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[3] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance of nonorthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Process. Lett., vol. 21, no. 12, pp. 1501–1505, Dec. 2014. [4] Z. Ding, M. Peng, and H. V. Poor, “Cooperative non-orthogonal multiple access in 5G systems,” IEEE Commun. Lett., vol. 19, no. 8, pp. 1462–1465, Aug. 2015. [5] Z. Ding, F. Adachi, and H. V. Poor, “The application of MIMO to nonorthogonal multiple access,” IEEE Trans. Wireless Commun., vol. 15, no. 1, pp. 537–552, Jan. 2016. [6] J. Choi, “Non-orthogonal multiple access in downlink coordinated two-point systems,” IEEE Commun. Lett., vol. 18, no. 2, pp. 313–316, Feb. 2014. [7] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multiple access in 5G systems,” IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1647–1651, Oct. 2015. [8] B. Kim et al., “Non-orthogonal multiple access in a downlink multiuser beamforming system,” in Proc. IEEE Mil. Commun. Conf., 2013, pp. 1278–1283. [9] Z. Ding, R. Schober, and H. V. Poor, “A general MIMO framework for NOMA downlink and uplink transmission based on signal alignment,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4438–4454, Jun. 2016. [10] Q. Sun et al., “Sum rate optimization for MIMO non-orthogonal multiple access systems,” in Proc. IEEE WCNC, 2015, pp. 747–752. [11] L. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “Robust cognitive beamforming with partial channel state information,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4143–4153, Aug. 2009. [12] J. Wang, G. Scutari, and D. P. Palomar, “Robust MIMO cognitive radio via game theory,” IEEE Trans. Signal Process., vol. 59, no. 3, pp. 1183–1201, Mar. 2011. [13] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U. K.: Cambridge Univ. Press, 2004. [14] J. R. Magnus and H. Neudecker, Matrix Differential Calculus With Applications in Statistics and Econometrics. New York, NY, USA: Wiley, 1988. [15] Y. C. Eldar and N. Merhav, “A competitive minimax approach to robust estimation of random parameters,” IEEE Trans. Signal Process., vol. 52, no. 7, pp. 1931–1946, Jul. 2004. [16] A. Beck and Y. Eldar, “Strong duality in nonconvex quadratic optimization with two quadratic constraints,” SIAM J. Opt., vol. 17, no. 3, pp. 844–860, Jul. 2006. [17] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U. K.: Cambridge Univ. Press, 2005.
Secrecy Outage on Transmit Antenna Selection/Maximal Ratio Combining in MIMO Cognitive Radio Networks Hui Zhao, Youyu Tan, Gaofeng Pan, Member, IEEE, Yunfei Chen, Senior Member, IEEE, and Nan Yang, Member, IEEE
Abstract—This paper investigates the secrecy outage performance of transmit antenna selection (TAS)/maximal ratio combining (MRC) in multiple-input–multiple-output (MIMO) cognitive radio networks (CRNs) over Rayleigh fading channels. In the considered system, a secondary user (SU-TX) equipped with NA (NA ≥ 1) antennas uses TAS to transmit confidential messages to another secondary user (SU-RX), which is equipped with NB (NB ≥ 1) antennas and adopts the MRC scheme to process multiple received signals. Meanwhile, an eavesdropper equipped with NE (NE ≥ 1) antennas also adopts the MRC scheme to overhear the transmitted information between SU-TX and SU-RX. SU-TX adopts the underlay strategy to guarantee the quality of service (QoS) of the primary user (PU) without spectrum sensing. In this paper, we derive the exact and asymptotic closed-form expressions for the secrecy outage probability (SOP). Simulations are conducted to validate the accuracy of the analysis. Index Terms—Cognitive radio networks (CRNs), maximal ratio combining (MRC), multiple-input–multiple-output (MIMO), secrecy outage probability (SOP), transmit antenna selection (TAS).
