Secure transmission in underlay cognitive radio network with outdated channel state information

Journal Pre-proof Secure transmission in underlay cognitive radio network with outdated channel state information Shilpa Thakur, Ajay Singh

PII: DOI: Reference:

S1874-4907(18)30638-4 https://doi.org/10.1016/j.phycom.2019.100890 PHYCOM 100890

To appear in:

Physical Communication

Received date : 25 October 2018 Revised date : 29 September 2019 Accepted date : 4 October 2019 Please cite this article as: S. Thakur and A. Singh, Secure transmission in underlay cognitive radio network with outdated channel state information, Physical Communication (2019), doi: https://doi.org/10.1016/j.phycom.2019.100890. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof

Secure Transmission in Underlay Cognitive Radio Network with Outdated Channel State Information Shilpa Thakur and Ajay Singh

pro of

Department of Electrical Engineering, Indian Institute of Technology Jammu, Jammu and Kashmir, India

Abstract

re-

This paper investigates the secrecy performance of an underlay cognitive radio network (CRN) with outdated channel state information (CSI) for two practical scenarios, i.e., scenario 1: the CSI of an eavesdropper’s channel is unavailable at the secondary transmitter and scenario II: the CSI of an eavesdropper is available at the secondary transmitter. The selection combining (SC) and maximal ratio combining (MRC) techniques are utilized at the secondary receiver while MRC is also employed at both eavesdropper and primary receiver. For the scenario I, the compact expressions of exact and asymptotic secrecy outage probability (SOP), intercept probability and - outage secrecy capacity are derived, and simulations are conducted to verify analytical results. Our results reveal that the secrecy performance of the system degrades when the CSI of the main channel is outdated. Additionally, in scenario II, a comprehensive analysis of average secrecy capacity is performed, and new closed-form expressions for exact and asymptotic average secrecy capacity are derived which are valid for an arbitrary number of antennas at secondary receiver and eavesdropper. Finally, we investigate the high signal to noise ratio (SNR) slope and high SNR power offset which characterize the impact of the outdated CSI of the main channel and eavesdropper’s channel on average secrecy capacity.

lP

Keywords: Average secrecy capacity, Cognitive radio network, Diversity combining techniques, Secrecy outage capacity, Physical layer security 1. Introduction

Jo

urn a

In the digital era, the protection of wireless networks and facilities from unauthorized and malicious access attempts have become increasingly significant to perform secure communication. Typically, wireless network security rises rapidly due to the distributed nature of the broadcasting channel [1] and further increased to the front line in the spectrum sharing networks. Also, the security in wireless network mainly relies on the higher layer cryptographic authentication and technologies to determine information security through massive computing. However, these cryptographic techniques are costly and vulnerable to attacks [2][3]. In the view of factors as mentioned above, keen interest has been developed recently in the area of the physical layer (PHY) security to secure communication without the need of complex cryptographic operations. In particular, the time-variable properties (fading, thermal noise, etc.) of channels are utilized to secure data at PHY [4], 1

Preprint submitted to Elsevier

[5], [6] and it also supports exiting cryptographic protocols. The basic principle of PHY is to enhance the signal to noise ratio (SNR) of the main channel in comparison to the wiretap channel to achieve perfect secrecy. With the advancement in the multiple-antennas system and cooperative communication, PHY security has become promising techniques for data security in wireless system [7].

Besides, cognitive radio is the most critical radio resource for fifth-generation wireless systems [8]-[9], and used to alleviate the scarcity of radio frequency spectrum. In cognitive radio network (CRN), there are three spectrum sharing approaches namely underlay, overlay, and interweave approach through which all the unlicensed users can occupy the bandwidth without causing interference to the licensed users [10]. Security is an essential requirement for future fifth-generation systems, and cognitive radio is no exception. In particular, the security of CRN is critical as it is easily exposed to external hazards. October 1, 2019

Journal Pre-proof

with the TAS scheme was studied in [27]. The effect of imperfect CSI on MIMO CRN with TAS and receive MRC was studied in [28]. However, the main limitation of [28] is that both legitimate receiver and eavesdropper are deployed with a single antenna, and hence, the received diversity gain has not been understood well. The general-order TAS scheme with outdated CSI for both passive and active eavesdropping scenarios was proposed in [29]. However, [29] has only considered the impact of the MRC technique on the secrecy performance of the CRN.

pro of

1.1. Related Works It is well known that the capacity of the wireless system severely degraded due to the fading effect. To overcome this, multiple-input-multiple-output (MIMO) network and diversity combining techniques have been proposed as an efficient tool to reduce wireless fading and enhance the secrecy capacity of wireless transmission [11]-[12]. [13] investigated the PHY security of wireless channels over a quasi-static fading environment. Maximal ratio combining (MRC) diversity scheme has used to secure transmission in [14], and a closed-form expression for secrecy outage probability (SOP) and the probability of non-zero secrecy capacity has been investigated. The impact of the multi-antenna transmitter with transmit antenna selection (TAS) scheme on secrecy performance of CRN has studied in [15]. The improvement in secrecy performance of wireless channels through cooperative relays was studied in [16]- [17]. In [18], the TAS scheme has employed at the secondary transmitter, and both secondary receiver ( also called legitimate receiver) and eavesdropper have utilized with either MRC or selection combining (SC). In [19], authors have examined the secrecy performance of multi-user multi-eavesdropper CRN, which is composed of multiple cognitive users transmitting to a typical cognitive base station, while multiple eavesdroppers may collaborate to intercept the data transmissions. The physical layer attacks like primary user emulation, sensing falsification, intelligence compromise, jamming, and eavesdropping attacks have studied in [20]. In the works as mentioned above, perfect channel state information (CSI) has considered. In a practical scenario, CSI may be outdated due to the complexity of electromagnetic wave spreading, and transmitting delay, which causes estimation error at the receiver [21][22]. The physical layer security in CRN with outdated CSI was considered in [20-[26]. [23] has analyzed the capacity in the low SNR regime of the MRC scheme for Rayleigh fading with CSI estimation errors. In [24], Shrestha.et.al., assumed that the legitimate receiver and eavesdropper both are employed with non-ideal MRC which was designed in terms of power correlation between the estimated and actual fading. Secure cooperative transmissions in a dual-hop MIMO relay system using a combined TAS/MRC scheme over the Nakagamim fading channel has studied in [25]. The secrecy performance of SIMO CRN assuming both MRC and SC at both the legitimate receiver and eavesdropper with imperfect CSI was investigated, and closed-form expression for SOP has derived in [26]. The impact of outdated CSI on secrecy performance of SIMO CRN

