Power beacon-assisted energy harvesting in a half-duplex communication network under co-channel interference over a Rayleigh fading environment: Energy efficiency and outage probability analysis

ene gies A icle Powe Beacon-Assis ed Ene gy Ha es ing in a Hal -Duplex Communica ion Ne wo k unde Co-Channel In e e ence o e a Rayleigh Fading En i onmen : Ene gy E iciency and Ou age P obabili y Analysis Van-Duc Phan 1, Tan N. Nguyen 2,* , Minh T an 3, T an Thanh T ang 4, Mi osla Voznak 5, Duy-Hung Ha 2,5 and Thanh-Long Nguyen 6 1 Cen e o Excellence o Au oma ion and P ecision Mechanical Enginee ing, Nguyen Ta Thanh Uni e si y, Ho Chi Minh Ci y 70000, Vie nam 2 Wi eless Communica ions Resea ch G oup, Facul y o Elec ical & Elec onics Enginee ing, Ton Duc Thang Uni e si y, 19 Nguyen Huu Tho S ee , Ho Chi Minh Ci y 70000, Vie nam 3Op oelec onics 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 70000, Vie nam 4Na ional Key Labo a o y o Digi al Con ol and Sys em Enginee ing, Ho Chi Minh Ci y 70000, Vie nam 5Facul y o Elec ical Enginee ing and Compu e Science, Technical Uni e si y o Os a a, Os a a 70833, Czech Republic 6Cen e o In o ma ion Technology, Ho Chi Minh Ci y Uni e si y o Food Indus y, Ho Chi Minh Ci y 70000, Vie nam *Co espondence: [email p o ec ed] Recei ed: 30 May 2019; Accep ed: 1 July 2019; Published: 4 July 2019   Abs ac : In his ime, ene gy e iciency (EE), measu ed in bi s pe Wa , has been conside ed as an impo an eme ging me ic in ene gy-cons ained wi eless communica ion ne wo ks because o hei ene gy sho age. In his pape , we in es iga e powe beacon assis ed (PB) ene gy ha es ing (EH) in hal -duplex (HD) communica ion ne wo k unde co-channel In e e e o e Rayleigh ading en i onmen . In his wo k, we in es iga e he model sys em wi h he ime swi ching (TS) p o ocol. Fi s ly, he exac and asymp o ic o m exp essions o he ou age p obabili y (OP) a e analyzed and de i ed. Then he sys em EE is in es iga ed and he in luence o he p ima y sys em pa ame e s on he sys em pe o mance. Finally, we e i y he co ec ness o he analy ical exp essions using Mon e Ca lo simula ion. Finally, we can s a e ha he simula ion and analy ical esul s a e he same. Keywo ds: powe beacon (PB); ene gy e iciency (EE); ou age p obabili y (OP); ene gy ha es ing (EH); co-channel in e e ence (CCI) 1. In oduc ion Nowadays, he In e ne o Things (IoT) is conside ed o be a ho esea ch opic. In his ne wo k, all sma de ices can wo k and coope a e o e he In e ne en i onmen . The links be ween he digi al and eal wo lds help sma de ices o eac like humans (sma de ices can hea , see, hink, make decisions, and pe o m complica ed asks). The c i ical ad an age o he IoT is ha sma de ices can au oma ically do hei asks wi hou human in ol emen . Nowadays, he IoT is in ol ed in all aspec s o ci ilian li e, including anspo a ion, sma g ids, secu i y and public sa e y, ag icul u e, logis ics, and e-heal h. Fu he mo e, ene gy ha es ing (EH), in which adio equency (RF) signals can ans