Secrecy performance enhancement for underlay cognitive radio networks employing cooperative multi-hop transmission with and without presence of hardware impairments

entropy Article Secrecy Performance Enhancement for Underlay Cognitive Radio Networks Employing Cooperative Multi-Hop Transmission with and without Presence of Hardware Impairments Phu Tran Tin 1,2 , Dang The Hung 3, Tan N. Nguyen 4,* , Tran Trung Duy 5 and Miroslav Voznak 1 1VSB—Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Poruba, Czech Republic; [email protected] (P.T.T.); miroslav[email protected] (M.V.) 2Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City 71408, Vietnam 3Faculty of Radio-Electronics Engineering, Le Quy Don Technical University, Hanoi 11917, Vietnam; [email protected] 4Wireless Communications Research Group, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 72912, Vietnam 5Department of Telecommunications, Posts and Telecommunications Institute of Technology, Ho Chi Minh City 71007, Vietnam; [email protected] *Correspondence: [email protected] Received: 2 January 2019; Accepted: 20 February 2019; Published: 24 February 2019   Abstract: In this paper, we consider a cooperative multi-hop secured transmission protocol to underlay cognitive radio networks. In the proposed protocol, a secondary source attempts to transmit its data to a secondary destination with the assistance of multiple secondary relays. In addition, there exists a secondary eavesdropper who tries to overhear the source data. Under a maximum interference level required by a primary user, the secondary source and relay nodes must adjust their transmit power. We first formulate effective signal-to-interference-plus-noise ratio (SINR) as well as secrecy capacity under the constraints of the maximum transmit power, the interference threshold and the hardware impairment level. Furthermore, when the hardware impairment level is relaxed, we derive exact and asymptotic expressions of end-to-end secrecy outage probability over Rayleigh fading channels by using the recursive method. The derived expressions were verified by simulations, in which the proposed scheme outperformed the conventional multi-hop direct transmission protocol. Keywords: physical-layer security; underlay cognitive radio; cooperative multi-hop transmission; secrecy outage probability; hardware impairments 1. Introduction Security is one of the most important issues in wireless communication because of the broadcast nature of wireless medium. Conventionally, encryption/decryption algorithms that generate public/private keys are used to guarantee the security [ 1 , 2 ]. Recently, a security framework for the physical layer, called the wiretap channel or physical-layer security (PLS) [ 3 – 11 ], has been introduced as a potential solution. In PLS, difference between Shannon capacity of the data link and that of the eavesdropping link, named secrecy capacity, is commonly used to evaluate secrecy performance such as average secrecy capacity (ASC), secrecy outage probability (SOP) and probability of non-zero secrecy capacity (PNSC). Hence, to enhance the secrecy performance for wireless systems, researchers Entropy 2019,21, 217; doi:10.3390/e21020217 www.mdpi.com/journal/entropy Entropy 2019,21, 217 2 of 16 proposed efficient communication methods to increase channel capacity of the data links, and/or decrease that of the eavesdropping links. Indeed, in [ 12 – 14 ], opportunistic