senso s
A icle
Sec ecy Ou age P obabili y o a NOMA Scheme and
Impac Impe ec Channel S a e In o ma ion in
Unde lay Coope a i e Cogni i e Ne wo ks
Tan-Phuoc Huynh 1,2 , Duy-Hung Ha 1, Cong T uong Thanh 1, Peppino Fazio 1and
Mi osla Voznak 1,*
1Depa men o Telecommunica ions, VSB-Technical Uni e si y o Os a a, 17. lis opadu 2172/15,
708 00 Os a a, Czech Republic; [email p o ec ed] (T.-P.H.); duy[email p o ec ed] (D.-H.H.);
[email p o ec ed] (C.T.T.); [email p o ec ed] (P.F.)
2Depa men o Compu e Ne wo ks and Da a Communica ions, Eas e n In e na ional Uni e si y (EIU),
Nam Ky Khoi Nghia, Binh Duong 75000, Vie nam
*Co espondence: mi osla [email p o ec ed]; Tel.: +420-603-565-965
Recei ed: 29 Decembe 2019; Accep ed: 3 Feb ua y 2020; Published: 7 Feb ua y 2020
Abs ac :
Secu i y pe o mance and he impac o impe ec channel s a e in o ma ion (CSI) in
unde lay coope a i e cogni i e ne wo ks (UCCN) is in es iga ed in his pape . In he p oposed
scheme, elay R uses non-o hogonal mul iple access (NOMA) echnology o ans e messages
e1
,
e2
om he sou ce node S o Use 1 (U
1
) and Use 2 (U
2
), espec i ely. An ea esd oppe (E) is also
p oposed o wi e ap he messages o
U1
and U
2
. The ansmission’s secu i y pe o mance in he
p oposed sys em was analyzed and pe o med o e Rayleigh ading channels. Th ough nume ical
analysis, he esul s showed ha he p oposed sys em’s sec ecy pe o mance became mo e e icien
when he ea esd oppe node E was a he away om he sou ce node S and he in e media e
coope a i e elay R. The sec ecy pe o mance o U
1
was also compa ed o he sec ecy pe o mance o
U2. Finally, he simula ion esul s ma ched he Mon e Ca lo simula ions well.
Keywo ds:
non-o hogonal mul iple access; physical laye secu i y (PLS); coope a i e communica ion;
successi e in e e ence cancella ion (SIC); decode-and- o wa d (DF); cogni i e adio (CR); channel s a e
in o ma ion; ou age p obabili y
1. In oduc ion
The UCCN is known as he CR which is a p omising echnology and inno a i e solu ion
o dealing wi h he adio spec um alloca ion and p ecise equi emen s issues [
1
]. CR pe mi s
seconda y use s (SUs o unlicensed use s) o access he do man equency spec um wi hou causing
in e up ion o he p ima y use s (PUs o licensed use ). Due o SUs being accep ed o he PUs a
he same ime, he SUs ha e o keep hei ansmi powe s wi hin he accep able le els. Besides ha ,
wi h apidly ex ending wi eless senso ne wo ks (WSNs) in many a eas o indus y, he secu i y o
in o ma ion ans e becomes a mo e se ious p oblem. Many esea che s in es iga ed PLS o help
secu i y ansmission be ween he sou ce node and he des ina ion node o imp o e and enhance he
sec ecy o WSNs.
Recen ly, many solu ions and echnologies ha e been in es iga ed o he pu poses o speeding up
mobile da a ansmission, ex ending wi eless communica ion ange, and assis ing use s in connec ing
secu i y oge he . Examples o hese echnologies include ampli y-and- o wa d (AF), o hogonal
mul iple access (OMA), and ene gy ha es ing [
2
–
4
]. NOMA echnology, howe e , is a p omising
me hod and has a ac ed signi ican a en ion in ecen yea s [5–10].
NOMA echnology has g adually become one o he mos e icien solu ions in de eloping he
i h-gene a ion mobile ne wo k (5G). In he NOMA echnique, he use s can sha e bo h ime and
Senso s 2020,20, 895; doi:10.3390/s20030895 www.mdpi.com/jou nal/senso s
Senso s 2020,20, 895 2 o 17
equency esou ces and only adjus hei powe alloca ion a ios. The use s wi h be e channel
condi ions can se e as elays o enhance he sys em pe o mance by using SIC [
9
]. This echnology
imp o es he limi a ion o o hogonal mul iple access (OMA). I mee s he needs o end use s in
p o iding access o da a quickly and secu ely. NOMA and PLS a e he e o e e y impo an echniques
in da a ans e . They assis in ansmi ing signals om he sou ce node o des ina ion node wi h high
speed, e iciency, and da a con iden iali y.
