Half-Duplex and Full-Duplex AF and DF Relaying with Energy-Harvesting in Log-Normal Fading

Half-Duple x and Full-Duple x A F and DF Relaying with Ener gy-Harv esting in Log -Normal F ading Khaled M. Rabie, Member , IEEE, Bamidele Adebisi, Senior Member , IEEE, Mohamed-Slim Alouini, F ellow , IEEE Abstract — Energy-har vesting (EH) and wireless powe r trans- fer in cooperativ e relaying networks hav e r ecentl y attracted a considerable amount of research attenti on. Most of the existing work on this topic howev er focuses on Rayleigh fading channels, which represent outdoor enviro nments. In contrast, th is paper is dedicated to analyze the performa nce of d ual-hop relaying systems with EH over indoor channels characterized by log- normal fading. Both half-duplex (HD) and full-duplex (FD) relaying mechanisms are stud ied in this work wi th d ecode-and- fo rward (DF) and amplify-and-forward (AF) relaying protoco ls. In addition, three EH schemes are in vestigated, namely , time switching relaying, power splittin g relaying and ideal relaying recei ver which ser ves as a lower b ound. The system perf ormance is ev aluated in terms of the er godic outage probability f or which we deriv e accurate analytical expr essions. Monte Carlo simulations are p ro vided throughout to va lidate th e accuracy of our analysis. Results re veal that, in both HD and FD scenarios, AF r el aying perf orms only slightly worse than DF relaying which can make th e former a more effi cient solution when the processing energy cost at the DF relay is taken into account. It is also shown th at FD relaying systems can generally outp erf orm HD relaying sch emes as long as the loop-back interference in F D is r elativ el y small. Furthermore, increasing the v ariance of the log-normal channel has shown to deteriorate the performance in all the relaying and EH protocols consi dered. Index T erms — Amplify-and -for ward relay , decode-and-forwa rd relay , er godic outage probability , full-duplex, half-dup lex, energy- harv estin g p rotocols, log-normal fading, wireless power transfer . I . I N T RO D U C T I O N In conventional energy-con strained wireless networks, the network connectivity and o perability is traditionally main- tained from manually recharging or replace ment of batter ies which can b e in many scenario s inconvenient or even impos- sible in others. Scavenging energy fro m the surroundin g en- vironm ent, also common ly known as energy h arvesting ( EH), using for instanc e solar power , thermal energy or wind ene rgy seems to offer a promising and cost-effecti ve solution that can prolon g the life-time of energy-co nstrained wireless d evices. Howe ver, such uncon trollable natural energy resources can n ot guaran tee a constant amount of e nergy which may pu t reliab le commun ications at risk. T o overcom e this limitation, EH from man-mad e radio-f requen cy (RF) signals has recen tly been propo sed since such signa ls not o nly can car ry info rmation but can also serve as a co nv en ient energy so urce; this is referred to as wireless p ower transfer . Khaled M. Rabie and Bamidele Adebisi are with the school of E lectri cal Engineeri ng, Manc hester Metrop olitan Univ ersity , Manche ster , M15 6BH, UK. (e-mail s: k.rabie@mmu.ac.uk; b .adebisi@mmu.ac.uk,). Mohamed-Sli m Alouini is with King Abdulla h Univ ersity of Scienc e and T echnol ogy (KA U ST), Thuwal, Mekkah Province, Saudi Arabia. (e-ma il: slim.alouin i@kaust.edu .sa). This concept has particularly generated considerable re- search interest in the so-called simu ltaneous wire less in - formation and power transfer (SWIPT) n etworks. Althoug h many studies h ave ana lyzed the p erform ance o f po int-to-p oint SWIPT based sy stems [1]–[5], coo perative relayin g SWIPT networks, where an intermediate relaying node is used to forward th e source’ s information to the intended destination, have been by far more extensively inves tigated in the literatu re, see e.g., [6]– [8] and the reference ther ein. More specifically , the au thors in [7 ] examined the perfo rmance of a h alf-du plex (HD) amplify -and- forward (AF) relaying network with EH where a greedy switching policy was deployed . Later on, the authors of [8] e valuated the performanc e o f a one-way AF relaying system w ith three d ifferent EH protoco ls, namely , time-switching re laying (T SR), power-splitting relaying ( PSR) and ideal relaying receiv er ( IRR). Furtherm ore, [9] consid ered the outage pr obability an d ergo dic capacity analysis o f a two- way EH r elay ne twork. Decode- and-fo rward (DF) relay ing with TSR and PSR EH was studied in [10] where the au- thors d erived exact analytical expr essions for the achiev able throug hput and ergodic capacity . Other works study ing DF EH systems have also ap peared in [1 1]–[1 4]. The per forman ce of energy-con strained multiple-relay networks with relay selec- tion is examined in [15 ]. Physical later security (PLS) HD EH- based multi-an tenna AF relay ne tworks was studied in [16]. In this work, the auth ors exploited the artificial noise signal, which is traditionally used to improve the secr ecy r ate, to assist in powering the energy-c onstrained relay . Similar works combinin g PLS an d E H have been rep orted in [17 ]–[19 ]. The af oremen tioned relaying SWIPT systems have been limited to HD relaying costing 50% loss in spectral efficiency . Therefo re, full- duplex (FD) r elaying mechan ism, wh ich ex- ploits the scarce frequency spectrum more efficiently b y supportin g simultan eous sign al tra nsmission and receptio n over the same frequency band [20]– [22], h as recently b een implemented in SWIPT network s, see e.g. , [23]– [28] . All the existing works on SWIPT , including th e o nes ab ove, have be en limited to a restricted nu mber of fading models such as Rayleigh , Nakagami-m and Rician, wh ich are valid to model the outdoo r wir eless channel. On the other hand , the analysis of SWIPT systems over indoo r log-nor mal fadin g channels is scarce; in fact, to the auth ors’ best knowledge, only one study has rec ently appeared in the literatur e inv estiga ting the perfo rmance of a HD SWIPT network with AF relayin g over the log-n ormal fading channel [29]. It is worth noting that log-no rmal fading is usually used to stud y the com municatio n perfor mance in many refere nce scenarios. For instanc e, it can accurately ch aracterize shadowing from obstacles and moving human b odies in indoo r en v ironme nts, and log-no rmal 2 distribution offers a b etter fit for modeling fading fluctuations in in door wireless ch annels [3 0]–[3 4]; it is also used to model small-scale fading for indoor u ltra-wideb and (UWB) commun ications [35], [36]. Fur thermor e, empir ical fading channel measurements have shown that sho rt-term and lon g- term fading effects over the slo wly-varying indoo r channe l tend to get m ixed and log -norm al statistics beco me d ominate; hence, it describ es th e distribution of the channel pa th gain [33], [37], [38]. RF s ignals in indoor wireless channels may strongly atten uate d ue to obstacles such as object mobility and building walls w hich necessitates the use of r elays [39] , [40]. Motiv ated by the ab ove consideration s, this pap er is there- fore d edicated to study the perfor mance of r elaying SWIPT systems over ind oor log-no rmal chann els with HD and FD; both AF a nd DF relaying schemes a re ad opted along with TSR, PSR and IRR EH proto cols. The system perfor mance is ev aluated in terms o f the ergo dic outag e prob ability 1 . Therefo re, the contribution of th is p aper is as follows. First, we derive an alytical expressions of the ergodic outage probab ility for a d ual-ho p HD SWIPT sy stem in log-norm al fading with both DF and AF relaying. In this respect, T SR, PSR and IRR EH pro tocols a re in vestigated f or each case; hence, six distinct system con figuration s a re studied resulting in six different analytical expressions. The second part of this work deals with FD SWIPT in log -nor mal fadin g for b oth DF and AF relaying. The other con tribution resides in examinin g the impact of lo g-no rmal fading p arameters on the ergodic outage pr obability as well as co mparing the per forman ce of DF and AF relaying in various EH p rotocols. Furth ermore, the optimization prob lem of the EH time factor an d power - splitting factor in the TSR and PSR schemes is addressed . Results show th at when the p rocessing energy cost of the DF r elay is ignored , DF-based systems always offer better perfor mance com pared to th at of AF relay ing. It is also demonstra ted th at incre asing the log-no rmal fading c hannel variance leads to performance degradation in all systems under study . In addition, the FD systems tend to ou tperfo rm the HD on es g iv en that the loop -back interferen ce, cau sed by FD relaying, is re lativ ely small. The following notation s are u sed in th is paper . f X ( · ) , F X ( · ) and ¯ F X ( · ) den ote the probab ility density function (PDF), the c umulative distribution f unction (CDF) an d the compleme ntary CDF (CCDF) of th e random variable (R V) X , respectively . E {·} and m in {·} den ote the expectation operato r and the min imum argum ent, respectively . The rest o f this pap er is organized as follows. Section I I briefly describes the HD and FD system mo dels. Section III analyzes the ergodic o utage pro bability perfo rmance of the dual-ho p HD SWIPT network with DF and AF relaying , and TSR, PSR a nd IRR EH proto cols. Section IV is dedicated to stud y FD with DF and AF relaying over log -norm al fading channels. Num erical examples and simulation results are presented an d discussed in Sectio n V. Finally , conclusio ns are d rawn in Section VI. 1 Note that part of thi s paper was present ed at the IEEE (GLOBECOM 2016) [41]. (a) HD relaying. (b) FD relaying. Figure 1: Basic bock diagram of the HD and FD rela ying systems under considera tion. I I . S Y S T E M M O D E L Fig. 1 illustrates the b asic system diagrams of th e considered dual-ho p HD and FD systems which co nsist of a sou rce nod e, relay n ode and destination node. The source first tran smits its data, with power P s , to th e destination via an intermediate energy-con strained relay . The relay can be either b ased on DF or AF . It is assumed that there is no direct link between the end nodes a nd that th e relay is powered entirely from h arvesting the energy signal transmitted b y the sou rce node. Th e source- to-relay and relay-to -destination ch annel coefficients are d e- noted by h 1 and h 2 with d 1 and d 2 being the corresp onding distances, respectively . Althoug h the p ower con sumed by the circuitry to process d ata at th e r elay is n eglected in our deriv atio ns, this will be d iscussed in depth in the r esults section. In the HD scenario , see Fig. 1(a), the relay h as a single- antenna a nd h ence the source-to- destination inform ation trans- mission is accomplished over two p hases. On the oth er hand , in FD relaying, Fig. 1(b), the relay is eq uipped with two antennas which allows simultaneous informatio n reception and transmission at th e relay with a loop-b ack interference channel denoted as ( g ) . It is im portant to mention that real chan nels are considered through out. Note that in both HD and FD cases, the sour ce and destination no des are equipped with a single- antenna each; this config uration has been adopted in several studies dealin g with log- normal fading channels [35 ], [42], [43]. In FD r elaying, one antenn a is dedicated to harvesting energy , and is only used f or th is p urpo se. This configur ation is chosen not o