Energy-Harvesting Decode-and-Forward Relaying Under Hardware Impairments

Abstract

In this paper, we propose a decode-and-forward relaying scheme under the effects of hardware impairments. In this proposed scheme, a cooperative relay uses a power-splitting receiver to harvest energy from the radio-frequency signals of a source node and applies a decode-and-forward technique to process the arriving signals. The system performance of the proposed protocol is analyzed in terms of the exact outage probability. Monte Carlo simulation is used to verify the theoretical analysis. Performance evaluations show that the proposed protocol performs best at an optimal power splitting ratio, and outperforms a direct transmission scheme when the relay is far to the destination. The optimal power splitting ratio is obtained more precisely using the golden section search method. In addition, serious effects of hardware impairments robustly diminish the system performance of the proposed protocol and the direct transmission protocol. Finally, the outage probability expressions match the simulation values well.

Appendices

Appendix 1: Proof of Lemma 1

The CDF of the random variable $X_i$ is expressed as

$$\begin{aligned} {F_{{X_i}}}\left( x \right) &=\Pr \left[ {{X_i}< x} \right] = \Pr \left[ {\frac{{a{g_i}}}{{b{g_i} + c}}< x} \right] = \Pr \left[ {{g_i}\left( {a - bx} \right)< cx} \right] \\&= \left\{ \begin{array}{ll} 1,&a - bx \le 0\\ {F_{{g_i}}}\left( {\frac{{cx}}{{a - bx}}} \right), &a - bx > 0 \end{array} \right. = \left\{ \begin{array}{ll} 1,&x \ge {a / b}\\ 1 - {e^{ - {\lambda _i}\left( {\frac{{cx}}{{a - bx}}} \right) }},&x < {a / b} \end{array} \right.\end{aligned}$$

(32)

Hence, the pdf of $X_i$ is inferred as

$$\begin{aligned} {f_{{X_i}}}\left( x \right) = \frac{{\partial {F_{{X_i}}}\left( x \right) }}{{\partial x}} = \left\{ \begin{array}{ll} 0,&x \ge {a / b}\\ \frac{{ac{\lambda _i}}}{{{{\left( {a - bx} \right) }^2}}}{e^{ - {\lambda _i}\left( {\frac{{cx}}{{a - bx}}} \right) }},&x < {a/b} \end{array} \right. \end{aligned}$$

(33)

From (32) and (33), Lemma 1 is completely proven.

Appendix 2: Proof of Lemma 2

The joint CDF of the random variable Z and the condition ${g_1} \ge {\theta _1}$ is expressed as

$$\begin{aligned} {F_{Z,{g_1} \ge {\theta _1}}}\left( z \right)&= \Pr \left[ {Z< z,{g_1} \ge {\theta _1}} \right] = \Pr \left[ {\frac{{\rho \eta \gamma {g_1}{g_2}}}{{\rho \eta \gamma {K^2}{g_1}{g_2} + 1 + \mu }}< z,{g_1} \ge {\theta _1}} \right] \\&= \Pr \left[ {\rho \eta \gamma {g_1}{g_2}\left( {1 - {K^2}z} \right) < \left( {1 + \mu } \right) z,{g_1} \ge {\theta _1}} \right] \end{aligned} $$

(34)

From (34), when $\left( {1 - {K^2}z} \right) \ge 0$ or $z \ge {K^{ - 2}}$, then ${F_{Z,{g_i} \ge {\theta _1}}}\left( z \right) = 1$.

Considering $z < {K^{ - 2}}$, ${F_{Z,{g_i} \ge {\theta _1}}}\left( z \right)$ in (34) is obtained as

$$\begin{aligned} {F_{Z,{g_i} \ge {\theta _1}}}\left( z \right)&= \Pr \left[ {{g_2} < \underbrace{{{\left( {1 + \mu } \right) } / {\left( {\rho \eta \gamma } \right) }}}_{{\theta _2}} \times \frac{1}{{{g_1}}} \times \frac{z}{{1 - {K^2}z}},{g_i} \ge {\theta _1}} \right] \\&= \int \limits _{{\theta _1}}^\infty {{f_{{g_1}}}\left( t \right) \times {F_{{g_2}}}\left( {\frac{{{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}} \right) dt} \\&= \int \limits _{{\theta _1}}^\infty {{f_{{g_1}}}\left( t \right) } \times \left\{ {1 - {e^{ - {\lambda _2}\left( {\frac{{{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}} \right) }}} \right\} dt\\&= 1 - {F_{{g_1}}}\left( {{\theta _1}} \right) - {\lambda _1}\int \limits _{{\theta _1}}^\infty {{e^{ - {\lambda _1}t - \frac{{{\lambda _2}{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}}}dt} \\&= {e^{ - {\lambda _1}{\theta _1}}} - {\lambda _1}\left\{ {\int \limits _0^\infty {{e^{ - {\lambda _1}t - \frac{{{\lambda _2}{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}}}dt} - \int \limits _0^{{\theta _1}} {{e^{ - {\lambda _1}t - \frac{{{\lambda _2}{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}}}dt} } \right\} \\&= {e^{ - {\lambda _1}{\theta _1}}} + {\lambda _1}\int \limits _0^{{\theta _1}} {{e^{ - {\lambda _1}t - \frac{{{\lambda _2}{\theta _2}}}{t} \times \frac{z}{{1 - {K^2}z}}}}dt} - 2\sqrt{\frac{{{\lambda _1}{\lambda _2}{\theta _2}z}}{{1 - {K^2}z}}} \times {K_1}\left( {2\sqrt{\frac{{{\lambda _1}{\lambda _2}{\theta _2}z}}{{1 - {K^2}z}}} } \right)\end{aligned}$$

(35)

where $K_v(x)$ is a v-order modified Bessel function of the second kind [22, Eq. (8.432.6)].

From (34) and (35), Lemma 1 is proven.