Secrecy Analysis of Amplify-and-Forward Relay Networks with Beamforming

Abstract

This paper analyzes the secrecy performance of an amplify-and-forward (AF) relay network, where a multi-antenna eavesdropper attempts to overhear the transmitted message from a multi-antenna source to a multi-antenna destination with a single antenna relay. Firstly, we derive the approximate analytical expressions for the secrecy outage probability (SOP) and average secrecy rate (ASR) of the relay network. Then, asymptotic expressions of SOP and ASR at high main-to-eavesdropper ratio (MER) are also provided to reveal the diversity gain of the secure communication. Finally, numerical results are given to verify the theoretical analysis and show the effect of the number of antennas in the considered relay network.

1. Introduction

Relay communication has been considered as a promising means to enhance the throughput and the power efficiency of wireless networks[1]. It also improves performance gain and reduces transmission loss as compared with direct transmission. In addition, relay systems show advantages in overcoming the shadowed fading. In relay communication systems, relay nodes are capable of employing various relaying protocols to assist information transmission from a source to its destination. Among the existing relaying protocols, amplify-and-forward (AF) and decode-and-forward (DF) are two most popular ones [2], [3]. The authors in [4] derived a closed-form bit error rate (BER) expression for their proposed scheme over Rayleigh fading channels, showing that the full diversity can be achieved by the new scheme. Also, the BER performance of the known coded cooperation is provided in [4] for the purpose of comparison. In [5], the authors studied the opportunistic cooperation with an AF relay transmission and proposed an opportunistic AF scheme where the relay transmission mode is adopted only when the source to relay channel is in a relatively good condition. It was shown that the reliability of wireless transmission can be significantly improved by using cooperative relay techniques.

On the other hand, some multi-antenna techniques, such as maximal ratio combining (MRC), transmit antennas selection (TAS), transmit beamforming, have also been proposed in literature to enhance the overall system performance [6], [7]. In [6], the authors investigated the impact of multi-antenna relay on the end-to-end error performance with the threshold-based MRC and the threshold-based selection combining (SC) at the relay. In [7], the authors considered the optimal signal-to-noise ratio (SNR)-based TAS at the source and the relay with AF protocol. Additionally, the maximal-ratio transmission (MRT) at the relay as well as the partial relay selection were also considered for performance analysis.

Recently, secure communication over the wireless medium at the physical layer (PHY) has achieved considerable attention. Compared with the traditional encryption method in secure communication, physical layer security communication with multi-antenna techniques have the advantages of stronger anti-eavesdrop ability. The security performance for cooperative systems in the presence of multiple eavesdroppers was investigated in [8], [9]. Specifically, the authors in [8] introduced the relay-eavesdropper channel and offered an outer-bound on the rate-equivocation region over several cooperation strategies. Moreover, the optimal relay weights to maximize the achievable secrecy rate for the multiple relays with several relay protocols were studied in [9]. In [10], the authors analyzed the performance of the external eavesdropper, and proposed two anti-eavesdropping schemes for IA-based networks. When the channel state information (CSI) of the eavesdropper is available, zero-forcing scheme can be utilized. Furthermore, a more generalized artificial noise (AN) scheme is proposed for IA-based networks without the knowledge of eavesdropper's CSI. In [11], the authors investigated the secure relay beamforming problems for the AFrelay network, where the transmitter sends information to the receiver with the help of a multiple-antenna relay in the presence of an eavesdropper. The authors in [12] proposed an anti-jamming scheme by aligning the jamming signal together with interference among users cooperation when an adversarial jammer exists. Then an AN scheme is proposed, in which the external eavesdropping is disrupted by AN without introducing any additional interference to the legitimate network. To further analyze the potential threat, a collusive eavesdropping scheme by some hostile IA users in the network is also proposed. The secrecy outage probability (SOP) of a dual hop AF relay system was evaluated in [13], showing that the SOP decreases with an increase in the number of relays and increases with an increase in the required secrecy rate. In [14], the authors discussed the secrecy rate of a two-hop multi-antenna relay network in the presence of an eveasdropper. However, to the best of our knowledge, the impact of multi-antenna transmission and reception with a single antenna relay while a multi-antenna eavesdropper exists has not been studied sufficiently.