Jo

urn a

lP

re-

1.2. Motivation and Contribution Distinct from all prior works ([26],[28],[29]), we have proposed SC and MRC technique at the legitimate receiver for secure data transmission in Rayleigh fading. In the MRC diversity scheme, the signals coming from all the diversity paths are co-phased to provide coherent addition which maximizes the SNR at the receiver. On the other hand, the receiver with the SC technique, which is a widely used combining technique in many practical applications, selects a single diversity path with maximum SNR. Thus, MRC is an optimum technique, and it performs better than SC, but SC has a lower complexity. Keeping this in mind, we have examined the physical layer security of an underlay CRN when SC and MRC schemes are employed at a legitimate receiver. The performance gap between SC and MRC schemes have studied when all the channels are outdated for two practical scenarios, i.e., scenario I: passive eavesdropping i.e., CSI of eavesdropper’s channel is not available at the secondary transmitter, and scenario II: active eavesdropping i.e., the CSI of eavesdropper is available at the secondary transmitter. The MRC technique is utilized at eavesdropper and primary receiver which maximizes the instantaneous SNR of the eavesdropper and primary receiver (i.e., worst case scenario). For the scenario I, SOP, intercept probability and, - outage secrecy capacity are recognized as main performance metrics due to the unavailability of the CSI of eavesdropper’s channel. For scenario II, when CSI of the eavesdropper is available, secrecy performance is measured in term of average secrecy capacity, since the transmitter adapts its transmission rate based on the global CSI. The main contributions of this work are as follow: • Scenario I: Passive Eavesdropping 1. To investigate the impact of diversity combining techniques and outdated CSI on the secrecy performance of the CRN, the compact expressions of

2

Journal Pre-proof

SOP, intercept probability and - outage secrecy capacity for SC/MRC and MRC/MRC schemes are derived and also verified by simulations. 2. The new compact expression for the probability of non-zero secrecy capacity (PNZC) has been derived in this paper.

pro of

3. The new closed-form asymptotic expressions for the SOP of two combining techniques have been investigated, from which secrecy array gain and secrecy diversity order have been derived. The most interesting thing is that both SC/MRC and MRC/MRC techniques achieve the same secrecy diversity order.

Figure 1: System Model with primary receiver (PR), secondary transmitter (Alice), secondary/legitimate receiver (Bob) and eavesdropper (Eve)

4. Compared with [26], we examine the secrecy performance of two diversity combining techniques with interference power constraint and also study two important metrics called intercept probability and - outage secrecy capacity for passive eavesdropping.

re-

Both MRC and SC techniques are employed at the legitimate receiver, and their performance gap is studied in terms of high SNR slope and high SNR power offset.

The rest of paper is organized as follows. Section 2 explains the system model and also gives the expressions for PDF and CDF of MRC and SC scheme. The performance matrices of scenario I and scenario II are studied in section 3 & 4. Section 5 highlights numerical results and their interpretation. Finally, the conclusion is drawn in section 6.

lP

5. In [30], the closed-form expression for ergodic capacity has been derived. Compared with [30], three important performance matrices SOP, intercept probability and - outage secrecy capacity is investigated for an underlay CRN with MRC scheme at primary receiver and compared with [24], SC scheme is also considered at legitimate receiver and performance gap between MRC and SC is measured in terms of secrecy diversity order and array gain.

2. System Model In this work, we are considering an underlay CRN, where both secondary network and primary network are transmitting concurrently in the same spectrum band as long as the interference level at the primary network remains acceptable. The secondary network consists of a secondary transmitter ( called Alice from now onward), a legitimate receiver (called Bob from now onward) and an eavesdropper (called Eve) as shown in figure.1. In this work, It is assumed that the primary transmitter is not in the proximity of Alice while the primary receiver (PR) is in the vicinity of Alice [31]-[32]. The secret messages are transmitted from Alice to Bob in the presence of Eve and Eve tries to decode these messages from its received vector. We assume that all nodes PR, Alice, Bob, and Eve are equipped with NR , NA , NB , and NE antennas, respectively. It is assumed that the interference channel (the channel between PR and Alice), the main channel (the channel between Alice and Bob), and an eavesdropper’s channel (the channel between Alice and Eve) undergo spatially independent Rayleigh fading with average channel SNR γR , γ1 , and γ2 , respectively.

urn a

• Scenario II: Active Eavesdropping

1. A new closed-form expressions for average secrecy capacity for MRC/MRC, SC/MRC schemes with outdated CSI in a Rayleigh fading environment have been derived.

Jo

2. In order to investigate the impact of main channel and wiretap channel on average secrecy capacity, we find compact expressions for asymptotic average secrecy capacity with MRC/MRC and SC/MRC schemes which facilitates the characterization of high SNR slope for two combining techniques, and demonstrates that high SNR slope is unity and independent of system parameters such as the number of antenna at both legitimate receiver and eavesdropper, channel estimation coefficients, etc. 3. Compared with [29], an underlay CRN with interference power constraint has been considered. 3

Journal Pre-proof

density distribution (CDF) and probability density function (PDF) of H˜ lMRC with estimation error for MRC scheme can be written as [24]   Nl uX l −1 X  xa  − γx   F H˜ lMRC (x) = Al (ul ) 1 − e l (5) a!γla  u =1 a=0

The channel fading coefficient with estimation error between Alice-PR, Alice-Bob, and Alice-Eve is given by [33] p √ h˜ ill = ρl hill + 1 − ρl gill (1)

where l ∈ (PR, Bob and Eve), hill represents the channel gain between Alice and the ith antenna at l is a Gaussian random variable with zero mean and variances ΩR , Ω1 and Ω2 respectively. gill is the channel estimate error, which is a complex Gaussian random variable with zero mean. Both hill and gill have same variance. ρl is the power correlation coefficient varies from 0 to 1. The instantaneous SNR at PR, Bob and Eve is written as PA ˜ Hl N0

pro of

γl =

l

fH˜ lMRC (x) =

Al (ul )

ul =1

−

x

xul −1 e γl (ul − 1)!γ1ul

(6)

respectively, where Al (ul ) =

(2)

! Nl − 1 (1 − ρl )Nl −ul ρl ul −1 u1 − 1

2.2. Selection combining scheme In this work, we are also adopting SC at legitimate receiver Bob. The channel gain of main channel with SC can be expressed as H˜ lS C = maxul ∈{1,2......Nl } |h˜ ul l |2 . The CDF and Pdf of H˜ lS C is written as

re-

where N0 is the noise variance, PA is the Alice’s transmit power and H˜ l represents the combined channel gains between Alice-PR, Alice-Bob and Alice-Eve. In underlay CRN, in order to reliable communication PA should be less than peak interference power threshold (IP ). Hence, Alice is a power-limited transmitter with maximum transmit power is PT . PA is strictly constrained by PT and IP at PU and given by ! IP PA = min PN k , PT (3) R ˜ 2 k=1 |h p |

lP

F H˜ lS C (x) =

where |h˜ kp |2 is the channel gain of interference channel. In this work, we assume that MRC technique is used at PR , which is the most worst case to limit the transmit power at Alice. From (3), the instantaneous SNR at Bob and Eve can be rewritten as γ  γ  p p γ M = min , γ0 X M , γE = min , γ0 XE (4) Y Y

N l −1 X ul =0

fH˜ lS C (x) =

respectively, where φl =

 xξ φl  − l 1 − e γl ξl

N l −1 X ul =0

urn a

(7)

φl − xξγ l e l γl

Nl (−1)ul

N −1

(8)

l

ul

(1 − ρl )ζl 1 ρl . ul + 1 and ξl = − 1 − ρl ζl (1 − ρl )2

, ζl =

ρl + 1 − ρl

3. Scenario I : Passive Eavesdropping In this section, we are considering passive eavesdropping i.e., CSI of Eve’s channels is unavailable at Alice. In this scenario, we investigate three important parameter metrics SOP, Intercept probability, and - outage secrecy capacity to evaluate the secrecy performance of the proposed system.

