Security-Reliability Tradeoff Analysis of Artificial Noise Aided Two-Way Opportunistic Relay Selection

26 Pages·2016·2.03 MB·English

IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016 1
Security-Reliability Tradeoff Analysis of Artificial
1
Noise Aided Two-Way Opportunistic Relay Selection
2
XiaojinDing,StudentMember,IEEE,TiechengSong,Member,IEEE,YulongZou,SeniorMember,IEEE,
3
XiaoshuChen,andLajosHanzo,Fellow,IEEE
4
5 Abstract—Inthispaper,weinvestigatethephysical-layersecu- andfast-fadingeffectsofwirelesschannels.Therearetwopop- 39
6 rity of cooperative communications relying on multiple two-way ular relaying protocols, namely the amplify-and-forward (AF) 40
7 relays using the decode-and-forward (DF) protocol in the pres- [1],[2]aswellasthedecode-and-forward(DF)[3],[4].Inthe 41
8 ence of an eavesdropper, where the eavesdropper appears to tap
9 thetransmissionsofboththesourceandoftherelay.Thedesign case of AF relaying, the selected relay multiplies its received 42
10 tradeofftoberesolvedisthatthethroughputisimprovedbyin- signalsbyagainfactorandthenforwardthemtothedestination 43
11 vokingtwo-wayrelaying,butthesecrecyofwirelesstransmissions [1],[2].Bycontrast,theDFrelaydecodes itsreceived signals 44
12 maybedegraded,sincetheeavesdroppermayoverhearthesignals and then the selected relay forward its decoded signal to the 45
13 transmitted by both the source and relay nodes. We conceive an destination[3],[4].Additionally,in[5],bothAFandDFrelay- 46
14 artificialnoiseaidedtwo-wayopportunisticrelayselection(ANaT-
15 WORS) scheme for enhancing the security of the pair of source ing schemes are investigated. In general, closer to the source, 47
16 nodescommunicatingwiththeassistanceofmultipletwo-wayre- DFrelayinghasahighprobabilityofsuccessfuldecodingand 48
17 lays. Furthermore, we analyze both the outage probability and flawless retransmission from the relay to the destination from 49
18 interceptprobabilityoftheproposedANaTWORSscheme,where areduceddistance[6].Bycontrast,closetothedestinationthe 50
19 thesecurityandreliabilityarecharacterizedintermsoftheinter- DFrelayhasjustasbadreceptionasthedestinationitself,hence 51
20 ceptprobabilityandthesecurityoutageprobability.Forcompari-
21 son,wealsoprovidethesecurity–reliabilitytradeoff(SRT)analysis itofteninflictserrorpropagation.Fortunatelyinthevicinityof 52
22 ofboththetraditionaldirecttransmissionandoftheone-wayre- thedestinationAFrelyingtendstooutperformDFrelaying[6]. 53
23 layingschemes.ItisshownthattheproposedANaTWORSscheme Additionally, [7] also shows that adaptive DF outperforms AF 54
24 outperformsboththeconventionaldirecttransmission,aswellas intermsofitsframeerrorrate(FER). 55
25 theone-wayrelaymethodsintermsofitsSRT.Morespecifically, Atthetimeofwritingthispaper,physical-layersecurity[8], 56
26 in the low main-user-to-eavesdropper ratio (MUER) region, the
27 proposedANaTWORSschemeiscapableofguaranteeingsecure [9]incooperativerelaynetworksisreceivingagrowingresearch 57
28 transmissions, whereas no SRT gain is achieved by conventional attentionasbenefitofitscapabilityofprotectingwirelesscom- 58
29 one-wayrelaying.Infact,theone-wayrelayingschememayeven municationsagainsteavesdroppingattacks.In[10]and[11],the 59
30 beinferiortothetraditionaldirecttransmissionschemeinterms physical-layer security of MIMO-aided relaying networks has 60
31 ofitsSRT. beenexplored,demonstratingthatthesecrecycapacitycanin- 61
32 Index Terms—Artificial noise, opportunistic relay selection, deed be improved by using MIMO-aided relays. Additionally, 62 Q1
33 physical-layer security, security-reliability tradeoff (SRT), TekinandYener[12]proposedthecooperativejammingphilos- 63
34 two-wayrelay. ophy,andstudiedtheattainablesecrecyratewiththeobjectiveof 64
improvingthephysical-layersecurity.Asafurtherdevelopment, 65
35 I. INTRODUCTION Long et al. [13] investigated cooperative jamming schemes in 66
bidirectionalsecrecycommunications.In[14]and[15],beam- 67
36 COOPERATIVE relaying has attracted substantial re- formingtechniqueshavebeeninvestigatedandsignificantwire- 68
37 search interests from both the academic and industrial less secrecy capability improvements were demonstrated with 69
38 community,sinceitiscapableofmitigatingboththeshadowing theaidofbeamformingtechniques.Additionally,theimpactof 70
antennaselectiononsecuretwo-wayrelayingcommunications 71
Manuscript received January 21, 2016; revised May 9, 2016 and July 22, hasbeenanalyzedin[16]. 72
2016;acceptedAugust12,2016.Dateofpublication;dateofcurrentversion.
ThisworkwassupportedinpartbytheNationalNaturalScienceFoundation As a design alternative, relay selection schemes may also 73
ofChinaunderGrant61401223,Grant61522109,Grant61271207,andGrant be used for improving the physical-layer security of wireless 74
61372104;inpartbytheNaturalScienceFoundationofJiangsuProvinceunder communications.One-wayrelayinghasbeenanalyzedin[17]– 75
GrantBK20140887andGrantBK20150040;andinpartbytheKeyProjectof
NaturalScienceResearchofHigherEducationInstitutionsofJiangsuProvince [24]. Specifically, hybrid relaying and jamming schemes are 76
underGrant15KJA510003.ThereviewofthispaperwascoordinatedbyProf. exploredin[17]–[22].In[17]–[19],jointAFrelayingandjam- 77
M.C.Gursoy. merselectionschemeshavebeeninvestigated.Additionally,hy- 78
X.Ding,T.Song,andX.ChenarewiththeNationalMobileCommunications
Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: brid cooperative beamforming and cooperative jamming have 79
[email protected];[email protected];[email protected]). beenproposedin[20]and[21].In[22],jointDFrelayingand 80
Y.ZouiswiththeSchoolofTelecommunicationsandInformationEngineer- cooperative jamming schemes have been investigated. More- 81
ing, Nanjing University of Posts and Telecommunications, Nanjing 210003,
China(e-mail:[email protected]). over, in [23], the AF- and DF-based optimal relay selection 82
L. Hanzo is with the Department of Electronics and Computer Sci- schemeshavebeenproposed.Theassociatedinterceptprobabil- 83
ence, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: itieshavealsobeenanalyzedinthecontextofbothAF-andDF- 84
[email protected]).
DigitalObjectIdentifier10.1109/TVT.2016.2601112 basedone-wayrelayingschemes,whereaneavesdropperisonly 85
0018-9545©2016IEEE.Personaluseispermitted,butrepublication/redistributionrequiresIEEEpermission.
Seehttp://www.ieee.org/publicationsstandards/publications/rights/index.htmlformoreinformation.
2 IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016
86 capableofwiretappingthetransmissionsoftherelays.Bycon-
87 trast, in [24], an eavesdropper was tapping the transmissions
88 of both the source and of the relays. Moreover, the security-
89 reliabilitytradeoff(SRT)hasbeenexploredinthecontextofthe
90 proposedopportunisticrelayselectionschemeinthehighmain-
91 user-to-eavesdropper ratio (MUER) region, where the MUER
92 is defined as the ratio of the average channel gain of the main
93 links(spanningfromthesourcetothedestination)tothatofthe
94 wiretap links (spanning from the source to the eavesdropper).