Se e al s udies ha e examined NOMA and PLS in wi eless sys ems [
11
–
13
]. In [
11
], he au ho s
conside ed a coope a i e elaying sys em using he NOMA echnique o enhance he e iciency o he
ansmi ed signal. The esea che s in [
13
] in es iga ed he e ec i eness o new schemes ha combined
pa ial elay selec ion and NOMA in AF elaying sys ems o inc ease da a ansmission a es o 5G
mobile ne wo ks.
A conside able amoun o li e a u e has been published on PLS [
14
–
16
]. In [
14
], he au ho s
analyzed he sec ecy pe o mance o coope a i e p o ocols wi h elay selec ion me hods in luenced by
co-channel in e e ence. The au ho s in [
15
] inspec ed he impac o co ela ed ading on he sec ecy
pe o mance o mul iple DF elaying ha uses he op imal elay selec ion me hod. Some esea che s
ha e also combined he NOMA echnique wi h PLS [
17
–
19
]. In [
17
], he au ho s esol ed he p oblem
o maximizing he minimum con iden ial in o ma ion a e in use s subjec o he sec ecy ou age
cons ain and ins an aneous ansmi powe cons ain . Coope a i e NOMA sys ems wi h PLS in
bo h AF and DF we e s udied by he au ho s in [18].
The applica ion o NOMA echniques and secu i y p inciples in unde lay cogni i e adio ne wo ks
we e also sugges ed by some au ho s in [
20
–
24
]. In [
20
], he au ho s discussed a coope a i e
ansmission scheme o a downlink NOMA in CR sys ems. This esea ch exploi ed maximum
spa ial di e si y. The esea che s in [
24
] conside ed secu e communica ion in cogni i e DF elay
ne wo ks in which a pai o cogni i e elays we e oppo unis ically selec ed o secu i y p o ec ion
agains ea esd oppe s.
Channel s a e in o ma ion (CSI) has a i al ole in wi eless communica ion sys ems. I desc ibes
how a signal p opaga es om he sou ce node o he elay, such as sca e ing, ading, and powe
decay o e dis ance. Du ing a ecei e ’s se -up pe iod, he CSI is e alua ed and ans e ed o ela ed
nodes in he sys em h ough a media access con ol p o ocol. In [
25
], he au ho s esea ched he e ec
o impe ec channel CSI on seconda y use s in an unde lay DF cogni i e ne wo k wi h mul iple
p ima y ecei e s. In [
26
], scien is s s udied he e ec o impe ec CSI on a DF coope a i e unde lay
cogni i e adio NOMA ne wo k in o de o de e mine he op imal powe alloca ion ac o s o di e en
use dis ances.
In mos o he li e a u e epo ed abo e, he combina ion o NOMA and PLS in a UCCN in luenced
by CSI was no p oposed. Mo i a ed and inspi ed by he abo e ideas, a coope a i e scheme is
sugges ed in his pape . In his scheme, a p oposed UCCN using NOMA is equi ed o bo h decode
and o wa d he messages
e1
and
e2
om node S o wo des ina ion nodes (U
1
and U
2
) unde he e ec
o CSI and an ea esd oppe . The sec ecy pe o mance o he communica ions
e1
and
e2
in he p oposed
sys em we e hen examined and es ima ed in e ms o sec ecy ou age p obabili y o e Rayleigh ading
channels o imp o e spec al e iciency and secu e communica ion.
The main con ibu ions o he pape a e summa ized as ollows:
-
A s udy o he impac o impe ec CSI and he sec ecy pe o mance o a UCCN applying he
NOMA echnique o imp o e sys em pe o mance in a 5G wi eless ne wo k.
-
Sec ecy ou age p obabili y (SOP) is pe o med o e Rayleigh ading channels and e i ied wi h
Mon e Ca lo simula ions.
-
The esul s achie ed by he p oposed scheme demons a e he secu i y pe o mance o U
1
and U
2
.
