Secrecy capacity analysis of untrusted relaying energy-harvesting systems with hardware impairments

Abstract

In this paper, we study the impact of hardware impairments, which can act as one of the factors that cause degradation in the performance of communication systems, on the secrecy capacity of an untrusted relaying wireless energy-harvesting (WEH) system. In the system, the energy-constrained relay is an untrusted node which can overhear the source’s confidential signal while assisting the source-destination communication. The relay operates in the amplify-and-forward (AF) mode and uses the power-splitting (PS) protocol for harvesting energy. The destination sends an artificial noise (AN) signal during the source-relay communication. The AN signal acts as an additional energy source and an interference source at the relay. In our study, we derive an approximation of the average secrecy capacity (ASC) for the high-power-regime approximation in order to evaluate the secrecy performance of the proposed system, which is also the upper bound for the ASC. The analytical results are confirmed via Monte Carlo simulations. The numerical results provide valuable insights into the effect of the various system parameters, such as the power-splitting ratio, the relay’s location, the trade-off between the source’s power and the destination’s power, and the level of hardware impairments, on the secrecy performance.

Appendices

Appendix A:: Proof of Proposition 1

From Eq. (16), $F_{\gamma _{\{r,d\}}}\left (\gamma \right )|_{\gamma < \tfrac {1}{\mathcal AC}}$ can be rewritten as

$$ \begin{array}{@{}rcl@{}} F_{\gamma_{\{r,d\}}}\left( \gamma \right)|_{\gamma < \tfrac{1}{\mathcal AC}} = \int\limits_{0}^{\infty} {{f_{{X_{2}}}}\left( t \right)} {F_{{X_{1}}}}\left( {t\frac{{\omega {\mathcal B}}}{{{{\left( {{\mathcal C}\gamma } \right)}^{- 1}} - {\mathcal A}}}} \right)dt. \end{array} $$

(27)

Using the PDF of X₂ given by $f\left ({{m_{2}},{{\bar \gamma }_{2}};x} \right )$ and the CDF of X₁ given by $F\left ({{m_{1}},{{\bar \gamma }_{1}};x} \right )$, Eq. (16) can be rewritten as

$$ \begin{array}{@{}rcl@{}} F_{\gamma_{\{r,d\}}}\left( \gamma \right)|_{\gamma < \tfrac{1}{\mathcal AC}} &=& 1 - \frac{{{{\bar \gamma }_{2}}^{{m_{2}}}}}{{\Gamma \left( {{m_{2}}} \right)}}\sum\limits_{i = 0}^{{m_{1}} - 1} {\frac{1}{{i!}}} {\left( {\frac{{{{\bar \gamma }_{1}}\omega {\mathcal B}}}{{{{\left( {{\mathcal C}\gamma } \right)}^{- 1}} - {\mathcal A}}}} \right)^{i}} \\ & \times &\int\limits_{0}^{\infty} {{t^{{m_{2}} + i - 1}}{e^{- t\left( {{{\bar \gamma }_{2}} + \frac{{{{\bar \gamma }_{1}}\omega {\mathcal B}}}{{{{\left( {{\mathcal C}\gamma } \right)}^{- 1}} - {\mathcal A}}}} \right)}}} dt. \end{array} $$

(28)

With the help of [17, Eq. (3.381.4)], Eq. (28) can be expressed as given in Eq. (17).

Appendix B: Proof of Proposition 2

2.1 B.1 The calculation of $\mathcal {J}_{1}$

In case 1 ($\mu {\mathcal C}=1$), ${\mathcal I}\left (\gamma \right )$ can be decomposed by using the partial-fraction expansion method as

$$ \begin{array}{@{}rcl@{}} {\mathcal I}\left( \gamma \right) = \frac{{{{\left( {\gamma - {\mathcal AC}} \right)}^{{m_{2}}}}}}{{\gamma {{\left( {1 + \gamma } \right)}^{{m_{2}} + i + 1}}}} = \frac{{{b_{1}}}}{\gamma } + \sum\limits_{j = 1}^{{m_{2}} + i + 1} {\frac{{{b_{2,j}}}}{{{{\left( {\gamma + 1} \right)}^{j}}}}}, \end{array} $$