e bo h in o ma ion and ene gy a he same ime, can be conside ed o be a p omising solu ion Ene gies 2019,12, 2579; doi:10.3390/en12132579 www.mdpi.com/jou nal/ene gies Ene gies 2019,12, 2579 2 o 14 o ex ending he li e ime o ene gy-cons ained IoT de ices [ 1 – 10 ]. EH communica ion sys ems commonly use he wo adi ional ime swi ching (TS) and powe spli ing (PS) p o ocols, wi h he compa ison o hese p o ocols in di e en sys em ne wo ks. Mo eo e , we can see ha he loss o in o ma ion in he ha es ing phase in he TS p o ocol and he low co e age a ea in he PS p o ocol a e he main disad an ages o hese p o ocols. In addi ion, he complica ed ha dwa e s uc u e o he PS p o ocol compa ed wi h he simple ha dwa e in he TS p o ocol is also a main disad an age o he PS p o ocol. [ 11 – 16 ]. In his esea ch, we selec ed he hal -duplex (HD) model because o i s simplici y and he possibili y o applying i in eal-wo ld scena ios. In eali y, no all wi eless nodes a e equipped wi h ull-duplex capaci y due o limi a ions in ha dwa e o implemen a ion cos . F om his poin o iew, we conside ed he TS p o ocol and he HD model o ou model sys em. In he nex gene a ion o communica ion sys ems, billions o wi eless de ices will be connec ed o each o he ia he IoT sys em, which will lead o an ene gy consump ion p oblem. The e o e, ene gy e iciency (EE), measu ed in bi s pe wa , is conside ed o be an impo an eme ging me ic in ene gy-cons ained wi eless communica ion ne wo ks due o hei ene gy sho age [ 17 – 19 ]. The au ho s o [ 20 ] in es iga ed he EE op imiza ion p oblems in he SWIPT senso ne wo ks, which employs bo h he PS and he TS p o ocols. In [ 21 ], he au ho s in es iga ed an EE esou ce alloca ion algo i hm o SWIPT in an OFDMA sys em, whe e he ecei e s employed he PS scheme o ha es ene gy. The au ho s o [ 22 ] in es iga ed OFDMA sys ems wi h he PS SWIPT, and p oposed a esou ce alloca ion (ResAll) scheme o maximize EE. Wi hou conside ing SWIPT, o he MIMO wo-way AF elay channels, he la es EE beam o ming scheme was de i ed in [ 23 ], which only conside s a single-s eam ansmission pe use and canno be applied o mul iple da a s eam scena ios because o he di e en objec i e exp essions. In [ 24 ], he op imal global EE pe o mance was in es iga ed in in e e ence-limi ed ne wo ks whe e he SWIPT scheme was no employed. The EE in SWIPT is based on he IoT dis ibu ed an enna sys em (DAS), as s udied in [ 25 ]. In [ 26 ], a ime-slo ed la ge-scale MIMO sys em o ene gy ha es ing and in o ma ion ansmission was s udied, whe e an ene gy-e icien op imiza ion scheme was p oposed by join ly op imizing bo h ans e du a ion and ansmi ed powe . The au ho s o [ 27 ] de eloped a dis ibu ed i e a ion algo i hm o powe alloca ion, PS a io, and elay selec ion o maximize EE in clus e ed wi eless senso ne wo ks wi h SWIPT. The au ho s o [ 28 ] conside ed he EE maximiza ion p oblem in a wi eless powe ed communica ion ne wo k (WPCN), whe e mul iple use s ha es ene gy om a dedica ed powe s a ion and