IET Communications Research Article Secrecy analysis of amplify-and-forward relaying networks with zero forcing ISSN 1751-8628 Received on 9th March 2017 Revised 1st June 2017 Accepted on 29th June 2017 E-First on 27th September 2017 doi: 10.1049/iet-com.2017.0215 www.ietdl.org Abdelhamid Salem1 , Khairi Ashour Hamdi1 1School of Electrical and Electronic Engineering, the University of Manchester, Manchester, M13 9PL, UK E-mail: [email protected] Abstract: In this study, the ergodic secrecy capacity and the corresponding outage probability of two-hop amplify-and-forward relaying system in the presence of a passive eavesdropper are analysed. In order to improve the security, in this study, zeroforcing (ZF) scheme is implemented at different locations in the system. The effect of the ZF-based scheme on the system security is considered for three different scenarios, based on where the ZF scheme is applied, namely, (i) ZF receivers at the relay and destination nodes, (ii) ZF precoders at the source and relay nodes, and (iii) ZF precoders/receivers at the relay nodes. For each scenario, explicit analytical expressions for the ergodic secrecy capacity and secrecy outage probability are derived. Monte Carlo simulations are also provided to validate the analysis. Results show that increasing the number of source, relay and/or destination nodes can be favourable or unfavourable to the system security and the significance of this enhancement/ degradation depends on the particular scenario deployed. In addition, the system security improves with increasing the source and/or relay power. 1 Introduction The broadcast nature of wireless channels makes wireless networks more vulnerable to the eavesdropping attack. Therefore, attention to the issue of the security in wireless communications has increased rapidly. Physical layer security, which is based on information theory, has attracted considerable attention in this context. The concept of physical layer security it was first developed by Wyner [1], who introduced the wiretap channel for single point-to-point communication before it was extended to broadcast channels by Csiszar and Korner in [2]. From these works, it is reported that achieving secure communications is possible if the destination channel quality is better than the eavesdropper channel. Due to the fact that multiple-input multiple-output (MIMO) networks have become a main element in the physical layer of wireless communication, security of such systems has particularly attracted a considerable amount of attention. For instance, very recently, the secrecy capacity of a 2 × 2 MIMO system with an eavesdropper having either one or two antennas is studied in [3, 4], respectively. Later on, in [5, 6] an optimisation problem is formulated to solve the secrecy capacity of a general MIMO scenario with an eavesdropper equipped with multiple antennas. The authors of [7] showed that antenna selection and combining techniques enhance the secrecy over such channels. Moreover, it is reported that the implementation of cooperative relays can also enhance system secrecy. For instance, it was presented in [8] that better security can be achieved by simply forwarding artificial noise by the relay to confuse the eavesdropper. Different relaying schemes are studied in [9, 10] to maximise the secrecy capacity, while minimising the total transmit power. Joint cooperative beamforming and jamming technique are discussed in [11] in the context of amplify-and-forward (AF) relaying systems with passive eavesdroppers. All these studies, the authors of [8–11] considered only single-antenna source and destination nodes. In contrast, the authors in [12–14] studied the security of MIMO relay wiretap channels in the presence of an eavesdropper when all the nodes are equipped with multiple antennas. However, in MIMO relaying systems, in order to maximise the achievable secrecy capacity a complex optimization problem, which can be inconveniently hard to handle, usually needs to be solved [15]. Alternatively and to reduce the complexity, other linear precoding/ IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 decoding techniques can be effectively implemented as in [15]. In addition, the authors