Performance analysis of jamming technique in energy harvesting cognitive radio networks

Abstract

This paper proposes a jamming technique which employs a self-powered secondary jammer to interfere a wire-tapper, who eavesdrops communications between a self-powered secondary source and a secondary destination in energy harvesting cognitive radio networks (EHCRNs). For generality, interference from a primary source, maximum transmit power constraint and interference power constraint are considered in analyzing security performance of the proposed jamming technique in terms of security-reliability compromise. Towards this end, exact expressions of detection/eavesdropping outage probabilities at the destination/the wire-tapper are first proposed and then verified by computer simulations. Finally, results are provided to demonstrate the efficacy of the jamming technique and the key effects (interference from the primary user, power constraints, interference power distribution factor, and time splitting factor) on security performance of EHCRNs.

Appendices

Appendix A: Exact closed-form representations of special integrals

This appendix defines integrals whose exact closed forms are expressed as (34)–(43).

The first integral is

$$\begin{aligned} {\mathcal{J}_1}\left( {a,b,c} \right) = \int \limits _a^\infty {{e^{ - \frac{b}{z} - cz}}\mathrm{d}z}. \end{aligned}$$

(46)

Applying the series expansion to ${e^{ - \frac{b}{z}}}$, one can rewrite (46) as

$$\begin{aligned} {\mathcal{J}_1}\left( {a,b,c} \right)= & {} \int \limits _a^\infty {{e^{ - \frac{b}{z}}}{e^{ - cz}}\mathrm{d}z} \nonumber \\= & {} \int \limits _a^\infty {\left[ {\sum \limits _{v = 0}^\infty {\frac{1}{{v!}}{{\left( { - \frac{b}{z}} \right) }^v}} } \right] {e^{ - cz}}\mathrm{d}z} \\= & {} \int \limits _a^\infty {{e^{ - cz}}\mathrm{d}z} + \sum \limits _{v = 1}^\infty {\frac{{{{\left( { - b} \right) }^v}}}{{v!}}\int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{{z^v}}}\mathrm{d}z} }.\nonumber \end{aligned}$$

(47)

By defining the below integral

$$\begin{aligned} {\mathcal{J}_{10}}\left( {u,p,n} \right) = \int \limits _u^\infty {\frac{{{e^{ - pz}}}}{{{z^{n + 1}}}}\mathrm{d}z}, \end{aligned}$$

(48)

one easily represents (47) in an exact closed form as (34).

By using [34, eq. (358.4)], one also expresses (48) in an exact closed form as (43).

The next integral is

$$\begin{aligned} {\mathcal{J}_2}\left( {a,b,c,g} \right) = \int \limits _a^\infty {\frac{{{e^{ - \frac{b}{z} - cz}}}}{{z + g}}\mathrm{d}z}. \end{aligned}$$

(49)

Applying the series expansion to ${e^{ - \frac{b}{z}}}$, one can rewrite (49) as

$$\begin{aligned} \begin{aligned} {\mathcal{J}_2}\left( {a,b,c,g} \right)&= \int \limits _a^\infty {{e^{ - \frac{b}{z}}}\frac{{{e^{ - cz}}}}{{z + g}}\mathrm{d}z} \\&= \int \limits _a^\infty {\left[ {\sum \limits _{v = 0}^\infty {\frac{1}{{v!}}{{\left( { - \frac{b}{z}} \right) }^v}} } \right] \frac{{{e^{ - cz}}}}{{z + g}}\mathrm{d}z} \\&= \int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{z + g}}\mathrm{d}z} + \sum \limits _{v = 1}^\infty {\frac{{{{\left( { - b} \right) }^v}}}{{v!}}} \int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{{z^v}\left( {z + g} \right) }}\mathrm{d}z}. \end{aligned} \end{aligned}$$

(50)

The second integral in (50) can be simplified by the partial fraction decomposition as

$$\begin{aligned} {\mathcal{J}_2}\left( {a,b,c,g} \right)= & {} \int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{z + g}}\mathrm{d}z} + {\sum \limits _{v = 1}^\infty {\frac{1}{{v!}}\left( {\frac{b}{g}} \right) } ^v}\int \limits _a^\infty \left( \frac{{{e^{ - cz}}}}{{z + g}} \right. \nonumber \\&\left. -\,\sum \limits _{u = 1}^v {{{\left( { - g} \right) }^{u - 1}}\frac{{{e^{ - cz}}}}{{{z^u}}}} \right) \mathrm{d}z. \end{aligned}$$

(51)

The integral

$$\begin{aligned} {\mathcal{J}_5}\left( {u,m,b} \right) = \int \limits _u^\infty {\frac{{{e^{ - mx}}}}{{x + b}}\mathrm{d}x} \end{aligned}$$

(52)

can be represented in an exact closed form as (38) with the help of [34, eq. (358.2)].

Using (48) and (52), one can represent (51) in an exact closed form as (35).

