Outage analysis of the power splitting based underlay cooperative cognitive radio networks

senso s A icle Ou age Analysis o he Powe Spli ing Based Unde lay Coope a i e Cogni i e Radio Ne wo ks Phu T an Tin 1, Van-Duc Phan 2, Tan N. Nguyen 3,* , Lam-Thanh Tu 4, Bui Vu Minh 5, Mi osla Voznak 6 and Peppino Fazio 6,7   Ci a ion: Tin, P.T.; Phan, V.-D.; Nguyen, T.N.; Tu, L.-T.; Minh, B.V.; Voznak, M.; Fazio, P. Ou age Analysis o he Powe Spli ing Based Unde lay Coope a i e Cogni i e Radio Ne wo ks. Senso s 2021,21, 7653. h ps://doi.o g/10.3390/ s21227653 Academic Edi o : Pe e Han Joo Chong Recei ed: 2 Oc obe 2021 Accep ed: 16 No embe 2021 Published: 18 No embe 2021 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2021 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). 1Facul y o Elec onics Technology, Indus ial Uni e si y o Ho Chi Minh Ci y, Ho Chi Minh Ci y 700000, Vie nam; [email p o ec ed] 2Facul y o Au omobile Technology, Van Lang Uni e si y, Ho Chi Minh Ci y 700000, Vie nam; [email p o ec ed] 3Communica ion and Signal P ocessing Resea ch G oup, Facul y o Elec ical and Elec onics Enginee ing, Ton Duc Thang Uni e si y, Ho Chi Minh Ci y 700000, Vie nam 4Ins i u e XLIM, Uni e si y o Poi ie s, 86000 Poi ie s, F ance; [email p o ec ed] 5Facul y o Au omo i e, Mechanical, Elec ical and Elec onic Enginee ing, Nguyen Ta Thanh Uni e si y, Ho Chi Minh Ci y 700000, Vie nam; [email p o ec ed] 6Facul y o Elec ical Enginee ing and Compu e Science, VSB-Technical Uni e si y o Os a a, 708 00 Os a a, Czech Republic; mi osla [email p o ec ed] (M.V.); [email p o ec ed] (P.F.) 7Depa men o Molecula Sciences and Nanosys ems, Ca’ Fosca i Uni e si y o Venice, Via To ino 155, 30123 Venezia VE, I aly *Co espondence: [email p o ec ed] Abs ac : In he p esen pape , we in es iga e he pe o mance o he simul aneous wi eless in o ma- ion and powe ans e (SWIPT) based coope a i e cogni i e adio ne wo ks (CCRNs). In pa icula , he ou age p obabili y is de i ed in he closed- o m exp essions unde he oppo unis ic pa ial elay selec ion. Di e en om he con en ional CRNs in which he ansmi powe o he seconda y ansmi e s coun me ely on he agg ega e in e e ence measu ed on he p ima y ne wo ks, he ansmi powe o he SWIPT-enabled ansmi e s is also cons ained by he ha es ed ene gy. As a esul , he ma hema ical amewo k in ol es mo e co ela ed andom a iables and, hus, is o highe complexi y. Mon e Ca lo simula ions a e gi en o co obo a e he accu acy o he ma hema ical analysis and o shed ligh on he beha io o he OP wi h espec o se e al impo an pa ame e s, e.g., he ansmi powe and he numbe o elays. Ou indings illus a e ha inc easing he ansmi powe and/o he numbe o elays is bene icial o he ou age p obabili y. Keywo ds: decode–and– o wa d; ou age p obabili y; elay selec ion; cogni i e adio ne wo k; SWIPT 1. In oduc ion Cogni i e adio ne wo ks (CRNs) a e conside ed one o he mos e ec i e solu ions o o e come he sca ci y o he equency spec um [ 1 , 2 ]. The p incipal idea o CRNs is o pe mi unlicensed use s o concu en ly ope a e wi h licensed use s while s ic ly gua an eeing he quali y-o -se ice (QoS) o p ima y use s. To ealize such ne wo ks wo popula p o ocols a e p oposed in