nly for its rela ti ve ease of implemen tation b ut also because, according to [44], it attains co mparab le perfo rmance to the case wh en th e two antenn as are exploited fo r EH. W e assume that h 2 1 and h 2 2 are indepen dent and identically distributed log-no rmal R Vs with p arameters LN  2 µ h 1 , 4 σ 2 h 1  and LN  2 µ h 2 , 4 σ 2 h 2  , respectively , where µ h i and σ h i (both in decib els) are respecti vely the m ean an d the stand ard devi- ation of 10 log 10 ( h i ) , i ∈ { 1 , 2 } . In addition , the loop- back interferen ce channel g 2 is assumed lo g-nor mally d istributed with p arameters LN  2 µ g , 4 σ 2 g  ; no te that this is a key p a- rameter determin ing the stren gth o f the loop- back in terferen ce and hence the overall per forman ce of FD relay ing. 3 As mentioned in the introduction, the system p erform ance is evaluated in terms of the ergodic outage prob ability . This probab ility is defin ed as the pr obability th at the instantaneou s capacity falls below a cer tain threshold value ( C th ) and ca n be calculated for the AF and DF relaying systems, respecti vely , as O ( C th ) = Pr { C d ( γ d ) < C th } , (1) and O ( C th ) = Pr { min { C r ( γ r ) , C d ( γ d ) } < C th } , (2) where C r and C d are the instantaneous capacities at the relay and destin ation nodes, respectively , wh ile γ r and γ d denote the correspo nding signal-to- noise ratios (SNR). The re ceiv ed signal at the re lay dur ing the E H ph ase in both HD and FD can be expr essed as y r ( t ) = s P s d m 1 h 1 s ( t ) + n a ( t ) , (3) where m is the path loss exponent, s ( t ) is th e infor mation signal no rmalized as E h | s | 2 i = 1 and n a ( t ) is narrowband Gaussian noise introduced by th e receiving antenn a at the relay with variance σ 2 a . W e n ext d erive analytical expr essions of the ergodic outa ge prob ability for the systems under study . I I I . H A L F - D U P L E X R E L AY I N G S Y S T E M This section analyzes the performan ce of HD re laying over log-no rmal channels with b oth DF an d AF relay ing. A. Half-Duplex with DF Rela ying Below , we deri ve an alytical expressions fo r the HD-D F system with TSR, PSR and IRR EH pro tocols. 1) HD-DF-TSR System : In the TSR protocol, the time re- quired to transmit one block from the source to the destination , also refer red to as the time fr ame ( T ) , is divided into three time slots as shown in Fig. 2. The first time p eriod is the EH time, τ T , during which th e re lay harvests the p ower signal broadc ast by the so urce no de, whe re 0 ≤ τ ≤ 1 is th e EH time factor . The rem aining time is divided in to two equ al time slots used for so urce-to- relay and relay-to -destination info rmation transmission. Therefo re, u sing (3), the harvested ene rgy at the relay for this system can b e written as E H = η τ T P s h 2 1 d m 1 , (4) where 0 < η < 1 is the EH efficiency determ ined mainly by the circ uitry . Now , the received signal at th e de stination no de can b e expressed as Source-Relay Inform ation Transmissi on Relay- Destination Inform ation Transmissi on 2 T (1 - 2 ) T/2 Energy- Harvesting (1 - 2 ) T/2 Figure 2: Time frame structure in the TSR protocol . y d ( t ) = s P r d m 2 h 2 ¯ s ( t ) + n d ( t ) , (5) where ¯ s ( t ) is th e decod ed version of the source signal, n d ( t ) = n a ( t ) + n c ( t ) is the overall noise at the destinatio n node with variance σ 2 d , n c ( t ) is the no ise ad ded by the informa tion receiver , and P r is th e rela y tra nsmit p ower which is re lated to th e har vested energy as P r = E H (1 − τ ) T / 2 = 2 η P s h 2 1 τ (1 − τ ) d m 1 . (6) Substituting (6 ) in to (5 ) y ields y d ( t ) = s 2 η τ P s (1 − τ ) d m 1 d m 2 h 1 h 2 ¯ s ( t ) + n d ( t ) . (7) Groupin g the in formation and n oise te rms in ( 3) an d (7 ), we obtain th e SNRs a t the r elay and destinatio n nod es, respectively , as f ollows γ r = P s h 2 1 d m 1 σ 2 r , (8) γ d = 2 η τ P s h 2 1 h 2 2 (1 − τ ) d m 1 d m 2 σ 2 d . (9) Since in th e TSR pro tocol informatio n transmission takes place only durin g the time fraction (1 − τ ) , th e instantaneou s capacity of th e first and second links can be given b y C H D − T S R i = (1 − τ ) 2 log 2 (1 + γ i ) (10) where i ∈ { r, d } an d the factor 1 2 is a r esult of HD relaying. T o deriv e the ergodic outage probab ility f or the HD- DF- TSR system , we first wr ite (2 ) as O T S R ( C th ) = Pr  min  C T S R r , C T S R d  < C th  = 1 − P r  min  C T S R r , C T S R d  ≥ C th  = 1 − P r  C T S R r ≥ C th , C T S R d ≥ C th  = 1 − P r  C T S R r ≥ C th  | {z } O T SR 1 ( C th ) + Pr  C T S R r ≥ C th , C T S R d < C th  | {z } . O T SR 2 ( C th ) (11) 4 It is clear that the ergodic o utage p robab ility requ ires calcu - lating two p robab ilities. Using (8) and (10), and substituting X = h 2 1 , th e first probab ility in ( 11) can be calculated as O T S R 1 ( C th ) = Pr  C T S R r ≥ C th  = Pr  (1 − τ ) 2 log 2  1 + P s X d m 1 σ 2 r  ≥ C th  = Pr  P s X d m 1 σ 2 r ≥ v  = Pr { X ≥ a 1 v } = 1 − F X ( a 1 v ) , (12) where v = 2 2 C th 1 − τ − 1 , a 1 = d m 1 σ 2 r /P s and F X ( · ) d enotes the CDF of the R V X . Since X is log- norma lly distrib uted, its CDF is g iv en by F X ( a 1 v ) = 1 − Q  ξ ln ( a 1 v ) − 2 µ h 2 2 σ h 2  , (13) where ξ = 10 / ln (10) is a scaling co nstant and Q ( · ) is the Gaussian Q - functio n, giv e n by Q ( x ) = ˆ ∞ x 1 √ 2 π exp  − t 2 2  d t. (14) Now , using (8) − ( 10), an d su bstituting Y = h 2 2 , the secon d probab ility in ( 11) can be determined as O T S R 2 ( C th ) = Pr  C T S R r ≥ C th , C T S R d < C th  = Pr  X ≥ a 1 v , 2 η τ P s X Y (1 − τ ) d m 1 d m 2 σ 2 d < v  = Pr n X ≥ a 1 v , Y < a 2 v X o , (15) where a 2 = (1 − τ ) d m 1 d m 2 σ 2 d / 2 η τ P s . Using the PDF and CDF o f the log-nor mally distributed R Vs X a nd Y , we ca n calcu late the seco