95 Additionally,two-wayrelayinghasbeenexploredin[25]–[31].
96 Specifically, Mo et al. [25] investigated two-way AF relaying
97 schemesrelyingoneithertwoslotsorthreeslotsdemonstrated
98 that the three-slot scheme performs better than the two-slot
99 scheme, when the transmitted source powers approach zero.
Fig.1. WirelessnetworkconsistingofapairofsourceS1,S2,andN relays
100 In[26],DFrelayinghasbeeninvokedforimprovingthewire- inthepresenceofaneavesdropperE.
101 less security of bidirectional communications, where a relay
102 is invoked for transmitting artificial noise in order to perturb
103 the eavesdropper’s reception both in the first and in the sec- thethirdtimeslot,therelayforwardtheencodedsignalstoS1 142
104 ondtransmissionslot.In[27],jointrelayandjammerselection andS2. 143
105 oftwo-wayrelaynetworkshavebeenproposed.In[28],Wang Second,wepresentthemathematicalSRTanalysisofthepro- 144
106 et al. explored hybrid cooperative beamforming and jamming posed ANaTWORS scheme in the presence of artificial noise 145
107 oftwo-wayrelaynetworks.In[29],securerelayandjammerse- imposed both on the relays and on the eavesdropper for trans- 146
108 lectionwasconceivedforthephysical-layersecurityimprove- mission over Rayleigh fading channels. Moreover, we assume 147
109 mentofawirelessnetworkhavingmultipleintermediatenodes that the teletraffic of S1 and S2 is different. Closed-form ex- 148
110 andeavesdroppers,wherethelinksbetweenthesourceandthe pressions are obtained both for the OP and for the intercept 149
111 eavesdropper are not considered. In [30], three different cat- probability(IP)ofbothS1andS2. 150
112 egories of relay and jammer selection have been considered, Finally,itisshownthatastheimpactofartificialnoiseonthe 151
113 where the channel coefficients between the legitimate nodes main link is reduced and on the wiretap link is increased, the 152
114 andtheeavesdroppersareusedbothforrelayselectionandfor SRT of the proposed ANaTWORS scheme is improved. Fur- 153
115 jammerselection.In[31],awirelessnetworkconsistingoftwo thermore,ourperformanceevaluationsrevealthattheproposed 154
116 source nodes is considered and multiple DF relay nodes are ANaOTWRS scheme consistently outperforms both the tradi- 155
117 involved in the presence of a single eavesdropper. The outage tionaldirecttransmissionregimeandtheone-waytransmission 156
118 probability(OP)hasbeenanalyzedforthetwo-wayDFscheme scheme[24]intermsofitsSRT. 157
119 relyingonthreetransmissionslots. The organization of this paper is as follows. In Section II, 158
120 Motivatedbytheaboveconsiderations,weinvestigateawire- webrieflycharacterizethephysical-layersecurityofatwo-way 159
121 lessnetworksupportingapairofsourcenodeswiththeaidofN wireless network. In Section III, the SRT analysis of the con- 160
122 two-wayDFrelaysinthepresenceofaneavesdropper.Incon- ventionaldirecttransmissionschemeaswellasoftheproposed 161
123 trastto[17]–[24],weexploreatwo-wayrelayingaidedwireless ANaOTWRS scheme communicating over a Rayleigh chan- 162
124 network.Furthermore,weproposeanartificialnoiseaidedtwo- nel is carried out. Our performance evaluations are detailed in 163
125 way opportunistic relay selection (ANaTWORS) scheme, and SectionIV.Finally,inSectionV,weconcludethepaper. 164
126 analyzetheSRTofthewirelessnetworkinvestigated.Duetothe
127 channelstateinformation(CSI)estimationerror,itisimpossible II. SYSTEMMODELANDRELAYSELECTION 165
128 toguaranteethatnointerferenceisreceivedattherelaynodes,
129 causedbythespeciallydesignedartificialnoise.Moreover,the A. SystemModel 166
130 impact of the artificial noise both on the relays and on the As shown in Fig. 1, we consider a wireless network con- 167
131 eavesdropperischaracterized,whichwillbetakenintoaccount sisting of a pair of source nodes, denoted by S1 and S2, plus 168
132 whenevaluatingthewirelessSRToftheproposedANaTWORS N two-way DF relays, denoted by Ri, i∈{1,...,N}, which 169
133 scheme.Againstthisbackground,themaincontributionsofthis communicate in the presence of an eavesdropper E, where 170
134 paperaresummarizedasfollows. E is assumed to be within the coverage area of S1, S2, and 171
135 First,weproposeanANaTWORSschemeforprotectingthe Ri. All nodes are equipped with a single antenna. We assume 172
136 ongoingtransmissionsagainsteavesdropping.Tobespecific,in that there is no direct link between S1 and S2 due to the path 173
137 thefirsttimeslot,S1 transmitsitssignalstotherelays,andS2 loss. Furthermore, in the spirit of [21], both the main and the 174
138 transmits artificial noise in order to protect the signals trans- wiretaplinksaremodeledbyRayleighfadingchannels,where 175
139 mittedbyS1 againsteavesdropping. Similarlytothefirsttime the main and wiretap links are represented by the solid and 176
140 slot, S2 transmits its signals to the relays in the second time dashed lines in Fig. 1, respectively. Let hs1i, hs2i, hs1e, and 177
141 slotundertheprotectionofartificialnoisetransmittedbyS1.In hs2e,i∈{1,...,N},representtheS1−Ri,S2−Ri,S1−E, 178