-
The sec ecy pe o mance o he p oposed sys em imp o ed when he dis ance be ween he
ea esd oppe node E and he sou ce and coope a i e elay inc eased.
Senso s 2020,20, 895 3 o 17
The pape has i e sec ions. Sec ion 1in oduces he opic. Sec ion 2desc ibes he p oposed
scheme’s sys em model. Sec ion 3p esen s he esul s o an analysis o he sec ecy ou age p obabili y
a he sou ce nodes. Sec ion 4p esen s he simula ion esul s. Sec ion 5summa izes he conclusions.
2. Sys em Model
Figu e 1illus a es he in luence o impe ec CSI in a UCCN using NOMA and PLS. The sys em
model consis s o he sou ce nodes S ans e ing a supe imposed signal
e1
and
e2
o U
1
and U
2
,
espec i ely, h ough elay node R. One ea esd oppe node E is p oposed o wi e ap he signals
e1
,
e2
o he links S-U
1
, S-U
2
. In addi ion, he sys em model also consis s o a node Pu which is known as he
p ima y use ha ing he license. Due o he in e e ence cons ain a he Pu node in he UCCN, he
elay R and sou ce S adjus hei ansmi ing powe s. In his model, we assume ha he in e media e
elay node R ope a es in DF elaying me hod and applies he NOMA p inciple unde he in luence o
impe ec CSIs and PLS in UCCN. In addi ion, he a iances o Ze o-mean Whi e Gaussian Noises
(AWGNs) a e equal, gi en as
N0
. In his wo k, he co esponding dis ances o he links S-Pu, R-Pu,
S-R, S-E, R-E, R-Pu, R-U1, and R-U2in Figu e 1a e gi en as lSPu,lRPu,lSR,lSE,lRE,lRPu,l1, and l2.
Rega ding he sys em channels,
hi
ep esen s he Rayleigh ading channel coe icien ,
i∈(hSR,hSE,hSPu,hRE,hRPu, 1, 2)
. We assume ha he channels
hi
do no change du ing block
ime T and a e independen ly and iden ically dis ibu ed be ween wo consecu i e block imes [10].
Finally, all o nodes in he sys em model ha e a single an enna o ansmi ing and
ecei ing messages.
Figu e 1. Sys em model o NOMA and PLS unde impe ec CSI in a UCCN.
In p inciple, he e a e wo ime slo s in ol ed in each sys em communica ion p ocess, and a e
gi en as ollows:
A he i s ime slo , he sou ce node S ans e s he in o ma ion e
S
o he elay R and he
ea esd oppe node E, which is gi en by he ma h exp ession as
es=pβ1Pse1+pβ2Pse2, (1)
whe e Ps is he powe a sou ce node S,
e1
, and
e2
a e he messages o U
1
, U
2
, espec i ely, wi h
E{|ej|2}=
1,
j∈(1, 2)
, (
E{e}
being no a ed o he expec a ion p ocess o
e
). The
β1
and
β2
a e he
powe alloca ion coe icien s. Following he p inciple o he NOMA, we assume ha
β1>β2
wi h
β1+β2=1.
Senso s 2020,20, 895 4 o 17
Because o he es ima ion e o s o channels
hi
, he e alua ed ading channel coe icien s a he
nodes a e ep esen ed as ollows [25]:
b
hi=ρhi+q1−ρ2εi, (2)
whe e
b
hi
,
hi
, and
εi
a e modeled as he addi i e whi e Gaussian noise (AWGN) wi h he andom
a iable
i=|b
hi|2
. The co ela ion coe icien
ρ∈[0, 1]
is desc ibed as he a e age quali y o he
channel es ima ion.
No a ion: The Cumula i e Dis ibu ion Func ion (CDF) and p obabili y densi y unc ion (pd ) o he
andom a iable
i
is deno ed espec i ely as
F i(x) =
1
−e−1
λix
and
i(x) = 1
λie−1
λix
, whe e
λi=l−β
i
,
and βis a pa h-loss exponen .