Jo

IP PT respectively, where γ p = , γ0 = . Y = N0 N P NR ˜ k 2 PNB ˜ j 2 PNE ˜ i0 2 j=1 |h B | and XE = i=1 |hE | are the k=1 |h p | , X M = channel gains of interfernce channel, main and eavesdropper’s channels. For ease of exposition and matheΩ γ matical tractability, we denote γ1 = Ω1 γ0 = 1σ p and γp IP Ω γ γ2 = Ω2 γ0 = 2σ p and σ = = . γ0 PT Next, we introduce SC and MRC diversity combining techniques at Bob and MRC scheme at Eve. 2.1. Maximal Ratio Combining When MRC is utilized at Bob and Eve, the comP bined channel gain can be written as H˜ lMRC = uNll=1 |h˜ ul l |2 where {l ∈ (PR(R), Bob(B), Eve(E)}. The cumulative

Nl X

3.1. Secrecy outage probability Secrecy capacity is defined as the maximum possible rate that is required for secure communication. We are assuming slow block fading, in that case, secrecy capacity is given by     1+γ M   if γ M > γE , C M − C E = log2 1+γE (9) Cs =    0 if γ ≤ γ , M

4

E

Journal Pre-proof

where C M = log2 (1 + γ M ) is the main channel’s capacity and C E = log2 (1 + γE ) is the wiretap channel’s capacity. C s in (9) can be rewritten as 1 + γ  M < Rs (10) C s = log2 1 + γE

where,     γ0 ,  λ= γp     , X

fγE |(X=γE ) (γE ) =

and J2 =

Z

σ 0

∞ σ

Z

∞

0

Z

0

re-

(14)

FγM |(X=(γE )) ((γE )) fγE |(X=γE ) (γE ) fY (y)dγE dy (15)

∞

FγM |(X=(γE )) (∈ (γE )) fγE |(X=γE ) (γE ) fY (y)dγE dy (16)





(21)

(22)

SOPs for different combining techniques are calculated as follow:

where,

!a−n 1 1  a 2 R s − 1 α2 = × σγ1 (d − 1)!(r − 1)!(σγ2 )d a! n ! n 2Rs (n + d − 1)! , σγ1  2Rs + 1 n+d σγ1 σγ2 σm (r − 1 + a − n)! ∆1 =  Rs r+a−n−m 2 −1 m! + 1

Jo

3.1.1. MRC/MRC scheme at Bob and Eve with MRC scheme at PR The CDF at Bob with MRC scheme can be expressed as   NB b−1 X  (γ ) X ((γ ))a   E − λΩE  1  FγM |(X=(γE )) ((γE )) = A(b) 1 − e a!(λΩ1 )a  b=1

NB X NE X

lP

Z

(20)

Where, !a−n Rs !n 1 a 2Rs − 1 2 (n + e − 1)! α1 = n  Rs  2 1 n+e a! γ1 γ1 + γ γ 1 2   A(b) = NbB (1 − ρB )NB −b ρb−1 B   A(d) = NdE (1 − ρE )NE −d ρd−1 E   A(r) = NrR (1 − ρR )NR −r ρr−1 R Similarly, J2 can be written as " r−1 − σ NB X NE X NR X A(b)A(d)A(r) X e γR MRC/MRC J2 = γR h! h=0 b=1 d=1 r=1  r−1+a−n  !h # Rs X σ −σ 2 σγ−1 + γ1 R 1 ∆1 − α2 e γR m=0

urn a

J1 =

(19)

(d − 1)!(λΩ2 )d

Rs " − 2 γ −1 b−1 X 1 e J1 MRC/MRC = A(b)A(d) 1 − (d − 1)!γ2d a=0 b=1 d=1  #X !h  NR a r−1 X  σ X 1  σ −  α1 A(r) 1 − e γR h! γ R n=0 r=1 h=0

Where fY (y) is the probability density function (PDF) of Y, fγE |(Y=y) is the PDF of γE conditioned on Y, and FγM |Y=y ((γE )) is the cumulative distribution function (CDF) of γ M conditioned on Y. SOP given in (13) can be also written as [1]

where,

A(d)

d=1

By substituting (17), (19) and (20) in (15) and (16); J1 and J2 for MRC/MRC scheme is calculated as

(13)

Pout = J1 + J2

γE − d−1 λΩ 2 γE e

The PDF at PR with MRC scheme is given by y − NR r−1 X γ y e R fY (y) = A(r) (r − 1)!γRr r=1

which can be simplified to (13) Z ∞Z ∞ Pout = FγM |{Y=y} (∈ (γE )) fγE |{Y=y} (γE ) fY (y)dγE dy 0

NE X

pro of

(11)

For secure transmission, R s should be less than secrecy capacity C s , otherwise, it leads to the compromising of the information-theoretic security. Secrecy outage probability is defined as the probability that secrecy capacity C s falls under R s , and is expressed as

Pout = Pr C s < RS = Pr γ M ≤ γE

+Pr γ M > γE Pr C s < RS | γ M > γE (12)

0

(18)

The PDF at Eve with can be written as

which is analogous to

(γE ) = 2Rs (1 + γE ) − 1 > γ M

γp γ γ0p X≥ γ0 X≤

a=0

σγ1

γR

SOP for this case is expressed as

MRC/MRC Pout = J1 MRC/MRC + J2 MRC/MRC

(17)

5

(23)

Journal Pre-proof

3.2.1. Intercept probability with MRC/MRC Scheme The intercept probability for this case is calculated as

3.1.2. SC/MRC scheme at Bob and Eve with MRC scheme at PR In this subsection, we consider SC scheme at Bob and MRC scheme at both Eve and PR. The CDF at Bob with SC can be expressed as FγM |(X=(γE )) ((γE )) = NB (−1)b

N

b=0

B −1

 (γ )ξ φB  − E B 1 − e λΩ1 ξB

(24)

a=0

 1+

1

 γ2 a+d γ1

#

(28)

Remark 1: The expressions (28) and (29) can be used to investigate other performance metrics like PNZC and it can be expressed as

        ξ B (2R s −1)     NX NE − γ B −1 X    1 ΦB  e   1 − = A(d)   !   d   Rs ξ   B 2 ξ B γ2   b=0 d=1    +1    γ1  !h  NR r−1 X X  1 σ  − γσ   A(r) 1 − e R h! γR  r=1 h=0

Pr(C s > 0) = Pr(γ M > γE ) = 1 − Pout (0) = 1 − Pint (30)

3.3. Asymptotic SOP The asymptotic nature of SOP in high SNR regime of γ1 is a key point in consideration in this subsection. As γ1 −→ ∞, it means Bob is located very near to Alice as comparison to the eavesdropper. When γ1 −→ ∞, the asymptotic CDF of γ M with MRC and SC can be written as

=

NX NE X NR X r−1 B −1 X

lP

J2

γ2 γ1

!a

3.2.2. Intercept probability with SC/MRC Scheme Intercept Probability for SC at Bob and MRC at Eve is calculated as   NX NE   B −1 X 1 ΦB A(d)   S C/MRC 1 −  Pint = d  (29)  ξB γ2 ξ B b=0 d=1 γ1 + 1



(25)

S C/MRC

!