The integral

$$\begin{aligned} {\mathcal{J}_3}\left( {c,a,b} \right) = \mathop \int \limits _c^\infty \frac{{{e^{ - \frac{a}{x}}}}}{{x(x + b)}}\mathrm{d}x \end{aligned}$$

(53)

can be simplified by performing the variable change $x=1/y$ as

$$\begin{aligned} \begin{aligned} {\mathcal{J}_3}\left( {c,a,b} \right)&= \mathop \int \limits _{1/c}^0 \frac{{{e^{ - ay}}}}{{\left( {b + 1/y} \right) /y}}\left( { - \frac{1}{{{y^2}}}} \right) \mathrm{d}y \\&= \frac{1}{b}\mathop \int \limits _0^{1/c} \frac{{{e^{ - ay}}}}{{y + 1/b}}\mathrm{d}x \\&= \frac{1}{b}\left( {\mathop \int \limits _0^\infty \frac{{{e^{ - ay}}}}{{y + 1/b}}\mathrm{d}x - \mathop \int \limits _{1/c}^\infty \frac{{{e^{ - ay}}}}{{y + 1/b}}\mathrm{d}x} \right) . \end{aligned} \end{aligned}$$

(54)

Applying (52) for integrals in the last equality of (54), one can express ${\mathcal{J}_3}\left( {c,a,b} \right)$ in an exact closed form as (36).

The integral

$$\begin{aligned} {\mathcal{J}_4}\left( {a,b} \right) = \int \limits _0^\infty {\frac{{{e^{ - \frac{a}{x}}}}}{{x\left( {x + b} \right) }}\mathrm{d}x} \end{aligned}$$

(55)

can be represented through ${\mathcal{J}_3}\left( {c,a,b} \right)$ as

$$\begin{aligned} {\mathcal {J}_4}\left( {a,b} \right) = \mathop {\lim }\limits _{c \rightarrow 0 } {\mathcal{J}_3}\left( {c,a,b} \right) . \end{aligned}$$

(56)

Using the exact closed form of ${\mathcal{J}_3}\left( {c,a,b} \right)$ in (36), one can express (56) as (37).

The integral

$$\begin{aligned} {\mathcal{J}_6}\left( {c,a,b} \right) = \int \limits _c^\infty {\frac{{{e^{ - \frac{a}{x}}}}}{{{x^2}\left( {x + b} \right) }}} \mathrm{d}x \end{aligned}$$

(57)

can be simplified by performing the variable change $x=1/y$ as

$$\begin{aligned} {\mathcal{J}_6}\left( {c,a,b} \right)= & {} \int \limits _{1/c}^0 {\frac{{{e^{ - ay}}}}{{{{\left( {1/y} \right) }^2}\left( {b + 1/y} \right) }}\left( { - \frac{1}{{{y^2}}}} \right) } \mathrm{d}y \nonumber \\= & {} \frac{1}{b}\int \limits _0^{1/c} {\frac{y}{{y + 1/b}}{e^{ - ay}}} \mathrm{d}y \nonumber \\= & {} \frac{1}{b}\left( {\int \limits _0^{1/c} {{e^{ - ay}}} \mathrm{d}y - \frac{1}{b}\int \limits _0^{1/c} {\frac{{{e^{ - ay}}}}{{y + 1/b}}} \mathrm{d}y} \right) \\= & {} \frac{1}{b}\left( \frac{{1 - {e^{ - {a}/{c}}}}}{a} - \frac{1}{b}\left[ \int \limits _0^\infty {\frac{{{e^{ - ay}}}}{{y + 1/b}}} \mathrm{d}y \right. \right. \nonumber \\&\left. \left. -\,\int \limits _{1/c}^\infty {\frac{{{e^{ - ay}}}}{{y + 1/b}}} \mathrm{d}y \right] \right) .\nonumber \end{aligned}$$

(58)

Applying (52) for integrals in the last equality of (58), one can express ${\mathcal{J}_6}\left( {c,a,b} \right)$ in an exact closed form as (39).

The integral

$$\begin{aligned} {\mathcal{J}_7}\left( {a,b,c} \right) = \int \limits _a^\infty {\frac{{{e^{ - \frac{b}{z} - cz}}}}{{{z^2}}}\mathrm{d}z} \end{aligned}$$

(59)

can be simplified by applying the series expansion to ${{e^{ - \frac{b}{z}}}}$ as

$$\begin{aligned} \begin{aligned} {\mathcal{J}_7}\left( {a,b,c} \right)&= \int \limits _a^\infty {{e^{ - \frac{b}{z}}}\frac{{{e^{ - cz}}}}{{{z^2}}}\mathrm{d}z} \\&= \int \limits _a^\infty {\left[ {\sum \limits _{v = 0}^\infty {\frac{1}{{v!}}{{\left( { - \frac{b}{z}} \right) }^v}} } \right] \frac{{{e^{ - cz}}}}{{{z^2}}}\mathrm{d}z} \\&= \sum \limits _{v = 0}^\infty {\frac{{{{\left( { - b} \right) }^v}}}{{v!}}} \int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{{z^{v + 2}}}}\mathrm{d}z}. \end{aligned} \end{aligned}$$

(60)

Using (48) for the last integral in (60), one can represent (60) in an exact closed form as (40).

