Secrecy Outage of Dual-Hop Amplify-and-Forward Relay System with Diversity Combining at the Eavesdropper

Abstract

In this paper, a dual-hop amplify-and-forward relay system is considered with an eavesdropper where each link undergoes independent, non-identical, flat Rayleigh fading. The eavesdropper is capable of diversity combining the direct and relayed communication from the source using maximal ratio combining (MRC) and selection combining (SC). Closed-form upper and lower bounds on secrecy outage probability are derived. Closed-form approximate secrecy outage probability and ergodic secrecy rate is also obtained when source–relay link average signal-to-noise ratio (SNR) is high. Asymptotic analysis is presented when dual-hop links have equal or unequal average SNR. It is found that SC has both the secrecy outage and ergodic secrecy rate performances are better than MRC. To achieve the same secrecy outage performance of SC, MRC requires relatively higher SNR at lower rate. MRC also requires relatively higher SNR to achieve same secrecy rate performance of SC when eavesdropper link quality degrades. It is observed that lower bound for secrecy outage is tight and tends towards secrecy outage as SNR increases. It is interesting to find that either one of the dual-hop link can limit the performances even if the other link average SNR is infinitely high.

Appendices

Appendix 1: Proof of Proposition 1

Proof

CDF and the PDF of the random variable T can be found in [16, 17]. For the convenience of the reader we deduce the the CDF of the RV T conditioned on $X, F_{T|X}(t|x)$, here again. From the definition of CDF we get

$$\begin{aligned} F_{T|X}(t|x)&= \mathbb {P}[\min (x,y)\le t|x]\nonumber \\&=1-\mathbb {P}[\min (x,y)> t|x]\nonumber \\&=1-\mathbb {P}[x> t|x]\mathbb {P}[y> t|x]\nonumber \\&=\left\{ \begin{array}{ll} 1-\mathbb {P}[y> t|x]&{} \quad t < x\\ 1 &{} \quad t \ge x\\ \end{array} \right. . \end{aligned}$$

(57)

We can write (57) from the fact that

$$\begin{aligned} \mathbb {P}[x> t|x]=\left\{ \begin{array}{ll} 1 &{} \quad t < x\\ 0 &{} \quad t \ge x\\ \end{array} \right. . \end{aligned}$$

(58)

As X and Y are independent, by simply using CDF of RV Y i.e. CDF of exponential distribution in (57) we get the CDF of the a RV T as

$$\begin{aligned} F_{T|X}(t|x)=\left\{ \begin{array}{ll} 1-e^{-\beta _yt} &{} \quad t < x\\ 1 &{} \quad t \ge x\\ \end{array} \right. . \end{aligned}$$

(59)

By differentiating (59) with respect to t we get the PDF as

$$\begin{aligned} f_{T|X}(t|x)=\left\{ \begin{array}{ll} \beta _ye^{-\beta _yt} &{} \quad t<x\\ \delta (t-x)e^{-\beta _yx} &{}\quad t \ge x\\ \end{array} \right. . \end{aligned}$$

(60)

From the definition, MGF expressed in (9) can be found by simply evaluating the following integrals

$$\begin{aligned} M_{T|X}(s|x)&=\int _0^x \beta _y e^{-(\beta _y-s)t}dt + \int _ x^\infty e^{-\beta _y x} e^{st} \delta (t-x)dt. \end{aligned}$$

(61)

Appendix 2: Proof of Corollary 1

Proof

Directly following the proof of proposition 1, the CDF is obtained in [16, 17] as

$$\begin{aligned} F_T(t|x) =\left\{ \begin{array}{ll} 1-e^{-2\beta _{y}t} &{} \quad t < \frac{x}{2}\\ 1 &{} \quad t \ge \frac{x}{2}\\ \end{array} \right. . \end{aligned}$$

(62)

The corresponding PDF is derived in [16, 17] by differentiating (62) as

$$\begin{aligned} f_T(t|x) =\left\{ \begin{array}{ll} 2\beta _{y} e^{-2\beta _{y}t} &{} \quad t < \frac{x}{2}\\ \delta \left(t-\frac{x}{2}\right)e^{-\beta _{y}x} &{} \quad t \ge \frac{x}{2}\\ \end{array} \right. . \end{aligned}$$

(63)

Using standard derivation of MGF as in (61), we can find MGF of RV T in (10).