The ecei ed signal a R om sou ce node S o decode
e1
unde impac impe ec CSIs is gi en
as ollows:
ye1
SR =hSRes+σR. (3a)
Replace hSR om o mula (2), he signal ye1
SR is calcula ed as
ye1
SR =pβ1Pse1bhSR−√1−ρ2εSR
ρ+pβ2Pse2bhSR−√1−ρ2εSR
ρ+nR
=√β1Pse1bhSR
ρ+√β2Pse2bhSR
ρ−√β1Pse1√1−ρ2εSR
ρ−√β2Pse2√1−ρ2εSR
ρ+nR,
(3b)
whe e nRdeno es he AWGNs a he elay R wi h he same a iance N0.
Because o applying NOMA echnology, hanks o he deploymen o SIC in NOMA p inciple,
i s ly, he elay R decodes he signal
e1
om o mula (3b) and emo es i , hen he signal
e2
will be
decoded wi hou he componen
√β1Pse1bhSR
ρ
in o mula (3b). The e o e, he signal
e2
ecei ed a R om
sou ce S a e emo ing he signal e1is exp essed as ollows:
ye2
SR =pβ2Pse2bhSR
ρ−pβ1Pse1p1−ρ2εSR
ρ−pβ2Pse2p1−ρ2εSR
ρ+nR. (4)
Simila ly, he node E also wi e aps he packe s
e1
and
e2
om S, espec i ely, and he ecei ed
signals a node E a e ob ained as ollows:
ye1
SE =pβ1Pse1bhSE
ρ+pβ2Pse2bhSE
ρ−pβ1Pse1p1−ρ2εSE
ρ−pβ2Pse2p1−ρ2εSE
ρ+nE(5)
ye2
SE =pβ2Pse2bhSE
ρ−pβ1Pse1p1−ρ2εSE
ρ−pβ2Pse2p1−ρ2εSE
ρ+nE, (6)
whe e nEdeno es he AWGNs a he E wi h he same a iance N0.
In he second ime slo , a e he ecei ed signals, he elay R sends hem o he sou ce nodes U
1
and U2. Hence, he ecei ed signals a he des ina ion node U1, U2a e gi en espec i ely as
ye1
RU1=pβ1PRe1b
h1
ρ+pβ2PRe2b
h1
ρ−pβ1PRe1p1−ρ2ε1
ρ−pβ2PRe2p1−ρ2ε1
ρ+nU1(7)
ye2
RU2=pβ2PRe2b
h2
ρ−pβ1PRe1p1−ρ2ε2
ρ−pβ2PRe2p1−ρ2ε2
ρ+nU2, (8)
whe e
nU1
,
nU2
deno e he AWGNs a he des ina ion U
1
, U
2
wi h he same a iance
N0
, and
PR
is a
ansmi powe o he elay R.
Senso s 2020,20, 895 5 o 17
In he p oposed scheme, unde he in e e ence cons ain a he node
Pu
, he sou ce node S and
elay node R ha e o adjus hei ansmi ing powe s so ha he in e e ence powe a he Pu mus
be less han a h eshold alue, which is assumed as
I h
. The maximum powe s o nodes S and R a e
gi en, espec i ely,
PS=I h
|d
hSR|2=I h
SR
. (9a)
PR=I h
|d
hRPu|2=I h
RPu
. (9b)
Because he node E connec s o he elay R di ec ly, so i wi e aps he packe s
e1
and
e2
om elay
R. The e o e, he ecei ed signals a E h ough he link R-E a e exp essed as
ye1
RE =pβ1PRe1bhRE
ρ+pβ2PRe2bhRE
ρ−pβ1PRe1p1−ρ2εRE
ρ−pβ2PRe2p1−ρ2εRE
ρ+nE. (10)
ye2
RE =pβ2PRe2bhRE
ρ−pβ1PRe1p1−ρ2εRE
ρ−pβ2PRe2p1−ρ2εRE
ρ+nE. (11)
We de ine he ecei ed Signal- o-In e e ence and Noise Ra ios (SINRs) as
γ=
E|signal|2/
E|o e all noise|2.
Fi s ly, we calcula e he ecei ed Signal- o-In e e ence and Noise Ra ios (SINRs) o decoding he
in o ma ion signal e1.
Thus, om o mula (3b), he SINR a he elay R wi h he link S-R is ob ained as ollows:
γe1
SR =
β1PS|d
hSR|2
ρ2
β2PS|d
hSR|2
ρ2+β1PS(1−ρ2)λSR
ρ2+β2PS(1−ρ2)λSR
ρ2+N0
=β1PS SR
β2PS SR+PS(1−ρ2)λSR(β1+β2)+ρ2N0.
