Secrecy Performance of Wirelessly Powered Wiretap Channels

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1 Secrecy Perforance of Wirelessly Powered Wiretap Channels Jiang, X, Zhong, C, Chen, X, Duong, T Q, Tsiftsis, T, & Zhang, Z 6 Secrecy Perforance of Wirelessly Powered Wiretap Channels IEEE Transactions on Counications DOI: 9/TCOMM65959 Published in: IEEE Transactions on Counications Docuent Version: Peer reviewed version Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal Publisher rights 6 IEEE Personal use of this aterial is peritted Perission fro IEEE ust be obtained for all other uses, in any current or future edia, including reprinting/republishing this aterial for advertising or prootional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted coponent of this work in other works General rights Copyright for the publications ade accessible via the Queen's University Belfast Research Portal is retained by the authors and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requireents associated with these rights Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output Every effort has been ade to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact openaccess@qubacuk Download date:3 Jul 8

2 Secrecy Perforance of Wirelessly Powered Wiretap Channels Xin Jiang, Student Meber, IEEE, Caijun Zhong, Senior Meber, IEEE, Xiaoing Chen, Senior Meber, IEEE, Trung Q Duong, Senior Meber, IEEE, Theodoros Tsiftsis, Senior Meber, IEEE, and Zhaoyang Zhang, Meber, IEEE Abstract This paper considers a wirelessly powered wiretap channel, where an energy constrained ulti-antenna inforation source, powered by a dedicated power beacon, counicates with a legitiate user in the presence of a passive eavesdropper Based on a siple tie-switching protocol where power transfer and inforation transission are separated in tie, we investigate two popular ulti-antenna transission schees at the inforation source, naely axiu ratio transission MRT and transit antenna selection TAS Closed-for expressions are derived for the achievable secrecy outage probability and average secrecy rate for both schees In addition, siple approxiations are obtained at the high signal-to-noise ratio SNR regie Our results deonstrate that by exploiting the full knowledge of channel state inforation CSI, we can achieve a better secrecy perforance, eg, with full CSI of the ain channel, the syste can achieve substantial secrecy diversity gain On the other hand, without the CSI of the ain channel, no diversity gain can be attained Moreover, we show that the additional level of randoness induced by wireless power transfer does not affect the secrecy perforance in the high SNR regie Finally, our theoretical clais are validated by the nuerical results Index Ters Physical layer security, wireless power transfer, secrecy outage probability, average secrecy rate I INTRODUCTION Recently, the rapidly increasing deands for high data rate wireless services have put a treendous pressure on the energy consuption of battery-powered obile devices Hence, how to prolong the lifetie of these energy-constrained Manuscript received February 4, 6, revised April 4, 6, and June 4, 6, accepted July, 6 This work was supported by the National Key Basic Research Progra of China No CB364, the National High-Tech R&D Progra of China under grant 4AAA7 and 4AAA75, the Zhejiang Provincial Natural Science Foundation of China LR5F, and the Fundaental Research Funds for Central Universities 6QNA54 The work of X Chen was supported by the National Natural Science Foundation of China No 63 The work of T Q Duong was supported by the UK Royal Acadey of Engineering Research Fellowship under Grant RF45\4\ The editor coordinating the review of this paper and approving it for publication was Prof Z Ding X Jiang, C Zhong and Z Zhang are with the Institute of Inforation and Counication Engineering, Zhejiang University, China eail: caijunzhong@zjueducn X Chen is with the College of Electronic and Inforation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Chinaeail: chenxiaoing@nuaaeducn T Q Duong is with ECIT, Queen s University Belfast, Belfast, UKeail: trungqduong@qubacuk T A Tsiftsis is with the School of Engineering, Nazarbayev University, Astana, Kazakhstan and with TEI of Central Greece, Laia 35, Greece eail: Theodorostsiftsis@nuedukz This paper was presented in part at the European Signal Processing Conference, Budapest, Hungary, August 6 obile devices has becoe a critical proble to be addressed Responding to this, energy harvesting techniques, which scavenge energy fro abient environent such as wind and solar have been proposed as a proising solution Nevertheless, harvesting energy fro nature resources depends heavily on the locations and weather conditions, which fails to generate stable energy output, hence ay not be suitable to power wireless devices with strict quality of service requireents As a result, a new energy harvesting paradig, generally referred to as wireless power transfer WPT, has gained considerable attentions By exploiting the radio frequency RF signals as a eans for energy transportation, WPT enables reliable and stable energy supplies to obile devices Since RF signals are widely used as a ediu for inforation transission, incorporating the feature of WPT into wireless counications networks has eerged as a hot topic, generally referred to as siultaneously wireless inforation and power transfer SWIPT systes, and significant research effects have been devoted to understand the fundaental perforance liitation as well as design efficient SWIPT systes, see for instances [] [9] and references therein However, SWIPT systes are also vulnerable to potential security issues This is because RF signals are shared by ultiple nodes, which ight be potential eavesdroppers Recent research results show that copared to conventional cryptographic approaches, physical layer security is a better choice in energy and coputation constrained systes, such as SWIPT systes [] The basic concept behind physical layer security is to exploit the physical layer characteristics of wireless channels to provide perfect secrecy The work was pioneered by Wyner [], which confired that perfect secrecy can be achieved when the quality of the wiretap channel is a degraded version of the ain channel Recently, ensuing security in SWIPT systes have gained increasing attentions In [], [3], the authors presented the optial beaforing design and power allocation schee for ultiple-input single-output MISO systes in the presence of passive eavesdroppers, later in [4], the issue of uncertain eavesdroppers was tackled, where joint optiization of inforation and energy beaforing and power allocation were studied Latest works have considered the security issue in ore sophisticated SWIPT systes, such as relay [5], ulticast [6], cognitive radio [7] and OFDMA systes [8] A coon of these works is that they consider hybrid network architecture, where the inforation source also acts as the energy source However, as analyzed in [9], the