I. I NTRODUCTION Recently, cognitive radio (CR) has received great attention as a promising solution to the inadequacy of spectrum [1]. In CR, a secondary user (SU) can share the spectrum with the primary user (PU) by using overlay, interweave, or underlay methods in order not to affect the quality of service (QoS) of the PU. Among them, underlay is easy to realize as the SU needs only to adjust its transmit power within a threshold that PU can tolerate [2]. Thus, underlay has been investigated in several works [3]–[5]. On the other hand, due to the broadcast nature of wireless links, it is difficult to prevent eavesdroppers from overhearing wireless communications. To address this concern, physical layer security has been widely considered as an effective technology to prevent information from being intercepted, which was first investigated in [6] and recently in [7]–[11]. Security issues play an important role in wireless networks, Manuscript received August 31, 2015; revised November 5, 2015 and December 29, 2015; accepted February 9, 2016. Date of publication February 12, 2016; date of current version December 14, 2016. This work was supported in part by the National Science Foundation under Grant 61401372 and Grant 61531016, the Research Fund for the Doctoral Program of Higher Education of China under Grant 20130182120017, the Natural Science Foundation of CQ CSTC under Grant cstc2013jcyjA40040, and the Fundamental Research Funds for the Central Universities under Grant XDJK2015B023. The work of N. Yang was supported by Australian Research Council’s Discovery Project DP150103905. The material of this paper was presented in part at the Seventh International Conference on Wireless Communications and Signal Processing (WCSP), Nanjing, Jiangsu, China, October 2015. The review of this paper was coordinated by Prof. S.-H. Leung. H. Zhao, Y. Tan, and G. Pan are with the School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China (e-mail:
[email protected]). Y. Chen is with the School of Engineering, University of Warwick, CV4 7AL, Coventry, U.K. (e-mail:
[email protected]). N. Yang is with the Research School of Engineering, The Australian National University, Canberra, A.C.T. 0200, Australia (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2016.2529704
0018-9545 © 2016 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.
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particularly in CR networks (CRNs) for which the licensed frequency band is shared among the PUs and SUs, leading to an increased possibility of eavesdropping of the transmitted information for both PU and SU [12]–[16]. However, very few research studies have considered the secrecy performance of multiantenna diversity, which is one of the most effective technologies to improve the transmission rate in CRNs. In [17] and [18], the secrecy outage performance of a single-input–multiple-output (SIMO) system using maximal ratio combining (MRC)/selection combining (SC) in CRNs was investigated. However, in [17], only an eavesdropper with a single antenna was considered. In [18], only the SC technique was considered. It is well known that MRC has better performance than SC. Motivated by these observations, in this paper, we analyze the secrecy outage performance of multiple-input–multiple-output (MIMO) CRNs, where an SU (SU-TX) equipped with NA (NA ≥ 1) antennas uses transmit antenna selection (TAS)1 to transmit confidential messages to another SU (SU-RX), which is equipped with NB (NB ≥ 1) antennas and adopts MRC to process multiple copies of the received signal. Meanwhile, an eavesdropper (Eve), which is equipped with NE (NE ≥ 1) antennas, adopts MRC for successful eavesdropping. SUTX adopts underlay strategy in order not to degrade the QoS of PU. The main contributions of our work are summarized as follows. • Compared with [19] and [20] in which only the physical layer security for a conventional non-CR system was considered, this paper considers the physical layer security for an underlay MIMO CRN. Because a CR system has a shared frequency band and therefore lower security, such an analysis of the secrecy performance for the CR system is important. Our proposed analytical model can bring out an insight into the secrecy outage performance of SU systems, which cannot be obtained from [19] and [20] for non-CR systems. • Compared with [18] in which the physical layer security for CR using SC was considered, this paper considers the physical layer security for CR by using MRC, as MRC not only can improve the secrecy performance of the desired user but also can increase the eavesdropping capability of the eavesdropper. Thus, it is important to identify the effect of MRC. • We study the secrecy outage performance of CRNs and derive accurate and asymptotic closed-form expressions for secrecy outage probability (SOP). Our asymptotic results accurately predict the secrecy diversity order and secrecy array gain. II. S YSTEM M ODEL We consider a MIMO wiretap channel in CRNs, as shown in Fig. 1. SU-TX is equipped with NA (NA ≥ 1) antennas, and TAS is used to encode the confidential messages into transmitted codeword x = [x(1), x(2), . . . , x(n)], which is subject to an average power constraint 2 (1/n) n i=1 E[|x(i)| ] ≤ PS . SU-TX transmits x to SU-RX, which is equipped with NB (NB ≥ 1) antennas, and adopts MRC to improve its received SNR, whereas Eve, which is equipped with NE (NE ≥ 1) antennas, also adopts MRC to promote successful eavesdropping. hP = [hp1 , hp2 , . . . , hpNA ] is the channel vector of the SU-TX-PU link. HD and HE are the channel gain matrices of the SU-TX-SU-RX and SU-TX-Eve links, respectively. We assume that all channels experience independent Rayleigh fading and additive white Gaussian noise (AWGN) with variance of N0 . We also assume that the channel state information (CSI) of the 1 TAS is a low-cost and low-complexity method for exploiting spatial diversity in multiple antenna settings [19].