DINGetal.:SECURITY-RELIABILITYTRADEOFFANALYSISOFARTIFICIALNOISEAIDEDTWO-WAYOPPORTUNISTICRELAYSELECTION 3
179 and S2−E channel gains, respectively. We assume that the From (1), the achievable rate of the S1−Ri link can be 231
180 channelcoefficientshs1i,hs2i,hs1e,andhs2e aremutuallyinde- expressedas 232
181 pendentzero-meancomplexGaussianrandomvariables(RVs) (cid:3) (cid:4)
118832 woviethr,vwaeriaanscsuesmoeftσhas21tit,hσes2S2i1,σ−s21Re,iaannddσSs222e−,rResipelicntkivsealyre.Mreocirpe-- Cs1i = 31log2 1+ α|h|hs2si1|i2|γ2sγs+2 (2)
184 rocal,i.e.,wehave,hs1i =his1 andhs2i =his2.Forsimplicity,
111888567 wσnes2el2eags=asiunαmsse2oefσσs2et21h,iew=mhαearisen1iσσlim2m2nk,asσns2ad2nidσ=e2oαrfestp2hrieeσsm2wen,irtσets2tha1eep=alvineαkrsas1,geeσree2cs,hpaaenncd-- wtfirmohmeeresSlto1httesofaSarce2tovreriaq1uR/i3rie.adrifsoerscforommpltehtienfgacthtethsaitgtnharleetraonrtshmoigsosinoanl 222333534
188 tively.Moreover,letλme =σm2 /σe2,whichisreferredtoasthe Naturally,theartificialnoiseisspeciallydesignedtointerfere 236
189 MUER. withtheeavesdropper.However,itsperturbationimposedonthe 237
190 ThethermalnoiseofanynodeismodeledasacomplexGaus- eavesdropper may be imperfect due to CSI estimation errors, 238
191 sian random variable with a zero mean and a variance of N0, which is characterized by β. Hence, the signals received at E 239
192 denoted by ns1, ns2, ni, and ne, respectively. Following [31], fromS1canbeexp(cid:2)ressedas (cid:2) 240
193 theoperationofthetwo-wayDFschemerelyingonopportunis-
y =h P /2x +h βP /2ω +n . (3)
194 ticrelayselectionissplitintothreetimeslots.Weassumethat s1e s1e s s1 s2e s s2 e
195 the nodes in the network are synchronized with each other. In From (3), the achievable rate of the S1−E link can be 241
196 thefirsttimeslot,S1 transmitsitssignal,denotedbyxs1 tothe formulatedas 242
119987 roeulsalyys.,IanntdhethseencoSn2dtrtaimnsemsiltost,thSe2atrrtainfiscmiailtsnoitissesiωgns2alsxims2ulttoanthee- Cs = 1log (cid:3)1+ |hs1e|2γs (cid:4). (4)
199 relays and S1 transmits artificial noise simultaneously. In the s1e 3 2 β|hs2e|2γs +2
200 thirdtimeslot,theselectedrelayforwardthesignalxr toboth
220012 SX1ORanodpSer2a,twiohne.rFeuwrtehehramvoerxe,rt=hexpsr1o⊕poxsesd2,raenlady⊕sedleecntoiotensctahne nodInest,heansdecSo1ndsitmimueltasnloeto,uSs2lytrtarnansmsmitistsitasrstiifignciaallsntooitshee. rSeilmay- 224434
203 becoordinatedbyrelyingonadistributedpattern(governedby ilarly, the signals received at Ri transmitted by S2 can be 245
204 atimer).Withoutlossofgenerality,weassumeE[|xsj|2]=1, expressedas (cid:2) (cid:2) 246
220056 E[F|ωusrjth|2e]rm=oNre0,,wje=al1s,o2a.ssumethatS1 andS2 havetoconvey ys2i =hs2i Ps/2xs2 +hs1i αPs/2ωs1 +ni. (5)
220078 dcoifmfepreanrits-orant,etthreafofince,-dweanyoteredlabyyinRgs1scahnedmRes(2O,rReSsp)eocftiv[2el4y].cFaonr Using(5),theachievabl(cid:3)erateoftheS2−Ri l(cid:4)inkisgivenby 247
209 besimplyextendedtoatwo-wayscenariorelyingonfourtime C = 1log 1+ |hs2i|2γs . (6)
210 slots. To be specific, S1 transmits its signals to the relays in s2i 3 2 α|hs1i|2γs +2
211 the first time slot, S2 transmits its signals to the relays in the
212 second time slot, and the selected relay forward the decoded Similarly,thesignalsreceivedatEfromS2canberepresented 248
213 signals to S2 and S1 in the third time slot and the fourth time as (cid:2) (cid:2) 249
214 slot,respectively. y =h P /2x +h βP /2ω +n , (7)
s2e s2e s s2 s1e s s1 e
whiletheachievablerateoftheS2−E linkis 250
215 B. Two-WayRelayingScheme (cid:3) (cid:4)
1 |h |2γ
216 In this section, we first consider the physical-layer security Cs = log 1+ s2e s . (8)
217 of the two-way relaying scheme. We then propose our ANaT- s2e 3 2 β|hs1e|2γs +2
218 WORSarrangement.
2) Decoding Set: In this section, we analyze the suc- 251
219 1) S1andS2Transmit: Inthefirsttimeslot,S1transmitsits cessful decoding set of the wireless network portrayed in 252
220 signaltotherelaysundertheprotectionofartificialnoisetrans-
Fig. 1. As shown in [24], the resultant successful de- 253
221 mitted by S2. For the sake of a fair power consumption com- coding set of the ORS scheme is given by Ω, where 254
222222342 ptthhaeerittsrooatnnalswmtriatihtnpsbomowtihteprthsoewofdeiSrre1ocfatnStdr1aSnan2smdariSess2dioiesnncoaotnenddsttbrhayeinPOesRd/S2to.sAcPhsse,mmtheeunss-, Ωslaeyt=sa,n{ndφ∈,ΦD{n11,r,De2p,2r.,e..s..e,.n2,tsNDtn−h,e.1.n}.t.h,TDnh2oeNns−eum1c}pc,etyφsssfduuelbnsdeoettceoosdfittnhhgeeseNemtsproteyf- 222555576
225 tionedabove,itisimpossibletoguaranteethattheartificialnoise
222267 ptheerfuebctilqyuiltioesusinCtShIeensutilmlsaptiaocneeorfrothr,ehSe1nc−eRleiadcihnagnntoelas,cdeuretation tdheecordeilnagysxdse1fiannedd axss2thaorseedtehnaotteadrebcyapΩab1leanodf sΩu2c,cersessfpuelcly- 225598
tively. Consequently, the set of the relays that successfully 260
222232908 iRSn1itecirsafneqrubeanenceteixfirpeerdceesbisvyeedαda.asTthReis.iTghnealismrpecaecitvoefdthateRarititfiracniaslmniottiesdeobny dlΨaetce=oddΩaes1bΨ∩otΩ=h2.x{φs1,Φan1,dΦx2s,2..i.s,dΦenn,o.t.e.d,Φby2NΨ−1,}w,whihcehreiswfeorhmavue- 222666321