he li e a u e, namely, he o e lay and unde lay p o ocols [ 3 , 4 ]. The o me allows seconda y de ices o oppo unis ically occupy he empo a ily unused spec um in e ms o space, ime, and equency. Rega ding he unde lay p o ocol, on he o he hand, seconda y use s a e always g an ed pe mission o access he licensed spec um p o ided ha he agg ega e in e e ence c ea ed by seconda y ne wo ks measu ed on he p ima y de ices is below he p ede ined h eshold. Compa ed o he o e lay p o ocol, he unde lay p o ocol is p e e able due o i s high a ailabili y and, hus, is sui able o u gen se ices, ideo con e ences, ideo gaming, and so o h. Ne e heless, he cons o his p o ocol a e ha i does no suppo a long ansmission and/o high-quali y se ices owing o he low ansmi powe o a oid exceeding he Senso s 2021,21, 7653. h ps://doi.o g/10.3390/s21227653 h ps://www.mdpi.com/jou nal/senso s Senso s 2021,21, 7653 2 o 17 in e e ence h eshold o he p ima y ne wo ks. As a consequence, in o de o employ he unde lay CRNs in p ac ice, he combina ion wi h o he echniques is necessa y. Fo una ely, elaying o coope a i e communica ions is a good complemen o he unde lay CRNs [ 5 – 7 ]. Mo e p ecisely, coope a i e communica ions ne wo ks a e wi eless ne wo ks whe e one o se e al elays a e deployed among end-use s o help hem exchange in o ma ion. Wi h he help o he elay, he ansmission dis ance is d ama ically dec eased, imp o ing he sys em pe o mance. Addi ionally, elaying echnology has also been p o en o be an e ec i e way o ex end he co e age a ea. The spec al e iciency and QoS issues can be add essed in a s aigh o wa d manne by employing he coope a i e cogni i e adio ne wo ks (CCRNs) [ 8 , 9 ]. None heless, he e s ill exis s an u gen issue, which is o imp o e he ene gy e iciency o wi eless ne wo ks. The p oblem escala es se iously in ei he he In e ne o Things (IoTs), low powe wide a ea ne wo ks (LPWAN) [ 10 ], o 5G and beyond ne wo ks owing o he exponen ial g ow h o wi eless-connec ed de ices accompanied by hei powe -hung y applica ions. In ac , imp o ing ene gy e iciency is one o he highes p io i ies in wi eless communica ions. Fo una ely, simul aneous wi eless in o ma ion and powe ans e (SWIPT) has ecen ly been p oposed o deal wi h his issue [ 11 – 13 ]. In pa icula , SWIPT ansmi s bo h in o ma ion and ene gy on he same ca ie equency, hus boos ing bo h he spec al and ene gy e iciency. Consequen ly, in he p esen pape , we in es iga e he pe o mance o he SWIPT-based coope a i e cogni i e adio ne wo ks (CCRNs) wi h he help o mul iple elays. Be o e highligh ing ou no el con ibu ions, he s a e-o - he-a is gi en as ollows. The pe o mance o CCRNs ne wo ks was s udied widely in [ 14 – 20 ]. Speci ically, he ou age p obabili y (OP) o he cogni i e adio non-o hogonal mul iple access (NOMA) ne wo ks was de i ed in [ 14 ]. The au ho s in [ 15 ], on he o he hand, in es iga ed he sec ecy pe o mance o he unde lay coope a i e mul ihop CRNs. In pa icula , he sec ecy ou age p obabili y (SOP) was de i ed in he app oxima ed closed- o m exp ession. Lee e al. in [ 16 ] add essed he minimiza ion o he numbe o eedback bi s equi ed in o de o sa is y he QoS in bo h p ima y and seconda y ne wo