Secrecy Outage Performance for SIMO Underlay ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2016.2536801, IEEE Transactions on Vehicular Technology 1

Secrecy Outage Performance for SIMO Underlay Cognitive Radio Systems with Generalized Selection Combining over Nakagami-m Channels Hongjiang Lei, Huan Zhang, Imran Shafique Ansari, M ember, IEEE , Chao Gao, Yongcai Guo, Gaofeng Pan, M ember, IEEE , and Khalid A. Qaraqe, SeniorM ember, IEEE

Abstract—This paper considers a single-input multiple-output underlay cognitive wiretap system over Nakagami-m channels with generalized selection combining, where confidential messages transmitted from a single-antenna transmitter to a multipleantennas legitimate receiver are overheard by a multipleantennas eavesdropper. Passive eavesdropping scenario is considered in this work, while the channel state information of the eavesdropping channel is not available at the secondary transmitter. We derived the closed-form expression for the exact secrecy outage probability. Simulations are conducted to validate the accuracy of our proposed analytical models. Index Terms—Cognitive radio, multiple antennas, physical layer security, Nakagami-m fading, secrecy outage probability.

I. I NTRODUCTION A. Background

I

N recent years, cognitive radio networks (CRN) have become widely interesting since it is envisioned as one of the most promising technique to solve radio spectrum scarcity [1], [2]. In CRN, all the unlicensed users (secondary users, SUs) are permitted to share the licensed band with the licensed users (primary users, PUs) through spectrum underlay, overlay, and interweave approaches [3], [4]. In Manuscript received. The work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61471076, 61401372, the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1299, Natural Science Foundation Project of CQ CSTC (cstc2013jcyjA40040, cstc2015jcyjB0536), the special fund of Chongqing Key Laboratory (CSTC), Research Fund for the Doctoral Program of Higher Education of China under Grant 20130182120017, and the Fundamental Research Funds for the Central Universities under Grant XDJK2015B023. Parts of this publication, specifically Sections [I], [III], and [IV], were made possible by PDRA (PostDoctoral Research Award) grant PDRA1-122713029 from the Qatar National Research Fund (QNRF) (a member of Qatar Foundation (QF)). H. Lei and H. Zhang is with Chongqing Key Lab of Mobile Communications Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China, and with Key Laboratory of Optoelectronic Technology and Systems of the Education Ministry of China, Chongqing University, Chongqing 400044, China. (e-mail: [email protected]; [email protected]). I. S. Ansari and K. A. Qaraqe are with the Department of Electrical and Computer Engineering (ECEN), Texas A&M University at Qatar (TAMUQ), Education City, Doha, Qatar (email: {imran.ansari, khalid.qaraqe}@qatar.tamu.edu). C. Gao and Y. Guo are with the Key Laboratory of Optoelectronic Technology and Systems of the Education Ministry of China, Chongqing University, Chongqing 400044, China. (e-mail: {gaoc, [email protected]) G. Pan is with the School of Electronic and Information Engineering, Southwest University, Chongqing, 400715, China. (e-mail: [email protected])

underlay cognitive spectrum sharing networks, PU and SU are allowed to transmit concurrently in the same spectrum under a peak interference power threshold to guarantee a reliable communication at PU. In such complex environments, security issues become more serious. The security of traditional wireless communications mainly relies on the cryptographybased encryption technologies in upper-layer, such as AES (Advanced Encryption Standard) and DES (Data Encryption Standard) etc., the essence of which is to obtain information security through massive computing. With the development of semiconductor technologies and the emergence of quantum computing and cloud computing technology with a powerful parallel processing capability of data, the computing capability of the computer has been rapidly improved. Thus, the way of relying on the computational complexity degree to ensure information security is facing enormous challenges. Recently, with the development of multi-antennas, cooperative communications and coding technologies, secure information transmission in physical layer has become a hot issue of academic researches [5]. Differing from the traditional encryption technologies, no encryption key is employed in physical layer security technologies whereas the time-variable characteristics of wireless channels are employed to realize secure communication [6], [7], [8]. B. Related Works Good amount of literature is available focusing on the physical layer security issues in CRN [9] - [16]. A comprehensive review on physical-layer attacks in CR networks was presented in [9], [10]. The close relationship between the multi-antennas cognitive radio transmission problem and the secrecy transmission problem was explored in [11]. Ref. [12] studied the secrecy performance for a multiple-input and single-output (MISO) CRN containing a multi-antennas SU transmitter in presence of an eavesdropper, and two numerical approaches were proposed to compute the secrecy capacity and the capacity-achieving transmit covariance matrix. Refs. [13], [14], [15] investigated the secrecy performance of single-input multiple-output (SIMO) CRN and the closedform expression of secrecy outage probability (SOP) was derived. Physical layer security in multiple-input multipleoutput (MIMO) cognitive wiretap channels has been analyzed in [16] and the closed-form expression for the SOP of transmit antenna selection (TAS) /maximal ratio combining (MRC)