" b−1 X A(b)A(d) 1 −

re-

J1 S C/MRC

b=1 d=0

a+d−1 d−1

ρB b , ζB = + b + 1 and (1 − ρB )ζB 1 − ρb 1 ρB . ξB = − 1 − ρB ζB (1 − ρB )2 By substituting (24), (19) and (20) in (15) and (16); J1 and J2 for SC/MRC scheme is calculated as

where, φB =

NB X NE X

pro of

NX B −1

MRC/MRC Pint =

−

b=0 d=1 r=1 h=0   ξ (2R s −1) − B γ + γσ

e

"

σ γR

!h

σr

R

1

σ

ΦB e γR A(d)A(r) ξB h!

− #

S C/MRC Pout

urn a

!d !r−h 1 2Rs ξB γ2 ξB (2Rs − 1) +1 + γ1 σγ1 γR (26)

= J1

S C/MRC

+ J2

S C/MRC

Fγ∞M MRC =

(1 − ρB )NB −1 γ M γ1

Fγ∞M S C =

NX B −1 b=0

ΦB

γM γ1

(31)

(32)

The asymptotic SOP with MRC/MRC scheme at Bob and Eve can be calculated as

(27)

NE X A(d) γ1 d=1  " !h  NR r−1 X X  1 σ  − γσ Rs Rs   A(r) 1 − e R 2 − 1 + 2 dγ2 h! γR  r=1 h=0 !h NE X NR r−1  X X 1 σ − γσ R s γ2 P 2 d + A(d)A(r)e γ1 h=0 h! γR d=1 r=1 # r X 1 1 r! h 1−z  Rs + (2 − 1) σ γR σγ1 Γ(r) z! z=0 MRC/MRC P∞ = (1 − PB )NB −1 out

Jo

3.2. Intercept Probability

An eavesdropper can easily decode the source message when C M falls below C E . In [34], intercept probability is defined as probability that the capacity of legitimate link falls below that of eavesdroppers link i.e Pint = P(C M < C E ). In contrast, SOP is defined the probability of difference between C M and C E .Thus, intercept probability is a special case of SOP for R s = 0 which implies ε(γE ) = γE .

(33)

6

Journal Pre-proof

• MRC/MRC and SC/MRC schemes has same diversity order of unity. G D is independent of NB , NE and NR .

The asymptotic SOP with SC/MRC at Bob and Eve can be expressed as "  ΦB A(d)  Rs 2 − 1 + 2Rs dγ2 γ1 b=0 d=1  !h  NB −1 X NE NR r−1 X X  1 σ  X − γσ  R A(r) 1 − e  + h! γR r=1 h=0 b=0 d=1 !h NR r−1  X X σ γ2 1 σ − ΦB A(d)A(r)e γP 2Rs d γ1 h=0 h! γR r=1 r # X 1 r! 1 h 1−z Rs + σ γR (2 − 1) σγ1 Γ(r) z! z=0 NX NE B −1 X

• MRC/MRC scheme offers low SOP than SC/MRC scheme. It is explained by the fact that G AMRC/MRC ≥ GSAC/MRC . • The performance difference between MRC and SC at legitimate receiver Bob can be characterized by ratio of their array gain and it can be written as −1  G AMRC/MRC  (1 − ρB )NB −1  (40) =  PN  B GSAC/MRC b=0 φ B

pro of

S C/MRC P∞ = out

(34)

From (40), it is clear that for the same SOP MRC/MRC scheme is better than SC/MRC   (1 − ρB )NB −1   scheme by an SNR gap of −10 log  PN B φ B b=0 dB.

Eqs.(33) & (34) can be written as    MRC/MRC −G DMRC/MRC −G DMRC/MRC MRC/MRC γ P∞ G = γ + O 1 out A 1 S C/MRC P∞ out

re-

(35)

3.4. ε-Outage Secrecy Capacity ε- outage secrecy capacity (Cout (ε)) is an important metric to investigate the secrecy performance of the wiretap channels for delay-limited wireless communications systems. It is defined as a largest secrecy rate R s,max , such that SOP is equal to ε. Mathematically, it can be written as

 S C/MRC   −GS C/MRC −G = GSAC/MRC γ1 D + O γ1 D

(36)

G DMRC/MRC = 1 GSDC/MRC = 1

(37)

and the secrecy array gain G A is "

d=1



A(d) 2Rs − 1 + 2Rs dγ2

urn a

G AMRC/MRC = (1 − ρB )NB −1

NE X

lP

where the secrecy diversity order G D is

(38)

GSAC/MRC =

( NX NE B −1 X b=0 d=1

" NR X ΦB A(d) 2Rs − 1 + 2Rs dγ2 A(r)

0

where, fγ1 (γ M ) is the PDF of main channel. For MRC and SC scheme, it can be expressed as

r=1

 !h  NB −1 X NE X NR r−1  σ X 1 σ  X − σ 1 − e− γR ΦB A(d)A(r)e γP  + h! γ R h=0 b=0 d=1 r=1 #)−1 !h r−1 r  X X 1 σ 1 Rs 1 r! h 1−z  R s γ2 2 d + (2 − 1) σ γR γ1 h=0 h! γR σ Γ(r) z! z=0

Jo

(41)

where Pout (R s,max ) = ε. Thus, substituting (23) and (27) into (41), ε- outage secrecy capacity for MRC/MRC scheme and SC/MRC technique can be calculated by using numerical evaluation. Alternatively, the Gaussian approximation is used to find the closed expressions for ε-outage secrecy capacity in order to avoid numerical roots finding. For this, we have to calculate the kth order moment of γ M and γE . Firstly, we calculate the kth order moment of γ M with MRC and SC scheme as Z ∞ E[γkM ] = (γ M )k fγ1 (γ M )dγ M (42)



 !h  NE X NR r−1 X  1 σ  X − σ − σ A(d)A(r)e γP A(r) 1 − e γR  + h! γ R r=1 h=0 d=1 r=1 !h !#−1 r r−1  X X γ2 1 r! h 1−z  1 σ 1 2Rs d + (2Rs − 1) σ γR γ1 h=0 h! γR σ Γ(r) z! z=0 NR X

Cout (ε) = R s,max

γ

fγMRC (γ M ) 1

=

(39)

fγSMC (γ M ) =

Remark 2: Based on equations (37),(38) and (39), following observations can be made.

NB − M X (γ M )b−1 e γ1 b=1

(b − 1)!γ1b

NX B −1

φB − γMγ ξB e 1 γ1

b=0

7

(43)

(44)

Journal Pre-proof

Using (43) and (44) in (42), kth order moment of γ1 with MRC and SC schemes is written as NB X

A(b)

b=1

E S C [γkM ] =

(k + b − 1)! k γ (b − 1)! 1

φB k!γ1k

(45)

  ! σ2M σ2E  √  µ M  − 2 + 2  + 2 log2 e Cout (ε) = log2 e ln µE 2µ M 2µE   2 2 2 σ σ σ2   σ M  2 − M2 + 2E − E2  er f c−1 (2 − 2ε) µ M 4µ M µE 4µE (53)

(46)

ξBk+1

pro of

E MRC [γkM ] =

where erfc is complementary error function. Thus, according to definition of ε-outage secrecy capacity and using (52), we have

Similarly, kth order moment of γ2 is expressed as E[γkE ] =

NE X

A(d)

d=1

(k + d − 1)! k γ (d − 1)! 2

where, µl = 1 + E[γl ], l{M, E} and σ2l = E[γl2 ] − E 2 [γl ]

(47)

Resorting to [35], the channel capacity of main channel C M can be expanded in Taylor series in term of the γ1 as  
C M (γ1 ) = log2 1 + E γ1 + log2 (e)  
∞ X γ1 − E γ1 h (−1)h−1 (48)  
h 1 + E γ1 h h=1

4. Scenario II : CSI of Eve’s channel is available at Alice

re-

In this section, we assume an active eavesdropping i.e., the CSI of Eve’s channel is available at Alice. In this case, average secrecy capacity is considered as a major performance metric, Alice can adapt transmission rate according to both CSI of the main channel and Eve’s channel to attain perfect secrecy.