nd pr obability in (15 ) as O T S R 2 ( C th ) = ∞ ˆ a 1 v f X ( z ) F Y  a 2 v z  d z , (16) where f X ( z ) = ξ z q 8 π σ 2 h 1 exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! (17) and F Y  a 2 v z  = 1 − Q ξ ln  a 2 v z  − 2 µ h 2 2 σ h 2 ! . (18) Finally , sub st ituting (1 2) an d (16) in to (11) yields the ergodic outag e pro bability of the HD-DF-TSR system, giv en by Source-Relay Information Transm ission ((1 - ! ) P s ) Relay-Destination Information Transmission T T/2 T/2 Energy-Harvesting ( ! P s ) Figure 3: Time frame structure of the PSR protocol . O T S R ( C th ) = 1 − Q  ξ ln ( a 1 v ) − 2 µ h 2 2 σ h 2  + ξ q 8 π σ 2 h 1 × ∞ ˆ a 1 v 1 z exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! × 1 − Q ξ ln  a 2 v z  − 2 µ h 2 2 σ h 2 !! d z . (19) 2) HD-DF-PSR System : I n the PSR protoco l, the block time, T , is divided evenly fo r the source-to -relay and relay- to- destination tran smissions as illustrated in Fig. 3. Du ring the first half, the relay allocates a portion of the receiv ed signal power , ρP , to the energy -harvester where as th e remain i ng power , (1 − ρ ) P , is used fo r infor mation transmission, wh ere 0 ≤ ρ ≤ 1 is the power -splitting factor . Therefo re, in the first time slot the recei ved signal at the input of th e energy- harvester is expressed as √ ρy r ( t ) = s ρP s d m 1 h 1 s ( t ) + √ ρ n a ( t ) . (20) Using (20), the harvested energy at the relay nod e c an be simply wr itten as E H = η ρP s h 2 1 T 2 d m 1 . (21) On the o ther hand, the base-band signal at the infor mation receiver , √ 1 − ρy r ( t ) , is given b y p 1 − ρy r ( t ) = s (1 − ρ ) P s d m 1 h 2 s ( t ) + n r ( t ) , (22 ) where n r ( t ) = √ 1 − ρ n a ( t ) + n c ( t ) is the overall noise at the relay with variance σ 2 r = √ 1 − ρ σ 2 a + σ 2 c . In the second time slo t, th e relay decodes th e sig nal in (22), re-mod ulates and forwards it using the h arvested energy in (21). There fore, the received signal at the destination node can b e expressed as y d ( t ) = s P r d m 2 h 2 ¯ s ( t ) + n d ( t ) , (23) 5 where P r is the re lay tran smit p ower which is related to the harvested energy as P r = E H T / 2 = η ρP s h 2 1 d m 1 . (24) Now , sub stituting (24 ) in to (2 3) pr oduces y d ( t ) = s η ρP s d m 1 d m 2 h 1 h 2 ¯ s ( t ) + n d ( t ) . (25) Using ( 22) and (2 5), the SNRs at the relay a nd destination nodes can r espectively b e expressed as γ r = (1 − ρ ) P s h 2 1 d m 1 σ 2 r , (26) γ d = η ρP s h 2 1 h 2 2 d m 1 d m 2 σ 2 d . (27 ) The instan taneous capacity at the r elay and destination for the HD-DF-PSR system can b e d etermined using C P S R i = 1 2 log 2 (1 + γ i ) , (28) where i ∈ { r , d } . Similar to the HD- DF-TSR sy stem, the ergodic outage probab ility of the HD- DF-PSR appr oach can be c alculated as O P S R ( C th ) = 1 − Pr  C P S R r ≥ C th  | {z } O P SR 1 ( C th ) + Pr  C P S R r ≥ C th , C P S R d < C th  | {z } . O P SR 2 ( C th ) (29) Using (2 6) and (28 ), the first p robab ility in (29) can be written as O P S R 1 ( C th ) = Pr  C P S R r ≥ C th  = Pr  1 2 log 2  1 + (1 − ρ ) P s X d m 1 σ 2 r  ≥ C th  = Pr  (1 − ρ ) P s X d m 1 σ 2 r ≥ R  = Pr { X ≥ b 1 R } = 1 − F X ( b 1 R ) , (30) where R = 2 2 C th − 1 , b 1 = d m 1 σ 2 r / (1 − ρ ) P s and F X ( · ) represents the CDF of X wh ich is gi ven in this case as F Y ( b 1 R ) = 1 − Q  ξ ln ( b 1 R ) − 2 µ h 2 2 σ h 2  . (31) Using (26) − (28) , the second p robab ility in (29) can be calculated as follows O P S R 2 ( C th ) = Pr  C P S R r ≥ C th , C P S R d < C th  = Pr  X ≥ b 1 R, η ρP s X Y d m 1 d m 2 σ 2 d < R  = Pr  X ≥ b 1 R, Y < b 2 R X  , (32) where b 2 = d m 1 d m 2 σ 2 d /η ρP s . Now , using the PDF and CDF of the R Vs X and Y , we ca n express O P S R 2 ( C th ) as O P S R 2 ( C th ) = ∞ ˆ b 1 R f X ( z ) F Y  b 2 R z  d z , (33) where f X ( z ) = ξ z q 8 π σ 2 h 1 exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! (34) and F Y  b 2 R z  = 1 − Q ξ ln  b 2 R z  − 2 µ h 2 2 σ h 2 ! . (35) Finally , sub stituting (3 0) and (33 ) in to (29) p roduce s the ergodic outage p robab ility of the HD-DF- PSR system in log- normal fadin g, expressed as O P S R ( C th ) = 1 − Q  ξ ln ( b 1 R ) − 2 µ h 2 2 σ h 2  + ξ q 8 π σ 2 h 1 × ∞ ˆ b 1 R 1 z exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! × 1 − Q ξ ln  b 2 R z  − 2 µ h 2 2 σ h 2 !! d z . (36) 3) HD-DF-IRR System : Un like the TSR and PSR protocols, the IRR scheme is capable o f concurren tly pr ocessing infor- mation and h arvesting energy fro m the same recei ved signal. Therefo re, the signal received at the informa tion receiv er of the re lay can be expressed as y r ( t ) = s P s d m 1 h 1 s ( t ) + n r ( t ) , ( 37) and the harvested en ergy and the relay transmit power can therefor e be g iv en respectively as E H = η P s h 2 1 d m 1 ( T / 2 ) , (38) P r = 2 E H T = η P s h 2 1 d m 1 . (39) Using (39) , the recei ved signa l at the destination can be written as 6 y d ( t ) = s η P s d m 1 d m 2 h 1 h 2 ¯ s ( t ) + n d ( t ) . (40) Now , using (3 7) and (4 0), the SNR s at the re lay and des- tination n odes in the HD-DF-I RR system can be respectively expressed as γ r = P s h 2 1 d m 1 σ 2 r (41) and γ d = η P s h 2 1 h 2 2 d m 1 d m 2 σ 2 d . (42) The ergodic outage p robab ility of th e HD-DF-T SR system can b e calculated as O I RR ( C th ) = 1 − Pr  C I RR r ≥ C th  | {z } O I RR 1 ( C th ) + Pr  C I RR r ≥ C th , C I RR d < C th  | {z } . O I RR 2 ( C th ) (43) The fir st p robab ility O I RR 1 ( C th ) ca n be g iv en by O I RR 1 ( C th ) = Pr  C I RR r ≥ C th  = Pr  1 2 log 2  1 + P s X d m 1 σ 2 r  ≥ C th  = Pr  P s X d m 1 σ 2 r ≥ R  = Pr { X ≥ c 1 R } = 1 − F X ( c 1 R ) , (44) where c 1 = d m 1 σ 2 r /P s and F X ( · ) rep resents the CDF of X expressed in this case as F Y ( c 1 R ) = 1 − Q  ξ ln ( c 1 R ) − 2 µ h 2 2 σ h 2  . (45) Using ( 28), (41 ) and (42), the seco nd prob ability in (43) can b e written a s O I RR 2 ( C th ) = Pr  C I RR r ≥ C th , C I RR d < C th  = Pr  X ≥ c 1 R, η P s X Y d m 1 d m 2 σ 2 d < R  = Pr  X ≥ c 1 R, Y < c 2 R X  , (46) where c 2 = d m 1 d m 2 σ 2 d /η P s . Using th e PDF an d CDF of the R Vs X and Y , we get O I RR 2 ( C th ) = ∞ ˆ c 1 R f X ( z ) F Y  c 2 R z  d z , (47) where f X ( z ) = ξ z q 8 π σ 2 h 1 exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! (48) and F Y  c 2 R z  = 1 − Q ξ ln  c 2 R z  − 2 µ h 2 2 σ h 2 ! . (49) Finally , substituting (44) and (47) into (43) we obtain the ergodic outage prob ability of th e HD-DF-IRR system as O I RR ( C th ) = 1 − Q  ξ ln ( c 1 R ) − 2 µ h 2 2 σ h 2  + ξ q 8 π σ 2 h 1 × ∞ ˆ c 1 R 1 z exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! × 1 − Q ξ ln  c 2 R z  − 2 µ h 2 2 σ h 2 !! d z . ( 50) B. Half-Duplex with AF Re laying In this section , we analy ze the ergod ic outage p robab ility of the HD s ystem with AF relaying f or different EH protoco ls. Although so me of th e r esults below have been derived in [2 9], we feel that presenting them here is importan t to make the current work self-co ntained. 1) HD-AF-TSR System : In this system, the sign al received at the relay d uring the EH time is given by (3), and theref ore the harvested energy is equa l to (4 ). After the baseband processing and amplification at th e relay , the relay tran smit signal can be written as r ( t ) = s P s P r d m 1 G h 1 s ( t ) + p P r G n r ( t ) , (51) where P r is given b y (6), n r ( t ) = n a ( t ) + n c ( t ) with variance σ 2 r = σ 2 a + σ 2 c and G is the relay g ain given b y G = 1 q P s d m 1 h 2 1 + σ 2 r . The received signal at the destination can then be expre ssed as y d ( t ) = s P s P r d m 1 d m 2 Gh 1 h 2 s ( t ) + s P r d m 2 G h 2 n r ( t ) + n d ( t ) . (52) Using (6) and (52), along with some b asic alg ebraic manip- ulations, the SNR a t th e de stination can b e ob tained as γ d = 2 η τ P s h 2 1 h 2 2 2 η τ d m 1 σ 2 r h 2 2 + (1 − τ ) d m 1 d m 2 σ 2 d , (53) which can also b e simplified to γ d = a 1 h 2 1 h 2 2 a 3 h 2 2 + a 2 , (54) 7 where a 1 = 2 η τ P s , a 2 = 2 η τ d m 1 σ 2 r and a 3 = (1 − τ ) d m 1 d m 2 σ 2 d . Now , substituting (5 4) into (1), we can determin e the ergodic outa ge prob ability for the HD-AF-TSR system as O T S R ( C th ) = Pr  a 1 X Y a 3 Y + a 2 < v  = Pr  Y < a 2 v a 1 X − a 3 v  . (55) Since Y is alw a ys a positi ve v alue, we c an express the probab ility above as O T S R ( C th ) =    Pr  Y < va 2 a 1 X − ν a 3  , X < va 3 a 1 Pr  Y > va 2 a 1 X − v a 3  = 1 , X > va 3 a 1 . (56) The e rgodic outag e prob ability can now be calculated as O T S R ( C th ) = va 3 a 1 ˆ 0 f X ( z ) d z + ∞ ˆ va 3 a 1 f X ( z ) Pr  Y ≤ v a 2 a 1 z − v a 3  | {z } F Y ( · ) d z , (57) where f X ( · ) is the PDF of X and the probability in the second integral represen ts the CDF o f Y , i.e., F Y ( · ) ; the se PDF an d CDF can be given respectively a s f X ( z ) = ξ z q 8 π σ 2 h 1 exp − ( ξ ln ( z ) − 2 µ h 1 − ξ ln ( a 1 )) 2 8 σ 2 h 1 ! (58) and F Y (Γ) = 1 − Q  ξ ln (Γ) − 2 µ h 2 2 σ h 2  , (59) where Γ = va 3 z − v a 2 . Finally , substituting (58) and (59) into (57) gi ves the ergodic outage p robability of th e HF-AF-TSR system as O T S R ( C th ) = 1 − ξ q 8 π σ 2 h 1 ∞ ˆ va 2 1 z Q  ξ ln (Γ) − 2 µ h 2 2 σ h 2  × exp − ( ξ ln ( z ) − (2 µ h 1 + ξ ln ( a 1 ))) 2 8 σ 2 h 1 ! d z . (60) 2) HD-AF-PSR System : In this system, the received signal at the inpu t of th e energy -harvester can also be given by ( 20), and there fore the h arvested energy is eq ual to (21). In light o f this, the re lay tran smit sign al can b e expressed as r ( t ) = s (1 − ρ ) P s P r d m 1 G h 1 s ( t ) + p P r G n r ( t ) , (61) where n r ( t ) = √ 1 − ρ n a ( t ) + n c ( t ) , P r is defined in (24) and G is g i ven for the HD-AF-PSR system as G = 1 r (1 − ρ ) P s d m 1 h 2 1 + σ 2 r . W e can no w expr ess the signal received at the d estination node as y d ( t ) = s (1 − ρ ) P s P r d m 1 d m 2 G h 1 h 2 s ( t )+ s P r d m 2 G h 2 n r ( t )+ n d ( t ) . (62) Substituting (24) into (6 2) an d with some basic a lgebraic manipulatio ns, we o btain the SNR a t th e de stination as γ d = η ρ (1 − ρ ) P s h 2 1 h 2 2 η ρ d m 1 σ 2 c h 2 2 + η ρ (1 − ρ ) d m 1 σ 2 a h 2 2 + (1 − ρ ) d m 1 d m 2 σ 2 d . (63) Using b 1 = η ρ (1 − ρ ) P s , b 2 = η ρ d m 1 σ 2 c , b 3 = η ρ (1 − ρ ) d m 1 σ 2 a and b 4 = ( 1 − ρ ) d m 1 d m 2 σ 2 d , we c an rewrite (63) in th e fo llowing simplified fo rm γ d = b 1 h 2 1 h 2 2 b 2 h 2 2 + b 3 h 2 2 + b 4 . (64) Follo w ing the same p rocedu re as in the HD- AF-TSR sys- tem, it is straightforward to sho w that the ergod ic outage probab ility of the HD- AF-PSR system can be calculated as O P S R ( C th ) = 1 − ξ q 8 π σ 2 h 1 ∞ ˆ υ ( b 2 + b 3 ) 1 z Q  ξ ln (Λ) − 2 µ h 2 2 σ h 2  × exp − ( ξ ln ( z ) − 2 µ h 1 − ξ ln ( b 1 )) 2 8 σ 2 h 1 ! d z , (65) where Λ = υ b 4 z − υ b 2 − υ b 3 . For more d etails the rea der may refer to [2 9]. 