hen communica e wi h an in o ma ion ecei ing s a ion. In his esea ch, we p opose and in es iga e powe beacon-assis ed (PB) EH in an HD communica ion ne wo k unde co-channel in e e ence o e a Rayleigh ading en i onmen . We conside ed he model sys em o be he TS p o ocol. Fi s ly, he exac and asymp o ic o m exp essions o he ou age p obabili y (OP) we e de i ed. Then we in es iga ed he EE o he model sys em and he in luence o he p ima y sys em pa ame e s on he pe o mance o he p oposed sys em. Finally, he accu acy o he analy ical exp essions was de i ed and e i ied using a Mon e Ca lo simula ion in connec ion wi h he p ima y sys em pa ame e s. F om he esul s, we de e mined ha he analy ical ma hema ical and simula ed esul s a e he same. The main con ibu ions a e summa ized as ollows: 1. The sys em model o a PB EH in an HD communica ion ne wo k unde co-channel in e e ence o e a Rayleigh ading en i onmen . 2. The exac and asymp o ic o m exp essions o he OP we e de i ed. 3. The EE o he model sys em and he in luence o he p ima y sys em pa ame e s on he pe o mance o he p oposed sys em we e in es iga ed. 4. A Mon e Ca lo simula ion was conduc ed o e i y he analysis esul s using he p ima y sys em pa ame e s. The es o his pape is o ganized as ollows. Sec ion 2p esen s he sys em model. Sec ion 3 in es iga es he sys em pe o mance and he sys em EE. Sec ion 4gi es he esul s, and some discussions a e p o ided. Finally, some conclusions a e d awn in Sec ion 5. Ene gies 2019,12, 2579 3 o 14 2. Sys em Model In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we deno e ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe beacon-assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le 1 and 2 ep esen he I-S and I-D in e e ence channels, espec i ely. Le hand gdeno e he PB-S and S-D channels. Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he whole ansmission block (T) can be di ided in o pa s. Le T and 0 < α <1 deno e he whole symbol du a ion and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal ecei ed om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 −α)T is spen on signal ansmission om node S o node D, and he signal om node I o node D. No e ha all he ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D. Ene gies 2019, 12, x FOR PEER REVIEW 3 o 15 2. Sys em Model In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we deno e ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe beacon- assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le 1 and 2 ep esen he I-S and I-D in e e ence channels, espec i ely. Le h and g deno e he PB-S and S-D channels. Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he whole ansmission block (T) can be di ided in o pa s. Le T and 0 < α < 1 deno e he whole symbol du a ion and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal ecei ed om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 − α)T is spen on signal ansmission om node S o node D, and he signal om node I o node D. No e ha all he ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D. IT EH PB S D I Figu e 1. Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node; 1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel. In o ma ion ansmission (IT) SD ID EH a S αT(1-α)T T Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block. 2.1. Ene gy Ha es ing In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as: 1 s bis yhx sn=++ (1) whe e b x is he ansmi signal a he PB, { } 2 bB x PΕ= , { } Ε•is he expec a ion ope a o , PB is he ansmi powe o he powe beacon, s n is he addi i e whi e Gaussian noise (AWGN) wi h ze o- mean and a iance N0, Si is he ansmi signal a he in e e e , { } 2 iI s PΕ= , and PI is he ansmi powe o he in e e e . The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as: 22 1sB B I I EhPT PT η αη α =+ (2) whe e { } 0,1 BI ηη << is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely. We assume ha BI η ηη ==. Hence, he a e age ansmi powe a node S could be ob ained by he ollowing equa ion: Figu e 1. Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node; 1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel. Ene gies 2019, 12, x FOR PEER REVIEW 3 o 15 2. Sys em Model In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we deno e ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe beacon- assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le 1 and 2 ep esen he I-S and I-D in e e ence channels, espec i ely. Le h and g deno e he PB-S and S-D channels. Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he whole ansmission block (T) can be di ided in o pa s. Le T and 0 < α < 1 deno e he whole symbol du a ion and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal ecei ed om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 − α)T is spen on signal ansmission om node S o node D, and he signal om node I o node D. No e ha all he ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D. IT EH PB S D I Figu e 1. Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node; 1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel. In o ma ion ansmission (IT) SD ID EH a S αT(1-α)T T Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block. 2.1. Ene gy Ha es ing In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as: 1 s bis yhx sn=++ (1) whe e b x is he ansmi signal a he PB, { } 2 bB x PΕ= , { } Ε•is he expec a ion ope a o , PB is he ansmi powe o he powe beacon, s n is he addi i e whi e Gaussian noise (AWGN) wi h ze o- mean and a iance N0, Si is he ansmi signal a he in e e e , { } 2 iI s PΕ= , and PI is he ansmi powe o he in e e e . The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as: 22 1sB B I I EhPT PT η αη α =+ (2) whe e { } 0,1 BI ηη << is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely. We assume ha BI η ηη ==. Hence, he a e age ansmi powe a node S could be ob ained by he ollowing equa ion: Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block. 