3 harvested energy fro hybrid networks is general infeasible to power larger devices such as sartphones, tablets and laptops Responding to this, a novel network architecture was proposed in [], where a dedicated station called power beacon PB is incorporated into the wireless network to power obile devices Very recently, the secrecy perforance of device-todevice DD counications in energy harvesting cognitive cellular networks has been investigated in [], where the DD transitter first harvests energy fro PBs, then perfors secure transission to the desired DD receiver Thus far, secure counications in SWIPT systes with dedicated PB reain largely an uncharted area Motivated by this, we consider a point-to-point four-node wirelessly powered wiretap channel consisting of a dedicated PB, an energy constrained inforation source and a legitiate user in the presence of a passive eavesdropper It is assued that the source has no transit power of its own, hence entirely relies on the external energy charging via wireless power transfer fro the PB For such systes, we present a coprehensive analysis on the achievable secrecy perforance It is worth pointing out that, unlike in the conventional counications systes, where the transit power is constant, the use of WPT effectively akes the available source transit power a rando variable In addition, since the transit power affects both the signals observed at the legitiate user and eavesdropper, the effective signal-to-noise ratios SNRs of the ain and wiretap channels becoe correlated, aking the secrecy perforance analysis uch ore challenging The ain contributions of this work can be suarized as follows: For enhancing wireless security, we propose siple diversity transission schees at the inforation source, naely, axiu ratio transission MRT and transit antenna selection TAS In particular, for the TAS schee, depending on the required channel state inforation CSI, three different selection criteria are devised For all schees, closed-for expressions for secrecy outage probability are derived, which enable efficient evaluation of the achievable secrecy perforance Furtherore, siple and inforative high SNR approxiations are presented The analytical results suggest that the achievable secrecy perforance depends heavily on the available CSI at the source With the CSI of the ain channel, the syste attains full secrecy diversity gain, while only unit secrecy diversity order can be achieved with only the CSI of the wiretap channel In addition, the best perforance is achieved when both the CSI of ain channel and wiretap channel are available For all schees, closed-for expressions for average secrecy rate and high SNR approxiations are also derived Our results indicate that, all the schees attain the sae high SNR slope of one and distinct high SNR power offset Moreover, increasing the nuber of transit antennas iproves the secrecy rate However, the gain gradually diinishes when the nuber of transit antennas is oderately large Based on the siple high SNR expressions, the optial tie switching ratio θ is studied It was shown that there exists a unique θ axiizing the secrecy throughput For the special single-antenna source case, closed-for expression for the optial θ is obtained We show that, the randoness of source transit power induced by WPT does not affect the secrecy diversity order and the high SNR slope The reainder of the paper is organized as follows Section II introduces the syste odel and proposes several transission schees Section III provides an analytical study on the achievable secrecy outage probability of the proposed schees, while Section IV investigates the average secrecy rate of the syste Nuerical results and discussions are presented in Section V Finally, Section VI concludes the paper and suarizes the key findings Notation: We use bold lower case letters to denote vectors and lower case letters to denote scalars; h denotes the Frobenius nor; E{x} stands for the expectation of the rando variable x and [x] + denotes ax,x; T denotes the transpose operator and denotes the conjugate operator I k is the identity atrix of size k Γx is the gaa function [, Eq 83], Γα, x is the upper incoplete gaa function [, Eq 835] and γα,x is the lower incoplete gaa function [, Eq 835] ψx denotes the Euler psi function [, Eq 836] and K v x is the v- th order odified Bessel function of the second kind [, Eq 847] S a,b x denotes the Loel function [, Eq 857] and G p,q,n x denotes the Meijer G-function [, Eq 93] II SYSTEM MODEL We consider a four-node wirelessly powered wiretap channel consisting of one PB, one inforation source Alice and one legitiate user Bob in the presence of one eavesdropper Eve as shown in Fig We assue that the source is equipped with N antennas, while the other three nodes are equipped with a single antenna Quasi-static fading is assued, such that the channel coefficients reain unchanged during each transission block but vary independently between different blocks We adopt the tie-sharing protocol proposed in [] Hence, a coplete transission slot with tie duration oft is divided into two orthogonal sub-slots, ie, the first one for power transfer with tie duration of θt with θ < θ < being the tie switching ratio, and the second one for inforation transission with tie duration of θt During the first phase, the PB sends an energy signal to Alice, and the received energy signal at Alice y s can be expressed as y s = P S h P x s +n s, where P S denotes the transit power of the PB, x s is the energy signal with unit power,n s is ann-diensional additive The considered ulti-antenna source odel is relevant to the scenarios where an energy constrained ulti-antenna sensor node [?] transits confidential inforation to a single-antenna fusion center or an energy constrained ulti-antenna transitter [4] perfors secrecy transission to a singleantenna receiver