relay selection protocols are considered to enhance the quality of the data channels in one-hop and dual-hop relaying networks. In [ 15 – 18 ], the authors considered cooperative jamming approaches to reduce the data rate received at the eavesdroppers. The authors of [ 19 – 25 ] considered the secrecy performance enhancement for underlay cognitive radio (UCR) networks in which transmit power of secondary users (SUs) is limited by maximum interference levels required by primary users (PUs). The authors of [ 26 – 29 ] proposed secure communication protocols for two-way relay networks. In [ 30 – 33 ], the end-to-end secrecy performance of multi-hop relaying systems is investigated. Thus far, most published works related to performance evaluation assume that transceiver hardware of wireless terminals is perfect. However, in practice, it suffers from impairments due to phase noises, amplifier–amplitude non-linearity and in phase and quadrature imbalance (IQI) [34–36], which significantly degrade the performance of wireless communication systems. In [ 37 , 38 ], the authors proposed various relay selection methods to compensate the impact of the hardware imperfection. The authors of [ 39 ] studied the outage performance of partial relay selection and opportunistic relay selection schemes in the UCR networks under the joint of hardware imperfection and interference constraint. To the best of our knowledge, several published works evaluate the secrecy performance under the impact of imperfect transceiver hardware. In [ 40 ], the authors first studied the impact of the hardware imperfection on the secrecy capacity. In particular, the work in [ 40 ] considers the effects of IQI in one-hop OFDMA communication systems. The authors of [ 41 ] designed a secure massive MIMO system in the presence of a passive multiple-antenna eavesdropper and the hardware impairments. Reference [ 42 ] provided a power-efficient resource allocation algorithm for secure wireless-powered communication networks with the hardware noises. Taking hardware imperfection into account, the authors of [ 43 ] proposed an optimal power allocation strategy to maximize the instantaneous secrecy rate of a cooperative amplify-and-forward (AF) relaying scheme. In [ 44 ], we calculated PNSC of multi-hop relay networks over Nakagamim fading channels in presence of the hardware impairments. The results in [ 44 ] show that the hardware impairments significantly affect on the PNSC performance. However, there is no published work related to cooperative multi-hop PLS in the UCR networks. This motivated us to propose such a scheme and evaluate its performance. In the proposed protocol, named Cooperative Multi-Hop Transmission Protocol (CMT), a secondary source sends its data to a secondary destination via multiple secondary relays. In addition, in the secondary network, a secondary eavesdropper overhears the source data transmitted by the source and relay nodes. In addition, the secondary transmitters must adjust the transmit power to satisfy the interference constraint required by a PU and a maximal power threshold. The operation of the proposed scheme can be realized via one or many orthogonal time slots. At each time slot, the current transmitter finds an intended receiver that is nearest to the destination, and can receive the data securely and successfully. If this receiver is the destination, the data transmission ends. Otherwise, the procedure