ks. The op imal powe alloca ion o non-o hogonal ampli y–and– o wa d (AF) elaying o unde lay CRNs was p o ided in [ 17 ] o maximize he sys em h oughpu . In [ 18 ], he capaci y o oice o e IP (VoIP) in CRNs was analyzed and maximized by modelling he VoIP a ic and channel coe icien s as a Ma ko -modula ed Poisson p ocess. Liu e al. in [ 19 ] in es iga ed he spec um sensing p oblem in he ull-duplex coope a i e spec um sensing CRNs. I was no ed ha spec um sensing is impo an in o e lay CRNs o gua an ee he oppo unis ic access o seconda y de ices while no in e up ing he p ima y de ice’s ansmission. Meanwhile, he pe o mance o SWIPT-enabled ne wo ks was in es iga ed in [21–29] . The ene gy-e icien op imiza ion o SWIPT-assis ed elaying ne wo ks was add essed in [ 21 ]. Tan e al. in [ 22 ] de i ed he OP and e godic capaci y o he powe spli ing (PS) based elaying ne wo ks unde he asymme ic channel, i.e., he Nakagami- m and Rayleigh channels. The e o p obabili y and ou age p obabili y o SWIPT-based NOMA ne wo ks we e de i ed in [ 23 ]. In pa icula , he pai wise e o p obabili y was compu ed in closed- o m exp ession in [ 23 ]. The asymp o ic amewo k unde a high SNRs egime was p o ided as well. The wo k in [ 24 ], di e en ly, de i ed he symbol e o a e (SER) o SWIPT-enabled elaying ne wo ks, whe e he noncohe en modula ion was employed in place o he con en ional phase-shi keying (PSK) and/o quad a u e ampli ude modula- ion (QAM). The au ho s in [ 25 ] also add essed he noncohe en modula ion. Speci ically, his wo k i s de i ed he momen s and momen gene a ing unc ion (MGF) o he end- o- end (e2e) signal- o-noise a ios (SNRs). Based on he MGF, hey hen compu ed he ou age p obabili y, he amoun o ading, and he sys em h oughpu . The au ho s in [ 26 , 27 ] deal wi h he physical laye secu i y (PLS) issue o he simul aneous wi eless in o ma ion and powe ans e assis ed elaying ne wo ks. Fu he mo e, he pe o mance o SWIPT-based cellula ne wo ks wi h and wi hou u ilizing millime e wa e (mmWa e) was p o ided in [28,29]. Senso s 2021,21, 7653 3 o 17 Despi e he ex ensi e s udy o ei he he CCRNs o he SWIPT-aided ne wo ks, he pe - o mance o SWIPT-enabled unde lay CCRNs is s ill in he in ancy s age. In pa icula , he e only a ew wo ks add essing his combina ion [ 30 – 33 ]. Speci ically, P a hima e al. in [ 30 ] s udied he pe o mance o he p ima y ne wo ks wi h he help o he seconda y use s ha also ac as he elay o p ima y use s. This wo k, howe e , concen a ed on he pe o mance o p ima y ne wo ks. Mo eo e , i solely conside ed wo elay nodes ins ead o he gene al scena io. The wo k in [ 31 ], di e sely, add essed he sec ecy pe o mance o he SWIPT-assis ed cogni i e elaying ne wo ks. The powe alloca ion and anscei e design we e in es iga ed in [32]. In his pape , di e en om he abo emen ioned wo ks, we ocus on he pe o mance o he seconda y ne wo ks as well as he eliabili y o SWIPT-based unde lay coope a i e cogni i e adio ne wo ks wi h ega d o anscei e design, powe alloca ion, and physical laye secu i y. In pa icula , he p incipal no el con ibu ions a e summa ized as ollows: • We conside a single-inpu single-ou pu (SISO) unde lay coope a i e cogni i e adio ne wo k wi