4 3 atch bea, as such, the received signal y M at Bob can be written as P Fig : A scheatic diagra of the syste odel consisting of one PB, one inforation source Alice, one legitiate user Bob and one eavesdropper Eve white Gaussian noise AWGN vector with E{n s n s } = N I The N vector h P denotes the power transfer channel fro PB to Alice Due to relatively short distance between the power beacon and the source, it is likely that the line-ofsight propagation exists Hence, the Nakagai- distribution is used to odel the power transfer channel, ie, the aplitude of each eleent of h P follows Nakagai- distribution with shape paraeter and average power λ P Therefore, at the end of the first phase, the total harvested energy within duration θt can be expressed as M W E = ηp S h P θt, where η < η < denotes the energy conversion efficiency Since the source counicates with the legitiate user during the second phase with duration θt, the transit power can be coputed as P = E θt = ηp S h P θ θ 3 To exploit the benefits of ultiple antennas at Alice, different transission schees can be adopted In this work, we consider two popular transission schees, naely MRT and TAS The ipleentation of MRT and TAS requires different types of CSI For MRT, only the CSI of the ain channel is required While for TAS, partial CSI of the ain channel or the wiretap channel is required In practice, the CSI of the ain channel can be estiated at Bob, and then feed back to Alice On the other hand, the CSI of the wiretap channel can be obtained when the eavesdropper is active in the network, a scenario that is particularly applicable in the networks cobining ulticast and unicast transissions where the terinals play dual roles as legitiate receivers for soe signals and eavesdroppers for others [5], [6] A Maxiu Ratio Transission MRT For the MRT schee, Alice ais at axiizing the reception quality of the ain channel by aking use of a channel- In the presence of line-of-sight effect, Rician fading is coonly used in literature However, the analysis with Rician fading is uch ore involved As such, for atheatical tractability, we adopt the Nakagai- fading odel, since the Nakagai- distribution provides very accurate approxiation to the Rician distribution y M = Ph T Mwx t +n M, 4 where x t denotes the inforation sybol with unit energy; N vector h M denotes the ain channel fro Alice to Bob, whose eleents are circularly syetric coplex Gaussian rando variables RVs with zero ean and variance ; n M denotes the AWGN with zero ean and variance N ; w is the MRT vector given by w = h M h M Siilarly, the signal received at Eve y W can be expressed as y W = Ph T W wx t +n W, 5 where the N vector h W denotes the wiretap channel fro Alice to Eve, whose eleents are circularly syetric coplex Gaussian RVs with zero ean and variance, and n W denotes the AWGN with zero ean and variance N As such, the instantaneous SNR at Bob γ M and at Eve γ W are given by and respectively γm MRT = ηp S h P h M θ N θ, 6 W = ηp S h P h T W h M h M θ θ, 7 γ MRT N B Transit Antenna Selection TAS TAS is another low-coplexity transission schee In this work, we consider three different selection criteria as elaborated below Criterion : In this case, the antenna with the axiu gain of ain channel is selected, ie, k = arg ax i=,,n h id, 8 where h id is the i-th eleent of ain channel h M It is worth noting that best antenna selection according to the above criterion iplies a rando antenna selection for the wiretap channel because the ain channel is independent of the wiretap channel Criterion : Instead of axiizing the gain of ain channel, we now intend to iniize the gain of wiretap channel As such, the best antenna is selected according to the following criterion: k = arg in i=,,n h ie, 9 where h ie is the i-th eleent of the wiretap channel h W 3 Criterion 3: Since the secrecy perforance of syste depends on the quality of both the ain channel and wiretap channel, we now propose the third selection criterion which picks the antenna axiizing the ratio of ain channel gain and wiretap channel gain, ie, k = arg ax i=,,n hid h ie

5 4 Hence, the instantaneous SNR at Bob γ M and at Eve γ W can be expressed as and γm TAS = ηp S h P h M,k θ N θ, γw TAS = ηp S h P h W,k θ N θ, whereh M,k denotes the channel coefficient of the link between the k-th antenna of the source and legitiate user, while h W,k denotes the channel coefficient of the link between the k-th antenna of the source and eavesdropper C Secrecy Perforance For wiretap channels, the secrecy rate C S is given by the difference of the ain channel capacity and the wiretap channel capacity [5] { log+γ C S = M log+γw γ M > γ W, γm γ W, 3 where {MRT,TAS} In this work, we consider two different counication scenarios In the first scenario, Alice uses a constant transission rate R S to counicate with Bob According to [], perfect secrecy is achievable when R S < C S, otherwise, secrecy is coproised In this case, secrecy outage probability becoes an appropriate perforance etric In the second scenario, we assue that Alice adapts its transission rate according to C S, as such, ergodic secrecy rate becoes the appropriate perforance easure In the following sections, we present a detailed analysis of the achievable secrecy perforance of both MRT and TAS schees III SECRECY OUTAGE PROBABILITY In this section, we investigate the secrecy outage perforance of the considered syste For both transission schees, new closed-for expressions for the exact and asyptotic secrecy outage probability are presented Based on which, the ipacts of ultiple antennas on the secrecy perforance are characterized in ters of the secrecy outage diversity order and the secrecy outage array gain According to the definition, the secrecy outage probability can be expressed atheatically as A MRT P out R S = PC S < R S 4 We start with the MRT schee, and we have the following key result: Theore : The exact secrecy outage probability of the MRT schee can be expressed in closed-for as out R S = ΓN P MRT k k λ P N+p N