The integral

$$\begin{aligned} {\mathcal{J}_8}\left( {a,b} \right) = \int \limits _0^\infty {\frac{{{e^{ - \frac{a}{x}}}}}{{{x^2}\left( {x + b} \right) }}} \mathrm{d}x \end{aligned}$$

(61)

can be represented through ${\mathcal{J}_6}\left( {c,a,b} \right)$ as

$$\begin{aligned} {\mathcal {J}_8}\left( {a,b} \right) = \mathop {\lim }\limits _{c \rightarrow 0 } {\mathcal{J}_6}\left( {c,a,b} \right) . \end{aligned}$$

(62)

Using the exact closed form of ${\mathcal{J}_6}\left( {c,a,b} \right)$ in (39), one can express (62) as (41).

The integral

$$\begin{aligned} {\mathcal{J}_9}\left( {a,b,c} \right) = \int \limits _a^\infty {\frac{{{e^{ - \frac{b}{z} - cz}}}}{z}\mathrm{d}z} \end{aligned}$$

(63)

can be simplified by applying the series expansion to ${{e^{ - \frac{b}{z}}}}$ as

$$\begin{aligned} {\mathcal{J}_9}\left( {a,b,c} \right)= & {} \int \limits _a^\infty {{e^{ - \frac{b}{z}}}\frac{{{e^{ - cz}}}}{z}\mathrm{d}z} \nonumber \\= & {} \int \limits _a^\infty {\left[ {\sum \limits _{v = 0}^\infty {\frac{1}{{v!}}{{\left( { - \frac{b}{z}} \right) }^v}} } \right] \frac{{{e^{ - cz}}}}{z}\mathrm{d}z} \\= & {} \sum \limits _{v = 0}^\infty {\frac{{{{\left( { - b} \right) }^v}}}{{v!}}} \int \limits _a^\infty {\frac{{{e^{ - cz}}}}{{{z^{v + 1}}}}\mathrm{d}z}.\nonumber \end{aligned}$$

(64)

Using (48) for the last integral in (64), one can represent (64) in an exact closed form as (42).

Appendix B: The pdfs of $P_s$ and $P_j$

This appendix derives the pdfs of $P_s$ in (19) and $P_j$ in (20).

The cdf of $P_s$ is defined as

$$\begin{aligned} {F_{{P_s}}}\left( x \right) = \Pr \left\{ {{P_s} \le x} \right\} . \end{aligned}$$

(65)

Inserting (19) into (65), one can expand (65) as

$$\begin{aligned} \begin{aligned} {F_{{P_s}}}\left( x \right)&= {\varXi _{{P_{sm}}}}\left\{ {\Pr \left\{ {\left. {\min \left( {\frac{{\beta {I_m}}}{{{{\left| {{h_{sr}}} \right| }^2}}},{P_{sm}}} \right) \le x} \right| {P_{sm}}} \right\} } \right\} \\&= {\varXi _{{P_{sm}}}}\left\{ {\Pr \left\{ {\left. {\frac{{\beta {I_m}}}{{{{\left| {{h_{sr}}} \right| }^2}}} \le x,\frac{{\beta {I_m}}}{{{{\left| {{h_{sr}}} \right| }^2}}}< {P_{sm}}} \right| {P_{sm}}} \right\} } \right. \\&\quad +\,\left. {\Pr \left\{ {\left. {{P_{sm}} \le x,\frac{{\beta {I_m}}}{{{{\left| {{h_{sr}}} \right| }^2}}}> {P_{sm}}} \right| {P_{sm}}} \right\} } \right\} \\&= {\varXi _{{P_{sm}}}}\left\{ {\Pr \left\{ {\left. {\frac{{\beta {I_m}}}{x} \le {{\left| {{h_{sr}}} \right| }^2},\frac{{\beta {I_m}}}{{{P_{sm}}}} < {{\left| {{h_{sr}}} \right| }^2}} \right| {P_{sm}}} \right\} } \right. \\&\quad +\,\left. {\Pr \left\{ {\left. {{P_{sm}} \le x,\frac{{\beta {I_m}}}{{{P_{sm}}}}> {{\left| {{h_{sr}}} \right| }^2}} \right| {P_{sm}}} \right\} } \right\} \\&= {\varXi _{{P_{sm}}}}\left\{ {\Pr \left\{ {\left. {{P_{sm}}> x,\frac{{\beta {I_m}}}{x} \le {{\left| {{h_{sr}}} \right| }^2}} \right| {P_{sm}}} \right\} } \right. \\&\quad +\,\left. {\Pr \left\{ {\left. {{P_{sm}} \le x,\frac{{\beta {I_m}}}{{{P_{sm}}}} > {{\left| {{h_{sr}}} \right| }^2}} \right| {P_{sm}}} \right\} } \right\} \\&= {\varXi _{{P_{sm}}}}\left\{ {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}}\left[ 1 - \mathcal {U}\left( {x - {P_{sm}}} \right) \right] \right. \\&\quad \left. +\,\left( {1 - {e^{ - \frac{{\beta {I_m}}}{{{P_{sm}}{\mu _{sr}}}}}}} \right) \mathcal {U}\left( {x - {P_{sm}}} \right) \right\} . \end{aligned} \end{aligned}$$