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


was derived. Obviously, all the channels in [12]-[16] were assumed to experience Rayleigh fading and only considered selection combining (SC) or MRC technique at destinations, respectively. Compared with Rayleigh fading, Nakagami-m model provides a good match to various empirically obtained measurement data [17] and is widely used for modeling wireless fading channels, including Rayleigh (m = 1) and one-sided Gaussian distribution (m = 0.5) as special cases. The closed-form expressions for the probability of non-zero secrecy capacity (PNSC), SOP, and average secrecy capacity were derived over the non-identically and identically independent Nakagami-m fading single-input multi-output (SIMO) channels in [18]. The impact of TAS/GSC for the physical layer security over MIMO wiretap channels has been analyzed in [19]. But all the results are not applicable to the CRN scenarios. The secrecy capacity was studied for the SIMO CRN over Nakagami-m fading channels in [20]. The SOP and PNSC of a basic underlay cognitive radio unit over Nakagami-m fading channels were presented in [21], which only considered the interference power at PU. Generalized selection combining (GSC) is a hybrid combining scheme that has been extensively studied in [22], [23]. This approach bridges the performance gap between MRC and SC, maximizes the performance and minimizes the complexity [22], and reduces the power consumption and the cost of radio frequency (RF) electronics at the receiver [23]. The impact of GSC and the outdated channel state information (CSI) on the outage probability in interferencelimited spectrum sharing networks was examined in [24]. The cognitive decode-and-forward relay network with TAS/GSC over Nakagami-m channels was taken into account in [25] and new exact closed-form expressions for the outage probability, the symbol error rate, and the ergodic capacity were derived, respectively. However, very few research has considered the security performance of the CRN with GSC scheme. For instance, physical layer security of TAS/GSC for MIMO CRN was studied in [26] and closed-form expressions for the exact and the asymptotic SOP were derived, while all the channels were assumed to experience Rayleigh fading. C. Motivation and Contributions The objective of this work is to examine the physical layer security for SIMO underlay CRN with GSC combining method over Nakagami-m channels. The main contributions of our work are listed as follows: 1) We analysis the secrecy performance for SIMO underlay CRN with GSC over Nakagami-m fading channels, and derive the closed-form expression of SOP, which is verified by simulations. 2) Compared with [21], in which only the the interference constraint at PU was considered and all the destinations were equipped with a single antenna, a more generalized system is considered in this work, as both the the interference constraint and the maximum transmission power constraint are considered while all the destinations are equipped with multiple antennas.

1'

' 3

6 1(

( Interfering channel to PU Main channel Wiretap channel

Fig. 1. System model signifying multiple antennas at the destination as well as the eavesdropper.

3) Compared with [13] and [14], in which SC or MRC scheme was used on D and E over Rayleigh channels, respectively, we consider GSC scheme at destinations over Nakagami-m channels, since Nakagami-m model includes Rayleigh (m = 1) and one-sided Gaussian distribution (m = 0.5) as special cases. Furthermore, GSC can offer a performance/implementation tradeoff between MRC and SC [22]. Moreover, SC and MRC schemes are the special cases of GSC. Thus, our proposed models can be applied to the scenarios wherein the eavesdropper adopts MRC or SC scheme by setting appropriate parameters. D. Structure The rest of the paper is organized as follows. In Section II, the system model considered in our work is described. The analysis on the SOP for SIMO underlay CRN over Nakagamim channels is proposed in Section III. In Section IV, we present and discuss the numerical results and the Monte-Carlo simulations. Finally, Section V concludes the paper. II. S YSTEM M ODEL In this work, we consider a SIMO cognitive wiretap system that consists of a PU (P ), a transmitter (S), a legitimate receiver (D), and an eavesdropper (E), as shown in Fig. 1. It is assumed that D and E are equipped with ND ≥ 1 and NE ≥ 1 antennas, whereas S and P are equipped with a single antenna. Both the primary channels and the secondary channels are assumed to experience independent and identically distributed (i.i.d.) quasi-static Nakagami-m fading with fading parameters mP , mD , and mE , and with average channel power gains ΩP , ΩD , and ΩE , respectively. The probability density function (PDF) and cumulative distribution function (CDF) of the channel gains between S and P can be given as mP m y mP −1 mP − Py e ΩP , (1) fYP (y) = Γ (mP ) ΩP 1 mP FYP (y) = 1 − Γ mP , y , (2) Γ (mP ) ΩP respectively, where Γ (·) is the Gamma R ∞ function, as defined by Eq. (8.310) of [27], Γ (a, x) = x e−t ta−1 dt is the upper