In this paper, we consider an AF relay network in the presence of an eavesdropper. First of all, we derive approximate expressions of secrecy outage probability and average secrecy rate of the considered network in the Rayleigh fading environment. Then, the asymptotic analysis at high main-to-eavesdropper ratio (MER) is presented to reveal the diversity order and array gain of the relay network. Next, we verify our analytical results by comparing with Monto Carlo simulation results. It is shown that the secrecy outage probability reduces with the increase of antennas in destination, at the same time the average secrecy rate increases with the number of destination antennas.

Notation: Bold letters denote the vectors, |⋅| epresents the absolute value, E[⋅] the expectation, CN(μ,σ22) the complex Gaussian distribution with mean μ and variance σ2, (⋅)H denotes the conjugate transpose operator, [a]+ denotes max(a,0), Kv(⋅) stands for the v-th modified Bessel function of the second kind, 2F1(a,b;c:d) the hypergeometric function, and

 

2. System Model

As illustrated in Fig. 1, we consider a relay network consisting of a source (S), a destination (D), a relay (R), and an eavesdropper (E) who attempts to overheat the confidential message between S and D. The S, D and E are equipped with NS, ND and NE antennas, respectively, whereas R has a single antenna. We assume that E is close to D, and the direct link between S and D and that between S and E are unavailable due to heavy shadowing. All the communication links are assumed to undergo Rayleigh fading. In the considered relay network, the AF protocol is adopted at R, and the overall communication occurs during two time slots. In the first time slot, S performs transmit beamforming with the weight vector ωS and then sends its confidential signal x with E[|x|2] = 1 to R. As such, the received signal at R is given by

where PS is the transmit power at S, gSR is the NS×1 channel vector of the S-R link, is the additive white Gaussian noise (AWGN) with zero mean and variance By adopting the MRT, the transmit beamformer ωS is chosen as ωS = gSR/||gSR||F. In the second time slot, R amplifies the received signal y with a variable gain G as [16] and then broadcasts the signal to D. The signal received at D can be expressed as

where PR denotes the transmit power at R, gRD the channel vector of the R-D link, the AWGN at D. To maximize the received SNR, MRC is used at D, and the receive beamformer ωD is defined as ωD = gRD/||gRD||F. Meanwhile, due to the broadcast nature of the wireless communication, the signal overheard by E can be written as

where gER is an NE×1 channel vector of the R-E link, ωE = gRE/||gRE||F is the receive beamformer of the MRC at E and is the AWGN at E. After some algebraic manipulations, the instantaneous received SNRs at D and E can be, respectively, obtained as

where

Fig. 1.Diagram of the System model

According to the definition of physical layer secure communication [2], the achievable secrecy rate is formulated as the difference of the capacities between the main channel and the wiretap channel, i.e.,

where the coefficient 12denotes that two time slots are required to complete the transmission process. In the following, we will analyze two important secrecy performance metrics of the considered relay network, namely SOP and ASR, and further derive the asymptotic SOP and ASR expressions at high MER to provide insights into the diversity order and array gain.

 

3. Performance Analysis

3.1 Preliminaries

To analyze the secrecy performance of the considered relay network, we need to know the statistical properties of each link. Suppose that the S-R, R-D and R-E links are subject to Rayleigh fading, the probability density function (PDF) of γαβ, α∈{S,R}, β∈{R,E,D} is given by [16]

where j∈{S,D,E} and represents the average received SNR of each link. By using (7), the corresponding cumulative distribution function (CDF) of γαβ can be obtained as

3.2 Secrecy Outage Probability

According to [13], SOP is defined as the probability that the achievable secrecy rate CS is below the target secrecy rate RS≥0, namely,

By substituting (6) into (9), one can obtain

Note that, CS=0 when γD<γE. Hence, we can obtain Pr(Cs

Since deriving the exact expression of the term Pr(Cs

and obtain

where L=22RS and Q(L,y,z) = (L-1)yz/(y-Lz).