(12a)
Replacing PS=I h
SR in (9a), and se ing P=I h
N0,γe1
SR is ew i en as
γe1
SR =Pβ1 SR
Pβ2 SR +P(1−ρ2)λSR +ρ2 SPu
, (12b)
Simila ly, wi h he o mula in (7), we also calcula e
γe1
RU1
, and his is achie ed by ma hema ical
exp ession as
γe1
RU1=Pβ1 1
Pβ2 1+P(1−ρ2)λ1+ρ2 RPu
, (13)
whe e P=I h
N0.
Applying o mulas (5) and (10), he ecei ed SINRs a he ea esd oppe node E wi h he link S-E
and R-E a e gi en, espec i ely, as ollows:
γe1
SE =β1Ps SE
β2Ps SE +Ps(1−ρ2)λSE +ρ2N0
=Pβ1 SE
Pβ2 SE +P(1−ρ2)λSE +ρ2 SPu
.
(14)
γe1
RE =Pβ1 RE
Pβ2 RE +P(1−ρ2)λRE +ρ2 RPu
. (15)

Senso s 2020,20, 895 6 o 17
In he second, simila o decoding he in o ma ion signal
e1
, we ind he ecei ed SINRs o
decoding he in o ma ion signal e2as ollows.
We apply o mulas (4) and (6), he ecei ed SINRs a he nodes R wi h he link S-R, and a he
ea esd oppe node E wi h he link S-E a e exp essed, espec i ely, as ollows:
γe2
SR =β2Ps SR
β1Ps(1−ρ2)λSR +β2Ps(1−ρ2)λSR +ρ2N0
=Pβ2 SR
P(1−ρ2)λSR +ρ2 SPu
.(16)
γe2
SE =β2Ps SE
Ps(1−ρ2)λSE +ρ2N0
=Pβ2 SE
P(1−ρ2)λSE +ρ2 SPu
.(17)
Simila ly, wi h o mulas (8) and (12), he ecei ed SINRs a he nodes U
2
and E om elay R a e
in e ed, espec i ely, as ollows:
γe2
RU2=Pβ2 2
P(1−ρ2)λ2+ρ2 RPu
. (18)
γe2
RE =Pβ2 RE
P(1−ρ2)λRE +ρ2 RPu
. (19)
Applying he Shannon capaci y o mula, he achie able a es o he links X–Y a e o mula ed as
Rej
XY =1
2log2(1+γej
XY ). (20)
whe e he a io 1/2 ep esen s he ac ha da a ansmission is spli in o wo ime slo s,
X∈{S,R}
,
and
Y∈{E,U1,U2}
. The sec ecy capaci y o he UCCN sys ems wi h DF-based NOMA o he S-U
j
communica ion can be exp essed as
SCj=hSCej
Uj−SCej
Ei+, (21)
whe e
[x]+=max (0, x)
;
SCei
SR
and
SCej
RUj
a e he sec ecy capaci ies om he sou ce node S o he elay
R and om he elay R o he des ina ion Uia e gi en, espec i ely, as
SCej
SR =max(0, Rej
SR −Rej
SE). (22)
SCej
RUi=max(0, Rej
RUi−Rej
RE). (23)
3. Sec ecy Ou age P obabili y Analysis
In his sec ion, he sec ecy ou age p obabili y o ea esd opping he signals o U
1
and U
2
in he
p oposed scheme a e analyzed. We assume ha a node success ully and sa ely decodes he ecei ed
packe i i s achie able sec ecy capaci y is la ge han a h eshold sec ecy capaci y SC h.
3.1. Sec ecy Ou age P obabili y o U1.
The sec ecy ou age p obabili y o U
1
occu ing when U
1
does no ecei e a signal sa ely om he
sou ce node S unde he malicious a emp o he ea esd oppe E is exp essed as ollows:
OPU1=P [min(SCe1
SR,SCe1
RU1)<SC h]
=1−P [SCe1
SR ≥SC h,SCe1
RU1≥SC h].(24)
Senso s 2020,20, 895 7 o 17
Replacing
SCe1
SR =max(
0,
Re1
SR −Re1
SE)
a o mula (22) and
SCe1
RU1=max(
0,
Re1
RU1−Re1
RE)
a (23)
in o mula (24), he OPU1is ew i en as ollows:
OPU1=1−P Re1
SR −Re1
SE ≥SC h
| {z }
P 1.1
×P hRe1
RU1−Re1
RE ≥SC hi
| {z }
P 1.2
(25)
P oposi ion 1. The p obabili y o he P 1.1, and P 1.2 in (25) is gi en as
P 1.1 =










0a≤θb
(1/λSPu)e−ψ11/λSR
(1/λSPu)+(ψ2/λSR)−(1/λSRλSPu)I1a>θb.