h he assis ance o mul iple elays. Addi ionally, he ansmi powe o he elay nodes elies only on he ha es ed ene gy om he ansmi e S. The pa ial elay selec ion is adop ed o bo h enhance he sys em pe o mance and educe he complexi y compa ed o he ully elay selec ion. • Di e en om he con en ional unde lay CRNs whe e he ansmi powe o he seconda y ansmi e conside s me ely he in e e ence powe , he ansmi powe o he conside ed ne wo ks is cons ained by bo h he in e e ence powe and he ha es ed ene gy. The ma hema ical amewo k, hus, is o highe complexi y owing o dealing wi h mo e co ela ed andom a iables. None heless, we a e s ill able o de i e he ou age p obabili y in he closed- o m exp essions. • Simula ion esul s a e p esen ed o co obo a e he exac ness o ou analysis and o iden i y he beha io o OP wi h espec o se e al impo an pa ame e s, namely, he ansmi powe , he numbe o elays, he powe spli ing a io, and so on. Ou indings show ha bo h inc easing he ansmi powe and numbe o elays a e bene icial o he OP. Addi ionally, an op imal alue o he powe spli ing a io exis s ha minimizes he OP. The emainde o his pape is o ganized as ollows. The sys em model is gi en in Sec ion 2. The de i a ion o he OP is p o ided in Sec ion 3. Nume ical esul s a e shown in Sec ion 4. Sec ion 5concludes he pape . 2. Sys em Model Le us conside SWIPT-based unde lay cogni i e adio ne wo ks as shown in Figu e 1. In pa icula , he seconda y ne wo ks comp ise one sou ce node deno ed by S, one des ina- ion deno ed by D, and M elay nodes deno ed by Ri , i∈{1, . . . , M} , while he p ima y ne wo ks a e ep esen ed by a p ima y ecei e deno ed by P. He e, P measu es he agg e- ga e in e e ence c ea ed by he seconda y ne wo ks on he p ima y ne wo ks. 2.1. Channel Modeling Conside ing a gene ic ansmission om node X o node Y, he channel coe icien s deno ed by hXY , X∈{S, Ri} , Y∈{Ri, D} a e ollowed by a Rayleigh dis ibu ion. As a esul , he channel gain deno ed by γXY =|hXY|2 is ollowed by an exponen ial dis ibu ion wi h pa ame e λXY whose cumula i e dis ibu ion unc ion (CDF) and p obabili y densi y unc ion (PDF) a e gi en as ollows [6]: FX(x) = 1−exp(−λXYx), (1) X(x) = ∂FX(x) ∂x=λXY exp(−λXYx). (2) Senso s 2021,21, 7653 4 o 17 He e λXY is also he la ge-scale pa h loss om X o Yand is o mula ed as ollows: λXY =(dXY)β, (3) whe e dXY is he Euclidean dis ance be ween node X and Y and β∈(2, . . . , 6) is he pa h loss exponen . Addi ionally, he block ading is aken in o conside a ion in his wo k, hence he channel coe icien s emain cons an s o he whole ansmission T and change independen ly be ween each ansmission. Figu e 1. SWIPT-based cogni i e adio elaying ne wo ks. 2.2. PS-Based Relaying Ne wo ks In his wo k, we adop he powe -spli ing (PS) p o ocol a he elay node. To be mo e p ecise, he ecei ed powe a R is di ided in o wo sepa a e pa s acco ding o he powe - spli ing a io ρ , 0 <ρ< 1, i.e., one is pu in o he ene gy ha es e and ano he goes o he in o ma ion decode . ρ akes in o accoun all loss in oduced by he ene gy ha es ing ecei e , e.g., noise in oduced by he ecei ed an enna, loss due o he con e ing RF- o- DC ci cui , and so on [ 34 , 35 ]. Addi ionally, o ealize he powe -spli ing p o ocol, each SWIPT-enabled ecei e needs o be equipped wi h a powe spli e o spli he ecei ed powe in o wo pa s. The i s pa is sen o he con en ional in o ma ion decoding ci cui , and he emaining pa is sen o he ene gy ha es ing ci cui [35,36]. 