is repeated with the new selected transmitter. We also design a cooperative MAC method at each time slot for reversing the channel as well as selecting the potential receiver. For performance measurement, we first formulate the secrecy capacity under joint constraint of the limited interference and the hardware imperfection. When the hardware impairments are relaxed, we derive exact and asymptotic expressions of the end-to-end SOP over Rayleigh fading channels by using a recursive expression. Computer simulations were realized to verify the theoretical derivations as well as to show the advantages of the CMT method. The results show that the proposed scheme outperformed the conventional multi-hop direct transmission (MDT) protocol, and parameters such as the imperfect CSI estimations, the number of intermediate relays, the hardware impairment level and the position of the eavesdropper significantly affected the end-to-end SOP. The rest of this paper is organized as follows. System model of the proposed scheme is described in Section 2. In Section 3, exact and asymptotic expressions of the end-to-end SOP for the MDT and Entropy 2019,21, 217 3 of 16 CMT protocols are derived. The simulation results are presented in Section 4. Section 5presents our conclusions. 2. System Model As illustrated in Figure 1, we consider an M -hop secondary network, where the source (N0) communicates with the destination (NM) via M− 1 relay nodes denoted by N1 , N2 , ..., NM−1 . The relay nodes are numbered according to their distances to the destination, i.e., the relay NM−1 is nearest and the relay N1 is the furthest. In UCR, the source and the relay nodes must adapt the transmit power so that the co-channel interference levels caused by their transmission are below a threshold (Ith) given by a primary user (PU). Moreover, the transmit power of the secondary transmitters is also limited by a maximum power (Pth). In addition, in the secondary network, the eavesdropper (E) attempts to overhear the source data transmitted by the secondary transmitters. Before describing the operation of the proposed protocol, we give assumptions used in this paper. Entropy 2019,xx, 5 3 of 16 2. System Model 0 N PU E 1 N 1M NM N Source Destination Figure 1. System model of the proposed protocol. As illustrated in Figure 1, we consider an M -hop secondary network, where the source (N0) communicates with the destination (NM) via M -1 relay nodes denoted by N1 , N2 , ..., NM−1 . The relay nodes are numbered according to their distances to the destination, i.e., the relay NM−1 is nearest and the relay N1 is the furthest. In UCR, the source and the relay nodes must adapt the transmit power so that the co-channel interference levels caused by their transmission are below a threshold (Ith) given by a primary user (PU). Moreover, the transmit power of the secondary transmitters is also limited by a maximum power (Pth). In addition, in the secondary network, the eavesdropper (E) attempts to overhear the source data transmitted by the secondary transmitters. Before describing the operation of the proposed protocol, we give assumptions used in this paper. We assume that all of the relays are in the radio range of the source and destination nodes. We assume that all of the nodes have a single antenna, and the data transmission is hence split into orthogonal time slots. For ease of presentation and analysis, it is assumed that all of the nodes have the same structure, and the impairment levels are the same. We also assume that the eavesdropper is an active node, and hence the secondary nodes can estimate channel state information (CSI) between themselves and the node E [ 45 ]. Next, the data transmission between two secondary nodes is considered to be secure and successful if the obtained secrecy capacity is higher than a positive threshold (RS) . Otherwise, the data are assumed to be intercepted, which is referred to as a secrecy outage event. 