By applying expectation operator to (48), the approximation of C M is expressed as   log2 D γ1  
E(C M ) ≈ log2 1 + E γ1 − (49) 2 (1 + E(γ1 ))2   where D γ1 is the variance of γ1 . By expanding C 2M in Taylor series about γ1 and using expectation operator, the second moment of C M is approximated as  

E(C 2M ) = log2 1 + E γ1 2 ! D[γ1 ] log2 e e + log2 (50) 1 + E[γ1 ] (1 + E[γ1 ])2

lP

4.1. Average secrecy capacity

urn a

By recalling the definition of achieved secrecy rate defined in (9) , we have [36]

(log2 e)2 D[γ1 ] 2

(1 + E[γ1 ])

−

(log2 e)D2 [γ1 ] 4 (1 + E[γ1 ])4

Z

0

∞

Z

∞

γE



log2 (1 + γ M ) − log2 (1 + γE )



fγE (γE ) fγM (γ M )dγE dγ M

(54)

To solve above integral, first we perform integration by parts on inner integral, and apply some algebraic manipulation, the average secrecy capacity (ASC) can be calculted as ! Z ∞ Z ∞ Fγ2 (γE ) 1 Cs = fγ1 (γ M )d(γ M ) dγE (55) ln 2 0 1 + γE γE

According to (49) and (50), the variance of C M is written as D[C M ] =

Cs =

(51)

We can find the expectation and variance of C E by putting the corresponding parameters into (49) and (50) respectively. When C M > C E , C s is the Gaussian random variable due to the fact that it is a linear combination of the two independent Gaussian random variable. Hence, we have E[C s ] = E[C M ] − E[C E ] and D[C s ] = D[C M ] − D[C E ], where E[.] is the expectation and D[.] is the variance. Hence, Gaussian approximation of CDF of C s is ! 1 x − E[C s ] (52) FC s (x) ≈ 1 − er f c √ 2 2D[C s ]

Jo

where, FγE (.) is the CDF at Eve and fγ1 (.) is the PDF at Bob. 4.1.1. Average Secrecy capacity for MRC/MRC Scheme The CDF at Eve with MRC scheme is given by   d−1  γ X γa   − γE E   FγE (γE ) = A(d) 1 − e 2 a a!γ2  a=0 d=1 NE X

8

(56)

Journal Pre-proof

where,

By inserting (43) and (56) in (55) and using ([37],Eq.3.353.5), ASC for MRC/MRC scheme is calculated as " NE X NB X b−1 X 1 A(b)A(d) C MRC/MRC = (−1)(k−1) e γ1 k k!γ1 d=1 b=1 k=0 ! X !c ! X k d−1 1 1 1 k−c Ei − + (c − 1)!(−1) − + a γ1 γ a!γ 1 2 c=1 a=0   !! X a+k 1 1 1 +1 (t − 1)! (−1)a+k−1 e γ1 γ2 Ei − + + γ1 γ2 t=1 !−t !# 1 1 (−1)a+k−t + γ1 γ2 (57)

τ1 =

1 ln 2

τ2 =

1 ln 2

Z

∞

0

ln(1 + γ M ) fγ1 (γ M )dγ M

(61)

T γ2 (γE ) fγ (γ M )dγE dγ M 1 + γE 1

(62)

and ∞ 0

pro of

Z

Now, we investigate τ1 and τ2 in the high SNR regime respectively. Hence, γ M → ∞, ln(1 + γ M ) ≈ ln(γ M ). Hence, by substituting (43) in (61) and using ([37], Eq.4.352.1), we have N

τ∞ 1 =

where Ei is exponential integral.

N

B B X 1 X A(b)ψ(b) + A(b) log2 γ1 ln 2 b=1 b=1

d ln (Γ(b)) is the digamma function. where ψ(b) = db Similarly, according to ([36], Eq.19), the asymptotic expression for τ2 is written as Z ∞ T γ2 (γE ) 1 ∞ τ2 = dγE ln 2 0 1 + γE ! ! NE X d−1 A(d) 1 1 1 X exp Γ(1 + a)Γ −a, = ln 2 d=0 a=0 a!γ2a γ2 γ2

lP

re-

4.1.2. Average Secrecy capacity for SC/MRC Scheme By inserting (56) and (44) in (55) and using ([37],Eq.3.353.1), ASC for SC/MRC scheme is written as ! X j−1 NE NE X X ξB ΦB  ξγB  −e 1 Ei − + CS C/MRC = A(d) ξB γ1 d=1 d=1 a=0   " !! ξB 1 A(d) 1 ΦB ξ 1 + B a−1 γ1 γ2 (−1) e Ei − + a! γ2a ξB γ1 γ2 !−c # a X ξB 1 + (c − 1)!(−1)(a−c) + γ1 γ2 c=1

(64)

By substituting Eq.(63) and Eq.(64) in Eq.(60), asymptotic average secrecy capacity derived as

(58)

urn a

4.2. Asymptotic average secrecy capacity In order to study the impact of key system parameters on ASC, we examine the ASC in the high SNR regime and also provide two metrics i.e high SNR slope and the high SNR power offset which characterize the asymptotic behavior of ASC.

N

N

B B X 1 X A(b)ψ(b) + A(b) log2 γ M − ln 2 b=1 b=1 ! ! NE X d−1 1 X A(d) 1 a Γ(1 + l)Γ −a, exp ln 2 d=0 a=0 a!γ2a γ2 γ2

C∞ MRC/MRC =

(65)

We also examine the high SNR power offset and high SNR slope to characterize the asymptotic average SNR in high SNR regime like conventional non-secrecy network. The asymptotic ASC in Eq.(65) can be written as   MRC/MRC ˇ MRC/MRC log2 γ1 − Ł∞ C∞ (66) MRC/MRC = S∞

Jo

4.2.1. Asymptotic average secrecy capacity for MRC/MRC scheme P NE for d=1 A(d) = 1, (56) can be written as FγE (γE ) = 1 − T γE (γE ), where ! d−1 !a NE X γE X 1 γE (59) T γ2 (γE ) = A(d) exp − γ2 a=0 a! γ2 d=1

MRC/MRC where Sˇ∞ is the high SNR slope in bits/s/Hz MRC/MRC (3dB) and Ł∞ is the high SNR power offset in 3dB units. According to [29], Sˇ∞ can be calculated as

Taking this into consideration, asymptotic ASC can be expressed as # Z ∞ "Z γM 1 − T γ2 (γE ) 1 C MRC/MRC = dγE fγM (γ M )dγ M ln 2 0 1 + γE 0 = τ1 − τ2

(63)

MRC/MRC Sˇ∞ = lim

γ1 →∞

(60)

9

C∞ MRC/MRC log2 γ1

(67)