k k= p= K N p k k p p! +k k p+ k, 5 k λ P where k = ηps θ N θ and k = RS Proof: See Appendix A Theore presents an exact closed-for expression for the secrecy outage probability, which can be efficiently evaluated However, the expression is too coplicated to yield any insights Motivated by this, we now look into the asyptotic regie, where siple expressions can be obtained 3 For the asyptotic high SNR regie, we assue that with an arbitrary Such a scenario has been widely adopted in the literature, see for instance [7] [3] In practice, this occurs when the quality of the ain channel is uch better than wiretap channel, ie, Bob is relatively close to Alice while Eve is far away fro Alice or the wiretap channel undergoes severe sall-scale and large-scale fading effects In the following, we characterize the two key perforance paraeters governing the secrecy outage probability in the high SNR regie, ie, secrecy diversity order G d and secrecy array gain G a defined by [3] P out R S = G a G d 6 Proposition : In the high SNR regie, ie,, the secrecy outage probability of the MRT schee can be approxiated by N k N PMRTR ΓN k k k S = ΓN k k λ P k= 7 Proof: See Appendix B It is evident fro 7 that the syste achieves a secrecy diversity order of N In addition, we observe the intuitive effect of the position of nodes on the secrecy outage probability For instance, the secrecy outage probability decreases when the PB is close to the source, ie, large λ P It is also easy to see that the high SNR secrecy outage probability PMRT R S is a decreasing function with respect to Ps N, indicating that increasing the transit power of the PB is always beneficial B TAS Criterion We now ove to the TAS Criterion schee, and we obtain the following key result: Theore : The exact secrecy outage probability of TAS Criterion schee can be expressed in closed-for as N Pout TAS k N R S = k+ λm ΓN +k k + k +k k λ P N K N k= k +k k λ P 8 3 Although the energy transfer efficiency of state of the art technique is low, it is still of both theoretical and practical interests in wirelessly powered counications systes due to the following reasons: First, the energy transfer efficiency can be significantly iproved by adopting ultiple antenna technology and the PB assisted WPC architecture Second, the effective SNR could be still reasonably high even if the energy transfer efficiency is low Third, the key insights obtained fro high SNR analysis provide useful guidance for practical syste design

6 5 Proof: See Appendix C While Theore presents an exact closed-for expression for the secrecy outage probability, the expression is too coplicated to gather ore insights As such, we study the asyptotic behavior for the outage perforance Proposition : In the high SNR regie, ie,, the secrecy outage probability of TAS Criterion schee can be approxiated as PTAS R S = N k N N! ΓN k k k 9 ΓN k k λ P k= Proof: When, we have e x N = N +o N x x As such, following the sae steps as that of Proposition yields the desired result It is evident fro 9 that the syste also achieves a secrecy diversity order of N Coparing 7 and 9, we see that PMRC R S = P TAS RS N!, naely, the MRT schee outperfors the TAS Criterion schee by a factor of N! This is not surprising since the MRT schee has access to perfect CSI of h M, while TAS schee only utilizes partial knowledge of h M Recall in the conventional wirelessly powered syste without secrecy constraint, the outage probability decays in a uch slower speed, ie, ρ N lnρ instead of ρ N due to the randoness of the transit power, where N is the nuber of antennas, and ρ is the SNR [3], [33] In the current work, we notice that the randoness of the transit power does not affect the scaling behavior of the secrecy outage probability This is because that in wiretap channels, the secrecy perforance is deterined by the difference between the ain and wiretap channel capacities, and the transit power affects both channels Hence, the effect of transit power randoness is canceled C TAS Criterion We now consider the TAS Criterion schee, and we have the following key result: Theore 3: The exact secrecy outage probability of TAS Criterion schee can be expressed in closed-for as N ΓN N +k λ W N k K N k λ P P TAS out R S = k k λ P Proof: See Appendix D Having obtained the exact outage probability of TAS Criterion schee, we now look into the high SNR regie, and derive a siple analytical approxiation for the outage probability of the syste Proposition 3: In the high SNR regie, ie,, the secrecy outage probability of TAS Criterion schee can be approxiated as P TASR S = N + N k k k k λ P Proof: See Appendix E Different fro the previous two cases which achieve a diversity order of N, TAS Criterion schee only attains unit diversity order This is also intuitive since TAS Criterion schee ais to iniize the received SNR of the eavesdropper and the selected antenna serves as a rando transit antenna for the ain channel As such, no secrecy diversity gain can be realized, and increasing the nuber of antennas N only yields soe secrecy array gain D TAS Criterion 3 We now analyze the secrecy outage probability of the syste with the TAS Criterion 3 schee Theore 4: The secrecy outage probability of TAS Criterion 3 schee can be approxiated by P TAS3 out k k + λm N Proof: See Appendix F Having obtained the outage probability of TAS Criterion 3 schee, we now look into the asyptotic regie Proposition 4: In the high SNR regie, ie,, the secrecy outage probability of TAS Criterion 3 schee can be approxiated as P TAS3 = k N 3 Proof: The proof is straightforward, hence oitted As expected, the syste achieves a secrecy diversity order of N Recall the high SNR outage probability of the MRT schee, and noticing that N + N k= ΓN k ΓN k= ΓN k ΓN k k k λ P k = k k k λ P k >, we observe that the TAS Criterion 3 schee outperfors the MRT schee This is reasonable since the TAS Criterion 3 schee considers both the CSI of h M and h W, while only the CSI of the h M is utilized in the MRT schee E Optiization of the Tie Switching Ratio θ Fro the analytical expressions derived in previous subsections, we are ready to study the optiization of tie switching ratio θ Specifically, we adopt the effective secrecy throughput as the perforance easure as in [34] Hence, when the source transits at a constant rater S, the average secrecy throughput can be evaluated by R = P out R S θ Also, due to space liitation and for illustrative purpose, we only focus on the MRT schee, while other cases will be nuerically illustrated in Section V Using the high SNR approxiation given in 7, the effective secrecy throughput of the MRT schee can be expressed