2.1. Channel and Hardware Impairment Models Let dNi,Nj , dNi,PU and dNi,E denote distances of the Ni→Nj , Ni→PU and Ni→ E links, respectively, where i , j∈{0, 1, ..., M−1, M} . We also denote hNi,Nj , hNi,PU and hNi,E as channel coefficients of Ni→Nj , Ni→PU and Ni→ E links, respectively. Because the channels experience a Rayleigh fading distribution, the channel gains such as γi,j=|hNi,Nj|2 , γi,P =|hNi,PU|2 and γi,E = |hNi,E|2 follow exponential distributions. To take path-loss into account, we can model the parameters of the random variables (RVs) γi,j , γi,P and γi,E as [ 46 ]: λi,j=dβ Ni,Nj , λi,P=dβ Ni,PU and λi,E=dβ Ni,E , where βis path-loss exponent. Considering the data transmission between the transmitter X and the receiver Y (X ∈ {N0,N1, ..., NM−1}, Y ∈{N1,N2, ..., NM, E, PU}), the received data at Y is given as in [34–36]: y=pPXhX,Y (x0+ηt,X)+ηr,Y +νY, (1) where x0 is the source data, PX is the transmit power of X, hX,Y is channel coefficient of the X-Y link, ηt,X and ηr,Y are hardware noises at X and Y, respectively, and νYis Gaussian noise at Y. Figure 1. System model of the proposed protocol. We assume that all of the relays are in the radio range of the source and destination nodes. We assume that all of the nodes have a single antenna, and the data transmission is hence split into orthogonal time slots. For ease of presentation and analysis, it is assumed that all of the nodes have the same structure, and the impairment levels are the same. We also assume that the eavesdropper is an active node, and hence the secondary nodes can estimate channel state information (CSI) between themselves and the node E [ 45 ]. Next, the data transmission between two secondary nodes is considered to be secure and successful if the obtained secrecy capacity is higher than a positive threshold (RS) . Otherwise, the data are assumed to be intercepted, which is referred to as a secrecy outage event. 2.1. Channel and Hardware Impairment Models Let dNi,Nj , dNi,PU and dNi,E denote distances of the Ni→Nj , Ni→PU and Ni→ E links, respectively, where i , j∈{0, 1, ..., M−1, M} . We also denote hNi,Nj , hNi,PU and hNi,E as channel coefficients of Ni→Nj , Ni→PU and Ni→ E links, respectively. Because the channels experience a Rayleigh fading distribution, the channel gains such as γi,j=|hNi,Nj|2 , γi,P =|hNi,PU|2 and γi,E = |hNi,E|2 follow exponential distributions. To take path-loss into account, we can model the parameters of the random variables (RVs) γi,j , γi,P and γi,E as [ 46 ]: λi,j=dβ Ni,Nj , λi,P=dβ Ni,PU and λi,E=dβ Ni,E , where βis path-loss exponent. Considering the data transmission between the transmitter X and the receiver Y (X ∈ {N0,N1, ..., NM−1}, Y ∈{N1,N2, ..., NM, E, PU}), the received data at Y is given as in [34–36]: y=pPXhX,Y (x0+ηt,X)+ηr,Y +νY, (1) where x0 is the source data, PX is the transmit power of X, hX,Y is channel coefficient of the X-Y link, ηt,X and ηr,Y are hardware noises at X