in [16], studied a multiple-relay MIMO network using a decode and forward (DF) relaying strategy, and a joint scheme, which joints the relay selection and the antenna selection at the base station and the selected relay has been proposed. In [17] a novel scheme combining relay selection with artificial noise for AF relaying system in the presence of an eavesdropper was designed. The secrecy of a distributed relaying system has been investigated in [18], where all nodes are equipped with multiple-antennas and the relay nodes cooperate to imitate a relay node with multiple antennas. In [19] the secrecy rates of untrusted one-way and two-way half-duplex AF relaying protocols were considered. The distributed beam-forming for MIMO fullduplex relay networks was considered in [20]. Furthermore, in [21] the authors studied the cooperative secure transmission for a DF system, where an opportunistic relay node is adopted to forward the confidential message and the other relay nodes transmit artificial noise to confuse the eavesdroppers. In [22] a joint source– relay precoding scheme was proposed to secure an AF MIMO wireless relay network in the existence of a multi-antenna eavesdropper. The authors of [23] investigated the security issue of AF multiuser peer-to-peer relay networks, where a secure user transmits the confidential message in the presence of a multiantenna eavesdropper, while the other unclassified users transmit unclassified messages. The source transmit power and relay beamformer have been designed to maximise the achievable secrecy rate under the minimum received signal-to-interference-plus-noise ratio (SINR) requirement at each destination. For more details, we refer the reader to [24] where an overview of the recent research on enhancing wireless transmission secrecy via cooperation was discussed. Unlike other studies, in this paper we analyse the security of MIMO AF relaying systems with zero-forcing (ZF) processing for various scenarios based on the ZF design strategy [We select ZF scheme and not others such as minimum mean square error (MMSE), because ZF is simple and ease of implementation.]. In light of this, the ergodic secrecy capacity and secrecy outage probability of the cooperative MIMO system are investigated for the following scenarios, (i) when ZF receivers are implemented at the relay and destination nodes, (ii) when the ZF precoders are implemented at the source and relay nodes, and (iii) when the ZF precoders/receivers are implemented at the relay nodes. 2181 can be straight forward applied to the cases where the nodes are antennas. Practical example of this model is the communication between the base station and users through relay station, in up-link (the first scenario, ZF receivers at the relay and destination nodes) or downlink (the second scenario, ZF precoders at the source and relay nodes) or multi-pair communication through relay station (the third scenario, ZF precoders/receivers at relay nodes). Therefore, the received signal vector at the relay nodes, yr = [y1, …, yNr]T, can be expressed as yr = as G1Ws x + nr Fig. 1 Block diagram of a two-hop AF relay system with ZF processing in the presence of one eavesdropper Throughout this paper, Monte Carlo simulations are presented to confirm the correctness of the analysis. The results reveal that increasing the number of source, relay and/or destination nodes can be generally favourable or unfavourable to the system security depending on the practical scenario. It will also be demonstrated that increasing the source/relay power can considerably enhance the secrecy capacity and secrecy outage probability of the systems under consideration. The notations used in this paper are: Bold uppercase and bold lowercase letters denote matrices and vectors, respectively. Transpose operation, conjugate operation and conjugate transpose are denoted by . T, . ∗ and . H, respectively. The notation ∥ . ∥ denotes Euclidean norm and . represents the absolute value of a scalar. log . represents logarithm of base-2; metric, complex Gaussian distribution with mean μ and variance σ 2 is denoted by CN μ, σ 2 ; I identity matrix and det A denotes the determinant of matrix A. diag{a} is a diagonal matrix whose diagonal