7 6 as τθ = P MRTθR S θt T = P MRT θr S θ 4 Hence, the optial θ is the solution of the following optiization proble: θ = argax τθ θ st < θ < 5 To this end, we have the following key result: Proposition 5: Consider a polynoial N k= a k θ θ k + k =, 6 θ k N where a k = ΓN k k N k ΓN ηp Sk λ P Then, the optial θ is the unique root of the polynoial in, Proof: Substituting 7 into 4, we obtain τθ = N k= a k θ θ k b θ, 7 where b = R S Thus, the derivative of τθ with respect to θ can be expressed as dτθ N k θ = a k b + k b 8 dθ θ θ k= dτθ dθ It is easy to show that is a onotonically decreasing function with respect toθ Whenθ approaches, dτθ dθ + N dτθ and when θ approaches, dθ b b b < Therefore, there exists a unique θ, with dτθ dθ =, where τθ attains its axiu value In general, due to the coplexity of the involved expression, deriving a closed-for solution for θ is very challenging However, for sall N, closed-for expressions for θ can be obtained Reark : When N =, the optial θ is given by θ = a +a a 9 Reark : When N =, the optial θ is given by θ = 3 q q p q q p 3, where p = 3a a a +a a and q = a a +a a F Coparison of the Proposed Protocols We now present a ore detailed perforance coparison for the proposed schees at the high SNR regie as suarized in Table I on the top of the next page In general, the secrecy perforance depends heavily on the available CSI at the source The ore CSI available, the better the secrecy perforance However, in ters of high SNR protocols investigated shown in Table I We list the CSI that is required for each protocol and investigate the secrecy diversity order and the outage perforance according to the asyptotic secrecy outage probability IV AVERAGE SECRECY RATE In this section, we focus on the average secrecy rate perforance of the syste For both transission schees, new closed-for expressions for the exact and asyptotic average secrecy rate are presented Based on which, the ipacts of ultiple antennas on the secrecy perforance are characterized in ters of the high SNR slope and the high SNR power offset Starting fro the definition, the average secrecy rate can be expressed as C = E{[log +γ M log +γ W] + } 3 We now study the achievable secrecy rate of different transission schees in the following A MRT Theore 5: The exact average secrecy rate of MRT schee can be expressed in closed-for as 3 on the top of the next page Proof: See Appendix G While 3 provides the exact average secrecy rate, the expression are too coplex to yield insightful inforation Motivated by this, we now look into the high SNR regie, where the secrecy rate is dictated by two key paraeters known as the high SNR slope S and the high SNR power offset L [3], ie, C = S log L 33 To this end, we have the following key result: Proposition 6: For the MRT schee, the high SNR slope is given by S MRT S MRT and the high SNR power offset L MRT L MRT = ln k λ P ΓN G3 3 =, 34 is given by +ψn+ψn + k λ P N,, 35 Proof: See Appendix H We note that the high SNR slope is also known as the axiu ultiplexing gain or the nuber of degrees of freedo [35] According to 34, the high SNR slope is one because we assue that the legitiate user only eploys a single antenna in this syste In addition, we observe that the wiretap channel only reflects the high SNR power offset It is also easy to see that the asyptotic average secrecy rate is an increasing function with respect to k and λ P, indicating that increasing the transit power of the PB is always beneficial and the secrecy rate increases when PB is close to the source

8 7 C MRT = TABLE I: Coparison of the proposed schees Schee CSI requireent Secrecy diversity order Outage perforance MRT h M N Second best TAS Criterion Index of the entry of h M N Third best TAS Criterion Index of the entry of h W Worst TAS Criterion 3 h M and h W N Best ΓN N k= N k k k= p= k λ P k+p+p! k k p ΓN k +p+ k λ P G 3 3 k λ P N k,, G k λ P p+ + λm + ] 3 N k,, B TAS Criterion We now consider the TAS Criterion schee, and we have the following key result: Theore 6: The exact average secrecy rate of TAS criterion schee can be expressed in closed-for as 36 on the top of the next page Proof: The proof follows siilar lines as that of Theore 5, hence is oitted We now look into the high SNR regie, and present the high SNR etrics in the following proposition Proposition 7: For the TAS criterion schee, the high SNR slope S TAS is given by S TAS and the high SNR power offset L TAS N L TAS = ln k λ P + N k k k= ΓN G3 3 =, 37 is given by lnk +ψ+ψn+ k λ P N,, 38 Proof: The proof follows siilar lines as that of Proposition 6, hence is oitted C TAS Criterion We now analyze the average secrecy rate of the syste with the TAS Criterion schee, and we have the following key result: Theore 7: The exact average secrecy rate of TAS criterion schee can be expressed in closed-for as C TAS = ΓN G 3 3 k λ P N,, + N N,, [ G 3 3 k λ P ] 39 Proof: The proof follows siilar lines as that of Theore 5, hence is oitted Having obtained the exact average secrecy rate of TAS Criterion schee, we now look into the asyptotic regie Proposition 8: For the TAS criterion schee, the high SNR slope S TAS is given by and the high SNR power offset L TAS L TAS = ln k λ P ΓN G3 3 S TAS =, 4 is given by +ψ+ψn + N k λ P N,, 4 Proof: The proof follows siilar lines as that of Proposition 6, hence is oitted D TAS Criterion3 We now ove to the TAS Criterion 3 schee, and we obtain the following key result: Theore 8: The average secrecy rate of TAS criterion3 schee can be approxiated by C TAS3 G 3 33 N k= + Γk +ΓN k N,+k N,+k N,+k N,k N 4 Proof: See Appendix I We now look into the high SNR regie, and present the high SNR etrics in the following proposition Proposition 9: For the TAS criterion3 schee, the high SNR slope S TAS3 is given by S TAS3 =, 43 and the high SNR power offset L TAS3 is given by L TAS3 = ln + N N N + N k k Proof: See Appendix J k= k 44