(66)

Inserting (12) into (66) and averaging the result over ${{{\left| {{h_{ts}}} \right| }^2}}$, (66) is simplified as

$$\begin{aligned} {F_{{P_s}}}\left( x \right)= & {} {\varXi _{{{\left| {{h_{ts}}} \right| }^2}}}\left\{ {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}} + \left( {1 - {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}}} \right) \mathcal {U}\left( x\right. \right. \nonumber \\&\left. \left. -\,\frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}{{\left| {{h_{ts}}} \right| }^2} + \sigma _s^2} \right] \right) \right. \nonumber \\&-\,\left. {e^{ - \frac{{\beta {I_m}}}{{{P_{sm}}{\mu _{sr}}}}}}\mathcal {U}\left( x - \frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}{{\left| {{h_{ts}}} \right| }^2} + \sigma _s^2} \right] \right) \right\} \nonumber \\= & {} {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}} \nonumber \\&+\,\left( {1 - {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}}} \right) \int \limits _0^\infty \mathcal {U}\left( x\right. \nonumber \\&\left. -\,\frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}y + \sigma _s^2} \right] \right) \frac{1}{{{\mu _{ts}}}}{e^{ - \frac{y}{{{\mu _{ts}}}}}}\mathrm{d}y\nonumber \\&-\,\int \limits _0^\infty {e^{ - \frac{{\left( {1 - \alpha } \right) \beta {I_m}}}{{\alpha {\eta _s}\left( {{P_t}y + \sigma _s^2} \right) {\mu _{sr}}}}}}\mathcal {U}\left( x \right. \nonumber \\&\left. -\,\frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}y + \sigma _s^2} \right] \right) \frac{1}{{{\mu _{ts}}}}{e^{ - \frac{y}{{{\mu _{ts}}}}}}\mathrm{d}y. \end{aligned}$$

(67)

It is recalled from the definition of the step function that ${\mathcal {U}\left( {x - \frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}y + \sigma _s^2} \right] } \right) }$ equals 1 when $\frac{{\left( {1 - \alpha } \right) x}}{{\alpha {\eta _s}{P_t}}} - \frac{{\sigma _s^2}}{{{P_t}}} \ge y$. Also, $y = {\left| {{h_{ts}}} \right| ^2}\ge 0$ and hence, ${\mathcal {U}\left( {x - \frac{{\alpha {\eta _s}}}{{1 - \alpha }}\left[ {{P_t}y + \sigma _s^2} \right] } \right) }$$=1$ when $\frac{{\left( {1 - \alpha } \right) x}}{{\alpha {\eta _s}{P_t}}} - \frac{{\sigma _s^2}}{{{P_t}}} \ge y$ and $x \ge \frac{{\alpha {\eta _s}\sigma _s^2}}{{1 - \alpha }}$. Using this fact in (67), one can simplify it as

$$\begin{aligned} \begin{aligned}&{F_{{P_s}}}\left( x \right) = {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}} + \left( {1 - {e^{ - \frac{{\beta {I_m}}}{{{\mu _{sr}}x}}}}} \right) \mathcal {U}\left( x \right. \\&\quad \left. -\,\frac{{\alpha {\eta _s}\sigma _s^2}}{{1 - \alpha }} \right) \int \limits _0^{\frac{{\left( {1 - \alpha } \right) x}}{{\alpha {\eta _s}{P_t}}} - \frac{{\sigma _s^2}}{{{P_t}}}} {\frac{1}{{{\mu _{ts}}}}{e^{ - \frac{y}{{{\mu _{ts}}}}}}\mathrm{d}y} \\&\quad -\,\mathcal {U}\left( {x - \frac{{\alpha {\eta _s}\sigma _s^2}}{{1 - \alpha }}} \right) \int \limits _0^{\frac{{\left( {1 - \alpha } \right) x}}{{\alpha {\eta _s}{P_t}}} - \frac{{\sigma _s^2}}{{{P_t}}}} {{e^{ - \frac{{\left( {1 - \alpha } \right) \beta {I_m}}}{{\alpha {\eta _s}\left( {{P_t}y + \sigma _s^2} \right) {\mu _{sr}}}}}}\frac{1}{{{\mu _{ts}}}}{e^{ - \frac{y}{{{\mu _{ts}}}}}}\mathrm{d}y}. \end{aligned} \end{aligned}$$