2.3. Oppo unis ic Pa ial Relaying (OPR) P o ocol In his pape , he oppo unis ic pa ial elaying (OPR) p o ocol is adop ed. In pa - icula , only he elay n deno ed by R n , which has he highes channel gain om S o all elay nodes, is selec ed o help exchange in o ma ion be ween S and D. O he elay nodes, as a esul , keep silen in o de o sa e ene gy consump ion and a oid c ea ing co-channel in e e ence. Rn:γSRn=max |{z} m=1,2,...,M {γSRm}(4) Compa ed wi h he scena io whe e all elays pa icipa e in he ansmission, ou adop ed p o ocol is simple since i does no equi e pe ec channel s a e in o ma ion (CSI) o all nodes o he ne wo ks a he des ina ion and pe ec synch oniza ion among elays [ 37 – 39 ]. To be mo e p ecise, he adop ed OPR p o ocol can be employed as ollows. Each elay is equipped wi h a ime , and he alue o he ime is se in e sely wi h he channel gain om S o elay. Thus, he bes elay is he one ha ing he smalles ime . When he ime ends, he bes elay o wa ds he sou ce’s signal o he des ina ion. O he elays sense he a ailabili y o he medium and keep silen once he medium is occupied. Senso s 2021,21, 7653 5 o 17 2.4. In o ma ion T ansmission The whole ansmission akes place in wo phases. In he i s phase, sou ce S b oad- cas s i s signals o all elay nodes. He e, we assume ha he di ec channel be ween S and D does no exis due o he long ansmission dis ance and deep ades; hus, des ina ion D does no ecei e he b oadcas signal om S. Al hough all elays a e ecei ed signals sen by S, only elay R n is selec ed o assis he ansmission om S o D. The c i e ia o selec ing R n is gi en in Sec ion 2.3. A elay R n , pa s o he incoming signals a e sen o he in o ma ion decode o decode he in o ma ion sen by S and a e gi en as yRn=p1−ρpPShSRnxS+nRn, (5) whe e nRn is he addi i e whi e Gaussian noise (AWGN) a elay R n , which ollows a complex Gaussian dis ibu ion wi h ze o mean and N 0 a iance, nRn∼ CN (0, N0) ; xS is he ansmi ed signal o S and E n|xS|2o= 1; E {•} is he expec a ion ope a o ; and PS is he ansmi powe o S and is de ined in he sequel. The emaining pa o he incoming signals om S is pu in o he ene gy ha es ed ecei e . The amoun o ha es ed ene gy deno ed by ERna e hen o mula ed as ERn=ηρ(T/2)PS|hSRn|2, (6) whe e η is he ene gy con e sion coe icien [ 34 , 40 ]; he ac o T/ 2 implies ha he ene gy ha es ing only akes place in hal o he whole ansmission p ocedu e. A he end o he i s phase, elay R n decodes he in o ma ion sen by S and o wa ds he e-encoded e sion o he des ina ion D in he second phase. The whole ansmission p ocedu e is shown in Figu e 2. The ecei ed signals a D is hen o mula ed as yD=pPRnhRnDxRn+nD, (7) whe e nD is he AWGN noise a D wi h ze o mean and N 0 a iance; xRn is he ansmi ed signals o he elay R n wi h E n|xRn|2o= 1, and PRn is he ansmi powe o R n and is de ined in Sec ion 2.5. I is no ed ha he ecei ed signal a Rn and D in Equa ions (5) and (7) is a unc ion o he la ge-scale pa h loss ia he channel