9 8 C TAS = ΓN N N k+ k k= [ G 3 k 3 k λ P N,, G 3 3 k + k λ P N,, ] 36 V NUMERICAL RESULTS In this section, we present nuerical results to verify the theoretical expressions Unless otherwise stated, we set the source transission rate as R S = bit/s/hz, the energy conversion efficiency as η = 8 and the tie switching ratio as θ = 5 The Nakagai- paraeter is set to be = 4, which corresponds to a Rician factor of K = 3+ The transit power of the PB to the noise ratio as PS N = db, the channel variance as λ P = and = Also, we set ρ = PS N and ρ = PS N to denote the average SNR of the ain channel and the wiretap channel, respectively Secrecy Outage Probability N = 3 4 Monte Carlo Siulation Analytical High SNR Approxiation ρ db Fig : Secrecy outage probability versus ρ with different N for the MRT schee Fig plots the secrecy outage probability versus ρ with different N for the MRT schee As illustrated, the analytical results are in exact agreeent with the Monte Carlo siulations, which deonstrates the correctness of the analytical expression In addition, the high SNR results accurately predict the secrecy diversity order and the secrecy array gain, and increasing N substantially enhances the secrecy outage perforance by achieving a higher secrecy diversity gain Fig 3 illustrates the secrecy outage probability of three different TAS schees Once again, we observe that the analytical curves are in perfect agreeent with the Monte Carlo siulation results and the high SNR approxiation are sufficiently tight for all curves As expected, the TAS Criterion schee only attains unit diversity order, while the other two TAS schees achieves a full diversity order of N In addition, it is observed that the TAS Criterion 3 schee yields the best secrecy outage perforance However, the TAS Criterion schee tends to outperfor the TAS Criterion schee in the low SNR regie, and then becoes inferior as the operating SNR becoes sufficiently high Fig 4 exaines the average secrecy rate of the MRT schee We observe that the high SNR slope of all the curves is one, which corroborates the theoretical analysis presented Secrecy Outage Probability TAS Criterion TAS Criterion TAS Criterion 3 Monte Carlo Siulation Analytical High SNR Approxiation ρ db Fig 3: Secrecy outage probability versus ρ for different TAS schees with N = 3 Average secrecy rate bits/s/hz Monte Carlo Siulation Analytical High SNR Approxiation N = ρ db Fig 4: Average secrecy rate versus ρ with different N for the MRT schee in previous section Nevertheless, increasing N iproves the average secrecy rate by decreasing the high SNR power offset However, the benefit of increasing the nuber of antennas N gradually diinishes when N is sufficiently large Fig 5 investigates the ipact of distance between Alice and Eve on the average secrecy rate for different TAS schees As expected, the TAS Criterion 3 schee always attains the best perforance At the low ρ regie, naely, relatively large distance between Alice and Eve or sall, the TAS Criterion schee outperfors the TAS Criterion schee However, the opposite holds when the distance decreases, ie, becoes large This phenoenon is quite intuitive, since when the average channel gain of the wiretap channel is better than that of the ain channel, selecting the worst antenna as per the TAS Criterion schee substantially degrades the capacity of the wiretap channel, thereby resulting in a larger secrecy perforance gain than the TAS Criterion schee

10 9 Average secrecy rate bits/s/hz TAS Criterion TAS Criterion TAS Criterion 3 = = ρ db Fig 5: Average secrecy rate versus ρ for different TAS schees with N = 3 Average secrecy rate bits/s/hz TAS Criterion TAS Criterion TAS Criterion 3 N = ρ db 55 5 N = 3 Fig 6: Average secrecy rate versus ρ for different TAS schees Fig 6 copares the average secrecy rate for different TAS schees Once again, we observe that the high SNR slopes are one The TAS Criterion schee outperfors the TAS Criterion schee with a large In addition, the nuber of antennas N has a significant ipact on the perforance of TAS schees For sall N, ie, N = 3, all the three TAS schees attain siilar secrecy rate While for large N, ie, N =, the secrecy rate difference becoes uch ore pronounced Fig 7 depicts the ipact of tie split paraeter θ on the secrecy perforance of different schees Specifically, we adopt the effective throughput as the perforance easure as in [34] Hence, when the source transits at a constant rate R S, the average throughput can be evaluated by R = P out R S θ, while when the source transits at a varying rate adapting to the secrecy rate, the throughput is given by R = θ C It is observed that, in all cases, the effective throughput first increases along with θ, and then start to decrease after reaching the axiu point, indicating that there exists a unique optial tie split paraeter θ In addition, we see that, with optiized tie split paraeter θ, the MRT schee achieves the highest throughput as expected Also, it is observed that the TAS Criterion schee outper- Throughput bits/s/hz Throughput bits/s/hz MRT TAS Criterion TAS Criterion θ a Rθ = P outr S θ MRT TAS Criterion TAS Criterion θ b Rθ = θ C Fig 7: Effective throughput versus θ for different schees with N = and = fors the TAS Criterion The reason is that we have set = and = in the siulations, a scenario where the average gain of the ain channel is worse than the wiretap channel Hence, the ain channel capacity enhanceent due to selection of the strongest channel in TAS Criterion schee is rather insignificant copared to the wiretap channel capacity degradation due to the selection of the weakest channel in TAS Criterion schee VI CONCLUSION In this paper, we have investigated the secrecy perforance of the wirelessly powered wiretap channels For both MRT and TAS schees, exact analytical expressions and asyptotic approxiations are presented, which facilitate the extraction of key insights of the achievable secrecy outage probability and average secrecy rate perforance The findings of the paper suggest that, with CSI of the ain channel eg, MRT and TAS Criteria and 3, the syste can achieve substantial secrecy diversity gain On the other hand, without the CSI of the ain channel eg, TAS Criterion, no diversity gain can be attained, which indicates the critical iportance of CSI in the design of practical systes