3) HD-AF-IRR System : The harvested energy and relay transmit po we r in this s ystem can a lso b e gi ven b y (38) and (39), respectively . Recalling that th e relay has a gain of G , we can wr ite the rec eiv e d signal at the destination as y d ( t ) = s P s P r d m 1 d m 2 Gh 1 h 2 s ( t ) + s P r d m 2 G h 2 n r ( t ) + n d ( t ) , (66) 8 where G = 1 q P s d m 1 h 2 1 + σ 2 r . Substituting (3 9) into (66), and then g roup ing th e inf orma- tion and n oise signals, the SNR at the destinatio n nod e ca n be giv en by γ d = η P s h 2 1 h 2 2 η d m 1 σ 2 r h 2 2 + d m 1 d m 2 σ 2 d , (67) which can also b e written, for mor e convenience, as γ d = c 1 h 2 1 h 2 2 c 2 h 2 2 + c 3 , (68 ) where c 1 = η P s , c 2 = η d m 1 σ 2 r and c 3 = d m 1 d m 2 σ 2 d . Follo w ing the same pro cedure as in Sec. III-B-3, we can show that the ergodic o utage probab ility of the HD-AF-IRR system can b e calcu lated a s O I RR ( C th ) = 1 − ξ q 8 π σ 2 h 1 ∞ ˆ υ c 2 1 z Q  ξ ln (Υ) − 2 µ h 2 2 σ h 2  × exp − ( ξ ln ( z ) − 2 µ h 1 − ξ ln ( c 1 )) 2 8 σ 2 h 1 ! d z , (69) where Υ = υ c 3 z − υ c 2 . I V . F U L L - D U P L E X R E L A Y I N G S Y S T E M This section is dedicated to analyze the per forman ce of a dual-ho p FD network with DF and AF relaying based on th e TSR EH pro tocol. Recall th at u nlike HD, in FD the r elay is equ ipped with two antenn as and this allows simultaneous reception and transmission o f in formatio n at the r elay , see Fig. 1(b). T o begin with, in the first time slot, the recei ved signal at the r elay c an also be g iv en by (3 ) a nd the relay tran smit power will h av e the fo llowing form P r = E H (1 − τ ) T = η τ P s h 2 1 (1 − τ ) d m 1 . (70) In the second time slot (informatio n tran smission p hase), the received signal at the relay is given by y r ( t ) = s P s d m 1 h 1 s ( t ) + g r ( t ) + n a ( t ) , (71) where g is the loop -back in terference chan nel and r ( t ) is the loop-b ack interfe rence signal due to FD relaying and it satisfies the following E n | r ( t ) | 2 o = P r . It sh ould be pointed o ut that in the FD system, since the relay k nows its o wn signal, it usually ap plies interference cancellation to reduce the lo op-b ack interference . T herefo re, we can now write the post-can cellation signal at the r elay as y r ( t ) = s P s d m 1 h 1 s ( t ) + ˆ g ˆ r ( t ) + n a ( t ) , (72) where ˆ g is the residual loop -back inte rference channel c aused by im perfect in terferenc e ca ncellation an d E n | ˆ r ( t ) | 2 o = P r . Below , we derive the ergodic outage prob ability expressions for both FD-DF and FD-AF relay ing systems. A. Full-Duplex with DF Re laying (F D-DF-TSR) When DF relay ing is ap plied, the re lay will deco de and forward the source signal; hence, the receiv e d signal at the destination can be expressed as y d ( t ) = s P r d m 2 h 2 ¯ s ( t ) + n d ( t ) . (73) Using (70) , (72) and (7 3), the SNRs at the relay an d desti- nation in the FD-DF-TSR sy stem can be written r espectively as γ r = P s h 2 1 P r d m 1 g 2 = 1 − τ η τ g 2 , (74) and γ d = P r h 2 2 d m 2 σ 2 d = η τ P s h 2 1 h 2 2 (1 − τ ) d m 1 d m 2 σ 2 d . (75) The instan taneous capacity o f the sou rce-to-r elay and relay- to-destination link s ca n n ow be expressed as C F D − D F i = (1 − τ ) log 2 (1 + γ i ) , (76) where i ∈ { r, d } . Compar ing (10) and (76), it is o bvious th at the factor 1 2 is no long er pr esent in (76) due to the FD nature of th e relay . Now , the ergodic ou tage pro bability for this system can be determined as O F D − D F ( C th ) = Pr  min  C F D − D F r , C F D − D F d  < C th  . (77) Using (74)-(76) , and substituting Z = h 2 1 h 2 2 and W = g 2 , the p robability in (7 7) c an b e expressed as O F D − D F ( C th ) = Pr  min  1 − τ η τ W , η τ P s Z (1 − τ ) d m 1 d m 2 σ 2 d  < v  , (78) where v = 2 C th 1 − τ -1. Using th e fact that the R Vs Z and W are indep endent, the ergodic outage pro bability of the FD-DF-TSR system can be giv en by 9 O F D − D F ( C th ) = 1 − ¯ F W  η τ 1 − τ v  ¯ F Z  (1 − τ ) d m 1 d m 2 σ 2 d η τ P s v  , (79) where ¯ F W ( · ) an d ¯ F Z ( · ) are the CCDFs of W and Z , respectively . The first C CDF , ¯ F W ( · ) , is straightforward to obtain since the R V W is log-n ormally d istributed. On the o ther hand, the R V W is a p roduct of two log-normally distrib u ted R Vs. Usin g the prop erties of the log-no rmal distribution, it can be shown that the CCDF o f Z is ¯ F Z ( · ) = Q  ξ ln (∆) − 2 ( µ h 1 + µ h 2 ) √ 2 ( σ h 1 + σ h 2 )  , (80) where ∆ = (1 − τ ) d m 1 d m 2 σ 2 d ητ P s v . W ith this in mind , we can fin ally expr ess the ergodic outage probab ility of the FD-D F-TSR sy stem as O F D − D F ( C th ) = 1 − Q   ξ ln  ητ 1 − τ v  + 2 µ g 2 σ g   × Q    ξ ln (∆) − 2 P i ∈{ 1 , 2 } µ h i √ 2 P i ∈{ 1 , 2 } σ h i    . (81) B. Full-Duplex with AF R elaying ( FD-AF-TSR ) In the case of FD-AF system, the rece iv ed signal at the relay (72) will be amplified and then forwarded to the destination node. Hence, the received signal at the destination can be expressed as y r ( t ) = s P s P r d m 1 d m 2 h 1 h 2 G s ( t ) + s P r d m 2 h 2 g G r ( t ) + s P r d m 2 h 2 Gn a ( t ) + n d ( t ) , (82) where P r is giv e n by (7 0) and G is the relay gain defined as G = 1 q P s d m 1 h 2 1 + ˆ g 2 P r + σ 2 r . (83) Groupin g the informatio n signa l and n oise terms in (82), we can wr ite the SNR at the destination as γ d = P s h 2 1 h 2 2 d m 1 d m 2 g 2  P s h 2 1 σ 2 r P r g 2 d m 1 + P r h 2 2 d m 2 + σ 2 r  . (84 ) Note that th e instantan eous capacity of the FD-AF-TSR system is d etermined using C F D − AF d = (1 − τ ) log 2 (1 + γ d ) . (85) Similar to the DF scenario, comparing (10 ) and (85), it ca n evident that th e factor 1 2 is no longer p resent (85 ) d ue to the FD r elaying. Now , using the definition in (1) a long with (84) an d (85), we ca n calculate the ergo dic outag e p robab ility of th e FD-AF- TSR system as O F D − AF ( C th ) = Pr n C F D − AF d ( γ d ) < C th o , = Pr    P s h 2 1 h 2 2 d − m 1 d − m 2 g 2  P s h 2 1 σ 2 r P r g 2 d m 1 + P r h 2 2 d m 2 + σ 2 r  < v    , (86) where v = 2 C th 1 − τ − 1 . Using th e sub stitutions k = η τ / (1 − τ ) , Z = h 2 1 h 2 2 and W = g 2 , along with some algeb raic man ipulation s, we can rewrite (86) as O F D − AF ( C th ) = Pr ( Z < d m 1 d m 2 v σ 2 r  1 k + W  P s − P s k v W ) . (8 7) W e know th at Z is al ways a positive value; hence, the probab ility in ( 87) can be rewritten in th e following form O F D − AF ( C th ) =        Pr  Z ≤ d m 1 d m 2 vσ 2 r ( 1 k + W ) P s − P s k v W  , W < 1 kv Pr  Z > d m 1 d m 2 vσ 2 r ( 1 k + W ) P s − P s k v W  = 1 , W > 1 kv . (88) This proba bility can be c alculated as f ollows O F D − AF ( C th ) = 1 − 1 kv ˆ 0 f W ( z ) ¯ F Z d m 1 d m 2 v σ 2 r  1 k + z  P s − P s k v z ! d z , (89) where f W ( · ) is th e PDF o f W given b y f W ( z ) = ξ z q 8 π σ 2 g exp − ( ξ ln ( z ) − 2 µ g ) 2 8 σ 2 g ! (90) and ¯ F Z ( · ) is the CCDF of Z de fined as ¯ F Z ( z ) = Q      ξ ln  d m 1 d m 2 vσ 2 r ( 1 k + z ) P s − P s k v z  − 2 P i ∈{ 1 , 2 } µ h i √ 2 P i ∈{ 1 , 2 } σ h i      . (91) Finally , substituting (90 ) and (9 1) into ( 89), we can express the ergodic outage probability of the FD-AF-TSR system as in (92 ), shown at the top o f the next page, whe re Γ = d m 1 d m 2 vσ 2 r ( 1 k + z ) P s − P s k v z . 