coe icien hSRn and hRnD , espec i ely. The signal- o-noise a ios a Rnand D a e hen o mula ed as ollows: γRn=(1−ρ)γSRnPS N0 , γD=PRn|hRnD|2 N0 . (8) Figu e 2. Ene gy ha es ing (EH) and in o ma ion ansmission (IT) p ocesses. EH akes place only in he i s hal o he ansmission du a ion, while IT akes place du ing he whole ansmis- sion du a ion. Senso s 2021,21, 7653 6 o 17 2.5. T ansmi Powe a Sou ce and Relay Nodes In he unde lay cogni i e adio ne wo ks, all seconda y ansmi e s, i.e., he sou ce node S and all elay R, ha e o adjus hei ansmi powe o s ic ly sa is y he in e e ence powe h eshold deno ed by IP (in Wa ) on he p ima y ne wo ks, i.e., he p ima y node P. As a esul , he ansmi powe o S is hen gi en as ([15], Equa ion (5)) PS=IP |hSP|2, (9) Rega ding he ansmi powe o Rn, we ha e PRn=IP |hRnP|2, (10) Addi ionally, he ansmi powe o he elay is also cons ained by he amoun o he ha es ed ene gy in he i s phase and is o mula ed as ([22], Equa ion (2)) PRn=ERn T/2 =ηρPS|hSRn|2. (11) As a consequence, PRncan be ew i en as ollows PRn=min IP |hRnP|2,ηρPS|hSRn|2!=minIP γRnP ,ηρPSγSRn(a) =IPmin1 γRnP ,ηργSRn γSP ,(12) whe e (a)is ob ained by subs i u ing PSin (9). 2.6. End- o-End Signal- o-Noise Ra ios a D Since he decode and o wa d (DF) p o ocol is employed, he e2e SNRs is hen com- pu ed as γe2e =min{γRn,γD} (a) =Ψmin(1−ρ)γSRn γSP , minγRnD γRnP ,ηργSRnγRnD γSP , (13) whe e Ψ=Ip N0;(a)is held by subs i u ing PSand PRnin (9) and (12) in o (8). Th ough di ec inspec ion (13) , we obse e ha he e2e SNR o he conside ed sys em is mo e challenging han o he wo k desc ibed in he li e a u e. Mo e p ecisely, he SNR is he composi e o wo minimum unc ions ins ead o only one. Addi ionally, he an- dom a iables inside hese minimum unc ions a e ully co ela ed as well. As a esul , he p oposed ma hema ical amewo k is no el and mo e complica ed han o he s. 3. Ou age P obabili y (OP) Analysis In his sec ion, we in es iga e one o he mos impo an me ics o a wi eless com- munica ions sys em, namely, he ou age p obabili y which measu es he quali y-o -se ice o he whole ne wo k. The OP e e s o he p obabili y ha he e2e SNRs a D is be- low a p ede ined h eshold. Ma hema ically speaking, i is o mula ed as ollows ([ 12 ], Equa ion (23)): OP =P {γe2e <γ h}=P {min(γRn,γD)<γ h},=1−P {γRn≥γ h,γD≥γ h}, (14) whe e γ h = 2 2R− 1, and R is he a ge ed a e [in bps/Hz]. In o de o compu e OP in (14), we i s de i e Lemma 1as ollows: Senso s 2021,21, 7653 7 o 17 Lemma 1. Gi en N independen and iden ically dis ibu ed (i.i.d.) exponen ial andom a iables (RVs) wi h pa ame e s Ω deno ed by Ym , m∈{1, . . . , N} . The CDF and PDF o he maximal RV deno ed by Ymax =max m∈{1,...,N}{Ym}a e gi en as ollows: FYmax (x)=1+ N ∑ m=1 (−1)mCm Nexp(−mΩx)(15) Ymax (x)=Ω N −1 ∑ m=1 (−1)mCm N −1exp(−(m+1)Ωx) whe e Ck N=N! k!