(26)
whe e
ψ11 =φcλSE
(a−φb);ψ2=φρ2
(a−φb)
I1=
∞
Z0
∞
Z
(ψ11+ψ2x)
e−1
λSPu x+1
λSR ye−1
λSE ζ1dxdy ,
ζ1=cλSE +ρ2xay −φby +cλSR +ρ2x
[φb+ (φ+1)a] [by +cλSR +ρ2x]−aby ,
P oo : See Appendix A.
P 1.2 =










0a≤θb
(1/λRPu)e−ψ12/λ1
(1/λRPu)+ψ2/λ1−(1/λ1λRPu)I2a>θb
(27)
whe e
ψ12 =φcλ1
(a−φb);ψ2=φρ2
(a−φb)
I2=
∞
Z0
∞
Z
(ψ12+ψ2x)
e−1
λRPu x+1
λ1ye−1
λRE ζ2dxdy ,
ζ2=cλRE +ρ2xay −φby +cλ1+ρ2x
[φb+ (φ+1)a] [by +cλ1+ρ2x]−aby ,
P oo : See Appendix B.
F om o mulas in (26) and (27), he sec ecy ou age p obabili y o he U1is ob ained as
OPU1=



















1a≤φb
1−

(1/λSPu)e−ψ11/λSR
(1/λSPu)+(ψ2/λSR)−(1/λSRλSPu)×I1
×1/λRPue−ψ12/λ1
1/λRPu+ψ2/λ1−(1/λ1λRPu)×I2

a>φb
(28)
Senso s 2020,20, 895 8 o 17
3.2. Sec ecy Ou age P obabili y o U2
Simila o U1, he SOP o U2can be exp essed as
OPU2=P hmin SCe2
SR,SCe2
RU2<SC hi=1−P hSCe2
SR ≥SC h,SCe2
RU2≥SC hi. (29)
P oposi ion 2. The sec ecy ou age p obabili y o U2in (26) is gi en as
OPU2=1−1−1
λSPuλSE ×I3×1−1
λRPuλRE
I4(30)
whe e
ζ3=φ(cλSR+ρ2x)
b+(φ+1)(cλSR+ρ2x)
(cλSE+ρ2x)y,
ζ4=φ(cλ2+ρ2x)
b+(φ+1)(cλ2+ρ2x)
(cλRE+ρ2x)y,
I3=
∞
Z0
∞
Z0e−1
λSPu x+1
λSE y×1−e−ζ3
λSR dxdy,
I4=
∞
Z0
∞
Z0e−1
λRPu x+1
λRE y×1−e−ζ4
λ2dxdy.
P oo : See Appendix C.
The in eg als
I1
and
I2
in (28) and
I3
and
I4
in (30) a e complex in eg als and a e di icul o esol e
p ac ically. In his pape , howe e , he alue o
I1
,
I2
,
I3
and
I4
can be ound using nume ical me hods.
4. Simula ion Resul s
In his sec ion, he sec ecy pe o mance o a NOMA scheme and he impac o impe ec CSI
in a UCCN we e examined, analyzed, and e alua ed. The heo e ical esul s o he analyses we e
e i ied wi h Mon e Ca lo simula ions. The coo dina es o S, R, U
1
, U
2
, Pu, and E we e se o
S(
0, 0
)
,
R(xR, 0
),
U1xU1,yU1=(1, 0)
,
U2xU2,yU2=(0.75, −0.5)
,
Pu (xPu,yPu )
,
E(xE,yE)
,
espec i ely, in he wo-dimensional plane and sa is ying
(xi>0)
. Hence,
lSR =xR
,
lRU1=xU1−xR
,
lRU2=qy2
U2+xU2−xR)2
,
lRPu =qy2
Pu +(xPu −xR)2
,
lRE =qy2
E+(xE−xR)2
,
lSE =qy2
E+x2
E
,
and
lSPu =qx2
Pu +y2
Pu
. We assume ha he a ge sec ecy capaci y
SC h =
0.5 (bi /s/Hz) and he
exponen βis se o a cons an β=3.