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Fig. 1. System model.
SU-TX-PU and SU-TX-SU-RX links are available at SU-TX, whereas the CSI of the SU-TX-Eve link is unavailable at SU-TX.2 The CSI from PU to SU-TX can be obtained by using channel reciprocity [5]. For simplification, we denote hP , hD , and hE as the average channel power gains of the SU-TX-PU, SU-TX-SU-RX, and SU-TX-Eve links, respectively. Because the CSI of the SU-TX-Eve link is not available at SU-TX, making use of the CSI among SU-TX-SU-RX, the “best” transmit antenna, which can maximize the total received signal power at SURX, is chosen to deliver the information. If such CSI is available, it would be also interesting to consider the selection of the worst antenna for the PU and eavesdropper. This can be a future research topic. In order not to degrade the QoS of PU, when the uth (u = 1, 2, . . . , NA ) antenna of SU-TX is selected to transmit messages, the transmit power Pt at SU-TX should be limited at a given threshold IP that PU equipped with a single antenna can tolerate IP , PS ≥ gIPP ⇒ gP ≥ PIPS (1) Pt = g P PS , PS < gIPP ⇒ gP < PIPS where gP = |hpu |2 is the channel power gain between the uth antenna of SU-TX and PU, and PS is the maximum transmitting power available at SU-TX. The received signal vectors of SU-RX and Eve from the uth transmit antenna at time t are yD (t) = hDu x(t) + nD
(2)
yE (t) = hEu x(t) + nE
(3)
where hDu and hEu are the channel vectors between the uth transmit antenna and SU-RX and Eve whose elements are independent and identically distributed (i.i.d.) under Rayleigh fading, and nD and nE are the AWGN vectors at SU-TX and Eve, respectively. This is a reasonable assumption, as the channels among SU-TX and each antenna at a terminal, like SU-RX or Eve, are close to each other. Let λP = 1/hP , λD = 1/hD , and λE = 1/hE . When the uth antenna of SU-TX is selected to transmit information, the probability density functions (PDF) of the MRC-combined channel power gain of SU-RX and Eve are given by [4] fgDu (gDu ) =
NB −1 B exp(−λD gDu )λN gDu D , (NB − 1)!
gDu ≥ 0
(4)
fgEu (gEu ) =
NE −1 E exp(−λE gEu )λN gEu E , (NE − 1)!
gEu ≥ 0
(5)
where gDu = hDu 2 and gEu = hEu 2 , in which · denotes the Euclidean norm, respectively. 2 In this scenario, SU-TX has no choice but to encode the confidential data into code words of a constant rate RS [18].
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III. S ECRECY P ERFORMANCE A NALYSIS A. Preliminaries The TAS/MRC-combined channel power gain gD of SU-RX can be given by gD =
max
u=1,2,...,NA
{gDu }.