incomplete Gamma function, as defined by Eq. (8.350.2) of [27]. In this work, GSC is adopted at D and E, which means that receivers combine Lk (1 ≤ Lk ≤ Nk ) (k ∈ {D, E}) strongest receive antennas at D and E benefiting from perfect CSI estimation via pilot signals transmitted by S [19], [25]. Based on the pilot signals, receivers 2 perfectly estimate CSI, then arrange channel gains, hki (i = 1, ..., Nk ), in descending 2 2 2 order as hk1 ≥ hk2 ≥ ... ≥ hkNk . Note that prior to the transmission process, the selected number of antennas Lk (k ∈ {D, E}) at the receivers are determined by the limited number of RF chains due to the limitations on the hardware size and implementation complexity [25]. The instantaneous signal noise ratio (SNR) at both D and E can be expressed as Lk X PS (3) γk = γik = 2 Yk , k ∈ {D, E} , σ i=1 where PS is the transmit power at S and σ 2 is the noise Lk P hk 2 (k ∈ {D, E}) are the channel gains variance, Yk = i i=1

of S-D and S-E links. Lemma 1: The CDF and PDF of Yk are given by FYk (y) = Ak

P

Tk P

Bk

n=0

Sk

fYk (y) = Ak

X

Bk

n

Tk X

n −ηk y

(µnk

−

(4)

(5)

ηkn y)

, k ∈ {D, E} ,

respectively, where Sk (k ∈ {D, E}) denotes a set of (2mk + 1) tuples satisfying the condition: Sk = mP k −1 Φ Φ F F nΦ nk,0 , ..., nk,mk −1 , nk,0 , ..., nk,mk k,i = Lk − 1, i=0 m Pk F nk,i = Nk − Lk (k ∈ {D, E}), Tk = i=0

mk Lk ! + bF Ak ! = k (k ∈ {D, E}), mk Lk k Nk mk Lk k! (k ∈ {D, E}), = l!(k−l)! Γ(mk ) Ωk l Lk bFk Φ F F Γ(mk +bk +bk ) mk , Bk = aΦ (k ∈ {D, E}), k ak mk +bΦ +bF Ω k k k Lk nFk,i m k Q (Nk −Lk )! −1 aF = (k ∈ {D, E}), mk k (i−1)! Q i=0

bF k

i=1 nF k,i !

=

cF k

=

(Lk −1)! mk −1 Q nΦ k,i !

mQ k −1 i=0

mk Ωk

m Pk

i=1 Φ 1 nk,i (k i!

m Pk i=1

(i − 1) nF k,i (k ∈ {D, E}),

nF k,i (k ∈ {D, E}), ∈ {D, E}),

aΦ k bΦ k

= =

i=0

mP k −1 i=0

inΦ k,i (k ∈ {D, E}).

Ck (n−1) (ρk −n)! , 1 ≤ n Dk (n−ρk −1) (Tk −n)! , ρk

,

≤ ρk + 1 ≤ n ≤ Tk

0, n = 0 ρk − n, 1 ≤ n ≤ ρk , Tk − n, ρk + 1 ≤ n ≤ Tk 0, n = 0 mk , Ωk , 1 ≤ n ≤ ρk ςk , ρk + 1 ≤ n ≤ Tk ! n−i−1 i ζk + n − 1 P n−1 mk (−1) where Ck (i) = Ωk n n=0 ! F −ζk −n j ρk + n − 1 P ck n−1 n−j−1 , Dk (j) = (−1) ςk Lk n n=0 F −ρk −n c F , and ζk = mk + bΦ − Lkk k + bk . Proof: The proof is given in Appendix A. In underlay cognitive relay networks, the transmit power at S must meet the following two conditions: 1) the proportional interference power constraint; and 2) the fixed interference power constraint [28], [29]. Thus, PS can be expressed as IP PS = min Pmax , , (6) X where IP is the peak interference power at P , X = |hP | is the channel gains between S and P , and Pmax is the maximal transmit power at S. III. S ECRECY O UTAGE P ROBABILITY A NALYSIS In this work, we consider passive eavesdropping, where S does not has the CSI of the eavesdropper’s channel and has no choice but to encode the confidential data into codewords of a constant rate Rs > 0. If Rs is less than secrecy capacity, perfect secrecy can be achieved. Otherwise, information-theoretic security is compromised [30], [31]. Compared to the outage probability in conventional underlay CRN, which is defined as the probability that the mutual information is less than the required data rate, SOP is defined as the probability that the instantaneous secrecy capacity is less than Rs and used to evaluate the reliability of the information security transmission [6], [13]-[16]. The instantaneous secrecy capacity is given by [6] Cs (γD , γE ) = max {ln (1 + γD ) − ln (1 + γE ) , 0} ,