(68)

The last integral in (68) makes the expression of ${F_{{P_s}}}\left( x \right)$ complicated. However, various results in Sect. 4 illustrate that its effect is negligible. Therefore, after ignoring it and using notations ($L_s$, $M_s$, $N_s$) in (23), (24), (25), one can rewrite (68) in a compact form as

$$\begin{aligned} {F_{{P_s}}}\left( x \right)= & {} {e^{ - \frac{{{N_s}}}{x}}} + \left( {1 - {e^{ - \frac{{{N_s}}}{x}}}} \right) \left( 1\right. \nonumber \\&\left. - {e^{{M_s}}}{e^{ - {L_s}{M_s}x}} \right) \mathcal {U}\left( {x - 1/{L_s}} \right) . \end{aligned}$$

(69)

Taking the derivative of ${F_{{P_s}}}\left( x \right)$ with respect to x, one obtains the pdf of $P_s$ as

$$\begin{aligned} \begin{aligned} {f_{{P_s}}}\left( x \right)&= \frac{{{N_s}}}{{{x^2}}}{e^{ - \frac{{{N_s}}}{x}}} + \left[ {{L_s}{M_s}{e^{{M_s}}}{e^{ - {L_s}{M_s}x}} - \frac{{{N_s}}}{{{x^2}}}{e^{ - \frac{{{N_s}}}{x}}} } \right. \\&\quad +\,{e^{{M_s}}}\frac{{{N_s}}}{{{x^2}}}{e^{ - \frac{{{N_s}}}{x} - {L_s}{M_s}x}}\\&\quad -\,\left. {{L_s}{M_s}{e^{{M_s}}}{e^{ - \frac{{{N_s}}}{x} - {L_s}{M_s}x}}} \right] \mathcal {U}\left( {x - 1/{L_s}} \right) . \end{aligned} \end{aligned}$$

(70)

Similarly, the pdf of $P_j$ can be expressed as

$$\begin{aligned} \begin{aligned} {f_{{P_j}}}\left( x \right)&= \frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x}}} + \left[ {{L_j}{M_j}{e^{{M_j}}}{e^{ - {L_j}{M_j}x}} - \frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x}}}} \right. \\&\quad +\,{e^{{M_j}}}\frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x} - {L_j}{M_j}x}} \\&\quad -\,\left. {{L_j}{M_j}{e^{{M_j}}}{e^{ - \frac{{{N_j}}}{x} - {L_j}{M_j}x}}} \right] \mathcal {U}\left( {x - 1/{L_j}} \right) . \end{aligned} \end{aligned}$$

(71)

where $L_j$, $M_j$, and $N_j$ are defined in (28), (29), and (30), respectively.

Appendix C: Proof of theorem 1

Inserting (7) into (21) and after some manipulations, one obtains

$$\begin{aligned} \begin{aligned} {OP_D}&= \Pr \left\{ {\frac{{{P_s}{{\left| {{h_{sd}}} \right| }^2}}}{{{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2}} \le {\gamma _0}} \right\} \\&= {\varXi _{{P_s}}}\left\{ {\Pr \left\{ {\left. {\frac{{{P_s}{{\left| {{h_{sd}}} \right| }^2}}}{{{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2}} \le {\gamma _0}} \right| {P_s}} \right\} } \right\} \\&= {\varXi _{{P_s}}}\left\{ {\varXi _{{\left| {{h_{td}}} \right| ^2}}}\left\{ \Pr \left\{ \left. \left| {{h_{sd}}} \right| ^2\right. \right. \right. \right. \\&\le \left. \left. \left. \left. \frac{{{\gamma _0}}}{{{P_s}}}\left( {{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2} \right) \right| {P_s},{{\left| {{h_{td}}} \right| }^2} \right\} \right\} \right\} \\&= {\varXi _{{P_s}}}\left\{ {{\varXi _{{{\left| {{h_{td}}} \right| }^2}}}\left\{ {{F_{{{\left| {{h_{sd}}} \right| }^2}}}\left( {\frac{{{\gamma _0}}}{{{P_s}}}\left[ {{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2} \right] } \right) } \right\} } \right\} . \end{aligned} \end{aligned}$$

(72)

Using (1) for ${{F_{{{\left| {{h_{sd}}} \right| }^2}}}\left( {\frac{{{\gamma _0}}}{{{P_s}}}\left[ {{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2} \right] } \right) }$, (72) is further simplified as

$$\begin{aligned} \begin{aligned} O{P_D}&= {\varXi _{{P_s}}}\left\{ {{\varXi _{{{\left| {{h_{td}}} \right| }^2}}}\left\{ {\left. {1 - {e^{ - \frac{{{\gamma _0}}}{{{P_s}{\mu _{sd}}}}\left( {{P_t}{{\left| {{h_{td}}} \right| }^2} + \sigma _d^2} \right) }}} \right| {P_s}} \right\} } \right\} \\&= 1 - {\varXi _{{P_s}}}\left\{ {\int \limits _0^\infty {{e^{ - \frac{{{\gamma _0}}}{{{P_s}{\mu _{sd}}}}\left( {{P_t}x + \sigma _d^2} \right) }}\frac{1}{{{\mu _{td}}}}{e^{ - \frac{x}{{{\mu _{td}}}}}}\mathrm{d}x} } \right\} \\&= 1 - \mathrm{Z}, \end{aligned} \end{aligned}$$