Figu es 2and 3g aph he SOP o he wo Use s U
1
and U
2
ia SNR (dB) wi h
SC h =
0.5
(bi /s/Hz). The elay R, Pu, U
1
, U
2
, and ea esd oppe E a e loca ed in posi ions R
(xR, 0)=(0.5, 0)
,
Pu (xPu,yPu)=(0.5, −1)
,
U1xU1,yU1=(1, 0)
,
U2xU2,yU2=(0.75, −0.5)
,
E(xE,yE)=(0.5, 1)
,
espec i ely. F om he esul s in Figu e 2, we can see he e ec o he ea esd opping node E o he
SOP when SNR is changed om 0 dB o 20 dB. Wi h
ρ=
0.95, he SOP alues o Use U
1
a e g ea e
han Use U
2
when SNR < 2.5 dB. Ne e heless, when he SNR inc eases om 2.5 dB o 30 dB, he
SOP o Use U
2
is be e han Use U
1
, and bo h also inc ease when he SNR inc eases as a esul o
la ge ansmi ing powe . Besides ha , i is no ed ha impe ec CSI deg ades he SOP o he signal.
Senso s 2020,20, 895 9 o 17
SNR(dB)
0 5 10 15 20 25 30
Sec ecy Ou age P obabili y(SOP)
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
Simula ion
Theo y-Use U1
Theo y-Use U2
Figu e 2. The SOP o U1and U2 e sus SNR (dB).
SNR(dB)
0 5 10 15 20 25 30
Sec ecy Ou age P obabili y(SOP)
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
Simula ion
Theo y-Use U1
Theo y-Use U2
ρ=0.9
ρ=0.95
Figu e 3. The SOP o U1and U2 e sus SNR (dB) when ρ=0.9 and ρ=0.95.
In Figu e 3, we obse e he ob ious a ec ion o he channel es ima ion coe icien
ρ
o he SOP.
The SOP o he wo use s in case
ρ=
0.95 ou pe o ms he SOP in case
ρ=
0.9. I means ha he
sys em has been impac ed by impe ec CSI. We also can see ha he sec ecy pe o mance o he wo
Senso s 2020,20, 895 16 o 17
OPU2=1−P Re2
SR −Re2
SE ≥SC h
| {z }
Ω2.1
×P hRe2
RU2−Re2
RE ≥SC hi
| {z }
Ω2.2
(A18)
Fi s ly, we calcula e he p obabili y o Ω2.1 as ollows:
Ω2.1 =P Re2
SR −Re2
SE ≥SC h=1−P Re2
SR <SC h +Re2
SE
=1−P 1
2log21+γe2
SR<SC h +1
2log21+γe2
SE
=1−P " SR <φcλSR +ρ2 SPu
b+(φ+1)cλSR +ρ2 SPu
(cλSE +ρ2 SPu) SE#(A19)
Applying he pd o he andom a iables SPu and SE, (A19) is w i en as
Ω2.1 =1−
∞
Z0
∞
Z0"P " SR <φcλSE +ρ2x
b+(φ+1)cλSR +ρ2x
(cλSE +ρ2x)y#× SPu (x) SE (y)#dxdy
=1−1
λSPuλSE
∞
Z0
∞
Z0
e−1
λSPu x+1
λSE y×
1−e−1
λSR φ(cλSR+ρ2x)
b+(φ+1)(cλSR+ρ2x)
(cλSE+ρ2x)y

dxdy
(A20)
Ω2.2 is calcula ed simila ly as (A20) and is gi en as
Ω2.2 =1−1
λRPuλRE
∞
Z0
∞
Z0


e−1
λRPu x+1
λRE y×

1−e−1
λ2 φ(cλ2+ρ2x)
b+(φ+1)(cλ2+ρ2x)
(cλRE+ρ2x)y!



dxdy (A21)
Finally, wi h o mulas (A20) and (A21), he sec ecy ou age p obabili y o U
2
is ob ained by he
exp ession as (30).
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c
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