(29)

where the values of b₁ and b_2,j are determined by

$$ \begin{array}{@{}rcl@{}} {b_{1}} = \gamma {{\mathcal I}_{1}}\left( \gamma \right){|_{\gamma = 0}} = {\left( { - {\mathcal AC}} \right)^{{m_{2}}}}, \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} && {b_{2,j}} =\frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}}\left( {{{\left( {\gamma + 1} \right)}^{{m_{2}} + i + 1}}{{\mathcal I}_{1}}\left( \gamma \right)} \right){|_{\gamma = - 1}} \\ && = \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}}\frac{{{{\left( {\gamma - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}}}{\gamma }{|_{\gamma = - 1}} \\ &&= \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}}\left( {\sum\limits_{n = 0}^{{m_{2}}} \binom{m_{2}}{n} {{\left( { - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}} - n}}{\gamma^{n - 1}}} \right){|_{\gamma = - 1}}\\ &&= \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\sum\limits_{n = 0}^{{m_{2}}} \binom{m_{2}}{n} {\left( { - {\mathcal A}{\mathcal C}} \right)^{{m_{2}} - n}} \\ && \!\times\! \left\{\begin{array}{lll} {-\left( {m_{2}} + i + 1 - j \right)!}&;{\textup{ if }} \textbf{1}\left( {n = 0} \right) \\ {\frac{{\left( {n - 1} \right)!}}{{\left( {n + j - {m_{2}} - i - 2} \right)!}}{{\left( { - 1} \right)}^{n + j - {m_{2}} - i - 2}}}&;{\textup{ if }} \textbf{1} \left( n \!\ge\! 1,n - 1 \!\ge\! {m_{2}} + i + 1 - j\right)\\ 0&;{\textup{ if } \textbf{1} \left( {n \!\ge\! 1,n - 1 \!<\! {m_{2}} + i + 1 - j}\right)} \end{array}\right.. \end{array} $$

(31)

The expression of b_2,j in Eq. (31) can be simplified by Eq. (22).

Then, $\mathcal J$ can be calculated by

$$ \begin{array}{@{}rcl@{}} {{\mathcal J}_{1}} &=& \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\mathcal I} d\gamma = \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{1}}}}{\gamma }} d\gamma + \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{2,1}}}}{{\gamma + 1}}} d\gamma + \sum\limits_{j = 2}^{{m_{2}} + i + 1} {\int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{2,j}}}}{{{{\left( {\gamma + 1} \right)}^{j}}}}} d\gamma } \\ &=& \underbrace {{b_{1}}\mathop {\lim }\limits_{M \to \infty } \ln \left( {\frac{M}{{M + 1}}} \right)}_{= 0} + {b_{1}}\ln \left( {1 + {{\left( {{\mathcal A}{\mathcal C}} \right)}^{- 1}}} \right) \\ && + \sum\limits_{j = 2}^{{m_{2}} + i + 1} {\frac{{{b_{2,j}}}}{{j - 1}}{{\left( {{\mathcal A}{\mathcal C} + 1} \right)}^{1 - j}}}. \end{array} $$

(32)

Finally, ${\mathcal J}_{1}$ can be expressed as in Eq. (21).

2.2 B.2 The calculation of $\mathcal {J}_{2}$

In case 2 ($\mu {\mathcal C}=0$), ${\mathcal I} \left (\gamma \right )$ can be decomposed by using the partial-fraction expansion method as

$$ \begin{array}{@{}rcl@{}} {\mathcal I}\left( \gamma \right) = \frac{{{{\left( {\gamma - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}}}{{\left( {1 + \gamma } \right){\gamma^{{m_{2}} + i + 1}}}} = \frac{{{b_{3}}}}{{\gamma + 1}} + \sum\limits_{j = 1}^{{m_{2}} + i + 1} {\frac{{{b_{4,j}}}}{{{\gamma^{j}}}}}, \end{array} $$