(cid:2) (cid:2) Forexample,thedecodingsetsofΩj andΨhavebeenshown 264
ys1i =hs1i Ps/2xs1 +hs2i αPs/2ωs2 +ni. (1) asTableI,wherewehaveN =3andj ∈{1,2}. 265
4 IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016
TABLEI ANaTWORSscheme,whichcanbegivenby 283
DECODINGSETSOFΩj ANDΨ,WHENN =3ANDWHENj∈{1,2}
o= argmax[min(C (i),C (i))]
i∈Φn (cid:7) (cid:5) is1 is2 (cid:6)(cid:8)
= argmax min |h |2,|h |2 (17)
i∈Φn is1 is2
where o denotes the selected optimal relay. Moreover, from a 284
morepracticalpointofview,theCSIs|his1|2and|his2|2canbe 285
estimatedinpracticalwirelesscommunications,usingchannel 286
estimationschemes[32]. 287
5) ConditionofInterceptEvent: IntheΦ=φcase,aneaves- 288
dropper can successfully wiretap the signal transmitted by S1, 289
whenCss1e >Rs1. 290
In the Φ=Φn and Css1e >Rs1 case, an eavesdropper can 291
successfullywiretapthesignaltransmittedbyS1. 292
In the Φ=Φn and Css1e <Rs1 scenario, if Css2e <Rs2, an 293
eavesdropper cannot successfully wiretap the signal transmit- 294
ted by S1. If Css2e >Rs2, the signal received at E can be 295
266 Asmentionedabove,theeventofΦ=φcanbecharacterized rewrittenas 296
(cid:2)
267 as
y =h P x +n . (18)
oe oe s s1 e
C <R orC <R , i∈{1,2,...,N} (9)
s1i s1 s2i s2 TheachievablerateoftheRo −E linkcanbeformulatedas 297
268 whiletheeventofΦ=Φn canbeexpressedas 1 (cid:5) (cid:6)
C = log 1+|h |2γ . (19)
C >R andC >R , i∈Φ oe 3 2 oe s
s1i s1 s2i s2 n
Cs1j <Rs1 orCs2j <Rs2, j ∈Φ¯n (10) canColenalrylys,uicnctehsesfΦul=lywΦinreatnadpCthsse1esi<gnRalst1racnassme,iattnedeabvyesSd1rowphpeenr 229998
269 whereΦ¯n representsthecomplementarysetofΦn. Css2e >Rs2 andCoe >Rs1. 300
270 3) RelayTransmits: Withoutlossofgenerality,hereweas- Similarly,wecanformulatetheconditionofaneavesdropper 301
271 sumethatRiisselectedfromthesetΦn.Thentheselectedrelay successfullywiretappingthesignaltransmittedbyS2as 302
272 Ri broadcaststheencodedsignalxr toS1 andS2.Thesignals IntheΦ=φcase,aneavesdroppercansuccessfullywiretap 303
273 receivedatS1froyms1R(ii)c=anhbies1w(cid:2)riPttsexnra+s ns1. (11) tshuecIcnseitgshsnefauΦllltyr=awnΦsimrneitatatnpeddtChbesys2seSig>2n,apRlrotsr2vasindcseemdnaittrhtieaodt,Cabnyss2eSea2v>.esRdrso2.ppercan 333000465
274 ThesourceS1mayinvokesuccessiveinterferencecancelation casIen,athneeaΦve=sdrΦonp,peCrscs2aen<suRccse2s,sCfussl1lye >wirRetsa1p, tahnedsiCgnoeal>traRnss2- 330078
275 (SIC),thus,(18)canbewrittenas
(cid:2) mittedbyS1. 309
y (i)=h P x +n . (12)
s1 is1 s s2 s1
III. SECURITY–RELIABILITYTRADEOFFANALYSIS 310
276 TheachievablerateoftheRi −S1linkcanbeexpressedas OVERRAYLEIGHFADINGCHANNELS 311
(cid:5) (cid:6)
C = 1log 1+|h |2γ . (13) Inthissection,weanalyzeboththeOPandIPoftheproposed 312
is1 3 2 is1 s ANaTWORSschemesoverRayleighfadingchannels. 313
277 Similarly, S2 can also invoke SIC, thus the signals received
278 atS2fromRi canbewrittenas A. SRTAnalysisoftheProposedANaTWORSScheme 314
(cid:2)
y (i)=h P x +n . (14) 1) SRTAnalysisofS1: IntheANaTWORSscheme,arelay 315
s2 is2 s s1 s2 willonly be chosen fromthe set Φn.Withthe aid of Shannon 316
279 TheachievablerateoftheRi −S2linkcanbeobtainedas [33]andthelawoftotalprobability[34],theOPoftheS1 →S2 317
(cid:5) (cid:6) linkrelyingontheANaTWORSschemecanbeformulatedas 318
1
C = log 1+|h |2γ . (15)
is2 3 2 is2 s Psingle = Pr(C <R ,Φ=φ)
outs1 os2 s1
280 ThesignalsreceivedatE fromRi canbewrittenas 2(cid:9)N−1
(cid:2) (cid:2) + Pr(C <R ,Φ=Φ ). (20)
y =h P x +n =h P (x ⊕x )+n . (16) os2 s1 n
ie ie s r e ie s s1 s2 e n=1
281 4) An Optimal Two-Way Relay Selection Criterion: In In the case of Φ=φ, no relay is chosen for forwarding the 319
282 this section, we present the relay selection criterion of the signals,whichleadstoCos2 =0forΦ=φ.Thus,(20)canbe 320
DINGetal.:SECURITY-RELIABILITYTRADEOFFANALYSISOFARTIFICIALNOISEAIDEDTWO-WAYOPPORTUNISTICRELAYSELECTION 5
⎛ ⎛ (cid:3) (cid:4) ⎞⎞⎞
321 rewrittenas (cid:9) 329
×⎝1−exp⎝− Δ1 + Δ1 − Δ1 − Δ1 ⎠⎠⎠
Posuintgsle1 =Pr(Φ=φ)+2n(cid:9)N=−11Pr(Cos2 <Rs1,Φ=Φn). (21) +2|Φ(cid:9)n|−1−1⎛⎝(−1)|Ajn∈(Amn)(|⎛⎝mσ) 2 σj2s(cid:9)2 σ(cid:3)j2s11 + σi12s1(cid:4)+σσi2si22s2+1⎞⎠−1
is2 σ2 σ2 σ2
322 Basedon(9)and(10),(21)canbeexpressedas m=1 j∈An(m) js2 js1 is1
⎛ ⎛ (cid:3) (cid:4) ⎞⎞⎞⎞
(cid:3) (cid:3) (cid:4)
(cid:9)
Posuintgsle1 = (cid:10)iN=1 1−Pr α|hs|2hi|s21γi|s2+2 >Δ1 ×⎝1−exp⎝−j∈An(m) σΔj2s12 + σΔj2s11 − σΔi2s11 − σΔi2s12⎠⎠⎠⎠.
(cid:3) (cid:4)(cid:4)
(24)
|h |2
×Pr (cid:3)α|hs1i|s(cid:3)22γis +(cid:3)2 >Δ2 (cid:4) ISnubosutrituAtNinagT(W23O)RanSds(c2h4e)minet,oa(n22e)a,vPesosduintrgsole1pcpaenr bcaenobotvaeirnheeda.r 333301
2(cid:9)N−1 (cid:10) |h |2 thesignalstransmittedbyS1,S2,andRi.Usingthelawoftotal 332
+ Pr α|h |s21γi +2 >Δ1 probability[34]andthedefinitionofaninterceptevent,wecan 333
n=1(cid:3) i∈Φn s2i (cid:4)s(cid:4) expresstheIPoftheS1 →E linkas 334
(cid:15) (cid:16)
×Pr α|h|h|s22γi|2+2 >Δ2 Pisnitngsl1e = Pr Css1e >Rs1,D=φ
(cid:3) s1i (cid:3)s (cid:4) 2(cid:9)N−1 (cid:15) (cid:16)
× (cid:10) 1−Pr α|h|hs|21jγ|2+2 >Δ1 + n=1 Pr Css1e >Rs1,Φ=Φn
j∈Φ¯(cid:3)n s2j s(cid:4)(cid:4) +2(cid:9)N−1Pr(cid:15)Cs <R ,Cs >R ,C >R ,Φ=Φ (cid:16).