(7)

where ln (1 + γD ) and ln (1 + γE ) are the capacity of the main and eavesdropper channels, respectively. Using Eqs. (3) and (6), the SOP of CRN can be expressed as Pout = Pr {Cs (γD , γE ) ≤ Rs } = Pr {Cs (γD , γE ) ≤ Rs , PS = Pmax } | {z } ∆

F − 1) − bΦ λn = k = 0, When ck = 0 or ρk = mk (L nk n k −Tk ςk , n=0 0,n=0 n , µnk = 0,n=0 n−1,n≥1 , ηk = ςk ,n≥1 , −ςk n−Tk −1 /(n−1)!, n≥1

ςk =

cF 6 0 and ρ 6= 0, = k −ρk k mk ςk −ζk , n = 0 Ωk

2

n

λnk y µk −1

n=1

Sk

×e

n

λnk y µk e−ηk y , k ∈ {D, E} ,

When     n λk =       µnk =      n ηk =  

cF k Lk

+

mk Ωk .

=P

1 IP + Pr Cs (γD , γE ) ≤ Rs , PS = . X | {z }

(8)

∆

=P2



where H (x) =

In the following, we will derive P1 and P2 , respectively.

Using Eqs. (4) and (5), we can obtain H (x) = AD AE

 (14) µiD −n i (θ−1)x ηD i θ − 1 . xµD −n e− β × θn Ξ β n=0 X

(9)

Then substituting Eq. (14) into Eq. (13) and using Eq. (3.326.2) of [27], we can obtain P2 = Λ

−

j ηE Γ(n+µjE +1)

. j i θ+η j i θ+η j n+µE +1 (ηD (ηD E) E) Considering a more practical scenario, when E combines all its antennas under the MRC scheme (LE = NE ), we obtain j E

i

I1 = AD AE

µD TD X XX

BD λiD θn

SD i=0 n=0

! µiD ×e Γ (mE NE + n) n −mE NE −n ! µiD −n θ−1 mE i × + ηD θ , α ΩE η i (θ−1) − Dα

mE NE 0 E . where AE = Γ(mE1 NE ) m ΩE Using Eq. (2), we have 1 mP IP I2 = 1 − Γ mP , . Γ (mP ) ΩP Pmax

(11)

(12)

Then, P1 can be obtained by substituting Eqs. (10) or (11) and (12) into Eq. (9). B. The derivation of P2 When PS = IXP , making use of Eq. (7), we have IP P2 = Pr Cs ≤ Rs , PS = X X IP = Pr YD ≤ θYE + (θ − 1) , X > β Pmax Z ∞ = H (x) fX (x) dx, IP Pmax

TD X TE XXX

BD BE λiD λjE

SD SE i=0 j=1

µjE Γ(n+µjE )

0

BD BE λiD λjE

SD SE i=0 j=1

Rs . where α = Pσmax 2 , θ = e Using Eqs. (4) and (5), and Eq. (3.326.2) of [27], we have Z ∞ θ−1 I1 = FYD θy + fYE (y) dy α 0 TE TD X i (θ−1) XXX ηD BD BE λiD λjE e− α = AD AE (10) SD SE i=0 j=1  ! µi −n µiD X µiD θ−1 D  n × , Ξθ α n n=0

n+µ

TD X TE XXX

µiD

I2

where Ξ =

0

FYD θy + (θ − 1) βx fYE (y) dy, β =

IP σ2 .