(6)
As gP = |hpu |2 ∼ exp(1/hP ), the items Pr{gP ≥ IP /PS } and Pr{gP ≤ IP /PS } in (11) can be easily obtained as IP λP I P Pr gP ≥ = exp − (12) PS PS IP λP I P = 1 − exp − (13) Pr gP ≤ PS PS respectively. Next, we derive SOP1 (Cth ) and SOP2 (Cth ) to calculate the overall SOP in (11).
According to [19], the PDF of gD can be derived as fgD (x) = NA [FgDu (x)]NA −1 fgDu (x)
B. Derivation of SOP1 (Cth )
NA −1 B N A − 1 N A λN D = (−1)l (NB − 1)! l=0 l
N −1 l B λiD xi × exp [−λD (l + 1)x] xNB −1 i! i=0
When Pt = IP /gP , we can write SOP1 (Cth ) as [12] α−1 gP ≥ gD − αgE Pr{CS ≤ Cth } = Pr ρ
where FgDu (·) is the cumulative probability density function of gDu given by NB −1
FgDu (x) = 1 − exp(−λD x)
i=0
λiD xi i!
where α = 2Cth , and ρ = IP /N0 . Let Z1 = ((α − 1)/ρ)gP , Z2 = gD − αgE , and X = αgE . The PDFs of X and Z1 can be derived as3
(8)
NA −1 B NA − 1 N A λN D (−1)l (NB − 1)! l=0 l
×
n1 n0 n1 =0 n2 =0
nR−1 =0
R−1
ni−1 1 ni −ni+1 n i × λD ni i! i=1
∞ fZ2 (z2 ) =
fgD (z2 + x)fX (x)dx.
(17)
0
(9)
· · · +nR−1 , R = NB , n0 = l, where M = N +NB −1, N = n1 +n2 + n0 i = 1. and nR = 0. When R = 1 and N = 0, [ R−1 i=0 (λD x) (1/i!)] As the transmit antenna index is optimum for the SU-TX-SU-RX link, Eve is not able to exploit any additional diversity from the multiple transmit antennas at SU-TX. Thus, the PDF of the combined channel power gE at Eve is given by fgE (x) = fgEu (x). The instantaneous secrecy capacity is given by
gE , gD > gE log2 1 + Pt gND0 − log2 1 + Pt N 0 CS = (10) 0, gD ≤ gE .
Substituting the PDFs of gD and X into (17) and after some mathematical manipulations, we can derive fZ2 (z2 ) as (18), shown at the bottom of the next page. We can rewrite (14) as ∞ SOP1 (Cth ) =
0 fZ1 (z1 )
B
fZ2 (z2 ) dz2 dz1
−∞
I1
∞ +
SOP is defined as the probability that the instantaneous secrecy capacity is below a target secrecy rate (Cth , Cth ≥ 0). Different from [18], we can calculate SOP under the two cases of Pt suggested by (1) as IP SOP(Cth ) = Pr gP ≥ SOP1 (Cth ) PS IP + Pr gP < SOP2 (Cth ) PS
(16)
respectively, where A = 1/ exp(−λP IP /PS ). Using [22, eqs. (6–55)] when Z2 ≥ 0, we can write the PDF of Z2 as4
× xM exp [−λD (l + 1)x]
(α − 1)N0 =B PS
z1 ≥
nR−2
···
(15)
ρλP z1 AρλP exp − α−1 α−1
fZ1 (z1 ) =
Using [21, eq. (9)], we can have fgD (x) =
E xNE −1 exp(−λE x/α)λN E N α E (NE − 1)!
fX (x) = .
(14)
(7)
z1 fZ1 (z1 )
B
fZ2 (z2 ) dz2 dz1 . 0
(19)
I2
To facilitate the following analysis, we define an integral I3 as follows: ∞ I3 =
(11)
where SOP1 (Cth ) and SOP2 (Cth ) refer to the SOP when Pt = IP /gP and Pt = PS , respectively.
fZ2 (z2 ) dz2 .