|h |2 s1e s1 s2e s2 oe s1 n
×Pr s2j >Δ n=1
(cid:5) α|hs1j|2γs +(cid:6)2(cid:6) 2 (25)
×Pr |hos2|2 <Δ1 (22) Using(4),(8),and(19),(25)canbeexpressedas 335
(cid:3) (cid:3) (cid:4)
(cid:10)N |h |2
332243 (w2h3e·Rres2 −w1)e/γs.have Δ1 =(23·Rs1 −1)/γs, and Δ2 = Pisnitngsl1e = i=1 1−Pr α|hs2i|s21γis +2 >Δ1
(cid:3) (cid:4)(cid:4)
332256 expBraessesdedaosn Appendix A, Pr(α|h|sh2si|12iγ|2s+2 >Δ1) can be ×Pr α|h|h|s22γi|2+2 >Δ2
(cid:3) (cid:4) (cid:3) (cid:4) (cid:3) s1i s (cid:4)
Pr α|hs|2hi|s21γi|s2+2 >Δ1 =Δ1αγsσσss2212ii +σs21i exp −2σΔs21i1 . ×Pr β|hs|2hes|21eγ|s2+2 >Δ1
(23) (cid:17) (cid:3) (cid:3) (cid:4)
327 According to Appendix B, Pr(|hos2|2 <Δ1)) can be +2(cid:9)N−1 (cid:10) Pr |hs1i|2 >Δ
328 expressedas α|h |2γ +2 1
(cid:3)(cid:3) (cid:3) (cid:4)(cid:4) n=1(cid:3) i∈Φn s2i (cid:4)s(cid:4)
(cid:5) (cid:6) (cid:9) Δ |h |2
Pr |h |2<Δ = 1−exp − 1 ×Pr s2i >Δ
+2|Φ(cid:9)n|o−s12−1(−1)|1An(m)i|∈⎛⎝Φnσi2s1 (cid:9) (cid:3)σ12σi2s+2 σ12 (cid:4)+1⎞⎠−1 × (cid:10) (cid:3)α1|h−s1iP|2rγ(cid:3)s +α|2h|h|s21γi|22+2 >Δ1(cid:4)
(cid:3)m=1 (cid:3) (cid:4)(cid:4) j∈An(m) js2 js1 j∈Φ¯(cid:3)n s2i s(cid:4)(cid:4)
Δ |h |2
× 1−exp − 1 ×Pr s2i >Δ
−2|Φ(cid:9)n|−1−1⎛⎝(−1σ)|i2As2n(m)|⎛⎝σ2 (cid:9) (cid:3) 1 + 1 (cid:4)+1⎞⎠−1 ×Pr(cid:3)α|hs|1his|21eγ|s2+2 >Δ2(cid:4)(cid:18)
is1 σ2 σ2 β|h |2γ +2 1
m=1 j∈An(m) js2 js1 s2e s
×⎛⎝σi2s2j∈A(cid:9)n(m)(cid:3)σj12s2 + σj12s1(cid:4)+ σσii22ss21 +1⎞⎠−1 +2n(cid:9)N=−11(cid:17)i(cid:10)∈Φn (cid:3)Pr(cid:3)α|hs|2hi|s21γi|s2+2 >Δ1(cid:4)
6 IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016
(cid:3) (cid:4)(cid:4)
336 ×Pr (cid:3)α|hs|1hi|s22γi|(cid:3)s2+2 >Δ2 (cid:4) andC3)lPeiSasnirtRnlgsyTl1e,.Panosuiantlgsyle2siasndofPSisn1itngsal2encdanS2b:eoTbhteaiInPedansidmOilaPrloyftothPeospuintagsile1r 333444567
× (cid:10) 1−Pr |hs1i|2 >Δ of sources is defined as the average IP and OP of S1 and S2, 348
α|h |2γ +2 1 respectively: 349
j∈Φ¯(cid:3)n s2i s(cid:4)(cid:4) Psingle +Psingle
|h |2 Psingle = ints1 ints2 (31)
×Pr s2i >Δ int 2
α|h |2γ +2 2
(cid:3) s1i s (cid:4) and 350
|h |2 |h |2 Psingle +Psingle
×Pr β|h s|21eγ +2 <Δ1,β|h s|22eγ +2 >Δ2 Posuintgle = outs1 2 outs2 . (32)
(cid:5) s2e s (cid:6)(cid:8) s1e s
×Pr |h |2 >Δ . (26) IV. PERFORMANCEEVALUATION 351
oe 1
Forcomparison,theSRTanalysisoftheconventionaldirect 352
337 Accord(cid:3)ingtoAppendixC, (cid:4) transmissionschemeoperatingwithoutrelaysisalsoprovided. 353
|h |2 |h |2 The total IP and OP of S1 and S2 with the traditional direct 354
Pr β|h s|21eγ +2 <Δ1,β|h s|22eγ +2 >Δ2 transmissionschemeisdefinedas 355
s2e s s1e s Pdirect+Pdirect
Pdirect = ints1 ints2 (33)
338 canobtainedas int 2
(cid:3) (cid:4)
Pr |hs1e|2 <Δ , |hs2e|2 >Δ and 356
β|h |2γ +2 1 β|h |2γ +2 2 Pdirect +Pdirect
s2e s s1e s Pdirect = outs1 outs2, (34)
(cid:19) (cid:20) (cid:19) (cid:20) out 2
Δ γ βσ2 2Δ
334309 lateAdPccaros(cid:5)r|d=hing|12to−>AΔΔpp2(cid:6)γesn=βd2iσxs(cid:9)s21De,+s⎡⎣2Peσ⎛⎝rs(22|1he+oe|e22|xD>(cid:9)pn|−Δ1−−11)σ(−cs22ae12n)|bA.en(fmor)(m|2u7-) aPs(re2proesd2euipRrcteetsscgic12tvitve=−ivleyen11.ly)M−/,bγoyeswrx.ehpoPNe(vir−dneoeitrrtie,σsincn1sΛ2wt1g1=se2P)htiedhn,aixtaravesptnce1t(d,−σΛPs21Pσ2oΛdss2=uii1d1n1rte,eitrsce()s2tc2σ,2t=,2s2R1Pes1P,i1dn−oitd−ruaeisrtcne2etsc1dx1t=),p/σ(γe−as2xs2npeσad(sΛn2−2ad2srP2eσΛ)Λos2d,u22i2trtereh=esc)e2t-, 333335556678901
⎛⎝ oσei2s2σi2s1 1 (cid:9)i∈Dn(cid:3) 1 + m1=(cid:4)1 +1⎞⎠−1⎞⎟⎠ erreexsspIpuneelcttcsthteiidfvsoesrlevycat.htliueoenst,rawodefitpiotrhneesaelnRdtVibrseoctht|hotsru2asrn1ns|2mu,mis|eshrisoic1nea,|l2aa,nsadwnsdeimll|uhalsas2teifo|o2nr, 333366662354
σi2s2 +σi2s1 j∈An(m) σj2s2 σj2s1 the ORS [24] and for the ANaTWORS schemes in terms of 366
(cid:19) (cid:20)(cid:24) their SRTs. Moreover, the analytic IP versus OP results of the 367
×exp −Δ1 . (28) directtransmissionandANaTWORSschemesareobtainedby 368
σ2 plotting(33),(34),(31),and(32),respectively.Itispointedthat 369
ie
theIPversusOPresultsoftheORSschemearecalculatedfrom 370
341 Substituting(27)and(28)into(26),Pisnitngsl1ecanbeobtained. (27) and (19) of [24], where α is rewritten as (24Rd −1)/γs. 371
342 2) SRTAnalysisofS2: SimilarlytoS1,theOPofS2canbe Throughout this performance evaluation, we assumed αs1i = 372
343 expressedas αs2i =αs1e =αs2e =αs1s2 =1. 373
We first consider the effect of different MUERs. Fig. 2 de- 374
2(cid:9)N−1
Psingle =Pr(Φ=φ)+ Pr(C <R ,Φ=Φ ). (29) pictstheSRTsofboththedirecttransmission,oftheORS[24] 375
outs2 os1 s2 n and of the ANaTWORS schemes for different MUERs. Both 376
n=1
the numerical and simulation results characterizing the SRT 377
344 Meanwhile,theIPofS2canbeshowntoobey of the ANaTWORS scheme are provided in this figure. Ob- 378
Psingle = Pr(cid:15)Cs >R ,D=φ(cid:16) serve from Fig. 2 that as the MUER decreases, all the IPs of 379
ints2 s2e s2 the direct transmission, of the ORS and of the ANaTWORS 380
2(cid:9)N−1 (cid:15) (cid:16) schemes are increased, which can be explained by observing 381
+ Pr Css2e >Rs2,Φ=Φn that upon decreasing the MUER, an eavesdropper can achieve 382
n=1 a higher achievable rate. Moreover, Fig. 2 also illustrates that 383
2(cid:9)N−1 (cid:15) (cid:16) the proposed ANaTWORS scheme generally has a lower IP 384
+ Pr Css2e <Rs2,Css1e >Rs1,Coe >Rs2,Φ=Φn . than the traditional direct transmission and ORS regime for 385
n=1 MUER=3 dB and MUER=0 dB. Additionally, the dif- 386
(30) ferencebetweentheanalyticandsimulatedIPversusOPcurves 387
DINGetal.:SECURITY-RELIABILITYTRADEOFFANALYSISOFARTIFICIALNOISEAIDEDTWO-WAYOPPORTUNISTICRELAYSELECTION 7
Fig. 2. IP versus OP of the direct transmission, ORS, and ANaTWORS Fig. 4. IP versus OP of the direct transmission, ORS, OSJ-MMISR, and
schemes for different MUERs λme and for N =8, which were calculated ANaTWORS schemes for different α and β associated with an MUER of
from[24,(33),(34)and[27]],[(24),(19)],and(31)and(32). λme = 0dB,N =8,whichwerecalculatedfrom[24,(33),(34)and[27]],
[(24),(19)],and(31)and(32).