A. The derivation of P1 When PS = Pmax , making use of Eq. (8), we have P1 = Pr {Cs ≤ Rs , PS = Pmax } θ−1 IP = Pr YD ≤ θYE + Pr X ≤ α Pmax Z ∞ θ−1 FYD θy + fYE (y) dy = α |0 {z } I1 IP × Pr X ≤ , Pmax {z } |

R∞

(13)

µiD

×

µiD n

X n=0

where Λ = mP ΩP

mP AD AE (mP −1)! ΩP i ηD (θ−1) . β

! θ m P

n

θ−1 β

,Ψ=

µiD −n



(15)

ΞΨ ,

GIP Γ µiD −n+mP , Pmax i

GµD −n+mP

, and

G= + Generally, when the MRC scheme is applied at E (LE = NE ), we obtain ! i TD µD i µ 0 XXX D P2 = Λ BD λiD θn Γ (mE NE + n) n SD i=0 n=0 ! µi −n −mE NE −n θ−1 D mE i × + ηD θ Ψ , β ΩE (16) m P 0 0 mP AD AE where Λ = (mP −1)! ΩP . Finally, SOP is obtained by substituting Eqs. (9) and (15) or (16) into Eq. (8). IV. N UMERICAL RESULTS In this section, Monte Carlo simulation results are presented to verify the proposed analytical models. The main parameters used in simulation and analysis are set as Pmax = 1 w, Rs = 0.1 bit/s/Hz, and mP = mD = mE = m. We plot the curves for various ΩP , ΩD , and ΩE for comparison purposes while IP varying. In Fig. 2, we compare the simulation and analytical results for SOP versus IP for various combinations of ΩP and m in CRN. Figs. 3 and 4 plot SOP versus IP with various combinations of ΩE and LD , combinations of ΩE and m, respectively. It is clear that analysis results match very well with Monte Carlo simulation curves. It is evident that for a fixed m or ΩE , SOP decreases with increasing IP , which increases the transmitting power at S. One can also see that, there exists a floor for SOP in the high IP region. It is because Ps = Pmax when Pmax → ∞, which means that in this case the transmitting power remains constant. Further, it can be observed from Fig. 2 that the SOP can be improved for a lower ΩP since the scenario with a lower ΩP outperforms the one with a higher ΩP as a higher ΩP represents that the condition between S and P is enhanced. We can also see from



0

10

m=1 −1

m=2

10

SOP

m=3

−2

10

ΩP = 1 Simualtion ΩP = 2 Simualtion ΩP = 3 Simualtion Analysis

−3

10

−30

−25

−20

−15

−10 −5 IP (dB)

0

5

10

15

Fig. 2. SOP versus IP with ΩD = 2, ΩE = 1, ND = 6, LD = 3, and NE = LE = 2.

0

We can find from Fig. 3 that SOP is improved while LD increases, since it means that more antennas are adopted to combine the received signals at D. Moreover, we can see from Figs. 3 and 4 that the SOP for a lower ΩE outperforms the one for a higher ΩE , since increasing ΩE means that the eavesdropper channel condition is enhanced. When PS = IP /X (Pmax → ∞), the system model falls into cognitive model. When LD = LE = 1 and m = 1, our results match the results of [13]. When ND = LD , NE = LE = 1, and m = 1, our results match the results of [14]. When ND = NE = LD = LE = 1, our results match the results of [21]. When PS = Pmax , it means that S adopts its maximum transmitting power to deliver the information to D. Obviously, the system model falls into non-cognitive model in this case. When ND = LD and NE = LE , our results match the results of [18]. When ND = LD , NE = LE , and m = 1, our results match the results of [32]. When NE = LE and m = 1, our results match the results of [33]. When ND = LD = 1 and LE = 1, our results match the results of [34].

10

V. C ONCLUSION

ΩE = 3 −1

10

SOP

ΩE = 2

−2

10

ΩE = 1

LD = 3 Simualtion LD = 4 Simualtion LD = 5 Simualtion Analysis

−3

10

−30

−25

−20

−15

−10 −5 IP (dB)

0

5

10

15

Fig. 3. SOP versus IP with ΩD = 2, ΩP = 3, ND = 6, NE = LE = 2, and m = 2.

0

10

VI. A PPENDICES

−1

10

SOP

ΩE = 3 ΩE = 2 −2

10

ΩE = 1

m = 1 Simualtion m = 2 Simualtion m = 3 Simualtion Analysis

−3

10

−30

In this paper, the analytical model for the SOP for SIMO underlay cognitive radio systems over Nakagami-m channels was presented, and verified by simulations. The model considered in this work is more general, which can be applied in various practical application scenarios, such as cognitive radio cellular wireless communication systems, where a singleantenna secondary mobile station (MS) shares the frequency band with the primary base-station (BS) to communicate with a BS with multiple antennas, while another BS equipped with multiple antennas wants to overhear the information from the MS of the legitimate BS. Our results provide a unified model to analyse the SOP performance over SIMO Nakagami-m fading channels in CRN and can be readily applied to practical CRN design, in which physical layer security issue is considered. Our model and results, especially the derivation of PDF and CDF, can be easily extended to the scenarios with multiple transmitters and multiple PUs in the presence of multiple eavesdroppers, which is part of our further works.