(73)

where

$$\begin{aligned} \mathrm{Z} = {\varXi _{{P_s}}}\left\{ {\frac{{{e^{ - {B_d}/P_s}}}}{{{A_d}/P_s + 1}}} \right\} , \end{aligned}$$

(74)

with $A_d$ and $B_d$ defined in (26) and (27), respectively.

Using the definition of the statistical average, (74) is rewritten as

$$\begin{aligned} \mathrm{Z} = \int \limits _0^\infty {\frac{{{e^{ - {B_d}/x}}}}{{{A_d}/x + 1}}} {f_{{P_s}}}\left( x \right) \mathrm{d}x. \end{aligned}$$

(75)

Inserting (70) into (75) and after some manipulations, one obtains

$$\begin{aligned} \begin{aligned} \mathrm{Z}&= {N_s}\int \limits _0^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x}}}}}{{x\left( {x + {A_d}} \right) }}\mathrm{d}x} \\&\quad +\,{L_s}{M_s}{e^{{M_s}}}\int \limits _{1/{L_s}}^\infty {\frac{x}{{x + {A_d}}}} {e^{ - \frac{{{B_d}}}{x} - {L_s}{M_s}x}}\mathrm{d}x \\&\quad -\,{N_s}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x}}}}}{{x\left( {x + {A_d}} \right) }}} \mathrm{d}x + {e^{{M_s}}}{N_s}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}}}{{x\left( {x + {A_d}} \right) }}\mathrm{d}x} \\&\quad -\,{L_s}{M_s}{e^{{M_s}}}\int \limits _{1/{L_s}}^\infty {\frac{x}{{x + {A_d}}}} {e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}\mathrm{d}x. \end{aligned} \end{aligned}$$

(76)

Performing the partial fraction decomposition, (76) is further simplified as

$$\begin{aligned} \begin{aligned} \mathrm{Z}&= {N_s}\int \limits _0^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x}}}}}{{x\left( {x + {A_d}} \right) }}\mathrm{d}x} +\, {L_s}{M_s}{e^{{M_s}}}\int \limits _{1/{L_s}}^\infty {{e^{ - \frac{{{B_d}}}{x} - {L_s}{M_s}x}}\mathrm{d}x} \\&\quad -\,{L_s}{M_s}{e^{{M_s}}}{A_d}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{B_d}}}{x} - {L_s}{M_s}x}}}}{{x + {A_d}}}} \mathrm{d}x\\&\quad -\,{N_s}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x}}}}}{{x\left( {x + {A_d}} \right) }}} \mathrm{d}x \\&\quad +\,\frac{{{e^{{M_s}}}{N_s}}}{{{A_d}}}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}}}{x}\mathrm{d}x} \\&\quad -\,\frac{{{e^{{M_s}}}{N_s}}}{{{A_d}}}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}}}{{x + {A_d}}}\mathrm{d}x} \\&\quad -\,{L_s}{M_s}{e^{{M_s}}}\int \limits _{1/{L_s}}^\infty {{e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}\mathrm{d}x} \\&\quad +\,{L_s}{M_s}{e^{{M_s}}}{A_d}\int \limits _{1/{L_s}}^\infty {\frac{{{e^{ - \frac{{{N_s} + {B_d}}}{x} - {L_s}{M_s}x}}}}{{x + {A_d}}}} \mathrm{d}x. \end{aligned} \end{aligned}$$

(77)

Expressing the integrals in (77) in terms of special functions in (46), (49), (53), (55), (63), one can reduce (77) to

$$\begin{aligned} \mathrm{Z}= & {} {N_s}{\mathcal{J}_4}\left( {{N_s} + {B_d},{A_d}} \right) + {L_s}{M_s}{e^{{M_s}}}{\mathcal{J}_1}\left( {L_s^{ - 1},{B_d},{L_s}{M_s}} \right) \nonumber \\&-\,{L_s}{M_s}{e^{{M_s}}}{A_d}{\mathcal{J}_2}\left( {L_s^{ - 1},{B_d},{L_s}{M_s},{A_d}} \right) \nonumber \\&-\,{N_s}{\mathcal{J}_3}\left( {L_s^{ - 1},{N_s} + {B_d},{A_d}} \right) \nonumber \\&+\,{e^{{M_s}}}{N_s}A_d^{ - 1}{\mathcal{J}_9}\left( {L_s^{ - 1},{N_s} + {B_d},{L_s}{M_s}} \right) \nonumber \\&-\,{e^{{M_s}}}{N_s}A_d^{ - 1}{\mathcal{J}_2}\left( {L_s^{ - 1},{N_s} + {B_d},{L_s}{M_s},{A_d}} \right) \nonumber \\&-\,{L_s}{M_s}{e^{{M_s}}}{\mathcal{J}_1}\left( {L_s^{ - 1},{N_s} + {B_d},{L_s}{M_s}} \right) \nonumber \\&+\,{L_s}{M_s}{e^{{M_s}}}{A_d}{\mathcal{J}_2}\left( {L_s^{ - 1},{N_s} + {B_d},{L_s}{M_s},{A_d}} \right) .\nonumber \\ \end{aligned}$$