(33)

where the values of b₃ and b_4,j are determined by

$$ \begin{array}{@{}rcl@{}} {b_{3}} = \left( {\gamma + 1} \right){{\mathcal I}_{1}}\left( \gamma \right){|_{\gamma = - 1}} = {\left( { - 1} \right)^{i + 1}}{\left( {1 + {\mathcal A}{\mathcal C}} \right)^{{m_{2}}}}, \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} && {b_{4,j}} = \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}}\left( {{\gamma^{{m_{2}} + i + 1}}{{\mathcal I}_{1}}\left( \gamma \right)} \right){|_{\gamma = 0}}\\ && = \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}}\frac{{{{\left( {\gamma - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}}}{{1 + \gamma }}{|_{\gamma = 0}}\\ && = \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\frac{{{d^{{m_{2}} + i + 1 - j}}}}{{d{\gamma^{{m_{2}} + i + 1 - j}}}} \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} &&\quad\times\left( {\sum\limits_{n = 0}^{{m_{2}}} \binom{m_{2}}{n} {{\left( { - 1 - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}} - n}}{{\left( {\gamma + 1} \right)}^{n - 1}}} \right){|_{\gamma = 0}}\\ && = \frac{1}{{\left( {{m_{2}} + i + 1 - j} \right)!}}\sum\limits_{n = 0}^{{m_{2}}} \binom{m_{2}}{n} {\left( { - 1 - {\mathcal A}{\mathcal C}} \right)^{{m_{2}} - n}}\\ && \!\times\! \left\{\begin{array}{lll} {{\left( { - 1} \right)}^{{m_{2}} + i + 1 - j}}\left( {{m_{2}} + i + 1 - j} \right)!&;{\textup{ if }} \textbf{1}\left( {n = 0} \right) \\ {\frac{{\left( {n - 1} \right)!}}{{\left( {n + j - {m_{2}} - i - 2} \right)!}}}&;{\textup{ if }} \textbf{1} \left( n \!\ge\! 1,n - 1 \!\ge\! {m_{2}} + i + 1 - j\right)\\ 0&;{\textup{ if } \textbf{1} \left( {n \!\ge\! 1,n - 1 \!<\! {m_{2}} + i + 1 - j}\right)} \end{array}\right.. \end{array} $$

(36)

The expression of b_4,j in Eq. (35) can be simplified by Eq. (24).

Then, $\mathcal J$ can be calculated by

$$ \begin{array}{@{}rcl@{}} {{\mathcal J}_{2}} &= &\int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\mathcal I} d\gamma = \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{3}}}}{{\gamma + 1}}} d\gamma + \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{4,1}}}}{\gamma }} d\gamma\\ &&+ \sum\limits_{j = 2}^{{m_{2}} + i + 1} {\int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{4,j}}}}{{{\gamma^{j}}}}} d\gamma} \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} &= &\underbrace {{b_{3}}\mathop {\lim }\limits_{M \to \infty } \ln \left( {\frac{{M + 1}}{M}} \right)}_{= 0} - {b_{3}}\ln \left( 1 + \frac{1}{{\mathcal AC}} \right)\\ &&+ \sum\limits_{j = 2}^{{m_{2}} + i + 1} {\frac{{{b_{4,j}}}}{{j - 1}}{{\left( {{\mathcal AC}} \right)}^{1 - j}}}. \end{array} $$

(38)

Finally, ${\mathcal J}_{2}$ can be expressed as in Eq. (23).

2.3 B.3 The calculation of $\mathcal {J}_{3}$

In ccase 3 ($\mu {\mathcal C} \notin \{0,1\}$), ${\mathcal I}\left (\gamma \right )$ can be decomposed by using the partial-fraction expansion method as

$$ \begin{array}{@{}rcl@{}} {\mathcal I}\left( \gamma \right) &=& \frac{{{{\left( {\gamma - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}}}{{\gamma \left( {1 + \gamma } \right){{\left( {\gamma + \mu {\mathcal C}} \right)}^{{m_{2}} + i}}}}\\ &=& \frac{{{b_{5}}}}{\gamma } + \frac{{{b_{6}}}}{{\gamma + 1}} + \sum\limits_{j = 1}^{{m_{2}} + i} {\frac{{{b_{7,j}}}}{{{{\left( {\gamma + \mu {\mathcal C}} \right)}^{j}}}}}, \end{array} $$