2.1. Ene gy Ha es ing In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as: ys=hxb+ 1si+ns(1) whe e xb is he ansmi signal a he PB, En|xb|2o=PB , E{•} is he expec a ion ope a o , P B is he ansmi powe o he powe beacon, ns is he addi i e whi e Gaussian noise (AWGN) wi h ze o-mean and a iance N 0 ,S i is he ansmi signal a he in e e e , En|si|2o=PI , and P I is he ansmi powe o he in e e e . The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as: Es=ηB|h|2PBαT+ηI 1 2PIαT(2) whe e 0 <ηB,ηI<1 is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely. Ene gies 2019,12, 2579 4 o 14 We assume ha ηB=ηI=η . Hence, he a e age ansmi powe a node S could be ob ained by he ollowing equa ion: Ps=Es (1−α)T=η|h|2PBαT+η 1 2PIαT (1−α)T=κPB|h|2+ 1 2PI(3) whe e κ=ηα 1−α. 2.2. In o ma ion T ansmission In he second phase, node S ansmi s he signal o node D, and he ecei ed signal, y D , a he des ina ion is o mula ed as: yD=gxs+ 2si+nd=gxs |{z} signal + 2si+nd | {z } noise (4) whe e we ha e En|xs|2o=Ps, and ndis he AWGN wi h ze o-mean and a iance N0. F om Equa ion (4), he end o end signal o in e e ence plus noise a io (SINR) was calcula ed as he ollowing: γe2e= Esignal 2 En|noise|2o=g 2Ps  2 2PI+N0 . (5) Subs i u ing Equa ion (3) in o Equa ion (5) and assuming ha he powe o in e e e noise is e y la ge, so PB≈PI, we ha e: γe2e= κg 2PB|h|2+ 1 2PI  2 2PI+N0 = κg 2∆|h|2+ 1 2  2 2∆+1 (6) whe e ∆=PB N0=PI N0. 3. The Sys em Pe o mance 3.1. Ou age P obabili y Based on he sys em model in he abo e sec ion, we de i ed he OP h oughpu pe o mance and EE o he p oposed sys em. F om Equa ion (6) we ob ain he OP o he model sys em as he ollowing: OP =P (γe2e< γ0)=P "κ|g|2∆|h|2+| 1|2 | 2|2∆+1< γ0# =P (XY < γ0)= ∞ R0 FYγ0 XX=x X(x)dx (7) whe e γ0= 2 2R− 1 is he h eshold o he sys em, R is he sou ce a e, X=|h|2+ 1 2=ϕ1+ ϕ2,Y=κ|g|2∆ | 2|2∆+1=κϕ3∆ 1+∆ϕ4, and ϕ1=|h|2,ϕ2= 1 2,ϕ3=g 2,ϕ4= 2 2. In o de o calcula e he p obabili y in Equa ion (7), we ha e o de e mine he p obabili y densi y unc ion (PDF) and he cumula i e densi y unc ion (CDF) o Xand Y, espec i ely, as in Lemma 1 and Lemma 2. Ene gies 2019,12, 2579 5 o 14 Lemma 1. The CDF o Y can be compu ed as: FY(a) = P (Y<a)=P κϕ3∆ 1+∆ϕ4<a=P ϕ3<a κ∆+aϕ4 κ = ∞ R0 ϕ4(ϕ4)dϕ4 a κ∆+aϕ4 κ R0 ϕ3(ϕ3)dϕ3= ∞ R0 Fϕ3a κ∆+aϕ4 κ ϕ4(ϕ4)dϕ4 = ∞ R0 ϕ4(ϕ4)dϕ4−1 λ4 ∞ R0 exp−a κ∆λ3−aϕ4 κλ3exp−ϕ4 λ4dϕ4 =1−κλ3exp−a κ∆λ3 aλ4+κλ3 (8) whe e λ3,λ4a e he mean o he andom a iables (RVs) ϕ3,ϕ4, espec i ely. Lemma 2. The CDF o X can be exp essed as: FX(a) = P [(ϕ1+ϕ2)<a]=P [ϕ1<a−ϕ2] = a R0 ϕ2(ϕ2)dϕ2 a−ϕ2 R0 ϕ1(ϕ1)dϕ1= a R0 Fϕ1(a−ϕ2) ϕ2(ϕ2)dϕ2 =1 λ2 a R0h1−exp−a−ϕ2 λ1iexp−ϕ2 λ2dϕ2 =1−exp−a λ2−exp−a λ1 λ2 a R0 expϕ2h1 λ1−1 λ2idϕ2 (9) whe e λ1,λ2a e he mean o he RVs ϕ1,ϕ2, espec i ely. In his si ua ion, we in es iga ed wo cases as ollows: Case 1: We assume ha λ1=λ2=λ. F om Equa ion (9), we ob ain: FX(a) = 1−exp−a λ−aexp−a λ λ. (10) The e o e, he PDF can be ob ained as: X(a) = ∂FX(a) ∂a=aexp−a λ λ2(11) Case 2: We assume ha λ1,λ2. Simila ly, we can ob ain he CDF and he PDF o Xas ollows: FX(a) = 1−exp−a λ2−λ1 λ2−λ1"exp−a λ2−exp −a λ1!