elements are the elements of the vector a; E . denotes expectation; Tr . is the trace of a matrix; A k is the kthcolumn in matrix A and A k, k is the element k, k in matrix A. 2 System model We consider a MIMO AF relay system consisting of Ns source nodes transmitting independent messages to Nd destination nodes via Nr relay nodes in the presence of a single antenna passive eavesdropper node to eavesdrop a specific message in the system, as shown in Fig. 1. As indicated in the figure, the channels between the nodes are denoted as G1 ∼ CNNr, Ns 0Nr × Ns, INr × Ns , G2 ∼ CNNd, Nr 0Nd × Nr, INd × Nr , and h ∼ CN1, Nr 01 × Nr, INr . In order to focus our study on the cooperative phase, i.e. relayto-destination link, we assume that all communications are performed through the relaying nodes and that there are no direct links between the source and destination/eavesdropper due to the deep shadowing. This assumption is well studied in the literature for the relay systems [25–27]. The assumption refers to the systems where the source communicates with the relay by a local connection [28], and it also refers to the relay systems in which the broadcast phase is secure. This situation occurs in systems where the source and the relay are located in one cluster, while the destination and the eavesdropper are located in another and the communication can only rely on the relay. Therefore, transmission between the source nodes and destination nodes is achieved as follows. In the first phase, the source nodes transmit their independent messages to the relay, and in the second phase the relay forward the received messages to the legitimate destination nodes. In this paper, we also assume that there is full cooperation between the nodes where the ZF is performed, i.e. the cooperating nodes exchange all the information with each other. Our analysis 2182 (1) where Ws is the Ns × Ns source weight matrix, x is the Ns × 1 transmitted signal vector with variance INs, nr is an Nr × 1 additive white Gaussian noise (AWGN) vector at the relay node with variance INrσr2 and as is the source normalisation constant which is designed to constrain the total transmit power at the source Ps [29, 30], and is given by as = Ps Tr E WsWsH . (2) Consequently, the received signal vector at the destination nodes, yd = [y1, …, yNd]T, can be written as yd = as ar Wd G2WrG1Ws x + ar Wd G2Wrnr + Wd nd (3) where Wd is the Nd × Nd destination weight matrix, Wr is the Nr × Nr relay weight matrix, nd is an Nd × 1 AWGN vector at the destination node with variance INdσd2 and ar is the relay normalisation constant which is designed to constrain the relay transmit power Pr , and is expressed as [29, 30] ar = (Pr /σr2) Tr E WsWsH (Ps /σr2) Tr E Q + Tr E WsWsH Tr E WrWrH (4) where Q = Wr G1 Ws WsH G1H WrH. On the other hand, the received signal at the eavesdropper is ye = asarh WrG1Ws x + arhWrnr + ne (5) where ne is the AWGN at the eavesdropper with variance σe2. Assuming that full-channel state information (CSI) is unknown at the transmitter, the ergodic secrecy capacity can be obtained as [31, page 4692] [32, Eq. (5)] C̄s = E Cd − E Ce + (6) where Cd and Ce are the destination and eavesdropper capacities given by Cd = 1/2 log 1 + γd and Ce = 1/2 log 1 + γe , respectively, with γd and γe denote the SINRs at the destination and eavesdropper, respectively. The factor 1/2 is resulted from the fact that two time slots are required for transmitter-to-receiver data transmission [9]. It should be highlighted here that when the transmitter has only the CSI of the main channel, (6) will represent the lower bound of secrecy capacity [33, 34]. In case the transmitter either knows both channels or only the legitimate channel the power allocation scheme can be applied to further improve the secrecy performance. In this case, non-zero ergodic secrecy capacity can be attained by allocating the power over the channel conditions for which the destination channel is better than the eavesdropper channel [31]. The other performance measure that we will be looking into in this paper is the secrecy outage probability Pout which is defined as the probability that the secrecy capacity is less than the target secrecy rate R0 and is expressed mathematically as [35, 36] IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 Pout Rs = Pr Cs < R0 . (7) Conventionally, to achieve optimal secrecy capacity of the system under consideration, a complex optimisation problem should be solved. However, in order to reduce complexity, we implement linear ZF scheme instead at various location as in the following sections. 