11 APPENDIX A PROOF OF THEOREM We start by expressing the SNR given in 6 and 7 as γ MRT M = k y hp y hm, and γ MRT W = k y hp y hw, 45 where k = ηps N θ θ, y h P = h P, y hm = h M and y hw = ht W h M h M It is straightforward to show that the probability density function pdf of y hp follows a gaa distribution with shape paraeter N and scale paraeter λ P / given by [36] N f yhp x = x N e λ x P, 46 ΓN λ P and the pdf of y hm follows a chi-square distribution with N degrees of freedo given by [37] f yhm x = xn λ N M ΓNe x 47 In addition, according to [38], y hw follows an exponential distribution with pdf f yhw x = e x, 48 and is independent of y hm As such, the secrecy outage probability can be written as Pout MRT +k y hp y hm R S = P k, 49 +k y hp y hw where k = RS Conditioned on y hp and y hw, with the help of [, Eq 335], we obtain P MRT out R S y hp,y hw = = e k k y k y h W hp N k= k k y hp +k y hw xn λ N M ΓNe x dx k k y hp + k y hw k 5 By applying the binoial expansion x + x n = k x x n k, 5 can be further expressed as n n k= k N Pout MRT R S y hp,y hw = e k k y hp k k y hp e k k k x k x k= p= p e k y h W ΓN k p!k p! k p k y hw 5 λ P Noticing that the RV y hp is decoupled with y hw, the expectation can be taken separately Hence, with the help of [, Eq 3479], we obtain p x N N e λ x P dx = ΓN k k λ P N+p K N p k k λ P 5 Siilarly, invoking [, Eq 336], we have k p e k x k x e x dx = k p! k k p +k k p+ 53 To this end, pulling everything together yields the desired result APPENDIX B PROOF OF PROPOSITION Starting fro 49, conditioned on y hp and y hm, we have P MRT out R S y hp,y hm = Prob y hm > k k y hp = Prob y hm > k k y hp k= y h M k k k k y hp e yh M e x dx k k λ + W k k y hp 54 k k y hp = With the help of [, Eq 335] and e k k, k y hp conditioned on yhp, the outage probability can be expressed as P MRT out R S y hp = e k k y hp e k N k y hp k= k=n k k + k y hp k k k y hp + λm k N k 55 Then, averaging overy hp, with the help of [, Eq 3479], the secrecy outage probability can be coputed as 56 on the top of next page Expanding the Bessel function by [, Eq 8446] and oitting the high order ites yield P MRT R S = k=n N k= k k λ P k k k λ P + λm k N k k ΓN k + ΓN ΓN k 57 ΓN By oitting the high order ites, the desired result can be obtained APPENDIX C PROOF OF THEOREM Based on and, we set y hm = h M,k and y hw = h W,k In this case, since the selected antenna corresponds to a rando transit antenna for the eavesdropper, the pdf of y hw can be expressed as f yhw x = e x Now, with soe siple algebraic anipulations, the cuulative distribution function cdf of y hm can be shown as x F yhm x = N e t dt = e x N 58

12 P MRT out R S = k=n k k λ P k ΓN N k= k k λ P k k k λ P N k + λm k N k ΓN K N k k k λ P k + k λ P N k K N k k 56 k λ P Hence the pdf of y hm can be obtained with a siple derivative as follows: f yhm x = N N e e x λ x M 59 Then, the desired result can be obtained by following siilar lines as in the proof of Theore APPENDIX D PROOF OF THEOREM 3 In this case, the strongest antenna corresponds to a rando transit antenna for the legitiate user, as such, the pdf of y hm can be expressed as f yhm x = e x Then, with soe siple algebraic anipulations, the cdf of y hw can be shown as N F yhw x = e t dt = e N λ x W x 6 Taking derivation of 6, the pdf of y hw can be expressed as f yhw x = N e N x 6 Then, the desired result can be obtained by following siilar lines as in the proof of Theore APPENDIX E PROOF OF PROPOSITION 3 By following the sae steps as in the proof of Proposition, we obtain N N +k ΓN N k K N k λ P P TAS out R S = k k λ P 6 Expanding the Bessel function and oitting the high order ites yield N out R S = N +k k +o N k λ P P TAS λ M 63 Then, the desired result can be obtained along with soe siple algebraic anipulations APPENDIX F PROOF OF THEOREM 4 The exact secrecy outage probability is difficult to characterize, instead, we apply the following approxiation as in [39] [ ] + +γm C S = log [ log +γ W ] + = γm γ W [ log hm,k ] + 64 h W,k It is worth pointing out that, such approxiation is reasonably tight and becoes asyptotically exact in the high SNR regie Then, we can define a new variable as follows X = ax i=,,n hid h ie 65 Along with soe siple algebraic anipulations, the cdf of X can be coputed as N F X x = e t e y dtdy y λ x W N x = 66 x+ λm Taking derivation of 66, the pdf of X can be expressed as f X x = N x N x+ λm N+ 67 To this end, the desired result can be expressed as F X k APPENDIX G PROOF OF THEOREM 5 Starting fro 3, conditioned on y hp, by following the sae steps as in [3], we forulate the average secrecy rate as C MRT = = k y hp +k y hp x F y hw x k y Γ hp e x +k y hp x F yhm x dx x N, ΓN dx 68 Invoking [, Eq 3383] and expanding the incoplete Gaa function, the average secrecy rate can be coputed

13 C MRT = N k= k k e k y hp N k y hp Γ k= p=, k k+p k y hp e k y hp + λ M k y hp k p! Γ, k y hp + k y hp p+ k y hp + p+ 69 as 69 on the top of the page The next step is to average over y hp Here, we set A = N k k N k ΓN λ P k= x N k e λ x+ P k x Γ, k x dx 7 With the help of [, Eq 83533] and [, Eq 3479], 7 can be further expressed as A = N k= k k k ΓN λ P 4x N k+ x + K N k x k λ P Invoking [, Eq 65657], we have A = N k= k λ P k N k ΓN k + ΓN N+k dx 7 k N k+ k λ P S N+k,N k k λ P 7 With the help of [, Eq 9346] and [, Eq 935], we obtain A = N k k= G 3 3 k ΓN k λ P k λ P N k,, 73 To this end, by following the sae steps and utilizing [, Eq 336], the desired result can be obtained APPENDIX H PROOF OF PROPOSITION 6 Capitalizing on the general fraework proposed in [3] for the evaluation of asyptotic average secrecy rate, conditioned on y hp, we have C = A B = k y hp +k y hp x lnk y hp xf yhm xdx F yhw x dx 74 With the help of [, Eq 435], A can be coputed as A = log + ln k λ P +ψn+ψn 75 The calculation of B is exactly the sae as C above Then we have B = ΓN G3 3 k λ P 76 N,, The desired result then follows iediately APPENDIX I PROOF OF THEOREM 8 Starting fro the definition, we have C TAS3 = Substituting 67 into 77 yields C TAS3 = N λ M logxf X xdx 77 lnx x N x+ λm N+ dx 78 Making a change of variable t = x and utilizing [4, Eq 8465], 78 can be alternatively written as C TAS3 = N = N λ M λ M t+ N lnt+ N+ dt t++ λm G t, t+ N, t++ λm N+ dt 79 Applying the binoial expansion and utilizing [, Eq 785] yield the desired result APPENDIX J PROOF OF PROPOSITION 9 By taking the general for, we obtain C = A B = lnxf yhm xdx lnyf yhw ydy, 8 where the two pdfs f yhm x and f yhw x have been derived in [4] as f yhm x = NxN x λ N Γ N,, 8 M