#. (12) X(a) = 1 λ2−λ1"exp−a λ2−exp −a λ1!#. (13) 3.1.1. Exac Analysis Case 1: λ1=λ2=λ. Ene gies 2019,12, 2579 6 o 14 Combining Equa ion (8), Equa ion (11) and applying hem o Equa ion (7) allows he OP o be compu ed as: OP1= ∞ R0 FYγ0 XX=x X(x)dx =1− ∞ R0 κλ3exp−γ0 xκ∆λ3 γ0 xλ4+κλ3 ×xexp(−x λ) λ2dx =1−κλ3 λ2 ∞ R0 x2 κλ3x+γ0λ4×exp−γ0 κ∆λ3x×exp−x λdx . (14) Case 2: λ1,λ2. Combining Equa ion (8) and Equa ion (13), we can ob ain he OP, in his case as ollows: OP2=1− ∞ R0 κλ3exp−γ0 xκ∆λ3 γ0 xλ4+κλ3 ×n1 λ2−λ1hexp−x λ2−exp−x λ1iodx =1−κλ3 λ2−λ1 ∞ R0 xexp−γ0 κ∆λ3x κλ3x+γ0λ4×nexp−x λ2−exp−x λ1odx . (15) 3.1.2. Asymp o ic Analysis The high SINR egime and he end o end SINR om Equa ion (6) can be app oxima ed as: γ∞ e2e≈ κg 2|h|2+ 1 2  2 2=ZX (16) whe e X=|h|2+ 1 2=ϕ1+ϕ2,Z=κ|g|2 | 2|2=κϕ3 ϕ4. Lemma 3. The CDF o Z can be calcula ed using he equa ion: FZ(a) = P (Z<a)=P κϕ3 ϕ4<a=P ϕ3<aϕ4 κ = ∞ R0 ϕ4(ϕ4)dϕ4 aϕ4 κ R0 ϕ3(ϕ3)dϕ3 =1−1 λ4 ∞ R0 exp−aϕ4 κλ3×exp−ϕ4 λ4dϕ4 =1−κλ3 aλ4+κλ3 . (17) Case 1: λ1=λ2=λ. F om Equa ions (11), (16), and (17), he OP can be compu ed as: OP∞ 1=P γ∞ e2e< γ0=P "κ|g|2|h|2+| 1|2 | 2|2< γ0# =P (XZ < γ0)= ∞ R0 FZγ0 XX=x X(x)dx =1−1 λ2 ∞ R0 x2 x+γ0λ4 κλ3 ×exp−x λdx . (18) Ene gies 2019,12, 2579 7 o 14 Applying Equa ion (3.353,5) om he able o in eg als [29], Equa ion (18) can be ew i en as: OP∞ 1=γ0λ4λ κλ3 + γ0λ4 κλ3λ!2 exp γ0λ4 κλ3λ!Ei −γ0λ4 κλ3λ!(19) whe e Ei(−z)=− ∞ Rz e− −1d is he exponen ial in eg al unc ion. Case 2:λ1,λ2. In his case, based on Equa ions (13), (16), and (17), he OP can be calcula ed as: OP∞ 2=1−1 λ2−λ1 ∞ R0 x x+γ0λ4 κλ3 ×nexp−x λ2−exp−x λ1odx =1−1 λ2−λ1 ∞ R0 xexp−x λ2 x+γ0λ4 κλ3 dx +1 λ2−λ1 ∞ R0 xexp−x λ1 x+γ0λ4 κλ3 dx. (20) Simila o he p e ious case, we apply Equa ion (3.353,5) om he able o in eg als [ 29 ], hus Equa ion (19) can be o mula ed as ollows: OP∞ 2=1−γ0λ4 κλ3(λ2−λ1)expγ0λ4 κλ3λ2Ei−γ0λ4 κλ3λ2−λ2 λ2−λ1 +γ0λ4 κλ3(λ2−λ1)expγ0λ4 κλ3λ1Ei−γ0λ4 κλ3λ1+λ1 λ2−λ1 =γ0λ4 κλ3(λ2−λ1)nexpγ0λ4 κλ3λ1Ei−γ0λ4 κλ3λ1−expγ0λ4 κλ3λ2Ei−γ0λ4 κλ3λ2o . (21) 3.2. Ene gy E iciency Analysis The EE is de ined as a a io o o al in o ma ion a e Cand o al powe consump ion ETin [18]: EE =C ET (22) whe e C=(1−α)T 2log2( 1 +γe2e) and ET=[2αT+ (1−α)T]P ε+PCT , in which P=PB=PI , and ε and PC deno e he powe ampli ie e iciency o he powe beacon and ci cui powe consump ion, espec i ely. Finally, we ob ain he EE as shown in he equa ion below: EE = (1−α) 2log2(1+γe2e) (T+αT)P/ε+PCT= (1−α) 2log2(1+γe2e) (1+α)P/ε+PC . (23) In o de o analyze he EE, we conside ed calcula ing he a e age o al in o ma ion a e in wo cases: λ1=λ2=λand λ1,λ2. The a e age o al in o ma ion a e can be exp essed as [30]: Ca g =(1−α) 2 ln 2 ∞ Z 0 1−Fγe2e(γ0) 1+γ0 dγ0. (24) Case 1: λ1=λ2=λ. a. Exac Analysis Ene gies 2019,12, 2579 8 o 14 Subs i u ing Equa ion (14) in o Equa ion (24), we ob ain Ca g as ollows: C1_a g =(1−α) 2 ln 2 ∞ R0 1−Fγe2e(γ0) 1+γ0dγ0 =(1−α)κλ3 2λ2ln 2 ∞ R0 ∞ R0 x2 (κλ3x+γ0λ4)(1+γ0)×exp−γ0 κ∆λ3x×exp−x λdxdγ0 . (25) Nex , subs