By employing the PDFs of γRD and γRE as given in (7), I1 in (14) can be rewritten as

By using variable replacement t = y-Lz and the binomial theorem, one can obtain

With the help of the following identifies [19, eq. 28], [17, eq. 24]

I1 can be further expressed as

where λ = ND+NE+i, μ = k-j+1, By substituting (19) into (14), one can obtain

Next, we derive the expression of Pr(γRD<γRE). By applying the PDFs of γRD and γRE in (7), one can obtain,

By substituting (20) and (21) into (11), we can eventually obtain the approximate expression of SOP as

3.3 Average Secrecy Rate

ASR is another fundamental performance metric for secure communications [22], which is defined as the average of the secrecy rate CS as given by

By substituting (6) into (23), we have

By using the integration by parts, I2 can be rewritten as

Substituting (25) into (24) yields

To derive the CDFs of γD and γE, we use the approximate formula (12) and obtain

where FγRD(y) is the CDF of γRD. By substituting (7) into (27), one can obtain

In a similar manner, the CDF of γE can be expressed as

Substituting (28) and (29) into (26) and employing (18), we can rewrite the ASR as

where

 

5. Asymptotic Analysis

To provide further insights into diversity order and array gain, we drive the asymptotic SOP expression at high MER of the considered relay network, where MER defined as the ratio of average channel from the relay to destination to that from the relay to the eavesdropper. Letting and using the following asymptotic relations for 2F1(a,b;c;d), x→1[18]

where B(a,b) = Γ(a)Γ(b)/Γ(a)+b) and Ψ(x) is the digamma function, the asymptotic expression of SOP at high MER can be written as

where O(•) represents the higher order terms, and the diversity order Gd and array gain Ga are respectively given by

and

Substituting (31) into (30), we finally obtain the asymptotic expression of ASR as

 

6. Numerical Results

As the first example, Fig. 2 plots the approximate and asymptotic SOP versus MER for different values of ND and NE. In this example, we assume RS = 0.5bits/s/Hz, NS=2, The approximate curves are obtained from (22), and the asymptotic curves are calculated by (32). As observed from the figure despite the use of approximation (12), the approximate curves match well with the Monte Carlo simulation results. It is also observed that the SOP increases with NE and decreases with ND. This is because the increase of ND improve the gain of the main channel, while the decrease of NE makes the main channel better quality than the wiretap channel. In addition, the diversity order and the array gain reflected from the asymptotic curves are consistent with the theoretical analysis results. Coincidentally, the secrecy diversity order is equal with ND.

Fig. 2.Secrecy outage probability with different ND and NE

Fig. 3 depicts the ASR versus MER for different values of ND with The approximate and asymptotic ASR curves are obtained from (30) and (35), respectively. As observed from the figure, regardless of the approximation, the approximate curves match well with Monte Carlo simulations, and the asymptotic result converges to the approximate value at high MER. It is also observed that the ASR increases with increasing ND, which can be explained by the fact that increasing ND improves the diversity order of the considered system. Moreover, the ASR increases with increasing MER. This is due to the fact that the increasing ND makes the main channel have better quality than the wiretap channel.

Fig. 3.Average secrecy rate versus the MER for different ND

 

7. Conclusion

In this paper, we have investigated the secrecy performance of the AF relay network with MRT/MRC in Rayleigh fading channels. Firstly, we have derived the SNRs of the main link and the wiretap link, respectively. Then, we have presented the approximate expressions for the SOP and ASR of the considered relay network. To further investigate the diversity gain and array gain, the asymptotic expressions of SOP and ASR at high MER have been derived. Finally, Monte Carlo simulation has been conducted to demonstrate the validity of the analytical results.