Journal Pre-proof

Putting (65) into (67) and performing some mathematical calculations,we have NB X

A(b) = 1

10-1

10-2

(68)

Secrecy Outage Probability

MRC/MRC = Sˇ∞

100

b=1

=τ∞ 2

1 ln 2

=

1 ln 2

and

Z

∞

0 NX B −1 b=1

= 0.6

B

B

Simulation MRC/MRC for

B

B

=0.6

=0.6

=0.2

=0.6

5

10

15

20

25

30

35

 S C/MRC  CS∞C/MRC = Sˇ∞ log2 γ1 − ŁS∞C/MRC

(70)

40

45

(75)

S C/MRC

where Sˇ∞ is the high SNR slope in bits/s/Hz (3dB) and ŁS∞C/MRC is the high SNR power offset in 3dB units. According to [29], Sˇ∞ can be calculated as

lP for

CS∞C/MRC

S C/MRC Sˇ∞ = lim

γ1 →∞

log2 γ1

=

NX B −1 b=1

ΦB =1 ξB

(76)

Hence, for SC/MRC high SNR slope is unity, that means it is independent of correlation coefficients and number of antenna at Bob and eavesdropper. High SNR power offset is expressed as   CS∞C/MRC   S C/MRC NB NE Ł∞ = lim log2 γ1 − S C/MRC  = Ł∞ + Ł∞ γ1 →∞ ˇS∞ (77)

(71)

where,

NB Ł∞ =−

NB −1 1 X (ψ(1) − ln(ξB )) ln 2 b=1

NE Ł∞ =Ω∞ 2

Jo

(73)

The asymptotic ASC is expressed as NB −1 NX B −1 1 X ΦB ΦB (ψ(1) − ln(ξB )) + log2 γ1 ln 2 b=1 ξB ξB b=1 ! ! NE X d−1 1 X A(d) 1 1 − exp Γ(1 + a)Γ −a, ln 2 d=1 a=0 a!γ2a γ2 γ2

CS∞C/MRC =

B

=0.6

Figure 2: Secrecy outage probability versus γ1 dB with γR = 0dB, γ2 = 10dB, ρE = .1, σ = 0.8 and ρP = .2

(72)

=

B

Average SNR of main channel ( 1 ) dB

NX B −1 ΦB ΦB (ψ(1) − ln(ξB )) + log2 γ1 ξB ξB b=1

τ∞ 2

=0.2

=0.6

Simulation MRC/MRC for

-7

0

ln(γ M ) fγ1 (γ M )dγ M

Ω∞ 2

Exact MRC/MRC for

10-6

B

B

10-8

urn a

Ω∞ 1 =

Exact SC/MRC for

which can be written as

4.2.2. Asymptotic average secrecy capacity SC/MRC scheme Asymptotic ASC for SC/MRC is written as

where,

=0.2

Simulation SC/MRC for

From above analysis, we conclude that key parameters of main channel such as NB , ρB has positive impact on ASC whereas key parameters of Eve’s channel like NE , ρE has negative impact on C s .

∞ CS∞C/MRC = Ω∞ 1 − Ω2

B

Exact MRC/MRC for

re-

NE Ł∞

Exact SC/MRC for

10-5

10

N

B 1 X A(b)ψ(b) ln 2 b=1

= 0.2

10-4

Simulation SC/MRC for

It is clear that Ł∞ characterize the effect of main channel and Eve’s channel on C s . By substituting (65) and (68) in (77), we have NB Ł∞ =−

B

pro of

Eq. (68) demonstrates that the high SNR slope is independent of key parameters like correlation coefficient, number of antennas at Bob and Eve. High SNR power offset is expressed as ! C∞ s NB NE MRC/MRC = Ł∞ + Ł∞ (69) Ł∞ = lim log2 γ1 − γ1 →∞ Sˇ∞

10-3

(74)

10

(78)

Remark 3 : High power offset of SC/MRC scheme is always greater than MRC/MRC scheme for given value of NB and ρB . This means that the average secrecy capacity of MRC/MRC scheme is always greater than SC/MRC for constant parameters. 5. Numerical Results The impact of outdated CSI and different diversity combining techniques on the secrecy performance is

Journal Pre-proof

100

100

10-1

B

SOP and Intercept Probability

= 0.2

10-3

10-4

10

-5

Exact SC/MRC for

=0.6 B

Exact MRC/MRC for Exact SC/MRC for

10-6

B

B

=0.6

=0.2

Exact MRC/MRC for

B

B

=0.6 B

Simulstion MRC/MRC for

10-7

B

Simulation MRC/MRC for Simulation SC/MRC for

1

2

E

=0.9

Exact SOP MRC/MRC for

E

=0.9

Exact Intercept Probability SC/MRC for

E

Exact Intercept Probability MRC/MRC for Exact SOP SC/MRC for

E

=0.9 E

=0.9

=0.1

Exact Intercept Probability SC/MRC for Exact SOP MRC/MRC for

10-4

E

E

Simulation SC/MRC for

E

=0.1

=0.1

Exact Intercept Probability MRC/MRC for

E

=0.1

=0.9

B B

Simulation SOP MRC/MRC for

= 0.6

10

=0.6

/

E

E

E

E

4

5

6

7

8

0.1

0.2

E

=0.9

=0.1

=0.1

Simulation Intercept Probability MRC/MRC for

0

=0.9

=0.9

=0.1

Simulation Intercept Probability MRC/MRC for

10-6

1

E

Simulation SOP MRC/MRC for

=0.2

3

Simulation SC/MRC for

-5

Simulation Intercept Probability SC/MRC for

=0.2

10-8 0

Exact SOP SC/MRC for

10-3

Simulation Intercept Probability SC/MRC for

=0.2

Simulation SC/MRC for

10-2

pro of

Intercept Probability

10

10-1

-2

0.3

0.4

E

=0.1

0.5

0.6

0.7

Exact SC/MRC Exact MRC/MRC

0.8

Simulation MRC/MRC Simulation SC/MRC

10-2

R

10-4

=0.01

Exact SC/MRC for

R

=0.2

Asymptotic SC/MRC for

10-6

Exact SC/MRC for

R

R

Exact MRC/MRC for

=0.2

= 0.01

Asymptotic SC/MRC for R

R

=0.01

=0.2

Asymptotic MRC/MRC for Exact SC/MRC for

10-8

R

R

=0.2

=0.01 =0.01 R

Asymptotic SC/MRC for Simulation SC/MRC for Simulation SC/MRC for

R R

Simulation MRC/MRC for

=0.2 =0.01 R

Simualgion MRC/MRC for

=0.2

R

=0.01

10-10 0

5

10

15

20

25

30

SNR of the main channel,

1

lP

Secrecy Outage Probability

=0.2 R

35

40

E

10-2

-3

-10

-5

Jo

-5

0

5

10

Exact SC/MRC for

E

E

Exact MRC/MRC for Exact SC/MRC for

0.45

5

10

15

E E

MRC/MRC for SC/MRC for 5

2

MRC/MRC for SC/MRC for

=0.3

25

30

35

2

2

=5dB

= 5dB 2

=15 dB

=15dB

4 5 dB

3

2

=0.7

=0.7 E

Simulation MRC/MRC for Simulation SC/MRC for Simulation SC/MRC for

20

20

Peak Interference Power, (IP) dB

6

=0.3

Simulation MRC/MRC for

15

0.55

Figure 7: SOP versus Peak interference power for ρB = 0.5, ρE = 0.8, ρR = 0.1, γ1 = 8dB, γ2 = γr = 0dB, PT = 15dB, NA = 2, NR = 2, NE = 5 and NB = 2