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15 4 [4] A P Prudnikow, Y A Brychkov, and O I Marchev, Integrals and Series: More Special Functions, vol 3, New York, NY, USA: Gordon and Breach, 99 [4] K Tourki, F A Khan, K A Qaraqe, H Yang, and M Alouini, Exact perforance analysis of MIMO cognitive radio systes using transit antenna selection, IEEE J Sel Areas Coun, vol 3, no 3, pp , Mar 4 Xin Jiang S 6 received the BS degree in inforation and counication engineering fro Zhejiang University, Hangzhou, China, in 5 He is currently working towards his MS degree in the college of inforation science and electronic engineering at Zhejiang University His research interests include physical layer security and wireless powered counications Trung Q Duong S 5, M, SM 3 received his PhD degree in Telecounications Systes fro Blekinge Institute of Technology BTH, Sweden in Since 3, he has joined Queen s University Belfast, UK as a Lecturer Assistant Professor His current research interests include physical layer security, energy-harvesting counications, cognitive relay networks He is the author or co-author of 9 technical papers published in scientific journals and presented at international conferences Dr Duong currently serves as an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, IEEE COMMUNICATIONS LETTERS, IET COMMUNICATIONS, WILEY TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES, and ELECTRONICS LETTERS He has also served as the Guest Editor of the special issue on soe ajor journals including IEEE JOURNAL IN SELECTED AREAS ON COMMUNI- CATIONS, IET COMMUNICATIONS, IEEE WIRELESS COMMUNICATIONS MAGAZINE, IEEE COMMUNICATIONS MAGAZINE, EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING, EURASIP JOUR- NAL ON ADVANCES SIGNAL PROCESSING He was awarded the Best Paper Award at the IEEE Vehicular Technology Conference VTC-Spring in 3, IEEE International Conference on Counications ICC 4 He is the recipient of prestigious Royal Acadey of Engineering Research Fellowship 5- Caijun Zhong S 7-M -SM 4 received the BS degree in Inforation Engineering fro the Xi an Jiaotong University, Xi an, China, in 4, and the MS degree in Inforation Security in 6, PhD degree in Telecounications in, both fro University College London, London, United Kingdo Fro Septeber 9 to Septeber, he was a research fellow at the Institute for Electronics, Counications and Inforation Technologies ECIT, Queens University Belfast, Belfast, UK Since Septeber, he has been with Zhejiang University, Hangzhou, China, where he is currently an associate professor His research interests include assive MIMO systes, full-duplex counications, wireless power transfer and physical layer security Dr Zhong is an Editor of the IEEE TRANSACTIONS ON WIRELESS COM- MUNICATIONS, IEEE COMMUNICATIONS LETTERS, EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING, and JOURNAL OF COMMUNICATIONS AND NETWORKS He is the recipient of the 3 IEEE CoSoc Asia-Pacific Outstanding Young Researcher Award He and his coauthors has been awarded a Best Paper Award at the WCSP 3 He was an Exeplary Reviewer for IEEE TRANSACTIONS ON COMMUNICATIONS in 4 Theodoros A Tsiftsis S, M 4, SM was born in Laia, Greece, in 97 He received the BSc degree in physics fro the Aristotle University of Thessaloniki, Greece, in 993, the MSc degree in digital systes engineering fro the Heriot-Watt University, Edinburgh, UK, in 995, the MSc degree in decision sciences fro the Athens University of Econoics and Business, Greece, in, and the PhD degree in electrical engineering fro the University of Patras, Greece, in 6 He joined the Departent of Electrical Engineering at the Technological Educational Institute of Central Greece in February Currently, he is Associate Professor of Counication Technologies in the Departent of Electrical and Electronic Engineering at the School of Engineering of the Nazarbayev University, Astana, Kazakhstan His research interests include the broad areas of cooperative counications, counication theory, wireless counications, and optical wireless counication systes Dr Tsiftsis acts as reviewer for several international journals and he was eber of the Editorial Boards of IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and IEEE COMMUNICATIONS LETTERS Currently he is an Area Editor for Wireless Counications II of the IEEE TRANSACTIONS ON COMMUNICATIONS Xiaoing Chen M -SM 4 received the BSc degree fro Hohai University in 5, the MSc degree fro Nanjing University of Science and Technology in 7 and the Ph D degree fro Zhejiang University in, all in electronic engineering He is currently with the College of Electronic and Inforation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China Since Feberuary 5, he is a Huboldt Resarch Fellow at the Institute for Digital Counications, University of Erlangen-Nürnberg, Erlangen, Gerany His research interests ainly focus on ultiple-antenna techniques, wireless security, interference network and wireless power transfer, etc Dr Chen serves as an Associate Editor for the IEEE ACCESS and an Editor for the IEEE COMMUNICATIONS LETTERS He was honoured as an Exeplary Reviewer of the IEEE Counications Letters in 4 and the IEEE Transactions on Counications in 5 Zhaoyang Zhang M received his PhD degree in counication and inforation systes fro Zhejiang University, China, in 998 He is currently a full professor with the Departent of Inforation Science and Electronic Engineering, Zhejiang University His research interests are ainly focused on inforation theory and coding theory, signal processing for counications and in networks, and their applications in the next generation wireless obile counication systes He has co-authored ore than 5 refereed international journal and conference papers as well as two books in the above areas He was a corecipient of several conference Best Paper Awards / Best Student Paper Award He is currently serving as Editor for IEEE TRANSACTIONS ON COMMUNICATIONS, IET COMMUNICATIONS and soe other international journals He has served or is serving as TPC Co-Chair or Syposiu Co-Chair for any international conferences like WCSP 3, ICUFN//3, and Globeco 4 Wireless Counications Syposiu, etc