Fig. 4 illustrates the IP versus OP of both the direct trans- 401
mission, as well as of the ORS, of the optimal selection 402
withjammingwithmax–mininstantaneoussecrecyrate(OSJ- 403
MMISR) [30] and of the ANaTWORS schemes for differ- 404
ent self-interference and interference factors, where (β,α)= 405
(0.95,0.06)and(β,α)=(0.99,0.02)areconsidered.Observe 406
fromFig.4thatastheartificialnoiseparametersof(0.95,0.06) 407
are changed to (0.99,0.02), the IP versus OP performance 408
of the ANaTWORS scheme improves. Furthermore, Fig. 4 409
alsoillustratesthattheproposedANaTWORSschemeoutper- 410
forms the direct transmission, the ORS and the OSJ-MMISR 411
schemes in terms of its IP versus OP tradeoff for both the 412
(β,α)=(0.95,0.06)and(β,α)=(0.99,0.02)cases,sincethe 413
CSI of the eavesdropper links cannot be readily acquired, the 414
CSIsofthewiretaplinksarenottakenintoaccountinthepro- 415
posedANaTWORSscheme.Forthesakeofafaircomparison, 416
the CSIs of the wiretap links in the OSJ-MMISR scheme [30] 417
Fig. 3. IP versus OP of the direct transmission, ORS and ANaTWORS arenotconsideredeither. 418
schemesfordifferentnumberofrelaysassociatedwithanMUERofλme = Fig.5showstheIPversusOPofthedirecttransmission,ofthe 419
0 dB,whichwerecalculatedfrom[24,(33),(34)and[27]],[(24),(19)],and ORSandoftheANaTWORSschemesfordifferenttele-traffic 420
(31)and(32).
ratiosofS1 andS2,namely,forRs1/Rs2 =0.5,Rs1/Rs2 =1, 421
and Rs1/Rs2 =2. Observe from Fig. 5 that the ANaTWORS 422
388 of the ANaTWORS scheme is negligible, demonstrating the scheme performs best for Rs1/Rs2 =1. Moreover, the dif- 423
389 accuracyofourSRTanalysis. ference remains modest for asymmetric traffic ratios of both 424
390 InFig.3,weshowtheIPverusOPperformanceofboththedi- Rs1/Rs2 =0.5 and Rs1/Rs2 =2. This is due to the fact that 425
391 recttransmission,aswellasoftheORSandoftheANaTWORS for a fixed power allocation case, some of the power will be 426
392 schemefordifferentnumberofrelaysN.Wecanobservefrom wasted,whentheinstantaneouschannelgainissufficientlyhigh 427
393 Fig. 3 that as the number of relays N increases from N =4 andthetrafficdemandislow.Additionally,nobeneficialrelia- 428
394 to 8, the IP of all schemes is reduced at a specific OP, which bilityimprovementisachieved,despitedegradingthesecurity. 429
395 meansthatincreasingthenumberofrelaysimprovesthesecurity This is interesting, hence we will adopt an adaptive power al- 430
396 versus reliability tradeoff of wireless transmissions. Addition- location scheme for improving the security of wireless trans- 431
397 ally, Fig. 3 also demonstrates that IP versus OP performance missions in our future research. Finally, Fig. 5 also illustrates 432
398 oftheproposedANaTWORSschemeisbetterthanthatofthe thattheproposedANaTWORSschemeperformsbetterthanthe 433
399 directtransmissionandoftheORSschemesforalltheN values directtransmissionandORSschemesforallthreetraffic-ratios 434
400 considered. considered. 435
8 IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016
specially designed artificial noise was projected onto the null 450
space of Ri, hence resulting in a certain amount of interfer- 451
enceimposedontherelays.Hence,theself-interferenceandthe 452
interference factors were taken into account for characterizing 453
thewirelessSRTsoftheproposedANaTWORS,wherethese- 454
curity and reliability are quantified in terms of the IP and OP, 455
respectively.ItwasalsoillustratedthattheANaTWORSscheme 456
outperforms both the conventional direct transmission and the 457
ORS schemes in terms of its (IP x OP) product. Furthermore, 458
asthenumberofrelaysincreases,theSRToftheANaTWORS 459
schemeimproves. 460
Here, we only explored the allocation of a fixed power to 461
thesourcenodesandrelaysnodes.Inourfuturework,wewill 462
adopt an adaptive power allocation scheme in this scenario. 463
Specifically,thepowercanbedynamicallyallocatedaccording 464
tothenearinstantaneouschannelgainandthetrafficdemands 465
ofusers. 466
Fig. 5. IP versus OP of the direct transmission, ORS and ANaTWORS
schemesfordifferenttrafficassociatedwithanMUERofλme =0dB,N =8,
whichwerecalculatedfrom[24,(33),(34)and[27]],[(24),(19)],and(31)and APPENDIXA 467
(32).