= 0.7

Exact MRC/MRC for

10-4

0.6

0.5

Cout = Rs,max nats/s/Hz

10-1

0.65

45

urn a

E

= 0.3

0.7

dB

Figure 4: Exact and asymptotic SOP versus γ1 dB with R s = 1, γR = 0dB, γ2 = 10dB, ρE = 0.1, σ = 0.8 and ρB = .6 100

Secrecy Outage Probability

re-

0.75

Probability of non-zero secrecy capacity

1

0.85

100

10

0.9

Figure 6: SOP and Intercept probability versus ρB for different value of ρE

Figure 3: Intercept probability versus γ1 dB with γR = 0dB, γ2 = 10dB, ρE = .1, σ = 0.8 and ρP = .01

10

0.8

B

2

25

E E E

=0.3

15 dB 1

=0.7

=0.7 =0.3

0

30

0

SNR of main channel, 1 dB

2

4

6

8

10

12

14

16

18

20

SNR of the main channel, 1 dB

Figure 5: Probability of non-zero secrecy capacity as a function of γ1 for varying ρB with ρB = 0.3,γ2 = 5dB, NB = 4, NR = 2 and NE = 5

Figure 8:  outage secrecy capacity versus SNR of main channel for  = 0.9, ρB = 0.9 and ρE = 0.4

11

Journal Pre-proof

investigated and presented by numerical results in this section. Furthermore, Monte Carlo simulation is done to check the validity of our results. The main parameter R s is set to be 1 nats/s/Hz.

20

Exact MRC/MRC

18

Asymptotic MRC/MRC Simulation MRC/MRC

16

Average secrecy capacity (nats/s/Hz)

Exact SC/MRC

Asymptotic MRC/MRC

12

10

8

pro of

Figure.2 and figure.3 plot exact SOP and intercept probability of two combining techniques (SC/MRC and MRC/MRC) for different value of ρB . It is clear from the figure.2 and figure.3 that both SOP and intercept probability decreases with increasing the average SNR of the main channel γ1 . This is due to that fact that the increasing γ1 will increase the capacity of the main channel, which in turn increases the secrecy capacity and reduces the outage probability. From figure.2 and figure.3, it is clear that SOP and intercept probability decreases with increasing ρB . It is because channel estimation also improves with increasing ρB . Figure.4 plots exact and asymptotic SOP versus γ1 for different value of ρR . Figure.4 depicts that at high SNR regime, parallel lines of asymptotic SOP approximate the exact SOP. These parallel lines of asymptotes confirm that secrecy diversity order is independent of γE , ρE and NE as indicated in (37). It is also clear from figure.4 that SOP decreases with decreasing ρR . It is because decreases value of ρR will decrease the channel estimation at PR, which results in more power utilization at Alice. Figure.5 plots non-zero secrecy capacity versus γ1 for varying ρB . From figure.5, it is observed that non-zero secrecy capacity exists if the SNR of Eve’s channel is greater than SNR of the main channel. Figure.6 shows SOP and intercept probability versus ρB for various ρE . Figure. 6 depicts that secrecy performance of the system is degraded with increasing ρE . It is because of the estimation of Eve’s channel improves with increasing ρE . In addition, as shown in figure.6, the SOP and intercept probability improve with MRC scheme over the SC scheme as ρB increases. Figure.7 depicts SOP versus peak power interference IP . The SOP is decreasing with increasing IP . This is due to the peak interference power constraint IP = σPT , which increases the transmit power of Alice, as shown in eq.(3). Figure.8 plots ε-outage secrecy capacity versus γ1 for different value of γ2 and for Pout = 0.8 for two combining techniques. It is depicted from fig. 8 that the εoutage secrecy capacity is non-zero even when γ M ≤ γE as long as Pout ≥ 0.5. Figure.9 plots average secrecy versus γ1 for two combining techniques. From fig.9, it is clear that the average secrecy capacity increases with increasing γ1 and average secrecy capacity of MRC/MRC scheme is

Simulation SC/MRC 14

6

4

2

0

0

5

10

SNR of the main channel,

15

1

20

25

db

Figure 9: Average secrecy capacity versus γ1 for NB = NE = 2, ρB = ρE = .9

re-

higher than SC/MRC. It is because high power offset of SC/MRC scheme is always greater than MRC/MRC scheme for the given value of NB and ρB . 6. Conclusion

Jo

urn a

lP

In this work, we have examined the impact of the outdated CSI on the secrecy performance of the underlay CRN in a Rayleigh fading environment for two distinct scenarios. The compact expressions for all performance matrix have been investigated for the scenario I, which reveal that secrecy performance of the system is increased with ρB . Furthermore, the asymptotic SOP expressions reveal that there is a certain relationship exist between the secrecy array gain of two combining techniques that achieve the same secrecy diversity order. For scenario II, closed-form expressions for the exact, and asymptotic average secrecy capacity has been derived, which facilitate us to characterize the impact of the main channel and eavesdropper channel on a power offset. We can extend this work by considering a massive MIMO system with low-cost transceiver hardware with an arbitrary array configurations (e.g., co-located or distributed antennas) [36],[37] at both Alice and Bob. Moreover, it will be interesting to examine the collective impact of non-ideal CSI and non-ideal hardware in future work. References [1] M. Elkashlan, L. Wang, T. Q. Duong, G. K. Karagiannidis, A. Nallanathan, On the security of cognitive radio networks, IEEE Transactions on Vehicular Technology 64 (2015) 3790– 3795.

