} \right) }}\sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\frac{{\varGamma \left( {k + {m_6}} \right) }}{{t!}} } } \\&\qquad \times \left( {\begin{array}{c} t \\ k \\ \end{array}} \right) \frac{{{e^{ - {N_0}{\beta _3}\left( {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right) /{P_s}}}{{\left( {{P_p}{\beta _3}\left\{ {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right\} /{P_s}} \right) }^t}}}{{{{\left( {{N_0}/{P_p}} \right) }^{k - t}}{{\left( {{\beta _6} + {P_p}{\beta _3}\left\{ {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right\} /{P_s}} \right) }^{k + {m_6}}}}} \\&\quad = 1 - \frac{{\beta _6^{{m_6}}}}{{\varGamma \left( {{m_6}} \right) }}\sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\frac{{{{\left( {x + 1 - {2^{ - {C_0}}}} \right) }^t}}}{{t!{e^{{N_0}{\beta _3}\left( {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right) /{P_s}}}}} } } \\&\qquad \times {\left( {\frac{{{P_p}{\beta _3}}}{{{P_s}{2^{ - {C_0}}}}}} \right) ^t}\left( {\begin{array}{c} t \\ k \\ \end{array}} \right) \frac{{{{\left( {{N_0}/{P_p}} \right) }^{t - k}}\varGamma \left( {k + {m_6}} \right) }}{{{{\left( {{\beta _6} + {P_p}{\beta _3}\left\{ {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right\} /{P_s}} \right) }^{k + {m_6}}}}}. \end{aligned} \end{aligned}$$
(58)

Performing the binomial expansion to \({{{\left( {x + 1 - {2^{ - {C_0}}}} \right) }^t}}\) and after some simplifications, one can write (58) in a compact form as

$$\begin{aligned} \begin{aligned}&{F_{{\varPsi _d}}}\left( {\left. {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right| {P_s}} \right) \\&\quad = 1 - \sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\frac{{\sum \limits _{u = 0}^t {\left( {\begin{array}{c} t \\ u \\ \end{array}} \right) {x^u}{{\left( {1 - {2^{ - {C_0}}}} \right) }^{t - u}}} }}{{t!{e^{{N_0}{\beta _3}\left( {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right) /{P_s}}}}} } } \\&\qquad \times {\left( {\frac{{{P_p}{\beta _3}}}{{{P_s}{2^{ - {C_0}}}}}} \right) ^t}\left( {\begin{array}{c} t \\ k \\ \end{array}} \right) \frac{{\beta _6^{{m_6}}{{\left( {{N_0}/{P_p}} \right) }^{t - k}}\varGamma \left( {k + {m_6}} \right) }}{{{{\left( {{\beta _6} + {P_p}{\beta _3}\left\{ {{2^{{C_0}}}\left[ {1 + x} \right] - 1} \right\} /{P_s}} \right) }^{k + {m_6}}}\varGamma \left( {{m_6}} \right) }} \\&\quad = 1 - \sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\sum \limits _{u = 0}^t {{Q_1}{{{{\bar{Q}}}}_2}{Q_3}} \frac{{{x^u}{e^{ - {{{{\bar{Q}}}}_5}x}}}}{{{{\left( {x + {{{{\bar{Q}}}}_4}} \right) }^{k + {m_6}}}}}} }, \end{aligned} \end{aligned}$$
(59)

where \(Q_1\) and \(Q_3\) are respectively given by (33) and (35), and

$$\begin{aligned} {{{{\bar{Q}}}}_2}&= P_s^{k + {m_6} - t}{e^{ - {N_0}{\beta _3}\left( {{2^{{C_0}}} - 1} \right) /{P_s}}}, \end{aligned}$$
(60)
$$\begin{aligned} {{{{\bar{Q}}}}_4}&= 1 - {2^{ - {C_0}}} + {{{\beta _6}{P_s}}}/\left( {{{P_p}{\beta _3}{2^{{C_0}}}}}\right) , \end{aligned}$$
(61)
$$\begin{aligned} {{{{\bar{Q}}}}_5}&= {{{N_0}{\beta _3}{2^{{C_0}}}}}/{{{P_s}}}. \end{aligned}$$
(62)