Upon introducing the notation of X1 =|hs1i|2 and X2 = 468
|hs2i|2, noting that RVs |hs1i|2 and |hs2i|2 are exponentially 469
distributed and independent of each other. Thus, the proba- 470
bility density functions (PDFs) of X1 and X2 are fX1(x1)= 471
σ21 exp(−σx21 )andfX2(x2)= σ21 exp(−σx22 ),respectively. 472
s1i s1i s2i s2i
Hence,Pr(α|h|sh2si|12iγ|2s+2 <Δ1)canbeexpressedas 473
(cid:3) (cid:4)
|h |2
Pr s1i <Δ
α|h |2γ +2 1
s2i s
=Pr[x <(x αγ Δ +2Δ )]
1 2 s 1 1
(cid:3) (cid:4)(cid:3) (cid:3) (cid:4)(cid:4)
(cid:25)
∞ 1 x 2Δ +Δ αγ x
= exp − 2 1−exp − 1 1 s 2 dx
σ2 σ2 σ2 2
0 s2i s2i s1i
(cid:3) (cid:4)
σ2 2Δ
=1− s1i exp − 1 (A.1)
Δ αγ σ2 +σ2 σ2
1 s s2i s1i s1i
Fwiigth.6λ.meIP=x0OdBPoanfdthNed=ire8c,twtrhaincshmwisesrieonc,aOlcRulSateadndfrAomNa[T24W,O(3R3S),s(3ch4e)maneds whereσs21i andσs22i aretheexpectedvaluesofRVs|hs1i|2 and 474
[27]],[(24),(19)],and(31)and(32). |hs2i|2,respectively. 475
436 Fig.6illustratesthe(IPxOP)productofthedirecttransmis- APPENDIXB 476
437 sion,oftheORS,andoftheANaTWORSschemesfordifferent Using the law of total probability [34], the term 477
443389 SthNeRscsh.eOmbessercvaenferxohmibiFtiagn.(6IPthxaOt uPp)opneaikn,cbreuatstihnegmthaexiSmNuRm,(aIlPl Pr(|hos2(cid:5)|2 <Δ1)canb(cid:6)erewrittenas 478
440 xOP)productoftheproposedANaTWORSschemeissmallest Pr |h |2 <Δ
441 ofthethreeschemes,whichdemonstratesitssuperiority. (cid:9) os(cid:19)2 1 (cid:5) (cid:6)
= Pr |h |2 <Δ , max min |h |2,|h |2
442 V. CONCLUSION i∈Φn (cid:5) is2 1(cid:6)j(cid:6)∈Φn−{i} js2 js1
443 In this paper, we proposed an ANaTWORS scheme for a
444 wirelessnetworkconsistingofthepairofsourcenodesS1 and <min |his2|2,|his1|2
445 S2, and multiple two-way relays Ri, i∈{1,2,...,N}, com- (cid:9) (cid:26) (cid:19) (cid:5) (cid:6)
444467 mSRuTnipceartifnogrminanthceeporfebsoetnhctehoefAaNnaeTavWesOdRroSppanerd.oWfethaentarlaydzietidotnhael = i∈Φn Pr |his2|2 <Δ1,j∈(cid:6)mΦna−x{i}min |hjs2|2,|hjs1|2
448 directtransmissionschemes.Moreover,duetothepresenceof
449 CSI estimation errors, it was impossible to guarantee that the < |his1|2,|his1|2 <|his2|2
DINGetal.:SECURITY-RELIABILITYTRADEOFFANALYSISOFARTIFICIALNOISEAIDEDTWO-WAYOPPORTUNISTICRELAYSELECTION 9
(cid:19) (cid:5) (cid:6)
479 Substituting(B.4)into(B.3)yields 498
+Pr |h |2 <Δ , max min |h |2,|h |2
is2 1 j∈Φn−{i} js2 js1 (cid:25) (cid:3) (cid:4)(cid:3)(cid:25) (cid:3) (cid:4)
<|his2|2,|his2|2 <|his1|2(cid:6)(cid:8). (B.1) Υ0 = 0Δ1 σi12s2 exp −σxi2s2 0x σi12s1 exp −σyi2s1
⎛
480 Denoting 2|Φ(cid:9)n|−1−1
×⎝1+ (−1)|An(m)|exp
Υ =Pr(|h |2 <Δ , max min(|h |2,|h |2)<|h |2, m=1
0 is2 1 j∈Φn−{i} js2 js1 is1 ⎡ (cid:3) (cid:4)⎤⎞ ⎞
(cid:9)
|h |2 <|h |2) × ⎣− y + y ⎦⎠dy⎠dx
is1 is2 σ2 σ2
j∈An(m) js2 js1
481 and (cid:3) (cid:4) (cid:3) (cid:3) (cid:4)(cid:4)
Υ1=Pr(|his2|2 <Δ1,j∈mΦna−x{i}min(|hjs2|2,|hjs1|2)<|his2|2, = 1−exp −σΔi2s12 − σi2s2σ+i2s1σi2s1 1−exp −σΔi2s12 − σΔi2s11
|his2|2 <|his1|2),Pr(|hos2|2 <Δ1) +2|Φ(cid:9)n|−1−1(−1)|An(m)|⎛⎝σ2 (cid:9) (cid:3) 1 + 1 (cid:4)+1⎞⎠−1
is1 σ2 σ2
482 yields m=1 j∈An(m) js2 js1
(cid:5) (cid:6) (cid:9) (cid:3) (cid:3) (cid:4)(cid:4)
Pr |h |2 <Δ = (Υ +Υ ). (B.2) × 1−exp −Δ1
os2 1 0 1 σ2
448834 |hiDs2e|2n,otainndgVX=j =mamxijn∈(Φ|nh−js{i2}|2X,|jh,i∈jssΦi1nn|2c)e,thYat=RV|hsis|1h|i2s,1|2Xan=d −2|Φm(cid:9)n=|−11−1⎛⎜⎝(−1)|Ains2(m)|⎛⎝σi2s1j∈A(cid:9)n(m)(cid:3)σj21s2 + σj21s1(cid:4)+1⎞⎠−1
444888765 |oThfhiuse2sa|,2chtohbeoetyhPeDerxFwpsoitnohefntXhtiealamdniedsatnrYisbuoatfiroeσni2fsaX1nd(axnt)hde=yσi2σas2ri21s,e2rieenxsdppee(pc−etniσvdix2eesl2ny)t. ×⎛⎝σi2s2j∈A(cid:9)n(m)(cid:3)σj21s2 + σj21s1(cid:4)+ σσii22ss21 +1⎞⎠−1
448889 arenwdriftYte(nya)s= σi21s1 exp(−σi2ys1), respectively. Thus, Υ0 can be ×⎛⎝1−exp⎛⎝− (cid:9) (cid:3) Δ1 + Δ1 (cid:4)− Δ1 − Δ1 ⎞⎠⎞⎠⎞⎠
(cid:25) Δ1 (cid:19)(cid:25) x (cid:19)(cid:25) y (cid:20) (cid:20) j∈An(m) σj2s2 σj2s1 σi2s1 σi2s2
Υ0 = fX (x) fY (y) fV (v)dv dy dx (B.5)
0 0 0
(cid:25) (cid:19)(cid:25) (cid:19) (cid:19) (cid:20)(cid:20) (cid:20)
= Δ1fX (x) xfY (y) Pr max Xj <y dy dx where|Φn|denotesthecardinalityofthesetΦn. 499
0 ⎛ 0 ⎛ j∈Φn−{i} ⎞ ⎞ NowΥ1canberewrittenas 500
(cid:25) (cid:25)
Δ1 ⎝ x ⎝ (cid:10) ⎠ ⎠ (cid:25) (cid:19)(cid:25) (cid:19)(cid:25) (cid:20) (cid:20)
= fX (x) fY (y) Pr(Xj <y) dy dx. Δ1 ∞ x
0 0 j∈Φn−{i} Υ1 = fX (x) fY (y) fV (v)dv dy dx
0 x 0
(B.3) (cid:25) Δ1 (cid:19)(cid:25) ∞ (cid:19) (cid:19) (cid:20)(cid:20) (cid:20)
= f (x) f (y) Pr max X <x dy dx
X Y j
490 Noting that RVs |hjs1|2 and |hjs2|2 are exponentially 0 ⎛ x ⎛ j∈Φn−{i} ⎞ ⎞
449921 d[(cid:27)1is8t]r,ibwuteedhavaendPr(inXdjep<enyd)en=t 1o−fexepac(−h σ2yoth−er,σ2yba)s.edThuosn, = (cid:25) Δ1fX (x)⎝(cid:25) ∞fY (y)⎝ (cid:10) Pr(Xj <x)⎠dy⎠dx.