(20)
0 3 In this case, g ≥ I /P . The PDF of g can be obtained by f gP (x) = P P S P AλP exp(−λP x). 4 To simplify the analysis, we need not calculate the PDF of Z < 0 directly. 2
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Substituting γB = PS gD /N0 , γ B = PS hM /N0 , γE = PS gE /N0 , and γ E = PS hE /N0 into [20, eq. (25)], where mB = mE = 1,5 we can calculate SOP2 (Cth ) as (26), shown below, where ψu = NB −1 iu , i0 = p, and iNE = 0 as follows: u=1
I3 can be easily calculated as (21) as follows: I3 =
−(NE +k) M λE M + λD (l + 1) Γ(NE + k) k α Ω k=0 × Γ(M − k + 1) [λD (l + 1)]−(M −k+1) .
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NA NA 1 p(α−1) (−1)p−1 exp − Γ(NE ) p=1 p γB ⎡ ⎤ NB −1 iu−1
iu−1 1 iu −iu+1 1 ψu ⎣ ⎦ × iu u! γB u=1 i =0
(21)
SOP2 (Cth ) = 1−
Therefore, it is easy to observe that ∞ fZ1 (z1 ) · (1 − I3 ) dz1 = 1 − I3 .
I1 =
(22)
u
B
We can rewrite I2 as (23), shown at the bottom of the page, x where Υ(n, x) = 0 exp(−t)tn−1 dt is the lower incomplete gamma function [23]. Using [23, eq. (8.352.1)] to expand Υ(., .) in I4 into the form of series, the integral in (23) can be derived as (24), shown where Λ = λD (l + 1) + (ρλP /(α − 1)) and Γ(n, x) = below, ∞ n−1 exp(−t)t dt is the upper incomplete gamma function [23] as x follows: I4 = (M − k)!
α−1 ρλP B exp − ρλP α−1 −
M −k m=0
m
[λD (l + 1)] m!
(26)
Finally, SOP can be obtained by substituting (12), (13), (25), and (26) into (11). IV. A SYMPTOTIC S ECRECY O UTAGE P ROBABILITY
Γ(m + 1, ΛB) Λm+1
.
(24)
Finally, we can derive SOP1 (Cth ) as SOP1 (Cth ) = 1 − I3 +
ψu 1 NE ψu t × α (α − 1)ψu −t γE t t=0 αp 1 −(t+NE ) × Γ(t + NE ) + . γB γE
M
ΦI4
(25)
Ω k=0
where I4 is given in (24), ΣΩ is given in (18), and Φ is given in (23).
Here, we will present the asymptotic SOP analysis when λD → 0, i.e., γ 1 = PS /(N0 λD ) → ∞ in [18], motivated by the fact that the secrecy diversity order and secrecy array gain govern the SOP at high SNR at SU-RX. Another aim of deriving asymptotic SOP is that the asymptotic expression is normally more concise than the exact expression. A. Derivation of Asymptotic SOP∞ 1 When λD → 0, by applying binomial combination and first-order Maclaurin series expansion and then keeping the first two terms in the
C. Derivation of SOP2 (Cth ) When gP < IP /PS , Pt = PS . It means that SU-TX only adopts its maximum transmitting power to deliver information to SU-RX. Obviously, the target system model becomes a non-CR model in this case.
5 The closed-form expression for SOP in [20] was derived in Nakagami-m fading scenarios. Then, we can easily obtain the closed-form expression for SOP over Rayleigh fading channels by substituting mB = mE = 1 into [20, eq. (25)].