and Y, respectively, and νYis Gaussian noise at Y. Entropy 2019,21, 217 4 of 16 Similar to the work in [ 34 – 36 ], ηt,X , ηr,Y and νY are modeled as Gaussian random variables (RVs) with zero-mean and their variances are given, respectively, as var {ηt,X}=τ2 t, var {ηr,Y}=τ2 rPX|hX,Y|2, var {νY}=σ2 0, (2) where τ2 tand τ2 rare levels of the hardware impairments at X and Y, respectively. From Equations (1) and (2) , the instantaneous signal-to-interference-plus-noise ratio (SINR) is formulated by ΨX,Y =PX|hX,Y|2 τ2 t+τ2 rPX|hX,Y|2+σ2 0 =PX|hX,Y|2 κPX|hX,Y|2+σ2 0 , (3) where κ=τ2 t+τ2 ris the total hardware impairment level. Let us consider the transmit power PX of the node X in the underlay CR network. Firstly, PX is below the maximum transmit power, i.e., PX≤Pth . Secondly, the interference caused at the PU due to the transmission of the node X must be below the interference threshold Ith, i.e., PX≤Ith (1+κ)|hX,PU|2. (4) Therefore, PXcan be given as PX=min Pth,Ith (1+κ)|hX,PU|2 =Pth min 1, µ (1+κ)|hX,PU|2, (5) where µ=Ith/Pth is assumed to be a constant. Combining Equations (3) and (5) yields ΨX,Y = Pmin 1, µ (1+κ)|hX,PU|2|hX,Y|2 κPmin 1, µ (1+κ)|hX,PU|2|hX,Y|2+1 , (6) where P=Pth/σ2 0. From Equation (6) , we can formulate the SINR for the Ni→Nj and Ni→ E links, where i,j∈{0, 1, ..., M}, respectively, as Ψi,j=Pmin (1, µ/γi,P)γi,j κPmin (1, µ/γi,P)γi,j+1, Ψi,E =Pmin (1, µ/γi,P)γi,E κPmin (1, µ/γi,P)γi,E +1. (7) Moreover, when the transceiver hardware of all the nodes is perfect, i.e., κ=κ2 t=κ2 r= 0, we can rewrite Equation (7) as Ψi,j=Pmin 1, µ γi,P γi,j, Ψi,E =Pmin 1, µ γi,P γi,E. (8) Entropy 2019,21, 217 5 of 16 Hence, the secrecy capacity obtained at Njdue to the transmission of Niis calculated as Ri,j=max 0, log21+Ψi,j−log2(1+Ψi,E) =log21+Ψi,j 1+Ψi,E + , (9) where [x]+=max (0, x). From Equations (7) and (9) , because Ψi,j P→+∞ ≈ 1 /κ and Ψi,E P→+∞ ≈ 1 /κ , the secrecy capacity at high Pregime can be given as Ri,j P→+∞ ≈log21+1/κ 1+1/κ+ =0. (10) Moreover, as κ=0, we have Ri,j=log21+Pmin (1, µ/γi,P)γi,j 1+Pmin (1, µ/γi,P)γi,E + P→+∞ ≈log2γi,j γi,E + . (11) 2.2. Operation of the Proposed Protocol Next, we describe the operation of the proposed protocol, in which a MAC layer operation is designed to reverse the channel. Similar to the CoopMAC proposed in [ 47 ], in the first time slot, before transmitting the data, the source sends a request-to-send (RTS) message to the destination and all of the relays. By receiving this message, all of the nodes can estimate CSI between themselves and the source, calculate the instantaneous secrecy capacity by using Equation (9) , and compare with RS . It is assumed that the source can exactly estimate the channel coefficients of the interference and eavesdropping links, and include these values into the RTS message. If the destination can receive the source data securely and successfully, i.e., R0,M≥RS , it will feedback a clear-to-send (CTS) message to inform. In this case, the source directly sends the data to the destination without using the relays. In the case where R0,M<RS , the destination has to generate a non-CTS message to request the help of the relays. Now, let us denote U1=nN11,N12, ..., N1r1o as set of the potential relays which can receive the data securely and successfully, i.e., R0,1u≥RS , where u= 1, 2, ..., r1 , 0 ≤r1≤M− 1, N1u∈{N1,N2, ..., NM−1} . To select the relay for the retransmission, we also propose a distributed relay selection