10 O F D − AF ( C th ) = 1 − ξ q 8 π σ 2 h 1 1 kv ˆ 0 1 z Q    ξ ln ( Γ) − 2 P i ∈{ 1 , 2 } µ h i √ 2 P i ∈{ 1 , 2 } σ h i    exp − ( ξ ln ( z ) − 2 µ h 1 ) 2 8 σ 2 h 1 ! d z (92) V . R E S U LT S A N D D I S C U S S I O N S In this sectio n, we pre sent some nu merical exam ples of the derived expression s above alo ng with Mon te Carlo simu la- tions. Unless specified oth erwise, we use in ou r ev aluation s the following s ystem parame ters: P s = 1 W , η = 1 , m = 2 , d 1 = d 2 = 5 m, C th = 2 bps/Hz, σ 2 1 = σ 2 2 = 4 dB, µ 1 = µ 2 = 3 d B and σ 2 r = σ 2 d = 2 σ 2 r a = 2 σ 2 r c = 0 . 005 W . A. The I mpact of τ and ρ on the P erforman ce of HD-DF and HD-AF Systems This section discusses and co mpares the effect of the EH parameters τ (EH time factor in the T AR app roach) an d ρ (power splitting factor in th e PSR ap proach ) on the ergodic outage prob ability of the HD-DF-TSR, HD-DF-PSR, HD- AF- TSR a nd HD-AF-PSR systems. T o achieve this, we plot in Fig. 4 the an alytical and simulated ergodic outage proba bility for the four systems as a function of τ and ρ . N ote that the analytical results ar e obtained fr om (19), (36 ), ( 60) and (6 5) for these systems, respectively . The goo d ag reement b etween the analytical and simulate d results clearly in dicates the accuracy of our an alysis. It can be seen f rom the re sults in Fig. 4 that DF relay ing, in both TSR and PSR systems, tends to offer slightly better perfor mance in com parison to th at of AF relayin g. This is mainly because the processing energy cost of DF relay ing, which is of cou rse higher than that of AF relaying , is ignored here; this will be investigated in more details later . In addition , it is worthwh ile poin ting out that this impr ovement is more pronounc ed in the TSR -based system compa red to that in the PSR-based appro ach. Ano ther interesting ob servation one can see fr om these r esults is the significant deterioration in the ergodic o utage probability when τ or ρ app roach either zero o r one due to the fact th at the harvested energy be comes either too small or unnecessarily too large (lead ing to no resources left fo r data tran smission). Therefo re, optim izing these parame ters is of great importan ce to ma ximize the system pe rforma nce; this phen omena is in vestigated th oroug hly below . It is worth mentioning that it is possible to im prove the perf orman ce by allocating th e c hannel unequ ally dep ending on the relative chann el distributions in both TSR a nd PSR protocols. B. P erforma nce Optimization an d Impact of Log-normal F a d- ing P a rameters In th is section , we address the optimization problem o f the EH time factor and the p ower -sp litting factor for the TSR and PSR systems. In addition , we also d iscuss the impact of log -norm al f ading, i.e., the impact of the d istribution parameters, on the behavior of the dif f erent protocols deployed in this work. In this respect, we show in Fig. 5 numerical 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ / ρ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ergodic Outage Probability HD-AF-TSR (analytical) HD-DF-TSR (analytical) HD-AF-PSR (analytical) HD-DF-PSR (analytical) simulation Figure 4: Ergodic outage probabi lity performance with respect to τ and ρ for the HD-DF-TSR, HD -DF-PSR, HD-AF-TS R and HD-AF-PSR systems. results of the min imum achiev able ergodic outage p robability as a fun ction o f th e sou rce-to-r elay and relay-to-destinatio n channel v arian ces for the HD- DF-TSR, H D-DF-PSR, HD- AF-TSR an d HD-AF-PSR systems with different transmit power values; these r esults are obtained using (19), (36), (60) and ( 65), resp ectiv e ly . Note that although it is very difficult to get th e solution in closed-fo rm, it does no t pose any difficulty to obtain nume rical solutions using software tools. For comparison’ s sak e , results for the HD-DF-IRR and HD- AF-IRR are a lso included o n this fi gure, which are o btained from ( 50) an d (69 ), respectively . A number of observations can b e seen from this figu re. For instan ce, as opposed to the Rayleigh fading case, it is observed th at increasing the channel variance will always degrade p erform ance for all the considered systems. Similar to the previous sectio n, results in Fig. 5 indicate tha t DF-based systems a lways outperfor m the AF-based ones through out the channel variance spectrum . It is also n oticeable that th e op timized PST schemes per form better than the optimized TSR systems a nd that t he IRR system serves as a lower bound. Furtherm ore, comparing Figs. 5 (a) and 5(b), it can be noted that increasing the source transmit power con siderably enh ances the e rgodic outage probability for all th e system co nfiguratio ns, which is intuitive. C. I mpact o f Pr o cessing a nd Cha nnel E stimation Energy Cost Although the assumption s of pe rfect chan nel state infor- mation a vailability and the zero processing energy co st at 11 σ 2 h1 = σ 2 h2 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Optimized Ergodic Outage Probability 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Optimized HD-DF Optimized HD-AF IRR TSR PSR (a) P s = 1 W . σ 