nR−2 R−1 NA −1 n1 n0 B NE NA − 1
ni−1 1 ni −ni+1 n N A λN l D λE i fZ2 (z2 ) = N ··· λD (−1) l α E (NE − 1)!(NB − 1)! l=0 ni i! n1 =0 n2 =0 nR−1 =0 i=1 ΣΩ
×
k=0
I2 =
−(NE +k) λE M + λD (l + 1) z2M −k exp (−λD (l + 1)z2 ) Γ(NE + k) k α
M
(18)
−(NE +k) M λE M AρλP + λD (l + 1) [λD (l + 1)]−(M −k+1) Γ(NE + k) k α α−1 Ω k=0 Φ
ρλP z1 exp − Υ (M − k + 1, λD (l + 1)z1 ) dz1 α−1 B ∞
×
I4
(23)
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B. Derivation of Asymptotic SOP∞ 2
Maclaurin series expansion, we can rewrite (7) as fgD (x) =
A NB N A λN D A NB (27) xNA NB −1 + o λN D N −1 (NB − 1)!(NB !) A
where o(·) denotes higher order terms. To derive the asymptotic analysis, (14) can be rewritten as α−1 gP + αgE ≥ gD . SOP1 = Pr ρ
where Ga2 is defined as (28)
Let Z3 = ((α − 1)/ρ)gP + αgE and a = (λE /α) − (ρλP /(α − 1)). Using [23, eq. (1.111)], we can derive the PDF of Z3 as fZ3 (z3 ) =
(α −
E AρλP λN E N 1)α E (NE
× (−1)
NE −q−1
− 1)!
q=0
×
(29)
xn exp(ax)dx = exp(ax)
∞ =
(30)
z3 fZ3 (z3 )
=
fgD (gD )dgD dz3 0
B A NB λN D (NB !)NA
∞
A NB . fZ3 (z3 )z3NA NB dz3 + o λN D
where the achieved secrecy array gain is G−1 a1 , where Ga1 is defined as NE E /((α − 1)α (NE − 1)!(NB !)NA ), (33), in which Θ = AρλP λN E as follows: NE −1
Θ
Γ(NE + p)
N0 . PS
(35)
λP I P A NB GN · exp − a1 PS 1 NA NB λP I P NA NB · 1 − exp − . + Ga2 PS
(37)
Note that the SOP expressions derived in the previous section allow for the direct determination of the effects of all the important system parameters on the secrecy performance. This eliminates the need for tedious simulations to exhaust all possible values of the system parameters and is therefore useful. Moreover, the asymptotic results in (32) and (36) are only polynomial functions, the simplest possible form, to give the direct insights into the effects of diversity order and array gain. They all represent important contributions. V. N UMERICAL AND S IMULATION R ESULTS
Substituting the PDF of Z3 into this equation and using the closedform expression of Q2 given in the Appendix, we can derive the closed-form expression of the asymptotic SOP∞ 1 as
NA NB A NB (32) + o λN SOP∞ 1 = (Ga1 · λD ) D
Ga1 =
1 A NB
(31)
B
N
p
−1 where the achieved secrecy diversity gain is NA NB , and G−1 a . Ga is the secrecy array gain, in which Ga is derived as
Ga =
When λD → 0, using (27), we can write the asymptotic SOP1 as SOP∞ 1
αPS (α − 1)λE N0
Finally, we can derive the closed-form expression of the asymptotic SOP as
A NB (36) SOP∞ = (Ga · λD )NA NB + o λN D
[Q1 (NE − q − 1, a, z3 )
n n(n − 1) · · · (n − p + 1) n−p xn + (−1)p x . a ap+1 p=1
NA NB NA NB (α − 1)NA NB (NB !)NA Γ(NE ) p=0 p
Ga2 =
NE − 1 q
where Q1 (·, ·, ·) is defined as (given by [24, eq. (1.3.2.6)])
·
NE −1
− Q1 (NE − q − 1, a, B)] λE z3 × z˙3q exp − α
Q1 (n, a, x) =
When gP < IP /PS , the target system model becomes a non-CR model in this case. Thus, considering [20, eqs. (26) and (27)], we can obtain the closed-form expression for SOP∞ 2
∞ NA NB A NB SOP2 = (Ga2 · λD ) + o λN (34) D
q=0
NE − 1 (−1)NE −q−1 q
× Q2 − Q1 (NE − q − 1, a, B) λE B × Γ NA NB + q + 1, α 1 NA NB α NA NB +q+1 × . λE
(33)