493 j∈Φn−{i}Pr(Xj <y)canbeexpandedasjs2 js1 0 x j∈Φn−{i}
(cid:3) (cid:3) (cid:4)(cid:4) (B.6)
(cid:10) (cid:10)
Pr(X <y)= 1−exp − y − y (cid:27)
j∈Φn−{i} j j∈Φn−{i} σj2s2 σj2s1 Similarlyto(B.4), j∈Φn−{i}Pr(Xj <x)canbeexpressed 501
⎡ (cid:3) (cid:4)⎤ as 502
=1+2|Φ(cid:9)n|−1−1(−1)|An(m)|exp⎣− (cid:9) σy2 + σy2 ⎦ (cid:10) 2|Φ(cid:9)n|−1−1
m=1 j∈An(m) js2 js1 Pr(X <x)= 1+ (−1)|An(m)|
j
(B.4) j∈Φn−{i} m=1
⎡ (cid:3) (cid:4)⎤
444999564 aawnnhdder|σAej2Ans2(nma(mr)e|)dtrheeenporeetxesspeenthctsteetdchaervmdailtnuhaelnsiotyonfeomfRtVphtesyss|huubjbsss1ee|tt2AoafnnΦ(dmn|h−).jσs{2ji2|}s21,, ×exp⎣−j∈A(cid:9)n(m) σxj2s2 + σxj2s1 ⎦.
497 respectively. (B.7)
10 IEEETRANSACTIONSONVEHICULARTECHNOLOGY,VOL.00,NO.00,2016
⎛ ⎛
503 Substituting(B.7)into(B.6)yields 2|Φ(cid:9)n|−1−1 (cid:9) 505
+ ⎝(−1)|An(m)|⎝σ2
(cid:25) (cid:3) (cid:3) (cid:4)(cid:3)(cid:25) (cid:3) (cid:4) (cid:4) is2
Δ1 1 x ∞ 1 y m=1 j∈An(m)
Υ = exp − exp − dy (cid:3) (cid:4) (cid:4)
1 0⎛ σi2s2 σi2s2 x σi2s1 σi2s1 × 1 + 1 + σi2s2 +1 −1
2|Φ(cid:9)n|−1−1 σj2s2 σj2s1 σi2s1
×⎝1+ (−1)|An(m)|exp ⎛ ⎛ (cid:3) (cid:4) ⎞⎞⎞
(cid:9)
⎡ m=1(cid:3) (cid:4)⎤⎞⎞ ×⎝1−exp⎝− σΔ21 + σΔ21 − σΔ21 − σΔ21 ⎠⎠⎠.
×⎣− (cid:9) x + x ⎦⎠⎠dx j∈An(m) js2 js1 is1 is2
σ2 σ2 (B.9)
j∈An(m) js2 js1
(cid:25) Δ1(cid:3) 1 (cid:3) x (cid:4)(cid:3) (cid:3) x (cid:4)(cid:4) Substituting (B.9) into (B.2), Pr(|hos2|2 <Δ1) can be 506
= exp − exp − obtained. 507
σ2 σ2 σ2
0 is2 is2 is1
⎛
2|Φ(cid:9)n|−1−1 APPENDIXC 508
×⎡⎝1+ m=(cid:3)1 (−1)|An(m(cid:4))|e⎤x⎞p⎞ tivLeleyt. NXo1tinagndthaXt2RVdseno|htes1e||h2s1aen|2d |ahnsd2e|2|hsa2ree|2,exrpeosnpeenc-- 550190
⎣−j∈A(cid:9)n(m)(cid:3) σxj2s2 +(cid:3)σxj2s1 ⎦⎠⎠dx(cid:4)(cid:4) tmXiae1lalynasnddoisfXtrσi2bs2u1aetreedanfadXn1dσ(xs22i1ne),d=erpeesσpns211edecetenixvtpeol(y−f. σeHxs2a11ecenh)ceoa,nthtdheerfXwP2Di(txhF2s)tho=ef 555111321
= σi2s1 1−exp − Δ1 − Δ1 σ21 exp(−σx22 ), respectively. Due to X1 and X2 are inde- 514
σ2 +σ2 σ2 σ2 s2e s2e
is2 is1⎛ ⎛is2 is1 pendent of each other, thus fX1X2(x1,x2)=fX1(x1)fX2(x2). 515
+2|Φ(cid:9)n|−1−1⎝(−1)|An(m)|⎝σi2s2 (cid:9) Pr(β|h|sh2es1|2e(cid:3)γ|2s+2 <Δ1,β|h|sh1es2|2eγ|2s+2 >Δ2)canbeobtained(cid:4)as 516
m=1 j∈An(m) |h |2 |h |2
(cid:3) (cid:4) (cid:4) Pr s1e <Δ , s2e >Δ
× 1 + 1 + σi2s2 +1 −1 β|hs2e|2γs +2 1 β|hs1e|2γs +2 2
σ2 σ2 σ2 (cid:25) (cid:25)
⎛ js2 ⎛js1 (cid:3)is1 (cid:4) ⎞⎞⎞ = ∞ (x2−2Δ2)/Δ2βγs f (x ,x )dx dx
(cid:9) X1X2 1 2 1 2
×⎝1−exp⎝− Δ1 + Δ1 − Δ1 − Δ1⎠⎠⎠. (cid:25)2Δ2 0 (cid:3)(cid:25) (cid:4)
j∈An(m) σj2s2 σj2s1 σi2s1 σi2s2 = ∞ f (x ) (x2−2Δ2)/Δ2βγs f (x )dx dx
(B.8) 2Δ2 X2 2 0 X1 1 1 2
(cid:19) (cid:20) (cid:19) (cid:20)
Δ γ βσ2 2Δ
504 Using(B.5)and(B.8),Υ0+Υ1canbeexpressedas = 1− Δ γ β2σs2 +s2eσ2 exp −σ2 2 . (C.1)
(cid:3) (cid:4) 2 s s1e s2e s2e
Υ +Υ =1−exp − Δ1 APPENDIXD 517
0 1 σ2
+2|Φ(cid:9)n|−1−1(−1)|An(m)|⎛⎝σi2s2 (cid:9) (cid:3) 1 + 1 (cid:4)+1⎞⎠−1 beUwsriinttg(cid:5)enthaeslaw of(cid:6)total probability [34], Pr(|hoe|2 >Δ) can 551189
is1 σ2 σ2 Pr |h |2 >Δ
m=1 j∈An(m) js2 js1 oe
(cid:3) (cid:3) (cid:4)(cid:4) (cid:9) (cid:19) (cid:5) (cid:6)
×−2|Φ1(cid:9)n−|−1−ex1⎛⎝p(−−1σ)Δ|i2As12n(m)|⎛⎝σ2 (cid:9) (cid:3) 1 + 1 (cid:4)+1⎞⎠−1 = i<∈Φmn Pinr(cid:5)|h|hisie2||22,>|hiΔs11|2,j(cid:6)∈(cid:6)Φmna−x{i}min |hjs2|2,|hjs1|2
×⎛⎝σm2=1 (cid:9) (cid:3) 1 + 1is1j(cid:4)∈A+n(σmi2)s2 +σj21s⎞⎠2 −1σj2s1 = i(cid:9)∈Φn Pr(cid:5)(cid:5)|hie|2 >Δ1(cid:6)(cid:6)P(cid:6)r(cid:19)j∈mΦna−x{i}min(cid:5)|hjs2|2,|hjs1|2(cid:6)
⎛is2j∈An⎛(m) σj2s2 (cid:3)σj2s1 σi2s1(cid:4) ⎞⎞⎞ <min |his2|2,|his1|2 . (D.1)
(cid:9)
×⎝1−exp⎝− Δ1 + Δ1 − Δ1 − Δ1 ⎠⎠⎠ We Denote Xj =min(|hjs2|2,|hjs1|2), Y =min(|his2|2, 520
j∈An(m) σj2s2 σj2s1 σi2s1 σi2s2 |his1|2), and V maxj∈Φn−{i}Xj. As mentioned above, RVs 521