2 h1 = σ 2 h2 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Optimized Ergodic Outage Probability 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Optimized HD-DF Optimized HD-AF PSR TSR IRR (b) P s = 5 W Figure 5: Minimum achie va ble ergodic outage probabilit y ve rsus the source-to -relay and relay-to-dest inatio n channe l varianc e for the HD-DF- TSR, HD-DF-PSR, HD-DF-IRR, HD-AF-TSR, HD-AF-PSR an d HD- AF-IRR s ystems when P s = 1 W and 5 W . the DF relay ha ve simplified our analysis in this work, such assumptions are unr ealistic in practice. W e therefore dedicate this section to examin e the imp act of the energy co st at th e DF relay d ue to informatio n processing and ch annel estimation; the combined power cost will be referred to a s ( P c ) . The focus here will particularly be kept on the IRR p rotoco l a nd the en d-to-e nd distance is fixed at 30m. W ith this in mind, Fig. 6 dem onstrates the ergodic outage p robab ility for both the HD-AF-IRR and HD-DF-I RR with respect to the source - to-relay distance with dif fer ent values of P c ; specifically , we consider P c = 0 , 0 . 0 1 P r , and 0 . 02 P r W , which corresp ond to 0%, 1% an d 2 % of th e harvested power . As o ne ca n re adily obser ve from this figu re, the ideal HD- DF-IRR scheme, i.e. , P c = 0 W , al ways outperf orms the HD- AF-IRR system regardless of the position of the relay . How- ev e r , when the energy consumption at the DF relay is taken into account, the perfo rmance of both DF and AF rela ying systems will become very comparable, e. g., P c = 0 . 0 1 P r W . Source-to-Relay Distance (d 1 ) 10 12 14 16 18 20 Ergodic Outage Probability 0.48 0.49 0.5 0.51 0.52 0.53 0.54 HD-AF-IRR HD-DF-IRR ( P p = 0 W ) HD-DF-IRR ( P p = 0.01 P r ) HD-DF-IRR ( P p = 0.02 P r ) Figure 6: E rgodi c outage probabi lity versus the source-to-rela y distance for the HD-DF-IRR and HD-AF-IRR systems with diffe rent v alues of P c at the DF relay when d 1 + d 2 = 30 m. Howe ver, wh en P c increases further, i.e., P c = 0 . 02 P r W , AF relaying can start to perform better than DF r elaying making the former more a ttractiv e in some p ractical scen arios. Another in teresting remark o n the r esults in Fig. 6 is that the poor est p robab ility perfo rmance, f or AF an d DF relayin g alike, is experienced when the relay is place d at the mid point between the source and destination nodes. This is simply justified by the f act th at at this p oint the relay will need more time h arvesting energy and th is co nsequen tly impacts the inf ormation transmission time ; henc e, high ergo dic outage probab ility occurs. D. FD versus HD Relaying The p erform ance of the FD an d HD s ystems with DF and AF relaying is discussed in this section. Fig. 7 d epicts the ergodic outag e pro bability as a f unction o f the transmission rate th reshold f or th e FD- DF an d FD- AF system s with various values o f the loop-b ack in terferen ce chan nel variance and the source transm it power; more specifically , σ 2 g = { 2 , 5 dB } an d P s = { 1 , 10 W } . Results for the HD-DF and HD-AF schemes are also includ ed in this figure. Note that th e analytical results of th e FD- DF an d FD-AF system s ar e o btained f rom (81) and (92), respectively . The system parameters used in her e are: d 1 = d 2 = 5 m , τ = 0 . 01 , σ 2 1 = σ 2 2 = 4 dB and µ 1 = µ 2 = µ g = 3 dB. I t is can be seen fro m Fig. 7(a) that wh en th e loo p-back interf erence chann el variance is σ 2 g = 5 dB, the FD schemes always ou tperfor m the HD ones for the same transmission rate irrespectiv e of the value of P s . On the other hand, howe ver, it is d emonstrated in Fig. 7(b) that when the loop-back inter ference chann el variance σ 2 g = 2 d B, HD outperf orms FD. The final r emark on these results is that FD-DF relaying has better perfo rmance than FD-AF re laying in all the protoco ls deployed. V I . C O N C L U S I O N S This paper stud ied the ergodic outage prob ability of HD and FD relayin g in EH ne tworks over log-no rmal fadin g channels. 12 Threshold Value 0 1 2 3 4 5 6 7 Ergodic Outage Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 HD-AF (analytical) HD-DF (analytical) FD-AF (analytical) FD-DF (analytical) simulation P s =1W P s =10W (a) σ 2 g = 5 dB Threshold Value 0 1 2 3 4 5 6 7 Ergodic Outage Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 HD-AF (analytical) HD-DF (analytical) FD-AF (analytical) FD-DF (analytical) simulation P s =1W P s =10W (b) σ 2 g = 2 dB Figure 7: Ergodic outage probabil ity with respect to the transmissio n rate value for the FD-DF , FD-AF , HD-DF and HD-AF systems with dif ferent v alues of σ 2 g and P s . All results in this fi gure are based on the TS R protocol. More sp ecifically , we investigated the p erform ance of bo th AF and DF relayin g with thre e well-k nown EH pr otocols, n amely , TSR, PSR and IRR. Accurate analytical expressions for the ergodic outage prob ability for these systems were d erived and then validated with computer simulations. I t was d emonstrated that DF r elaying is able to always offer b etter pr obability perfor mance comp ared to AF r elaying when the processing energy cost for the former at the relay is ig nored . Ho wever , when th e pr ocessing e nergy cost is taken into acco unt AF relaying may outperfo rm DF relaying . It was also sh own that increasing th e variance of th e log-no rmal fadin g ch annel will degrade the perf ormanc e. Comparing the perform ances of FD and HD r elaying systems, it was fo und that FD relaying can considerably enh ance the system performance as long as th e lo op-b ack interferenc e due FD relayin g is relativ ely low . Howev er , if this interfer ence increases, perfor mance may se verely degrad e and conseque ntly HD relaying can p erform better . 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