Physical Layer Security for Wireless Powered Massive MIMO Decode and Forward Relay Systems with Hardware Impairments: Performance Analysis

Abstract

In this paper, the issue of secrecy capacity of wireless powered massive MIMO dual hop relay system with a single antenna eavesdropper having non ideal hardware is addressed. The relay harvests energy in a proportionate manner and passes it to destination through beamforming with classical decode and forward relaying protocol. The relay has no channel state information (CSI) of passive eavesdropper but has CSI of the legitimate channel. The work presented in this paper focuses on the analysis of the difference in system performance with ideal and non ideal hardware, bounded by strict outage probability. The performance (in terms of secrecy outage capacity) is studied with hardware impairments (HWIs) defined for all network elements, i.e., source, relay, destination and passive eavesdropper. It is also observed that compared to ideal hardware, there is significant degradation in performance due to HWIs.

Appendix

Proof of Theorem 1

$$\begin{aligned} \epsilon= & {} Pr(R_{SOC}>C_D-C_E) \,\,\,=Pr\left( min(C_{S,R},C_{R,E})>C_D-R_{SOC}\right) \\= & {} Pr\left( min\left( \frac{(1-\theta )P_S N_R\alpha _{S,R}}{(1-\theta )P_S N_R(\kappa _{1}^2+\kappa _{2}^2)+(1-\theta )N_0+\mu N_0},\right. \right. \\&\left. \left. \frac{\theta \eta \alpha _{S,R}\alpha _{R,E}P_S N_R \left| \frac{h^H_{R,E}{\hat{h}}_{R,D}}{||{\hat{h}}_{R,D}||^2}\right| ^2}{\theta \eta \alpha _{S,R}\alpha _{R,E}P_S N_R \left| \frac{h^H_{R,E}{\hat{h}}_{R,D}}{||{\hat{h}}_{R,D}||^2}\right| ^2(\kappa _{3}^2+\kappa _{4}^2)+N_0+\mu N_0}\right) > 2^{\frac{C_D-R_{SOC}}{W}}-1\right) \end{aligned}$$

Let $p=(1-\theta )P_S N_R\alpha _{S,R}$,  $\varDelta _p=(1-\theta )N_0+\mu N_0$,  $\kappa _p=\sqrt{\kappa _{1}^2+\kappa _{2}^2}.$ Let $q=\theta \zeta \alpha _{S,R}\alpha _{R,E}P_S N_R,$  $\varDelta _q=N_0+\mu N_0$,  $\kappa _q=\sqrt{\kappa _{3}^2+\kappa _{4}^2}$. The term $\left| \frac{h^H_{R,E}{\hat{h}}_{R,D}}{||{\hat{h}}_{R,D}||^2}\right| ^2$ is $\chi ^2$ Random Variable with two Degrees of Freedom. Let $y\sim \left| \frac{h^H_{R,E}{\hat{h}}_{R,D}}{||{\hat{h}}_{R,D}||^2}\right| ^2$ and $x = 2^\frac{C_D-R_{SOC}}{W}-1$.

$$\begin{aligned} \epsilon&= {} Pr\left( min\left( \frac{p}{p\kappa _p^2+\varDelta _p},\frac{q y}{qy\kappa _q^2+\varDelta _q}\right)> x\right) \\&= {} Pr\left( \frac{p}{p\kappa _p^2+\varDelta _p}\le \frac{qy}{qy\kappa _q^2+\varDelta _q}\right) \times Pr\left( \frac{p}{p\kappa _p^2+\varDelta _p}> x\right) \\&\quad +\, Pr\left( \frac{p}{p\kappa _p^2+\varDelta _p}\ge \frac{qy}{qy\kappa _q^2+\varDelta _q}\right) \times Pr\left( \frac{qy}{qy\kappa _q^2+\varDelta _q}> x\right) \end{aligned}$$

As there is no eavesdropping on S-R link, we have $Pr\left( C_{S,R}>C_{D}-R_{SOC}\right) =1$. Then,

$$\begin{aligned} \epsilon & = {} Pr \left( \frac{p \times \varDelta _q }{pq \left( \kappa _p^2-\kappa _q^2\right) + q \times \varDelta _p} \le y \right) \times 1\,+ \left( 1-Pr \left( \frac{p \times \varDelta _q }{pq \left( \kappa _p^2-\kappa _q^2\right) + q \times \varDelta _p} \le y \right) \right) \\&\quad \times \, Pr \left( \frac{q y}{q y\kappa _q^2+\varDelta _q} > x \right) \end{aligned}$$

The statistics of Pr $\left( \frac{q y}{q y\kappa _q^2+\varDelta _q} > x \right)$   are obtained as follows:

$$\begin{aligned} {Pr \left( \frac{A y}{By+C}< x \right) } = \left\{ \begin{array}{ll} {1-exp{ \left( {\frac{{-Cx}}{{A - Bx}}} \right) }},&{}\quad A-Bx > {0}\\ {{1} },&{}\quad A-Bx < {0} \end{array} \right. \end{aligned}$$

Using above relations,

$$\begin{aligned} \epsilon= & {} \exp {\left( -\frac{p \times \varDelta _q }{pq \left( \kappa _p^2 - \kappa _q^2\right) + q \times \varDelta _p} \right) }+ \left( 1-\exp \left( -\frac{p \times \varDelta _q }{pq \left( \kappa _p^2- \kappa _q^2\right) + q \times \varDelta _p} \right) \right) \times \exp \left( -\frac{\varDelta _qx}{q-q \kappa _q^2x}\right) \\\Rightarrow & {} \exp \left( -\frac{\varDelta _qx}{q-q \kappa _q^2x}\right) =\left( \epsilon - \exp \left( -\frac{p \times \varDelta _q }{pq \left( \kappa _p^2-\kappa _q^2\right) + q \times \varDelta _p} \right) \right) \Bigg / \left( 1-\exp \left( -\frac{p \times \varDelta _q }{pq \left( \kappa _p^2-\kappa _q^2\right) + q \times \varDelta _p}\right) \right) \\ \end{aligned}$$

Letting $\hat{\epsilon } = \exp \left( -\frac{\varDelta _qx}{q-q \kappa _q^2x}\right)$ and rearranging further we get,

$$\begin{aligned} R_{SOC}=C_D-Wlog_2\left( 1+\frac{\ln {\hat{\epsilon }}\times q}{\ln {\hat{\epsilon }}q \kappa _q^2-\varDelta _q}\right) \end{aligned}$$

Hence, the theorem is proved. $\square$