3 Scenario 1: ZF at the relay and destination nodes In this scenario, we analyse the secrecy capacity and secrecy outage probability when ZF receivers are implemented at the relay and destination nodes. It is assumed that the relay and destination know G1 and G2, respectively. It is also assume that Nr > Ns and Nd > Nr. According to [30, 37], the weights at the nodes are given by Ws = INs Wr = P G1H G1 −1 G1H G2H γdk = C̄s1 = E Cd − E Ce E Ce = ar2 G1H G1 −1 k, k Wd WdH σd2 k, k (9) 2 + σd2 G2H G2 . −1 ∑ i = 1, i ≠ k N as ar ∑i =s 1, i ≠ k 2 2 (10) Ns (17) (18) 1 + γdk <υ 1 + γek (19) (20) Pout = 1 − Pr γdk > υ + υ γek − 1 ∣ γek = Φ ∫ −∫ (1 − υ)/ υ 0 ∞ (1 − υ)/ υ 2 2 2 . 2 2 hWr g1i + ar ∥ hWr ∥ σr + σe Ps Ns E Cd = (13) 1 2 ln 2 × ∫ 0 ∞ IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 f γek Φ dΦ (22) F̄ γdk υ + υ Φ − 1 f γek Φ dΦ 2 zc/a Nd − Nr + 1 e−(z / a) c + b Γ Nd − Nr + 1 × × Nr − Ns ∑ zb/a p! p=0 ((q − p + 1)/2) b c p Nd − Nr + p ∑ Nd − Nr + p q q=0 Jp − q − 1 (23) 2z bc a where F̄ γdk is the complementary cumulative distribution function (CCDF) of γdk, which is given by (23), shown at the top of the next page, where z = υ + υ Φ − 1, a = PsPr /σs2σr2, b = Ns(Pr /σr2), and c = (Ps /σs2)Nr + (Ns2 /(Nr − Ns)) and f γek is the probability density function (PDF) of γek and given by 2 γr z 1 1 − e− z γrs z (1 + N d − N r)/2 2z F̄ γdk z = (12) The normalisation constants at the source and relay nodes of this scenario are determined simply by substituting the weights (8) into (2) and (4) to obtain the following: (21) and consequently hWr g1i xi + ar hWr nr + ne (11) as2 ar2 h Wr g1k 2 as = 0 Pout = Pr γdk < υ + υ γek − 1 . where a = as ar, g1k and g1i are the kth and the ith columns in the matrix G1, respectively. Hence, the SINR of the kth transmitted signal at the eavesdropper can be given as γek = −1 + Ns + ψ where υ = 22R0. In addition, Pout = 1 − Ns 0 which can also be expressed in terms of the destination and eavesdropper SINRs as k, k On the other hand, the received signal at the eavesdropper of the kth transmitted signal is expressed as yek = a h Wr g1k xk + a 1 −ψ 2ln 2 By conditioning on γek, (20) can be written as as2 ar2 σ k, k r (15) Pout = Pr Cd − Ce < R0 where A = Wd G2 Wr G1 Ws and B = Wd G2 Wr. Now substituting (8) into (9) yields γdk = + E Cd and E Ce are given by (16) and (17), shown at the top of the next page, respectively, where γrs = ar2 as2 /σd2 , γr = ar2 σr2 /σd2 and ψ 0 . is the Poly-gamma function. (see (16)) Pout = Pr as ar A A ar2 B BH σr2 + and Ns /(Nr − Ns) = Tr E [38, Lemma 1]. The ergodic secrecy capacity of this scenario can be obtained as H 2 Nr = Tr E Q , WrWrH (8) where P is an INr × Ns matrix to ensure that the Nr signals are transmitted at the relay nodes. From (3), the SINR of the kth transmitted signal at the destination can be written as follows: 2 Ns = Tr E WsWsH , where (14) Proof: The proof is provided in Appendix 1. □ The secrecy outage probability of this scenario can be obtained as follows. Based on (7), Pout can also be written as −1 Wd = G2H G2 (Pr /σr2) Ns Nr − Ns (Ps /σr2) Nr Nr − Ns + Ns2 ar = (1 + N r − N s)/2 J1 + Nr − Ns 2 γr z Γ Nr − Ns + 1 J1 + N d − N r 2 z dz . Γ Nd − Nr + 1 (16) 2183 3.1 Numerical results In this subsection, numerical results of the secrecy capacity and secrecy outage probability for system 1 are presented. In all our evaluations from this point onward, Monte Carlo simulations are included and the channel coefficients are randomly generated in each simulation run. It should also be mentioned that, the integrals are efficiently evaluated by using numerical integration. To start with, Fig. 2 depicts the ergodic secrecy capacity as a function of Ns for Pr = 2, 4 , 6, and 8 dBw when Ps = 10 dBw, Nd = 50, Nr = 42 and noise power at all nodes is set σr2 = σd2 = σe2 = 10 dBm. One can see that the analytical results obtained from (15) and the simulated ones are in good agreement. It is also apparent that the secrecy capacity degrades with increasing Ns and this is because increasing Ns leads to decrease the normalisation constant at the source as. In addition, the secrecy capacity enhances as Pr is increased, and this enhancement becomes less significant when Ns is larger than 35. To illustrate the impact of Nr and Ps on the secrecy capacity, we plot in Fig. 3 the ergodic secrecy capacity versus Nr for several values of Ps when Pr = 2 dBw, Ns = 10, Nd = 50 and the noise power at all nodes is again set as σr2 = σd2 = σe2 = 10 dBm. The first observation one can see here is that as Ps increases the ergodic secrecy capacity enhances and this enhancement becomes less significant when Nr is relatively high. The other interesting observation from this figure is the fact that for each value of Ps, there exist an optimal Nr value that maximises the ergodic secrecy capacity. This can be explained by the fact that, increasing Nr will lead to increase ar to an optimal value after that increasing Nr will result in smaller values of ar. Therefore, adding more relay nodes might be detrimental to the system performance. As for the secrecy outage probability, it is plotted in Fig. 4 versus the threshold value of the secrecy rate for different values of Nd when Ps = − 13 dBw, Pr = − 13 dBw, Nr = 11, Ns = 10, and σr2 = σd2 = σe2 = 0 dBm. It is clearly visible that the analytical results agree well with the simulated ones. It is also observed that this probability deteriorates as Nd is increased which is due to the fact that increasing Nd will result in increasing the diversity in the second phase. Similar trend is observed in [30], in terms of outage probability. Fig. 2 Ergodic secrecy capacity versus Ns for various values of Pr Fig. 3 Ergodic secrecy capacity versus Nr for various values of Ps 4 Scenario 2: ZF at the source and relay nodes In this section, the ergodic secrecy capacity and the corresponding secrecy outage probability are analysed when the ZF precoders are implemented at the source and relay nodes. Similarly, as in the previous section, it is assumed here that the source nodes know G1 and the relay nodes know G2, and that Ns > Nr, Nr > Nd. The weights at the legitimate nodes in this scenario are [30, 37] Ws = G1H G1 G1H −1 Wr = G2H G2 G2H −1 P1 P2 Wd = I N d where P1 is an INr × Ns matrix to ensure that Nr out of Ns messages are sent at the source, and P2 is an INd × Nr matrix to also ensure that Nd out of Nr messages are sent at the relay. By substituting (25) into (9), we can obtain Fig. 4 Secrecy outage probability of system 1 f γek Φ = − 1 + Φ − Ns 1 − Ns . (24) Proof: The proof is provided in Appendix 2. □ Finally, the secrecy outage probability is obtained by substituting (23) and (24) into (22). 2184 (25) γdk = as2 ar2 . ar σr2 + σd2 2 (26) On the other hand, the received signal at the eavesdropper node of the kth signal is IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 where Nr /(Ns − Nr) = Tr E WsWsH , WrWrH Nd /(Nr − Nd) = Tr E Q , and Nd /(Nr − Nd) = Tr E [38]. The ergodic secrecy capacity of this scenario can be obtained as C̄s2 = E Cd − E Ce + (31) E Cd and E Ce are given by (32) and (33), respectively, where N1 = Nr − Nd + 1, N2 = Nr − Nd + 2, and N = Nr − Nd E Cd = 1 log 1 + γdk . 2 (32) (see (33)) Proof: The proof is provided in Appendix 3. □ As for the secrecy outage probability of this scenario, the PDF of the SINR at the eavesdropper, f γek Φ , in (22) cannot be easily expressed in a closed-form. However, numerical solution of this problem does not introduce any computational difficulties and hence, only simulation results are presented below: Fig. 5 Ergodic secrecy capacity versus Nd for various values of Pr 4.1 Numerical results Fig. 6 Ergodic secrecy capacity versus Nr for various values of Ps yek = a h Wr k xk + a Nd ∑ i = 1, i ≠ k h Wr i xi + ar hWr nr + ne (27) where a = asar. Therefore, the SINR at the eavesdropper of the kth message is given by γek = N as ar ∑i =d 1, i ≠ k 2 2 as2 ar2 h Wr h Wr i 2 k 2 + ar ∥ hWr ∥2 σr2 + σe2 2 (28) Again, by substituting (25) into (2) and (4), we can find the normalisation constants as as = ar = Ps Ns − Nr Nr (Pr /σr2) Nr Nr − Nd . (PS /σr2)Nd Ns − Nr + Nr Nd E Ce = 1 ln 2 − ∫ ∞ Nd − 1 0 (1/2) + N1 − (N2 /2) Nd IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 We discuss here the secrecy capacity and secrecy outage probability of system 2. Fig. 5 presents the ergodic secrecy capacity versus Nd for Pr = 2, 4, 6, and 8 dBw when Ns = 50, Nr = 42, Ps = 10 dBw and σr2 = σd2 = σe2 = 10 dBm. From this figure, we can observe that, the secrecy capacity decreases as the number of destination nodes increases. However, this decreasing becomes more significant when Nd is larger than 25. It is also noted that when Nd approaches Nr, i.e. 42, the ergodic secrecy capacity seriously deteriorates. The last remark on these results is that increasing Pr will always result in enhancing the secrecy capacity. Similar to the previous system, and in order to show the impact of Nr on this capacity of this system, we plot in Fig. 6 the ergodic secrecy capacity as a function of Nr for various values of Ps when Nd = 10, Ns = 45, Pr = 2 dBw and σr2 = σd2 = σe2 = 10 dBm. In general, it is evident that as Ps increases the secrecy is enhanced and this is because increasing Ps leads to increase the interference power at the eavesdropper. It is also worthwhile pointing out that for each value of Ps there exists an optimal Nr value that will maximise the ergodic secrecy capacity and therefore the latter should be carefully chosen. This could be explained by the fact that increasing Nr will increase the normalisation constant at the relay ar while decreasing the normalisation constant at the source as. Therefore, as in the previous system, adding more relay nodes might not be always beneficial to the system secrecy. Furthermore, the secrecy outage probability results of system 2 are presented in Fig. 7 when Nr = 10, Nd = 5, Ns = 15, 20, 25, Pr = 5 dBw, Ps = 6 dBw, and σr2 = σd2 = σe2 = 30 dBm. It is clear that this probability decreases as the number of the source nodes increases, since increasing Ns leads to increase the normalisation constant at the source as. (29) 5 Scenario 3: ZF at the relay node (30) In this scenario, the ergodic secrecy capacity and secrecy outage probability are analysed when ZF precoders/receivers are deployed at the relay node only. It is assumed here that the relay node knows both G1 and G2, and that Nr > Ns and Ns = Nd = N. The weights at the legitimate nodes of this scenario are [30, 37] (1/2) + N1 − (N2 /2) Z (1/2) −1 + N2 J N2 − 1, 2 z N! Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z dz . z N! Nd − 1 Z (33) 2185 Dik, j = { Nr − N + i − 2 ! Nr − N + i + j − 3 !, j=k Nr − N + i − 1 ! Nr − N + i + j − 2 !, j≠k (40) At high SNR, last equation (39) is reduced to ar = (Pr /σr2) Nr − Nd . (Ps /σr2) (41) The ergodic secrecy capacity of this scenario can be obtained as C̄s3 = E Cd − E Ce E Cd = 1 2ln 2 ∫ 0 ∞ + 1 1 − e− zt e− zb z 2 z(1 + Nr − Ns)/2 J 1 + Nr − Ns, 2 z dz Γ Nr − Ns + 1 Fig. 7 Secrecy outage probability of system 2 Ws = INs Wr = G2H G2 G2H −1 G1 G1H −1 G1H (34) Wd = INd . Substituting these weights into (9) produces γdk = as2 ar2 ar2 G1H G1 −1 2 σ k, k r + σd2 . (35) On the other hand, the received signal at the eavesdropper node of the kth message is given by yek = a hWr G1 k xk +a Nd ∑ i = 1, i ≠ k hWr G1 i xi + arhWr nr + ne (36) where a = as ar. Consequently, the SINR at the eavesdropper node of the kth transmitted signal can be written as γek = N as2 ar2 hWr G1 as2 ar2 ∑i =d 1, i ≠ k hWr G1 i 2 k 2 + ar2∥ hWr ∥2 σr2 + σe2 . (37) Now, by substituting the weights (34) into (2), the normalisation constants at the source node becomes as = Ps Ns (38) where Ns = Tr E WsWsH . Similarly, substituting these weights into (4) yields [30, 39] ar = (Pr /σr2) Nr − Nd Ξ N (Ps /σr2) Ξ + Nr − Nd ∑ k = 1det Dk (39) N while Ξ = ∏l = 1 Nr − l ! N − l ! Nr − l ! and Dk is N × N matrix the elements of which are given by E Ce = 1 2 ln 2 − 2186 ∫ ∞ 2 Nd − 1 0 N2 1 + N1 − 2 2 2 Nd (1/2) + N1 − (N2 /2) (42) (43) (see (44)) E Cd and E Ce are given by (43) and (44), respectively, where t = as2 /σr2 and b = σd2 /σr2 ar2. Proof: The proof is provided in Appendix 4. □ Similarly as in the previous scenario, the secrecy outage probability cannot be expressed in a closed-form. Hence, only numerical results will be presented in the following section using software tools. 