−25

−20

−15

−10 −5 IP (dB)

0

5

10

15

Fig. 4. SOP versus IP with ΩD = 2, ΩP = 1, ND = 6, LD = 3, and NE = LE = 2.

Figs. 2 and 4 that SOP is improved while m increases. This is because a higher m means that the weaker the channel fading is.

We use the same method with [19] to derive the statistic of the SNR of the GSC over a single link, but our results are more general and fit all the integer m of the Nakagami-m fading. From [19], we can obtain the expression of moment generating function (MGF) (define as Φγ (s) = E [e−γs ]) for SNR γD after GSC as bΦD +mD (1−LD ) D s+ m X ΩD ΦγD (s) = AD BD mD +bΦD +bFD , (17) cF mD D SD s + LD + ΩD ! mD LD ND LD mD where AD = , Γ(mD ) ΩD LD bFD Φ F F Γ(mD +bD +bD ) mD BD = aΦ a , SD denotes a D D mD +bΦ +bF ΩD LD

D

D

set of (2mk + 1)-tuples satisfying the condition: SD =



−1 D mP B. Case II nΦ D,i =LD − 1 , i=0 mQ When c 6= 0 and a 6= b, using the partial fraction expansion, D −1 Φ (LD −1)! 1 nD,i , we have mD −1 i! Q Φ

Φ F F nΦ D,0 , ..., nD,mD −1 , nD,0 , ..., nD,mD

m PD i=0

nF D,i

= ND − LD , aΦ D

=

nD,i !

i=0

bΦ D bF D =

mP D −1

=

i=0 m PD i=1

inΦ k,i ,

aF D

=

i=0

nFD,i mD (ND −LD )! Q −1 , m D (i−1)! Q i=1 nF D,i !

1 c

=

d

s(s + a) (s + b)

1 ac bd

s

+

c−1 X i=0

i=0

F (i − 1) nF D,i , and cD =

mD ΩD

m PD i=1

nF D,i .

where

Using the definition of the MGF, we obtain ΦγD (s) FγD (x) = L−1 s  bΦD +mD (1−LD )  mD s + X   ΩD  = AD BD L−1  Φ +bF ,  m +b D D D cF m SD s s + LDD + ΩDD (18) where FγD (x) is the CDF of γD , respectively, L−1 [·] means the inverse Laplace transform. For simplification purposes, we define h i F (x) = L−1 s(s+a)1c (s+b)d , (19)

C (i) =

k=0

D (j) =

= (mD − 1)

mX D −1

!

c+k−1 k

!

1 ac bd

s

+ e−bx

=

+

C (i)

i=0

(s + a)

c−i

where

=

c+d X

−e

+



d−1 X

D (j) d−j

j=0

(s + b)

d−1 X

(21)

c+d n−1−(c+d) X b xn−1 (n − 1)! n=1



D (j) xd−j−1 (d − j − 1)!

c+d X

c+d X D (n − c − 1) d−n+c x (d − n + c)! n=c+1

λn xµn e−ηn x ,

n=0

When c = 0 or a = b, we have " # 1 −1 F (x) = L c+d s(s + b) " # c+d n−(c+d)−1 −(c+d) X b −1 b =L − n s (s + b) n=1

.

c X C (n − 1) c−n 1 −ax x = c d +e a b (c − n)! n=1

nΦ k,i

A. Case I

−c−k

(a − b)

d

c−1 X

λn

=

      

=b

b

C (i) 1 + e−ax xc−i−1 ac bd (c − i − 1)! i=0

j=0

(20)

Then we have c ≥ 0.

,

c−1 X

+ e−bx

= (mD − 1) (LD − 1) ≤ mD (LD − 1) .

−bx

k−1 k−j−1

(−1)

−d−k

(b − a)

#

c

= L−1 

i=0

−(c+d)

a

s(s + a) (s + b) 

=

k−1 k−i−1

(−1)

1

F (x) = L−1

(mD − 1) nΦ k,i

i=0

d+k−1 k

Then we have "

mD Φ D where a = m ΩD , b = LD + ΩD , c = mD (LD − 1) − bD , and F d = mD + bΦ D + bD > 0. Since mX D −1 bΦ = inΦ D k,i

≤

j P k=0

cF D

i=0 mX D −1

i P

d−1 X D (j) + , c−i d−j (s + a) j=0 (s + b) (22)

C (i)

1 , n= ac bd C(n−1) 1 (c−n)! , D(n−c−1) (d−n+c)! ,

(23) 0 ≤n≤c

,

c+1≤n≤c+d  0, n = 0   c − n, 1 ≤ n ≤ c µn = , and   d − n + c, c + 1 ≤ n ≤ c + d    0, n = 0 a, 1 ≤ n ≤ c ηn = .   b, c + 1 ≤ n ≤ c + d We can obtain F (x) from the derivation of Eqs. (21) and (23). Then substituting F (x) into Eq. (18), we can obtain the CDF and PDF of γD , respectively. Similarly, we can obtain the CDF and PDF of γE as Eqs. (5) and (6), respectively.