Here, we run Monte Carlo simulation to validate our analytical expressions of the SOP over Rayleigh fading channels. In each simulation case, SU-TX sends 106 bits to SU-RX. To present the increasing secrecy diversity of using TAS at SU-TX, Fig. 2 shows the SOP of TAS/MRC in a MIMO wiretap system, as compared with the SOP of MRC in a SIMO wiretap system, i.e., NA = 1, versus IP for various NE over Rayleigh fading channels in CRNs. It is shown that SOP increases with increasing NE as the MRC diversity gain increases at Eve. It is evident that for a fixed NE , SOP decreases with increasing IP , which increases peak transmit power at PU, and this increases the transmitting power at SU-TX. We can also see that there exists a floor for SOP in the high-IP region because Pt = PS when IP → ∞, which means that, in this case, the transmitting SNR remains constant. Moreover, it can be also seen that the secrecy performance of the TAS/MRC scheme greatly outperforms the secrecy performance of the MRC scheme because the secrecy diversity order of TAS/MRC is NA NB , whereas the secrecy diversity order of MRC is NB . Figs. 3 and 4 show the SOP versus ω = 1/λD . In Fig. 3, SOP decreases with increasing NA . This can be explained by the fact that NA increases the transmitting diversity at SU-TX. The diversity gain of MRC at SU-RX can be improved as NB increases. This offers
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Fig. 2. SOP versus IP for PS = 1 dB, λP = 2, λD = 2, λE = 4, Cth = 0.1 bit/s/Hz, NA = 2, NB = 4, and N0 = 1.
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Fig. 4. SOP versus ω = 1/λD for PS = 3 dB, IP = 1 dB, λP = 3, Cth = 0.1 bit/s/Hz, NA = 1, and N0 = 1.
SC schemes, i.e., NA = 1. Obviously, apart from the scenario of (NB , NE ) = (3, 6), the secrecy outage performance of MRC outperforms the secrecy outage performance of SC among the other three scenarios, i.e., (NB , NE ) = (3, 3), (6, 6), and (6, 3), respectively, although the secrecy diversity orders of MRC and SC schemes are same. Furthermore, simulation and analytical results match very well with each other, which verify our proposed analytical models.
VI. C ONCLUSION In this paper, we have studied the physical layer security in MIMO cognitive wiretap channels and investigated the secrecy outage performance over Rayleigh fading channels by deriving closed-form expressions for the exact and asymptotic SOPs.
A PPENDIX Fig. 3. SOP versus ω = 1/λD for PS = 3 dB, IP = 1 dB, λP = 2, Cth = 1 bit/s/Hz, NE = 2, and N0 = 1.
an explanation of why SOP declines while NB is rising. It is also worth noting that the secrecy diversity order for (NA , NB ) = (3, 4) is the highest at NA NB = 12, followed by the secrecy diversity order for (NA , NB ) = (3, 3), whereas the secrecy diversity order for (NA , NB ) = (2, 3) is the lowest, at 6, which means that the secrecy outage performance of (NA , NB ) = (3, 4) is the best among the three (NA , NB ) combinations. Moreover, the obtained asymptotic results match very well with the exact results and accurately predict the secrecy diversity order and secrecy array gain in the high-ω region, i.e., in the high-γ 1 region in [18]. Furthermore, we can also observe that the asymptotic SOP presents the upper bound of the exact SOP. Because the secrecy performance of the SC scheme in CRNs was only considered in [18], whereas the secrecy performance of the MRC scheme has not yet been investigated in previous studies,6 Fig. 4 compares the secrecy outage performance between the MRC and 6 In [17], it has been only considered that Eve is equipped with a single antenna and the transmit power restriction at SU-TX is incomplete; hence, the contribution of [17] is significantly limited.
We consider the following integral: ∞ λE NA NB +q z3 dz3 . (38) exp − Q2 = Q1 (NE −q−1, a, z3 )· z3 α B
Substituting Q1 (·, ·, ·) into this equation and using[23, eq. (3.351.2)], we can derive Q2 as Q2 =
1 a
α−1 ρλP
NE +NA NB
NE −q−1 ρλP B + × Γ NE + NA NB , (−1)p α−1 p=1 (NE − q − 1)(NE − q − 2) · · · (NE − q − p) ap+1 α−1 NE +NA NB −p ρλP B . · Γ NE +NA NB − p, ρλP α−1 (39) ×
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