(78)

Inserting (78) into (73), one reduces (73) to (44), completing the proof.

Appendix D: Proof of theorem 2

Inserting (8) into (22) and after some manipulations, one obtains

$$\begin{aligned} \begin{aligned} O{P_W}&= \Pr \left\{ {\frac{{{P_s}{{\left| {{h_{sw}}} \right| }^2}}}{{{P_t}{{\left| {{h_{tw}}} \right| }^2} + {P_j}{{\left| {{h_{jw}}} \right| }^2} + \sigma _w^2}} \le {\gamma _0}} \right\} \\&= \Pr \left\{ {{{\left| {{h_{sw}}} \right| }^2} \le \frac{{{\gamma _0}}}{{{P_s}}}\left( {{P_t}{{\left| {{h_{tw}}} \right| }^2} + {P_j}{{\left| {{h_{jw}}} \right| }^2} + \sigma _w^2} \right) } \right\} \\&= {\varXi _{{P_s},{P_j},{{\left| {{h_{tw}}} \right| }^2},{{\left| {{h_{jw}}} \right| }^2}}}\left\{ {1 - {e^{ - \frac{{{\gamma _0}}}{{{P_s}{\mu _{sw}}}}\left( {{P_t}{{\left| {{h_{tw}}} \right| }^2} + {P_j}{{\left| {{h_{jw}}} \right| }^2} + \sigma _w^2} \right) }}} \right\} \\&= 1 - {\varXi _{{P_s},{P_j}}}\left\{ {{\varPsi _t}{\varPsi _j}{e^{ - \frac{{{B_w}}}{{{P_s}}}}}} \right\} , \end{aligned} \end{aligned}$$

(79)

where

$$\begin{aligned} {\varPsi _t}= & {} {\varXi _{{{\left| {{h_{tw}}} \right| }^2}}}\left\{ {{e^{ - \frac{{{\gamma _0}{P_t}{{\left| {{h_{tw}}} \right| }^2}}}{{{P_s}{\mu _{sw}}}}}}} \right\} , \end{aligned}$$

(80)

$$\begin{aligned} {\varPsi _j}= & {} {\varXi _{{{\left| {{h_{jw}}} \right| }^2}}}\left\{ {{e^{ - \frac{{{\gamma _0}{P_j}{{\left| {{h_{jw}}} \right| }^2}}}{{{P_s}{\mu _{sw}}}}}}} \right\} . \end{aligned}$$

(81)

and $B_w$ is given by (33).

The ${\varPsi _t}$ term can be expressed in an exact closed form as

$$\begin{aligned} {\varPsi _t} = \int \limits _0^\infty {{e^{ - \frac{{{\gamma _0}{P_t}x}}{{{P_s}{\mu _{sw}}}}}}} \frac{1}{{{\mu _{tw}}}}{e^{ - \frac{x}{{{\mu _{tw}}}}}}\mathrm{d}x = \frac{1}{{{A_w}/{P_s} + 1}}, \end{aligned}$$

(82)

where $A_w$ is given by (32).

Similarly, the ${\varPsi _j}$ term can be expressed in an exact closed form as

$$\begin{aligned} {\varPsi _j} = \frac{1}{{{P_j}C/{P_s} + 1}}, \end{aligned}$$

(83)

where C is given by (31).

Inserting (82) and (83) into (79), one can rewrite (79) as

$$\begin{aligned} O{P_W}= & {} 1 - {\varXi _{{P_s}}}\left\{ {\frac{{{e^{ - {B_w}/{P_s}}}}}{{{A_w}/{P_s} + 1}}\underbrace{{\varXi _{{P_j}}}\left\{ {\frac{1}{{{P_j}C/{P_s} + 1}}} \right\} }_{\varPhi \left( {{P_s}} \right) }} \right\} \nonumber \\= & {} 1 - \int \limits _0^\infty {\frac{{{e^{ - {B_w}/x}}}}{{{A_w}/x + 1}}\varPhi \left( x \right) {f_{{P_s}}}\left( x \right) \mathrm{d}x}. \end{aligned}$$

(84)