λn xµn e−ηn x ,

n=0

(

1 , n= bc+d bn−c−d−1 − (n−1)! , (

where λn = ( 0, n = 0 , and ηn = n − 1, n ≥ 1

0, b,

VII. ACKNOWLEDGE 0 n≥1 n=0 . n≥1

,

µn

=

We would like to thank Dr. Stefano Tomasin and the anonymous reviewers for their useful suggestions and constructive comments, which helped us to improve the quality of our manuscript.



R EFERENCES [1] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] I. F. Akyildiz, L. Won-Yeol, M. C. Vuran, and S. Mohanty, “A survey on spectrum management in cognitive radio networks,” IEEE Commun. Mag., vol. 46, no. 4, pp. 40-48, Apr. 2008. [3] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum gridlock with cognitive radios: An information theoretic perspective,” Proc. IEEE, vol. 97, no. 5, pp. 894-914, May 2009. [4] J. Lee, H. Wang, J. G. Andrews, and D. Hong, “Outage probability of cognitive relay networks with interference constraints,” IEEE Trans. Wireless Commun., vol. 10, no. 2, pp. 390-395, Feb. 2011. [5] N. Yang, L. Wang, G. Geraci, M. Elkashlan, J. Yuan, and M. D. Renzo, Safeguarding 5G wireless communication networks using physical layer security,” IEEE Commun. Mag. , vol. 53, no. 4, pp. 20-27, Apr. 2015. [6] M. Bloch, J. Barros, M. R. Rodrigues, and S. W. McLaughlin, “Wireless information-theoretic security,” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2515-2534, Jun. 2008. [7] G. Pan, C. Tang, X. Zhang, T. Li, Y. Weng, and Y. Chen, “Physical layer security over non-small scale fading channels,” IEEE Trans. Veh. Technol., DOI: 10.1109/TVT.2015.2412140. [8] G. Pan, C. Tang, T. Li, and Y. Chen, “Secrecy performance analysis for SIMO simultaneous wireless information and power transfer systems,” IEEE. Trans. Commun., vol. 63, no. 9, pp. 3423-3433, Sep. 2015. [9] Z. Shu, Y. Qian, and S. Ci, “On physical layer security for cognitive radio networks,” IEEE Network, vol. 27, no. 3, pp. 28-33, May 2013. [10] Y. Zou, J. Zhu, L. Yang, Y.-C. Liang, and Y.-D. Yao, “Securing physicallayer communications for cognitive radio networks,” IEEE Commun. Mag., vol. 53, no. 9, pp. 48-54, Sep. 2015. [11] L. Zhang, R. Zhang, Y.-C. Liang, Y. Xin, and S. Cui, “On the relationship between the multi-antenna secrecy communications and cognitive radio communications,” IEEE Trans. Commun., vol. 58, no. 6, pp. 1877-1886, Jun. 2010. [12] Y. Pei, Y.-C. Liang, L. Zhang, K. C. Teh, and K. H. Li, “Secure communication over MISO cognitive radio channels,” IEEE Trans. Wireless Commun., vol. 9, no. 4, pp. 1494-1502, Apr. 2010. [13] M. Elkashlan, L. Wang, T. Duong, G. Karagiannidis, and A. Nallanathan, “On the security of cognitive radio networks,” IEEE Trans. Veh. Technol., vol. 64, no. 8, pp. 3790-3795, Aug. 2015. [14] H. Zhao, D. Wang, C. Tang, Y. Liu, G. Pan, T. Li, and Y. Chen, “Physical layer security of underlay cognitive radio using maximal ratio combining,” Frontiers of Information Technology & Electronic Engineering, to be published. [15] H. Liu, H. Zhao, H. Jiang, C. Tang, G. Pan, T. Li, et al., “Physical-layer secrecy outage of spectrum sharing CR systems over fading channels,” Science China Information Sciences, DOI: 10.1007/s11432-015-5451-2. [16] H. Zhao, Y. Tan, G. Pan, Y. Chen, and N. Yang, “Secrecy outage on transmit antenna selection/maximal ratio combining in MIMO cognitive radio networks,” IEEE Trans. Veh. Technol., DOI 10.1109/TVT.2016.2529704. [17] Z. Kang, K. Yao, and F. Lorenzelli, “Nakagami-m fading modeling in the frequency domain for OFDM system analysis,” IEEE Commun. Lett., vol. 7, no. 10, pp. 484-486, Oct. 2003. [18] M. Z. I. Sarkar and T. Ratnarajah, “On the secrecy mutual information of Nakagami-m fading SIMO channel,” in Proc. IEEE ICC 2010, Cape Town, South Africa, May 2010, pp. 1-5. [19] L. Wang, M. Elkashlan, J. Huang, R. Schober, and R. Mallik, “Secure transmission with antenna selection in MIMO Nakagami-m fading channels,” IEEE Trans. Wireless Commun., vol. 13, no. 11, pp. 6054-6067, Nov. 2014. [20] Y. Yang, X. Liu, Q. Wu, and J. Wang, “Outage capacity analysis for SIMO Nakagami-m fading channel in spectrum sharing environment,” in Proc. ICITIS 2010, Beijing, Dec. 2010, pp. 1058-1063. [21] C. Tang, G. Pan, and T. Li, “Secrecy outage analysis of underlay cognitive radio unit over Nakagami-m fading channels,” IEEE Wireless Commun. Lett., vol. 3, no. 6, pp. 609-612, Dec. 2014. [22] A. Annamalai, G. Deora, and C. Tellambura, “Analysis of generalized selection diversity systems in wireless channels,” IEEE Trans. Veh. Technol., vol. 55, no. 6, pp. 1765-1775, Nov. 2006. [23] X. Cai and G. B. Giannakis, “Performance analysis of combined transmit selection diversity and receive generalized selection combining in Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1980-1983, Nov. 2004. [24] Y. Deng, M. Elkashlan, N. Yang, P. Yeoh, and R. Mallik, “Impact of primary network on secondary network with generalized selection