method. Similar to the work in [48], the relay N1uwill set a timer given as ω1u=A λ1u,M , (12) where A is a predetermined constant. Then, the relay whose timer expires first will broadcast the CTS message, and it be selected to retransmit the data to the destination. We can observe from Equation (12) that the selected relay is nearest to the destination. It is worth noting that, if the set U1 is empty ( r1= 0), no relay node can retransmit the data to the destination, and this case is considereda secrecy outage event. In the case where r1≥1, the operation will be repeated with the new source. Generally, at the k th time slot (k≥1) , assume that the current source is Nik , ik∈{0, 1, ..., M−1} and i1= 0. Let Wk=Nik+1,Nik+2, ..., NM denote set of relays from the node Nik+1 to the destination. Similarly, Nik sends the RTS message to all of the nodes belonging to Wk . Then, if Rik,M≥RS , the destination generates the CTS message, and Nik will directly transmit the data to NM . Otherwise, the potential relay which belongs to Wk and is nearest to the destination will become the new source and repeat the process that Nik did. Indeed, we denote Uk as the set of the potential relays, i.e., Entropy 2019,21, 217 6 of 16 Uk=nNk1,Nk2, ..., Nkrko , where Uk⊂ Wk , 0 ≤rk≤M−ik . In addition, let us denote Zk= nNkrk+1,Nkrk+2, ..., NM−iko as set of the nodes that cannot receive the data securely, where krk+1< krk+2< ... <kM−ik and NkM−ik≡NM . Then, assume that k1<k2< ... <krk and rk≥ 1, using the relay selection method described above, the relay Nkr will become the new source at the (k+1) th time slot. This process is only stopped when NM can securely and successfully receive the data or there is no relay between the current source and the destination that can securely and successfully receive the data. It is noted that, to avoid the eavesdropper and combine the received data with maximal ratio combining (MRC) technique, the source and the selected relays use randomize-and-forward (RF) method [49,50]. In the proposed protocol, to select the successful relay at each time slot correctly, the CSI estimations over the data, interference and eavesdropping links are assumed to be perfect. However, in practice, the estimations may not be correct due to the time variation of the channel, finite number of pilot symbols and noises. Hence, we will discuss this problem in the next sub-section. 2.3. Imperfect Channel Estimation In this subsection, we consider the imperfect channel estimation at the transmitter Ni and the receiver Nj. From Equation (9), if Njwants to calculate the secrecy capacity Ri,j, it has to estimate the channel coefficient hNi,Nj correctly. In addition, Ni has to estimate the channel coefficients hNi,PU and hNi,E, which are then sent to Njthrough the RTS message. Let he Ni,Nj , he Ni,PU and he Ni,E denote the estimated CSIs of hNi,Nj , he Ni,PU and hNi,E , respectively; the correlation between he Ni,Nj and hNi,Nj ; he Ni,PU and hNi,PU ; and he Ni,E and hNi,E can be expressed, respectively as in [51]: he Ni,Nj=φDhNi,Nj+q1−φ2 DεD, he Ni,PU =φPhNi,PU +q1−φ2 PεP, he Ni,E =φEhNi,E +q1−φ2 EεE, (13) where φD , φP and φE are channel correlation factors, and εD , εP and εE are estimation errors. We can observe that if φD=φP=φE= 1, all of the channel estimations are perfect. If φD< 1, φP< 1, φE< 1, the channel estimations have errors, and the estimated secrecy capacity in Equation (9) is written by Re i,j=  log2   1+Pmin 1, µ γe i,P γe i,j 1+Pmin 1, µ γe i,P γe i,E      + , (14) where γe i,j=|he Ni,Nj|2 , γe i,P =|he Ni,PU|2 and γe i,E =|he Ni,E|2 . Again, we note that the CSI estimation errors may lead to the incorrect relay selection, which would degrade the system performance. 