5.1 Numerical results Fig. 8 shows the ergodic secrecy capacity as function of Ns and Nd for Pr = 2, 4, 6, and 8 dBw when Nr = 50, Ps = 10 dBw, and σr2 = σd2 = σe2 = 10 dBm. As we can see from this result, the ergodic secrecy capacity deteriorates as Ns and Nd are increased for all values of Pr. The other observation is that, for given values of Ns and Nd, increasing Pr results in enhancing the secrecy capacity and this enhancement becomes less significance when Pr is larger than 6 dBw. Moreover, the corresponding secrecy outage probability for this system is shown in Fig. 9 for different Nr values with Ns = Nd = 5, Ps = 4 dBw, Pr = 3 dBw, and σr2 = σd2 = σe2 = 30 dBm. It is seen that this probability degrades as Nr is increased and this is because increasing Nr will generally increase the system diversity in the first phase, as it is found in [30], in terms of outage probability. 6 Conclusion In this paper, the ergodic secrecy capacity and the secrecy outage probability of MIMO AF relay systems in the existence of a passive eavesdropper are analysed for different scenarios: (i) when ZF receivers are implemented at the relay and destination nodes, (ii) when the ZF precoders are implemented at the source and relay nodes, and (iii) when the ZF precoders/receivers are implemented at the relay nodes. In each scenario, the ergodic secrecy capacity and secrecy outage probability are investigated. Results showed that the number of source, relay and/or destination nodes can control on both the ergodic secrecy capacity and secrecy outage probability, based on the ZF design strategy. It was also shown that the ergodic secrecy capacity and secrecy outage probability can be further improved by increasing the source and/or the relay transmit power. Z (1/2) −1 + N2 J N2 − 1, 2 z N! Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z dz z N! Nd − 1 Z (44) IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 [12] [13] [14] [15] [16] [17] [18] [19] Fig. 8 Ergodic secrecy capacity versus Ns /Nd for various values of Pr [20] [21] [22] [23] [24] [25] [26] [27] [28] Fig. 9 Secrecy outage probability of system 3 7 Acknowledgments This paper was an extension of the authors' previous work published in [41]. The authors extend their work by studying the impact of other parameters on the secrecy capacity and considering the secrecy outage probability. 8 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Wyner, A.: ‘The wire-tap channel’, Bell Syst. Tech. 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To begin with, (10) can be rewritten as γrs γr X + Y γdk = k, k X = G1H G1 −1 k, k , and . Using lemma 1 in [40], the destination ergodic 1 2 ln 2 ∫ 0 1 ℳR z − ℳ γrs + R z dz z ℳR z = ∫ ∞ −∞ − zr e f R r dr (47) while f R r is the PDF of R. Since X and Y are independent random variables, we can write [42] ℳR z = ℳγr X z ℳY z (49) ∫ 0 ∞ 1 1 − e− z γrs ℳR z dz . z ℳX z = ∫ 0 e− zx −1/ x e dx Γ N r − N s + 1 X Nr − Ns + 2 (51) where Γ . is the Gamma function. Using the identities in [42, 43], we can get ℳγr X (z) = 2 γr z (1 + N r − N s)/2 J1 + Nr − Ns 2 γr z Γ Nr − Ns + 1 (52) where J . denotes the modified Bessel function of the second kind. Similarly, the MGF of Y can be found to be ℳY z = 2 z(1 + Nd − Nr)/2 J1 + Nd − Nr 2 z . Γ Nd − Nr + 1 (53) Substituting (52) and (53) into (48) and then into (50), we find the destination ergodic capacity. Now to derive the eavesdropper ergodic capacity, (12) can be written as 2188 0 (55) i 2 . Using lemma 1 in [40], 1 ℳY z − ℳ X + Y z dz . z (56) − Ns − 1 − Ns (57) . (58) Now by substituting (57) and (58) into (56), the eavesdropper ergodic capacity can be found as in (17). 