Inserting (59) and (52) into (46), one obtains

$$\begin{aligned} \begin{aligned} \varUpsilon \left( {{P_s}} \right)&= \sum \limits _{l = 0}^{{m_4} - 1} {\sum \limits _{n = 0}^l {{H_1}\left( {{{\bar{H}}_5}\int \limits _0^\infty {\frac{{{x^l}{e^{ - {{\bar{H}}_3}x}}}}{{{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7} + 1}}}}dx} } \right. } } \\&\quad +{{{{\bar{H}}}}_6}\int \limits _0^\infty {\frac{{{x^l}{e^{ - {{\bar{H}}_3}x}}}}{{{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7}}}}}dx} - {{{{\bar{H}}}}_2}\int \limits _0^\infty {\frac{{{x^{l - 1}}{e^{ - {{\bar{H}}_3}x}}}}{{{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7}}}}}dx} \\&\quad + \sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\sum \limits _{u = 0}^t {{Q_1}{{{{\bar{Q}}}}_2}{Q_3}\left\{ {{{\bar{H}}_2}\int \limits _0^\infty {\frac{{{x^{u + l - 1}}{e^{ - \left( {{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) x}}}}{{{{\left( {x + {{\bar{Q}}_4}} \right) }^{k + {m_6}}}{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7}}}}}dx} } \right. } } } \\&\quad - {{{{\bar{H}}}}_5}\int \limits _0^\infty {\frac{{{x^{u + l}}{e^{ - \left( {{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) x}}}}{{{{\left( {x + {{{{\bar{Q}}}}_4}} \right) }^{k + {m_6}}}{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7} + 1}}}}dx}\\&\quad \left. { - \left. {{{{{\bar{H}}}}_6}\int \limits _0^\infty {\frac{{{x^{u + l}}{e^{ - \left( {{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) x}}}}{{{{\left( {x + {{{{\bar{Q}}}}_4}} \right) }^{k + {m_6}}}{{\left( {x + {{{{\bar{H}}}}_4}} \right) }^{n + {m_7}}}}}dx} } \right\} } \right) . \end{aligned} \end{aligned}$$
(63)

The integrals in (63) are solved with the aid of Lemmas 2 and 3, reducing (63) to

$$\begin{aligned} \begin{aligned}&\varUpsilon \left( {{P_s}} \right) = \sum \limits _{l = 0}^{{m_4} - 1} {\sum \limits _{n = 0}^l {{H_1}\left[ {{{{{\bar{H}}}}_5}\varTheta \left( {l,{{{{\bar{H}}}}_3},{{{{\bar{H}}}}_4},n + {m_7} + 1} \right) } \right. } } \\&\quad + {{{{\bar{H}}}}_6}\varTheta \left( {l,{{{{\bar{H}}}}_3},{{{{\bar{H}}}}_4},n + {m_7}} \right) - {{{{\bar{H}}}}_2}\varTheta \left( {l - 1,{{{{\bar{H}}}}_3},{{{{\bar{H}}}}_4},n + {m_7}} \right) \\&\quad + \sum \limits _{t = 0}^{{m_3} - 1} {\sum \limits _{k = 0}^t {\sum \limits _{u = 0}^t {{Q_1}{{{{\bar{Q}}}}_2}{Q_3}\left\{ {{{{{\bar{H}}}}_2}\varOmega \left( {u + l - 1,{{{{\bar{Q}}}}_4},k + {m_6},{{{{\bar{H}}}}_4},n + {m_7},{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) } \right. } } } \\&\quad - {{{{\bar{H}}}}_5}\varOmega \left( {u + l,{{{{\bar{Q}}}}_4},k + {m_6},{{{{\bar{H}}}}_4},n + {m_7} + 1,{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) \\&\quad - \left. {\left. {{{{{\bar{H}}}}_6}\varOmega \left( {u + l,{{{{\bar{Q}}}}_4},k + {m_6},{{{{\bar{H}}}}_4},n + {m_7},{{{{\bar{H}}}}_3} + {{{{\bar{Q}}}}_5}} \right) } \right\} } \right] . \end{aligned} \end{aligned}$$
(64)

Inserting \({{\bar{H}}}_2\), \({{\bar{H}}}_3\), \({{\bar{H}}}_4\), \({{\bar{H}}}_5\), \(\bar{H}_6\), \({{\bar{Q}}}_2\), \({{\bar{Q}}}_4\), \({{\bar{Q}}}_5\) in (53), (54), (55), (56), (57), (60), (61), (62) into (64) and then using the new notations of \(Q_2\), \(Q_4\), \(Q_5\), \(H_2\), \(H_3\) in (34), (36), (37), (39), (40), respectively, one can reduce (64)–(31), finishing the proof.