combining,” IEEE Trans. Veh. Technol., vol. 64, no. 7, pp. 3280-3285, Jul. 2015. [25] Y. Deng, L. Wang, M. Elkashlan, K. J. Kim, and T. Q. Duong, “Generalized selection combining for cognitive relay networks over Nakagami-m fading,” IEEE Trans. Signal Process., vol. 63, no. 8, pp. 1993-2006, Apr. 2015. [26] N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and J. Yuan, “MIMO wiretap channels: Secure transmission using transmit antenna selection and receive generalized selection combining,” IEEE Commun. Lett., vol. 17, no. 9, pp. 1754-1757, Sep. 2013. [27] I. Gradshteyn and I. Ryzhik, Table of integrals, series and products, 7th ed. San Diego, CA: Academic Press, 2007. [28] I. S. Ansari, M. M. Abdallah, M.-S. Alouini, and K. Qaraqe, “Outage performance analysis of underlay cognitive RF and FSO wireless channels,” in Proc. IWOW 2014, Funchal, Portugal, Sep. 2014, pp. 6-10. [29] I. S. Ansari, M. M. Abdallah, M.-S. Alouini, and K. Qaraqe, “A performance study of two hop transmission in mixed underlay RF and FSO fading channels,” in Proc. WCNC 2014, Istanbul, Turkey, Apr. 2014, pp. 388-393. [30] H. Lei, C. Gao, I. S. Ansari, Y. Guo, G. Pan, and K. Qaraqe, “On physical layer security over SIMO generalized-K fading channels,” IEEE Trans. Veh. Technol., DOI 10.1109/TVT.2015.2496353. [31] H. Lei, H. Zhang, I. S. Ansari, C. Gao, Y. Guo, G. Pan, and K. Qaraqe, “Performance analysis of physical layer security over generalized-K fading channels using a mixture Gamma distribution,” IEEE. Commun. Lett., vol. 20, no. 2, pp. 408-411, Feb. 2016. [32] M. Z. I. Sarkar and T. Ratnarajah, “Secrecy capacity and secure outage performance for Rayleigh fading SIMO channel,” in Proc. ICASSP 2011, Prague, May 2011, pp. 1900-1903. [33] L. Chen, Y. Yang, and G. Wei, “Physical layer security enhancement with generalized selection diversity combining,” in Porc. PIMRC 2013, Sep. 2013, pp. 518-521. [34] M. Z. I. Sarkar, T. Ratnarajah, and M. Sellathurai, “Secrecy capacity of Nakagami-m fading wireless channels in the presence of multiple eavesdroppers,” in Proc. ACSSC 2009 , Pacific Grove, CA, Nov. 2009, pp. 829-833.