9 Appendix 2 To drive the CDF of SINR at the destination for scenario 1, we first rewrite SINR at the destination node as a bX + cY γdk = a = as2 ar2, Y = G2H G2 −1 k, k b = ar2σr2, c = σd2 , (59) X = G1H G1 −1 k, k , and . The CDF of γdk is given by a ≤z bX + cY (60) = Pr χ a Υ − cz ≤ z bΥ (50) By using the PDF of X presented in [30, 37], we can derive the MGF of X as follows [42]: ∞ ∞ Fγdk z = Pr Therefore, (46) becomes (54) as2 ar2 X as2 ar2 Y ℳX + Y z = z + 1 where ℳ γrs + R z = e− z γrsℳR z . . Since Y and X + Y both have Gamma distribution, their MGFs are given by, respectively, (48) and 1 2 ln 2 ∫ 1 2 ln 2 E Ce = (46) where R = γr X + Y and ℳR z denotes the moment generating function (MGF) of R and calculated as E Cd = + ar ∥ hWr ∥2 σr2 + σe2 ℳY z = z + 1 ∞ 2 2 N capacity can be expressed as E Cd = k k where X = h k 2, Y = ∑i =s 1, i ≠ k h the ergodic eavesdropper capacity is (45) where γrs = ar2 as2 /σd2 , γr = ar2 σr2 /σd2 , −1 h 2 In interference limited systems, the noise power can be neglected in comparison with the interference power [44–46], which represents the worst case scenario and produces the upper bound of the eavesdropper capacity; hence, (54) becomes 8 Appendix 1 Y = G2H G2 as2 ar2 h N as ar ∑i =s 1, i ≠ k 2 (61) while χ = 1/ X and Υ = 1/Y. We get Fγdk z = ∫ 0 ∞ z by a y − cz Fχ f Υ y dy . (62) Simply we can rewrite (62) as Fγdk z = 1 − ∫ ∞ cz / a F̄ χ z by a y − cz f Υ y dy . (63) From [37, 47, 48], the CDF of χ and the PDF of Υ are given, respectively, by Fχ x = fΥ y = γ Nr − Ns + 1, x Γ Nr − Ns + 1 (64) yNd − Nre− y Γ Nd − Nr + 1 (65) where γ , is incomplete Gamma function. Now, by changing variables and substituting (64) and (65), we can get the CDF and then the CCDF of γdkas in (23). IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 To find the PDF of the SINR at the eavesdropper, we first derive the CDF, Fγek, which is defined as Fγek = Pr γek ≤ Φ . where N1 = Nr − Nd + 1, N2 = Nr − Nd + 2, and N = Nr − Nd. Similarly, ℳΥ z is obtained as in (66) ℳΥ z Substituting (55) into (66) we get Fγek = Pr X ≤ Y Φ . = (67) Fγek = Pr X ≤ Y Φ ∣ Y = y ∫ ∞ 0 ∫ ∞ 0 Nd − 1 Z FX yΦ f Y y dy . By substituting ℳΥ z and ℳ β z into (75), the eavesdropper ergodic capacity can be found. (69) 11 Appendix 4 1 − e− yΦ y Ns − 2 e − y dy . Γ Ns − 1 Fγek = 1 − 1 + Φ 1 − Ns . (71) − Ns 1 − Ns t γdk = where The PDF of γek is found by simply differentiating Fγek, therefore f γek Φ = − 1 + Φ To derive the destination ergodic capacity of the third scenario, (35) can be rewritten as (70) Using the identities in [43], (70) can be simplified to t = as2 /σr2 H ϕ = G1 G1 −1 k, k −1 H G1 G1 k, k b = σd2 /σr2 ar2. and (77) +b Now, by substitution and using lemma 1 in [40], the ergodic capacity at the destination can be expressed as E Cd = (72) 1 2ln 2 ∫ 0 ∞ 1 1 − e− zt e− zbℳϕ z dz . z (78) In order to find the MGF of ϕ, we can follow the same steps used to derive ℳX z in Appendix 1. Hence 10 Appendix 3 Since γdk is not random variable, E Cd can be simply expressed as 1 log 1 + γdk . 2 E Cd = (73) In order to derive the eavesdropper ergodic capacity in interference limited systems, which represents the worst-case scenario and gives the upper bound of the eavesdropper capacity, we can write (28) as as2 ar2 X γek = 2 2 as ar Υ N ∑i =d 1, i ≠ k 2 1 E Ce = 2 ln 2 ∫ 0 ∞ 2 1 ℳΥ z − ℳ β z dz . z (75) 1 + N1 − N2 2 Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z N! IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189 © The Institution of Engineering and Technology 2017 2 z(1 + Nr − Ns)/2 J 1 + Nr − Ns, 2 z . Γ Nr − Ns + 1 (79) Now, to calculate the eavesdropper ergodic capacity in interference limited systems, which represents the worst scenario and produces the upper bound of the eavesdropper capacity, (37) can be simplified as γek = where By using the PDF of the random variable β derived in [49], its MGF is found to be 2 Nd2 ℳϕ z = (74) where X = h Wr k , Υ = h Wr i , β = X + Υ. Aain, using lemma 1 in [40], we can express the eavesdropper ergodic capacity as ℳβ z = Z (1/2) −1 + N2 J N2 − 1, 2 N! (68) Since X and Y have exponential and Gamma distributions, respectively, (69) becomes Fγek = (1/2) + N1 − (N2 /2) . By conditioning on Y, we get = 2 Nd − 1 as2 ar2 h Wr1 2 as2 ar2 ∑ i = 1, i ≠ k Nd h Wr1 Wr1 = G2H G2 G2H N ∑i =d 1, i ≠ k k −1 . Let i (80) 2 ζ = h Wr 1 k 2 , 2 h Wr1 i and again by using lemma 1 [40], the ϱ= eavesdropper ergodic capacity can written as E Ce = 1 2 ln 2 ∫ 0 ∞ 1 ℳϱ z − ℳτ z dz z (81) where τ = ϱ + ζ; ℳϱ z and ℳτ z are exactly identical to ℳΥ z and ℳ β z derived in Appendix 3, respectively. Substituting these values in (81), the eavesdropper ergodic capacity for this scenario can be easily obtained. (76) 2189

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