i u ing Equa ion (25) in o Equa ion (22), inally, he a e age EE can be gi en as: EE1_a g = (1−α)κλ3 2λ2ln 2 ∞ R0 ∞ R0 x2 (κλ3x+γ0λ4)(1+γ0)×exp−γ0 κ∆λ3x×exp−x λdxdγ0 (1+α)P/ε+PC . (26) b. Asymp o ic Analysis Subs i u ing Equa ion (19) in o Equa ion (24), he a e age o al in o ma ion a e can be exp essed as: C∞ 1_a g =(1−α) 2 ln 2 ∞ Z 0 1−       γ0λ4λ κλ3 + γ0λ4 κλ3λ!2 exp γ0λ4 κλ3λ!Ei −γ0λ4 κλ3λ!       dγ0 1+γ0 . (27) Combining Equa ion (25) wi h Equa ion (27), he EE in Case 1 can be ob ained as EE∞ 1= (1−α) 2 ln 2 ∞ R01−γ0λ4λ κλ3+γ0λ4 κλ3λ2expγ0λ4 κλ3λEi−γ0λ4 κλ3λ dγ0 1+γ0 (1+µ)P/ε+PC . (28) Case 2: λ1,λ2. Simila o Case 1, we can calcula e he EE o he exac and he asymp o ic analyses, espec i ely, as ollows: EE2_a g = (1−α)κλ3 2(λ2−λ1)ln 2 ∞ R0 ∞ R0 xexp−γ0 κ∆λ3x (κλ3x+γ0λ4)(1+γ0)×nexp−x λ2−exp−x λ1odxdγ0 (1+α)P/ε+PC (29) EE∞ 2_a g = (1−α) 2 ln 2 ∞ R01−γ0λ4 κλ3(λ2−λ1)nexpγ0λ4 κλ3λ1Ei−γ0λ4 κλ3λ1−expγ0λ4 κλ3λ2Ei−γ0λ4 κλ3λ2o dγ0 1+γ0 (1+α)P/ε+PC (30) 4. Resul s and Discussion In his sec ion, we used he Mon e Ca lo simula ion o e i y he accu acy o he analysis exp essions om he p e ious sec ion. Fo each simula ion, we i s p o ided he g aphs o he OP and he EE ob ained by he analy ical o mulas. Secondly, we gene a ed plo s o he same OP and EE cu es ha esul ed om he Mon e Ca lo simula ion. To do his, we gene a ed 10 5 andom samples o each channel gain, which we e Rayleigh dis ibu ed. The analy ical cu e and he simula ion should ma ch oge he o e i y he accu acy o ou analysis [30–33]. In Figu e 3, we plo ed he e ec o he ime swi ching ac o α on he OP in wo cases—Case 1 and Case 2, espec i ely. In hese cases, we se he p ima y sys em pa ame e s as R ={1,3} bps/Hz, ∆ =5 dB, and η =0.8. F om he esea ch esul s, he OP o he p oposed sys em dec eased wi h inc easing α om 0 o 1, and he OP o R =1 bps/Hz was no be e han ha wi h R =3 bps/Hz. I can be seen ha he OP was be e wi h he highe R, and ha he OP o Case 2 was be e han ha o Case 1. Mo eo e , he OP e sus ∆ is illus a ed in Figu e 4wi h he exac and asymp o ic exp essions. In his Ene gies 2019,12, 2579 9 o 14 case, we se R =1 bps/Hz and α =0.5. Simila o Figu e 3, we conside ed wo cases, Case 1 and Case 2, in Figu e 4. We obse ed ha he OP dec eased signi ican ly wi h inc easing ∆ om − 5 o 15 dB and hen con e ged in o he asymp o ic OP wi h ∆ om 15 o 25, as shown in Figu e 4. Once again, we saw ha he OP o Case 2 was be e han ha o Case 1. F om he esul s in Figu es 3and 4, we can conclude ha all he simula ion and analy ical esul s a e he same o all alues o αand ∆. Ene gies 2019, 12, x FOR PEER REVIEW 9 o 15 () 04 04 04 04 04 0 32 1 31 31 32 32 0 0 2_ (1 ) 1expEiexpEi 2ln2 1 (1 ) / a g C d EE PP γ λ γ λ γ λ γ λ γ λ γ α κλ λ λ κλ λ κλ λ κλ λ κλ λ γ αε ∞ ∞       −  −−−−       −+          =++  (30) 4. Resul s and Discussion In his sec ion, we used he Mon e Ca lo simula ion o e i y he accu acy o he analysis exp essions om he p e ious sec ion. Fo each simula ion, we i s p o ided he g aphs o he OP and he EE