2.4. Multi-Hop Direct Transmission Protocol To show the advantages of the proposed protocol, we compared the secrecy performance of the proposed protocol with that of the conventional multi-hop direct transmission protocol (MDT) [ 44 ]. In the MDT scheme, the data are transmitted hop-by-hop from the source to the destination. Particularly, the data transmission is split into M orthogonal time slots. At the m th time slot, where m= 1, 2, ..., M , the node Nm transmits the source data to the node Nm+1 . If the communication between Nm and Nm+1 is secure and successful, Nm+1 will forward the data to the next hop in the next time slot. Otherwise, the data transmission is insecure and the secrecy outage event occurs. Similar to the MCT protocol, the source and relays in the MDT protocol use the RF technique. Entropy 2019,21, 217 7 of 16 3. Performance Analysis Firstly, we can formulate SOP of the Ni→Njlink as SOPDT i,j=Pr Ri,j<RS =Pr 1+Ψi,j 1+Ψi,E <ρ, (15) where ρ=2RS(ρ>1). From Equations (9) and (15), it is straightforward that, if κ>0, then SOPDT i,j P→+∞ ≈1. (16) When the transceiver hardware is perfect (κ=0) , we can derive the exact closed-form expression for SOPDT i,j. At first, setting x=γi,P, SOPDT i,jconditioned on xcan be given by SOPDT i,j(x)=Pr γi,j<ρ−1 Pmin (1, µ/x)+ργi,E. (17) Due to the independence of γi,jand γi,E, we can write SOPDT i,j(x)=Z+∞ 0fγi,E (y)Fγijρ−1 Pmin (1, µ/x)+ρydy. (18) Substituting probability density function (PDF) of the exponential RV γi,E fγi,E (y)=λi,E exp (−λi,Ey) , and the cumulative distribution function (CDF) of the exponential RV γi,jγi,E Fγi,j(y)=1−exp −λi,jyinto Equation (18), after some manipulations, we obtain SOPDT i,j(x)=1−λi,E λi,E +λi,jρexp −ρ−1 Pmin (1, µ/x). (19) Then, SOPDT i,jcan be obtained from SOPDT i,j(x)by SOPDT i,j=Z+∞ 0SOPDT i,j(x)fγi,P (x)dx. (20) Substituting Equation (19) and fγi,P (y)=λi,P exp (−λi,Py) into Equation (20) , we obtain an exact closed-form expression of SOPDT i,jas SOPDT i,j=Zµ 0 1−λi,E λi,E +λi,jρexp −ρ−1 P!λi,P exp (−λi,Px)dx +Z+∞ µ 1−λi,E λi,E +λi,jρexp −ρ−1 Pµx!λi,P exp (−λi,Px)dx =1−λi,E λi,E+λi,jρ"(1−exp(−λi,Pµ)) exp −λi,j ρ−1 P+λi,PPµ λi,PPµ+λi,j(ρ−1)exp −λi,Pµ−λi,j ρ−1 P#. (21) Furthermore, using the approximation in Equation (11) , an asymptotic closed-form expression for SOPDT i,jat high Pvalues can be provided by SOPDT i,j P→+∞ ≈Pr γi,j γi,E <ρ=1−λi,E λi,E +λi,jρ. (22) Entropy 2019,21, 217 8 of 16 3.1. Multi-hop Direct Transmission Protocol (MDT) Because the transmission on each hop is independent, the end-to-end SOP of the MDT protocol can be given as SOPMDT 0,M=1− M ∏ m=11−SOPDT m−1,m. (23) As κ= 0, substituting Equation (21) into Equation (23) , we obtain an exact closed-form expression for the end-to-end SOP of the MDT protocol as SOPMDT 0,M=1− M ∏ m=1   λm−1,E λm−1,E +λi,jρ  (1−exp (−λm−1,Pµ)) exp −λm−1,mρ−1 P +λm−1,PPµ λm−1,PPµ+λm−1,m(ρ−1)exp −λm−1,Pµ−λm−1,mρ−1 P    . (24) At high P regions, using Equation (22) , an approximate expression for Equation (24) can be obtained by SOPMDT 0,M P→+∞ ≈1− M ∏ m=1 λm−1,E λm−1,E +λm−1,mρ. (25) 3.2. Cooperative Multi-Hop Transmission Protocol (CMT) In the CMT protocol, the end-to-end SOP is expressed by a