12

Journal Pre-proof

[2] J. Huang, A. L. Swindlehurst, Cooperative jamming for secure communications in mimo relay networks, IEEE Transactions on Signal Processing 59 (2011) 4871–4884. [3] A. Mukherjee, A. L. Swindlehurst, Robust beamforming for security in mimo wiretap channels with imperfect csi, IEEE Transactions on Signal Processing 59 (2011) 351–361. [4] Y. Zou, J. Zhu, X. Wang, V. C. Leung, Improving physical-layer security in wireless communications using diversity techniques, IEEE Network 29 (2015) 42–48. [5] Z. Shu, Y. Qian, S. Ci, On physical layer security for cognitive radio networks, IEEE Network 27 (2013) 28–33. [6] H. V. Poor, Information and inference in the wireless physical layer, IEEE Wireless Communications 19 (2012). [7] N. Yang, L. Wang, G. Geraci, M. Elkashlan, J. Yuan, M. Di Renzo, Safeguarding 5g wireless communication networks using physical layer security, IEEE Communications Magazine 53 (2015) 20–27. [8] Z. Shu, Y. Qian, S. Ci, On physical layer security for cognitive radio networks, IEEE Network 27 (2013) 28–33. [9] A. Goldsmith, S. A. Jafar, I. Maric, S. Srinivasa, Breaking spectrum gridlock with cognitive radios: An information theoretic perspective, proc. IEEE 97 (2009) 894–914. [10] S. Srinivasa, S. A. Jafar, Cognitive radios for dynamic spectrum access-the throughput potential of cognitive radio: A theoretical perspective, IEEE Communications Magazine 45 (2007). [11] Y. Zou, J. Zhu, X. Wang, L. Hanzo, A survey on wireless security: Technical challenges, recent advances, and future trends, Proceedings of the IEEE 104 (2016) 1727–1765. [12] Y. Zou, J. Zhu, X. Wang, V. C. Leung, Improving physical-layer security in wireless communications using diversity techniques, IEEE Network 29 (2015) 42–48. [13] M. Bloch, J. Barros, M. R. Rodrigues, S. W. McLaughlin, Wireless information-theoretic security, IEEE Transactions on Information Theory 54 (2008) 2515–2534. [14] F. He, H. Man, W. Wang, Maximal ratio diversity combining enhanced security, IEEE Communications Letters 15 (2011) 509– 511. [15] H. Alves, R. D. Souza, M. Debbah, M. Bennis, Performance of transmit antenna selection physical layer security schemes, IEEE Signal Processing Letters 19 (2012) 372–375. [16] J. Huang, A. Mukherjee, A. L. Swindlehurst, Secrecy analysis of unauthenticated amplify-and-forward relaying with antenna selection, in: Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, IEEE, pp. 2481–2484. [17] L. Dong, Z. Han, A. P. Petropulu, H. V. Poor, Improving wireless physical layer security via cooperating relays, IEEE transactions on signal processing 58 (2010) 1875–1888. [18] N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, I. B. Collings, Transmit antenna selection for security enhancement in mimo wiretap channels, IEEE Transactions on Communications 61 (2013) 144–154. [19] Y. Zou, X. Li, Y.-C. Liang, Secrecy outage and diversity analysis of cognitive radio systems, IEEE Journal on Selected Areas in Communications 32 (2014) 2222–2236. [20] Y. Zou, J. Zhu, L. Yang, Y.-C. Liang, Y.-D. Yao, Securing physical-layer communications for cognitive radio networks, IEEE Communications Magazine 53 (2015) 48–54. [21] A. P. Liavas, Tomlinson-harashima precoding with partial channel knowledge, IEEE Transactions on Communications 53 (2005) 5–9. [22] M. Gans, The effect of gaussian error in maximal ratio combiners, IEEE Transactions on Communication Technology 19 (1971) 492–500. [23] J. Subhashini, V. Bhaskar, Capacity analysis of rayleigh fading

[24]

pro of

[25]

channels in low snr regime for maximal ratio combining diversity due to combining errors, IET Communications 7 (2013) 745–754. A. Shrestha, K. Kwak, On maximal ratio diversity with weighting errors for physical layer security, IEEE Communications Letters 18 (2014) 580–583. R. Zhao, H. Lin, Y.-C. He, D.-H. Chen, Y. Huang, L. Yang, Secrecy performance of transmit antenna selection for mimo relay systems with outdated csi, IEEE Transactions on Communications 66 (2018) 546–559. H. Lei, J. Zhang, K.-H. Park, I. S. Ansari, G. Pan, M.-S. Alouini, Secrecy performance analysis of simo underlay cognitive radio systems with outdated csi, IET Communications 11 (2017) 1961–1969. N. S. Ferdinand, D. B. da Costa, M. Latva-aho, Effects of outdated csi on the secrecy performance of miso wiretap channels with transmit antenna selection, IEEE Communications Letters 17 (2013) 864–867. J. Xiong, Y. Tang, D. Ma, P. Xiao, K.-K. Wong, Secrecy performance analysis for tas-mrc system with imperfect feedback, IEEE Transactions on Information Forensics and Security 10 (2015) 1617–1629. Y. Huang, F. S. Al-Qahtani, T. Q. Duong, J. Wang, Secure transmission in mimo wiretap channels using general-order transmit antenna selection with outdated csi, IEEE Transactions on Communications 63 (2015) 2959–2971. H. Zhao, Y.-y. Tan, G.-f. Pan, Y.-f. Chen, Ergodic secrecy capacity of mrc/sc in single-input multiple-output wiretap systems with imperfect channel state information, Frontiers of Information Technology & Electronic Engineering 18 (2017) 578–590. H. Lei, C. Gao, I. S. Ansari, Y. Guo, Y. Zou, G. Pan, K. A. Qaraqe, Secrecy outage performance of transmit antenna selection for mimo underlay cognitive radio systems over nakagamim channels, IEEE Transactions on Vehicular Technology 66 (2017) 2237–2250. X. Cai, G. B. Giannakis, Performance analysis of combined transmit selection diversity and receive generalized selection combining in rayleigh fading channels, IEEE Transactions on Wireless Communications 3 (2004) 1980–1983. Y. Hu, X. Tao, Secrecy outage on transmit antenna selection with weighting errors at maximal-ratio combiners, IEEE Communications Letters 19 (2015) 597–600. A. Singh, M. R. Bhatnagar, R. K. Mallik, Physical layer security of a multi-antenna based cr network with single and multiple primary users, IEEE Trans. Veh. Technol. 66 (2017) 11–011. J. P´erez, J. Ib´anez, L. Vielva, I. Santamaria, Closed-form approximation for the outage capacity of orthogonal stbc, IEEE Communications Letters 9 (2005) 961–963. L. Wang, N. J. Yang, M. Elkashlan, P. L. Yeoh, J. Yuan, Physical layer security of maximal ratio combining in two-wave with diffuse power fading channels., IEEE Trans. Information Forensics and Security 9 (2014) 247–258. I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic press, 2007.

[26]

[27]

[28]

re-

[29]

[30]

lP

[31]

[32]

urn a

[33] [34] [35] [36]

Jo

[37]

13

Journal Pre-proof

Conflicts of Interest Statement

pro of

Manuscript title: Secure transmission in underlay cognitive radio network with outdated channel state information

Jo

urn a

lP

Shilpa Thakur Research Scholar, Indian Institute of Technology, Jammu, (J&k), India

re-

I declare no conflict of interest.

Journal Pre-proof

pro of

Author’s biography

Shilpa Thakur received B.tech degree in Electronics and Communication Engineering from Himachal Pradesh University Shimla, Himachal Pradesh in 2009 and M.Tech degree in Electronics and Communication Engineering from Kurukshetra University, Haryana, India in 2013. Currently, she is perusing Ph.D from Indian Institute of Technology Jammu, Jammu and Kashmir, India. Her research interests include cognitive radio networks, and physical layer security.

•

Ajay Singh received B.Tech. degree in Electronics and Communication Engineering from the Kurukshetra University, Haryana, India, in 2003 and the M.E. degree in Electronics and Communication Engineering from Punjab University, Chandigarh, India, in 2007. He obtained Ph.D. degree from the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India, in 2012. From May 2013 to January 2016, he was with the faculty of the Department of Electronics and Telecommunication Engineering, National Institute of Technology Raipur, Chhattisgarh, India and from Jan 2016 to March 2018, he was with the faculty of the Department of Electronics and Communication Engineering, National Institute of Technology Hamirpur, H.P, India. Since April 2018, he has been with the faculty of the Department of Electrical Engineering, Indian Institute of Technology Jammu, Jammu and Kashmir, India, where he is currently working as Assistant Professor. His research interests include physical layer security, cognitive radio networks and molecular communication.

Jo

urn a

lP

re-

•

Jo

urn a

lP

re-

pro of

Journal Pre-proof