(N −k)!is he binomial coe icien . P oo . Le us begin wi h he de ini ion o he CDF as ollows: FYmax (x)=P Ymax =max m∈{1,...,N}{Ym}<x (a) = N ∏ m=1 FYm(x)(b) =(1−exp(−Ωx))N(16) (c) =1+ N ∑ m=1 (−1)mCm Nexp(−mΩx), whe e (a) is held owing o he independence p ope y be ween RVs; (b) is a ained by yielding he CDF o Ym ; and (c) is achie ed wi h he help o he binomial heo em. Taking he i s -o de de i a i e o he CDF wi h espec o x, we a ain he PDF as ollows: Ymax (x)=∂FYmax (x) ∂x=Ω N −1 ∑ m=1 (−1)mCm N −1exp(−(m+1)Ωx). (17) We close he p oo he e. Nex , he OP in (14) is ew i en as ollows: OP (a) =1−P (1−ρ)γSRnΨ γSP ≥γ h,ΨγRnD γRnP ≥γ h,ΨηργSRnγRnD γSP ≥γ h =1−P X≥γ h (1−ρ)Ψ,γRnD≥γ hγRnP Ψ,X≥γ h ΨηργRnD | {z } Ξ , (18) whe e (a) is a ained by subs i u ing (13) in o (14) , X=γSRn γSP . Ξ in (18) can be calcula ed as ollows: Ξ=P X≥γ h (1−ρ)Ψ,γRnD≥γ hγRnP Ψ,γ h (1−ρ)Ψ≥γ h ΨηργRnD | {z } Ξ1 +P X≥γ h ΨηργRnD ,γRnD≥γ hγRnP Ψ,γ h (1−ρ)Ψ<γ h ΨηργRnD | {z } Ξ2 . (19) Senso s 2021,21, 7653 8 o 17 Ξ1in (19) can be spli in o wo p obabili ies, i.e., Ξ11 and Ξ12, as ollows: Ξ1=P X≥γ h (1−ρ)Ψ,γRnD≥γ hγRnP Ψ,γRnD≥(1−ρ) ηρ  =P X≥γ h (1−ρ)Ψ,γRnD≥γ hγRnP Ψ,γ hγRnP Ψ≥(1−ρ) ηρ  | {z } Ξ11 (20) +P X≥γ h (1−ρ)Ψ,γRnD≥(1−ρ) ηρ ,γ hγRnP Ψ<(1−ρ) ηρ  | {z } Ξ12 , whe e Ξ11 is e alua ed as Ξ11 =P X≥γ h (1−ρ)Ψ | {z } P1 ×P γRnD≥γ hγRnP Ψ,γ hγRnP Ψ≥(1−ρ) ηρ  | {z } P2 , (21) Looking a (21), o compu e Ξ11, we i s compu e P1as ollows: P1=1−P γSRn γSP <γ h (1−ρ)Ψ=1− ∞ Z0 FγSRnγ hx (1−ρ)Ψ× γSP (x)dx, (a) = M ∑ k=1 (−1)k+1Ck M ∞ Z0 λSPexp−xkλSRγ h (1−ρ)Ψ+λSPdx (22) = M ∑ k=1 (−1)k+1Ck MλSP(1−ρ)Ψ kλSRγ h +λSP(1−ρ)Ψ, whe e (a)is achie ed by u ilizing Lemma 1. Nex , P2is calcula ed as P2=P (1−ρ)Ψ ηργ h ≤γRnP≤γRnDΨ γ h = ∞ Z (1−ρ)Ψ ηργ h γRnD(x)dx xΨ γ h Z (1−ρ)Ψ ηργ h γRnP(y)dy =λRD ∞ Z (1−ρ)Ψ ηργ h exp(−λRDx)exp−(1−ρ)ΨλRP ηργ h −exp−xΨλRP γ h dx =exp−(1−ρ)ΨλRP ηργ h ∞ Z (1−ρ)Ψ ηργ h λRD exp(−λRDx)dx (23) −λRD ∞ Z (1−ρ)Ψ ηργ h exp−xλRD +ΨλRP γ h dx =exp−(1−ρ)ΨλRP ηργ h −(1−ρ)ΨλRD ηργ h  −λRD λRD +ΨλRP γ h exp−(1−ρ)Ψ ηργ h λRD +ΨλRP γ h , Senso s 2021,21, 7653 9 o 17 F om (22) and (23), Ξ11 in (21) is hen compu ed as Ξ11 = M ∑ k=1 (−1)k+1Ck MλSP(1−ρ)Ψ kλSRγ h +λSP(1−ρ)Ψ × exp−ζΨ[λRP +λRD] γ h −exp−ζΨ γ h hλRD +ΨλRP γ h i 1+ΨλRP γ hλRD  , (24) whe e ζ=1−ρ ηρ . Ha ing ob ained he Ξ11, we now mo e o Ξ12. Le us i s ew i e Ξ12 as Ξ12 =P X≥γ h (1−ρ)Ψ,γRnD≥(1−ρ) ηρ ,γ hγRnP Ψ<(1−ρ) ηρ  =P X≥γ h (1−ρ)Ψ×P γRnD≥(1−ρ) ηρ ×P γRnP<Ψ(1−ρ) ηργ h (25) =1−FXγ h (1−ρ)Ψ×1−FγRnD(1−ρ) ηρ ×FγRnPΨ(1−ρ) ηργ h , (a) = M ∑ k=1 (−1)k+1Ck MλSP(1−ρ)Ψ kλSRγ h +λSP(1−ρ)Ψexp(−λRDζ)1−exp−λRPΨζ γ h . He e, (a) is held by employing he esul s o (22) . Ha ing Ξ11 and Ξ12 in hand, Ξ1 in (20) is hen compu ed as Ξ1= M ∑ k=1 (−1)k+1Ck MλSP(1−ρ)Ψ kλSRγ h +λSP(1−ρ)Ψexp−ζΨ[λRP +λRD] γ h  −exp−ζΨ γ h hλRD +ΨλRP γ h i 1+ΨλRP γ hλRD +exp(−λRDζ)−exp−ζλRD +λRPΨ γ h  . (26) A e ob aining Ξ1, we now compu e Ξ2in (19) as ollows: Ξ2=P X≥γ h ΨηργRnD ,γRnD≥γ hγRnP Ψ,γ h (1−ρ)Ψ<γ h ΨηργRnD =P X≥γ h ΨηργRnD ,γ hγRnP Ψ≤γRnD<ζ = ζΨ γ h Z0 γRnP(x)dx ζ Z γ hx Ψ γRnD(y)dy ∞ Z γ h Ψηρy X(z)dz, (27) (a) = M ∑ k=1 (−1)k+1Ck M ζΨ γ h Z0 γRnP(x)dx ζ Z γ hx Ψ λSPλRDΨηρy kλSRγ h +λSPΨηρyexp(−λRDy)dy | {z } Ξ21 Senso s 2021,21, 7653 16 o 17 12. Ha, D.H.; Nguyen, T.N.; T an, M.H.Q.; Li, X.; T an, P.T.; Voznak, M. Secu i y and Reliabili y Analysis o a Two-Way Hal -Duplex Wi eless Relaying Ne wo k Using Pa ial Relay Selec ion and Hyb id TPSR Ene gy Ha es ing a Relay Nodes. IEEE Access 2020 , 8, 187165–187181. [C ossRe ] 13. Nguyen, T.N.; T an, P.T.; Voznák, M. 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