(39)

where the values of b₅, b₆, and b_7,j are determined by

$$ \begin{array}{@{}rcl@{}} {b_{5}} &= &\gamma {{\mathcal I}_{1}}\left( \gamma \right){|_{\gamma = 0}} = \frac{{{{\left( { - {\mathcal A}} \right)}^{{m_{2}}}}}}{{{\mu^{{m_{2}} + i}}{{\mathcal C}^{i}}}}, \end{array} $$

(40)

$$ \begin{array}{@{}rcl@{}} {b_{6}} &= &\left( {\gamma + 1} \right){{\mathcal I}_{1}}\left( \gamma \right){|_{\gamma = - 1}} = \frac{{ - {{\left( { - 1 - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}}}{{{{\left( {\mu {\mathcal C} - 1} \right)}^{{m_{2}} + i}}}}, \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} {b_{7,j}} &= &\frac{1}{{\left( {{m_{2}} + i - j} \right)!}}\frac{{{d^{{m_{2}} + i - j}}}}{{d{\gamma^{{m_{2}} + i - j}}}}\\ &&\times\left( {{{\left( {\gamma + \mu {\mathcal C}} \right)}^{{m_{2}} + i}}{{\mathcal I}_{1}}\left( \gamma \right)} \right){|_{\gamma = - \mu {\mathcal C}}}\\ &= &\frac{1}{{\left( {{m_{2}} + i - j} \right)!}}\frac{{{d^{{m_{2}} + i - j}}}}{{d{\gamma^{{m_{2}} + i - j}}}}\\ &&\times\left( {{{\left( {\gamma - {\mathcal A}{\mathcal C}} \right)}^{{m_{2}}}}\left( {\frac{1}{\gamma } - \frac{1}{{\gamma + 1}}} \right)} \right){|_{\gamma = - \mu {\mathcal C}}} . \end{array} $$

(42)

By following similar calculations as in Eqs. (31) and (35), the expression of b_4,j in Eq. (42) can be simplified by Eqs. (26). Then, $\mathcal {J}$ can be calculated by

$$ \begin{array}{@{}rcl@{}} {{\mathcal J}_{3}} &=& \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{5}}}}{\gamma }} d\gamma + \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{6}}}}{{\gamma + 1}}} d\gamma + \int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{7,1}}}}{{\gamma + \mu {\mathcal C}}}} d\gamma\\ &&+ \sum\limits_{j = 2}^{{m_{2}} + i} {\int\limits_{{\mathcal A}{\mathcal C}}^{\infty} {\frac{{{b_{7,j}}}}{{{{\left( {\gamma + \mu {\mathcal C}} \right)}^{j}}}}} d\gamma } \\ &= &\underbrace {\mathop {\lim }\limits_{M \to \infty } \ln \left( {{M^{{b_{5}}}}{{\left( {M + 1} \right)}^{{b_{6}}}}{{\left( {M + \mu {\mathcal C}} \right)}^{{b_{7,1}}}}} \right)}_{= 0} \\ && - \ln \left( {{{\left( {{\mathcal A}{\mathcal C}} \right)}^{{b_{5}}}}{{\left( {{\mathcal A}{\mathcal C} + 1} \right)}^{{b_{6}}}}{{\left( {{\mathcal A}{\mathcal C} + \mu {\mathcal C}} \right)}^{{b_{7,1}}}}} \right)\\ &&+ \sum\limits_{j = 2}^{{m_{2}} + i} {\frac{{{b_{7,j}}}}{{\left( {j - 1} \right)}}{{\left( {{\mathcal A}{\mathcal C} + \mu {\mathcal C}} \right)}^{- \left( {j - 1} \right)}}} \end{array} $$

(43)

Finally, $\mathcal {J}_{3}$ can be expressed as in Eq. (25).