Appendix 2: Proof of Theorem 2

Using (5) to rewrite (8) as

$$\begin{aligned} {P_s} = \min \left\{ {A{g_1} + B,{I_{th}}/{g_5}} \right\} . \end{aligned}$$
(65)

According to the definition of the CDF, one obtains

$$\begin{aligned} \begin{aligned} {F_{{P_s}}}\left( x \right)&= \Pr \left\{ {{P_s} \le x} \right\} \\&= \Pr \left\{ {\min \left\{ {A{g_1} + B,{I_{th}}/{g_5}} \right\} \le x} \right\} \\&= 1 - \Pr \left\{ {\min \left\{ {A{g_1} + B,{I_{th}}/{g_5}} \right\} \ge x} \right\} \\&= 1 - \Pr \left\{ {A{g_1} + B \ge x} \right\} \Pr \left\{ {{I_{th}}/{g_5} \ge x} \right\} \\&= \left\{ {\begin{array}{ll} {1 - \Pr \left\{ {{g_1} \ge \left( {x - B} \right) /A} \right\} \Pr \left\{ {{g_5} \le {I_{th}}/x} \right\} ,} & {x \ge B} \\ {1 - \Pr \left\{ {{g_5} \le {I_{th}}/x} \right\} ,} & {x< B} \\ \end{array}} \right. \\&= \left\{ {\begin{array}{ll} {1 - \frac{{\varGamma \left( {{m_1},{\beta _1}\left[ {x - B} \right] /A} \right) }}{{\varGamma \left( {{m_1}} \right) }}\frac{{\gamma \left( {{m_5},{\beta _5}{I_{th}}/x} \right) }}{{\varGamma \left( {{m_5}} \right) }},} & {x \ge B} \\ {1 - \frac{{\gamma \left( {{m_5},{\beta _5}{I_{th}}/x} \right) }}{{\varGamma \left( {{m_5}} \right) }},} & {x < B} \\ \end{array}} \right. . \end{aligned} \end{aligned}$$
(66)

It is recalled that \(\gamma \left( {a,u} \right) = \int \limits _0^u {{e^{ - t}}{t^{a - 1}}dt}\) and \(\varGamma \left( {a,u} \right) = \int \limits _u^\infty {{e^{ - t}}{t^{a - 1}}dt}\) where u is a function of x. Applying the Leibnitz differentiation [28], the first derivatives of \(\gamma \left( {a,u} \right) \) and \(\varGamma \left( {a,u} \right) \) with respect to x are respectively given by

$$\begin{aligned} \frac{{d\gamma \left( {a,u} \right) }}{{dx}}&= {e^{ - u}}{u^{a - 1}}\frac{{du}}{{dx}}, \end{aligned}$$
(67)
$$\begin{aligned} \frac{{d\varGamma \left( {a,u} \right) }}{{dx}}&= - {e^{ - u}}{u^{a - 1}}\frac{{du}}{{dx}}. \end{aligned}$$
(68)

The PDF of \(P_s\) can be achieved by taking the derivative of \({F_{{P_s}}}\left( x \right) \) in (66) with respect to x as

$$\begin{aligned} {f_{{P_s}}}\left( x \right) = \left\{ {\begin{array}{ll} { - \frac{{\gamma \left( {{m_5},\frac{{{\beta _5}{I_{th}}}}{x}} \right) }}{{\varGamma \left( {{m_5}} \right) \varGamma \left( {{m_1}} \right) }}\frac{{d\varGamma \left( {{m_1},\frac{{{\beta _1}\left[ {x - B} \right] }}{A}} \right) }}{{dx}} - \frac{{\varGamma \left( {{m_1},\frac{{{\beta _1}\left[ {x - B} \right] }}{A}} \right) }}{{\varGamma \left( {{m_1}} \right) \varGamma \left( {{m_5}} \right) }}\frac{{d\gamma \left( {{m_5},\frac{{{\beta _5}{I_{th}}}}{x}} \right) }}{{dx}},} & {x \ge B} \\ { - \frac{{d\gamma \left( {{m_5},\frac{{{\beta _5}{I_{th}}}}{x}} \right) }}{{dx}},} & {x < B} \\ \end{array}} \right. \end{aligned}$$
(69)

By applying the results in (67) and (68) and after some simplifications, one reduces (69)–(41). This completes the proof.

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Do-Dac, T., Ho-Van, K. Energy harvesting cognitive radio networks: security analysis for Nakagami-m fading. Wireless Netw 27, 1561–1572 (2021). https://doi.org/10.1007/s11276-019-02132-1

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