Inserting (71) into the $\varPhi \left( {{P_s}} \right)$ term in (84), one can represent $\varPhi \left( {{P_s}} \right)$ as

$$\begin{aligned} {\varPhi \left( {{P_s}} \right) }= & {} \int \limits _0^\infty {\frac{{{f_{{P_j}}}\left( x \right) }}{{Cx/{P_s} + 1}}} \mathrm{d}x \nonumber \\&= \int \limits _0^\infty \frac{1}{{Cx/{P_s} + 1}} \left[ \frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x}}}\right. \nonumber \\&\quad \left. +\, \right. \left( {{L_j}{M_j}{e^{{M_j}}}{e^{ - {L_j}{M_j}x}} - \frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x}}}} \right. \nonumber \\&\quad +\,\left. \left. {e^{{M_j}}}\frac{{{N_j}}}{{{x^2}}}{e^{ - \frac{{{N_j}}}{x} - {L_j}{M_j}x}} \right. \right. \nonumber \\&\quad \left. \left. -\,{L_j}{M_j}{e^{{M_j}}}{e^{ - \frac{{{N_j}}}{x} - {L_j}{M_j}x}} \right) \mathcal {U}\left( {x - \frac{1}{{{L_j}}}} \right) \right] \mathrm{d}x \nonumber \\&= \frac{{{N_j}}}{C}{P_s}\int \limits _0^\infty {\frac{{{e^{ - {N_j}/x}}}}{{{x^2}\left( {x + {P_s}/C} \right) }}} \mathrm{d}x \nonumber \\&\quad +\,\frac{{{L_j}{M_j}}}{C}{e^{{M_j}}}{P_s}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {L_j}{M_j}x}}}}{{x + {P_s}/C}}} \mathrm{d}x \nonumber \\&\quad -\,\frac{{{N_j}}}{C}{P_s}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x}}}}{{{x^2}\left( {x + {P_s}/C} \right) }}} \mathrm{d}x \nonumber \\&\quad +\,\frac{{{N_j}}}{C}{e^{{M_j}}}{P_s}\underbrace{\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x - {L_j}{M_j}x}}}}{{{x^2}\left( {x + {P_s}/C} \right) }}} \mathrm{d}x}_\varUpsilon \nonumber \\&\quad -\,\frac{{{L_j}{M_j}}}{C}{e^{{M_j}}}{P_s}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x - {L_j}{M_j}x}}}}{{x + {P_s}/C}}} \mathrm{d}x. \end{aligned}$$

(85)

Applying the partial fraction decomposition, $\varUpsilon$ is rewritten as

$$\begin{aligned} \varUpsilon= & {} \frac{C}{{{P_s}}}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x - {L_j}{M_j}x}}}}{{{x^2}}}} \mathrm{d}x\nonumber \\&\quad - {\left( {\frac{C}{{{P_s}}}} \right) ^2}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x - {L_j}{M_j}x}}}}{x}} \mathrm{d}x \nonumber \\&\quad +\,{\left( {\frac{C}{{{P_s}}}} \right) ^2}\int \limits _{L_j^{ - 1}}^\infty {\frac{{{e^{ - {N_j}/x - {L_j}{M_j}x}}}}{{x + {P_s}/C}}} \mathrm{d}x. \end{aligned}$$

(86)

Inserting (86) into (85) and using the definitions of the integrals in (49), (52), (57), (59), (61), (63), one can express ${\varPhi \left( {{P_s}} \right) }$ in an exact closed form as

$$\begin{aligned} \begin{aligned} {\varPhi \left( {{P_s}} \right) }&= \frac{{{N_j}}}{C}{P_s}{\mathcal{J}_8}\left( {{N_j},{P_s}/C} \right) \\&\quad +\,\frac{{{L_j}{M_j}}}{{{e^{ - {M_j}}}C}}{P_s}{\mathcal{J}_5}\left( {L_j^{ - 1},{L_j}{M_j},{P_s}/C} \right) \\&\quad -\,\frac{{{N_j}}}{C}{P_s}{\mathcal{J}_6}\left( {L_j^{ - 1},{N_j},{P_s}/C} \right) \\&\quad -\,\frac{{{L_j}{M_j}}}{{{e^{ - {M_j}}}C}}{P_s}{\mathcal{J}_2}\left( {L_j^{ - 1},{N_j},{L_j}{M_j},{P_s}/C} \right) \\&\quad +\,{e^{{M_j}}}{N_j}{\mathcal{J}_7}\left( {L_j^{ - 1},{N_j},{L_j}{M_j}} \right) \\&\quad -\,{e^{{M_j}}}{N_j}C{\mathcal{J}_9}\left( {L_j^{ - 1},{N_j},{L_j}{M_j}} \right) /{P_s} \\&\quad +\,{e^{{M_j}}}{N_j}C\frac{{{\mathcal{J}_2}\left( {L_j^{ - 1},{N_j},{L_j}{M_j},{P_s}/C} \right) }}{{{P_s}}}. \end{aligned} \end{aligned}$$

(87)

Inserting (87) and (70) into (84), one obtains (45), completing the proof.