recursive expression as follows: SOPCMT Nik,Uk=∑ Uk Pr    1+Ψik,k1 1+Ψik,E ≥ρ,1+Ψik,k2 1+Ψik,E ≥ρ, ..., 1+Ψik,krk 1+Ψik,E ≥ρ, 1+Ψik,krk+1 1+Ψik,E <ρ,1+Ψik,krk+2 1+Ψik,E <ρ, ..., 1+Ψik,kM−ik 1+Ψik,E <ρ   =∑ Uk Pr                1+Pmin1,µ/γik,Pγik,k1 1+Pmin1,µ/γik,Pγik,E ≥ρ,1+Pmin1,µ/γik,Pγik,k2 1+Pmin1,µ/γik,Pγik,E ≥ρ, ..., 1+Pmin1,µ/γik,Pγik,krk 1+Pmin1,µ/γik,Pγik,E ≥ρ, 1+Pmin1,µ/γik,Pγik,krk+1 1+Pmin1,µ/γik,Pγik,E <ρ,1+Pmin1,µ/γik,Pγik,krk+2 1+Pmin1,µ/γik,Pγik,E <ρ, ..., 1+Pmin1,µ/γik,Pγik,kM−ik 1+Pmin1,µ/γik,Pγik,E <ρ                , (26) where SOPCMT Nik,Uk is SOP at k th time slot, k= 1, 2, ..., M . Then, the end-to-end SOP of the CMT protocol is given as SOPCMT 0,M=SOPCMT N0,U1. (27) Before calculating SOPCMT Nik,Uk, we give an example with M=3, where SOPCMT 0,3 is expressed by SOPCMT 0,3 =SOPCMT N0,{∅}+SOPCMT N0,{N1}+SOPCMT N0,{N2} +SOPCMT N0,{N1,N2}. (28) Equation (28) shows that there are 04 possible cases for the set U1 , i.e., U1={∅} , U1={N1} , U1={N2} , U1={N1,N2} . In Equation (28) , the terms SOPCMT N0,{∅} and SOPCMT N0,{N2} can be calculated as in (32). Considering the term SOPCMT N0,{N1}, which can be written by Entropy 2019,21, 217 9 of 16 SOPCMT N0,{N1}=SOPCMT N1,U2=SOPCMT N1,{∅}+SOPCMT N1,{N2}. (29) In Equation (29) , there are two possible cases for the set U2 , i.e., U2={∅} , U2={N2} , and SOPCMT N1,{∅} and SOPCMT N1,{N2} are SOP at the second time slots. In addition, SOPCMT N1,{∅} is calculated by Equation (32), while SOPCMT N1,{N2}is expressed by SOPCMT N1,{N2}=SOPDT 2,3 , (30) where, because the transmission between N2 and N3 is direct, Equation (21) is used to calculate SOPCMT N1,{N2}. Next, let us consider the term SOPCMT N0,{N1,N2} in Equation (28) , where the relay N2 will be selected for retransmitting the data to the destination. Similar to Equation (30), we have SOPCMT N0,{N1,N2}=SOPDT 2,3 . (31) Now, the recursive expression of SOPCMT Nik,Ukis given as in Lemma 1. Lemma 1. When κ=0,SOPCMT Nik,Ukcan be expressed as SOPCMT Nik,Uk=∑ Uk λik,E λik,E + rk ∑ t=1 λik,ktρ      exp − rk ∑ t=1 λik,kt(ρ−1) P1−exp −λik,Pµ +λik,PPµ λik,PPµ+ rk ∑ t=1 λik,kt(ρ−1) exp −λik,Pµ− rk ∑ t=1 λik,kt (ρ−1) P     +∑ Uk M−ik−rk ∑ v=1 (−1)vM−ik−rk ∑ Nj1,...,Njv∈Zk j1<j2<...<jv λik,E λik,E +v ∑ t=1 λik,jv+ rk ∑ t=1 λik,ktρ ×      exp −v ∑ t=1 λik,jv+ rk ∑ t=1 λik,ktρ−1 P1−exp −λik,Pµ +λik,PPµ λik,PPµ+ λik,E+ rk ∑ t=1 λik,kt!(ρ−1) exp −λik,Pµ−v ∑ t=1 λik,jv+ rk ∑ t=1 λik,ktρ−1 P      . (32) Proof. At first, we set x=γik,E and y=γik,P, and SOPCMT Nik,Ukconditioned on xand ycan be given by SOPCMT Nik,Uk(x,y) =∑ Uk"rk ∏ t=1 exp−λik,ktρ−1 Pmin (1, µ/y)+ρxM−ik−rk ∏ v=11−exp −λik,kvρ−1 Pmin (1, µ/y)+ρx# =∑ Uk exp − rk ∑ t=1 λik,ktρ−1 Pmin (1, µ/y)+ρx! +∑ Uk M−ik−rk ∑ v=1 (−1)vM−ik−rk ∑ Nj1,...,Njv∈Zk j1<j2<...<jv exp − v ∑ t=1 λik,jv+ rk ∑ t=1 λik,kt!ρ−1 Pmin (1, µ/y)+ρx!. (33) Entropy 2019,21, 217 16 of 16 39. Sharma, P.K.; Upadhyay, P.K. Cognitive relaying with transceiver hardware impairments under interference constraints. IEEE Commun. Lett. 2016,20, 820–823. [CrossRef] 40. Boulogeorgos, A.A.; Karas, D.S.; Karagiannidis, G.K. How Much Does I / Q Imbalance Affect Secrecy Capacity? IEEE Commun. Lett. 2016,20, 1305–1308. [CrossRef] 41. Zhu, J.; Ng, D.W.K.; Wang, N.; Schober, R.; Bhargava, V.K. Analysis and Design of Secure Massive MIMO Systems in the Presence of Hardware Impairments. IEEE Trans. Wirel. Commun. 2017 ,16, 2001–2016. [CrossRef] 42. 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