ob ained by he analy ical o mulas. Secondly, we gene a ed plo s o he same OP and EE cu es ha esul ed om he Mon e Ca lo simula ion. To do his, we gene a ed 105 andom samples o each channel gain, which we e Rayleigh dis ibu ed. The analy ical cu e and he simula ion should ma ch oge he o e i y he accu acy o ou analysis [30–33]. In Figu e 3, we plo ed he e ec o he ime swi ching ac o α on he OP in wo cases—Case 1 and Case 2, espec i ely. In hese cases, we se he p ima y sys em pa ame e s as R = {1,3} bps/Hz, Δ = 5 dB, and η = 0.8. F om he esea ch esul s, he OP o he p oposed sys em dec eased wi h inc easing α om 0 o 1, and he OP o R = 1 bps/Hz was no be e han ha wi h R = 3 bps/Hz. I can be seen ha he OP was be e wi h he highe R, and ha he OP o Case 2 was be e han ha o Case 1. Mo eo e , he OP e sus Δ is illus a ed in Figu e 4 wi h he exac and asymp o ic exp essions. In his case, we se R = 1 bps/Hz and α = 0.5. Simila o Figu e 3, we conside ed wo cases, Case 1 and Case 2, in Figu e 4. We obse ed ha he OP dec eased signi ican ly wi h inc easing Δ om −5 o 15 dB and hen con e ged in o he asymp o ic OP wi h Δ om 15 o 25, as shown in Figu e 4. Once again, we saw ha he OP o Case 2 was be e han ha o Case 1. F om he esul s in Figu es 3 and 4, we can conclude ha all he simula ion and analy ical esul s a e he same o all alues o α and Δ. Figu e 3. OP e sus α. OP, ou age p obabili y. Figu e 3. OP e sus α. OP, ou age p obabili y. Ene gies 2019, 12, x FOR PEER REVIEW 10 o 15 Figu e 4. OP e sus Δ. Fu he mo e, he OP o he model sys em e sus he ene gy con e sion coe icien η o he wo cases, Case 1 and Case 2, is shown in Figu e 5 wi h Δ = {1,5} dB, R = 1 bps/Hz and α = 0.5. As he esul s a e he same in he abo e igu es, we de e mined ha he OP o he p oposed sys em dec eased c ucially while η a ied om 0 o 1 and ha he OP o Case 2 was be e han ha o Case 1. Also, all simula ion esul s ag eed well wi h he analy ical esul s wi h a ying η. Figu e 5. OP e sus η. We in es iga ed he in luence o he λ1 = λ2 = λ on Case 1 o he OP o he model sys em, as illus a ed in Figu e 6. He e, we se Δ = {1,5,10} dB, η = 0.8, α = 0.5, and R = 1 bps/Hz. F om he esul s, we saw ha he OP ell wi h he ising λ, and he OP was be e wi h a highe alue o Δ. Mo eo e , he e ec o λ1 ≠ λ2 on he OP o he model sys em in connec ion wi h λ1 and λ2 is p esen ed in Figu e 7 wi h he p ima y sys em pa ame e s as Δ = {1,5,10} dB. In his si ua ion, bo h λ1 and λ2 a ied om 0 o 4. F om he esul s, we can conclude ha he OP o he sys em alls when λ1 and λ2 inc ease. The Figu e 4. OP e sus ∆. Fu he mo e, he OP o he model sys em e sus he ene gy con e sion coe icien η o he wo cases, Case 1 and Case 2, is shown in Figu e 5wi h ∆ ={1,5} dB, R =1 bps/Hz and α =0.5. As he esul s a e he same in he abo e igu es, we de e mined ha he OP o he p oposed sys em dec eased c ucially while η a ied om 0 o 1 and ha he OP o Case 2 was be e han ha o Case 1. Also, all simula ion esul s ag eed well wi h he analy ical esul s wi h a ying η.